Experimental Tests of Quantum Gravity and Exotic Quantum Field Theory Effects

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Guest Editors: Emil T. Akhmedov, Stephen Minter, Piero Nicolini, and Douglas Singleton Experimental Tests of Quantum Gravity and Exotic Quantum Field Theory Effects Advances in High Energy Physics

Experimental Tests of Quantum Gravity and Exotic Quantum Field Theory Effects

This is a special issue published in “Advances in High Energy Physics.” All articles are open access articles distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Editorial Board

Botio Betev, Switzerland Ian Jack, UK Neil Spooner, UK Duncan L. Carlsmith, USA Filipe R. Joaquim, Portugal Luca Stanco, Italy Kingman Cheung, Taiwan Piero Nicolini, Germany EliasC.Vagenas,Kuwait Shi-Hai Dong, Mexico Seog H. Oh, USA Nikos Varelas, USA Edmond C. Dukes, USA Sandip Pakvasa, USA Kadayam S. Viswanathan, Canada Amir H. Fatollahi, Iran Anastasios Petkou, Greece Yau W. Wah, USA Frank Filthaut, The Netherlands Alexey A. Petrov, USA Moran Wang, China Joseph Formaggio, USA Frederik Scholtz, South Africa Gongnan Xie, China Chao-Qiang Geng, Taiwan George Siopsis, USA Hong-Jian He, China Terry Sloan, UK Contents

Experimental Tests of Quantum Gravity and Exotic Quantum Field Theory Effects,EmilT.Akhmedov, Stephen Minter, Piero Nicolini, and Douglas Singleton Volume2014,ArticleID192712,2pages

Testing General Relativistic Predictions with the LAGEOS Satellites,RobertoPeron Volume 2014, Article ID 791367, 12 pages

Nonlocal Quantum Effects in Cosmology,YuriiV.Dumin Volume2014,ArticleID241831,8pages

Hawking-Unruh Hadronization and Strangeness Production in High Energy Collisions, PaoloCastorinaandHelmutSatz Volume2014,ArticleID376982,7pages

Amplifying the Hawking Signal in BECs, Roberto Balbinot and Alessandro Fabbri Volume2014,ArticleID713574,8pages

Theory and Phenomenology of Space-Time Defects, Sabine Hossenfelder Volume2014,ArticleID950672,6pages

Inflation and Topological Phase Transition Driven by Exotic Smoothness, Torsten Asselmeyer-Maluga and Jerzy Krol´ Volume2014,ArticleID867460,14pages

Does the Equivalence between Gravitational Mass and Energy Survive for a Composite Quantum Body?,A.G.Lebed Volume 2014, Article ID 678087, 10 pages

Quantum-Spacetime Scenarios and Soft Spectral Lags of the Remarkable GRB130427A, Giovanni Amelino-Camelia, Fabrizio Fiore, Dafne Guetta, and Simonetta Puccetti Volume2014,ArticleID597384,16pages

Kaluza-Klein Multidimensional Models with Ricci-Flat Internal Spaces: The Absence of the KK Particles, Alexey Chopovsky, Maxim Eingorn, and Alexander Zhuk Volume 2013, Article ID 106135, 6 pages

Closing a Window for Massive Photons,SergioA.HojmanandBenjaminKoch Volume 2013, Article ID 967805, 5 pages

The Final Stage of Gravitationally Collapsed Thick Matter Layers,PieroNicolini,AlessioOrlandi, and Euro Spallucci Volume 2013, Article ID 812084, 8 pages

The 5D Standing Wave Braneworld with Real Scalar Field, Merab Gogberashvili and Pavle Midodashvili Volume 2013, Article ID 873686, 6 pages

Why the Length of a Quantum String Cannot Be Lorentz Contracted, Antonio Aurilia and Euro Spallucci Volume 2013, Article ID 531696, 7 pages Hindawi Publishing Corporation Advances in High Energy Physics Volume 2014, Article ID 192712, 2 pages http://dx.doi.org/10.1155/2014/192712

Editorial Experimental Tests of Quantum Gravity and Exotic Quantum Field Theory Effects

Emil T. Akhmedov,1,2,3 Stephen Minter,4 Piero Nicolini,5,6 and Douglas Singleton7

1 Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudny, Moscow 141700, Russia 2 International Laboratory of Representation Theory and Mathematical Physics, National Research University Higher School of Economics and Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudny 141700, Russia 3 Institute for Theoretical and Experimental Physics, B. Cheremushkinskaya, No. 25, Moscow 117218, Russia 4 Vienna Center for Quantum Science and Technology, University of Vienna, Boltzmanngaße 5, 1090 Vienna, Austria 5 Frankfurt Institute for Advanced Studies (FIAS), Ruth-Moufang-Straße 1, 60438 Frankfurt am Main, Germany 6 Institut fur¨ Theoretische Physik, Johann Wolfgang Goethe-Universitat,¨ Max-von-Laue-Straße 1, 60438 Frankfurt am Main, Germany 7 Physics Department, California State University, Fresno, Fresno, CA 93740, USA

Correspondence should be addressed to Douglas Singleton; [email protected]

Received 9 June 2014; Accepted 9 June 2014; Published 23 June 2014

Copyright © 2014 Emil T. Akhmedov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

Physics at its core is an experimental pursuit. If one theory GRB130427A. The article by S. Hossenfelder addressed the does not agree with experimental results, then the theory question of whether spacetime is fundamentally continuous is wrong. However, it is becoming harder and harder to or discrete and postulates that in the case when spacetime is directly test some theories of fundamental physics at the high discrete it might have defects which would have important energy/small distance frontier exactly because this frontier is observational consequences. becoming technologically harder to reach. The Large Hadron Shortly after the formulation of general relativity Theodor Colliderisgettingnearthelimitofwhatwecandowith Kaluza proposed the idea that one could unify gravity and present accelerator technology in terms of directly reaching electromagnetism by allowing spacetime to have four spatial the energy frontier. The motivation for this special issue was dimensions rather than the three spatial dimensions which to try and collect together ideas and potential approaches to are normally observed. Depending on the size of the extra experimentally probe some of our ideas about physics at the dimensions, one might have interesting and observable con- high energy/small distance frontier. Some of the papers in this sequences connected with these extra dimensions. The article special issue directly deal with the issue of what happens to by M. Gogberashvili and P. Midodashvili investigates gravi- spacetime at small distance scales. In the paper by A. Aurilia tational and scalar field standing waves in a five-dimensional and E. Spallucci a picture of quantum spacetime is given spacetime (one time dimension and four spatial dimensions). based on the effects of ultrahigh velocity length contractions Their model gives rise to standing wave modes which could on the structure of the spacetime. The work of P.Nicolini et al. lead to observable consequences at accelerators. The paper further pursues the idea that spacetime has a minimal length. by A. Chopovsky et al. argues that there are constraints in The consequences of this minimal length are investigated certain extra dimensional models which lead to an absence in terms of the effects it would have on the gravitational of Kaluza-Klein excitations. collapse of a star to form a black hole. In the article by G. In the past decade or more, satellite experiments such Amelino-Camelia et al. the quantum structure of spacetime is asCOBE,WMAP,andPLACNKhavegivenusnewinsight studiedthroughtheFermiLATdataontheGammaRayBurst into the fundamental workings of the Universe. The results 2 Advances in High Energy Physics of these and other observational cosmological experiments inspire future investigation in this fundamental and develop- have shown that there is much in the Cosmos that we still ing field. do not understand (e.g., dark energy), but through studying these unknown quantities hopefully a better understanding Emil T. Akhmedov of the fundamental laws of physics will emerge. The work Stephen Minter of R. Peron studies and discusses how the LAGEOS satellite, Piero Nicolini launched mainly for geodetic and geodynamical purposes, Douglas Singleton is being used to perform studies of fundamental physics. In the paper by Y. V. Dumin the application of nonlocal quantum effects to the large scale structure of the Universe is investigated, using an analogy with superconducting multi- Josephson-junction loops and ultracold gases in periodic potentials.TheworkofT.Asselmeyer-MalugaandJ.Krol´ uses the idea of exotic smoothness of spacetime to propose that the initial inflationary phase of the Universe was caused by a topological phase transition. One of the first steps in the direction of formulating a quantum theory of gravity is Stephen Hawking’s results that black holes give off a thermal radiation at a temperature inversely proportional to the mass of the black hole. This result is the consequence of studying quantum fields in a curved spacetime background. However, astrophysical black holes have Hawking temperatures too small to detect with presenttechnology.Tothisendpeoplehavelookedatanalog systems which exhibit phenomenon similar to Hawking radiation. The work of P. Castorina and H. Satz looks at the idea that some unusual features of high energy collision can be explained by invoking thermal hadron production via Hawking-Unruh-like radiation. The paper by R. Balbinot and A. Fabbri looks at the Hawking effect in terms of the analog system of Bose-Einstein condensates, where one has acoustic black holes, with quanta of light (i.e., photons) replaced by quanta of sound (i.e., phonons) and with the black hole event horizonreplacedbyanacoustichorizon. Much of modern physics is based on symmetries (e.g., gauge symmetries and spacetime symmetries) and in some cases the breaking of the symmetry (e.g., the breaking of some of the gauge symmetries of the Standard Model of particle physics via the Higgs mechanism). The paper by S. A. Hojman and B. Koch looks at the gauge symmetry of E&M whichwould,ifunbroken,predictthatthephotonisexactly massless. S. A. Hojman and B. Koch derive an astrophysical lower bound on the photon mass, which when combined with the present experimental upper bound would imply that the photon is required to be exactly massless—the gauge symmetry of E&M is not broken. The work by A. G. Lebed studies potential violations of the Equivalence Principle, which forms the conceptual basis of general relativity via the superposition of quantum states. The violations of the Equivalence Principle proposed by A. G. Lebed lead to experiments which can be performed using satellites in Earth orbit. By editing the present volume, we offer a panoramic overview of the experimental and phenomenological aspects of quantum gravity and exotic quantumfield theory effects to Hindawi Publishing Corporation Advances in High Energy Physics Volume 2014, Article ID 791367, 12 pages http://dx.doi.org/10.1155/2014/791367

Review Article Testing General Relativistic Predictions with the LAGEOS Satellites

Roberto Peron1,2 1 Istituto di Astrofisica e Planetologia Spaziali (IAPS-INAF), Via del Fosso del Cavaliere 100, 00133 Roma, Italy 2 Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy

Correspondence should be addressed to Roberto Peron; [email protected]

Received 8 December 2013; Accepted 12 March 2014; Published 22 May 2014

Academic Editor: Douglas Singleton

Copyright © 2014 Roberto Peron. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

The spacetime around Earth is a good environment in order to perform tests of gravitational theories. According to Einstein’s view of gravitational phenomena, the Earth mass-energy content curves the surrounding spacetime in a peculiar way. This (relatively) quiet dynamical environment enables a good reconstruction of geodetic satellites (test masses) orbit, provided that high-quality tracking data are available. This is the case of the LAGEOS satellites, built and launched mainly for geodetic and geodynamical purposes, but equally good for fundamental physics studies. A review of these studies is presented, focusing on data, models, and analysis strategies. Some recent and less recent results are presented. All of them indicate general relativity theory as a very good description of gravitational phenomena, at least in the studied environment.

1. Introduction Among the ways to test gravitational dynamics one of the simplest is to follow (track) the motion of an object orbiting ThegeneraltheoryofrelativitybyAlbertEinsteinisnowadays in the gravitational field produced by another, bigger one (the the most precise description of the gravitational dynamics we primary). The orbiting object should be as close as possible have at our disposal. Notwithstanding its precise accounting to a point mass, in order not to perturb in a significant way of gravitational interaction as the effect of curved spacetime the gravitational field of the primary; it should be what is on the dynamics of matter and the other fundamental called a test mass. A suitable modellization (analytical or fields, it is challenged by several theoretical ideas, mainly numerical) of this system gives a prediction for the resulting related to the search for a quantum theory of gravitation orbit which can be compared with experimental tracking and to the unification of gravitation itself with the other data. Such a scheme is rather general and could be applied known fundamental interactions of nature. These issues to a variety of experimental situations. We describe here a areultimatelyrelatedtothequestionmarkonthesmall- particular such situation, given by the availability around scale structure of spacetime and to the appearance in the the Earth of objects (satellites) specifically designed to be as theory of singularities. At the astrophysical and cosmological much as possible close to the ideal concept of a test mass: levels, several unresolved problems may imply a revision of the LAGEOS satellites [1].These,aswellassimilarones, our knowledge of gravitational phenomena. All these issues have been designed, built and launched for geodetic and reflect themselves also on the smaller scale of Solar System, geodynamic purposes. In 2012 the LARES satellite has been in particular the near-Earth environment, where—thanks to launched and placed in orbit around Earth. The data from space exploration and more and more advanced experimental this new laser-ranged satellite, together with those of the techniques—many experimental setups can be conceived and LAGEOS, are expected to open the way to still more accurate put in place. tests of general relativity; See, for example, [2]. The study of 2 Advances in High Energy Physics their orbital motion, indeed, helps to characterize the fine A test mass orbiting around Earth is subjected in its details of the Earth gravitational field (and therefore of its motion to three main relativistic effects. The biggest contri- structure and composition) and to establish and maintain bution comes from the gravitoelectric curvature of spacetime a global reference frame with applications that range from induced by the Earth mass-energy: astronomy to navigation (see, e.g., [3, 4]). 𝐺𝑚 4𝐺𝑚 The LAGEOS are target for laser pulses sent from 𝑎 = E [( E − V2)𝑟+4(V ⋅𝑟) V]. Schw 𝑐2𝑟3 𝑟 (1) ground stations, used to calculate their instantaneous distance (range); the outstanding precision of this tracking This is called Einstein or Schwarzschild precession [7]. The technique, named satellite laser ranging (SLR), allows a satellite, in its motion around Earth, follows its revolution precise determination of their orbits. This can be done with in the spacetime curved by the Sun mass-energy; this (via dedicated procedures and a fine modelling of their dynamics. parallel transport of the normal to the satellite orbit) induces Along the years, the availability to the scientific community the de Sitter or geodetic precession [8]: of the ranging data allowed a variety of studies. Many of 3 𝐺𝑀 𝑋 them, as said above, are related to geodesy and geophysics. 𝑎 =2Ω×V Ω≈− (𝑉 −𝑉)× S ES . dS 2 E S 𝑐2𝑅3 (2) Atthesametime,however,itispossibletoexploitthesame ES data to perform fundamental physics tests, by comparing the (measured and reconstructed) orbit with the ones predicted In general relativity, unlike Newtonian physics, mass-energy by several, competing, gravitational theories. This very simple currents also cause effects, named gravitomagnetic (see [5]). objective requires a number of steps to be performed, which In particular, Earth intrinsic angular momentum curves will be described in the following. It has to be stressed that, spacetime and induces a further effect on the satellite orbit, in this quest, to better data better models must follow. This called Lense-Thirring effect [9, 10](alsotermeddragging of is especially true since the sought for signals often lie several inertial frames in a more general setting): orders of magnitude below the “competitive” signals. 2𝐺𝑚 3 𝑎 = E [ (𝑟×V)(𝑟⋅𝐽) + V ×𝐽]. LT 𝑐2𝑟3 𝑟2 (3) 2. Gravitational Physics Opportunities In the previous expressions, 𝑐 is the speed of light, 𝐺 the Newtonian gravitational constant, 𝑚 and 𝐽 are Earth mass As mentioned above, along the years the LAGEOS satellites E and angular momentum, 𝑟 and V are the test mass position turned out to be very good targets to be tracked. They andvelocityinthegeocentricframe,𝑀 is the Sun mass, 𝑉 materialize very finely (though not exactly) the ideal concept S E and 𝑉 aretheEarthandSungeocentricpositions,and𝑋 is of a test mass, which has to satisfy the following requirements S ES the geocentric Earth-Sun vector, with distance 𝑅 . [5]: ES Using the methods of celestial mechanics (in particular (i) no electric charge, first-order perturbation theory), the secular effects of relativistic corrections in the satellite Keplerian elements can be (ii) gravitational binding energy negligible with respect to evaluated (see, e.g., [11]). In first-order perturbation theory, rest mass-energy, two kinds of behavior for a given element can arise. The first is a term ∝ sin 𝑡 or cos 𝑡;thisiscalledperiodic.The (iii) angular momentum negligible, second is a term ∝𝑡(or higher powers); this is called secular, (iv)sufficientlysmalltoneglecttidaleffects. since it tends to accumulate over time. It turns out that the Schwarzschild term is mainly effective on the argument of An ideal test mass follows a purely gravitational orbit (a perigee geodesic in metric theories) and is therefore an appropriate 3/2 tool to study gravitational phenomena. 3(𝐺𝑀⊕) 𝜔̇ = , (4) Schw 𝑐2𝑎5/2 (1 − 𝑒2) 2.1. Relativistic Effects on Test Masses around Earth. General relativity, in its weak-field and slow-motion limit, provides an the de Sitter one on the longitude of the ascending node effective description of the gravitational phenomena around Ω̇ = |Ω| 𝜀 dS cos (5) Earth. The weak-field condition considers the spacetime curvature so small that the metric can be written as 𝑔𝜇] = (with 𝜀 obliquity of the ecliptic) and the Lense-Thirring one 𝜂𝜇] +ℎ𝜇] (Minkowski metric plus a “small” perturbation). on both node The slow-motion condition requires V ≪𝑐. Given the relative 2𝐺𝐽 Ω̇ = ⊕ smallnessofthemassesatplay,aswellasthatoftheirspeed LT 3/2 𝑐2𝑎3(1 − 𝑒2) (6) when compared with that of light, this approximation of the theory is sufficient for the purpose. A formulation of and perigee the relevant equations of motion in a geocentric noninertial reference system (nonrotating with respect to the barycentric −6𝐺𝐽⊕ 𝜔̇ = cos 𝐼. one)isgivenin[6], from which we quote the relevant LT 𝑐2𝑎3(1 − 𝑒2)3/2 (7) terms. The analyses described here are consistent with this formulation. Numerical values can be found in Table 1. Advances in High Energy Physics 3

Table 1: Rate (mas/yr) and orbital shift (over 14 days) of the different types of secular relativistic precession on LAGEOS and LAGEOS II longitude of ascending node and argument of pericenter, and their sum (1 mas/yr = 1 milli—arc—second per year).

Precession Rate (mas/yr) Shift (m) Δ𝜔̇Schw 3278.77 7.49 −2 LAGEOS ΔΩ̇LT 30.88 7.46 × 10 −2 Δ𝜔̇LT 32.00 7.31 × 10 Δ𝜔̇Schw 3351.95 7.60 −2 LAGEOS II ΔΩ̇LT 31.48 7.14 × 10 −1 Δ𝜔̇LT −57.00 −1.29 × 10

Are the expected values compatible with the uncertainty 2.2. Measurement Concept. Among the various techniques associated with tracking data? An estimate of the orbital shift used to track satellites, SLR is one of the most precise [18]. It due to each effect can be obtained for nearly circular orbits by usesthepropagationofacollimatedlaserpulsetomeasure Δ𝑥|14𝑑 ≃𝑎Δ𝛼|14𝑑;here𝑎 is the semimajor axis of the orbit and the instantaneous distance between a station on Earth and Δ𝛼 is the precession (on node or perigee) integrated over the a satellite. At the ground station a definite laser pulse is 14-day estimation period. The values can be seen in the fourth generated and—through a telescope—is sent towards the column of Table 1: given a typical SLR Normal Point precision satellite,whereitisreflectedbackinthesamedirectionby of ≃1mm,wecannoticethattheSchwarzschildsignaliswell optical elements called cube corner retroreflectors (CCR); it above the noise, while the gravitomagnetic one is barely above then comes back to the same station, and it is focused by it. the telescope and detected by a proper sensing device. By Another important issue is testing for the inverse-square precisely measuring the start and stop times of the pulses, it law behaviour of gravitation. On one side, this is useful to is then possible to recover the instantaneous station-satellite better characterize gravitation itself, especially in the short distance (range): and intermediate range. On another side, possible violations 𝑐Δ𝑡 Δ𝑠= . of this behaviour could be related to new interactions between 2 (10) bodies acting at macroscopic distances (new long range interaction (NLRI)). In addition, these NLRIs may be thought Thisisofcoursethebasicconceptofthemeasurement.In of as the residual of a cosmological primordial scalar field practice, things are made more complex from having to take related to the inflationary stage (dilaton scenario)12 [ ]. into account various phenomena, from the propagation of the Usually this supplementary interaction is modelled via pulse in the atmosphere to instrumental biases due to (among a Yukawa-type potential added to the Newtonian one, such other things) laser stability, detector, and timing device. A that, between two bodies of masses 𝑚1 and 𝑚2,respectively, hint into the complexities of each single measurement can at distance 𝑟 apart be found in [19]. Laser range observations from the various stationsontheglobearecollectedbytheInternationalLaser 𝑚 𝑚 1 2 −𝑟/𝜆 Ranging Service (ILRS) [18] and are publicly available. 𝑉 = −𝛼𝐺∞ 𝑒 . (8) 𝑟 Presently, the two (almost twin) LAGEOS satellites are among the best tracked ones through SLR. LAGEOS, 𝜆 Here the Yukawa-type part has a characteristic range launched by NASA (1976), and LAGEOS II, launched by beyond which it becomes negligible, and a relative strength NASA/ASI (1992), have been designed spherical in shape, 𝛼 𝐺 with respect to the Newtonian part ∞ is the Newtonian with high density and small area-to-mass ratio in order to 𝑟→∞ constant of gravitation in the limit .Thesuggestion minimize the effects of the nongravitational perturbations in the eighties of a possible “fifth force” [13]boostedfurther [20]. Their radius is just 30 cm and their mass about 407 kg. research on this (see also [14, 15]forreviewsand[16]for Their aluminum surface is covered with 426 CCRs. LAGEOS recent results). 𝑒 ≃ 0.004 has an almost circular orbit, with an eccentricity I , An adequate observable in order to test for such non- 𝑎 ≃ 12270 asemimajoraxis I km and an inclination over the Newtonian behaviour is the pericenter of a binary system. A 𝑖 ≃ 109.8∘ Earth’s equator I . The LAGEOS II corresponding perturbative analysis of pericenter shift has been performed 𝑒 ≃ 0.014 𝑎 ≃ 12162 𝑖 ≃ 52.66∘ elements are: II , II km and II . in [17]. The effect is maximum at a scale comparable with Their aluminum surface is covered with 426 CCRs. the system semimajor axis; therefore, in the Earth LAGEOS In the analyses described here, a multiarc technique has II case, the experiment would be sensitive mainly to an been employed [21]. The time period considered in the data 𝜆≃𝑎𝑎 interaction with ( being the semimajor axis of analysis has been divided into shorter periods, called arcs. LAGEOS II orbit). The maximum secular effect is given by For each arc, the tracking data are reduced, resulting in an estimate of the state vector (position and velocity) at ⟨Δ𝜔̇Yuk ⟩ ≃ 8.29 ⋅ 1011𝛼( / ) 2𝜋 mas yr (9) the beginning of the arc and of selected parameters for the dynamics. A very precise orbit is therefore obtained for each and it corresponds to the peak value at a range 𝜆 = 6082 km, arc, which can be expressed in terms of Keplerian elements. verycloseto1Earthradius. The arcs have a 1-day overlap, calculating the difference in 4 Advances in High Energy Physics

Table 2: Magnitude of the main disturbing effects on the LAGEOS II spacecraft (adapted from20 [ ]).

−2 Effect Estimate Magnitude (ms ) 𝐺𝑀⊕ Earth’s monopole 𝑟2 2.69 𝐺𝑀 𝑅 2 3 ⊕ ( ⊕ ) 𝐶 −1.1 × 10−3 Earth’s oblateness 𝑟2 𝑟 20 𝐺𝑀 𝑅 2 3 ⊕ ( ⊕ ) 𝐶 5.4 × 10−6 Low-order geopotential harmonics 𝑟2 𝑟 22 𝐺𝑀 𝑅 18 19 ⊕ ( ⊕ ) 𝐶 1.4 × 10−12 High-order geopotential harmonics 𝑟2 𝑟 18,18 𝐺𝑀 2 P 𝑟 2.2 ×−12 10 Moon perturbation 𝑡3 P 𝐺𝑀 2 ⨀ 𝑟 9.6 × 10−13 Sun perturbation 𝑟3 ⨀ 𝐺𝑀 𝐺𝑀 1 ⊕ ⊕ 9.8 × 10−10 General relativistic correction 𝑟2 𝑐2 𝑟 1 𝐴 𝐶 𝜌𝑉2 3.4 × 10−12 Atmospheric drag 2 𝐷 𝑀 𝐴 Φ 𝐶 ⨀ 3.2 × 10−9 Solar radiation pressure 𝑅 𝑀 𝑐 𝐴 Φ 𝑅 2 𝐶 ⨀ 𝐴 ( ⊕ ) 3.5 × 10−10 Albedo radiation pressure 𝑅 𝑀 𝑐 ⊕ 𝑟 Φ 4 𝐴 ⨀ Δ𝑇 −11 Thermal emission 𝛼 2.8 × 10 9 𝑀 𝑐 𝑇0 𝐺𝑀 𝑅 2 𝑅3 3𝑘 P ( ⊕ ) ⊕ 3.7 × 10−6 Dynamic solid tide 2 4 𝑟P 𝑟P 𝑟 −7 Dynamic ocean tide ∼0.1 of the dynamic solid tide 3.7 × 10 elements at the middle of this overlap provides time series of estimation, tracking instruments calibration, and many other residuals which contain information on the part of dynamics applications in the field of space geodesy. The software which has not been modelled (or has been mismodelled). numerically integrates the equations of motion of the satellite The fundamental observable being the range, strictly also the using the Cowell’s method (a predictor-corrector one, with residuals, in their meaning of “observed minus computed”,are a fixed time step). The equations of motion for the satellite range. The elements difference method used in these analyses are integrated in an inertial reference frame, which for retains the concept for the various Keplerian elements, as GEODYN is the mean equinox and equator of J2000. The shown in [22]. The analysis of the residuals time series allows orbit determination employs the least-squares solution of the recovering aposteriorithe signature of effects which have not range residuals: been modelled, as it was purposely done for the relativistic 𝜕𝑀 part. 𝑂 −𝑀 =−∑ 𝑖 𝑑𝑃 +𝑑𝑂, 𝑖 𝑖 𝑗 𝑖 (11) 𝑗 𝜕𝑃𝑗 2.3. Analysis Strategy. The tracking data contain the information associated with the satellite dynamics, as well as with the where 𝑂𝑖 are the range observations, 𝑀𝑖 are their modelled measurement procedure and the observational “constraints” values, 𝑑𝑃𝑗 are the corrections to the vector P of parameters (i.e., station positions, reference frames). This information to be estimated, and 𝑑𝑂𝑖 are the errors associated with each has to be extracted in some way from the data. The problem is observation. Concerning these errors, the 𝑑𝑂𝑖 account for not trivial, considering the relative magnitudes of the effects both the contribution from the noise in the observations, as involved (see Table 2). A direct comparison between the well as for the incompleteness of the mathematical model Normal Point precision and the average size of orbit shift included in the orbit determination software. The least- due to the relativistic effects shows that these effects could squares algorithm seeks to minimize the residuals 𝑂𝑖 −𝑀𝑖 be recovered once the satellite dynamics has been properly by adjusting at the same time the state vector at the epoch of modelled (for a description of the models employed see arc and the parameters selected for estimation. Section 3). The recovery of the information could be done A basic choice of the analysis has been to use the with the least-squares procedure, in which data are fit to a residuals in order to recover the relativistic effects. By model by a proper estimation of a set of selected parameters. construction, they provide a measure of the discrepancy FortheanalysestheNASA/GSFCsoftwareGEODYNII between experimental data and models; by purposely not [23, 24] has been used. This software is dedicated to satellite including relativity into the modelling set, the residuals time orbit determination and prediction, geodetic parameters series is expected to contain signatures of relativity itself. Advances in High Energy Physics 5

Table3:ModellingsetupasincludedinatypicalanalysisofLAGEOSsatellitesrangedata.

Model for Model type Reference Geopotential (static) EIGEN-GRACE02S, EGM96 [73, 74] Geopotential (time-varying, tides) Ray GOT99.2 [75] Geopotential (time-varying, nontidal) IERS Conventions (2003) [41] Third body JPL DE-403 [52] Relativistic correctionsa Parameterized post-Newtonian [6] Direct solar radiation pressure Cannonball [23] Earth albedo Knocke-Rubincam [76] Station positions ITRF 2000 [77, 78] Ocean loading Schernek and GOT99.2 tides [23, 75] Earth Rotation Parameters IERS EOP C04 [79] aIn fact, as explained in the text, these corrections have not been included in the modellization setup used in the analysis.

The basic observable being distance, the residuals are strictly The precision of the measurements is mainly related with −10 speaking on station-satellite distances. Being interested in the pulse width, which is usually ≈1×10 sdownto3× −11 effects related to individuals orbital elements, the method 10 s for the best laser ranging stations. In the case of the outlined in [22] has been employed in order to obtain derived two LAGEOS satellites, the NPs are characterized by a RMS residual time series for the various elements. This is the down to a few mm, that corresponds to an accuracy in the method that has been employed in the relativistic precessions orbit reconstruction at a few cm levels, when using the best measurements performed so far [25–33]. dynamical models. The strategy employed here could be considered as “minimal” or “conservative” in the following sense. The precisemodellingoftheorbitsrequirescomplexmodels, 3. Models which depend on thousands of parameters (see Section 3). The procedures for determining the satellite orbit at a level We underline that, while in general geodetic and geophys- comparable with the quality of tracking data require models ical problems often the majority of model parameters are not only for satellite dynamics but also for measurement estimated, in the analyses only few of them were estimated, procedure and reference frame transformations. The dynam- namely, those most directly related to the particular orbit of icsofLAGEOSsatellites,seenatthelevelenabledbythe the satellites; the other parameters were selected as consider accuracy of SLR data, is rather complex. Several gravitational parameters, that is, ones which are already known with and nongravitational effects are at work; estimates of their sufficient accuracy from other sources. magnitude are provided in Table 2 (see [20, 36]). This approach considerably simplifies the mathematical The models included in GEODYN are devoted to describe structure of the problem being solved, moreover, strongly notonlythesatellitedynamics,butalsothemeasurement lowering the chance of estimation biases. In particular, the so- procedure and the reference frame transformations. These called empirical accelerations have not been included in the models include (i) the geopotential (both in its static and set of models fitting the SLR data. These can bias the estimate dynamic part), (ii) lunisolar and planetary perturbations, procedure and corrupt, in particular, the argument of perigee (iii) solar radiation pressure and Earth’s albedo, (iv) Rubin- residuals [34]. cam and Yarkovsky-Schach effects (which need the satellite spin-axis orientation in order to be modelled), (v) drag 2.4. LAGEOS Range Data Sets. The basic products of SLR effects, (vi) SLR stations coordinates, (vii) ocean loading, observations are the Fullrate ranges. In the 1980s, a more (viii) Earth Orientation Parameters and (ix) measurement compact format has been introduced, called Normal Point procedure. Usually, the models implemented in the code also (NP), which is the one commonly used. A NP is basically include the general relativistic corrections in the so-called an “average” of the Fullrate observations over a defined time parameterized post-Newtonian (PPN) formalism [37–40]. In period (bin); for the LAGEOS the bin size amounts to 120 s. the analyses performed in order to solve for the relativistic 𝑖 𝑂 In the formation of NP for bin ,theobservation 𝑖 nearest secular precessions, such corrections were not included in the to the midpoint of the bin is located, and a fit residual FR𝑖 (a setup. residual from which systematic trends in the predictions have The particular models used for the analyses described beenremoved)iscalculated.TheNPisthencalculatedas here are listed in Table 3. For the relevant part, the Con-

NP𝑖 =𝑂𝑖 − FR𝑖 + FR𝑖, (12) ventions established by the International Earth Rotation and Reference Systems Service (IERS), which constitute where FR𝑖 denotes the mean value of FR𝑖.TheNPso the general framework for reference systems related issues calculated is characterized by the fact that its random error andmeasurementmodels,havebeenfollowedasmuchas isreducedtothatofthemeanofthebin.Moredetailscanbe possible. The reference version has been IERS Conventions found in [35]. (2003) [41]. 6 Advances in High Energy Physics

3.1. Gravitational Perturbations. The deviations of the Earth’s only of the total response to the external potentials, their gravitational field from the point mass one, due to the uncertainties are a factor of 10 larger than those of solid tides. inhomogeneous mass density distribution inside the Earth, The effect of third-body perturbations has been mod- are by far the most important source of perturbations in elled as well, using the well-established JPL Solar System the orbits of LAGEOS satellites. It is customary in geodesy ephemerides, DE-403 [52]. As discussed in Section 2.1,the andgeophysicstorepresentthegravitationalpotentialby relativistic corrections are consistent with the formulation expanding it in spherical harmonics (real basis): of [6]. In line with the chosen strategy of recovering the relativistic effects aposterioriin the residuals time series, in 𝐺𝑀 𝑈 (r) = ⊕ fact these corrections have been not included in the setup. 𝑟 3.2. Nongravitational Perturbations. An important part of the ∞ 𝑅 𝑙 𝑙 ×[1+ ∑( ⊕ ) ∑ 𝑃 (sin 𝜃) (𝐶 cos (𝑚𝜙)+𝑆 sin (𝑚𝜙))] . satellites dynamics is represented by the effects caused by 𝑟 𝑙𝑚 𝑙𝑚 𝑙𝑚 𝑙=1 𝑚=0 nongravitational forces. These, of various origin, are caused (13) by the interaction of the satellite body with the near-Earth radiation and particle environment. Such forces are typically See, for example, [36, 42, 43]. Here 𝑟, 𝜃,and𝜙 represent surface ones and depend in a complex way on the physical the polar coordinates of the point at which the potential properties of the satellite, as well as on its attitude. Even for 𝑈 is evaluated, 𝑃𝑙𝑚 are the normalized associated Legendre very simple satellites as the LAGEOS (spherical in shape, very functions, 𝑀⊕ is the Earth mass, and 𝑅⊕ is the Earth dense, and passive) these effects are relevant and, especially, mean equatorial radius. The normalized coefficients 𝐶𝑙𝑚 verydifficulttomodel.Awideliteratureisavailableonthe subject; see, for example, [20, 53–55]. and 𝑆𝑙𝑚,with𝑙 called degree and 𝑚 order, are function of the mass density distribution, and completely characterize The biggest contribution is given by the push of radiation the gravitational potential outside the distribution itself. In on the satellite surface (radiation pressure), in particular directvisibleradiationfromtheSun;alsoreflectedvisible practice, the series is truncated at some finite 𝑙max:themodel radiation from Earth (albedo) and infrared radiation emitted is then sensitive to inhomogeneities at the scale of 𝜋𝑅⊕/𝑙max. Thelowerdegreeharmonicsarerelatedtothechoiceofthe from Earth surface are important. They depend on the way reference frame in which the potential itself is expressed. Of this radiation is reflected, diffused, and absorbed by the paramount importance are the so-called zonal harmonics, satellitesurfaceandthereforeontheopticalpropertiesofthis that is, the ones with 𝑚=0: they represent the part of the surface. potentialwithrotationalsymmetryandplayanimportant The most important nongravitational effect is the direct role in the error budget of the measurements. Some care must solar radiation pressure. The resultant acceleration, for a body be put in dealing with the permanent tide. In GEODYN, a of spherical shape, can be modelled as “tide-free” geopotential is modelled, that is, one in which both 𝐴 Φ ⟨𝐷 ⟩ 2 the permanent part and the related deformation of the Moon a =−𝐶 ⊙ ( ⊙ ) ̂s, (14) ⊙ R 𝑚 𝑐 𝐷 and Sun tidal perturbations have been removed. The 𝐶2,1 and ⊙ 𝑆 2,1 coefficients describe the position of the Earth’s figure axis. where 𝐴 is the cross-sectional area of the satellite, 𝑚 is its TheEarthgravitationalfield,alsoseeninan“Earthfixed” mass, Φ⊙ is the solar radiation flux at 1 AU, 𝑐 is the speed frame, is not static: it varies in time due to a series of phenom- 𝐶 of light, and R (called radiation coefficient)summarizesthe ena, from tides to mass transport in the Earth/atmosphere optical properties of the satellite surface. The last squared system at various scales. The tidal deformations of the term is due to the modulation coming from the eccentricity Earth—both solid and ocean—and of its atmosphere are of the Earth orbit around the Sun (𝐷⊙ is the Earth-Sun of primary interest for our measurement because of their distance and ⟨𝐷⊙⟩ its average value) and ̂s is the Sun unit combined periodic variations in the gravitational attraction vector direction. Equation (14) corresponds to the so-called oftheplanetonthesatellite[44–49]. In particular, solid tides cannonball model for the direct solar radiation pressure from account for about 90% of the total response to the Moon and the Sun. This model is rather good for the LAGEOS satellites, Sun tidal disturbing potential. 𝐶 provided an estimate is done of the R parameter. Evidences A convenient way to describe these deformations is have been provided that LAGEOS II optical properties could 𝑓 through the so-called Love numbers (𝑘2,𝑚 ≃ 0.30,where𝑓 have been changed since the launch time [56]. represents the frequency of the tidal wave), which measure More subtle perturbing effects are due to the so-called the ratio between the response of the real Earth and the the- thermal forces; these are caused by an inhomogeneous oretical response of a perfect fluid sphere and are determined temperature distribution of the body (due to its finite thermal with very high accuracy because of their long-term effects on inertia), resulting in a thrust force due to emitted radiation. In geodetic satellites, as in the case of the two LAGEOS [50, 51]. particular, thermal forces depend on the satellite spin vector, In particular, in the case of solid tides, the degree 𝑙=2terms, giving different contributions on the orbit as a function of that is, those due to the quadrupole tidal potential, are the the spin orientation and rate; see, for example, [54]. We most important to be considered. Ocean tides are difficult have a seasonal-like Yarkovsky-Schach effect in the case of a to model because of the greater complexity of the involved rapidly spinning satellite, and a diurnal-like Yarkovsky-Schach phenomena. Indeed, even if ocean tides account for ≈10% effect when the fast rotation approximation is no more valid. Advances in High Energy Physics 7

0.25 20 0.2 15 0.15 10 Count

RMS (m) 0.1 5 0.05 0 49500 50000 50500 51000 51500 52000 52500 53000 53500 −0.015 −0.01 −0.005 0 0.005 0.01 Time (MJD) Residual (m) (a) (b)

Figure 1: LAGEOS II postfit weighted RMS (a) and residuals in range (b) computed for each of the arcs in which it has been divided the analysed time period.

The Yarkovsky-Rubincam effect [57–59], or Earth-Yarkovsky an estimate of the initial conditions (state vector) and of effect,isrelatedtotheinfraredradiationemittedbytheEarth’s selected parameters. The models employed (see Table 3) surface. enabledaverygoodfitofthedata,ascanbeseeninthe In order to model—as accurately as possible—the per- statistics. In particular, in Figure 1 the postfit weighted RMS turbing thermal thrust effects, and especially the Yarkovsky- and a histogram of the residuals in range are shown. These Schach effect, a detailed description of the evolution of the plots are related to runs dedicated to analyze the LAGEOS spin-axisiscrucial.Severalauthorshavefocusedonthis IIperigeebehavior(see[31]). The average RMS is somewhat problem and tried to explain the evolution of the LAGEOS higher than the “ideal” level that could be expected based satellites spin-axis, in either an analytical [60–62]oramore on data quality: this is due to the fact that in this analysis empirical approach [63]. no relativistic effects were inserted in the modellization set, thereby lowering the overall accuracy. In this way, however, 3.3. Empirical Accelerations. Amodellizationpiecethatis the residuals contain useful information, which is indeed often used in precise orbit determination is given by the so- related to relativity itself. This can be seen in the histogram called empirical accelerations. These are general acceleration of the residuals in range: their distribution appears close to terms added to the equations of motion and are aimed at but is not exactly Gaussian, indicating that some information modelling small otherwise unknown effects which may be is still present in the residuals themselves (more information relevant to the dynamics. They are usually decomposed in the onthiscanbefoundin[33]).Thesamereasoningappliesalso three Gauss directions 𝑟̂, ̂𝑡, 𝑤̂ (radial, transverse, and out-of- to the analysis reported in [29];inthatcase,however,onlythe plane), in the form (for each component) gravitomagnetic contribution was taken out. 𝐴 (𝑡) =𝐴 +𝐴 𝑀+𝐴 𝑀, 0 1 sin 2 cos (15) 5. Zonal Harmonics Related where 𝑀 represents the satellite mean anomaly, aiming thus Uncertainties and Combination Formula for at modelling constant or once-per-revolution accelerations. Lense-Thirring Measurements Thisorbitmodellingtoolisusefulwhenlongwavelengths Detailed error budget calculations show the importance of orbit errors, including secular disturbing effects, need to be the zonal harmonics uncertainties in the overall effective- removed, as well as for long-period resonances and also ness of the analysis procedure in extracting the relativistic nongravitational perturbations that are not included in the signals. This is especially true in the case of Lense-Thirring software dynamical model. Experience shows that, while they measurements employing the nodal residuals. In particular, areusefultoimprovethefitquality,theycaneasilybiasthe 𝐶 estimation of other quantities. the quadrupole coefficient 20 has been found to be the major We highlight once more that, in order to avoid the source of uncertainty. Its secular effect on the nodal longitude orbit corruption, in particular of the satellite argument of is given by pericenter,theyhavenotbeenusedduringthedatareduction. √ 2 ̇ 3 5 𝑅 cos 𝐼 Ω = 𝑛( ) 𝐶20. (16) class 2 𝑎 (1 − 𝑒2)2 4. Data Reduction In the analyses described here more than ten years of See, for example, [64, 65](weuseherethenormalized LAGEOS and LAGEOS II laser tracking data, provided by coefficients 𝐶𝑙0 instead of the nonnormalized 𝐶𝑙0 or the 𝐽𝑙; √ ILRS, have been reduced using the GEODYN II software. The we remember that 𝐽2 =−𝐶20 =− 5 𝐶20). Therefore the orbit selected period has been divided into 15-day arcs, with a 1- of the satellite is subject to a classical precession whose value day overlap. For each of them, the data reduction provides is much higher than the relativistic (gravitomagnetic) one to 8 Advances in High Energy Physics

−1 Table 4: Values (in mas yr ) of the nodal precession for LAGEOS and LAGEOS II orbits due to relativistic and classical gravitational effects.

Effect LAGEOS LAGEOS II Lense-Thirring 30.88 31.48 10 10 𝐶20 (EGM96) 2.702 × 10 −4.982 × 10 3 3 𝛿𝐶20 (EGM96) −3.240 × 10 5.975 × 10 3 3 𝛿𝐶20 (EIGEN-GRACE02S) −2.960 × 10 5.458 × 10

be measured; see Table 4 (for the purpose of this work we 4 consider only the quadrupolar part of this classical precession). Of course, what really matters is the unknown part of 2 this precession due to the uncertainty in the accepted value of 0 𝐶20; its value based on two geopotential models is shown in Table 4. This precession actually hides the relativistic one and

𝛿𝜇 −2 mustbehandledinsomeway.Tothisaim,in[66] a procedure based on the combination of orbital residuals from more than −4 one satellite to get rid of this masking precession has been developed; see also [67]. −6 The simplest case is that of two satellites, along with their two nodal longitudes. Let us consider a single arc of orbit −8 determination; the procedure allows obtaining a residual for 50000 51000 52000 53000 54000 each orbital element, in particular for the nodal longitude: Time (MJD) 𝛿Ω̇(in [22] it is shown that the residuals obtained with their method are in fact rates). If the modellization setup is Figure 2: Relativistic effect from the combined nodal longitude accurate enough, this residual, that is, the difference between residuals of LAGEOS and LAGEOS II (squares). The average value thecalculatedandtheobservedvalue,ismainlyfunction is 1.056, to be compared with the general relativistic prediction value, 1 (continuous line). MJD stands for “Modified Julian Day”; of two quantities, the classical quadrupole precession and the considered time period for the analysis starts on 1993. the relativistic (Lense-Thirring) one, so that the following equationcanbereasonablyconsideredtohold: 𝛿Ω=𝛿̇ Ω̇ +𝛿Ω̇ . The total uncertainty from (17) can therefore be written as class rel (17) 𝛿Ω=𝑁̇ 𝛿𝐶 + Ω̇ 𝛿𝜇. In writing this equation we neglect higher-degree multipoles 20 20 LT (20) and other sources of perturbation for the node. Regarding the two terms on the right-hand side of the previous equation, We see that each residual can be expressed as a function of 𝛿𝐶 𝛿𝜇 these can be expressed as follows. Following (16), the classical two uncertainties, 20 and .Assuch,(20)isnotmuch Ω̇ =𝑁𝐶 𝑁 useful. But adding a further observable (i.e., taking the nodal precession can be written as class 20 20,with 20 function of Earth equatorial radius and satellite orbit. Upon residualsoftwosatellites,asLAGEOSand LAGEOS II) one can construct a system of two equations: the assumption that the biggest uncertainty comes from 𝐶20, one can write 𝛿Ω̇I =𝑁I 𝛿𝐶 + Ω̇I 𝛿𝜇, 20 20 LT 𝜕Ω̇ (21) 𝛿Ω̇ = class 𝛿𝐶 =𝑁 𝛿𝐶 . 𝛿Ω̇II =𝑁II 𝛿𝐶 + Ω̇II 𝛿𝜇 class 20 20 20 (18) 20 20 LT 𝜕𝐶20 which can be solved to obtain 𝛿𝜇: Inthesameway,therelativisticprecessioncanbewrittenas ̇ ̇ ̇ Ωrel = Ω 𝜇;hereΩ comes from (6), while 𝜇 is an empirical I ̇II II ̇I LT LT 𝑁20𝛿Ω −𝑁20𝛿Ω parameter measuring the actual value of the relativistic effect 𝛿𝜇 = . (22) 𝑁I Ω̇II −𝑁II Ω̇I ((𝜇=0) in Newtonian physics, (𝜇=1) in general relativity). 20 LT 20 LT In fact, in the post-Newtonian framework 𝜇 = (1 + 𝛾)/2,with 𝛿𝜇 𝛿𝐶 𝛾 the PPN parameter quantifying how much space curvature This ,togetherwith 20, is just the right one to account ̇ 𝛿Ω̇I 𝛿Ω̇II is produced by unit rest mass; see [68]. In writing Ωrel this for the total residuals and . way, we in fact parameterize the general relativistic prediction It is worth emphasizing two things. First, the solution with 𝜇 andassumethattheerroriscontainedinthisempirical given by (22) clearly does not contain 𝛿𝐶20 and so the parameter (in fact, this equals to admit no aprioriknowledge expression overcomes the problem of the uncertainty in 𝐶20, on the amplitude of Lense-Thirring effect), so both static and time-dependent. Second, 𝛿𝜇 as expressed by (22)isrelatedtoasingle arc of orbit determination, so what ̇ 𝜕Ωrel one obtains is a time series covering the period of analysis. 𝛿Ω̇ = 𝛿𝜇 = Ω̇ 𝛿𝜇. (19) rel 𝜕𝜇 LT The outcome can be seen in Figure 2,inwhichtherelativistic Advances in High Energy Physics 9

0.4 0.2 0.3 ) 0.2 ) 0 −1 −1

0.1 −0.2 0 Residual (as yr (as Residual Residual (as yr (as Residual −0.1 −0.4 −0.2

50000 51000 52000 53000 54000 50000 51000 52000 53000 54000 Time (MJD) Time (MJD) (a) (b)

−1 −1 Figure 3: Nodal longitude residuals, LAGEOS (a) and LAGEOS II (b). The average values are 141.3 as yr and −167.3 as yr for LAGEOS and LAGEOS II, respectively. parameter 𝛿𝜇 fromLAGEOSandLAGEOSIIcombined In [30] a dedicated analysis has been performed, focused nodal residuals is plotted as a function of time. Notice the on the LAGEOS II perigee behavior. In that case, the residuals cancellation of the nonseasonal, “anomalous” change in Earth being analyzed were directly those of the Keplerian element: quadrupole (apparent in the nodal longitude residuals; see the combination was not necessary, since the overall mag- Figure 3), as reported in [29, 69]. nitude of the relativistic effects (Schwarzschild plus Lense- Thirring) is much bigger. A fit value Δ𝜔̇meas = 3306.58 mas/yr 6. A Review of Recent Measurements for the slope has been reported (see Figure 5). This value canbetakenasanestimateofthetotalrelativisticperigee The idea of using laser-ranged satellites in order to test precession, given by selected predictions of general relativity theory dates back to Δ𝜔̇rel ≃𝜀 Δ𝜔̇ +𝜀 Δ𝜔̇ . (24) the 1970s and 1980s. The test of Schwarzschild precession has Schw Schw LT LT been discussed in [70]. The measurement of Lense-Thirring The slope estimate has small variations depending on the effect has been suggested by [71]andproposedby[72]. We number of periodic effects which are fitted together with review here some recent results which come out from precise the linear trend. The following conservative result for the orbit determinations of LAGEOS and LAGEOS II satellites. magnitude of the total relativistic effect has been reported, at These analyses produced residuals time series of the satellites the post-Newtonian level: Keplerian elements in the way discussed in Section 2.2.The 𝜀 =1+(0.28 ± 2.14) ×10−3, expected relativistic signals, both in longitude of ascending 𝜔 (25) node and in argument of perigee, were of a secular type, so where 𝜀𝜔 =1in general relativity. Since the dominant they should appear as a secular trend upon time integration contribution in (24) comes from of the relevant time series. 2+2𝛾−𝛽 In [28, 29, 32]suchananalysishasbeenperformedon 𝜀 = , (26) LAGEOS and LAGEOS II tracking data, and an accuracy Schw 3 of about 10% was achieved in the test of the Lense-Thirring the estimate given by (25) is mainly a measurement of such effect as predicted by General Relativity. We concentrate here a combination of 𝛾 and 𝛽 PPN parameters. A preliminary on [29]. Given the limitations due to Earth gravitational error budget for the measurement, taking into account the quadrupole uncertainty, the node residuals of both satellites various systematics, estimated the error to be at 2% level [17]. have been combined as discussed in Section 5.Thecorre- A complete error budget has been reported in [33]: sponding combined time series have been fitted with a secular −3 −3 −2 trend plus a number of periodic terms (in order to account for 𝜖𝜔 − 1 = [−0.12 ⋅ 10 ±2.10⋅10 ] ± [1.74 ⋅ 10 ], (27) mismodelling in some perturbations). They report a value of whereinthefirstsquarebracketitisshowntheresultand 𝜇 = 0.984 ±5%−10% (23) thestatisticalerrorfromthebestfitandinthesecondsquare brackettheerrorbudgetduetothegravitationalandnon- using the EIGEN-GRACE02S as geopotential model (the gravitational systematic sources of error is represented. reportedvaluehasbeenobtainedfittingthecombinedresidu- The measured value for the argument of perigee preces- als with a secular trend plus ten periodic terms). See Figure 4 sion can also be used to constrain a non-Newtonian contri- for the related fit. bution to the satellite dynamics, as discussed in Section 2.1. 10 Advances in High Energy Physics

600 7. Conclusions

500 There is a great deal of interest in testing the experimental consequences of general relativity, given the many challenges to the theory. Alternative theories, devised to solve at least 400 in part some of these issues, have testable consequences in the weak-field and slow-motion conditions at work in the 300 Solar System, and in particular around Earth. Thanks to advances in experimental techniques, these consequences Milliarcsec 200 can be nowadays explored. In particular, laser ranging to geodetic satellites in orbit around Earth offers the possibility 100 of studying with high precision the motion of objects which can be considered very good approximations to a test mass. Their geodetic motion shows some peculiarities with respect 0 024681012to the Newtonian one; in particular, some of the Keplerian Years elements undergo a precession, also due to gravitomagnetism (the rotating Earth being the source). Figure 4: Figure 14 of [29]. A combination of LAGEOS and Analyses of LAGEOS satellites tracking data, aimed at a LAGEOS II node residuals together with their fit with a secular trend precise reconstruction of their orbits, have been discussed. plus ten periodic terms is shown. Digging in their dynamics down to the level of the small relativistic effects is made possible not only by the precise

4 laser range data, but also by the accurate modellization ×10 of their motion (gravitational and nongravitational parts, 5 reference frames, and measurement models). Some results 4.5 have been discussed, which provide confirmation of gen- 4 eral relativistic predictions (Schwarzschild precession, Lense- Thirring effect) and rule out to a high degree alternative 3.5 theories (NLRIs/Yukawa potential). Such investigations have 3 still a great potential of improvement and are being carried on in order to further constrain the space of possible theories. 2.5 2 Conflict of Interests 1.5 The author declares that there is no conflict of interests 1 regarding the publication of this paper. 0.5 0 Acknowledgments Argument of pericenter integrated residuals (mas) residuals integrated pericenter of Argument −0.5 0 1000 2000 3000 4000 5000 TheauthorwouldliketothankDavidM.Lucchesi(IAPS- Time (days) INAF) and an anonymous referee. The author also acknowl- edges the ILRS for providing high-quality laser ranging data Figure 5: Figure 1 of [30].Theperigeeresidualsareshowntogether of the two LAGEOS satellites. with their fit with a secular trend plus four periodic terms. References Indeed, the absence of such a signal in the residuals time [1] S. C. Cohen, R. W. King, R. Kolenkiewicz, R. D. Rosen, and B. series allows placing a strong constraint to the strength 𝛼 at E. Schutz, “LAGEOS scientific results,” Journal of Geophysical 𝜆≃𝑎.In[30] it has been reported the following upper bound: Research,vol.90,pp.9215–9438,1985. [2] I. Ciufolini, A. Paolozzi, E. Pavlis et al., “Testing General −12 |𝛼| ≃ |1.0 ± 8.9| ×10 , (28) Relativity and gravitational physics using the LARES satellite,” European Physical Journal Plus,vol.127,article133,2012. [3] G. Xu, Sciences of Geodesy—I: Advances and Future Directions, this result has been improved in [33]: Springer, Berlin, Germany, 2010.

−12 [4] G. Xu, Sciences of Geodesy—II: Innovations and Future |𝛼| ≃ |(0.5 ± 8.0) ±69| ⋅10 . (29) Developments, Springer, Berlin, Germany, 2013. [5] I.CiufoliniandJ.A.Wheeler,Gravitation and Inertia, Princeton These results represent a huge improvement with respect University Press, Princeton, NJ, USA, 1995. to previous constraints at this scale and are comparable with [6] C. Huang, J. C. Ries, B. D. Tapley, and M. M. Watkins, “Rela- the Lunar Laser Ranging results. tivistic effects for near-earth satellite orbit determination,” Advances in High Energy Physics 11

Celestial Mechanics and Dynamical Astronomy,vol.48,no.2, [26] I. Ciufolini, D. Lucchesi, F. Vespe, and F. Chieppa, “Meas- pp.167–185,1990. urement of gravitomagnetism,” Europhysics Letters,vol.39,no. [7] A. Einstein, “Die grundlage der allgemeinen relativitatstheorie,”¨ 4, pp. 359–364, 1997. Annals of Physics,vol.354,pp.769–822,1916. [27] I. Ciufolini, E. Pavlis, F. Chieppa, E. Fernandes-Vieira, and J. [8]W.deSitter,“OnEinstein’stheoryofgravitationandits Perez-Mercader,´ “Test of general relativity and measurement of astronomical consequences. Second paper,” Monthly Notices of the lense-thirring effect with two earth satellites,” Science,vol. the Royal Astronomical Society,vol.77,no.155,184pages,1916. 279, no. 5359, pp. 2100–2103, 1998. [9]J.LenseandH.Thirring,“Uber¨ den Einfluss der Eigenrotation [28] I. Ciufolini and E. C. Pavlis, “A confirmation of the general der Zentralkorper¨ auf die Bewegung der Planeten und Monde relativistic prediction of the Lens-Thirring effect,” Nature,vol. nach der Einsteinschen GravitationstheoriePhysikalische 431, no. 7011, pp. 958–960, 2004. Zeitschrift,” vol. 19, pp. 156–163, 1918. [29] I. Ciufolini, E. C. Pavlis, and R. Peron, “Determination of [10] B.Mashhoon,F.W.Hehl,andD.S.Theiss,“Onthegravitational frame-dragging using Earth gravity models from CHAMP and effects of rotating masses: the Thirring-Lense papers,” General GRACE,” New Astronomy,vol.11,no.8,pp.527–550,2006. Relativity and Gravitation,vol.16,no.8,pp.711–750,1984. [30] D. M. Lucchesi and R. Peron, “Accurate measurement in the [11] M. H. Soffel, Relativity in Astrometry, Celestial Mechanics and field of the earth of the general-relativistic precession of the Geodesy, Springer, Berlin, Germany, 1989. LAGEOS II pericenter and new constraints on non-newtonian gravity,” Physical Review Letters,vol.105,no.23,ArticleID [12] T. Damour, F. Piazza, and G. Veneziano, “Violations of the 231103, 2010. equivalence principle in a dilaton-runaway scenario,” Physical Review D,vol.66,no.4,ArticleID046007,15pages,2002. [31] R. Peron, “Einstein is still right: New tests of gravitational dynamics in the field of the Earth,” Nuovo Cimento C,vol.36, [13] E. Fischbach, D. Sudarsky, A. Szafer, C. Talmadge, and S. H. pp.125–134,2013. Aronson, “Reanalysis of the Eoumltvos¨ experiment,” Physical Review Letters,vol.56,no.1,pp.3–6,1986. [32] I. Ciufolini, A. Paolozzi, R. Koenig et al., “Fundamental physics and general relativity with the LARES and LAGEOS satellites,” [14] E. Fischbach, G. T. Gillies, D. E. Krause, J. G. Schwan, and C. Nuclear Physics B—Proceedings Supplements,vol.243-244,pp. Talmadge, “Non-Newtonian gravity and new weak forces: an 180–193, 2013. index of measurements and theory,” Metrologia,vol.29,no.3, pp.213–260,1992. [33] D. M. Lucchesi and R. Peron, “LAGEOS II pericenter general relativistic precession (1993–2005): error budget and [15]A.S.GoldhaberandM.M.Nieto,“Photonandgravitonmass constraints in gravitational physics,” Physical Review D,vol.89, limits,” Reviews of Modern Physics,vol.82,no.1,pp.939–979, no. 8, Article ID 082002, 2014. 2010. [34] D. M. Lucchesi, “The LAGEOS satellites orbital residuals [16] J. H. Gundlach, S. Schlamminger, and T. Wagner, “Laboratory determination and the way to extract gravitational and non- tests of the equivalence principle at the university of washing- gravitational unmodeled perturbing effects,” Advances in Space ton,” Space Science Reviews,vol.148,no.1–4,pp.201–216,2009. Research,vol.39,no.10,pp.1559–1575,2007. [17] D. M. Lucchesi, “LAGEOS II perigee shift and Schwarzschild [35] A. T. Sinclair, Data Screening and Normal Point Formation— gravitoelectric field,” Physics Letters A: General, Atomic and Re—Statement of Herstmonceux Normal Point Recommen- Solid State Physics,vol.318,no.3,pp.234–240,2003. dation, 1997, http://ilrs.gsfc.nasa.gov/products formats proce- [18]M.R.Pearlman,J.J.Degnan,andJ.M.Bosworth,“The dures/normal point/np algo.html. International Laser Ranging Service,” Advances in Space [36] O. Montenbruck and E. Gill, Satellite Orbits. Models, Methods Research,vol.30,no.2,pp.135–143,2002. and Applications, Springer, Berlin, Germany, 2000. [19] J. J. Degnan, “Satellite laser ranging: current status and future [37] K. Nordtvedt, “Equivalence principle for massive bodies. II. prospects,” IEEE Transactions on Geoscience and Remote Theory,” Physical Review,vol.169,no.5,pp.1017–1025,1968. Sensing, vol. 23, no. 4, pp. 398–413, 1985. [38] C. M. Will, “Theoretical frameworks for testing relativistic [20] A.Milani,A.M.Nobili,andP.Farinella,Non-Gravitational Per- gravity. II. Parametrized post-Newtonian hydrodynamics, and turbations and Satellite Geodesy,AdamHilger,Bristol,UK,1987. the Nordtvedt effect,” The Astrophysical Journal,vol.163,pp. [21] A. Milani and G. F. Gronchi, Theory of Orbit Determination, 611–628, 1971. Cambridge University Press, Cambridge, UK, 2010. [39] C. M. Will and K. Nordtvedt, Jr., “Conservation laws and [22] D. M. Lucchesi and G. Balmino, “The LAGEOS satellites preferred frames in relativistic gravity. I. Preferred-frame orbital residuals determination and the Lense-Thirring effect theories and an extended PPN formalism,” The Astrophysical measurement,” Planetary and Space Science,vol.54,no.6,pp. Journal,vol.177,pp.757–774,1972. 581–593, 2006. [40] K. Nordtvedt, Jr. and C. M. Will, “Conservation laws and pre- [23] D. E. Pavlis et al., GEODYN II Operations Manual. NASA ferred frames in relativistic gravity. II. Experimental evidence GSFC, 1998. to rule out preferred-frame theories of gravity,” The Astro- [24] B. Putney, R. Kolenkiewicz, D. Smith, P. Dunn, and M. physical Journal,vol.177,pp.775–792,1972. H. Torrence, “Precision orbit determination at the NASA [41] D. D. McCarthy and G. Petit, “IERS Conventions (2003),” IERS Goddard Space Flight Center,” Advances in Space Research,vol. Technical Note 32, IERS, 2004. 10, no. 3-4, pp. 197–203, 1990. [42] B. Bertotti, P. Farinella, and D. Vokrouhlicky,´ “Physics of [25] I. Ciufolini, D. Lucchesi, F. Vespe, and A. Mandiello, “Meas- the Solar System. Dynamics and Evolution, Space Physics, urement of dragging of inertial frames and gravitomagnetic and Spacetime Structure,” in Astrophysics and Space Science field using laser-ranged satellites,” Il Nuovo Cimento A,vol.109, Library, vol. 293, Kluwer Academic Publishers, Dordrecht, The no. 5, pp. 575–590, 1996. Netherlands, 2003. 12 Advances in High Energy Physics

[43] B. Hofmann-Wellenhof and H. Moritz, Physical Geodesy, [64] W. M. Kaula, Theory of Satellite Geodesy. Applications of Springer, Berlin, Germany, 2006. Satellites to Geodesy, Blaisdell, Waltham, Mass, USA, 1966. [44] P.Melchior, The Tides of the Planet Earth,PergamonPress,New [65] L. Iorio, “The impact of the static part of the earth’s gravity field York, NY, USA, 1981. on some tests of general relativity with satellite laser ranging,” [45] K. Lambeck, “Tidal dissipation in the oceans: astronomical, Celestial Mechanics and Dynamical Astronomy,vol.86,no.3, geophysical and oceanographic consequences,” Royal Society pp. 277–294, 2003. of London Philosophical Transactions Series A,vol.287,pp. [66] I. Ciufolini, “On a new method to measure the gravitomagnetic 545–594, 1977. field using two orbiting satellites,” Il Nuovo Cimento A,vol.109, [46] Schwiderski, E. W.: Ocean tides, and part i:, “Global ocean tidal no.12,pp.1709–1720,1996. equations,” Marine Geodesy,vol.3,no.1–4,pp.161–217,1980. [67] R. Peron, “On the use of combinations of laser-ranged satellites [47] E. W. Schwiderski, “Ocean tides, part ii: a hydrodynamical orbital residuals to test relativistic effects around Earth,” interpolation model,” Marine Geodesy,vol.3,no.1–4,pp. Monthly Notices of the Royal Astronomical Society,vol.432,pp. 219–255, 1980. 2591–2595, 2013. [48] J. M. Wahr, “A normal mode expansion for the forced response [68] C. M. Will, “The confrontation between general relativity and of a rotating Earth,” Geophysical Journal, Royal Astronomical experiment,” Living Reviews in Relativity,vol.4,article4,2001. Society,vol.64,no.3,pp.651–675,1981. [69] C. M. Cox and B. F. Chao, “Detection of a large-scale mass [49] J. M. Wahr, “Body tides on an elliptical, rotating, elastic and redistribution in the terrestrial system since 1998,” Science,vol. oceanless earth,” Geophysical Journal, Royal Astronomical 297, no. 5582, pp. 831–833, 2002. Society,vol.64,pp.677–703,1981. [70] D. P. Rubincam, “General relativity and satellite orbits: the [50] M. K. Cheng, C. K. Shum, and B. D. Tapley, “Determination motion of a test particle in the Schwarzschild metric,” Celestial of long-term changes in the Earth’s gravity field from satellite Mechanics,vol.15,no.1,pp.21–33,1977. laser ranging observations,” Journal of Geophysical Research: [71] L. Cugusi and E. Proverbio, “Relativistic effects on the motion Solid Earth,vol.102,no.10,pp.22377–22390,1997. of earth’s artificial satellites,” Astronomy and Astrophysics,vol. [51] B. Wu, P. Bibo, Y. Zhu, and H. Hsu, “Determination of love 69, pp. 321–325, 1978. numbers using satellite laser ranging,” The Geodetic Society of [72] I. Ciufolini, “Measurement of the Lense-Thirring drag on Japan,vol.47,pp.174–180,2001. high-altitude, laser-ranged artificial satellites,” Physical Review [52] E.M.Standish,X.X.Newhall,J.G.Williams,andW.M.Folkner, Letters,vol.56,no.4,pp.278–281,1986. “JPL Planetary and Lunar Ephemerides, DE403/LE403,” Tech. [73] C. Reigber, R. Schmidt, F. Flechtner et al., “An Earth gravity Rep. 314. 10–127, JPL IOM, 1995. field model complete to degree and order 150 from GRACE: [53] D. M. Lucchesi, “Reassessment of the error modelling of non- EIGEN-GRACE02S,” Journal of Geodynamics,vol.39,no.1,pp. gravitational perturbations on LAGEOS II and their impact in 1–10, 2005. the Lense-Thirring determination. Part I,” Planetary and Space [74] F.G.Lemoine,S.C.Kenyon,J.K.Factoretal.,“Thedevelopment Science,vol.49,no.5,pp.447–463,2001. of the joint nasa gsfc and the national imagery and mapping [54] D. M. Lucchesi, “Reassessment of the error modelling of non- agency (nima) geopotential model egm96,” Technical Paper gravitational perturbations on LAGEOS II and their impact NASA/TP-1998-206861, NASA GSFC, 1998. in the Lense-Thirring derivation. Part II,” Planetary and Space [75] R. D. Ray, “A Global Ocean Tide Model From TOPEX/ Science, vol. 50, no. 10-11, pp. 1067–1100, 2002. POSEIDON Altimetry: GOT99. 2,” Technical Paper NASA/ [55] J. I. Andres´ de la Fuente, [Ph.D. thesis], Delft University, 2007. TM-1999-209478, Goddard Space Flight Center, Greenbelt, [56]D.M.Lucchesi,I.Ciufolini,J.I.Andres´ et al., “LAGEOS II Md,USA,1999. perigee rate and eccentricity vector excitations residuals and [76] D. P.Rubincam, P.Knocke, V.R. Taylor, and S. Blackwell, “Earth the Yarkovsky-Schach effect,” Planetary and Space Science,vol. anisotropic reflection and the orbit of LAGEOS,” Journal of 52,no.8,pp.699–710,2004. Geophysical Research,vol.92,pp.11662–11668,1987. [57] D. P. Rubincam, “LAGEOS orbit decay due to infrared [77] Z. Altamimi, P. Sillard, and C. Boucher, “ITRF2000: new radiation from earth,” Journal of Geophysical Research,vol.92, release of the International Terrestrial Reference Frame for pp.1287–1294,1987. earth science applications,” Journal of Geophysical Research B: [58] D. P.Rubincam, “Yarkovsky thermal drag on LAGEOS,” Journal Solid Earth,vol.107,no.10,pp.2–19,2002. of Geophysical Research, vol. 93, no. 1113805, p. 13810, 1988. [78] C. Boucher, Z. Altamimi, P. Sillard, and M. Feissel-Vernier, [59] D. P. Rubincam, “Drag on the LAGEOS satellite,” Journal of “The ITRF2000,” IERS Technical Note 31, IERS, 2004. Geophysical Research,vol.95,pp.4881–4886,1990. [79] “International Earth Rotation Service: EOP Combined Series [60] B. Bertotti and L. Iess, “The rotation of LAGEOS,” Journal of EOP,” Tech. Rep. C04, IERS. Geophysical Research,vol.96,no.2,pp.2431–2440,1991. [61] S. Habib, D. E. Holz, A. Kheyfets, R. A. Matzner, W. A. Miller, and B. W. Tolman, “Spin dynamics of the LAGEOS satellite in support of a measurement of the Earths gravitomagnetism,” Physical Review D,vol.50,no.10,pp.6068–6079,1994. [62] S. E. Williams, The Lageos satellite: a comprehensive spin model andanalysis[Ph.D.thesis],NorthCarolinaStateUniversity, Raleigh, NC, USA, 2002. [63] J. Ries, R. Eanes, and M. M. Watkins M, “Spin vector influence on LAGEOS ephemeris,” presented at the Second Meeting of IAG Special Study Group 2. 130, Baltimore, Md, USA, 1993. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2014, Article ID 241831, 8 pages http://dx.doi.org/10.1155/2014/241831

Research Article Nonlocal Quantum Effects in Cosmology

Yurii V. Dumin1,2 1 Sternberg Astronomical Institute (GAISh) of Lomonosov Moscow State University, Universitetski Prosp. 13, Moscow 119991, Russia 2 Space Research Institute (IKI) of Russian Academy of Sciences, Profsoyuznaya Street 84/32, Moscow 117997, Russia

Correspondence should be addressed to Yurii V. Dumin; [email protected]

Received 7 January 2014; Revised 7 April 2014; Accepted 13 April 2014; Published 22 May 2014

Academic Editor: Stephen Minter

Copyright © 2014 Yurii V. Dumin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

Since it is commonly believed that the observed large-scale structure of the universe is an imprint of quantum fluctuations existing at the very early stage of its evolution, it is reasonable to pose the question: do the effects of quantum nonlocality, which are well established now by the laboratory studies, manifest themselves also in the early universe? We try to answer this question by utilizing the results of a few experiments, namely, with the superconducting multi-Josephson-junction loops and the ultracold gases in periodic potentials. Employing a close analogy between the above-mentioned setups and the simplest one-dimensional Friedmann- Robertson-Walker cosmological model, we show that the specific nonlocal correlations revealed in the laboratory studies might be of considerable importance also in treating the strongly nonequilibrium phase transitions of Higgs fields in the early universe. Particularly, they should substantially reduce the number of topological defects (e.g., domain walls) expected due to independent establishment of the new phases in the remote spatial regions. This gives us a hint on resolving a long-standing problem of the excessive concentration of topological defects, inconsistent with observational constraints. The same effect may be also relevant to the recent problem of the anomalous behavior of cosmic microwave background fluctuations at large angular scales.

1. Introduction disconnected regions). However, the existence of EPR correlations is well confirmed now by a lot of laboratory experi- The concept of quantum nonlocality originates actually from ments. In fact, these correlations look much less surprising the paper by Einstein et al. (EPR) [1] who posed the problem if we keep in mind the fact that both light cones include of correlation between the measurements of two physical thesamecommonsourceinthepast.Itmightbereasonable objects located in the causally disconnected regions of space, to emphasize also that despite of a “superluminal” character that is, beyond the light cones of each other. As regards of the EPR correlations, they cannot be employed for a the modern and most frequently used in the experiments faster-than-light communication because the outcomes of form, this phenomenon can be illustrated in Figure 1.Here, correlated measurements of 𝑠1 and 𝑠2 in the points 𝑥1 and 𝑥2 an original particle of zero spin decays at the instant 𝑡=0 are random. into two particles with equal but oppositely directed spins Turning attention to cosmology, we should first of all 𝑠1 and 𝑠2, which subsequently move from each other in the mention that it is widely believed now that the observed opposite directions. Next, if measurements of the spins of large-scale structure of the universe is an imprint of quantum both particles are performed in the remote spatial points 𝑥1 fluctuationsexistingattheveryearlystagesofcosmological and 𝑥2 at the same instant of time, their values turn out to be evolution. Therefore, it becomes interesting to pose the perfectly correlated (𝑠1 =−𝑠2) just because of the law of spin question: can the effects of quantum nonlocality, similar to conservation. EPR correlations, manifest themselves also in cosmology? In At first sight, such correlation is quite surprising because particular, it should be kept in mind that Higgs field, filling the measurements are performed in the spots of space-time the entire universe and giving masses to the elementary par- lying beyond the mutual light cones (i.e., in the causally ticles, represents actually a kind of Bose-Einstein condensate 2 Advances in High Energy Physics

t t t Future light Future light ∗ Instant of cone cone Symmetry-broken observation state of the field Observable part of the s2 s1 Detectors EPR correlation Domains universe of various phases Domain Past Past walls light cone light cone t 0 End of phase transition x2 Source x1 x Onset of 0 r phase transition Figure 1: Sketch of the typical laboratory ERP experiment. 𝜉 Seeds of new phase Symmetric state of the field Big Bang

(BEC), that is, a macroscopic quantum state in which the Figure 2: Sketch of development of the phase transition in the specific quantum correlations may naturally occur. expanding universe. The most important feature in temporal dynamics of the Higgs field is phase transition caused by the evolving temperature of the universe, which can finally result in the Strictly speaking, the scenario outlined in Figure 2 refers formation of the nontrivial states of the physical vacuum to the simplest case of Higgs field possessing Z2 symme- [2]. In fact, the problem of complex vacuum was recognized try group (i.e., admitting a discrete symmetry breaking). a long time ago just after the appearance of the idea of However, the same basic idea is applicable also to more spontaneous symmetry breaking in the quantum field theory. realistic Higgs fields with the continuous symmetries whose Particularly, at the conference on the occasion of the 400th breaking can lead to the formation of more complex defects anniversary of Galileo Galilei’s birth, held in Pisa in 1964, ofthevacuum,suchasthemonopolesandcosmicstrings Bogoliubov emphasized that “it is hard to admit, for example, (or vortices). In general, KZ mechanism gives the following that the “phases” are the same everywhere in the space. estimate for concentration of the topological defects: So it appears necessary to consider such things as “domain structure” of the vacuum” [3]. In the next decade, the 1 𝑛≈ , problem of formation of the domain walls in the course of 𝜉𝑑 (1) cosmological evolution was considered in much detail in the eff work by Zeldovich et al. [4] and later in papers by many other 𝜉 where eff istheeffectivecorrelationlength,thatis,atypical authors. (A quite comprehensive overview of the domain wall distance between the seeds of the new phase; and 𝑑=3, dynamics was given, e.g., in [5].) 2, and 1 for the monopoles, cosmic strings, and domain A commonly accepted scenario of formation of the walls, respectively. (Their concentrations refer evidently to domain walls by the phase transition in the early universe is unit volume, area, and length.) illustrated in Figure 2. A uniform initial (symmetric) state of An accurate calculation of the effective correlation length, the Higgs field, existing soon after the Big Bang, cools down in general, is a difficult task. However, its upper bound can be due to the expansion of the universe, so that the symmetric 𝜉 ≤𝑐𝑡 obtained from a simple causality argument: eff pt,where phase of the field becomes energetically unfavorable and the 𝑐 𝑡 is the speed of light and pt is the characteristic time from seedsofanew(symmetry-broken)phaseemergeinsome the Big Bang to the instant of phase transition. Next, the time spatial points. (For the sake of simplicity, we will assume that from the Big Bang in Friedmann-Robertson-Walker (FRW) 𝑟 these points are equally separated along the coordinate and cosmology is estimated as ∼1/𝐻,where𝐻 is the value of 𝑡=0 originate at the same instant of time .) Since the seeds Hubble parameter at the respective instant. Consequently, of the low-temperature phase emerge independently in the 𝑐 remote spatial regions, their states of the degenerate vacuum, 𝜉 ≲ . eff 𝐻 (2) in general, will be different from each other. pt Then, the domains of the new phase quickly grow in the course of time and at the instant 𝑡=𝑡0 collide with each At last, substituting (2)into(1), we get other. However, if the values of the symmetry-broken phase 𝑑 in two neighboring domains were different, they cannot 𝐻 𝑛≳( pt ) , (3) merge smoothly. Instead, a stable topological defect—domain 𝑐 wall(or“kink”)—shouldbeformedattheirboundary.So, 𝐻 if the initial separation between the seeds of the new phase where pt isthevalueofHubbleparameterattheinstantof was 𝜉, then the resulting concentration of the domain walls phase transition. (To avoid misunderstanding, let us mention canberoughlyestimatedas𝑛≈1/𝜉. This is the sothat most of laboratory experiments, aimed at the verification called Kibble-Zurek (KZ) mechanism for the formation of of KZ mechanism, measured a scaling relationship between topological defects after the strongly nonequilibrium phase the size of uniform domains and the quench rate rather than transformations [6, 7]. the absolute number of the defects. However, in cosmological Advances in High Energy Physics 3 applications it is more appropriate to discuss just the absolute Φ concentration of the defects.) Unfortunately, the lower theoretical bound (3) is inconsistent with the upper bounds following from observations (e.g., review [8]). One possible way to mitigate this disagreement is to modify Lagrangian of the field theory under consideration, typically, by the introduction of the “biased” vacuum I (thereby, aprioriremoving the degeneracy) [4, 5]. Yet another conceivable approach, which was not exploited before, is to take into account the nonlocal quantum correlations which Random phases Correlated phases mightmanifestthemselvesinthemacroscopicBEC. T>Tc However, exploiting the idea of macroscopic quantum correlations, one should bear in mind the following two TcJ

−𝐸𝐽 (𝛿𝑖) 𝑃(𝛿𝑖)∝exp [ ], (4) 𝑘𝐵𝑇 0 0 0.10.2 0.3 ,c where 𝐸𝐽 is the energy concentrated in the Josephson junc- Average contract of interference pattern 0 tion, 𝑇 is the temperature, and 𝑘𝐵 is the Boltzmann constant. So, the main conclusion following from the above exper- Figure 4: Fraction of the interference patterns showing at least one dislocation as a function of the interference contrast 𝑐0.Inset, iment is that the energy concentrated in the defects should examples of images with the increasing number of dislocations betakenintoaccountinthecalculationoftheprobabilityof (frombottomtotop).AdaptedbypermissionfromMacmillan realization of various field configurations even if the phase Publishers Ltd: Nature, vol. 441, no. 7097, pp. 1118–1121, © 2006. transformation develops independently in the remote parts of the system.

(dislocations) formed in the BEC of ultracold gas increases 2.2. Experiments with Ultracold Gases in Optical Lattices. The with temperature by qualitatively the same law as (4); see MJJL experiment, discussed in the previous section, gave the Figure 4 (a sharp outlier at 𝑐0 ≈ 0.2 is most probably an first hint about the importance of using the Gibbs law even experimental inaccuracy). for the systems composed of the apparently isolated parts. Therefore, both the MJJL and ultracold-atom experi- Unfortunately, this experiment did not enable us to check ments suggest that the probability of the realization of the particular functional dependence (4). (To do so, it would various configurations of the BEC order parameter should be be necessary to perform the same experiment with different calculated taking into account the total energy of the system types of superconductors, which has not been fulfilled by even if separate parts of this system do not interact with each now.) other during a particular physical process (e.g., the phase Nevertheless, a few years later it became possible to solve transformation). Of course, these parts of the system must this tack by using the BECs of ultracold gases in periodic be causally connected during its previous evolution. This is potentials (or the so-called optical lattices) formed by the always satisfied in the laboratory experiments but requires a intersecting laser beams. These systems represent a close special consideration in the cosmological context. analog of the multiple Josephson junctions and, as distinct In some sense, the above phenomenon can be interpreted from the solid-state setups, their parameters can be easily as analog of EPR correlation for the system that does not varied. A diagnostics of the phase jumps in such installations possess an exact conservation law. In such a case, just the is performed by a removal of the external potential, thereby energetic criteria should come into play (see also discussion enabling the pieces of BEC to expand and interfere with each in the end of Section 1). other. For example, an array of 30 BECs of the ultracold gas in a regular one-dimensional lattice was created in the experi- 3. Cosmological Implications ment [10]. Next, it was demonstrated that such condensates can well interfere with each other even if they were produced There is evidently a close similarity between the symmetry- independently, that is, “have never seen one another.” breaking phase transitions in the multi-Josephson-junction A further experiment of the same group [11]wasdevoted loop, depicted in Figure 3, and in the simplest one- to a detailed study of the phase defects. In particular, an dimensional (1D) Friedmann-Robertson-Walker (FRW) cos- efficiency of the defect formation was measured as a function mological model, schematically illustrated in Figure 5.For of temperature. (In fact, the determination of temperature the sake of definiteness, we will consider only the simplest is not an easy task in the experiments with ultracold gases. type of defects, namely, the domain walls or kinks. So, the primary independent parameter was taken to be To make the quantitative estimates, let us consider the the average contract of the interference pattern 𝑐0 which is, space-time metric roughly speaking, inversely proportional to the temperature 2 2 2 2 𝑇.) As a result, it was found that the number of defects 𝑑𝑠 =𝑑𝑡 −𝑎 (𝑡) 𝑑𝑥 , (5) Advances in High Energy Physics 5

t 𝜂 𝜂∗

Symmetry- broken phase 𝜂0

0 x

a(t) Δ𝜂

Symmetric phase

Figure 6: Conformal diagram of 1D FRW cosmological model. 0

2 Figure 5: Sketch of the phase transition in 1D FRW cosmological so that the light rays (𝑑𝑠 =0) will represent the straight lines model(thephysicaldistanceismeasuredalongthecircles). inclined at ±𝜋/4:

𝑥=±𝜂+const. (11)

𝑡 𝑥 𝑎(𝑡) The entire structure of the space-time can be conveniently where is the time, is the spatial coordinate, and is the described by the conformal diagram in Figure 6.Let𝜂=0 scale factor of FRW model. (From here on, we will assume and 𝜂=𝜂0 be the beginning and end of the phase transition, that 𝑐≡1.) Let this space-time be filled with the real scalar respectively, and let 𝜂=𝜂∗ be the instant of observation. (The field 𝜑 simulatingHiggsfieldinthetheoryofelementary instants 𝜂=0, 𝜂0,and𝜂∗ of the conformal time correspond to particles. Its Lagrangian the instants 𝑡=0, 𝑡0,and𝑡∗ of the physical time in Figure 2.) 2 2 Since it is commonly assumed that bubbles of the new phase 1 2 2 𝜆 2 𝜇 L (𝑥,) 𝑡 = [(𝜕𝑡𝜑) −(𝜕𝑥𝜑) ]− [𝜑 −( )] (6) grow at the rate close to the speed of light, their boundaries 2 4 𝜆 canbewelldepictedbythelightrays.Then,asfollowsfroma simple geometric consideration, possesses Z2 symmetrygroup,whichshouldbebrokenbythe phase transition. (𝜂 −𝜂 ) 𝜂 𝑁= ∗ 0 ≈ ∗ ( 𝑁) As is known, the stable low-temperature vacuum states of 𝜂 𝜂 at large (12) the field6 ( )are 0 0 𝜇 𝜑 =± , is the number of spatial subregions in the observable universe 0 √𝜆 (7) causally disconnected during the phase transition. Their final vacuum states can be conveniently denoted by the arrows, like and a transition region between them (domain wall) is spins. described as Let us calculate a probability of the phase transition with- 𝜇 out formation of the domain walls in the observable space- 0 𝜑 (𝑥) =𝜑0 tanh [( )(𝑥−𝑥0)] . (8) 𝑃 𝑁 √2 time (the past light cone) 𝑁,wheresubscript implies the number of subregions and superscript 0 implies the absence Such a domain wall contains the energy of domain walls. A trivial estimate can be obtained by taking a ratio of the number of field configurations without domain 2√2 𝜇3 2𝑁 𝐸= . (9) walls (which equals 2) to their total number ( ): 3 𝜆 2 1 𝑃0 = = . We will assume that the thickness of the wall, ∼1/𝜇,issmall 𝑁 2𝑁 2𝑁−1 (13) in comparison with a characteristic distance between them; thatis,thedomainwallscanbetreatedaspoint-likeobjects. This quantity evidently tends to zero very quickly at large 𝑁. Next, it is convenient to introduce the conformal time 𝜂= In other words, the observable part of the universe, repre- ∫ 𝑑𝑡/𝑎(𝑡). As a result, the space-time metric (5)willtakethe sented by the large upper triangle in Figure 6,willinevitably conformal flat form [12]: contain some number of the domain walls. Unfortunately, as recognized a long time ago [4, 5], a presence of the domain 2 2 2 2 𝑑𝑠 =𝑎 (𝑡) [𝑑𝜂 −𝑑𝑥 ], (10) wallsisincompatiblewithastronomicalobservations. 6 Advances in High Energy Physics

Apossibleresolutionofthisparadoxcanbebasedjust on taking into consideration the nonlocal Gibbs-like correlations (4).Firstofall,wemustensurethatsuchcorrelations canreallydevelop;thatis,thesubdomainsofthenewphase 1.0 were causally connected in the past. As seen in Figure 6,this is really possible if a sufficiently long interval of the conformal 0.8 time 0.6 Δ𝜂≥𝜂∗ (14) 0 PN 0.4 preceded the phase transition. Then, the lower shaded tri- 𝜂=0 anglewillcoverattheinstant the upper triangle 0.2 representing the observable part of the universe. The inequality (14) can be satisfied, particularly, in the 0 2.0 2 1.6 case of sufficiently long inflationary (de Sitter) stage. Really, 4 1.2 if 𝑎(𝑡) ∝ exp(𝐻𝑡),where𝐻 is the Hubble constant, then 6 0.8 N 8 0.4 E/T 1 𝜂∝− 𝑒−𝐻𝑡 + 󳨀→ − ∞ 𝑡󳨀→−∞, 10 0 𝐻 const at (15) Figure 7: The probability of phase transition without the formation 𝑃 0 so that the above-mentioned interval Δ𝜂 can be suffi- of the domain walls 𝑁 as a function of the number of disconnected ciently large. Let us be reminded that from the viewpoint subregions 𝑁 andtheratioofthedomainwallenergytothephase of elementary-particle physics, the de Sitter stage can be transition temperature 𝐸/𝑇. naturallyrealizedintheovercooledstateoftheHiggsfield immediately before its first-order phase transition; and just this idea was the starting point of the first inflationary models condensed-matter physics). Yet another method for the cal- [13]. culation of the same quantity, which is more straightforward Next, if the condition (14) is satisfied, then it is reasonable and pictorial but less informative, can be found in [14]; the to assume that the above-mentioned correlations (4)should approach outlined here was employed for the first time in our take place between all 𝑁 subdomains drawn in the conformal paper [16]. 𝑃 0 𝑁 𝐸/𝑇 diagram, Figure 6. (It is interesting to mention that in our old The quantity 𝑁 as a function of and is plotted 0 work [14], performed before the MJJL experiment, the same in Figure 7.Itisseenthat𝑃𝑁 drops very sharply with the Gibbs-like correlations were introduced on the basis of some increase in 𝑁 at small values of 𝐸/𝑇 (when the effect of 0 metaphysical speculations.) In such a case, the probability 𝑃𝑁 Gibbs-like correlations is insignificant), but it becomes a should be calculated taking into account Gibbs factors for the gently decreasing function of 𝑁 when the parameter 𝐸/𝑇 is field configurations involving the domain walls: sufficiently large. Some other plots illustrating the suppression of concentration of the domain walls by the nonlocal 0 2 𝑃𝑁 = , (16) correlations can be found in paper [17], devoted to the 𝑍 phase transformations in superfluids and superconductors. where Therefore, just the large energy concentrated in the domain wallsturnsouttobethefactorsubstantiallyreducingthe 𝑁 { 𝐸 𝑁 1 } probability of their creation. 𝑍=∑ ∑ exp {− ∑ (1 − 𝑠𝑗𝑠𝑗+1)} . (17) Next, as can be easily derived from (18), the probability 𝑠 =±1 𝑇 2 𝑖=1 𝑖 { 𝑗=1 } of absence of the domain walls in the observable universe becomes on the order of unity, for example 1/2, if 𝐸/𝑇 ≳ Here, 𝑠𝑗 is the spin-like variable denoting a sign of the vacuum 𝑁 𝑗 𝐸 ln . Taking into account (9)and(12), this inequality can be state in the ’th subdomain, is the domain wall energy, given rewritten as by (9), and 𝑇 is the characteristic temperature of the phase 𝜇3 𝜂 transition. (From here on, the temperature will be expressed ≳ ∗ . 𝑘 ln (19) in energetic units, so the Boltzmann constant 𝐵 will be 𝜆𝑇 𝜂0 omitted.) From a formal point of view, the statistical sum (17)isvery Because of the very weak logarithmic dependence in the similar to the sum for Ising model, well studied in the physics right-hand side, such a condition could be reasonably satis- of condensed matter, for example [15]. Using exactly the same fiedforacertainclassoffieldtheories. mathematical approach, we get the final result: Moreover, the situation becomes even more favorable in thecaseoftwo-orthree-dimensionalspace.Thepointis 0 2 that a well-known property of the 2D and 3D Ising models 𝑃𝑁 = [1+𝑒−𝐸/𝑇]𝑁 + [1−𝑒−𝐸/𝑇]𝑁 (18) is a tendency for aggregation of the domains with the same value of the order parameter when the temperature drops (attention should be paid to the appropriate choice of zero below some critical value 𝑇𝑐 ∼𝐸[18]. In the condensed- energy, which is different from the one commonly used in the matter applications, this corresponds, for example, to the Advances in High Energy Physics 7 spontaneous magnetization of a solid body. Regarding the that such anomalies are associated just with the nonlocal cosmological context, one can expect that probability of correlations in the early universe but, of course, a much more formation of the domain walls will be reduced dramatically elaborated analysis must be performed to draw a reliable at the sufficiently large values of 𝐸/𝑇; some illustrations of conclusion. this phenomenon can be found in [17]. (To avoid misunder- At last, we would like to mention a recent activity in the standing, let us emphasize that the above-mentioned Ising experiments with ultracold gases for simulation of various models are only the auxiliary mathematical constructions, dynamical phenomena in cosmology, for example, the so- describing a final distribution of the domain walls after the called Sakharov oscillations [21]. From our point of view, a phase transformation. So, the formal phase transitions in the more careful study of the nonlocal correlations may become 2D and 3D Ising models should not be associated with the an important branch in this rapidly growing research field. 4 physical phase transition in the original 𝜑 -field model (6); for more details, see Table 2 in [17].) Conflict of Interests The author declares that there is no conflict of interests 4. Discussion and Conclusions regarding the publication of this paper. As discussed in the present paper, a few laboratory experiments suggest the presence of the nonlocal Gibbs-like Acknowledgments correlations between the phases of BECs after the rapid phase The initial stage of this work was substantially supported transformations. Therefore, it can be reasonably conjectured by the ESF COSLAB (Cosmology in the Laboratory) Pro- that the same correlations should occur in the BEC of gramme. The author is grateful to a large number of Higgs field, which is formed in the course of evolution researchers with whom he discussed the problem of nonlocal of the universe. As follows from our quantitative estimates effects in cosmology since the early 2000s till now: E. for the simplest case of 1D FRW cosmological model, such Arimondo,V.B.Belyaev,R.A.Bertlmann,Yu.M.Bunkov, correlations can show a way to resolve the well-known A. M. Chechelnitsky, I. Coleman, J. Dalibard, V. B. Efimov, V. problem of the excessive concentration of the domain walls B.Eltsov,H.J.Junes,I.B.Khriplovich,T.W.B.Kibble,M. resulting from phase transitions in the early universe. In Knyazev, V. P. Koshelets, O. D. Lavrentovich, V. N. Lukash, fact,thisproblemwasrecognizedinthemid1970s[4]and A.Maniv,P.V.E.McClintock,L.B.Okun,G.R.Pickett, since that time a commonly used approach to its resolution E. Polturak, A. I. Rez, R. J. Rivers, M. Sakellariadou, M. was based on the introduction of the so-called “biased” (or Sasaki, R. Schutzhold,¨ V.B. Semikoz, M. Shaposhnikov, A. A. asymmetric) vacuum. As a result, under the appropriate Starobinsky,A.V.Toporensky,W.G.Unruh,G.Vitiello,G.E. choice of parameters, the regions of “false” (energetically Volovik, C. Wetterich, and W. H. Zurek. unfavorable) vacuum should quickly disappear, eliminating the corresponding domain walls [5]. Unfortunately, the concept of biased vacuum was not supported by independent References data in the physics of elementary particles. On the other hand, [1] A. Einstein, B. Podolsky, and N. Rosen, “Can quantum- the idea of nonlocal correlations, employed in the present mechanical description of physical reality be considered com- study, is supported by at least a few laboratory experiments. plete?” Physical Review,vol.47,no.10,pp.777–780,1935. Therefore, from our point of view, it looks more attractive. [2] A. D. Linde, “Phase transitions in gauge theories and cosmol- It should be emphasized again that a number of argu- ogy,” Reports on Progress in Physics,vol.42,no.3,pp.389–437, ments from the modern observational cosmology impose 1979. severe constraints on the concentration of domain walls. In [3] N. N. Bogoliubov, “Field-theoretical methods in physics,” Sup- particular, an appreciable number of the domain walls would plemento al Nuovo Cimento (Serie Prima),vol.4,no.2,pp.346– produce an unacceptable anisotropy of the cosmic microwave 357, 1966. background (CMB) radiation, change the overall rate of the [4] Ia.B. Zeldovich, I. Y. Kobzarev, and L. B. Okun, “Cosmolog- cosmological expansion, and so forth. Besides, it is com- ical consequences of a spontaneous breakdown of a discrete monly believed now that the primordial spectrum of density symmetry,” Soviet Physics—JETP,vol.40,no.1,pp.1–5,1975, fluctuations, responsible for the formation of the large-scale Translated from: ZhurnalEksperimental’noiiTeoreticheskoi structure, is formed by the Gaussian quantum fluctuations Fiziki, vol. 67, pp. 3–11, 1974. in the very early universe amplified by the subsequent [5] G. B. Gelmini, M. Gleiser, and E. W.Kolb, “Cosmology of biased inflationary stage, while contribution from the topological discrete symmetry breaking,” Physical Review D,vol.39,no.6, defects is quite insignificant. All these facts imply that there pp.1558–1566,1989. should be an efficient mechanism for the suppression of [6] T. W. B. Kibble, “Topology of cosmic domains and strings,” thedomainwalls,andthenonlocalquantumcorrelations Journal of Physics A: Mathematical and General,vol.9,no.8, discussed in the present paper might be a reasonable option. pp. 1387–1398, 1976. Yet another recent cosmological problem, recognized due [7] W. H. Zurek, “Cosmological experiments in superfluid to WMAP andconfirmedbythePlanck satellite data, is the helium?” Nature,vol.317,no.6037,pp.505–508,1985. anomalous behaviour of CMB fluctuations at large angular [8]H.V.Klapdor-KleingrothausandK.Zuber,Particle Astro- ∘ scales, approximately over 10 [19, 20]. It can be conjectured physics, Institute of Physics Publishing, Bristol, UK, 1997. 8 Advances in High Energy Physics

[9] R. Carmi, E. Polturak, and G. Koren, “Observation of spontaneous flux generation in a multi-Josephson-junction loop,” Physical Review Letters,vol.84,no.21,pp.4966–4969,2000. [10] Z. Hadzibabic, S. Stock, B. Battelier, V. Bretin, and J. Dalibard, “Interference of an array of independent Bose-Einstein condensates,” Physical Review Letters, vol. 93, no. 18, Article ID 180403, 4pages,2004. [11] Z. Hadzibabic, P. Kruger,M.Cheneau,B.Battelier,andJ.Dal-¨ ibard, “Berezinskii-Kosterlitz-Thouless crossover in a trapped atomic gas,” Nature, vol. 441, no. 7097, pp. 1118–1121, 2006. [12] C. W. Misner, “Mixmaster universe,” Physical Review Letters, vol. 22, no. 20, pp. 1071–1074, 1969. [13] A. D. Linde, “The inflationary universe,” ReportsonProgressin Physics,vol.47,no.8,pp.925–986,1984. [14] Y. V. Dumin, “On a probable role of EPR, (Einstein-Podolsky- Rosen) correlations in breaking the symmetry of Higgs fields in cosmological phase transitions,” in Proceedings of the Interna- tional Workshop on Hot Points in Astrophysics, pp. 114–120, Joint Institute for Nuclear Research, Dubna, Russia, 2000. [15] A. Isihara, Statistical Physics, Academic Press, New York, NY, USA, 1971. [16] Y. V. Dumin, “On the influence of Einstein-Podolsky-Rosen effect on the domain wall formation during the cosmological phase transitions,” in Frontiers of Particle Physics: Proceedings of the 10th Lomonosov Conference on Elementary Particle Physics, pp. 289–294, World Scientific Publishing, Singapore, 2003. [17] Y. V. Dumin, “Ultracold gases and multi-Josephson junctions as simulators of out-of-equilibrium phase transformations in superfluids and superconductors,” New Journal of Physics,vol. 11, Article ID 103032, 12 pages, 2009. [18] Y. B. Rumer and M. S. Ryvkin, Thermodynamics, Statistical Physics, and Kinetics, Izdat. Mir, Moscow, Russia, 1980. [19] A. Wright, “Across the Universe,” Nature Physics, vol. 9, no. 5, p. 264, 2013. [20] R. Hofmann, “The fate of statistical isotropy,” Nature Physics, vol. 9, no. 11, pp. 686–689, 2013. [21] J. Schmiedmayer and J. Berges, “Cold atom cosmology,” Science, vol. 341, no. 6151, pp. 1188–1189, 2013. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2014, Article ID 376982, 7 pages http://dx.doi.org/10.1155/2014/376982

Research Article Hawking-Unruh Hadronization and Strangeness Production in High Energy Collisions

Paolo Castorina1,2,3 and Helmut Satz4 1 Dipartimento di Fisica ed Astronomia, UniversitadiCatania,ViaSantaSofia64,95100Catania,Italy` 2 INFNSezionediCatania,ViaSantaSofia64,95100Catania,Italy 3 PHDepartment,THUnit,CERN,1211Geneva23,Switzerland 4 Fakultat¨ fur¨ Physik, Universitat¨ Bielefeld, Germany

Correspondence should be addressed to Paolo Castorina; [email protected]

Received 8 January 2014; Accepted 16 April 2014; Published 11 May 2014

Academic Editor: Piero Nicolini

Copyright © 2014 P. Castorina and H. Satz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

The thermal multihadron production observed in different high energy collisions poses many basic problems: why doeven + − elementary, 𝑒 𝑒 and hadron-hadron, collisions show thermal behaviour? Why is there in such interactions a suppression of strange particle production? Why does the strangeness suppression almost disappear in relativistic heavy ion collisions? Why in these collisions is the thermalization time less than ≃ 0.5 fm/c? We show that the recently proposed mechanism of thermal hadron production through Hawking-Unruh radiation can naturally answer the previous questions. Indeed, the interpretation of quark (q)- antiquark (𝑞) pairs production, by the sequential string breaking, as tunneling through the event horizon of colour confinement leads to thermal behavior with a universal temperature, 𝑇 ≃ 170 Mev, related to the quark acceleration, a,by𝑇=𝑎/2𝜋.The resulting temperature depends on the quark mass and then on the content of the produced hadrons, causing a deviation from full equilibrium and hence a suppression of strange particle production in elementary collisions. In nucleus-nucleus collisions, where the quark density is much bigger, one has to introduce an average temperature (acceleration) which dilutes the quark mass effect and the strangeness suppression almost disappears.

1. Introduction for hadrons containing one, two, or three strange quarks (or antiquarks), respectively. In high energy heavy ion collisions, Hadron production in high energy collisions shows remark- 𝑒+𝑒− strangeness suppression becomes less and disappears at high ably universal thermal features. In annihilation [1–3], energies [14, 15]. in 𝑝𝑝,in𝑝𝑝 [4], and more general ℎℎ interactions [3], as well as in the collisions of heavy nuclei [5–11], over an energy The origin of the observed thermal behaviour has been range from around 10 GeV up to the TeV range; the relative an enigma for many years and there is a still ongoing debate abundances of the produced hadrons appear to be those of an about the interpretation of these results [16–24]. Indeed, in ideal hadronic resonance gas at a quite universal temperature high energy heavy ion collisions multiple parton scattering 𝑒+𝑒− 𝑇𝐻 ≈ 160–170 MeV [12]. couldleadtokineticthermalization,but or elementary Thereis,however,oneimportantnonequilibriumeffect hadron interactions do not readily allow such a description. observed: the production of strange hadrons in elementary The universality of the observed temperatures, on the collisions is suppressed relative to an overall equilibrium. otherhand,suggestsacommonoriginforallhighenergy This is usually taken into account phenomenologically by collisions, and it was recently proposed [25]thatthermal introducing an overall strangeness suppression factor 𝛾𝑠 <1 hadron production is the QCD counterpart of Hawking- 𝛾 ,𝛾2 𝛾3 [13], which reduces the predicted abundances by 𝑠 𝑠 ,and 𝑠 Unruh (H-U) radiation [26, 27], emitted at the event horizon 2 Advances in High Energy Physics due to colour confinement. In the case of approximately Lorentz-invariant quantities such as multiplicities, one can massless quarks, the resulting hadronization temperature is introduce a simplifying assumption and thereby obtain a determined by the string tension 𝜎,with𝑇≃√𝜎/2𝜋 ≃ simple analytical expression in terms of a temperature. The 170 Mev [25]. Moreover in [28], it has been shown that key point is to assume that the distribution of masses and strangeness suppression in elementary collisions naturally charges among clusters is again purely statistical [3], so that, occurs in this framework without requiring a specific sup- as far as the calculation of multiplicities is concerned, the pression factor. The crucial role here is played by the nonneg- set of clusters becomes equivalent, on average, to a large ligible strange quark mass, which modifies the emission tem- cluster (equivalent global cluster)whosevolumeisthesum perature for such quarks: in the Hawking-Unruh approach of proper cluster volumes and whose charge is the sum of the temperature associated with strange particle production cluster charges (and thus the conserved charge of the initial is different from that one of nonstrange particles. Although colliding system). In such a global averaging process, the this conclusion has been obtained by a full analysis in the equivalent cluster generally turns out to be large enough in statistical hadronization model in [28], here we propose an massandvolumesothatthecanonicalensemblebecomes elementary scheme of understanding strangeness suppres- a good approximation. In other words, a temperature can sion which, moreover, opens the possibility of explaining why be introduced which replaces the apriorimore fundamental this suppression disappears in heavy ion collisions where the description in terms of an energy density. large parton density in causally connected space-time region To obtain a simple expression for our further discussion, of hadronization produces a unique temperature for strange we neglect for the moment an aspect which is important in and nonstrange particles. any actual analysis. Although in elementary collisions the In particular, in the second section, after recalling the conservation of the various discrete abelian charges (electric statistical hadronization model, one discusses the subsequent charge, baryon number, strangeness, heavy flavour) has to description in terms of H-U radiation in QCD, including be taken into account exactly [30], we here consider for the dependence of the radiation temperature on the mass the moment a grand-canonical picture. We also assume oftheproducedquarkandthestrangenesssuppressionin Boltzmann distributions for all hadrons. The multiplicity of elementary collisions. In Section 3 the dynamical mechanism a given scalar hadronic species 𝑗 then becomes for strangeness enhancement (i.e., no suppression) in rela- 2 𝑉𝑇𝑚 𝑛 𝑚 tivisticheavyionscatteringsisintroducedgivingacoarse ⟨𝑛 ⟩primary = 𝑗 𝛾 𝑗 𝐾 ( 𝑗 ) (1) grain evaluation of the Wroblewski factor [29], a parameter 𝑗 2𝜋2 𝑠 2 𝑇 which counts the relative number of strange and nonstrange 𝑚 𝑛 quarks and antiquarks. Section 4 is devoted to comments and with 𝑗 denoting its mass and 𝑗 thenumberofstrange conclusions. quarks/antiquarks it contains. Here primary indicates that it gives the number at the hadronisation point, prior to all subsequent resonance decay. The Hankel function 𝐾2(𝑥), 2. Hawking-Unruh Hadronization with 𝐾(𝑥) ∼ exp{−𝑥} for large 𝑥,givestheBoltzmann factor, while 𝑉 denotes the overall equivalent cluster volume. In this section we will recall the essentials of the statistical In other words, in an analysis of 4𝜋 data of elementary hadronization model and of the Hawking-Unruh analysis of collisions, 𝑉 is the sum of the all cluster volumes at all the string breaking mechanism. For a detailed descriptions different rapidities. It thus scales with the overall multiplicity see [3, 25, 28]. and hence increases with collision energy. A fit of production data based on the statistical hadronisation model thus 𝑇 2.1. Statistical Hadronization Model. The statistical hadr- involves three parameters: the hadronisation temperature , 𝛾 onization model assumes that hadronization in high energy the strangeness suppression factor 𝑠,andtheequivalent 𝑉 collisions is a universal process proceeding through the global cluster volume . formation of multiple colourless massive clusters (or fireballs) As previously discussed, at high energy the temperature of finite spacial extension. These clusters are taken to decay turns out to be independent of the initial configuration into hadrons according to a purely statistical law: every and this result calls for a universal mechanism underlying multihadron state of the cluster phase space defined by its the hadronization. In the next paragraph we recall the mass, volume, and charges is equally probable. The mass interpretation of the string breaking as QCD H-U radiation. distribution and the distribution of charges (electric, bary- onic, and strange) among the clusters and their (fluctuating) 2.2. String Breaking and Event Horizon. Let us outline the number are determined in the prior dynamical stage of the thermal hadron production process through H-U radiation + − process. Once these distributions are known, each cluster can for the specific case of 𝑒 𝑒 annihilation (see Figure 1). The be hadronized on the basis of statistical equilibrium, leading separating primary 𝑞𝑞 pair excites a further pair 𝑞1𝑞1 from the to the calculation of averages in the microcanonical ensemble, vacuum, and this pair is in turn pulled apart by the primary enforcing the exact conservation of energy and charges of constituents. In the process, the 𝑞1 shields the 𝑞 from its each cluster. original partner 𝑞,withanew𝑞𝑞1 string formed. When it is Hence, in principle, one would need the mentioned stretched to reach the pair production threshold, a further dynamical information in order to make definite quantitative pair is formed, and so on [31, 32]. Such a pair production predictions to be compared with data. Nevertheless, for mechanism is a special case of H-U radiation [33–37], emitted Advances in High Energy Physics 3

q 𝑚 ≃0 𝑚 q 3 massless quark ( 𝑞 ) and one of strange quark mass 𝑠, 2 the average acceleration becomes

q2 𝑤 𝑎 +𝑤𝑎 2𝜎 q1 q 0 0 𝑠 𝑠 1 𝑎0𝑠 = = . (7) 𝑤0 +𝑤𝑠 𝑤0 +𝑤𝑠 q From this the Unruh temperature of a strange meson is given q by 𝜎 𝑇 (0𝑠) ≃ , (8) 𝜋(𝑤0 +𝑤𝑠)

𝛾 with 𝑤0 ≃ √1/2𝜋𝜎 and 𝑤𝑠 given by (4)with𝑚𝑞 =𝑚𝑠. Figure 1: String breaking through 𝑞𝑞 pair production. Similarly, we obtain 𝜎 𝑇 (𝑠𝑠) ≃ , (9) 2𝜋𝑤𝑠 as hadron 𝑞1𝑞2 when the quark 𝑞1 tunnels through its event horizon to become 𝑞2. The corresponding hadron radiation forthetemperatureofamesonconsistingofastrangequark- 2 temperature is given by the Unruh form 𝑇𝐻 = 𝑎/2𝜋,where𝑎 antiquark pair (𝜙). With 𝜎≃0.2GeV ,(6)gives𝑇0 ≃ is the acceleration suffered by the quark 𝑞1 due to the force 0.178 GeV. A strange quark mass of 0.1 GeV reduces this to of the string attaching it to the primary quark 𝑄. This is 𝑇(0𝑠) ≃ 0.167 GeV and 𝑇(𝑠𝑠) ≃ 157 MeV, that is, by about 6% equivalent to that suffered by quark 𝑞2 due to the effective and 12%, respectively. force of the primary antiquark 𝑄.Hencewehave The scheme is readily generalized to baryons. The production pattern is illustrated in Figure 2 and leads to an average 𝜎 𝜎 𝑎𝑞 = = , of the accelerations of the quarks involved. We thus have 𝑤𝑞 2 2 (2) √𝑚𝑞 +𝑘𝑞 𝜎 𝑇 (000) =𝑇(0) ≃ (10) 2𝜋𝑤0 2 2 where 𝑤𝑞 = √𝑚𝑞 +𝑘𝑞 is the effective mass of the produced for nucleons, quark, with 𝑚𝑞 for the bare quark mass and 𝑘𝑞 the quark momentum inside the hadronic system 𝑞1𝑞1 or 𝑞2𝑞2.Since 3𝜎 𝑇 (00𝑠) ≃ (11) the string breaks [25]whenitreachesaseparationdistance 2𝜋 (2𝑤0 +𝑤𝑠) 2 𝜋𝜎 Λ Σ 𝑥 ≃ √𝑚2 +( ), (3) for and production, 𝑞 𝜎 𝑞 2 3𝜎 𝑇 (0𝑠𝑠) ≃ (12) the uncertainty relation gives us 𝑘𝑞 ≃1/𝑥𝑞 2𝜋 0(𝑤 +2𝑤𝑠)

for Ξ production, and 𝜎2 𝑤 = √𝑚2 + [ ] (4) 𝜎 𝑞 𝑞 2 𝑇 (𝑠𝑠𝑠) =𝑇(𝑠𝑠) ≃ (4𝑚𝑞 +2𝜋𝜎) (13) [ ] 2𝜋𝑤𝑠 fortheeffectivemassofthequark.Theresultingquark-mass for that of Ω’s.Wethusobtainaresonancegaspicturewith dependent Unruh temperature is thus given by five different hadronization temperatures, as specified by the strangeness content of the hadron in question: 𝑇(00) = 𝜎 𝑇(000), 𝑇(0𝑠), 𝑇(𝑠𝑠) = 𝑇(𝑠𝑠𝑠), 𝑇(00𝑠) 𝑇(0𝑠𝑠) 𝑇 (𝑞𝑞) ≃ . and . (5) 2𝜋𝑤𝑞 2.3. Strangeness Suppression in Elementary Collisions. In Note that here it is assumed that the quark masses for 𝑞1 and order to evaluate the primary hadron multiplicities, the pre- 𝑞2 are equal. For 𝑚𝑞 ≃0,(5)reducesto vious different species-dependent temperatures, fully determined by the string tension 𝜎 andthestrangequarkmass 𝜎 𝑚𝑠,havebeeninsertedintoacompletestatisticalmodel 𝑇 (00) ≃ √ , (6) 2𝜋 calculation [28]. Apart from possible variations of the quantities of 𝜎 and as obtained in [25]. 𝑚𝑠,thedescriptionisthusparameter-free.Asillustration,we 2 Iftheproducedhadron𝑞1𝑞2 consists of quarks of different show in Table 1 the temperatures obtained for 𝜎=0.2GeV masses, the resulting temperature has to be calculated as and three different strange quark masses. It is seen that in an average of the different accelerations involved. For one all cases, the temperature for a hadron carrying nonzero 4 Advances in High Energy Physics

p Which corresponds to a 𝛾𝑠 parameter given by u u p d 𝑇𝑘 𝐾2 (𝑚𝑘/𝑇𝑘) 𝛾𝑠 = . (16) 𝑇𝜋 𝐾2 (𝑚𝑘/𝑇𝜋) u u u d 2 For 𝜎 = 0.2 GeV , 𝑚𝑠 =0.1GeV, 𝑇𝜋 = 178 Mev, and 𝑇𝑘 = d 167 Mev (see Table 1), the crude evaluation by (16)gives𝛾𝑠 ≃ q 0.73. In conclusion, our picture implies that the produced hadrons are emitted slightly “out of equilibrium,”in the sense that the emission temperatures are not identical. As long q as there is no final state interference between the produced quarks or hadrons, we expect to observe this difference and hence a modification of the production of strange hadrons, in comparison to the corresponding full equilibrium values. 𝛾 Once such interference becomes likely, due to the Figure 2: Nucleon formation by string breaking through 𝑞𝑞 pair increases of parton density, as in high energy heavy ion production. collisions, equilibrium can be essentially restored producing a small strangeness suppression as we will discuss in the next section. Table 1: Hadronization temperatures for hadrons of different strangeness content, for 𝑚𝑠 = 0.075, 0.100, 0.125 GeV and 𝜎 = 0.2 GeV2. 3. Hawking-Unruh Strangeness Enhancement in Heavy Ion Collisions 𝑇𝑚𝑠 = 0.075 𝑚𝑠 = 0.100 𝑚𝑠 = 0.125 𝑇(00) 0.178 0.178 0.178 As previosuly discussed, in elementary collisions the produc- 𝑇(0𝑠) 0.172 0.167 0.162 tion of hadrons containing 𝑛 strange quarks or antiquarks is 𝑇(𝑠𝑠) 0.166 0.157 0.148 reduced in comparison to nucleus-nucleus scattering and this 𝑇(000) 0.178 0.178 0.178 reductioncanbeaccountedforbytheintroductionofthe 𝑇(00𝑠) 0.174 0.171 0.167 phenomenological universal strangeness suppression factor 𝛾𝑛 𝑇(0𝑠𝑠) 0.170 0.164 0.157 𝑠 . 𝑇(𝑠𝑠𝑠) 0.166 0.157 0.148 However,thehadronproductioninhighenergycollisions occurs in a number of causally disconnected regions of finite space-time size38 [ ]. As a result, globally conserved strangeness is lower than that of nonstrange hadrons and quantum numbers (charge, strangeness, and baryon number) this leads to an overall strangeness suppression in elementary must be conserved locally in spatially restricted correlation collisions, in good agreement with data [28], without the clusters. This provides a dynamical basis for understanding introduction of the ad-hoc parameter 𝛾𝑠. the suppression of strangeness production in elementary 𝑝𝑝 𝑒+𝑒− Although a complete statistical hadronization model interactions ( , )duetoasmallstrangenesscorrelation calculation has been necessary for a detailed comparison with volume [11, 30, 39, 40]. experimental results [28], a rather simple, coarse grain argu- The H-U counterpart of this mechanism is that in ment can illustrate why a reduction of the H-U temperature elementary collisions there is a small number of partons for strange particle production reproduces the 𝛾𝑠 effect. in a causally connected region and the hadron production 𝑞𝑞 To simplify matters, let us assume that there are only two comes from the sequential breaking of independent strings species: scalar and electrically neutral mesons, “pions” with with the consequent species-dependent temperatures which mass 𝑚𝜋, and “kaons” with mass 𝑚𝑘 and strangeness 𝑠=1. reproduce the strangeness suppression. According to the statistical model with the 𝛾𝑠 suppression In contrast, the space-time superposition of many colli- factor, the ratio 𝑁𝑘/𝑁𝜋 is obtained by (1)andisgivenby sionsinheavyioninteractionslargelyremovesthesecausality constraints [38], resulting in an ideal hadronic resonance gas 󵄨 2 𝑁 󵄨stat 𝑚 𝐾 (𝑚 /𝑇) in full equilibrium. 𝑘 󵄨 = 𝑘 𝛾 2 𝑘 󵄨 2 𝑠 (14) IntheH-Uapproach,theeffectofalargenumberof 𝑁𝜋 󵄨𝛾 𝑚𝜋 𝐾2 (𝑚𝜋/𝑇) 𝑠 causally connected quarks and antiquarks can be implemented by defining the average temperature of the system and because there is thermal equilibrium at temperature 𝑇. determining the hadron multiplicities by the statistical model On the other hand, in the H-U based statistical model with this “equilibrium” temperature. there is no 𝛾𝑠, 𝑇𝑘 =𝑇(0𝑠)=𝑇̸𝜋 =𝑇(00)=𝑇and therefore Indeed, the average temperature depends on the numbers 𝑁 𝑁 󵄨stat 𝑚2 𝐾 (𝑚 /𝑇 ) of light quarks, 𝑙,andofstrangequarks, 𝑠,which,in 𝑁𝑘 󵄨 𝑘 𝑇𝑘 2 𝑘 𝑘 󵄨 = . (15) turn, are counted by the number of strange and nonstrange 𝑁 󵄨 𝑚2 𝑇 𝐾 (𝑚 /𝑇 ) 𝜋 H-U 𝜋 𝜋 2 𝜋 𝜋 hadrons in the final state at that temperature. Advances in High Energy Physics 5

+ − A detailed analysis requires again a full calculation in the In our simplified world, for 𝑒 𝑒 annhilation, 𝜆=𝑁𝑘/𝑁𝜋 statistical model that will be done in a forthcoming paper; and is given by (15) with the species-dependent temperatures however the mechanism can be roughly illustrated in the in Table 1.Onegets𝜆 ≃ 0.26. worldof“pions”and“kaons”introducedinSection 2. To evaluate the Wroblewski factor in nucleus-nucleus Let us consider a high density system of quarks and collisions one has to consider the average “equilibrium” antiquarks in a causally connected region. Generalizing our temperature 𝑇 and the number of light quarks in the initial formulas in Section 2, the average acceleration is given by configuration. The latter point requires a realistic calculation in the statitical model which includes all resonances and 𝑁𝑙𝑤0𝑎0 +𝑁𝑠𝑤𝑠𝑎𝑠 𝑎 = . (17) stable particles. 𝑁𝑙𝑤0 +𝑁𝑠𝑤𝑠 However to show that one is on the right track, let us neglect the problem of the initial configurationa and let 𝑁 ≫𝑁 By assuming 𝑙 𝑠,by((6)–(8)), after a simple algebra, us evaluate the effect of substituting in15 ( )thespecies- the average temperature, 𝑇=𝑎/2𝜋,turnsouttobe dependent temperatures with the equilibrium temperature 𝑇. With this simple modification the Wroblewski factor 𝑁 𝑤 +𝑤 𝑇 (0𝑠) 𝑁 2 𝑇=𝑇(00) [1 − 𝑠 0 𝑠 (1 − )] + 𝑂 [( 𝑠 ) ]. increases, 𝜆 = 0.33. 𝑁𝑙 𝑤0 𝑇 (00) 𝑁𝑙 In other terms, the change from a nonequilibrium condi- (18) tion, with species-dependent temperatures, to an equilibrated system with the average temperature 𝑇 is able to reproduce Nowinourworldof“pions”and“kaons”onehas𝑁𝑙 =2𝑁𝜋 + part of the observed growing of the number of strange quarks 𝑁𝑘 and 𝑁𝑠 =𝑁𝑘 and therefore with respect to elementary interactions. 𝑁 𝑤 +𝑤 𝑇 (0𝑠) 𝑇=𝑇(00) [1 − 𝑘 0 𝑠 (1 − )] 2𝑁𝜋 𝑤0 𝑇 (00) 4. Conclusions and Comments (19) 𝑁 2 The Hawking-Unruh hadronization mechanism gives a +𝑂[( 𝑘 ) ]. dynamical basis to the universality of the temperature 𝑁𝜋 obtained in the statistical model, to the strangeness sup- On the other hand, in the H-U based statistical calculation pression in elementary collisions and to its enhancement in the ratio 𝑁𝑘/𝑁𝜋 depends on the equilibrium (average) nucleus-nucleus scatterings. temperature 𝑇;thatis, We are aware that the simplified formulation in Section 3 has to be checked by a full analysis in the hadron reso- 𝑁 𝑚2 𝐾 (𝑚 /𝑇) nance gas framework or by montecarlo simulations and that, 𝑘 = 𝑘 2 𝑘 , in addition, the interpretation of production of charmed 𝑁 𝑚2 (20) + − 𝜋 𝜋 𝐾2 (𝑚𝜋/𝑇) hadrons in 𝑒 𝑒 annihilations, which is very successfully described by the hadron resonance gas model with an exact and, therefore, one has to determine the temperature 𝑇 charm conservation, should be addressed through Hawking- in such a way that (19)and(20) are self-consistent. This Unruh radiation mechanism to check if there is a consistent condition implies the equation picture. However the proposed approach is also able to under- 2 [1 − 𝑇/𝑇 (00)]𝑤0 𝑚 𝐾2 (𝑚𝑘/𝑇) stand the extremely fast equilibration of hadronic matter. 2 = 𝑘 , Indeed, Hawking-Unruh radiation provides a stochastic [1−𝑇(0𝑠) /𝑇 (00)] (𝑤 +𝑤 ) 𝑚2 (21) 𝑠 0 𝜋 𝐾2 (𝑚𝜋/𝑇) rather than kinetic approach to equilibrium, with a randomization essentially due to the quantum physics of the color thatcanbesolvednumerically. horizon. The barrier to information transfer due to the event 2 For 𝜎 = 0.2 GeV , 𝑚𝑠 = 0.1,andthetemperaturesin horizon requires that the resulting radiation states excited Table 1,theaveragetemperatureturnsout𝑇 = 174 Mev and from the vacuum be distributed according to maximum one can evaluate the Wroblewski factor defined by entropy, with a temperature determined by the strength of the confining field. 2𝑁𝑠 The ensemble of all produced hadrons, averaged over 𝜆= , (22) 𝑁𝑙 all events, then leads to the same equilibrium distribution as obtained in hadronic matter by kinetic equilibration. In where 𝑁𝑠 is the number of strange and antistrange quarks in the case of a very high energy collision with a high average the hadrons in the final state and 𝑁𝑙 is the number of light multiplicity already one event can provide such equilibrium; quarks and antiquarks in the final state minus their number because of the interruption of information transfer at each of in the initial configuration. the successive quantum color horizons, there is no phase rela- The experimental value of the Wroblewski factor in high tion between two successive production steps in a given event. energy collisions is rather independent on the energy and is The destruction of memory, which in kinetic equilibration about 𝜆 ≃ 0.26 in elementary collisions and 𝜆 ≃ 0.5 for is achieved through sufficiently many successive collisions, nucleus-nucleus scattering. is here automatically provided by the tunnelling process. So 6 Advances in High Energy Physics

the thermal hadronic final state in high energy collisions is [15] P. Braun-Munzinger, D. Magestro, K. Redlich, and J. Stachel, obtained through a stochastic process. “Hadron production in Au-Au collisions at RHIC,” Physics Finally, high energy particle physics, and in particular Letters B,vol.518,no.1-2,pp.41–46,2001. hadron production, is, in our opinion, the promising sector [16] U. Heinz, “Primordial hadrosynthesis in the little bang,” Nuclear tofindtheanalogueoftheHawking-Unruhradiationfortwo Physics A,vol.661,pp.140–149,1999. main reasons: color confinement and the huge acceleration [17]R.Stock,“Thepartontohadronphasetransitionobservedin that cannot be reached in any other dynamical systems. Pb + Pb collisions at 158 GeV per nucleon,” Physics Letters B, vol. 456, no. 2-4, pp. 277–282, 1999. Conflict of Interests [18] A. Bialas, “Fluctuations of the string tension and transverse mass distribution,” Physics Letters B,vol.466,no.2-4,pp.301– The authors declare that there is no conflict of interests 304, 1999. regarding the publication of this paper. [19] H. Satz, “The search for the QGP: a critical appraisal,” Nuclear Physics B—Proceedings Supplements, vol. 94, no. 1–3, pp. 204– References 218, 2001. [20] J.Hormuzdiar,S.D.H.Hsu,andG.Mahlon,“Particlemultiplic- [1] F. Becattini, “A thermodynamical approach to hadron produc- ities and thermalization in high energy collisions,” International + − tion in 𝑒 𝑒 collisions,” Zeitschriftur f¨ Physik C,vol.69,pp.485– Journal of Modern Physics E,vol.12,no.5,pp.649–659,2003. 492, 1996. [21] V. Koch, “Some remarks on the statistical model of heavy ion [2] F. Becattini, “Universality of thermal hadron production in pp, collisions,” Nuclear Physics A,vol.715,p.108,2003. + − 𝑝𝑝 and 𝑒 𝑒 collisions,” in Universality Features in Multihadron [22] L. McLerran, “The quark gluon plasma and the color glass Production and the Leading Effect, pp. 74–104, World Scientific, condensate: 4 lectures,” http://arxiv.org/abs/hep-ph/0311028. Singapore, 1998, http://arxiv.org/abs/hep-ph/9701275. [23] Y. Dokshitzer, “Historical and futuristic perturbative and non- [3] F. Becattini and G. Passaleva, “Statistical hadronization model perturbative aspects of QCD jet physics,” Acta Physica Polonica and transverse momentum spectra of hadrons in high energy B, vol. 36, p. 361, 2005. collisions,” European Physical Journal C,vol.23,no.3,pp.551– [24] F. Becattini, “What is the meaning of the statistical hadroniza- 583, 2002. tion model?” Journal of Physics: Conference Series,vol.5,p.175, [4] F. Becattini and U. Heinz, “Thermal hadron production in pp 2005. and 𝑝𝑝 collisions,” Zeitschriftur f¨ Physik C,vol.76,no.268,p. 286, 1997. [25] P.Castorina, D. Kharzeev, and H. Satz, “Thermal hadronization and Hawking-Unruh radiation in QCD,” European Physical [5] J. Cleymans, H. Satz, E. Suhonen, and D. W. Von Oertzen, Journal C,vol.52,no.1,pp.187–201,2007. “Strangeness production in heavy ion collisions at finite baryon number density,” Physics Letters B,vol.242,no.1,pp.111–114, [26] S. W. Hawking, “Particle creation by black holes,” Communica- 1990. tions in Mathematical Physics,vol.43,no.3,pp.199–220,1975. [6] J. Cleymans and H. Satz, “Thermal hadron production in high [27] W. G. Unruh, “Notes on black-hole evaporation,” Physical energy heavy ion collisions,” Zeitschriftur f¨ Physik C,vol.57,pp. Review D,vol.14,no.4,pp.870–892,1976. 135–147, 1993. [28] F. Becattini, P. Castorina, J. Manninen, and H. Satz, “The + − [7]K.Redlich,J.Cleymans,H.Satz,andE.Suhonen,“Hadroni- thermal production of strange and non-strange hadrons in 𝑒 𝑒 sation of quark-gluon plasma,” Nuclear Physics A,vol.566,pp. Collisions,” European Physical Journal C,vol.56,no.4,pp.493– 391–394, 1994. 510, 2008. [8] P. Braun-Munzinger, J. Stachel, J. P. Wessels, and N. Xu, “Ther- [29] A. Wroblewski, “On the strange quark suppression factor in mal equilibration and expansion in nucleus-nucleus collisions high energy collisions,” Acta Physica Polonica B,vol.16,p.379, at the AGS,” Physics Letters B,vol.344,no.1–4,pp.43–48,1995. 1985. [9]F.Becattini,M.Gazdzicki,´ and J. Sollfrank, “On chemical [30] R. Hagedorn and K. Redlich, “Statistical thermodynamics in rel- equilibrium in nuclear collisions,” European Physical Journal C, ativistic particle and ion physics: canonical or grand canonical?” vol. 5, no. 1, pp. 143–153, 1998. Zeitschriftur f¨ Physik C,vol.27,pp.541–551,1985. [10] F. Becattini et al., “Features of particle multiplicities and [31] J. D. Bjorken, “Hadron final states in deep inelastic processes,” strangeness production in central heavy ion collisions between in Current Induced Reactions ,vol.56ofLecture Notes in Physics, 1.7A and 158A GeV/c,” Physical Review C,vol.64,ArticleID pp. 93–158, Springer, 1976. 024901, 2001. [11] P. Braun-Munzinger, K. Redlich, and J. Stachel, “Particle pro- [32] A. Casher, H. Neuberger, and S. Nussinov, “Chromoelectric- duction in heavy ion collisions,”in Quark-Gluon Plasma 3,R.C. flux-tube model of particle production,” Physical Review D,vol. Hwa and X. N. Wang, Eds., World Scientific, Singapore, 2003. 20, no. 1, pp. 179–188, 1979. [12] F.Becattini, “Statistical hadronisation phenomenology,” Nuclear [33] R. Brout, R. Parentani, and P. Spindel, “Thermal properties Physics A,vol.702,pp.336–340,2001. of pairs produced by an electric field: a tunneling approach,” [13] J. Letessier, J. Rafelski, and A. Tounsi, “Gluon production, Nuclear Physics B,vol.353,no.1,pp.209–236,1991. cooling, and entropy in nuclear collisions,” Physical Review C, [34] R. Parentani and S. Massar, “Schwinger mechanism, Unruh vol.50,no.1,pp.406–409,1994. effect, and production of accelerated black holes,” Physical [14] F. Becattini, J. Manninen, and M. Gazdzicki, “Energy and Review D,vol.55,no.6,pp.3603–3613,1997. system size dependence of chemical freeze-out in relativistic [35] K. Srinivasan and T. Padmanabhan, “Particle production and nuclear collisions,” Physical Review C,vol.73,ArticleID044905, complex path analysis,” Physical Review D,vol.60,no.2,Article 2006. ID 024007, 1999. Advances in High Energy Physics 7

[36] D. Kharzeev and K. Tuchin, “From color glass condensate to quark-gluon plasma through the event horizon,” Nuclear Physics A,vol.753,no.3-4,pp.316–334,2005. [37] S. P. Kim, “Schwinger mechanism and Hawking radiation as quantum tunneling,” Journal of the Korean Physical Society,vol. 53, pp. 1095–1099, 2008. [38] P. Castorina and H. Satz, “Causality constraints on hadron production in high energy collisions,” International Journal of Modern Physics E,vol.23,ArticleID1450019,2014. [39]I.Kraus,J.Cleymans,H.Oeschler,K.Redlich,andS.Wheaton, “Chemical equilibrium in collisions of small systems,” Physical Review C, vol. 76, Article ID 064903, 2007. [40]I.Kraus,J.Cleymans,H.Oeschler,andK.Redlich,“Particle production in p-p collisions and predictions for √𝑠=14TeV at the CERN Large Hadron Collider (LHC) ,” Physical Review C,vol.79,ArticleID014901,2009. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2014, Article ID 713574, 8 pages http://dx.doi.org/10.1155/2014/713574

Review Article Amplifying the Hawking Signal in BECs

Roberto Balbinot1,2 and Alessandro Fabbri2,3,4 1 Dipartimento di Fisica dell’UniversitadiBolognaandINFNSezionediBologna,ViaIrnerio46,40126Bologna,Italy` 2 MuseoStoricodellaFisicaeCentroStudieRicerche“EnricoFermi”,PiazzadelViminale1,00184Roma,Italy 3 Dipartimento di Fisica dell’Universita` di Bologna, Via Irnerio 46, 40126 Bologna, Italy 4 Departamento de F´ısica Teorica` and IFIC, Universidad de Valencia-CSIC, C. Dr. Moliner 50, 46100 Burjassot, Spain

Correspondence should be addressed to Alessandro Fabbri; [email protected]

Received 7 January 2014; Accepted 6 March 2014; Published 27 April 2014

Academic Editor: Piero Nicolini

Copyright © 2014 R. Balbinot and A. Fabbri. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

We consider simple models of Bose-Einstein condensates to study analog pair-creation effects, namely, the Hawking effect from acoustic black holes and the dynamical Casimir effect in rapidly time-dependent backgrounds. We also focus on a proposal by Cornell to amplify the Hawking signal in density-density correlators by reducing the atoms’ interactions shortly before measurements are made.

1. Introduction the condensate should appear for points situated on opposite sides with respect to the horizon [14].Thisisthe“smoking Analogue models in condensed matter systems are nowadays gun” of the Hawking effect. Soon after this proposal, Eric an active field of investigation, not only on the theoretical Cornell at the first meeting on “experimental Hawking radia- side but, more importantly, also on the experimental one. tion”heldinValenciain2009suggestedthatonecanamplify The underlying idea is to reproduce in a condensed matter this characteristic signal by reducing the interaction coupling context peculiar and interesting quantum effects predicted among the atoms of the BEC shortly before measuring the by Quantum Field Theory in curved space, whose experi- density correlations [15]. mental verification in the gravitational context appears at the Here we review in a simple pedagogical way, using toy moment by far out of reach. models, how the analogous of Hawking radiation occurs in Manyeffortsaredevotedtofindthemostfamousofthese a supersonic flowing BEC and how the corresponding char- effects, namely, the thermal emission by black holes predicted acteristic peak in the correlation function can be amplified by Hawking in 1974 [1]. Among the condensed matter systems according to Cornell’s suggestion. It should be stressed that under examination, Bose-Einstein condensates appear as nowadays correlation functions measurements are becoming the most promising setting to achieve this goal [2–13]. The the basic experimental tool to investigate Hawking-like radi- major problem one has to face experimentally is the correct ation in condensed matter systems. identification of the signal corresponding to the analogue of Hawking radiation, namely, a thermal emission of phonons as a consequence of a sonic horizon formation, since it can 2. BECs: The Gravitational Analogy be covered by other competing effects, like large thermal and Hawking Radiation fluctuations. ABosegasinthedilutegasapproximationisdescribedbya A major breakthrough to overcome this problem came field operator Ψ̂ with equal-time commutator (see, e.g., [16]) in 2008, when it was predicted that, as a consequence of Hawking radiation being a genuine pair creation process, [Ψ̂ (𝑡, 𝑥⃗) , Ψ̂† (𝑡, 𝑥⃗󸀠)] = 𝛿3 (𝑥−⃗ 𝑥⃗󸀠) a characteristic peak in the density correlation function of (1) 2 Advances in High Energy Physics

satisfying the time-dependent Schrodinger¨ equation horizon located at the surface where |V0|=𝑐.Thesame analysis performed by Hawking in the gravitational case can 2 ̂ ℏ ⃗2 ̂†̂ ̂ be repeated step by step, leading to the prediction [17]that 𝑖ℏ𝜕𝑡Ψ=(− ∇ +𝑉 +𝑔Ψ Ψ) Ψ, (2) 2𝑚 ext acoustic black holes will emit a thermal flux of phonons at the temperature 𝑚 𝑉 where is the mass of the atoms, ext is the external 𝜅 𝑔 𝑇 = , potential, and is the nonlinear atom-atom interaction 𝐻 2𝜋 (9) coupling constant. At sufficiently low temperatures a large 𝜅 = (1/2𝑐)(𝑑(𝑐2 − V⃗2)/𝑑𝑛)| 𝑛 fraction of the atoms condense into a common ground state where hor,with being the normal which is described, in the mean field approach, by a 𝑐-number to the horizon, is the horizon’s surface gravity. field Ψ0(𝑡, 𝑥)⃗. To consider linear fluctuations around this classical mac- 3. The Model roscopic condensate, one writes the bosonic field operator Ψ̂ as To simplify the mathematics involved in the process, we will consider a 1D configuration (in 1D one should more correctly Ψ∼Ψ̂ (1 + 𝜙)̂ , 0 (3) speak of quasi-condensation [18, 19]) in which 𝑛0 and V0 are ̂ ̂ constant and where the only nontrivial quantity is the speed where 𝜙 is a small (quantum) perturbation. Ψ0 and 𝜙 satisfy, of sound 𝑐. As explained in [20], this can be achieved by respectively, Gross-Pitaevski varying the coupling constant 𝑔 (and therefore 𝑐)andthe 𝑔𝑛 +𝑉 2 external potential but keeping the sum 0 ext constant. ℏ 2 𝑖𝑘0𝑥−𝑖𝑤0𝑡 𝑖ℏ𝜕 Ψ =(− ∇⃗ +𝑉 +𝑔𝑛 )Ψ , In this way, the plane-wave function Ψ0 = √𝑛0𝑒 , 𝑡 0 2𝑚 ext 0 0 (4) where V0 =ℏ𝑘0/𝑚 is the condensate velocity and ℏ𝑤0 = 2 2 ℏ 𝑘 /2𝑚 + 𝑉 +𝑔𝑛0,whereℏ𝑤0 is the chemical potential 𝑛 =|Ψ|2 0 ext where 0 0 is the number density, and Bogoliubov-de of the gas, is a solution of (4)everywhere.Notethatsucha Gennes equations stationary configuration is difficult to reach experimentally; nevertheless it gives results similar to those obtained by more ℏ2 ℏ2 ∇Ψ⃗ 𝑖ℏ𝜕 𝜙=−(̂ ∇⃗2 + 0 ∇)⃗ 𝜙+𝑚𝑐̂ 2 (𝜙+̂ 𝜙̂†), realistic configurations [21]. 𝑡 (5) ̂ 2𝑚 𝑚 Ψ0 The fluctuation operator 𝜙 is expanded in the usual form in terms of positive and negative norm modes as with 𝑐=√𝑔𝑛0/𝑚 being the speed of sound. 𝜙̂(𝑡, 𝑥) = ∑ [𝑎̂ 𝜙 (𝑡, 𝑥) + 𝑎̂†𝜑∗ (𝑡, 𝑥)], Contact with the gravitational analogy (see, e.g., [13]) is 𝑗 𝑗 𝑗 𝑗 (10) achieved in the (long wavelength) hydrodynamic approxima- 𝑗 tion, more easily realised by considering the density-phase ̂ ̂† 𝑖𝜃̂ where 𝑎𝑗 and 𝑎𝑗 are quasi-particle’s annihilation and creation representation for the Bose operator Ψ=̂ √𝑛𝑒̂ and the operators. From (5)anditsHermiteanconjugate,weseethat 𝑛=𝑛̂ + 𝑛̂ , 𝜃=𝜃̂ + 𝜃̂ 𝑛̂ , 𝜃̂ splitting 0 1 0 1 in which 1 1 represent the the modes 𝜙𝑗(𝑡, 𝑥) and 𝜑𝑗(𝑡, 𝑥) satisfy the coupled differential linear (quantum) density and phase fluctuations, respectively. ̂ ̂† equations In terms of 𝜙 and 𝜙 we have 𝜉𝑐 2 𝑐 𝑐 [𝑖 𝑡(𝜕 + V0𝜕𝑥)+ 𝜕𝑥 − ]𝜙𝑗 = 𝜑𝑗, † 𝑖 † 2 𝜉 𝜉 𝑛̂ =𝑛 (𝜙̂ + 𝜙̂ ) , 𝜃̂ =− (𝜙̂ −𝜙 ) . (6) 1 0 1 2 (11) 𝜉𝑐 2 𝑐 𝑐 [−𝑖 𝑡(𝜕 + V0𝜕𝑥)+ 𝜕𝑥 − ]𝜑𝑗 = 𝜙𝑗. Provided that the condensate density 𝑛0 and velocity V⃗0 = 2 𝜉 𝜉 ℏ∇𝜃0/𝑚 vary on length scales much bigger than the healing length 𝜉=ℏ/𝑚𝑐(the fundamental length scale of the The normalizations are fixed, via integration of the equal-time condensate), the BdG equation reduces to the continuity and commutator obtained from (1), namely, 𝑛̂ 𝜃̂ 1 Euler equations for 1 and 1 and these can be combined [𝜙̂(𝑡, 𝑥) , 𝜙̂† (𝑡, 𝑥󸀠)] = 𝛿 (𝑥−𝑥󸀠) , 𝜃̂ (12) to give a second-order differential equation for 1 which is 𝑛0 mathematically equivalent to a Klein-Gordon (KG) equation by ◻𝜃̂ =0, 1 (7) 𝛿 󸀠 ∗ ∗ 𝑗𝑗 ∫ 𝑑𝑥 [𝜙 𝜙 −𝜑 𝜑 󸀠 ] = . 𝑗 𝑗󸀠 𝑗 𝑗 𝑛 (13) where ◻ is the covariant KG operator from the acoustic metric 0 𝑛 In order to get simple analytical expressions, in the 𝑑𝑠2 = 0 [− (𝑐2 − V⃗2)𝑑𝑡2 −2V⃗𝑑𝑡𝑑𝑥+𝑑⃗ 𝑥⃗2]. 𝑚𝑐 0 0 (8) following we will consider simple models with step-like discontinuities in the speed of sound 𝑐 and impose the For a flow which presents a transition from a subsonic appropriate boundary conditions for the modes that are (|V0|<𝑐)flowtoasupersonicone(|V⃗0|>𝑐in some solutions to (11). For more general profiles a numerical region) the metric (8) describes an acoustic black hole, with analysis can be performed; see, for example, [22–25]. Advances in High Energy Physics 3

3.1. Acoustic Black Holes and the Hawking Effect. Asimple 1.0 analytical model of an acoustic black hole [26]canbe obtained by gluing two semi-infinite stationary and homo- 0.5 geneous 1D condensates, one subsonic (𝑥<0)andthe other supersonic (𝑥>0), along a spatial discontinuity at 𝑥=0(see [27], to which we refer for more detailed 0.0 𝜔 explanations throughout this paragraph, and the references therein): 𝑐(𝑥)𝑙 =𝑐 𝜃(−𝑥)𝑟 +𝑐 𝜃(𝑥).WetakeV0 <0;that is,theflowisfromrighttoleft,and𝑐𝑙 <|V0|<𝑐𝑟.We −0.5 denote the modes solutions in each homogeneous region and corresponding to the fields 𝜙 and 𝜑 as −1.0 −2 −1 0 1 2 𝜙 =𝐷(𝜔) 𝑒−𝑖𝑤𝑡+𝑖𝑘(𝜔)𝑥,𝜑=𝐸(𝜔) 𝑒−𝑖𝑤𝑡+𝑖𝑘(𝜔)𝑥, 𝜔 𝜔 (14) k so that (11)become Figure 1: Dispersion relation in the subsonic region (𝜔 is given in units of the chemical potential and 𝑘 in units of the healing length). 𝜉𝑐𝑘2 𝑐 𝑐 [(𝜔 − V 𝑘) − − ]𝐷(𝜔) = 𝐸 (𝜔) , 0 2 𝜉 𝜉 (15) 0.2 𝜉𝑐𝑘2 𝑐 𝑐 [− (𝜔 − V 𝑘) − − ]𝐸 𝜔 = 𝐷 𝜔 , 0 ( ) ( ) 𝜔 2 𝜉 𝜉 max 0.1

k4 while the normalization condition (13)gives k k3 ku 0.0 󵄨 󵄨 𝜔 1 󵄨 𝑑𝑘 󵄨 |𝐷 (𝜔)|2 − |𝐸 (𝜔)|2 = 󵄨 󵄨 . 󵄨 󵄨 (16) 2𝜋𝑛0 󵄨𝑑𝑤󵄨 −0.1 The combination of the two equations15 ( )givestheBogoli- ubov dispersion relation for a one-dimensional Bose liquid k −0.2 max flowing at constant velocity −2 −1 0 1 2 2 4 k 2 2 2 𝜉 𝑘 (𝜔−V0𝑘) =𝑐 (𝑘 + ) (17) 4 Figure 2: Dispersion relation in the supersonic region (again, 𝜔 is given in units of the chemical potential and 𝑘 in units of the healing containing the positive and negative norm branches 𝑤−V0𝑘= length). ±𝑐√𝑘2 +(𝜉2𝑘4/4) ≡ ±Ω(𝑘) which, for the subsonic and supersonic regions, are given, respectively, in Figures 1 and 2. 𝐷 𝐸 Moreover, inserting the relation between and from (15) 𝑘 ,𝑘 (∼ 1/𝜉) 𝑘 𝑘 into (16)wefindthemodenormalizations and 3 4 ,twoofwhich( 𝑢 and 4)belongtothe negative norm branch. 𝜔−V𝑘+(𝑐𝜉𝑘2/2) To find modes evolution for all 𝑥 one needs to write down 𝜙 𝜑 𝑙 𝐷 (𝜔) = 󵄨 󵄨, the general solutions for ( )intheleftsupersonic()andthe √ 2 󵄨 −1󵄨 𝑟 4𝜋𝑛0𝑐𝜉𝑘 󵄨(𝜔−V𝑘)(𝑑𝑘/𝑑𝜔) 󵄨 right subsonic ( ) regions (we restrict to the most interesting (18) case 𝜔<𝜔max) 𝜔−V𝑘−(𝑐𝜉𝑘2/2) 𝐸 (𝜔) =− , 󵄨 󵄨 𝜙𝑙 =𝑒−𝑖𝜔𝑡 √ 2 󵄨 −1󵄨 𝜔 4𝜋𝑛0𝑐𝜉𝑘 󵄨(𝜔−V𝑘)(𝑑𝑘/𝑑𝜔) 󵄨 𝑙 𝑙 𝑖𝑘𝑙 𝑥 𝑙 𝑙 𝑖𝑘𝑙 𝑥 𝑙 𝑙 𝑖𝑘𝑙 𝑥 𝑙 𝑙 𝑖𝑘𝑙 𝑥 ×[𝐷 𝐴 𝑒 V +𝐷 𝐴 𝑒 𝑢 +𝐷 𝐴 𝑒 3 +𝐷 𝐴 𝑒 4 ], where 𝑘=𝑘(𝜔)are the roots of the quartic equation (17)at V V 𝑢 𝑢 3 3 4 4 fixed 𝜔. 𝜙𝑟 =𝑒−𝑖𝜔𝑡 In the subsonic case (17) admits two real and two complex 𝜔 𝑘 (∼ solutions. Regarding the real solutions, Figure 1,wecall V 𝑟 𝑟 𝑖𝑘𝑟𝑥 𝑟 𝑟 𝑖𝑘𝑟 𝑥 𝑟 𝑖𝑘3 𝑥 𝑟 𝑖𝑘𝑟 𝑥 2 2 ×[𝐷 𝐴 𝑒 V +𝐷 𝐴 𝑒 𝑢 +𝑑 𝐴 𝑒 𝑑 +𝐺 𝐴 𝑒 𝑔 ] 𝜔/(V0 −𝑐)+𝑂(𝜉 )) and 𝑘𝑢(∼ 𝜔/(V0 +𝑐)+𝑂(𝜉 )) the ones corre- V V 𝑢 𝑢 𝜙 𝑑 𝜙 𝑔 sponding to, respectively, negative and positive group velocity (19) V𝑔 =𝑑𝜔/𝑑𝑘(the other two complex conjugated solutions 𝑘 correspond to, respectively, spatially decaying 𝑑 and growing (the expansions for 𝜑 are the same up to the replacement 𝑘 𝑔 modes). In the supersonic case (see Figure 2)weseethat, 𝐷→𝐸)andimpose,from(11), the matching conditions for the most interesting regime (𝜔<𝜔max ∼1/𝜉), there are now four real solutions, corresponding to four propagating 󸀠 󸀠 [𝜙]=0, [𝜙] = 0, [𝜑] = 0, [𝜑 ]=0, (20) modes: 𝑘V,𝑘𝑢 (present also in the hydrodynamical limit 𝜉=0) 4 Advances in High Energy Physics where []indicates the variation across the jump at 𝑥=0, and partners allowing us to write down the relations between left and right 󵄨 󵄨 1 𝐴 𝑀 󵄨 𝑢𝑙,out† 𝑢𝑙,out󵄨 󵄨 󵄨2 amplitudes through a scattering matrix scatt in the form ⟨0, 󵄨𝑎̂ 𝑎̂ 󵄨 0, ⟩ = 󵄨𝑆 󵄨 ∼ in 󵄨 𝜔 𝜔 󵄨 in 󵄨 𝑢𝑙,4𝑙󵄨 𝜔 (25) 𝐴𝑙 𝐴𝑟 V V follow an approximate (low-frequency) thermal (1/𝑤) spec- 𝑙 𝑟 𝐴 𝐴𝑢 trum [21], the proportionality factor allowing the identifica- ( 𝑢)=𝑀 ( ). 𝑙 scatt 𝑟 (21) tion of a Hawking temperature (∼1/𝜉)inthisidealisedset- 𝐴 𝐴𝑑 3 ting. We can understand the mechanism by which Hawking 𝑙 𝐴𝑟 𝐴4 𝑔 radiation is emitted by looking, using (6), at the equal-time density-density correlator This allows us to construct explicitly the decomposition of 𝜙̂ (2) 󸀠 1 󵄨 1 1 󸀠 󸀠 󵄨 the field operator in terms of the “in” and “out” bases. The 𝐺 (𝑡; 𝑥, 𝑥 )≡ lim ⟨0, in 󵄨𝑛̂ (𝑡, 𝑥) , 𝑛̂ (𝑡 ,𝑥 )󵄨 0, in⟩, in 2 󸀠 󵄨 󵄨 “in”basisisconstructedwith𝜙 modes propagating from the 𝑛0 𝑡→𝑡 asymptotic regions (𝑥→±∞)towardsthediscontinuity (26) (𝑥=0), while the “out” basis is constructed with modes 󸀠 𝜙out propagatingawayfromthediscontinuityto𝑥=±∞. whose main contribution in the 𝑥<0and 𝑥 >0sector Looking at Figures 1 and 2,weseethatunitamplitudemodes comes from the 𝑢𝑙-𝑢𝑟 term 𝑟 𝑙 𝑙 defined on the left moving 𝑘V and right moving 𝑘3,𝑘4(∼ 󵄨 1 󵄨 1/𝜉) momenta define the ingoing scattering states, while unit ⟨𝑛̂ 𝑛̂ ⟩󵄨 𝑟 𝑛2 1 1 󵄨 amplitude modes defined on the right moving 𝑘𝑢 and left 0 󵄨Hawking 𝑘𝑙 ,𝑘𝑙 moving V 𝑢 momenta define the outgoing scattering states. 𝑤max ∗ 𝑤,out 𝑤,out Onecanthenwritedownthe“in”decompositionintermsof ∼ Re ∫ 𝑑𝑤𝑆𝑢𝑙,4𝑙𝑆𝑢𝑟,4𝑙 (𝜙𝑢,𝑙 +𝜑𝑢,𝑙 ) the “in” scattering states 0 (27) ×(𝜙𝑤,out∗ +𝜑𝑤,out∗) 𝜔max 𝑢,𝑟 𝑢,𝑟 ̂ V,in V,in 3,in 3,in 4,in† 4,in 𝜙=∫ 𝑑𝜔[𝑎̂𝜔 𝜙𝜔 + 𝑎̂𝜔 𝜙𝜔 + 𝑎̂𝜔 𝜙𝜔 + ℎ.𝑐.] 0 󸀠 sin [𝜔max (𝑥 /(V0 +𝑐𝑟)−𝑥/(V0 +𝑐𝑙))] (22) ∼ . 󸀠 𝑥 /(V0 +𝑐𝑟)−𝑥/(V0 +𝑐𝑙) or, equivalently, on the basis of the “out” scattering ones. Note 𝑙 that since 𝑘4 belongs to the negative norm branch the corre- The existence of the peak at 𝜙in sponding “in” mode 4,𝑙 is multiplied by a creation operator 𝑥󸀠 𝑥 4,in† 𝑎̂𝜔 (the same thing happens, in the “out” decomposition, = (28) out V0 +𝑐𝑟 V0 +𝑐𝑙 for 𝜙𝑢,𝑙 ). Using (21)onecanconstructthe3×3S-matrix in out relating 𝜙 and 𝜙 modes was first pointed out in [14] in the hydrodynamical approximation using QFT in curved space techniques. The physical 𝜙V,in =𝑆 𝜙V,out +𝑆 𝜙𝑢𝑟,out +𝑆 𝜙𝑢𝑙,out, 𝜔 V𝑙,V𝑟 𝜔 𝑢𝑟,V𝑟 𝜔 𝑢𝑙,V𝑙 𝜔 picture that emerges is that Hawking quanta and partners are continuously created in pairs from the horizon at each time 𝜙3,in =𝑆 𝜙V,out +𝑆 𝜙𝑢𝑟,out +𝑆 𝜙𝑢𝑙,out, 𝜔 V𝑙,3𝑙 𝜔 𝑢𝑟,3𝑙 𝜔 𝑢𝑙,3𝑙 𝜔 (23) 𝑡, propagate on opposite directions at speeds V0 +𝑐𝑙 <0and 󸀠 V0 +𝑐𝑟 >0, and after time Δ𝑡 are located at 𝑥 and 𝑥 related 𝜙4,in =𝑆 𝜙V,out +𝑆 𝜙𝑢𝑟,out +𝑆 𝜙𝑢𝑙,out, 𝜔 V𝑙,4𝑙 𝜔 𝑢𝑟,4𝑙 𝜔 𝑢𝑙,4𝑙 𝜔 as in (28). The existence of the Hawking peak was nicely confirmed by numerical “ab initio” simulations with more which is not trivial since it mixes positive and negative norm realistic configurations performed in [20]. How correlation modes. As a consequence, the Bogoliubov transformation measurements can reveal the quantum nature of Hawking between “in” and “out” creation and annihilation operators radiation is discussed in [28–30]. is also not trivial because it mixes creation and annihilation operators. This has the crucial consequence that the “in” and “out” Hilbert spaces are not unitary related; in particular the 3.2. Analog Dynamical Casimir Effect. Adistincttypeofpair- corresponding vacua are different; that is, |0, in⟩ =|0,̸out⟩. creation takes place in time-dependent backgrounds, with The physical consequence is that if we prepare the system one important example being quantum particle creation in in the |0, in⟩ vacuum state, so that there are no incoming cosmology. We will study the analogue of these phenomena phonons at 𝑡=−∞,wewillhave,atlatetimes,outgoing in BECs with another simple model in which the speed of 𝑡=0 quanta on both sides of the horizon: the vacuum has sponta- sound has a step-like discontinuity in time at separating 𝑟 two “initial” and “final” infinite homogeneous condensates: neously emitted phonons, mainly in the 𝑘𝑢 channel (Hawking 𝑙 𝑐(𝑡) =𝑐 𝜃(−𝑡) +𝑐 𝜃(𝑡); see for more details [31, 32]. The quanta) and 𝑘 (partners). The analytical calculations show in fin 𝑢 modes solutions in the initial (𝑡<0)andfinal(𝑡>0)regions that the number of emitted Hawking quanta [26] arenowofthetype 󵄨 󵄨 1 󵄨 𝑢𝑟,out† 𝑢𝑟,out󵄨 󵄨 󵄨2 ⟨0, in 󵄨𝑎̂ 𝑎̂ 󵄨 0, in⟩=󵄨𝑆𝑢𝑟,4𝑙󵄨 ∼ (24) −𝑖𝑤(𝑘)𝑡+𝑖𝑘𝑥 −𝑖𝑤(𝑘)𝑡+𝑖𝑘𝑥 󵄨 𝜔 𝜔 󵄨 󵄨 󵄨 𝜔 𝜙𝑘 =𝐷(𝑘) 𝑒 ,𝜑𝑘 =𝐸(𝑘) 𝑒 , (29) Advances in High Energy Physics 5 for which (11)become One can easily construct “in” and “fin” decompositions for the 𝜙̂ 2 field by considering initial (final) unit amplitude positive 𝑐𝜉𝑘 𝑐 𝑐 in(fin) in(fin) [− (𝜔−V𝑘) + + ]𝐷(𝑘) =− 𝐸 (𝑘) , norm modes (𝐴 ( ) =1,𝐵 =0) 𝜙 (𝜑 ) modes for 2 𝜉 𝜉 in fin in 𝑘 𝑘 all 𝑡 using (36): (30) 𝑐𝜉𝑘2 𝑐 𝑐 ∞ [(𝜔−V𝑘) + + ]𝐸(𝑘) =− 𝐷 (𝑘) , 𝜙̂(𝑡, 𝑥)in(fin) = ∫ 𝑑𝑘[𝑎̂in(fin)𝜙in(fin) +𝑎in(out)†𝜑in(fin)∗]. 2 𝜉 𝜉 𝑘 𝑘 𝑘 𝑘 −∞ (38) while the normalization condition (13)yields The modes are related through a nontrivial Bogoliubov 2 2 1 |𝐷 (𝑘)| − |𝐸 (𝑘)| = (31) transformation mixing positive and negative norm modes 2𝜋𝑛0 𝜙in =𝛼𝜙fin +𝛽 𝜙out∗, giving 𝑘 𝑘 𝑘 −𝑘 (39) 𝜔−V𝑘+(𝑐𝜉𝑘2/2) where 𝐷 (𝑘) = , Ω +Ω Ω −Ω √4𝜋𝑛 𝑐𝜉𝑘2 |(𝜔−V𝑘)| 𝛼 = in fin ,𝛽= fin in , 0 𝑘 2√Ω Ω 𝑘 2√Ω Ω (40) (32) in fin in fin 𝜔−V𝑘−(𝑐𝜉𝑘2/2) 𝐸 (𝑘) =− . and, consequently, also the relation between “in” and “fin” 2 annihilation and creation operators will mix annihilation and √4𝜋𝑛0𝑐𝜉𝑘 |(𝜔−V𝑘)| creation operators, implying again that the two decompositions are inequivalent and in particular the two vacuum states Here, 𝜔=𝜔(𝑘)correspondstothetworealsolutionsto(17), |0, in⟩ and |0, fin⟩ are different. The physical consequence is which is quadratic in 𝜔 at fixed 𝑘.Theseread that the step-like discontinuity at 𝑡=0will induce particle creation, the features of which can be understood by looking 𝑐2𝑘4𝜉2 √ 2 2 𝜔+ (𝑘) = V𝑘+ 𝑐 𝑘 + ≡ V𝑘−Ω(𝑘) , at the time-dependent terms of the one-time density-density 4 correlator which in the hydrodynamical limit reads (33) 2 4 2 (2) 󸀠 󵄨 𝑐 𝑘 𝜉 𝐺 (𝑡, 𝑥, 𝑥 )󵄨 𝜔 (𝑘) = V𝑘−√𝑐2𝑘2 + ≡ V𝑘−Ω(𝑘) , 󵄨dyncas − 4 1 1 (41) ∼ + . 2 2 where 𝜔+(𝑘) corresponds to the positive norm branch and (2𝑐 𝑡−(𝑥−𝑥󸀠)) (2𝑐 𝑡−(𝑥󸀠 −𝑥)) fin fin 𝜔−(𝑘) corresponds to the negative norm one. To find modes 𝑡 evolution for all one first writes down the general solutions At 𝑡=0andeverywhereinspacecorrelatedpairsofparticles 𝜙 𝜑 intheinitialandfinalregionsfor ( ) with opposite momentum are created out of the vacuum state, with velocities V −𝑐 (left moving) and V +𝑐 (right 𝜙fin(in) =𝑒𝑖𝑘𝑥 fin fin 𝑘 moving). At time 𝑡 such particles are separated by a distance

+ −𝑖𝜔fin(in)(𝑘)𝑡 󵄨 󵄨 ×[𝐷 (𝑘) 𝐴 𝑒 + 󵄨𝑥−𝑥󸀠󵄨 =2𝑐 𝑡, fin(in) fin(in) (34) 󵄨 󵄨 fin (42)

− −𝑖𝜔fin(in)(𝑘)𝑡 +𝐷 (𝑘) 𝐵 𝑒 − ] which is indeed the correlation displayed in (41). This effect fin(in) fin(in) was recently observed in [33] by considering homogeneous 𝑉 (for 𝜑 we have the same expansion with 𝐷 replaced by 𝐸)and condensates with trapping potential ext rapidly varying in impose the matching conditions, from (11), time, where correlation functions in velocity/momentum space were measured. [𝜙] =0, [𝜑] =0, (35) 3.3. Amplification of the Hawking Signal in Density Corre- [] where now indicates the variation across the discontinuity lators. It has been argued by Cornell [15]thatawayto fin at 𝑡=0. They allow the final amplitudes 𝐴 to be related to amplify the Hawking signal in density-density correlators 𝐴in 𝑀 the initial ones through the matrix Bog: is to reduce the interactions shortly before measuring the density correlations. Since 𝑐=√𝑔𝑛0/𝑚,reducing𝑔 means 𝐴 𝐴 ( fin)=𝑀 ( in), that the speed of sound is also reduced. We will model 𝐵 bog 𝐵 (36) fin in this situation by matching our idealised acoustic black hole configuration of Section 3.1 with a final infinite homogeneous where 𝑐 <𝑐 < condensate characterised by a small sound velocity fin ( 𝑙 Ω +Ω Ω −Ω 𝑐𝑟); see Figure 3. To study this situation, in which a spatial 1 in fin fin in 𝑀 = ( ). (37) step-like discontinuity in 𝑐 at 𝑥=0is combined with a bog 2√Ω Ω Ω −Ω Ω +Ω in out fin in in fin temporal step-like discontinuity at some 𝑡=𝑡0,wewilluse 6 Advances in High Energy Physics

c the tools introduced in the previous two subsections. We will fin calculate the density-density correlator in the “in” vacuum by expanding the density operator in the “in” decomposition c1 cr 𝑛̂1 (𝑡, 𝑥)

𝜔max V,in in V,in 3,in in in ≃𝑛0 ∫ 𝑑𝑤[𝑎̂𝜔 (𝜙V,𝑟 +𝜑V,𝑟 )+𝑎̂𝜔 (𝜙3,𝑙 +𝜑3,𝑙) 0

4,in† in in + 𝑎̂𝜔 (𝜙4,𝑙 +𝜑4,𝑙)+ℎ.𝑐.] (43) to get Figure 3: Spacetime diagram sketch of a spatial step-like discontinuity (acoustic black hole-like) followed by a temporal one leading 1 󵄨 󵄨 to a final homogeneous configuration. ⟨0, 󵄨𝑛̂ 𝑛̂ 󵄨 0, ⟩ 2 in 󵄨 1 1󵄨 in 𝑛0

𝑤 󸀠 max 󸀠 𝑖(𝑐 −𝑐𝑟)𝑘 𝑡0 󸀠 in V,in in∗ V,in∗ with 𝛼 = ((𝑐𝑟 +𝑐 )/2√𝑐𝑟𝑐 )𝑒 fin , 𝛽 = ((𝑐 − = ∫ 𝑑𝑤 V[(𝜙 ,𝑟 +𝜑V,𝑟 )(𝜙V,𝑟 +𝜑V,𝑟 ) fin fin fin −𝑖(𝑐 +𝑐 )𝑘󸀠𝑡 0 (44) 𝑐 )/2 𝑐 𝑐 )𝑒 fin 𝑟 0 𝑟 √ 𝑟 fin . in in in∗ in∗ It is useful to rewrite (46)and(47)intermsof𝑤 to +(𝜙3,𝑙 +𝜑3,𝑙)(𝜙3,𝑙 +𝜑3,𝑙 ) compare with the standard result without the temporal step- in∗ in∗ in in like discontinuity +(𝜙4,𝑙 +𝜑4,𝑙 )(𝜙4,𝑙 +𝜑4,𝑙)] . 𝜙𝑤,out +𝜑𝑤,out By expressing the “in” modes in terms of the “out” modes 𝑢,𝑙 𝑢,𝑙 andintheabsenceofthetemporalstep-likediscontinuity(say 𝑤 −𝑖𝑤𝑡+𝑖(𝑤𝑥/(V+𝑐 )) 𝑤 𝑖(𝑤𝑥/(V+𝑐 )) 𝑡0 →+∞) the Hawking signal is given by ∼ √ 𝑒 𝑙 󳨀→ √ 𝑒 𝑙 𝑐 𝑐 󵄨 𝑙 fin 1 󵄨 󵄨 󵄨 ⟨0, 󵄨𝑛̂ 𝑛̂ 󵄨 0, ⟩󵄨 2 in 󵄨 1 1󵄨 in 󵄨 −𝑖((V+𝑐 )/(V+𝑐 ))𝑤𝑡 −𝑖((V−𝑐 )/(V+𝑐 ))𝑤𝑡 𝑛 󵄨 ×(𝛼𝑒 fin 𝑙 +𝛽𝑒 fin 𝑙 ), 0 󵄨Hawking (48) 𝑤max 𝑤,out 𝑤,out ∗ 𝑤,out 𝑤,out (45) 𝜙𝑢,𝑟 +𝜑𝑢,𝑟 ∼ Re ∫ 𝑑𝑤𝑆𝑢𝑙,4𝑙𝑆𝑢𝑟,4𝑙 (𝜙𝑢,𝑙 +𝜑𝑢,𝑙 ) 0 𝑤 −𝑖𝑤𝑡+𝑖(𝑤𝑥󸀠/(V+𝑐 )) 𝑤 𝑖(𝑤𝑥󸀠/(V+𝑐 )) ×(𝜙𝑤,out∗ +𝜑𝑤,out∗). ∼ √ 𝑒 𝑟 󳨀→ √ 𝑒 𝑟 𝑢,𝑟 𝑢,𝑟 𝑐 𝑐 𝑟 fin

Inthepresenceofthetemporalstep-likediscontinuitywe 󸀠 −𝑖((V+𝑐 )/(V+𝑐𝑟))𝑤𝑡 󸀠 −𝑖((V−𝑐 )/(V+𝑐𝑟))𝑤𝑡 out out out out ×(𝛼𝑒 fin +𝛽𝑒 fin ). need to evolve the relevant modes 𝜙𝑢,𝑙 (𝜑𝑢,𝑙 )and𝜙𝑢,𝑟 (𝜑𝑢,𝑟 ), at the same value of 𝑤,acrossthediscontinuityat𝑡=𝑡0.Going 󸀠 The standard result is, from (45), a stationary peak at to the 𝑘 basis and considering 𝑘, 𝑘 small (𝑤∼(V +𝑐𝑙)𝑘 ∼ (V +𝑐)𝑘󸀠 𝑟 )wehave 𝑥 𝑥󸀠 − =0 V +𝑐 V +𝑐 (49) 𝑘, 𝑘, |𝑘| −𝑖(V+𝑐 )𝑘𝑡+𝑖𝑘𝑥 𝑙 𝑟 𝜙 out +𝜑 out ∼ √ 𝑒 𝑙 𝑢,𝑙 𝑢,𝑙 𝑐 𝑙 weighted by the 𝑤-independent factor (see (24)and(25))

𝑤 ∗ |𝑘| 𝑖𝑘𝑥 −𝑖(V+𝑐 )𝑘𝑡 −𝑖(V−𝑐 )𝑘𝑡 𝑆 𝑆 . 󳨀→ √ 𝑒 (𝛼𝑒 fin +𝛽𝑒 fin ), 𝑢𝑙,4𝑙 𝑢𝑟,4𝑙 (50) 𝑐 √𝑐𝑙𝑐𝑟 fin

(46) The effect of the temporal step-like discontinuity at 𝑡=𝑡0 (see 𝑖(𝑐 −𝑐 )𝑘𝑡 (48)) is to modify this signal into a main signal located at 𝛼 = ((𝑐 +𝑐 )/2 𝑐 𝑐 )𝑒 fin 𝑙 0 𝛽 = ((𝑐 −𝑐)/ where 𝑙 fin √ 𝑙 fin , fin 𝑙 −𝑖(𝑐 +𝑐 )𝑘𝑡 󸀠 2 𝑐 𝑐 )𝑒 fin 𝑙 0 (V +𝑐 ) (V +𝑐 ) √ 𝑙 fin ,and 𝑥 𝑥 fin fin − =( − )(𝑡−𝑡0) (51) 󸀠 󸀠 V +𝑐𝑙 V +𝑐𝑟 V +𝑐𝑙 V +𝑐𝑟 𝑘 ,out 𝑘 ,out 𝜙𝑢,𝑟 +𝜑𝑢,𝑟 with strength 󸀠 𝑘 −𝑖(V+𝑐 )𝑘󸀠𝑡+𝑖𝑘󸀠𝑥󸀠 ∼ √ 𝑒 𝑟 𝑤𝛼𝛼󸀠 𝑐 (47) 𝑆 𝑆∗ 𝑟 𝑐 𝑢𝑙,4𝑙 𝑢𝑟,4𝑙 (52) fin 󸀠 𝑘 𝑖𝑘󸀠𝑥󸀠 󸀠 −𝑖(V+𝑐 )𝑘󸀠𝑡 󸀠 −𝑖(V−𝑐 )𝑘󸀠𝑡 󸀠 󳨀→ √ 𝑒 (𝛼 𝑒 fin +𝛽𝑒 fin ), and three smaller signals located at 𝑥/(V +𝑐𝑙)−𝑥/(V +𝑐𝑟) = 𝑐 ((V+𝑐 )/(V+𝑐)−(V−𝑐 )/(V+𝑐 ))(𝑡−𝑡 ) 𝑥/(V+𝑐)−𝑥󸀠/(V+𝑐 ) fin fin 𝑙 fin 𝑟 0 , 𝑙 𝑟 = Advances in High Energy Physics 7

((V−𝑐 )/(V+𝑐)−(V+𝑐 )/(V+𝑐 ))(𝑡−𝑡 ) 𝑥/(V+𝑐)−𝑥󸀠/(V+ fin 𝑙 fin 𝑟 0 ,and 𝑙 condensate,” Physical Review Letters, vol. 109, Article ID 195301, 𝑐𝑟) = ((V−𝑐 )/(V+𝑐𝑙)−(V−𝑐 )/(V+𝑐𝑟))(𝑡−𝑡0) with strengths 2012. fin fin 󸀠 given by (52)inwhich𝛼𝛼󸀠 is substituted, respectively, by 𝛼𝛽 , [6] R. Schley, A. Berkovitz, S. Rinott, I. Shammass, A. Blumkin, 󸀠 󸀠 𝛼 𝛽,and𝛽𝛽 . and J. Steinhauer, “Planck distribution of phonons in a Bose- We see immediately that we loose the stationarity of the Einstein condensate,” Physical Review Letters, vol. 111, Article ID Hawking signal (49) and that the main signal is multiplied by 055301, 2013. [7] C. Barcelo,´ S. Liberati, and M. Visser, “Towards the observation √𝑐 𝑐 (𝑐 +𝑐)(𝑐 +𝑐) of Hawking radiation in Bose-Einstein condensates,” Interna- 𝜂= 𝑟 𝑙 𝛼𝛼󸀠 = fin 𝑙 fin 𝑟 𝑐 4𝑐2 (53) tional Journal of Modern Physics A,vol.18,no.21,pp.3735–3745, fin fin 2003. with respect to the standard result. This results indeed in an [8] C. Barcelo,´ S. Liberati, and M. Visser, “Probing semiclassical analog gravity in Bose-Einstein condensates with widely tun- amplification (i.e., the above term is >1) when 𝑐 <𝑐𝑙,𝑐𝑟. fin −3 able interactions,” Physical Review A,vol.68,ArticleID053613, Being the Hawking peak expected to be of order 5×10 2003. for realistic experimental settings [20](where𝑐𝑟 =2𝑐𝑙 was [9] S. Giovanazzi, C. Farrell, T. Kiss, and U. Leonhardt, “Conditions considered), we obtain an amplification factor 𝜂 = 15/4 for 𝑐 =2𝑐 =4𝑐 for one-dimensional supersonic flow of quantum gases,” Physi- 𝑟 𝑙 fin. cal Review A,vol.70,no.6,ArticleID063602,2004. [10] R. Schutzhold,¨ “Detection scheme for acoustic quantum radia- 4. Conclusions tion in Bose-Einstein condensates,” Physical Review Letters,vol. 97, Article ID 190405, 2006. In this paper we have briefly reviewed the analysis of [11] S. Wuster¨ and C. M. Savage, “Limits to the analog Hawking the analog Hawking effect and of the analog dynamical temperature in a Bose-Einstein condensate,” Physical Review A, Casimir effect by considering simple analytical models of vol.76,ArticleID013608,2007. Bose-Einstein condensates in which the speed of sound has [12] Y. Kurita and T. Morinari, “Formation of a sonic horizon in step-like discontinuities. We focussed on the study of the isotropically expanding Bose-Einstein condensates,” Physical density-density correlators which show, in the former case, Review A,vol.76,no.5,ArticleID053603,2007. the existence of a characteristic stationary Hawking quanta- [13]C.Barcelo,S.Liberati,andM.Visser,“Analoguegravity,”Living partner peak located at (28) and, in the latter, that of a time- Reviews in Relativity,vol.8,article12,2005. dependent feature (42). Following a suggestion by Cornell, we [14] R. Balbinot, A. Fabbri, S. Fagnocchi, A. Recati, and I. Carusotto, combined these two analyses to construct a model in which “Nonlocal density correlations as a signature of Hawking radiation from acoustic black holes,” Physical Review A,vol.78, the atoms’ interactions are rapidly lowered (and so the speed no. 2, Article ID 021603, 2008. of sound) before the correlations are measured. This results [15] E. Cornell, “Towards the observation of Hawking radiation in an amplification of the main Hawking peak, now time- in condensed matter systems,” Talk given at the workshop, dependent and located at (51), by the factor (53)thatcould Valencia, Spain, 2009, http://www.uv.es/workshopEHR. be useful in the experimental search. [16] L. Pitaevskii and S. Stringari, Bose-Einstein Condensation,vol. 116 of InternationalSeriesofMonographsonPhysics,The Conflict of Interests Clarendon Press, Oxford, UK, 2003. [17] W. G. Unruh, “Experimental black-hole evaporation?” Physical The authors declare that there is no conflict of interests Review Letters,vol.46,no.21,pp.1351–1353,1981. regarding the publication of this paper. [18] Y. Castin, “Simple theoretical tools for low dimension Bose gases,” JournaldePhysiqueIVFrance,vol.116,pp.89–132,2004. [19] C. Mora and Y. Castin, “Extension of Bogoliubov theory to Acknowledgment quasicondensates,” Physical Review A,vol.67,ArticleID053615, The authors thank I. Carusotto for useful discussions. 2003. [20] I. Carusotto, S. Fagnocchi, A. Recati, R. Balbino, and A. Fabbri, “Numerical observation of Hawking radiation from acoustic References black holes in atomic Bose-Einstein condensates,” New Journal of Physics,vol.10,ArticleID103001,2008. [1] S. W. Hawking, “Particle creation by black holes,” Communica- [21] P.-E.´ Larre,A.Recati,I.Carusotto,andN.Pavloff,“Quantum´ tions in Mathematical Physics,vol.43,no.3,pp.199–220,1975. fluctuations around black hole horizons in Bose-Einstein con- [2]L.J.Garay,J.R.Anglin,J.I.Cirac,andP.Zoller,“Sonicblack densates,” Physical Review A,vol.85,no.1,ArticleID013621, holes in dilute Bose-Einstein condensates,” Physical Review A, 2012. vol. 63, no. 2, Article ID 023611, 2001. [22] J. Macher and R. Parentani, “Black-hole radiation in Bose- [3]L.J.Garay,J.R.Anglin,J.I.Cirac,andP.Zoller,“Sonicanalog Einstein condensates,” Physical Review A,vol.80,no.4,Article of gravitational black holes in Bose-Einstein condensates,” ID 043601, 2009. Physical Review Letters, vol. 85, no. 22, pp. 4643–4647, 2000. [23] S. Finazzi and R. Parentani, “Spectral properties of acoustic [4] O. Lahav, A. Itah, A. Blumkin et al., “Realization of a sonic black black hole radiation: broadening the horizon,” Physical Review hole analog in a Bose-Einstein condensate,” Physical Review D,vol.83,no.8,ArticleID084010,2011. Letters,vol.105,no.24,ArticleID240401,2010. [24] S. Finazzi and R. Parentani, “Hawking radiation in dispersive [5]I.Shammass,S.Rinott,A.Berkovitz,R.Schley,andJ.Stein- theories, the two regimes,” Physical Review D,vol.85,Article hauer, “Phonon dispersion relation of an atomic Bose-Einstein ID 124027, 2012. 8 Advances in High Energy Physics

[25]S.Bar-Ad,R.Schilling,andV.Fleurov,“Nonlocalityand fluctuations near the optical analog of a sonic horizon,” Physical Review A,vol.87,ArticleID013802,2013. [26]A.Recati,N.Pavloff,andI.Carusotto,“Bogoliubovtheory of acoustic Hawking radiation in Bose-Einstein condensates,” Physical Review A,vol.80,no.4,ArticleID043603,2009. [27] R. Balbinot, I. Carusotto, A. Fabbri, C. Mayoral, and A. Recati, “Understanding hawking radiation from simple models of atomic Bose-Einstein condensates,” in Analogue Gravity Phenomenology,vol.870ofLecture Notes in Physics,pp.181–219, 2013. [28] K. V.Kheruntsyan, J.-C. Jaskula, P.Deuar et al., “Violation of the Cauchy-Schwarz inequality with matter waves,” Physical Review Letters,vol.108,ArticleID260401,2012. [29] J. R. M. de Nova, F. Sols, and I. Zapata, “Violation of Cauchy- Schwarz inequalities by spontaneous Hawking radiation in resonant boson structures,” http://arxiv.org/abs/1211.1761. [30] S. Finazzi and I. Carusotto, “Entangled phonons in atomic Bose- Einstein condensates,” http://arxiv.org/abs/1309.3414. [31] I. Carusotto, R. Balbinot, A. Fabbri, and A. Recati, “Density correlations and analog dynamical Casimir emission of Bogoliubov phonons in modulated atomic Bose-Einstein condensates,” European Physical Journal D,vol.56,no.3,pp.391–404,2010. [32] C. Mayoral, A. Fabbri, and M. Rinaldi, “Steplike discontinuities in Bose-Einstein condensates and Hawking radiation: dispersion effects,” Physical Review D, vol. 83, Article ID 124047, 2011. [33] J. C. Jaskula, G. B. Partridge, M. Bonneau et al., “Acoustic analog to the dynamical casimir effect in a Bose-Einstein condensate,” Physical Review Letters, vol. 109, Article ID 220401, 2012. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2014, Article ID 950672, 6 pages http://dx.doi.org/10.1155/2014/950672

Review Article Theory and Phenomenology of Space-Time Defects

Sabine Hossenfelder

Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, 106 91 Stockholm, Sweden

Correspondence should be addressed to Sabine Hossenfelder; [email protected]

Received 5 December 2013; Accepted 12 February 2014; Published 20 March 2014

Academic Editor: Piero Nicolini

Copyright © 2014 Sabine Hossenfelder. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

Whether or not space-time is fundamentally discrete is of central importance for the development of the theory of quantum gravity. If the fundamental description of spacetime is discrete, typically represented in terms of a graph or network, then the apparent smoothness of geometry on large scales should be imperfect—it should have defects. Here, we review a model for space-time defects and summarize the constraints on the prevalence of these defects that can be derived from observation.

1. Introduction about the nature of the problem but the big breakthrough has left us waiting. Next to the technical difficulties, the reason A theory of quantum gravity is necessary to describe the for the slow progress is the lack of experimental guidance. quantum behavior of space and time and to understand The possibility that quantum gravitational phenomena might what happens in strong gravitational fields, when curvature be observable has not been paid much attention to till the reaches the Planckian regime. Finding this missing theory of late 90s, and even now the awareness that this possibility quantum gravity is one of the big open problems in theoretical existsisslowtosinkintothemindsofthecommunity. physics today, and it concerns the most fundamental ingredi- However, without making contact to observation, no theory ents of our existing theories: space-time and its curvature, the arena in which physics happens. of quantum gravity can ever be accepted as a valid description But general relativity still stands apart from the quantum of nature. field theories of the standard model as a classical theory. In the absence of a fully fledged theory, this search There exists to date no known way to consistently couple a for observable consequences proceeds by the development classical theory to a quantum theory and neither do we know of phenomenological models. Such models parameterize how to quantize gravity. While several theoretical approaches properties that the theory of quantum gravity could have are being pursued with success, this success has so far been with the purpose of allowing to experimentally test or at exclusively on the side of mathematical consistency, and the least constrain the presence of these properties. This in turn connection of these approaches to reality is still unclear. guides the development of the theory. General reviews of The problem how to resolve the tension between quantum phenomenological models for quantum gravity can be found field theory and general relativity is more than an aesthetic in [1, 2]. In this contribution to the AHEP special issue unease. This tension signals that our understanding of nature on “Experimental Tests of Quantum Gravity and Exotic is incomplete, but it also offers an opportunity to improve Quantum Field Theory Effects,” we will discuss a possible our theories. The missing theory of quantum gravity has the phenomenological consequence of quantum gravity that has potential to revolutionize our understanding of space, time, so far received very little attention—the existence of space- and matter. time defects. Progress on the theory of quantum gravity, however, has In many approaches to quantum gravity—such as causal been slow. The problem has been known since more than sets, spin foams, causal dynamical triangulation, loop quan- 80 years now. Since then, we gained a great many insights tum gravity, and emergent and induced gravity scenarios 2 Advances in High Energy Physics based on condensed matter analogies—space-time is funda- distance—these points are not local (or not points, depending mentally discrete and the smooth background geometry that on your perspective). But besides this, during the last decades, we see only emerges as an approximation at low energies it has also become increasingly clear that a resolution of and large distances [3]. In this case, one expects that the the black hole information loss problem requires some type apparently smooth background geometry is imperfect and of nonlocality [7]. Meanwhile, a completely different line of has defects, just because perfection would require additional investigation has led to the conclusion that requiring the explanation. Planck energy to be an observer-independent component of In the following, we will review the recently proposed a four-momentum necessitates giving up absolute locality. model for space-time defects [4, 5]andsummarizethe Instead, one might have to settle on a weaker locality constraints on the prevalence of these defects that can be requirement, which has been called “relative locality” [8]. derived from observation. We will then discuss which steps While Planck-scale nonlocality has received quite some can be taken to improve the model so that the constraints attention, for example, in the well-studied models for quan- touch on the interesting parameter range. tumfieldtheorywithaminimallength[9], here, we will focus on a feature whose phenomenology has so far received very little attention: macroscopic nonlocality, where macroscopic 2. Space-Time Defects means much larger than the Planck length. Whether or not space-time is fundamentally discrete is a Macroscopic nonlocality can be expected to arise in question of central importance for the development of the approaches towards quantum gravity in which space-time is theory. But this discreteness typically makes itself noticeable only to good approximation a smooth manifold but funda- atthePlancklength,whichishardifnotimpossibletoaccess mentally a network (graph) consisting of nodes and links experimentally. Thus, instead of searching for direct evidence which can carry additional charges or degrees of freedom. for the Planck scale discreteness, we here propose to look The reason is that the emergence of a manifold from the for imperfections in this discreteness. Such imperfections in networkwillnotbeperfect,butitwillhavedefects.And space-time will cause deviations from general relativity, since since our macroscopic notion of distance only emerges with general relativity rests on the assumption that space-time is thespace-timeandthemetricthatwedefineonit,there a manifold and locally smooth and differentiable. Because is no reason why these defects should respect the emergent general relativity is an extremely well-tested theory even the macroscopic locality. smallest deviations can become noticeable, making space- This has been demonstrated explicitly by Markopoulou time defects a promising phenomenological consequence to and Smolin in [10]forthecaseofspinnetworks.Their search for. Looking for space-time defects as evidence for the argumentcanbebrieflysummarizedasfollows. presence of a discrete geometry is akin to looking for specks States of the spin network describe spatial slices of space- of dirt as evidence for the presence of a window. time and they change in time by a set of allowed evolution That such defects should exist is a model-independent moves, which are local according to the locality of the expectation for all approaches to quantum gravity in which network.Certainspinnetworkstates,called“weavestates,” geometry has a fundamentally discrete structure. But the match to good precision (up to Planck scale corrections), prevalence, distribution, and properties of the defects will slowly varying classical spatial metrics. The structure of depend on the details of the underlying fundamental theory. nodes and links of the network carries information about This way, the phenomenological model can bridge the gap area and volumes and thus the geometry that the network between theory and experiment. In the following, we will fundamentally describes. It can then be shown that it is aim to parameterize the consequences of space-time defects possible to act with a large number of local evolution moves so that contact can be made to the underlying theory if the on a state without nonlocal links and by this create a state relevant parameters can be extracted. The defects considered that contains macroscopically nonlocal links. This is possible here are quantum gravitational effects in the sense that they without changing the classical state that it approximates; thus are relics of the underlying discrete structure. the existence of the classical approximation alone cannot be In contrast to defects in condensed matter systems, used to rule out these nonlocal links. analogues of which have also been considered for General A nonlocal link is, intuitively, a link in the network that Relativity [6], space-time defects are not only localized in does not respect the emergent macroscopic locality. More space but also in time. They do not have world lines but strictly, it can be identified by the number of nodes on the are space-time events. Space-time defects can come in two shortest closed loop that it is part of. For the nonlocal link, different versions, local defects and nonlocal defects. The thesewillbealargenumberofnodes,whileallthelocallinks general case would be a hybrid of both, but treating these two have a small number of nodes (depending on the valence of types separately is helpful to develop the theory. We will first the network). See Figure 1 for illustration. discuss the nonlocal defects. In the example by Markopoulou and Smolin, the states with nonlocal links are still allowed solutions and thus valid 2.1. Nonlocal Space-Time Defects. One of the reasons we semiclassical space-times but do not respect the macroscopic expect that quantum gravity necessitates nonlocality is that locality. Because of simple combinatorics, one finds that there the notion of the Planck length as a minimal length implies are in fact many more nonlocal states than local states. This that it is meaningless to distinguish points below this means that the locality of the state that we live in today is not Advances in High Energy Physics 3

where the choice of sign determines whether the translation is time-like or space-like. The interpretation of Λ is roughly speakingthattheparticlewillbetranslatedadistanceofabout A 𝛼 in the rest frame in which its energy is about Λ. The translation that the particle experiences when it encounters the nonlocal defect is then given by a probability distribution 𝑃𝑁𝐿 (𝛼, Λ) over the endpoints. By construction, this is all entirely Lorentz-invariant. The density 𝛽 together 𝑃 (𝛼, Λ) Figure 1: Schematic picture for nonlocal links on a regular lattice. with the distribution 𝑁𝐿 determines the phenomenol- The lattice represents space-time. The link, which represents a defect ogy of the model. One can further simplify this situation in the regularity of the background, is long according to the distance by approximating the probability distribution by a Gaussian measure of the background. Note that all nodes remain 4-valent. If with mean values ⟨𝛼⟩, ⟨Λ⟩ and variances Δ𝛼,andΔΛ.This space-time is fundamentally discrete, similar defects should be all leaves one with 5 parameters. These can be further reduced around us. by assuming that there is only one new length scale 𝐿∼⟨𝛼⟩ and that the width of the distributions is comparable to the mean values ⟨𝛼⟩ ∼ Δ𝛼 and ⟨Λ⟩ ∼ ΔΛ. One is then left perfect; nonlocal links should be all around us. This situation with two parameters, one length scale and one mass scale that was aptly dubbed “disordered locality” in [10]. can be constrained by experiment quite simply. While this The argument by Markopoulou and Smolin is an explicit might not be the most general case, it allows one to get a first example that one can have in mind. But the expectation for impression on what amount of nonlocality is compatible with the existence of nonlocal defects is more general than that; it observation. arises because perfection requires additional explanations or A massless particle that encounters a nonlocal defect will selection criteria that we do not have. bedeviatedfromthelightconeandontheaveragetravel Now, such macroscopic nonlocality might strike one as either faster or slower than the normal speed of light (depend- something to be avoided, since we do not seem to experi- ing on the choice of sign in (2)). It is possible to restrict ence it, but this is a question of experimental constraints. translations to be time-like and velocities to be subluminal To really understand the implications of such nonlocality, to avoid causality problems because such a restriction does one first needs to develop a phenomenological model that not violate Lorentz-invariance. The repeated scattering on parameterizes the effects and then contrast them with data. nonlocal defects creates a small effective mass for the photon Such a model was developed in [4]. This model does not that is the mass that a particle of the photon’s energy would start with an underlying discrete structure but instead deals have on the average trajectory that contains nonlocal links. with the defects in the local structure as deviations from the This is reflected in the average speed of the particle which, in smooth background geometry of general relativity. the presence of nonlocal defects, can deviate from the speed The central assumption for the model [4]isthatLorentz- of light because the translations that the particle experiences invariance is preserved on the average, since violations when it encounters a nonlocal defects may be space-like or of Lorentz-invariance are strongly disfavored by the data. time-like rather than light-like. Lorentz-invariance, maybe not so surprisingly, proves to be It is important to note that the distribution of translation very restrictive on the type of nonlocality that is allowed. The vectors in this model is not an independent property of distribution of nonlocal defects in this model is assumed to be space-time but depends on the wave function of the incident given by the only presently known Lorentz-invariant discrete particle. This is in the simplest case its momentum vector, distribution on Minkowski space, which is defined by the in the general case the average momentum vector and Poisson-process described in [11, 12]. With this distribution, the spatial width of the wave function (at the moment it the probability of finding 𝑁 points in a space-time volume 𝑉 encounters the defect). It is the dependence on the incident is particle that allows one to construct a normalizable and Lorentz-invariant distribution. While the full Lorentz group (𝛽𝑉)𝑁 (−𝛽𝑉) 𝑃 (𝑉) = exp , (1) is noncompact, using measurable properties of the incident 𝑁 𝑁! particle as reference prevents the need to introduce a Lorentz- invariance violating cutoff while still maintaining observer 4 where 𝛽=𝐿is a constant space-time density and 𝐿 a independence. parameter of dimension length. The central feature that distinguishes this model from A particle that encounters a nonlocal defect will expe- other random walk-like models for propagation in a quantum rience a translation in space-time. The translation vector is space-time is that the probability of the particle being affected parameterized in a probability distribution which, besides by the quantum properties, here the space-time defects, 𝐿, introduces a parameter of dimension mass, Λ,anda depends on the (Lorentz-invariant) world volume that is parameter of dimension length, 𝛼. Λ and 𝛼 (both real- sweptoutbytheparticle’swavefunction.Thus,thelarger valued and positive) quantify the translation, 𝑦](𝛼, Λ),that the position uncertainty of the particle and the longer its the particle experiences at the nonlocal defect. Consider propagation time, the more likely the particle to be affected by the space-time defects. This generically means that particles ] ] 2 𝑝]𝑦 =𝛼Λ,] 𝑦 =±𝛼, (2) of long wavelength are better suited to find phenomenological 4 Advances in High Energy Physics consequences than highly energetic ones, in contrast to the 1 phenomenology that arises, for example, within deformations −5 or violations of Lorentz-invariance [2, 13]. 10 With the use of this model for nonlocal space-time 10−10 defects, constraints on the density of defects and the param- (m) eters of the model can then be derived from various observ- L 10−15 ables. For example, we have good evidence that protons of ultra- 10−20 high energy which give rise to cosmic ray showers originate 10−25 in active galactic nuclei. The protons have a finite mean- 10−40 10−20 1 1020 1040 free path when traveling through the cosmic microwave 1/Λ (m) background (CMB) because they can scatter on the CMB photons and produce pions. In the presence of nonlocal Figure 2: Summary of constraints on nonlocal space-time defects, defects, the protons’ mean-free path can increase because the from [4]. The red shaded region is excluded. The dashed (black) line particles effectively do not travel the full distance. If the mean- indicates the value of the cosmological constant. free path increases substantially, this would be in conflict with observation and can thus be used to derive bounds on the density of defects. Other constraints come from the blurring derivative 𝜕+𝑒𝐴→ 𝜕+𝑒𝐴+𝑔𝜕𝑃.Here,𝐴 is some gauge of interference rings in images of distant quasars and from field with coupling constant 𝑒, 𝑔 is a coupling constant for the the close monitoring of single photons in a cavity. For details space-time defects, and 𝑃 is essentially the Fourier transform and references, please refer to [4]. The constraints on nonlocal of the distribution of the momentum that is stochastically defects can be visually summarized in Figure 2.Roughly transmitted by the defect. This then allows one to calculate the speaking, it can be concluded that the density of nonlocal probability for different scattering processes in an 𝑆-matrix 4 defects has to be less than one in a space-time volume of fm . expansion as usual. Importantly, this particular coupling to In the cosmological context, a natural scale is given the defects has the effect that an on-shell particle that scatters by the length scale associated with the measured value of on a defect is necessarily off-shell after scattering. the cosmological constant, which is about 1/10 mm. The For a massless particle with energy 𝐸 in 1+1dimensions, constraints on 𝐿 are presently about 10 orders of magnitude the assumption that the momentum has a Gaussian distribu- below this interesting parameter range. One can expect, tion over the model parameters leads to a space-time defect however, that the existing data is actually sensitive to larger that also has a Gaussian distribution in light cone coordinates values of 𝐿.Itisjustthatthemodelinitspresentformcannot 2 2 + − + − 𝑖(⟨𝑘+⟩𝑥 +⟨𝑘−⟩𝑥 ) be used to reliably analyze much of the existing cosmological + − (𝑥 ) (𝑥 ) 𝑒 𝑃(𝑥 ,𝑥 )=exp ( + ) , (3) data because it does not take into account background (2𝜎+)2 (2𝜎−)2 2𝜋√𝜎+𝜎− curvature. It is clearly desirable to generalize the model to at least a Friedmann-Robertson-Walker background to be with widths able to analyze cosmological data for evidence of space-time defects. + √2𝐸 − √2 𝜎 = ,𝜎= , (4) Δ𝑀2 𝐸

2 2.2. Local Space-Time Defects. A model for local defects can where Δ𝑀 is the variance of the distribution of the parame- 2 be developed based on a similar approach as that for the ter 𝑀 . One sees that the typical space-time patch covered by nonlocal defects [5]. Much like the fact that the nonlocal the defect is defects cause a statistically distributed translation in position + − 2 −1 space, local defects cause a statistically distributed translation 𝜎 𝜎 =2(Δ𝑀) . (5) in momentum space. In other words, the local defect stochastically changes the momentum of the incident particle, The defect has a Lorentz-invariant volume independent of 𝐸, thereby violating momentum conservation. Since the defect though it will deform under boosts that red- or blue-shift 𝐸 as transfers a distribution of momenta, it consequently has a one sees from (4). Care must be taken in higher dimensions finite size. The finite size of the defect, together withthe to properly normalize the momentum distribution. As with preservation of locality, makes the local defects much easier to the nonlocal defects, the normalization can be achieved using treatandincorporatingthemintoaquantumfieldtheoretical thesamemethodthatiscommonlyusedintheevaluation framework is relatively straightforward. of scattering amplitudes, by taking into account that, in The distribution of local defects is again assumed to reality, we strictly speaking never deal with plane waves. The be given by the Poisson sprinkling (1). The momentum finite spatial extension of the incident particle’s wave function nonconservation is then parameterized in the length scale 𝐿 serves to regularize the distribution. of the distribution and a mass scale 𝑀.Thesecouldapriori Massiveparticlescanbetreatedsimilartothemassless be different from the parameters relevant for local defects. ones.Again,itisofcentralrelevancethatthemomentum The coupling of quantum fields to the local defects distribution of the defect is a function of the properties of is incorporated by adding a term to the gauge covariant the incident particle. For massive particles, the distribution Advances in High Energy Physics 5 canbeassignedmosteasilyintherestframeoftheparticle, 1 and, in that rest frame, it will have an especially simple form.Thus,whilethedistributionisnotLorentz-invariant −10 in the sense that its expression changes under arbitrary 10

Lorentz-transformations, observer independence is main- (m) tained because all observers can use the incident particle’s 1/M −20 momentum as a reference and obtain the same result. 10 With this setup, constraints can be derived from processes normally forbidden in the standard model, which are now 10−30 allowed. The most important bounds come from long-lived 10−30 10−20 10−10 1 1010 particles that travel long distances and are the following. L (m)

(1) Photon decay: after scattering on a defect, a photon Figure 3: Summary of constraints on local space-time defects, from is off-shell and subsequently decays into a fermion [5]. The red (dark) shaded region is excluded. The peachy (light) pair. This effect is similar to pair production in the shaded region indicates a stronger constraint from photon decay presence of an atomic nucleus in standard quantum with the ad hoc assumption that the typical distance between defects increases with the cosmological scale factor. electrodynamics (QED). This process results in a finite photon lifetime and leads to excess electron- positron pairs.

(2)Photonmass:thepresenceofspace-timedefects 3. Discussion makes a contribution to the photon propagator and creates a small photon mass. (Gauge invariance is The preliminary work [4, 5] tested the potential of detecting violated.) space-time discreteness by the occurrence of defects in the background’sregularity.Themodelsusedinthispreliminary (3) Vacuum Cherenkov radiation: an electron can emit work can deliver only rough estimates. They are suitable for a (real) photon after scattering on a defect. This is these estimates but are theoretically unsatisfactory. Since the similar to QED Bremsstrahlung. estimates show that it seems possible to reach the interesting parameter ranges experimentally, a further investigation and Theconstraintsfromtheseeffectscanbesummarizedin improvement of the theory and phenomenology of space- Figure 3. Again, note that the existing bounds on 𝐿 are several time defects are desirable. In the following, we will propose orders of magnitude, but not too far, below the interesting some steps into this direction. parameter range. (Making 1/𝑀 smaller than shown in the First, the models proposed in [4, 5]areforflat3+1 plotmeansthatitbecomescomparabletothePlancklength dimensional Minkowski space. Since the best constraints on −35 ∼10 m and then it does not make sense any more to speak thepresenceoflocalandnonlocaldefectscomefromparticles of defects.) It would thus be highly desirable to improve the that have traveled long times and distances, one could derive model to tighten the bounds. better constraints when background curvature in general, and Not much work has been done on local space-time defects an expanding Friedmann-Robertson-Walker (FRW) metric prior to [5], except for the model proposed in [14]. The model in particular, can be taken into account. This would then in [14] differs from the one discussed here in three important allow one to use cosmological data, for example, from the ways. cosmic microwave background, to constrain the density of First, in [14], the interaction with the defect is mediated defects. by scalar field. Second, the treatment in [14]necessitatesthe First, steps towards a cosmology with nonlocal defects introduction of a cutoff in the momentum-space integration have been taken in [15, 16]basedonthequantumgraphity which breaks Lorentz-invariance and defeats the point of model developed by Konopka et al. [17]. It was assumed in using a Lorentz-invariant distribution of defects to begin [15] that the nonlocal connections lie within a time-like slice with. Such a cutoff is unnecessary in the model discussed thatisassumedtobeidenticaltothecosmologicaltime. here, where the regulator is essentially the spatial width of This of course violates Lorentz-invariance, but, in a time- the wave packet. Third, and most important, the coupling dependent background, this can be expected. However, this to the defect in [14] is different. The approach in[5]started specific violation of Lorentz-invariance is very strong and from the assumption that the defects originate in a distortion artificial: there is really no reason why the time evolution of space-time regularity and make themselves noticeable as of the network should be identical to the cosmological time, a modification in the covariant derivative. This leads toa which is only an approximation based on the assumption of a specific structure of the coupling terms, which was not used homogeneous matter distribution anyway. More realistically, in [14]. one would expect both types of slicing to differ, so that the In summary, it can be said that space-time defects are links would still have a spread in the time-like coordinate. pretty much unexplored territory, where not much previous This would alter phenomenological consequences. The model work has been done. That makes the topic very exciting as it for defects discussed here provides a good basis to study this harbors a potential for breakthrough. phenomenology. 6 Advances in High Energy Physics

Second, the model with nonlocal defects also so far Quantum Gravity,vol.28,no.2,ArticleID025002,11pages, only operates on a kinematical level and a full dynamical 2011. description in terms of a quantum field theoretical treatment [8] G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman, and L. is missing. It would be desirable to develop a quantum field Smolin, “Principle of relative locality,” Physical Review D,vol. theoretical model for this case and thus also be able to 84, no. 8, Article ID 084010, 2011. combine both local and nonlocal defects. One way to address [9] S. Hossenfelder, “Minimal length scale scenarios for quantum this point would be to use the dual nature of the local and gravity,” Living Reviews in Relativity,vol.16,p.2,2013. nonlocal defects and to make the idea mathematically precise [10] F. Markopoulou and L. Smolin, “Disordered locality in loop that nonlocal defects act like local defects, just in position quantum gravity states,” Classical and Quantum Gravity,vol.24, spaceratherthaninmomentumspace.Thiswouldallowone no.15,pp.3813–3823,2007. to express them as operators on the particles’ Hilbert space [11] F. Dowker, J. Henson, and R. D. Sorkin, “Quantum gravity and facilitate their incorporation into quantum field theory. phenomenology, Lorentz invariance and discreteness,” Modern Third, while the search for defects as a model-independ- Physics Letters A,vol.19,no.24,pp.1829–1840,2004. ent expectation for space-time discreteness is interesting in [12] L. Bombelli, J. Henson, and R. D. Sorkin, “Discreteness without its own right, contact to theoretical approaches to quantum symmetry breaking: a theorem,” Modern Physics Letters A,vol. gravity would serve as a better identification of the parame- 24,no.32,pp.2579–2587,2009. ters. The density of defects might, for example, be possible to [13] G. Amelino-Camelia, “Doubly-special relativity: facts, myths extract in approaches that display a phase transition from a and some key open issues,” Symmetry,vol.2,no.1,pp.230–271, pregeometrical phase to an approximately smooth geometry. 2010. In this case, the density of remaining defects should depend [14] M. Schreck, F. Sorba, and S. Thambyahpillai, “Simple model on the properties of the phase transition. of pointlike spacetime defects and implications for photon propagation,” Physical Review D,vol.88,no.12,ArticleID 125011, 28 pages, 2013. 4. Conclusion [15] C. Prescod-Weinstein and L. Smolin, “Disordered locality as an explanation for the dark energy,” Physical Review D,vol.80,no. If space-time is fundamentally of nongeometric origin, then 6, Article ID 063505, 2009. the smooth background geometry of general relativity should [16] F. Caravelli and F. Markopoulou, “Disordered locality and have defects. The consequences of these defects can be Lorentz dispersion relations: an explicit model of quantum parameterized and described in phenomenological models, foam,” Physical Review D,vol.86,no.2,ArticleID024019,11 whichallowonetoputconstraintsonthedensityofthe pages, 2012. defects and the strength of their effects. These models are in [17] T. Konopka, F. Markopoulou, and S. Severini, “Quantum their infancy and much remains to be done, but they harbor graphity: a model of emergent locality,” Physical Review D,vol. the possibility that a targeted search for space-time defects 77, no. 10, Article ID 104029, 15 pages, 2008. will allow us to find evidence for quantum gravity.

Conflict of Interests The author declares that there is no conflict of interests regarding the publication of this paper.

References

[1] S. Hossenfelder, “Experimental search for quantum gravity,” in Classical and Quantum Gravity: Theory, Analysis and Applica- tions,V.R.Frignanni,Ed.,chapter5,NovaPublishers,2011. [2] G. Amelino-Camelia, “Quantum-spacetime phenomenology,” Living Reviews in Relativity,vol.16,p.5,2013. [3] R. Loll, “Discrete approaches to quantum gravity in four dimensions,” Living Reviews in Relativity,vol.1,p.13,1998. [4] S. Hossenfelder, “Phenomenology of space-time imperfection. I: nonlocal defects,” Physical Review D, vol. 88, no. 12, Article ID 124030,13pages,2013. [5] S. Hossenfelder, “Phenomenology of space-time imperfection. II: local defects,” Physical Review D,vol.88,no.12,ArticleID 124031, 9 pages, 2013. [6] R. A. Puntigam and H. H. Soleng, “Volterra distortions, spinning strings, and cosmic defects,” Classical and Quantum Gravity,vol.14,no.5,pp.1129–1149,1997. [7] S. B. Giddings, “Nonlocality versus complementarity: a conservative approach to the information problem,” Classical and Hindawi Publishing Corporation Advances in High Energy Physics Volume 2014, Article ID 867460, 14 pages http://dx.doi.org/10.1155/2014/867460

Research Article Inflation and Topological Phase Transition Driven by Exotic Smoothness

Torsten Asselmeyer-Maluga1 and Jerzy Król2

1 German Aerospace Center (DLR), 10178 Berlin, Germany 2 Institute of Physics, University of Silesia, 40-007 Katowice, Poland

Correspondence should be addressed to Torsten Asselmeyer-Maluga; [email protected]

Received 5 December 2013; Accepted 17 January 2014; Published 19 March 2014

Academic Editor: Douglas Singleton

Copyright © 2014 T. Asselmeyer-Maluga and J. Krol.´ This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

We will discuss a model which describes the cause of inflation by a topological transition. The guiding principle is the choice of an 3 3 exotic smoothness structure for the space-time. Here we consider a space-time with topology 𝑆 × R.Incaseofanexotic𝑆 × R, there is a change in the spatial topology from a 3-sphere to a homology 3-sphere which can carry a hyperbolic structure. From the physical point of view, we will discuss the path integral for the Einstein-Hilbert action with respect to a decomposition of the space-time. The inclusion of the boundary terms produces fermionic contributions to the partition function. The expectation value of an area (with respect to some surface) shows an exponential increase; that is, we obtain inflationary behavior. We will calculate the amount of this increase to be a topological invariant. Then we will describe this transition by an effective model, the Starobinski 2 or 𝑅 model which is consistent with the current measurement of the Planck satellite. The spectral index and other observables are also calculated.

1. Introduction are going in this direction we will motivate the usage of the smooth manifold as our basic concept. General relativity (GR) has changed our understanding of When Einstein developed GR, his opinion about the space-time. In parallel, the appearance of quantum field importance of general covariance changed over the years. theory (QFT) has modified our view of particles, fields, In 1914, he wrote a joint paper with Grossmann. There, he and the measurement process. The usual approach for the rejected general covariance by the now famous hole argu- unification of QFT and GR, to a quantum gravity, starts ment. But after a painful year, he again considered general with a proposal to quantize GR and its underlying structure, covariance now with the insight that there is no meaning in space-time. There is a unique opinion in the community referring to the space-time point A or the event A,without about the relation between geometry and quantum theory: further specifications. Therefore the measurement of a point thegeometryasusedinGRisclassicalandshouldemerge withoutadetailedspecificationofthewholemeasurement from a quantum gravity in the limit (Planck’s constant tends process is meaningless in GR. The reason is simply the to zero). Most theories went a step further and try to get diffeomorphism invariance of GR which has tremendous a space-time from quantum theory. Then, the model of a consequences. Furthermore, GR do not depend on the smooth manifold is not suitable to describe quantum gravity. topology of space-time. All restrictions on the topology of But, there is no sign for a discrete space-time structure or the space-time were formulated using additional physical higher dimensions in current experiments. Hence, quantum conditions like causality (see [1]). This ambiguity increases gravity based on the concept of a smooth manifold should in the 80’s when the first examples of exotic smoothness also able to explain the current problems in the standard structures in dimension 4 were found. The (smooth) atlas of cosmological model (ΛCDM) like the appearance of dark a smooth 4-manifold 𝑀 is called the smoothness structure energy/matter or the correct form of inflation. But before we (unique up to diffeomorphisms). One would expect that there 2 Advances in High Energy Physics is only one smooth atlas for 𝑀; all other possibilities can All these restrictions on the representation of space-time be transformed into each other by a diffeomorphism. But in by the manifold concept are clearly motivated by physical contrast, the deep results of Freedman [2]onthetopology questions. Among these properties there is one distinguished of 4-manifolds combined with Donaldson’s work [3]gavethe element: the smoothness. Usually one starts with a topolog- first examples of nondiffeomorphic smoothness structures ical 4-manifold 𝑀 and introduces structures on them. Then 4 on 4-manifolds including the well-known R .Muchofthe one has the following ladder of possible structures: motivationcanbefoundintheFQXIessay[4, 5]. Here we will 󳨀→ 󳨀→ discuss another property of the exotic smoothness structure: Topology piecewise-linear (PL) Smoothness its quantum geometry in the path integral. 󳨀→ bundles, Lorentz, Spin, etc. (1) Diffeomorphism invariance is the most important property of the Einstein-Hilbert action with far reaching conse- 󳨀→ metric, geometry,.... quences [6]. One of our results is a close relation between geometry and foliation to exotic smoothness [7, 8]. In the We do not want to discuss the first transition, that is, the 4 particular example of the exotic R ,wediscussedtheexotic existence of a triangulation on a topological manifold. But we remark that the existence of a PL structure implies uniquely smoothness structures as a manifestation of quantum gravity R4 a smoothness structure in all dimensions smaller than 7 [19]. (by using string theory [9, 10]). This exotic has some Here we have to consider the following steps to define a space- interesting properties as first noted by Brans11 [ , 12]. More 4 time. importantly as shown by Sładkowski [13], the exotic R has 4 𝑀 a nontrivial curvature in contrast to the flat standard R . (i) Fix a topology for the space-time . 4 It was the first result that an exotic R canbeseenasa (ii) Fix a smoothness structure, that is, a maximal differ- source of gravity (or it must contain sources of gravity). entiable atlas A. Sładkowski [14–16] went further and showed a relation to (iii) Fix a smooth metric or get one by solving the Einstein particle physics also related to quantum gravity. But why equation. there is a relation to quantum gravity? In [17]wepresented the first idea to understand this relation which was further The choice of a topology never fixes the space-time uniquely; 3 extended in [18]. An exotic 4-manifold like 𝑆 × R is also thatis,therearetwospace-timeswiththesametopology characterized by the property that there is no smoothly which are not diffeomorphic. The main idea of the paper isthe embedded 3-sphere but a topological embedded one. This introduction of exotic smoothness structures into space-time. 3 topological 𝑆 is wildly embedded; that is, the image of the If two manifolds are homeomorphic but nondiffeomorphic, embedding must be triangulated by an infinite polyhedron. they are exotic to each other. The smoothness structure is In [18], we proved that the (deformation) quantization of a called an exotic smoothness structure. usual (or tame) embedding is a wild embedding which can In dimension four there are many examples of com- be seen as a quantum state. But then any exotic 4-manifold pact 4-manifolds with countable infinite nondiffeomorphic can be interpreted as a quantum state of the 4-manifold smoothness structures and many examples of noncompact 4- with standard smoothness structure. From this point of manifolds with uncountable infinite many nondiffeomorphic view, the calculation of the path integral in quantum gravity smoothness structures. But in contrast, the number of non- hastoincludetheexoticsmoothnessstructures.Usuallyit diffeomorphic smoothness structures is finite for any other dimension [19]. As an example, we will consider the space- is hopeless to make these calculations. But by using the 3 time 𝑆 × R having uncountable many nondiffeomorphic close relation of exotic smoothness to hyperbolic geometry, smoothness structures in the following. one has a chance to calculate geometric expressions like the expectation value of the surface area. In this paper we 3 will show that this expectation value has an inflationary 3. The Path Integral in Exotic 𝑆 × R behavior; that is, the area grows exponentially (along the For simplicity, we consider general relativity without matter time axis). Therefore quantum gravity (in the sense of exotic (using the notation of topological QFT). Space-time is a smoothness) can be the root of inflation. smooth oriented 4-manifold 𝑀 which is noncompact and without boundary. From the formal point of view (no diver- 2. Space-Time and Smoothness gences of the metric) one is able to define a boundary 𝜕𝑀 at infinity. The classical theory is the study of the existence and From the mathematical point of view, the space-time is uniqueness of (smooth) metric tensors 𝑔 on 𝑀 that satisfy the a smooth 4-manifold endowed with a (smooth) metric as Einstein equations subject to suitable boundary conditions. basic variable for general relativity. The existence question In the first order Hilbert-Palatini formulation, one specifies for Lorentz structure and causality problems (see Hawking an SO(1, 3)-connection 𝐴 together with a cotetrad field 𝑒 and Ellis [1]) give further restrictions on the 4-manifold: rather than a metric tensor. Fixing 𝐴|𝜕𝑀 at the boundary, causality implies noncompactness; Lorentz structure needs one can derive first-order field equations in the interior (now a nonvanishing normal vector field. Both concepts can be called bulk)whichareequivalenttotheEinsteinequations combined in the concept of a global hyperbolic 4-manifold provided that the cotetrad is nondegenerate. The theory is 𝑀 having a Cauchy surface S so that 𝑀=S × R. invariant under space-time diffeomorphisms 𝑀→𝑀. Advances in High Energy Physics 3

3 In the particular case of the space-time 𝑀=𝑆× R topology of the boundary as well on the diffeomorphism class (topologically), we have to consider a smooth 4-manifolds of the interior of 𝑀𝑖,𝑓. Therefore we will shortly write 𝑀𝑖,𝑓 as parts of 𝑀 whose boundary 𝜕𝑀𝑖,𝑓 =Σ𝑖 ⊔Σ𝑓 is the 󵄨 󵄨 󵄨 󵄨 disjoint union of two smooth 3-manifolds Σ𝑖 and Σ𝑓 to which ⟨Σ 󵄨𝑇 󵄨 Σ ⟩=⟨𝐴| 󵄨𝑇 󵄨 𝐴| ⟩ 𝑓 󵄨 𝑀󵄨 𝑖 Σ𝑓 󵄨 𝑀󵄨 Σ𝑖 (5) we associate Hilbert spaces H𝑗 of 3-geometries, 𝑗=𝑖,𝑓. 𝐴| These contain suitable wave functionals of connections Σ𝑗 . andconsiderthesumofmanifoldslike𝑀𝑖,ℎ =𝑀𝑖,𝑓 ∪Σ 𝑀𝑓,ℎ |𝐴| ⟩ 𝑓 We denote the connection eigenstates by Σ𝑗 .Thepath with the amplitudes integral, 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 ⟨Σℎ 󵄨𝑇𝑀󵄨 Σ𝑖⟩= ∑ ⟨Σℎ 󵄨𝑇𝑀󵄨 Σ𝑓⟩⟨Σ𝑓 󵄨𝑇𝑀󵄨 Σ𝑖⟩, 󵄨 󵄨 𝐴| (6) ⟨𝐴| 󵄨𝑇 󵄨 𝐴| ⟩ Σ𝑓 Σ𝑓 󵄨 𝑀󵄨 Σ𝑖 𝑖 (2) where we sum (or integrate) over the connections and frames = ∫ 𝐷𝐴 𝐷𝑒 ( 𝑆 [𝑒, 𝐴, 𝑀 ]) , Σ exp ℎ 𝐸𝐻 𝑖,𝑓 on ℎ (see [22]). Then the boundary term 𝐴|𝜕𝑀𝑖,𝑓 𝑆 [Σ ] = ∫ 𝜖 (𝑒𝐼 ∧𝑒𝐽 ∧𝐴KL) = ∫ 𝐻√ℎ𝑑3𝑥 𝜕 𝑓 𝐼𝐽KL (7) 𝐴 𝐴| Σ Σ is the sum over all connections matching 𝜕𝑀𝑖,𝑓 and 𝑓 𝑓 over all 𝑒. It yields the matrix elements of a linear map 𝐻 Σ𝑓 𝑇𝑀 : H𝑖 → H𝑓 between states of 3-geometry. Our basic is needed where is the mean curvature of corresponding 𝐼 𝐼𝐽 to the metric ℎ at Σ𝑓 (as restriction of the 4-metric). Therefore gravitational variables will be cotetrad 𝑒 and connection 𝐴 𝑎 𝑎 we have to divide the path integration into two parts: the on space-time 𝑀 with the index 𝑎 to present it as 1-forms contribution by the boundary (boundary integration) and the and the indices 𝐼, 𝐽 foraninternalvectorspace𝑉 (used for contribution by the interior (bulk integration). the representation of the symmetry group). Cotetrads 𝑒 are “square-roots” of metrics and the transition from metrics to Σ tetrads is motivated by the fact that tetrads are essential if 3.1. Boundary Integration. The boundary of a 4-manifold 𝐼 𝑀 one is to introduce spinorial matter. 𝑒 is an isomorphism can be understood as embedding (or at least as immer- 𝑎 sion). Let 𝜄:Σ󳨅→𝑀be an immersion of the 3-manifold between the tangent space 𝑇𝑝(𝑀) at any point 𝑝 and a fixed → Σ 𝑀 internal vector space 𝑉 equipped with a metric 𝜂𝐼𝐽 so that into the 4-manifold with the normal vector 𝑁.The 𝐽 𝑔 =𝑒𝐼 𝑒 𝜂 spin bundle 𝑆𝑀 of the 4-manifold splits into two subbundles 𝑎𝑏 𝑎 𝑏 𝐼𝐽.Hereweusedtheaction ± + 𝑆𝑀,whereonesubbundle,say𝑆𝑀,canberelatedtothespin bundle 𝑆Σ of the 3-manifold. Then the spin bundles are related 𝑆𝐸𝐻 [𝑒, 𝐴,𝑖,𝑓 𝑀 ,𝜕𝑀𝑖,𝑓] ∗ + by 𝑆Σ =𝜄𝑆𝑀 with the same relation 𝜙=𝜄∗Φ for the + 𝑀 Σ spinors (𝜙 ∈ Γ(𝑆Σ) and Φ∈Γ(𝑆𝑀)).Let∇𝑋 ,∇𝑋 be the = ∫ 𝜖 (𝑒𝐼 ∧𝑒𝐽 ∧ (𝑑𝐴 + 𝐴 ∧𝐴)KL) 𝐼𝐽KL covariant derivatives in the spin bundles along a vector field 𝑀 (3) 𝑖,𝑓 𝑋 as section of the bundle 𝑇Σ.Thenwehavetheformula

𝐼 𝐽 KL 𝑀 Σ 1 → → + ∫ 𝜖𝐼𝐽 (𝑒 ∧𝑒 ∧𝐴 ), KL ∇𝑋 (Φ) =∇𝑋𝜙− (∇𝑋 𝑁) ⋅ 𝑁 ⋅𝜙, (8) 𝜕𝑀𝑖,𝑓 2

𝜙 𝐼 with the obvious embedding 𝜙 󳨃→( )=Φof the spinor 𝜖𝐼𝐽 (𝑒 ∧ 0 in the notation of [20, 21]. Here the boundary term KL → 𝐽 KL 𝑒 ∧𝐴 ) is equal to twice the trace over the extrinsic curvature spaces. The expression ∇𝑋 𝑁 isthesecondfundamentalform → (or the mean curvature). For fixed boundary data,2 ( )is (∇ 𝑁)=2𝐻 Σ =Σ of the immersion, where the trace tr 𝑋 is related a diffeomorphism invariant in the bulk. If 𝑖 𝑓 are 𝐻 Σ=Σ =Σ H = H = to the mean curvature .Thenfrom(8)oneobtainsasimilar diffeomorphic, we can identify 𝑖 𝑓 and 𝑖 relation between the corresponding Dirac operators H𝑓;thatis,weclosethemanifold𝑀𝑖,𝑓 by identifying the two 󸀠 𝑀 3𝐷 boundaries to get the closed 4-manifold 𝑀 .Providedthatthe 𝐷 Φ=𝐷 𝜙−𝐻𝜙, (9) trace over H can be defined, the partition function, 3𝐷 with the Dirac operator 𝐷 of the 3-manifold Σ.Nearthe 󸀠 𝑖 boundary Σ, the 4-manifolds looks like Σ×[0,1]and a spinor 𝑍 (𝑀 ) = trH𝑇𝑀 = ∫ 𝐷𝐴 𝐷𝑒 exp ( 𝑆𝐸𝐻 [𝑒, 𝐴, 𝑀,] 𝜕𝑀 ) , ℎ Φ on this 4-manifold is a parallel spinor and has to fulfill the (4) following equation: 𝐷𝑀Φ=0; where the integral is now unrestricted, is a dimensionless (10) number which depends only on the diffeomorphism class of 󸀠 that is, 𝜙 yields the eigenvalue equation the smooth manifold 𝑀 .Incaseofthemanifold𝑀𝑖,𝑓,the path integral (as transition amplitude) ⟨𝐴|Σ |𝑇𝑀|𝐴|Σ ⟩ is the 3𝐷 𝑓 𝑖 𝐷 𝜙=𝐻𝜙, (11) diffeomorphism class of the smooth manifold relative to the boundary. But the diffeomorphism class of the boundary is with the mean curvature 𝐻 of the embedding 𝜄 as eigenvalue. unique and the value of the path integral depends on the See our previous work [23] for more details. 4 Advances in High Energy Physics

Now we will use this theory to get rid of the boundary diffeomorphisms which is in our case detected by the Eta integration. At first we will discuss the deformation of an invariant. For 3-manifolds there is a deep relation to the immersion using a diffeomorphism. Let 𝐼:Σ󳨅→𝑀 Chern-Simons invariant [27]whichwillbefurtherstudiedat be an immersion of Σ (3-manifold) into 𝑀 (4-manifold). our forthcoming work. 󸀠 󸀠 󸀠 A deformation of an immersion 𝐼 :Σ 󳨅→ 𝑀 is 𝑓:𝑀󸀠 →𝑀 𝑔:Σ →󸀠 Σ 𝑀 Σ diffeomorphisms and of and , 3.2. Bulk Integration. Nowwewilldiscussthepathintegralof respectively, so that the action 󸀠 𝑓∘𝐼=𝐼 ∘𝑔. (12) 𝑆 [𝑒, 𝐴, 𝑀] = ∫ 𝜖 (𝑒𝐼 ∧𝑒𝐽 ∧ (𝑑𝐴 + 𝐴 ∧𝐴)KL) 𝐸𝐻 𝐼𝐽KL (15) 𝑀 One of the diffeomorphisms (say 𝑓)canbeabsorbed 𝑀 into the definition of the immersion and we are left with in the interior of the 4-manifold .Thecontributionofthe one diffeomorphism 𝑔∈Diff(Σ) to define the deformation boundary was calculated in the previous subsection. In the of the immersion 𝐼.Butasstatedabove,theimmersionis (formal) path integral (2) we will ignore all problems (ill- directlygivenbyanintegraloverthespinor𝜙 on Σ fulfilling definiteness, singularities, etc.) of the path integral approach. 𝐷𝑒 the Dirac equation (11). Therefore we have to discuss the Next we have to discuss the measure of the path integral. action of the diffeomorphism group Diff(Σ) on the Hilbert Currently there is no rigorous definition of this measure and 2 space of 𝐿 -spinors fulfilling the Dirac equation. This case as usual we assume a product measure. Thenwehavetwopossiblepartswhicharemoreorless was considered in the literature [24]. The spinor space 𝑆𝑔,𝜎(Σ) on Σ depends on two ingredients: a (Riemannian) metric 𝑔 independent from each other: and a spin structure 𝜎 (labeled by the number of elements 𝐷𝑒 1 (i) integration 𝐺 over geometries; in 𝐻 (Σ, Z2)). Let us consider the group of orientation + (ii) integration 𝐷𝑒𝐷𝑆 over different differential structures preserving diffeomorphism Diff (Σ) acting on 𝑔 (by pullback ∗ ∗ parametrized by some structure (see below). 𝑓 𝑔)andon𝜎 (by a suitable defined pullback 𝑓 𝜎). The 2 Hilbert space of 𝐿 -spinors of 𝑆𝑔,𝜎(Σ) is denoted by 𝐻𝑔,𝜎. Nowwehavetoconsiderthefollowingpathintegral: 𝑓∈ +(Σ) Thenaccordingto[24], any Diff leads in exactly 𝑍 (𝑀) twowaystoaunitaryoperator𝑈 from 𝐻𝑔,𝜎 to 𝐻𝑓∗𝑔,𝑓∗𝜎. The (canonically) defined Dirac operator is equivariant with 𝑖 = ∫ 𝐷𝑒 (∫ 𝐷𝑒 ( 𝑆 𝑒, 𝑀 )) , respect to the action of 𝑈 and the spectrum is invariant under 𝐷𝑆 𝐺 exp 𝐸𝐻 [ ] Diff structures Geometries ℎ (orientation preserving) diffeomorphisms. In particular we (16) obtain for the boundary term andwehavetocalculatetheinfluenceofthedifferential √ 3 3𝐷 3 𝑆𝜕 [Σ𝑓,ℎ] = ∫ 𝐻 ℎ𝑑 𝑥=∫ 𝜙 𝐷 𝜙𝑑 𝑥, (13) structures first. At this level we need an example, an exotic Σ Σ 3 𝑓 𝑓 𝑆 × R. 2 |𝜙| = 3 with const. (see [25]). But then we can change the 3.3. Constructing Exotic 𝑆 × R. In [28], Freedman con- integration process from the integration over the metric class 3 structed the first example of an exotic 𝑆 × R of special type. ℎ on the 3-manifold Σ𝑓 with mean curvature to an integration 4 There are also uncountable many different exotic R having over the spinor 𝜙 on Σ𝑓.Thenweobtain 3 an end homeomorphic to 𝑆 × R but not diffeomorphic to 𝑖 it. But Freedman’s first example is not of this type (as an 𝑍(Σ )=∫ 𝐷ℎ ( 𝑆[Σ ,ℎ]) 4 𝑓 exp ℎ 𝑓 end of an exotic R ). Therefore to get an infinite number of 3 3 different exotic 𝑆 × R one has to see 𝑆 × R as an end of 4 4 4 𝑖 3𝐷 3 R R \𝐷 = ∫ 𝐷𝜙𝐷𝜙 ( ∫ 𝜙𝐷 𝜙𝑑 𝑥) (14) also expressible as complement of the 4-disk. A exp ℎ Σ𝑓 second possibility is the usage of the end-sum technique of 3 Gompf, so that the standard 𝑆 × R can be transformed into 𝑖𝜋𝜂(Σ )/2 3 4 = √det (𝐷3𝐷𝐷∗3𝐷)𝑒 𝑓 , an exotic 𝑆 × R by end-sum with an exotic R .Herewewill 3 concentrate on the first construction; that is, the exotic 𝑆 ×R 4 where 𝜂(Σ𝑓) is the Eta invariant of the Dirac operator at the is an end of an exotic R . 4 3-manifold Σ𝑓 (here we use a result of Witten; see [26]). Furthermore we will restrict on a subclass of exotic R 4 4 From the physical point of view, we obtain fermions at the called small exotic R (exotic R which can be embed- 4 boundary. The additional term with the Eta invariant reflects ded in a 4-sphere 𝑆 ). For this class there is an explicit 4 also an important fact. The state space of general relativity is handle decomposition. Small exotic R ’s are the result of the space of the (Lorentzian) metric tensor up to the group an anomalous behavior in 4-dimensional topology. In 4- of coordinate transformations. This group of coordinate manifold topology [2], a homotopy equivalence between two transformations is not the full diffeomorphism group; it is compact, closed, simply connected 4-manifolds implies a only one connected component of the diffeomorphism. That homeomorphism between them (the so-called h cobordism). is the group of diffeomorphisms connected to the identity. But Donaldson [29] provided the first smooth counterex- In addition, there is also the (discrete) group of global ample that this homeomorphism is not a diffeomorphism; Advances in High Energy Physics 5

− − − that is, both manifolds are generally not diffeomorphic to towers are attached along 𝜕 𝑇ℓ.Let𝑇𝑛 be (interior 𝑇𝑛)∪𝜕 𝑇𝑛 each other. The failure can be localized at some contractible and the Casson handle, submanifold (Akbulut cork) so that an open neighborhood of − 4 = ⋃𝑇 , this submanifold is a small exotic R .Thewholeprocedure CH ℓ (18) 4 ℓ=0 implies that this exotic R can be embedded in the 4-sphere 4 𝑆 . The idea of the construction is simply given by the fact is the union of towers (with direct limit topology induced that every smooth h-cobordism between nondiffeomorphic from the inclusions 𝑇𝑛 󳨅→ 𝑇𝑛+1). 4-manifolds can be written as a product cobordism except The main idea of the construction above is very simple: an for a compact contractible sub-h-cobordism 𝑉,theAkbulut 4 immersed disk (disk with self-intersections) can be deformed cork. An open subset 𝑈⊂𝑉homeomorphic to [0, 1] × R into an embedded disk (disk without self-intersections) by is the corresponding sub-h-cobordism between two exotic 4 4 4 sliding one part of the disk along another (embedded) disk R ’s. These exotic R ’s are called ribbon R ’s. They have to kill the self-intersections. Unfortunately the other disk can the important property of being diffeomorphic to open be immersed only. But the immersion can be deformed to 4 subsets of the standard R .In[30] Freedman and DeMichelis an embedding by a disk again and so forth. In the limit of 4 constructed also a continuous family of small exotic R .Now this process one “shifts the self-intersections into infinity” and 2 2 2 2 we are ready to discuss the decomposition of a small exotic obtains the standard open 2-handle (𝐷 × R ,𝜕𝐷 × R ).In 4 R by Bizacǎ and Gompf [31] using special pieces, the handles the proof of Freedman [2], the main complications come from forming a handle body. Every 4-manifold can be decomposed thelackofcontrolaboutthisprocess. 𝑘 4−𝑘 (seen as handle body) using standard pieces such as 𝐷 ×𝐷 , ACassonhandleisspecifiedupto(orientationpreserv- 𝑘 4−𝑘 the so-called 𝑘-handleattachedalong𝜕𝐷 ×𝐷 to the ing) diffeomorphism (of pairs) by a labeled finitely branching 3 4 0 4 4 ∗ boundary 𝑆 =𝜕𝐷of a 0-handle 𝐷 ×𝐷 =𝐷.The tree with base-point , having all edge paths infinitely 4 extendable away from ∗. Each edge should be given a label construction of the handle body for the small exotic R ,called 4 +or−. Here is the construction: tree → CH. Each vertex 𝑅 in the following, can be divided into two parts: corresponds to a kinky handle; the self-plumbing number of that kinky handle equals the number of branches leaving the 𝑅4 =𝐴 ⋃ R4. vertex. The sign on each branch corresponds to the sign of cork CH decomposition of small exotic 𝐷2 the associated self-plumbing. The whole process generates a (17) tree with infinitely many levels. In principle, every tree with a finite number of branches per level realizes a corresponding Casson handle. Each building block of a Casson handle, the The first part is known as the Akbulut cork, a contractible 4- “kinky” handle with 𝑛 kinks, is diffeomorphic to the 𝑛-times 1 3 manifold with boundary a homology 3-sphere (a 3-manifold boundary connected sum ♮𝑛(𝑆 ×𝐷 )(see Appendix A) with with the same homology as the 3-sphere). The Akbulut cork two attaching regions. The number of end connected sums is 𝐴 cork is given by a linking between a 1-handle and a 2-handle exactly the number of self-intersections of the immersed two 0 of framing .ThesecondpartistheCassonhandleCHwhich handle. One region is a tubular neighborhood of band sums will be considered now. ofWhiteheadlinksconnectedwiththepreviousblock.The Let us start with the basic construction of the Casson other region is a disjoint union of the standard open subsets handle CH. Let 𝑀 be a smooth, compact, simple-connected 𝑆1 ×𝐷2 𝑆1 ×𝑆2 =𝜕(♮𝑆1 ×𝐷3) 𝑓:𝐷2 →𝑀 in #𝑛 𝑛 (thisisconnectedwith 4-manifold and a (codimension-2) mapping. the next block). 𝐷2 𝑀 3 By using diffeomorphisms of and , one can deform the For the construction of an exotic 𝑆 × R, denoted mapping 𝑓 to get an immersion (i.e., injective differential) 𝑆3× R 𝑅4 \𝐷4 −1 by 𝜃 , we consider the complement or the genericallywithonlydoublepoints(i.e.,#|𝑓 (𝑓(𝑥))|) =2 decomposition: as singularities [32]. But to incorporate the generic location 3 4 4 4 of the disk, one is rather interesting in the mapping of a 2- 𝑆 ×𝜃 R =𝑅 \𝐷 =(𝐴 \𝐷 ) ⋃ CH. 2 2 2 2 cork (19) handle 𝐷 ×𝐷 induced by 𝑓×𝑖𝑑:𝐷 ×𝐷 →𝑀from 𝑓. 𝐷2 𝑓(𝐷2) Then every double point (or self-intersection) of leads 𝐴 \𝐷4 𝐷2 ×𝐷2 The first part cork contains a cobordism between the to self-plumbings of the 2-handle . A self-plumbing 𝑆3 𝜕𝐴 𝐷2 ×𝐷2 𝐷2 ×𝐷2 𝐷2,𝐷2 ⊂ 3-sphere andtheboundaryoftheAkbulutcork cork (a is an identification of 0 with 1 ,where 0 1 R4 \𝐷4 𝐷2 homology 3-sphere). The complement is conformally are disjoint subdisks of the first factor disk. In complex 𝑆3 × R 𝑅4 \𝐷4 (𝑧, 𝑤) 󳨃→(𝑤,𝑧) equivalent to . Equivalently, the complement coordinates the plumbing may be written as 𝑆3 × R or (𝑧, 𝑤) 󳨃→(𝑤, 𝑧) creating either a positive or negative is diffeomorphic to an exotic .Buttheexoticnessisnot 2 confined to a compact subset but concentrated at infinity (for (resp.,) double point on the disk 𝐷 ×0(the core). Consider 2 2 2 2 instance at +∞). In our case we choose a decomposition like the pair (𝐷 ×𝐷 ,𝜕𝐷 ×𝐷 ) and produce finitely many self- 2 2 plumbings away from the attaching region 𝜕𝐷 ×𝐷 to get a 𝑆3× R =𝑀(𝑆3,𝜕𝐴 ) ⋃ , − − 𝜃 cork CH (20) kinky handle (𝑘, 𝜕 𝑘),where𝜕 𝑘 denotes the attaching region 𝐷2 − of the kinky handle. A kinky handle (𝑘, 𝜕 𝑘) is a one-stage − − 3 3 tower (𝑇1,𝜕 𝑇1) and an (𝑛+1)-stage tower (𝑇𝑛+1,𝜕 𝑇𝑛+1) is an where 𝑀(𝑆 ,𝜕𝐴 ) is a cobordism between 𝑆 and 𝜕𝐴 . 𝑛 − cork cork 𝑛-stage tower union kinky handles ∪ℓ=1(𝑇ℓ,𝜕 𝑇ℓ),wheretwo For the Casson handle we need another representation 6 Advances in High Energy Physics

obtained by using Morse theory (see [33]). Every kinky The tower 𝑇ℓ has a hyperbolic geometry (with finite − handle (𝑘, 𝜕 𝑘) is given by 𝑛 pairs of 1-/2-handle pairs, where volume) and therefore fixed size; that is, it cannot be scaled 𝑛 is the number of kinks (or self-intersections). These handles by any diffeomorphism or conformal transformation. Then aregivenbythelevelsetsoftheMorsefunctions we obtain an invariant decomposition of the Casson handle into towers arranged with respect to a tree. Secondly, inside of 𝑓 =𝑥2 +𝑦2 +𝑧2 −𝑡2 1 , 1 for the -handle every tower 𝑇ℓ there is (at least one) an incompressible surface (21) also of fixed size. 𝑓 =𝑥2 +𝑦2 −𝑧2 −𝑡2 2 , 2 for the -handle In case of the tower 𝑇ℓ,oneknowstwoincompressible surfaces, the two tori coming from the complement of that is, by the sets 𝐿(𝑓𝑖,𝐶)= {(𝑥,𝑦,𝑧,𝑡)|𝑖(𝑥,𝑦,𝑧,𝑡)= 𝑓 𝐶= const.} for 𝑖=1,2. Now we represent the Casson handle by the Whitehead link (with two components) used in the the union of all 𝑛-stage towers construction. = ⋃ ( (𝑇 )∪𝜕−𝑇 ), 𝑆3 ×R CH int ℓ ℓ (22) 3.5. The Path Integral of the Exotic . Now we will discuss level ℓ of tree T the path integral using the decomposition arranged along the tree T. But every tower 𝑇ℓ is given by the 3 𝑆 ×𝜃 R union of pairs (𝑓1,𝑓2).Butwhatisthegeometryof𝑇ℓ (and better of int(𝑇ℓ))? Every level set 𝐿(𝑓1,𝐶) and 𝐿(𝑓2,𝐶) is a =𝑀(𝑆3,𝜕𝐴 ) ⋃ ( ⋃ ( (𝑇 )∪𝜕−𝑇 )) , hyperbolic 3-manifold (i.e., with negative curvature) and the cork int ℓ ℓ union of all level sets is a hyperbolic 4-manifold. A central 𝐷2 level ℓ of tree T point in our argumentation is Mostow rigidity, a central (23) property of all hyperbolic 3-manifolds (or higher) with finite volume explained in the next subsection. and we remark that the construction of the cobordism 𝑀(𝑆3,𝜕𝐴 ) cork requirestheusageofaCassonhandleagain, 𝑀(𝑆3,𝜕𝐴 )∪ 3.4. The Hyperbolic Geometry of 𝐶𝐻. The central element in denoted by cork CHcork. Therefore we have to clar- theCassonhandleisapairof1-and2-handlesrepresenting ify the role of the Casson handle. In the previous Section 3.4, we discussed the strong connection between geometry and a kinky handle. As we argued above this pair admits a 3 hyperbolic geometry (or it is a hyperbolic 3-manifold) having topology for hyperbolic manifolds. The topology of 𝑆 ×𝜃 R is negative scalar curvature. A 3-manifold admits a hyperbolic rather trivial but the smoothness structure (and therefore the structure in the interior if there is a diffeomorphism to differentialtopology)canbeverycomplicate. 3 H /Γ,whereΓ is a discrete subgroup Γ⊂SO(3, 1) of As stated above, the boundary terms can be factorized the Lorentz group and we have a representation of the from the terms in the interior. Formally we obtain fundamental group 𝜋1(𝑀) into SO(3, 1) (the isometry group 3 of the hyperbolic space H ). One property of hyperbolic 3- 𝑍(𝑆3× R)={∏𝑍(𝜕−𝑇 )𝑍(𝜕𝐴 )𝑍(𝑆3)} 𝜃 ℓ cork and 4-manifolds is central: Mostow rigidity. As shown by ℓ Mostow [34], every hyperbolic 𝑛-manifold 𝑛>2with finite (24) volumehasthisproperty:Every diffeomorphism (especially ×(∏𝑍( (𝑇 )) 𝑍 ( ), every conformal transformation) of a hyperbolic 𝑛-manifold int ℓ CHcork ℓ with finite volume is induced by an isometry. Therefore one cannot scale a hyperbolic 3-manifold with finite volume. and for an expectation value of the observable O Then the volume vol() and the curvature are topological 3 3 invariants but for later usages we combine the curvature and ⟨𝑆 ×𝜃 R |O| 𝑆 ×𝜃 R⟩, (25) the volume into the Chern-Simons invariant CS().Butmore is true: in a hyperbolic 3-manifold there are special surfaces but for the following we have to discuss it more fully. To which cannot be contracted, called incompressible surface. understand the time-like evolution of a disk (or a surface), we A properly embedded connected surface 𝑆⊂𝑁in a 3- have to describe a disk inside of a Casson handle as pioneered manifold 𝑁 is called 2-sided if its normal bundle is trivial and by Bizacǎ [35]. With the same arguments, one can also 1-sided if its normal bundle is nontrivial. The “sides” of 𝑆 then describe the modification of the 3-sphere into homology 3- correspond to the components of the complement of 𝑆 in a spheres Σ. But then we obtain (formally) an infinite sequence 𝑆×[0,1]⊂ 𝑁 tubular neighborhood .A2-sidedconnected of homology 3-spheres Σ1 →Σ2 →⋅⋅⋅with amplitudes 2 2 surface 𝑆 other than 𝑆 or 𝐷 is called incompressible if 3 󵄨 󵄨 󵄨 󵄨 for each disk 𝐷⊂𝑁with 𝐷∩𝑆 = 𝜕𝐷there is a disk 𝑍(𝑆 ×𝜃 R)=⟨Σ1 󵄨𝑇𝑀󵄨 Σ2⟩⟨Σ2 󵄨𝑇𝑀󵄨 Σ3⟩⋅⋅⋅ , (26) 󸀠 󸀠 󵄨 󵄨 󵄨 󵄨 𝐷 ⊂𝑆with 𝜕𝐷 =𝜕𝐷;thatis,theboundaryofthedisk 𝐷 can be contracted in the surface 𝑆.Theboundaryofa3- including the boundary terms. Every spatial section Σ𝑛 can manifold is an incompressible surface. More importantly, this be seen as an element of the phase space in quantum gravity. surface can be detected in the fundamental group 𝜋1(𝑁) of Therefore this change of transitions is a topological phase the 3-manifold, that is; there is an injective homomorphism transition which will be further investigated in our work. 𝜋1(𝑆) →1 𝜋 (𝑁). The consequence of all properties is the The choice of the boundary term has a kind of arbitrari- following conclusion. ness. We can choose the decomposition much finer to get Advances in High Energy Physics 7 more boundary terms. Therefore the path integral (14)must of this hyperbolic geometry, there are incompressible surfaces be extended away from the boundary. We will discuss this inside of the hyperbolic manifold as the smallest possible extension also in our forthcoming work. units of geometry. Then Mostow rigidity determines the Beforewegoaheadwehavetodiscussthefoliation behavior of this incompressible surface. At first we will 3 𝑀(𝑆3,𝜕𝐴 ) 𝑆3 structure of 𝑆 ×𝜃 R or the appearance of different time concentrate on the first cobordism cork between 𝜕𝐴 variables. As stated above, our space-time has the topology and the boundary cork oftheAkbulutcork.Theareaofa 3 of 𝑆 × R with equal slices parametrized by a topological time surface is given by 𝑡 TOP;thatis, 2 √ 𝑎 𝑏 3 3 𝐴 (𝑒,) 𝑆 = ∫ 𝑑 𝜎 𝐸 𝐸 𝑛𝑎𝑛𝑏, (28) (𝑆 ×𝜃 R) ={(𝑝,𝑡 )|𝑝∈𝑆,𝑡 ∈ R} 𝑆 TOP TOP TOP (27) 3 𝑎 ={𝑆 ×{𝑡 }|𝑡 ∈ {−∞,...,+∞}}, with the normal vector 𝑛𝑎 and the densitized frame 𝐸 = TOP TOP 𝑎 det(𝑒)𝑒 . The expectation value of the area 𝐴, 3 3 defined by the topological embedding 𝑆 󳨅→ 𝑆 ×𝜃 R.Itis 3 ⟨𝑆3 |𝐴 (𝑒,) 𝑆 | 𝜕𝐴 ⟩ the defining property of exotic smoothness that 𝑆 inside cork 𝑆3× R of 𝜃 is only a topological 3-sphere; that is, it is wildly 1 embedded and so only represented by an infinite polyhedron. = 𝑡 𝑍(𝑀(𝑆3,𝜕𝐴 )) (29) There is another possibility to introduce TOP which will point cork us to the smooth case. For that purpose we define a map 𝐹: 𝑖 𝑆3 × R → R (𝑥, 𝑡) 󳨃→𝑡 𝑡 = 𝐹(𝑝) 𝑝∈𝑆3 × R × ∫ 𝐷𝑒 𝐴 (𝑒,) 𝑆 ( 𝑆 [𝑒, 𝑀3 (𝑆 ,𝜕𝐴 ]) , by so that TOP for . exp 𝐸𝐻 cork 𝑡 ℎ In contrast one also has the smooth time Diff which we have to define now. Locally it is the smooth (physical coordinate) depends essentially on the hyperbolic geometry. As argued 𝑆3× R time. We know also that the exotic 𝜃 is composed by a above, this cobordism has a hyperbolic geometry but in Σ →Σ →Σ → ⋅⋅⋅ sequence 1 2 3 of homology 3-spheres the simplest case, the boundary of the Akbulut cork is or better by a sequence 𝑀(Σ1,Σ2)∪Σ 𝑀(Σ2,Σ3)∪Σ ... of 𝜕𝐴 = Σ(2, 5, 7) 2 3 the homology 3-sphere cork ,aBrieskorn (homology) cobordism between the homology 3-spheres. homology 3-sphere. Now we study the area of a surface Allsequencesareorderedandsoitisenoughtoanalyze where one direction is along the time axis. Then we obtain 𝑀(Σ ,Σ ) one cobordism 1 2 . Every cobordism between two a decomposition of the surface into a sum of small surfaces Σ Σ homology 3-spheres 1 and 2 is characterized by the so that every small surface lies in one component of the 3 existence of a finite number of 1-/2-handle pairs (or dually cobordism. Remember, that the cobordism 𝑀(𝑆 ,𝜕𝐴 ) is 𝐹 : 3 cork 2-/3-handle pairs). Now we define a smooth map cob decomposed into the trivial cobordism 𝑆 × [0, 1] and a 𝑀(Σ1,Σ2) → [0, 1] which must be a Morse function (i.e., Casson handle CH =∪ℓ𝑇ℓ. Then the decomposition of the it has isolated critical points) [33]. The number of critical surface points 𝑁 of 𝐹 is even, say 𝑁=2𝑘,where𝑘 is the number of 1-/2-handle pairs. These critical points are also denoted as 𝑆=∪ℓ𝑆ℓ, (30) naked singularities in GR (but of bounded curvature). Like 𝑡 in the case of topological time TOP we introduce the smooth corresponds to the decomposition of the expectation value of 𝑡 =𝐹 (𝑝) 𝑝∈𝑀(Σ,Σ )⊂𝑆3× R time by Diff cob for all 1 2 𝜃 .The the area 𝑡 𝑆3× R extension of Diff to the whole 𝜃 by the Morse function 3 𝐹:𝑆× R → R 𝑡 = 𝐹(𝑝) 𝑝∈ 2 √ 𝑎 𝑏 𝜃 is straightforward Diff for all 𝐴ℓ (𝑒,ℓ 𝑆 ) = ∫ 𝑑 𝜎 𝐸 𝐸 𝑛𝑎𝑛𝑏, (31) 3 3 𝑆 𝑆 ×𝜃 R.Thecobordism𝑀(Σ1,Σ2) is part of the exotic 𝑆 ×𝜃 R ℓ 3 and can be embedded to make it 𝑆 × [0, 1] topologically. 𝐹 so that Therefore function cob is a continuous function which is strictly increasing on future directed causal curves, so it is a − 󵄨 󵄨 − − 󵄨 󵄨 − ⟨𝜕 𝑇 󵄨𝐴 󸀠 (𝑒, 𝑆 󸀠 )󵄨 𝜕 𝑇 ⟩=⟨𝜕 𝑇 󵄨𝐴 (𝑒, 𝑆 )󵄨 𝜕 𝑇 ⟩𝛿 󸀠 , time function (see [36, 37]). But there is also another method ℓ 󵄨 ℓ ℓ 󵄨 ℓ+1 ℓ 󵄨 ℓ ℓ 󵄨 ℓ+1 ℓℓ 𝑡 3 − 󵄨 󵄨 − to construct Diff by using codimension-1 foliations. In [7]we ⟨𝑆 |𝐴 (𝑒,) 𝑆 | 𝜕𝐴 ⟩=∑ ⟨𝜕 𝑇 󵄨𝐴 (𝑒, 𝑆 )󵄨 𝜕 𝑇 ⟩. cork ℓ 󵄨 ℓ ℓ 󵄨 ℓ+1 uncovered a strong relation between codimension-1 foliations ℓ (alsousedtoconstructaLorentzstructureonamanifold) 4 (32) and exotic smoothness structures for a small exotic R .The coordinate of this codimension-1 submanifold is also the The initial value for ℓ=0is the expectation value 𝑡 smooth time Diff. This approach will be more fully discussed in our forthcoming work. − 󵄨 󵄨 − 2 ⟨𝜕 𝑇0 󵄨𝐴0 (𝑒,0 𝑆 )󵄨 𝜕 𝑇1⟩=𝑎0 , (33)

3 4. The Expectation Value of where 𝑎0 is the radius of the 3-sphere 𝑆 .Butbecauseof the Area and Inflation the hyperbolic geometry (with constant curvature because of Mostow rigidity) every further level scales this expectation In Section 3.4 we described the hyperbolic geometry origi- value by a constant factor. Therefore, to calculate the expec- 3 nated in the exotic smoothness structure of 𝑆 ×𝜃 R.Because tation value, we have to study the scaling behavior. 8 Advances in High Energy Physics

Consider a cobordism 𝑀(Σ0,Σ1) between the homology hyperbolic in most case. The first case is more complicated. 3-spheres Σ0,Σ1.AsshownbyWitten[38–40], the action, Here we need the smooth function to represent the jump in the curvature parameter 𝑘.Letuschoosethefunction 3 √ 3 𝑘:R → R ∫ 𝑅 ℎ𝑑 𝑥=𝐿⋅CS (Σ0,1) , (34) Σ 0,1 +1 0 ≤ 𝑡 𝑡 󳨃󳨀→ { −2 (42) for every 3-manifold (in particular for Σ0 and Σ1 denoted by 1−2⋅exp (−𝜆 ⋅ 𝑡 )𝑡>0, Σ0,1) is related to the Chern-Simons action CS(Σ0,1)(defined in Appendix B).Thescalingfactor𝐿 is related to the volume which is smooth and the parameter 𝜆 determines the slope of 3 this function. Furthermore the metric (40)isalsothemetric by 𝐿=√vol(Σ0,1) and we obtain formally of a hyperbolic space (which has to fulfill Mostow rigidity 𝑀(Σ ,Σ ) (Σ ) (Σ ) because the cobordism 1 2 is compact). 𝐿⋅ (Σ ,𝐴)=𝐿3 ⋅ CS 0,1 = ∫ CS 0,1 √ℎ𝑑3𝑥, In the following we will switch to quadratic expressions CS 0,1 𝐿2 𝐿2 Σ0,1 because we will determine the expectation value of the area. (35) Thenweobtain 𝑎2 by using 𝑑𝑎2 = 𝑑𝑡2 =𝜗𝑑𝑡2, (43) 𝐿2 3 √ 3 𝐿 = vol (Σ0,1)=∫ ℎ𝑑 𝑥. (36) with respect to the scale 𝜗.ByusingthetreeoftheCasson Σ 0,1 handle,weobtainacountableinfinitesumofcontributions Together with for (43).Beforewestartwewillclarifythegeometryofthe Cassonhandle.ThediscussionoftheMorsefunctionsabove 3 3𝑘 uncovers the hyperbolic geometry of the Casson handle (see 𝑅 = , (37) 𝑎2 also Section 3.4). Therefore the tree corresponding to the Casson handle must be interpreted as a metric tree with 2 2 2 2 2 one can compare the kernels of the integrals of (34)and(35) hyperbolic structure in H and metric 𝑑𝑠 =(𝑑𝑥 +𝑑𝑦 )/𝑦 . to get for a fixed time The embedding of the Casson handle in the cobordism is given by the following rules. 3𝑘 (Σ ) = CS 0,1 . 2 2 (38) (i) The direction of the increasing levels 𝑛→𝑛+1is 𝑎 𝐿 2 2 identified with 𝑑𝑦 and 𝑑𝑥 is the number of edges This gives the scaling factor forafixedlevelwithscalingparameter𝜗. 𝑎2 3 (ii) The contribution of every level in the tree is deter- 𝜗= = , mined by the previous level best expressed in the 𝐿2 (Σ ) (39) CS 0,1 scaling parameter 𝜗. 𝑛 whereweset𝑘=1in the following. The hyperbolic geometry (iii) An immersed disk at level needs at least one disk to of the cobordism is best expressed by the metric resolve the self-intersection point. This disk forms the level 𝑛+1but this disk is connected to the previous 2 2 2 2 𝑖 𝑘 disk.Soweobtainfor𝑑𝑎 |𝑛+1 at level 𝑛+1 𝑑𝑠 =𝑑𝑡 −𝑎(𝑡) ℎ𝑖𝑘𝑑𝑥 𝑑𝑥 , (40) 2 2󵄨 𝑑𝑎 | ∼𝜗⋅𝑑𝑎󵄨 also called the Friedmann-Robertson-Walker metric (FRW 𝑛+1 󵄨𝑛 (44) metric) with the scaling function 𝑎(𝑡) for the (spatial) 3- up to a constant. manifold. But Mostow rigidity enforces us to choose 2 2 2 2 By using the metric 𝑑𝑠 =(𝑑𝑥+𝑑𝑦)/𝑦 with the 2 2 𝑎̇2 1 embedding (𝑦 →𝑛+1, 𝑑𝑥 →𝜗)weobtainforthechange ( ) = , (41) 2 2 𝑎 𝐿2 𝑑𝑥 /𝑦 along the 𝑥-direction (i.e., for a fixed 𝑦) 𝜗/(𝑛+1).This change determines the scaling from the level 𝑛 to 𝑛+1;that 𝐿 in the length scale of the hyperbolic structure. But why is, is it possible to choose the FRW metric? At first we state 𝑛+1 2󵄨 𝜗 2󵄨 𝜗 2󵄨 that the FRW metric is not sensitive to the topology of 𝑑𝑎 󵄨 = ⋅𝑑𝑎󵄨 = ⋅𝑑𝑎󵄨 ; (45) thespace-time.Oneneedsonlyaspace-timewhichadmits 󵄨𝑛+1 𝑛+1 󵄨𝑛 (𝑛+1)! 󵄨0 𝑡 a slicing with respect to a smooth time Diff and a metric of constant curvature for every spatial slice. Then for the and after the whole summation (as substitute for an integral cobordism 𝑀(Σ1,Σ2) between Σ1 and Σ2 we have two cases: for the discrete values) we obtain for the relative scaling the curvature parameter 𝑘(Σ1) of Σ1 (say 𝑘(Σ1)=+1) ∞ ∞ 2 2󵄨 2 1 𝑛 jumps to the value 𝑘(Σ2) of Σ2 (say 𝑘(Σ2)=−1)orboth 𝑎 = ∑ (𝑑𝑎 󵄨 )=𝑎 ⋅ ∑ 𝜗 󵄨𝑛 0 𝑛! curvaturesremainconstant.Thesecondcaseistheusualone. 𝑛=0 𝑛=0 (46) Each homology 3-sphere Σ1,Σ2 has the same geometry (or =𝑎2 ⋅ (𝜗) =𝑎2 ⋅𝑙 geometric structure in the sense of Thurston41 [ ]) which is 0 exp 0 scale Advances in High Energy Physics 9

2 2 with 𝑑𝑎 |0 =𝑎0 . With this result in mind, we consider the first derivatives at these points). A homology 3-sphere has two expectation value where we use the constant scalar curvature critical points (located at the two poles). The Morse function 2 (Mostow rigidity). By using the normalization, many terms looks like ±‖𝑥‖ at these critical points. The transition 𝑦= areneglected(liketheboundaryterms): 𝜙(𝑥) represented by the (homology) cobordism 𝑀(Σ0,Σ1) 2 maps the Morse function 𝜓(𝑦) = ‖𝑦‖ on Σ0 to the ⟨𝑆3 |𝐴 (𝑒,) 𝑆 | 𝜕𝐴 ⟩ 2 2 cork Morse function Ψ(𝑥) = ‖𝜙(𝑥)‖ on Σ1.Thefunction−‖𝜙‖ represents also the critical point of the cobordism 𝑀(Σ0,Σ1). =(({∏𝑍(𝜕−𝑇 )𝑍(𝜕𝐴 )𝑍(𝑆3)} But as we learned above, this cobordism has a hyperbolic ℓ cork ‖𝜙(𝑥)‖2 ℓ geometry and we have to interpret the function not as Euclidean form but change it to the hyperbolic geometry ∞ so that − 󵄨 󵄨 − × ∑ ⟨𝜕 𝑇𝑛 󵄨𝐴𝑛 (𝑒,𝑛 𝑆 )󵄨 𝜕 𝑇𝑛+1⟩) 󵄨 󵄨 󵄩 󵄩2 2 2 2 −2𝜙 2 2 𝑛=0 (47) 󵄩 󵄩 1 −󵄩𝜙󵄩 =−(𝜙1 +𝜙2 +𝜙3)󳨀→−𝑒 (1 + 𝜙2 +𝜙3); (50) −1 ×({∏𝑍(𝜕−𝑇 )𝑍(𝜕𝐴 )𝑍(𝑆3)}) ) that is, we have a preferred direction represented by a single ℓ cork 𝜙 :Σ → R Σ → ℓ scalar field 1 1 . Therefore, the transition 0 Σ1 is represented by a single scalar field 𝜙1 :Σ1 → R and ∞ 󵄨 󵄨 we identify this field as the moduli. Finally we interpret this = ∑ ⟨𝜕−𝑇 󵄨𝐴 (𝑒, 𝑆 )󵄨 𝜕−𝑇 ⟩. 𝑛 󵄨 𝑛 𝑛 󵄨 𝑛+1 Morse function in the interior of the cobordism 𝑀(Σ0,Σ1) as 𝑛=0 thepotential(shiftedawayfromthepoint0 )ofthescalarfield 2 𝜙 with Lagrangian Finallyweobtainforthearea𝑎0 for the first level ℓ=0, 2 𝜌 𝐿=𝑅+(𝜕𝜙) − (1 − (−𝜆𝜙))2, ∞ 󵄨 󵄨 𝜇 2 exp (51) ⟨𝑆3 |𝐴 (𝑒,) 𝑆 | 𝜕𝐴 ⟩=∑ ⟨𝜕−𝑇 󵄨𝐴 (𝑒, 𝑆 )󵄨 𝜕−𝑇 ⟩ cork 𝑛 󵄨 𝑛 𝑛 󵄨 𝑛+1 𝑛=0 with two free constants 𝜌 and 𝜆.Forthevalue𝜆=√2/3 ∞ 𝑛 𝜌=3𝑀2 2 1 3 and we obtain the Starobinski model [42](bya =𝑎0 ⋅ ∑ ( ) (48) conformal transformation using 𝜙 and a redefinition of the 𝑛! CS (𝜕𝐴 ) 𝑛=0 cork scalar field [43])

2 3 1 =𝑎 ⋅ exp ( ), 𝐿=𝑅+ 𝑅2, 0 (𝜕𝐴 ) 2 (52) CS cork 6𝑀 𝑀≪𝑀 with the radius 𝑎0 of Σ0 and arrive at 𝑎 for Σ1.Fromthe with the mass scale 𝑃 much smaller than the physical point of view we obtain an exponential increase Planck mass. From our discussion above, the appearance of of the area; that is, we get an inflationary behavior. This this model is not totally surprising. It favors a surface to be derivation can be also extended to the next Casson handle incompressible (which is compatible with the properties of 𝜕𝐴 hyperbolic manifolds). In the next section we will determine but we have to determine the 3-manifold in which cork can change. It will be done below. this mass scale.

5. An Effective Theory 6. A Cosmological Model Compared to the Planck Satellite Results Nowwillaskforaneffectivetheorywheretheinfluenceof the exotic smoothness structure is contained in some moduli In this section we will go a step further and discuss the 3 (or some field). As explained above, the main characteris- path integral for 𝑆 × R, where we sum over all smoothness 3 tics is given by a change of the (spatial) 4-manifold (but structures. Furthermore we will assume that 𝑆 × R is the 4 without changing the homology). Therefore let us describe end of a small exotic R . But then we have to discuss this change (the so-called homology cobordism) between the parametrization of all Casson handles. As discussed two homology 3-spheres Σ0 and Σ1.Thesituationcanbe by Freedman [2], all Casson handles can be parametrized described by a diagram by a dual tree where the vertices are 5-stage towers (with three extra conditions). We refer to [2]orto[44]forthe Ψ Σ1 󳨀→ R, details of the well-known construction. This tree has one root 𝜙↓ ↻ ↕𝑖𝑑, (49) from which two 5-stage towers branch. Every tower has an 𝜓 Σ 󳨀→ R, attaching circle of any framing. Using Bizacas technique [35], 0 we obtain an attaching of a 5-tower along the sum 𝑃#𝑃 of 𝑃 which commutes. The two functions 𝜓 and Ψ are the Morse two Poincare spheres (for the two towers). Therefore for the universal case, we obtain two transitions function of Σ0 and Σ1,respectively,withΨ=𝜓∘𝜙.The Morse function over Σ0,1 is a function Σ0,1 → R having only 𝑆3 󳨀→cork 𝜕 𝐴 tower󳨀→ 𝑃 𝑃, (53) isolated, nondegenerated, critical points (i.e., with vanishing cork # 10 Advances in High Energy Physics with the scaling behavior, oneobtainsthesize 140 3 3 𝑎=𝐿 ⋅ ( + 90) ≈ 9.14 ⋅ 1024𝑚≈109𝐿𝑗. 𝑎=𝑎 ⋅ ( + ). 𝑃 exp (62) 0 exp 2⋅ (𝜕𝐴 ) 2⋅ (𝑃 𝑃) (54) 3 CS cork CS # Asexplainedabove,theeffectivetheoryistheStarobinsky It can be expressed by the expectation value model. This model is in very good agreement with results of 3 3 the Planck satellite [48] with the two main observables: ⟨𝑆3 |𝐴 (𝑒,) 𝑆 | 𝑃 𝑃⟩ =2 𝑎 ⋅ ( + ), # 0 exp (𝜕𝐴 ) (𝑃 𝑃) 𝑛 ∼ 0.96 CS cork CS # 𝑠 spectral index for scalar perturbations, (55) 𝑟 ∼ 0.004 tensor-to-scalar ratio, 3 for the transition 𝑆 →𝑃#𝑃. It is important to note that this but one parameter of the model is open, the energy scale expectation value is the sum over all smoothness structures 𝑀 in Planck units. In our model it is related to the second 3 of 𝑆 × R and we obtain also derivative of the Morse function, which is the curvature of 3 3 the critical point. In our paper [49], we determined also the ⟨𝑆 × R |𝐴 (𝑒,) 𝑆 | 𝑆 × R⟩ energy scale of the inflation by using a simple argument to

3 3 incorporate only the first 3 levels of the Casson handle. For = ∑ ⟨𝑆 ×𝜃 R |𝐴 (𝑒,) 𝑆 | 𝑆 ×𝜃 R⟩ the scale (56) diff structures 3 𝜗= , (63) 2 3 3 2⋅CS (Σ (2, 5, 7)) =𝑎0 ⋅ exp ( + ), CS (𝜕𝐴 ) CS (𝑃#𝑃) cork of the first transition, we obtain the scaling of the Planck 2 3 with 𝑎0 as the size of the 3-sphere 𝑆 at −∞.Withthe energy (associated with the Planck-sized 3-sphere at the argumentation above, the smoothness structure has a kind of beginning) universality so that the two transitions above are generic. 𝐸 𝐸 = Planck , In our model (using the exotic smoothness structure), we Inflation (1+𝜗+(𝜗2/2) + (𝜗3/6)) (64) obtain two inflationary phases. In the first phase we have a transition with the relative scaling 𝑆3 󳨀→ 𝜕 𝐴 , cork (57) 𝐸 1 𝛼= Inflation = ≈ 5.5325 ⋅ 10−5, 𝜕𝐴 = Σ(2, 5, 7) 𝐸 (1+𝜗+(𝜗2/2) + (𝜗3/6)) andforthesimplestcase cork ,aBrieskorn Planck homology3-sphere.Nowwewillassumethatthe3-spherehas (65) Planck-size leading to the energy scale of the inflation ℎ𝐺 𝑎 =𝐿 = √ ; (58) 𝐸 ≈ 6.7547 ⋅ 1014𝐺𝑒𝑉, 0 𝑃 𝑐3 Inflation (66) 19 by using 𝐸 ≈ 1.2209 ⋅ 10 𝐺𝑒𝑉.Weremarkthatthe then we obtain for the size Planck −5 relative scaling 𝛼 ≈ 5.5325 ⋅ 10 (in Planck units) above is 3 an energy scale for the potential in the effective theory (52). 𝑎=𝐿𝑃 ⋅ exp ( ). (59) 2 2⋅CS(Σ (2, 5, 7) Therefore we have to identify 𝑀=𝛼in the parameter 1/6𝑀 of the Starobinsky model (in very good agreement with the WecanusethemethodofFintushelandStern[45–47]tocal- measurements). Now we can go a step further and discuss the culate the Chern-Simons invariants for the Brieskorn spheres. appearance of the cosmological constant. The calculation can be found in Appendix C.Notethatthe Again we can use the hyperbolic geometry to state that relation (34)isonlytruefortheLevi-Civitaconnection.Then thecurvatureisnegativeandwehavetheMostowrigidity; the Chern-Simons invariant is uniquely defined to be the that is, the scalar curvature of the 4-manifold has a constant 𝜏() minimum, denoted by (see (B.4)).Thenweobtainforthe value, the cosmological constant Λ. If we assume that the 3- invariant (C.7)sothat sphere has the size of the Planck length (as above), then we 140 −15 obtain 𝐿𝑃 ⋅ exp ( )≈7.5⋅10 𝑚 (60) 3 1 3 3 Λ= ⋅ (− − ). 𝐿2 exp (Σ (2, 5, 7)) (𝑃 𝑃) (67) is the size of the cosmos at the end of the first inflationary 𝑃 CS CS # phase.Thissizecanberelatedtoanenergyscalebyusingitas With the values of the Chern-Simons invariants (C.7), we Compton length and one obtains 165 MeV, comparable to the obtain the value energy scale of the QCD. For the two inflationary transitions 2 280 −118 3 Λ⋅𝐿 = (− − 180) ≈ 5 ⋅ 10 , 𝑆 󳨀→ Σ (2, 5, 7) 󳨀→ 𝑃 #𝑃 (61) 𝑃 exp 3 (68) Advances in High Energy Physics 11

in Planck units. In cosmology, one usually relate the cosmo- correcting the value ΩΛ ≈ 0.88 to logical constant to the Hubble constant 𝐻0 (expressing the critical density) leading to the length scale ΩΛ ≈ 0.704. (80) 𝑐2 𝛾 𝐿2 = . But 0 depends on the gauge group and if one uses the value 𝑐 2 (69) 3𝐻0 [52]

The corresponding variable is denoted by ΩΛ and we obtain ln (3) 𝛾0 = , (81) 𝜋⋅√8 𝑐5 3 3 Ω = ⋅ (− − ), Λ 2 exp (70) 3ℎ𝐺𝐻0 CS (Σ (2, 5, 7)) CS (𝑃#𝑃) agreeing also with calculations in the spin foam models [53], then one gets a better fit in units of the critical density. This formula is in very good agreement with the WMAP results; that is, by using the value ΩΛ ≈ 0.6836, (82) for the Hubble constant km whichisingoodagreementwiththemeasurements. 𝐻0 =74 , (71) 𝑠⋅Mpc 7. Conclusion weareabletocalculatethedarkenergydensitytobe Ω = 0.729, The strong relation between hyperbolic geometry (of the Λ (72) space-time) and exotic smoothness is one of the main agreeing with the WMAP results. But it differs from the results in this paper. Then using Mostow rigidity, geometric Planck results [50]oftheHubbleconstant observables like area and volume or curvature are topological invariants which agree with the expectation values of these km observables (calculated via the path integral). We compared (𝐻0) =68 , (73) Planck 𝑠⋅Mpc the results with the recent results of the Planck satellite and found a good agreement. In particular as a direct result for which we obtain of the hyperbolic geometry, the inflation can be effectively described by the Starobinski model. Furthermore we also ΩΛ ≈ 0.88, (74) obtained a cosmological model which produces a realistic in contrast with the measured value of the dark energy cosmological constant.

(ΩΛ) = 0.683. (75) Planck Appendices But there is another possibility for the size of the 3-sphere at the beginning and everything depends on this choice. But we A. Connected and Boundary Connected canusetheentropyformulaofaBlackholeinLoopquantum Sum of Manifolds gravity Now we will define the connected sum # and the boundary 𝐴⋅𝛾 ♮ 𝑀, 𝑁 𝑛 𝑆= 0 ⋅2𝜋, connected sum of manifolds. Let be two -manifolds 4⋅𝛾⋅𝐿2 (76) with boundaries 𝜕𝑀, 𝜕𝑁.Theconnected sum 𝑀#𝑁 is the 𝑃 𝑛 procedure of cutting out a disk 𝐷 from the interior int(𝑀) \ 𝑛 𝑛 𝑛−1 𝑛−1 with 𝐷 and int(𝑁) \ 𝐷 with the boundaries 𝑆 ⊔𝜕𝑀and 𝑆 ⊔ 𝜕𝑁 ln (2) , respectively, and gluing them together along the com- 𝛾0 = , (77) 𝑛−1 𝜋⋅√3 mon boundary component 𝑆 .Theboundary𝜕(𝑀#𝑁) = 𝜕𝑀 ⊔ 𝜕𝑁 is the disjoint sum of the boundaries 𝜕𝑀, 𝜕𝑁.The according to [51] with the Immirzi parameter 𝛾,wherethe boundary connected sum 𝑀♮𝑁 is the procedure of cutting out 𝑛−1 𝑛−1 𝑛−1 extra factor 2𝜋 is given by a different definition of 𝐿𝑃 adisk𝐷 from the boundary 𝜕𝑀 \𝐷 and 𝜕𝑁 \𝐷 and ℎ ℎ 𝑛−2 replacing by . In the original approach of Ashtekar in Loop gluing them together along 𝑆 of the boundary. Then the 𝛾=1 quantum gravity one usually set .Ifwetakeitseriously boundary of this sum 𝑀♮𝑁 is the connected sum 𝜕(𝑀♮𝑁) = then we obtain a reduction of the length in (70) 𝜕𝑀#𝜕𝑁 of the boundaries 𝜕𝑀, 𝜕𝑁. 1 1 2⋅ (2) 2𝜋𝛾 1 󳨀→ ⋅ ln = 0 ≈ 0.80037 ⋅ , 2 2 2 2 (78) 𝐿𝑃 𝐿𝑃 √3 𝐿𝑃 𝐿𝑃 B. Chern-Simons Invariant or the new closed formula Let 𝑃 be a principal 𝐺 bundle over the 4-manifold 𝑀 with 𝜕𝑀 =0̸. Furthermore let 𝐴 be a connection in 𝑃 with the 𝑐5 3 3 Ω = ⋅𝛾 ⋅ (− − ), curvature, Λ 2 0 exp 3ℎ𝐺𝐻0 CS (Σ (2, 5, 7)) CS (𝑃#𝑃) (79) 𝐹𝐴 =𝑑𝐴+𝐴∧𝐴, (B.1) 12 Advances in High Energy Physics and Chern class, C. Chern-Simons Invariant of 1 Brieskorn Spheres 𝐶 = ∫ (𝐹 ∧𝐹 ), 2 8𝜋2 tr 𝐴 𝐴 (B.2) 𝑀 In [45–47] an algorithm for the calculation of the Chern- 𝑃 Simons invariant for the Brieskorn sphere Σ(𝑝,𝑞,𝑟) is for the classification of the bundle .ByusingtheStokes 𝛼: theorem we obtain presented. According to that result, a representation 𝜋1(Σ(𝑝, 𝑞, 𝑟) → SU(2) is determined by a triple of 3 numbers 2 ⟨𝑘, 𝑙, 𝑚⟩ with 0<𝑘<𝑝,0<𝑙<𝑞,0<𝑚<𝑟and the further ∫ tr (𝐹𝐴 ∧𝐹𝐴)=∫ tr (𝐴 ∧ 𝑑𝐴 + 𝐴∧𝐴∧𝐴), (B.3) 𝑀 𝜕𝑀 3 relations with the Chern-Simons invariant 𝑙 𝑚 + <1 𝑙mod 2=𝑚mod 2, 1 2 𝑞 𝑟 (𝜕𝑀, 𝐴) = ∫ (𝐴 ∧ 𝑑𝐴 + 𝐴∧𝐴∧𝐴). CS 2 tr (B.4) 8𝜋 𝜕𝑀 3 𝑘 𝑙 𝑚 + + >1, −1 𝑝 𝑞 𝑟 Now we consider the gauge transformation 𝐴→𝑔 𝐴𝑔 + −1 (C.1) 𝑔 𝑑𝑔 and obtain 𝑘 𝑙 𝑚 − + <1, −1 −1 𝑝 𝑞 𝑟 CS (𝜕𝑀, 𝑔 𝐴𝑔 +𝑔 𝑑𝑔) = CS (𝜕𝑀, 𝐴) +𝑘, (B.5) 𝑘 𝑙 𝑚 + − <1. with the winding number 𝑝 𝑞 𝑟 1 3 𝑘= ∫ (𝑔−1𝑑𝑔) ∈ Z, 2 (B.6) Then the Chern-Simons invariant is given by 24𝜋 𝜕𝑀 𝑒2 of the map 𝑔:𝑀. →𝐺 Thus the expression, (𝛼) = 1, CS 4⋅𝑝⋅𝑞⋅𝑟 mod (C.2) CS (𝜕𝑀, 𝐴) mod 1, (B.7) with is an invariant, the Chern-Simons invariant. Now we will calculate this invariant. For that purpose we consider the 𝑒=𝑘⋅𝑞⋅𝑟+𝑙⋅𝑝⋅𝑟+𝑚⋅𝑝⋅𝑞. (C.3) functional (B.4)anditsfirstvariationvanishes Now we consider the Poincaresphere´ 𝑃 with 𝑝=2,𝑞= 𝛿 (𝜕𝑀, 𝐴) =0, CS (B.8) 3, 𝑟 = 5.Thenweobtain 1 because of the topological invariance. Then one obtains the ⟨1, 1, 1⟩ = , equation CS 120 (C.4) 𝑑𝐴 + 𝐴 ∧ 𝐴 =0; 49 (B.9) ⟨1, 1, 3⟩ = , CS 120 that is, the extrema of the functional are the connections of vanishing curvature. The set of these connections up to andfortheBrieskornsphereΣ(2, 5, 7) gauge transformations is equal to the set of homomorphisms 81 𝜋1(𝜕𝑀) → SU(2) up to conjugation. Thus the calculation ⟨1, 1, 3⟩ = , CS 280 of the Chern-Simons invariant reduces to the representation theory of the fundamental group into SU(2).In[45]the 9 ⟨1, 3, 1⟩ = , authors define a further invariant CS 280 (C.5) 𝜏 (Σ) = { (𝛼) |𝛼:𝜋 (Σ) 󳨀→ (2)}, 169 min CS 1 SU (B.10) ⟨1, 2, 2⟩ = , CS 280 for the 3-manifold Σ. This invariants fulfills the relation 249 1 ⟨1, 2, 4⟩ CS = . 𝜏 (Σ) = ∫ (𝐹 ∧𝐹 ) , 280 2 tr 𝐴 𝐴 (B.11) 8𝜋 Σ×R In [45] the authors define a further invariant which is the minimum of the Yang-Mills action 󵄨 󵄨 𝜏 (Σ) = min {CS (𝛼) |𝛼:𝜋1 (Σ) 󳨀→ SU (2)} , (C.6) 󵄨 1 󵄨 1 󵄨 ∫ (𝐹 ∧𝐹 )󵄨 ≤ ∫ (𝐹 ∧∗𝐹 ), 󵄨 2 tr 𝐴 𝐴 󵄨 2 tr 𝐴 𝐴 (B.12) 󵄨8𝜋 Σ×R 󵄨 8𝜋 Σ×R for a homology 3-sphere Σ.For𝑃 and Σ(2, 5, 7) one obtains that is, the solutions of the equation 𝐹𝐴 =±∗𝐹𝐴.Thusthe 1 9 𝜏 (𝑃) = ,𝜏(Σ (2, 5, 7)) = . invariant 𝜏(Σ) of Σ corresponds to the self-dual and anti-self- 120 280 (C.7) dual solutions on Σ×R,respectively.Ortheinvariant𝜏(Σ) is the Chern-Simons invariant for the Levi-Civita connection. And we are done. Advances in High Energy Physics 13

Conflict of Interests [18] T. Asselmeyer-Maluga and J. Krol,´ “Quantum geometry and wild embeddings as quantum states,” International Journal The authors declare that there is no conflict of interests of Geometric Methods in Modern Physics,vol.10,ArticleID regarding the publication of this paper. 1350055, 2013. [19] R. Kirby and L. C. Siebenmann, Foundational Essays on Topo- logical Manifolds, Smoothings, and Triangulations, Annals of Acknowledgment Mathematics Studies, Princeton University Press, Princeton, NJ, The authors acknowledged all critical remarks of the referees USA, 1977. increasing the readability of the paper. [20] A. Ashtekar, J. Engle, and D. Sloan, “Asymptotics and Hamil- tonians in a first-order formalism,” Classical and Quantum Gravity,vol.25,no.9,p.095020,2008. References [21] A. Ashtekar and D. Sloan, “Action and Hamiltonians in higher- dimensional general relativity: first-order framework,” Classical [1] S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of and Quantum Gravity,vol.25,no.22,p.225025,2008. Space-Time, Cambridge University Press, Cambridge, UK, 1994. [22] S. W. Hawking, “The path-integral approach to quantum [2] M. H. Freedman, “The topology of four-dimensional mani- gravity,” in General Relativity. An Einstein Centenary Survey, folds,” Journal of Differential Geometry,vol.17,pp.357–3454, I. Hawking, Ed., pp. 746–789, Cambridge University Press, 1982. Cambridge, UK, 1979. [3] S.Donaldson,“Anapplicationofgaugetheorytothetopologyof [23] T. Asselmeyer-Maluga and H. Rose,´ “On the geometrization of 4-manifolds,” Journal of Differential Geometry,vol.18,pp.279– matter by exotic smoothness,” General Relativity and Gravita- 315, 1983. tion,vol.44,no.11,pp.2825–2856,2012. [4] T. Asselmeyer-Maluga, “A chicken-and-egg problem: which [24] L. Dabrowski and G. Dossena, “Dirac operator on spinors and came first, the quantum state or spacetime?” 2012, http://fqxi diffeomorphisms,” Classical and Quantum Gravity,vol.30,no. .org/community/forum/topic/1424. 1, Article ID 015006, 2013. [5] Fourth Prize of the FQXi Essay contest, “Questioning the [25] T. Friedrich, “On the spinor representation of surfaces in Foundations,” 2012, http://fqxi.org/community/essay/winners/ euclidean 3-space,” Journal of Geometry and Physics,vol.28,no. 2012.1. 1-2, pp. 143–157, 1998. [6] H. Pfeiffer, “Quantum general relativity and the classification of [26] E. Witten, “Global gravitational anomalies,” Communications in smooth manifolds,” Tech. Rep. DAMTP 2004-32, 2004. Mathematical Physics,vol.100,no.2,pp.197–229,1985. 𝜂 [7] T. Asselmeyer-Maluga and J. Krol,´ “Abelian gerbes, generalized [27] T. Yoshida, “The -invariant of hyperbolic 3-manifolds,” Inven- 4 geometries and exotic 𝑅 ,” http://arxiv.org/abs/0904.1276. tiones Mathematicae,vol.81,pp.473–514,1985. 3 4 𝑆 ×𝑅 [8]T.Asselmeyer-MalugaandR.Mader,“Exotic𝑅 and quantum [28] M. H. Freedman, “A fake ,” Annals of Mathematics,vol. field theory,” JournalofPhysics:ConferenceSeries,vol.343, 110,pp.177–201,1979. Article ID 012011, 2012. [29] S. Donaldson, “Irrationality and the h-cobordism conjecture,” 4 Journal of Differential Geometry,vol.26,pp.141–168,1987. [9] T. Asselmeyer-Maluga and J. Krol,´ “Exotic smooth 𝑅 and certain configurations of NS and D branes in string theory,” [30] S. DeMichelis and M. H. Freedman, “Uncountable many exotic 𝑅4 International Journal of Modern Physics A,vol.26,no.7-8,pp. ’s in standard 4-space,” Journal of Differential Geometry,vol. 1375–1388, 2011. 35, pp. 219–254, 1992. [31] Z.ˇ Bizacaˇ and R. Gompf, “Elliptic surfaces and some simple [10] T. Asselmeyer-Maluga and J. Krol,´ “Quantum D-branes and 4 4 R exotic smooth R ,” International Journal of Geometric Methods exotic ’s. ,” Journal of Differential Geometry,vol.43,no.3,pp. in Modern Physics,vol.9,no.3,p.1250022,18,2012. 458–504, 1996. [32] M. Golubitsky and V. Guillemin, Stable Mappings and Their [11] C. H. Brans, “Localized exotic smoothness,” Classical and Singularities,vol.14ofGraduate Texts in Mathematics,Springer, Quantum Gravity,vol.11,no.7,pp.1785–1792,1994. New York, NY, USA, 1973. [12] C. H. Brans, “Exotic smoothness and physics,” Journal of [33] J. Milnor, Morse Theory,vol.51ofAnnals of Mathematical Study, Mathematical Physics,vol.35,pp.5494–5506,1994. Princeton University Press, Princeton, NJ, USA, 1963. R4 [13] J. Sładkowski, “Gravity on exotic with few symmetries,” [34] G. D. Mostow, “Quasi-conformal mappings in n-space and the International Journal of Modern Physics D,vol.10,pp.311–313, rigidity of hyperbolic space forms,” Publications Mathematiques´ 2001. de l’Institut des Hautes Etudes´ Scientifiques, vol. 34, pp. 53–104, [14] J. Sładkowski, “Exotic smoothness, noncommutative geome- 1968. try and particle physics,” International Journal of Theoretical [35] Z.ˇ Bizaca,ˇ “An explicit family of exotic Casson handles,” Proceed- Physics, vol. 35, no. 10, pp. 2075–2083, 1996. ings of the American Mathematical Society,vol.123,no.4,pp. [15] J. Sładkowski, “Exotic smoothness and particle physics,” Acta 1297–1302, 1995. Physica Polonica B,vol.27,pp.1649–1652,1996. [36] A. N. Bernal and M. Sanchez,´ “Further results on the smootha- [16] J. Sładkowski, “Exotic smoothness, fundamental interactions bility of Cauchy hypersurfaces and Cauchy time functions,” and noncommutative geometry,” http://arxivabs/hep-th/ Letters in Mathematical Physics,vol.77,no.2,pp.183–197,2006. 9610093. [37] A. N. Bernal and M. Sanchez,´ “Globally hyperbolic spacetimes [17] T. Asselmeyer-Maluga and J. Krol,´ “Topological quantum D- can be defined as “causal” instead of ‘strongly causal’,” Classical 4 branes and wild embeddings from exotic smooth 𝑅 ,” Interna- and Quantum Gravity,vol.24,no.3,pp.745–750,2007. tionalJournalofModernPhysicsA,vol.26,no.20,pp.3421–3437, [38] E. Witten, “2+1dimensional gravity as an exactly soluble 2011. system,” Nuclear Physics B, vol. 311, pp. 46–78, 1988. 14 Advances in High Energy Physics

[39] E. Witten, “Topology-changing amplitudes in 2+1dimensional gravity,” Nuclear Physics B,vol.323,pp.113–140,1989. [40] E. Witten, “Quantization of Chern-Simons gauge theory with complex gauge group,” Communications in Mathematical Physics,vol.137,pp.29–66,1991. [41] W. Thurston, Three-Dimensional Geometry and Topology, Princeton University Press, Princeton, NJ, USA, 1st edition, 1997. [42] A. A. Starobinski, “Anew type of isotropic cosmological models without singularity,” Physical Letters B,vol.91,pp.99–102,1980. [43] B. Whitt, “Fourth order gravity as general relativity plus matter,” Physical Letters B,vol.145,pp.176–178,1984. [44] M. Freedman and F. Quinn, Topology of 4-Manifolds, Princeton Mathematical Series, Princeton University Press, Princeton, NJ, USA, 1990. [45]R.FintushelandR.J.Stern,“InstantonhomologyofSeifert fibred homology three spheres,” Proceedings of the London Mathematical Society,vol.61,pp.109–137,1990. [46] P. Kirk and E. Klassen, “Chern-Simons invariants of 3- manifolds and representation spaces of knot groups,” Mathema- tische Annalen,vol.287,no.1,pp.343–367,1990. [47] D. S. Freed and R. E. Gompf, “Computer calculation of Wit- ten’s 3-manifold invariant,” Communications in Mathematical Physics,vol.141,pp.79–117,1991. [48] P. Ade, N. Aghanim, C. Armitage-Caplan et al., “Planck 2013 results. XXII. Constraints on inflation,” http://arxiv.org/ abs/1303.5082. [49] T. Asselmeyer-Maluga and J. Krol,´ “On the origin of inflation by using exotic smoothness,” http://arxiv.org/abs/1301.3628. [50] P. Ade, N. Aghanim, C. Armitage-Caplan et al., “Planck 2013 results. XVI. Cosmological parameters,” http://arxiv.org/abs/ 1303.5076. [51] A. Ashtekar, J. Baez, A. Corichi, and K. Krasnov, “Quantum geometry and black hole entropy,” Physical Review Letters,vol. 80, no. 5, pp. 904–907, 1998. [52] M. Domagala and J. Lewandowski, “Black-hole entropy from quantum geometry,” Classical and Quantum Gravity,vol.21,no. 22, pp. 5233–5243, 2004. [53] M. H. Ansari, “Generic degeneracy and entropy in loop quantum gravity,” Nuclear Physics B,vol.795,no.3,pp.635–644, 2008. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2014, Article ID 678087, 10 pages http://dx.doi.org/10.1155/2014/678087

Research Article Does the Equivalence between Gravitational Mass and Energy Survive for a Composite Quantum Body?

A. G. Lebed1,2 1 Department of Physics, University of Arizona, 1118 E. 4th Street, Tucson, AZ 85721, USA 2 L. D. Landau Institute for Theoretical Physics, 2 Kosygina Street, Moscow 117334, Russia

Correspondence should be addressed to A. G. Lebed; [email protected]

Received 7 November 2013; Accepted 21 January 2014; Published 9 March 2014

Academic Editor: Douglas Singleton

Copyright © 2014 A. G. Lebed. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

We define passive and active gravitational mass operators of the simplest composite quantum body—a hydrogen atom. Although they do not commute with its energy operator, the equivalence between the expectation values of passive and active gravitational masses and energy is shown to survive for stationary quantum states. In our calculations of passive gravitational mass operator, we take into account not only kinetic and Coulomb potential energies but also the so-called relativistic corrections to electron motion in a hydrogen atom. Inequivalence between passive and active gravitational masses and energy at a macroscopic level is demonstrated to reveal itself as time-dependent oscillations of the expectation values of the gravitational masses for superpositions of stationary quantum states. Breakdown of the equivalence between passive gravitational mass and energy at a microscopic level reveals itself as unusual electromagnetic radiation, emitted by macroscopic ensemble of hydrogen atoms, moved by small spacecraft with constant velocity in the Earth’s gravitational field. We suggest the corresponding experiment on the Earth’s orbit to detect this radiation, which would be the first direct experiment where quantum effects in general relativity are observed.

1. Introduction Ifaphotonisconfinedinaboxwithmirrors,thenwehave a composite body at rest. In this case, as shown in [2], we 𝑔 Formulation of a successful quantum gravitational theory have to take into account a negative contribution to 𝑚𝑎 is considered to be one of the most important problems in from stress in the box walls to restore Einstein’s equation, 𝑔 2 physics and the major step towards the so-called “Theory 𝑚𝑎 =𝐸/𝑐. It is important that the latter equation is restored of Everything.” On the other hand, fundamentals of general only after averaging over time. A role of the classical virial relativity and quantum mechanics are so different that it is theorem in establishing the equivalence between averaged possible that these two theories will not be united in the feasi- over time active and passive gravitational masses and energy blefuture.Inthisdifficultsituation,itseemstobeimportant is discussed in detail in [3, 4]fordifferenttypesofclassical to suggest a combination of quantum mechanics and some composite bodies. In particular, for electrostatically bound nontrivial approximation of general relativity. In particular, two bodies, it is shown that gravitational field is coupled this is important in the case where such theory leads to to a combination 3𝐾+2𝑈,where𝐾 is kinetic and 𝑈 is meaningful physical results, which can be experimentally the Coulomb potential energies. Since the classical virial tested. theorem states that the following time average is equal to zero, A notion of gravitational mass of a composite body is ⟨2𝐾 + 𝑡𝑈⟩ =0,thenweconcludethataveragedovertime knowntobenontrivialingeneralrelativityandrelatedto active and passive gravitational masses are proportional to the the following paradoxes. If we consider a free photon with total amount of energy [3, 4], energy 𝐸 andapplytoittheso-calledTolman’sformulafor 𝑚𝑔 = active gravitational mass (see, e.g., [1]), we will obtain 𝑎 𝑔 ⟨3𝐾 + 2𝑈⟩ 𝐸 2 ⟨𝑚 ⟩ =𝑚 +𝑚 + 𝑡 = , (1) 2𝐸/𝑐 (i.e., two times bigger value than the expected one) [2]. 𝑎,𝑝 𝑡 1 2 𝑐2 𝑐2 2 Advances in High Energy Physics

2 where 𝑚1 and 𝑚2 are bare masses of the above considered from the value 𝐸/𝑐 . Our fourth result is that we suggest how bodies. the abovementioned inequivalence can be experimentally observed. In particular, we propose experimental detection of 2. Goal electromagnetic radiation, emitted by macroscopic ensemble of hydrogen atoms (in a real experiment—molecules), sup- The main goal of our paper is to study a quantum problem ported by and moving with constant velocity in the Earth’s about passive [5–7]andactivegravitationalmassesofa gravitational field, using small spacecraft or satellite. If such composite body. As the simplest example, we consider a experiment is done, to the best of our knowledge, it will hydrogen atom. We claim four main results in the paper. Our be the first direct experimental test of quantum effects in first result is that the equivalence between passive and active general relativity. We stress that so far only quantum effects gravitational masses and energy survives at a macroscopic in the Newtonian variant of gravity, where general relativity level for stationary quantum states. In the calculations of corrections are negligible, have been directly studied in the passive gravitational mass operator, we take into account famous COW [11]andILL[12] experiments. both nonrelativistic kinetic and Coulomb potential energies Most of the abovementioned results, related to passive and the so-called relativistic corrections (see, e.g., [8]) to an gravitational mass, have been recently published by us in [5– electron motion in a hydrogen atom, whereas in calcula- 7]. All results, related to active gravitational mass, and results, tions of active gravitational mass we take into account only related to the so-called relativistic corrections to passive gravitational mass, are new and, to the best of our knowledge, nonrelativistic kinetic and Coulomb potential energies. More have not been published. specifically, we show that the expectation values of passive and active gravitational masses of the atom are equivalent to its energy for stationary quantum states due to some 3. Equivalence of the Expectation Values of mathematical theorems. In the case of active gravitational Passive Gravitational Mass and Energy for mass, the corresponding theorem is known as the quantum Stationary States virial theorem (see, e.g., [9]),whereas,inthecaseofpas- sive gravitational mass, the corresponding theorem is more Let us use the standard weak field approximation to describe complicatedthanthatin[9]. In fact the latter is an extension spacetime outside the Earth (see, e.g., [13]). (We pay attention of the relativistic quantum virial theorem [10]forthecase to the fact that, to calculate the Hamiltonian in a linear with 2 of a particle with spin 1/2. We would like to draw attention respect to a small parameter 𝜙(𝑅)/𝑐 approximation, we do 2 2 to the fact that the abovementioned results are nontrivial. not need to keep the terms of the order of [𝜙(𝑅)/𝑐 ] in Indeed, below we define passive and active gravitational the metric (2), in contrast to the classical problem about 𝑚̂𝑔 𝑚̂𝑔 mass operators of an electron, 𝑝 and 𝑎 ,respectively,in perihelion precession of Mercury’s orbit [13].), the post-Newtonian approximation to general relativity. It is 2𝜙 2𝜙 important that these operators occur not to commute with 𝑑𝑠2 =−(1+ ) (𝑐𝑑𝑡)2 +(1− )(𝑑𝑥2 +𝑑𝑦2 +𝑑𝑧2), electron energy operator, taken in the absence of the field. 𝑐2 𝑐2 Therefore, from the first point of view, it seems that the 𝐺𝑀 equivalence between passive and active gravitational masses 𝜙=− , andenergyisbroken.Nevertheless,usingrathersophisticated 𝑅 mathematical tools, we show that the expectation values of (2) 𝑔 passive and active gravitational mass operators are ⟨𝑚̂𝑝⟩= 𝑔 2 where 𝐺 is the gravitational constant, 𝑐 is the velocity of light, ⟨𝑚̂ ⟩=𝑚𝑒 +𝐸𝑛/𝑐 for stationary quantum states in a 𝑎 𝑀 is the Earth’s mass, and 𝑅 is a distance from center of the hydrogen atom, where 𝑚𝑒 isthebareelectronmassand𝐸𝑛 Earth. Then, in the local proper spacetime coordinates: is the total electron energy of 𝑛th atomic energy level. Our second result is that the equivalence between elec- 𝜙 𝜙 𝑥󸀠 =(1− )𝑥,󸀠 𝑦 =(1− )𝑦, tron energy and its passive and active gravitational masses is 𝑐2 𝑐2 showntobebrokenforsuperpositionsofstationaryquantum (3) 𝜙 𝜙 states. More strictly speaking, we demonstrate that there exist 𝑧󸀠 =(1− )𝑧,󸀠 𝑡 =(1+ )𝑡, such quantum states where the expectation values of energy 𝑐2 𝑐2 are constant, whereas the expectation values of passive and active gravitational masses are oscillatory functions of time. the Schrodinger¨ equation for an electron motion in a hydro- Our third result is a breakdown of the equivalence between gen atom can be approximately written in the following passive gravitational mass and energy at a microscopic standard form: level. It is a consequence of the fact that passive electron 󸀠 󸀠 𝑚̂𝑔 𝜕Ψr ( ,𝑡 ) gravitational mass operator, 𝑝,doesnotcommutewithits 𝑖ℏ = 𝐻(̂ p̂󸀠, r󸀠)Ψ(r󸀠,𝑡󸀠), (4) energy operator, taken in the absence of the field. Therefore, 𝜕𝑡󸀠 an atom with a definite energy in the absence of gravitational ̂󸀠 󸀠 field, 𝐸, is not characterized by a definite passive gravitational where 𝐻(̂ p , r ) is the standard Hamiltonian. We stress that, mass in an external gravitational field. Passive gravitational in (4) and everywhere below, we disregard all tidal effects (i.e., mass is shown to be quantized and can significantly differ we do not differentiate gravitational potential with respect Advances in High Energy Physics 3

󸀠 to electron coordinates, r and r , corresponding to electron the so-called Eotv¨ os¨ method to measure gravitational mass positions in the center of mass coordinate systems). It is [13].). Note that, in (8), the first term corresponds to the bare possible to show that this means that we consider a hydrogen electron mass, 𝑚𝑒, and the second term corresponds to the atom as a point-like body and disregard all tidal terms in expected electron energy contribution to the gravitational electron Hamiltonian, which are usually very small and of the mass operator, whereas the third nontrivial term is the virial 2 −17 2 −26 order of (𝑟𝐵/𝑅0)|𝜙/𝑐 |∼10 |𝜙/𝑐 |∼10 in the Earth’s contribution to passive gravitational mass operator. It is gravitational field. (Here 𝑟𝐵 is the so-called Bohr’s radius and possible to make sure [5, 14]that(7)and(8)canbeobtained 𝑅0 is the Earth’s radius.) directly from the Dirac equation in a curved spacetime, corresponding to the weak centrosymmetric gravitational 3.1. Nonrelativistic Case. Letusfirstconsidernonrelativistic field (2) (see, e.g., (3.24) in [15]), if we disregard all tidal Schrodinger¨ equation for electron motion in a hydrogen terms). atom, where we take into account only kinetic and Coulomb Here, we discuss some important consequence of (7)and potential energies: (8). It is crucial that the operator (8)doesnotcommutewith

󸀠 󸀠 electron energy operator, taken in the absence of gravitational 𝜕Ψ (r ,𝑡 ) field. Therefore, it is not clear from the beginning that the 𝑖ℏ = 𝐻̂ (p̂󸀠, r󸀠)Ψ(r󸀠,𝑡󸀠), 𝜕𝑡󸀠 0 equivalence between electron passive gravitational mass and 2 (5) its energy exists. Toestablish the equivalence at a macroscopic p̂󸀠 𝑒2 level, we consider a macroscopic ensemble of hydrogen atoms 𝐻̂ (p̂󸀠, r󸀠)=𝑚𝑐2 + − , 0 𝑒 󸀠 with each of them being in a stationary quantum state 2𝑚𝑒 𝑟 with a definite energy 𝐸𝑛.Then,from(8), it follows that 󸀠 where 𝑒 is the electron charge, 𝑟 is a distance between elec- the expectation value of electron passive gravitational mass ̂󸀠 󸀠 tron and proton, and p = −𝑖ℏ𝜕/𝜕r is electron momentum operator per atom is operator in the local proper spacetime coordinates. Below, 2 2 𝐸 ⟨(2 (p̂ /2𝑚𝑒)) − (𝑒 /𝑟)⟩ we treat the weak gravitational field (2)asaperturbation ⟨𝑚̂𝑔⟩=𝑚 + 𝑛 + 𝑝 𝑒 𝑐2 𝑐2 in inertial coordinate system, corresponding to spacetime (9) coordinates (𝑥,𝑦,𝑧,𝑡) in (3)[3, 4],andcalculatethecorre- 𝐸 =𝑚 + 𝑛 , sponding Hamiltonian: 𝑒 𝑐2 2 2 ̂ 2 p̂ 𝑒 where the third term in (9) is zero in accordance with the 𝐻0 (p̂, r,𝜙)=𝑒 𝑚 𝑐 + − 2𝑚𝑒 𝑟 quantum virial theorem [9]. Therefore, we conclude that the (6) equivalence between passive gravitational mass and energy p̂2 𝑒2 𝜙 survives at a macroscopic level for stationary quantum states, +𝑚𝜙+(3 −2 ) . 𝑒 2 2𝑚𝑒 𝑟 𝑐 if we consider only pairings of nonrelativistic kinetic and Coulomb potential energies with an external gravitational From (6), it is clear that the Hamiltonian can be rewritten in field. Note that an important difference between our result (9) the following form: and the corresponding result in classical case [3, 4]isthatthe expectation value of passive gravitational mass corresponds 2 2 ̂ 2 p̂ 𝑒 𝑔 to averaging procedure over a macroscopic ensemble of 𝐻0 (p̂, r,𝜙)=𝑚𝑒𝑐 + − + 𝑚̂𝑒 𝜙, (7) 2𝑚𝑒 𝑟 hydrogen atoms, whereas in classical case we average over time. whereweintroducepassivegravitationalmassoperatorofan electron: 3.2. Relativistic Corrections. In this section, we study a more 2 2 general case, where the so-called relativistic corrections to an ((p̂ /2𝑚𝑒)−(𝑒 /𝑟)) 𝑚̂𝑔 =𝑚 + electron motion in a hydrogen atom are taken into account. 𝑝 𝑒 𝑐2 (8) As well-known [8], there exist three relativistic correction ((2 (p̂2/2𝑚 )) − (𝑒2/𝑟)) terms, which have different physical meanings. The total + 𝑒 , 2 Hamiltonianintheabsenceofgravitationalfieldcanbe 𝑐 written as which is proportional to its weight operator in the weak ̂ ̂ ̂ 𝐻 (p̂, r) = 𝐻0 (p̂, r) + 𝐻1 (p̂, r) , (10) gravitational field2 ( ). (Note that passive gravitational mass of a composite body, as a measurable quantity, has to be where represented in quantum mechanics by some Hermitian oper- Ŝ ⋅ L̂ ator. The most logical and straightforward way is to define 𝐻̂ (p̂, r) =𝛼p̂4 +𝛽𝛿3 (r) +𝛾 , (11) 1 𝑟3 the passive gravitational mass operator to be proportional to a weight operator in a weak gravitational field (i.e., the with the parameters 𝛼, 𝛽,and𝛾 being operator which couples with gravitational potential in a 1 𝜋𝑒2ℏ2 𝑒2 linear approximation in (6)). This extends definitions of 𝛼=− ,𝛽= ,𝛾= . 3 2 2 2 2 2 (12) gravitational mass in classical case [3, 4] and corresponds to 8𝑚𝑒 𝑐 2𝑚𝑒 𝑐 2𝑚𝑒 𝑐 4 Advances in High Energy Physics

3 (Here, 𝛿 (r) = 𝛿(𝑥)𝛿(𝑦)𝛿(𝑧) is a three-dimensional Dirac’s to the expectation value of passive gravitational mass and delta function.) Note that the first contribution in11 ( )iscalled energy, can be applied to stationary quantum states. Here, we the kinetic term, the second one is the so-called Darwin’s define the so-called virial operator [9]as term, and the third one is the spin-orbital interaction, 1 L̂ = −𝑖ℏ[r ×𝜕/𝜕r] 𝐺=̂ (pr̂ + rp̂) , where is electron angular momentum 2 (17) operator. In the presence of the weak gravitational field (2), the Schrodinger¨ equation for an electron motion in the and write the standard equation of motion for its expectation local proper spacetime coordinates (3) can be approximately value: written as 𝑑 𝑖 ⟨𝐺⟩̂ = ⟨[𝐻̂ (p̂, r) +𝐻 (p̂, r) , 𝐺]⟩̂ , (18) 𝑑𝑡 ℏ 0 1 𝜕Ψ (r󸀠,𝑡󸀠) 𝑖ℏ =[𝐻̂ (p̂󸀠, r󸀠)+𝐻̂ (p̂󸀠, r󸀠)] Ψ ( 󸀠, 󸀠). (13) [𝐴,̂ 𝐵]̂ 𝜕𝑡󸀠 0 1 r t where ,asusual,standsforacommutatoroftwo operators, 𝐴̂ and 𝐵̂. If we consider a stationary quantum state 󸀠 ̂ (Note that, as discussed above, we disregard everywhere all with a definite energy, 𝐸𝑛, then the derivative 𝑑⟨𝐺⟩/𝑑𝑡 in (18) tidal effects.) hastobezeroand,thus, Bymeansofthecoordinatestransformation(3), the ̂ ̂ ⟨[𝐻0 (p̂, r) +𝐻1 (p̂, r) , 𝐺]⟩ = 0, (19) corresponding Hamiltonian in inertial coordinate system (𝑥,𝑦,𝑧,𝑡) ̂ ̂ canbeexpressedas where the Hamiltonians 𝐻0(p̂, r) and 𝐻1(p̂, r) are defined by (5)and(11). By means of rather lengthy but straightforward 𝐻(̂ p̂, r,𝜙) calculations it is possible to show that 𝜙 4 ̂ ̂ [𝐻̂ (p̂, r) , 𝐺]̂ p̂2 𝑒2 [𝛼p̂ , 𝐺]̂ =[𝐻0 (p̂, r) + 𝐻1 (p̂, r)](1+ ) 0 4 𝑐2 (14) =2 − , =4𝛼p̂ , −𝑖ℏ 2𝑚𝑒 𝑟 −𝑖ℏ 2 2 ̂ ̂ p̂ 𝑒 4 3 S ⋅ L 𝜙 3 +(2 − +4𝛼p̂ +3𝛽𝛿 (r) +3𝛾 ) . [𝛽𝛿 (r) , 𝐺]̂ 1 Ŝ ⋅ L̂ Ŝ ⋅ L̂ 2𝑚 𝑟 𝑟3 𝑐2 =3𝛽𝛿3 (r) , [𝛾 , 𝐺]̂ = 3𝛾 , 𝑒 −𝑖ℏ −𝑖ℏ 𝑟3 𝑟3 For the Hamiltonian (14), passive gravitational mass operator (20) of an electron can be written in a more complicated form than wherewetakeintoaccountthefollowingequality:𝑥𝑖(𝑑𝛿(𝑥𝑖)/ (8): 𝑑𝑥𝑖)=−𝛿(𝑥𝑖). As directly follows from (19)and(20): 𝑔 𝑚̂ 2 2 𝑝 p̂ 𝑒 Ŝ ⋅ L̂ ⟨2 − +4𝛼p̂4 +3𝛽𝛿3 (r) +3𝛾 ⟩=0, 3 (21) 2𝑚𝑒 𝑟 𝑟 =𝑚𝑒 2 2 4 3 ̂ ̂ 3 and, therefore, (16)canberewritteninEinstein’sform: ((p̂ /2𝑚𝑒)−(𝑒 /𝑟) + 𝛼p̂ +𝛽𝛿 (r) +(𝛾((S ⋅ L)/𝑟 ))) + 𝐸󸀠 𝑐2 ⟨𝑚̂𝑔⟩=𝑚 + 𝑛 . (22) 𝑝 𝑒 𝑐2 (( 2 ( p̂2/2𝑚 )) − ( 𝑒 2/𝑟)+4𝛼p̂4 +3𝛽𝛿3(r)+(3𝛾((Ŝ ⋅ L̂)/𝑟3))) + 𝑒 . (Note that (21) extends the so-called relativistic quantum 𝑐2 virial theorem [10], derived for spinless particles, to the case (15) of particles with spin 1/2.) It is important that (22) directly establishes the equiv- Let us consider again a macroscopic ensemble of hydrogen alence between the expectation value of electron passive atoms with each of them being in a stationary quantum state gravitational mass and its energy in a hydrogen atom, 𝐸󸀠 𝐸󸀠 with a definite energy 𝑛,where 𝑛 takes into account the including the relativistic corrections, for the Eotv¨ os’¨ type of relativistic corrections (11)toelectronenergy.Inthiscase,the experiments [13]. We speculate that such equivalence exists expectation value of the electron mass operator (15)peratom also for more complicated quantum systems, including many- canbewrittenas body systems with arbitrary interactions of particles. These ⟨𝑚̂𝑔⟩ reveal and establish the physical meaning of a coupling of 𝑝 a macroscopic quantum test body with a weak gravitational 𝐸󸀠 field. =𝑚 + 𝑛 𝑒 𝑐2 4. Inequivalence between Passive Gravitational ⟨( 2 ( p̂2/2𝑚 )) − ( 𝑒 2/𝑟)+4𝛼p̂4 +3𝛽𝛿3(r)+(3𝛾((Ŝ ⋅ L̂)/𝑟3))⟩ + 𝑒 . Mass and Energy for Superpositions of 𝑐2 Stationary States (16) In the previous section, we have shown that the expectations Below, we show that the expectation value of the third values of passive gravitational mass and energy are equiv- term in (16) is zero and, therefore, Einstein’s equation, related alent to each other in stationary quantum states. Here, we Advances in High Energy Physics 5 investigate if such equivalence survives for superpositions of 5. Breakdown of the Equivalence between stationary quantum states. For this purpose, we consider the Passive Gravitational Mass and Energy at 𝑠 simplest superposition of the ground and first excited -wave a Microscopic Level states in a hydrogen atom, where electron wave function has the following form: In this section, we study how noncommutation of passive gravitational mass operators (8)and(15) and the correspond- 1 ing energy operators, taken in the absence of gravitational Ψ1,2 (𝑟,) 𝑡 = [Ψ1 (𝑟) exp (−𝑖𝐸1𝑡) +2 Ψ (𝑟) exp (−𝑖𝐸2𝑡)] . √2 field, results in a breakdown of the equivalence between (23) passive gravitational mass and energy. This conclusion does not depend on the relativistic corrections (11); therefore, It is important that wave function (23) is characterized by for certainty, below we consider passive gravitational mass the time-independent expectation value of energy, ⟨𝐸⟩ = operator in the form of (8). The physical meaning of the (𝐸1 +𝐸2)/2. Nevertheless, the expectation value of passive abovementionedbreakdownisthatanelectroninitsground 𝑚 𝑐2 +𝐸 gravitational mass operator (8)occurstobethefollowing state with a definite energy, 𝑒 1, is not characterized by time-dependent oscillatory function: a definite passive gravitational mass and, thus, measurements of the mass can give values, which are not related to electron 𝑔 2 𝐸 +𝐸 𝑉 (𝐸 −𝐸 )𝑡 energy by Einstein’s equation, 𝑚𝑝 =𝑚̸𝑒 +𝐸1/𝑐 .Asweshow ⟨𝑚̂𝑔⟩ =𝑚 + 1 2 + 1,2 [ 1 2 ] , 𝑝 𝑒 2 2 cos (24) below, the passive electron gravitational mass values in a 2𝑐 𝑐 ℏ 𝑔 2 hydrogen atom are quantized: 𝑚𝑝 =𝑚𝑒 +𝐸𝑛/𝑐 ,where𝐸𝑛 is energy corresponding to 𝑛th energy level. where 𝑉1,2 is a matrix element of the virial operator:

2 2 5.1. First Thought Experiment. Here, we describe the first ∗ p̂ 𝑒 3 𝑉1,2 = ∫ Ψ1 (𝑟) (2 − )Ψ2 (𝑟) 𝑑 r. (25) thought experiment, illustrating inequivalence of energy and 2𝑚𝑒 𝑟 passive gravitational mass at a microscopic level. Suppose that we create quantum state of a hydrogen atom with a Note that the above obtained result holds both for non- definite energy in the absence of a gravitational field and relativistic passive gravitational mass operator (8)andfor then adiabatically switch on the gravitational field (2). More the operator (15), which takes into account the so-called specifically, at 𝑡→−∞(i.e., in the absence of gravitational relativistic corrections (11). Therefore, we make a conclusion field),ahydrogenatomisinitsgroundstatewithwave that the oscillations of passive gravitation mass (24)directly function, demonstrate inequivalence of passive gravitational mass and 2 energy at a macroscopic level at any given moment of time. 𝑖𝑚𝑒𝑐 𝑡 𝑖𝐸1𝑡 Ψ1 (𝑟,) 𝑡 =Ψ1 (𝑟) exp (− − ) , (27) Letusdiscussarelativemagnitudeoftheoscillations(24). ℏ ℏ By using actual numbers for a hydrogen atom, it is possible to obtain the following numerical value of the matrix element whereas, in the vicinity of 𝑡=0(i.e.,inthepresenceofthe 2 of the virial operator: 𝑉1,2 = 5.7 eV. Since 𝑚𝑒𝑐 ≃ 0.5 MeV gravitational field2 ( )), it is characterized by the following and 𝑚𝑝 ≃ 1800𝑚𝑒,wecancometotheconclusionthat wave function: the oscillations (24) are weak but not negligible: 𝛿𝑚𝑒/𝑚𝑒 ∼ ∞ 2 10−5 𝛿𝑚 /𝑚 ∼10−8 𝑖𝑚𝑒𝑐 𝑡 𝑖𝐸𝑛𝑡 and 𝑒 𝑝 .Theycorrespondtothefollowing Ψ (𝑟,) 𝑡 = ∑𝑎𝑛 (𝑡) Ψ𝑛 (𝑟) exp (− − ). (28) 16 𝑛=1 ℏ ℏ angular and linear frequencies: 𝜔1,2 ≃ 1.6 × 10 Hz and ] ≃ 15 2.5×10 Hz, respectively. We hope that the abovementioned (Here, Ψ𝑛(𝑟) is a normalized electron wave function in a oscillations of passive gravitational mass are experimentally hydrogen atom in the absence of gravitational field, corre- measured, despite the fact that the quantum state (23)decays sponding to energy 𝐸𝑛 (due to symmetry of our problem, we with time. need to keep in (28) only wave functions, corresponding to On the other hand, if we average the oscillations (24) 𝑛𝑆 quantum states).) over time, we obtain the modified equivalence principle As follows from (7)and(8), adiabatically switched between the averaged over time expectation value of passive on gravitational field corresponds to the following time- gravitationalmassandtheexpectationvalueofenergyinthe dependent small perturbation: following form: 𝑈̂ (r,𝑅,𝑡) (𝐸 +𝐸 ) 𝐸 ⟨⟨𝑚̂𝑔⟩⟩ =𝑚 + 1 2 =⟨ ⟩. (26) 𝑝 𝑡 𝑒 2𝑐2 𝑐2 ((p̂2/2𝑚 )−(𝑒2/𝑟)) =𝜙(𝑅) [𝑚 + 𝑒 𝑒 𝑐2 We pay attention that the physical meaning of averaging (29) procedure in (26) is completely different from that for 2 2 ((2 (p̂ /2𝑚𝑒)) − (𝑒 /𝑟)) classical time averaging procedure (1) and does not have the + ] exp (𝜆𝑡) , corresponding classical analogs. 𝑐2 6 Advances in High Energy Physics where 𝜆→0.(Notethatourchoiceofadiabaticallyswitched the Schrodinger¨ equation, corresponding to the Hamiltonian on gravitational potential (29)with𝜆→0allows to avoid (7)and(8),canbewrittenas extra velocity-dependent terms (see, e.g., the Largangian 𝜙 3/2 ∞ 𝜙 in [3]).) The standard calculations by means of the time- Ψ (𝑟,) 𝑡 =(1− ) ∑𝑎 Ψ [(1 − )𝑟] 2 𝑛 𝑛 2 dependent quantum mechanical perturbation theory [8]give 𝑐 𝑛=1 𝑐 the following results: 2 2 −𝑖𝑚𝑒𝑐 (1 + 𝜙/𝑐 )𝑡 𝑖𝜙 (𝑅) 𝑚 𝑐2𝑡+𝑖𝜙(𝑅) 𝐸 𝑡 × exp [ ] (35) 𝑎 (𝑡) = [− 𝑒 1 ], ℏ 1 exp 𝑐2ℏ (30) 2 −𝑖𝐸𝑛 (1 + 𝜙/𝑐 )𝑡 𝜙 (𝑅) 𝑉 × exp [ ]. 𝑎 (0) =− 𝑛,1 ,𝑛=1,̸ ℏ 𝑛 2 (31) 𝑐 𝐸𝑛 −𝐸1 Wepayattentiontothefactthatwavefunction(35)isaseries where of eigenfunctions of passive gravitational mass operator (8), p̂2 𝑒2 if we take into account only linear terms with respect to small 𝑉 = ∫ Ψ∗ (𝑟) (2 − )Ψ (𝑟) 𝑑3r. 2 2 𝑛,1 𝑛 1 (32) parameter 𝜙/𝑐 .(Here,factor1−𝜙/𝑐 is due to a curvature of 2𝑚𝑒 𝑟 2 space, whereas the term 𝐸𝑛(1 + 𝜙/𝑐 ) represents the famous (Note that the perturbation (29) is characterized by the red shift in gravitational field. We also pay attention to the fact following selection rule. Electron from 1𝑆 ground state of a that the wave function (35) contains a normalization factor (1 − 𝜙/𝑐2)3/2 hydrogen atom can be excited only into 𝑛𝑆 excited state.) .) Let us discuss (30)–(32). It is important that (30)corre- In accordance with the basic principles of the quantum 𝑡>0 sponds to the well-known red shift of atomic ground state mechanics, probability that, at , an electron occupies 𝑚 𝑐2(1 + 𝜙/𝑐2)+𝐸 (1 + 𝜙/𝑐2) energy 𝐸1 in the gravitational field2 ( ). On the other hand, excited state with energy 𝑒 𝑛 is (31) demonstrates that there is a finite probability, 󵄨 󵄨2 𝑃𝑛 = 󵄨𝑎𝑛󵄨 , 2 2 󵄨 󵄨2 𝜙 (𝑅) 𝑉𝑛,1 𝑃 = 󵄨𝑎 (0)󵄨 =[ ] ( ) ,𝑛=1,̸ ∗ 𝜙 3 𝑛 󵄨 𝑛 󵄨 2 (33) 𝑎 = ∫ Ψ (𝑟) Ψ [(1 − )𝑟]𝑑 r 𝑐 𝐸𝑛 −𝐸1 𝑛 1 𝑛 𝑐2 (36) 𝜙 that, at 𝑡=0,electronoccupies𝑛th energy level. In fact, ∗ 󸀠 3 =−( ) ∫ Ψ1 (𝑟) 𝑟Ψ𝑛 (𝑟) 𝑑 r. this means that measurements of gravitational mass (8)in 𝑐2 a quantum state with definite energy27 ( )givethefollowing 𝑎 quantized values: (Note that it is possible to demonstrate that for 1 in (36) a linear term with respect to gravitational potential, 𝜙,is 𝐸 𝑚𝑔 (𝑛) =𝑚 + 𝑛 , zero, which is a consequence of the quantum virial theorem.) 𝑝 𝑒 𝑐2 (34) Taking into account that the Hamiltonian is a Hermitian operator,itispossibletoshowthat,for𝑛 =1̸, with the probabilities (33)for𝑛 =1̸. Note that although the 𝑉 probabilities (33) are quadratic with respect to gravitational ∗ 󸀠 3 𝑛,1 ∫ Ψ1 (𝑟) 𝑟Ψ𝑛 (𝑟) 𝑑 r = , (37) potential and, thus, small, the corresponding changes of (𝐸𝑛 −𝐸1) 2 gravitational mass (34) are large and of the order of 𝛼 𝑚𝑒, where 𝛼 isthefinestructureconstant.Itisimportantthat where 𝑉𝑛,1 is a matrix element of the virial operator given by the excited levels of a hydrogen atom spontaneously decay; (32). therefore, one can detect the above discussed quantization Let us discuss (36)and(37). We stress that they directly law of gravitational mass (34) by measuring electromagnetic demonstrate that there is a finite probability, radiation, emitted by a macroscopic ensemble of hydrogen 2 2 󵄨 󵄨2 𝜙 (𝑅) 𝑉𝑛,1 atoms. 𝑃 = 󵄨𝑎 󵄨 =[ ] ( ) ,𝑛=1,̸ 𝑛 󵄨 𝑛󵄨 2 (38) 𝑐 𝐸𝑛 −𝐸1 5.2. Second Thought Experiment. Let us discuss the second thought experiment, which directly demonstrates inequiv- that, at 𝑡>0, an electron occupies 𝑛th (𝑛 =1̸)energylevel, 𝑚𝑔 =𝑚 + alence between energy and passive gravitational mass at a which breaks the expected Einstein’s equation, 𝑝 𝑒 2 microscopic level. Suppose that, at 𝑡=0,wecreateaground 𝐸1/𝑐 . In fact, this means that quantum measurement of state wave function of a hydrogen atom, corresponding to passive gravitational mass (i.e., weight in the gravitational the absence of gravitational field (see27 ( )). Then, in the field (2)) in a quantum state with a definite energy27 ( ) presence of the gravitational field2 ( ),thewavefunction(27) gives the quantized values (see (34)), corresponding to the is not anymore a ground state of the Hamiltonian (7)and(8), probabilities (33)and(38), which are equal. (Note that, as where we treat gravitational field as a small perturbation in it follows from quantum mechanics, we have to calculate inertial coordinate system [3, 4, 15]. It is important that, for an wave function (35) in a linear approximation with respect 2 inertial observer, in accordance with (3), a general solution of to small parameter 𝜙/𝑐 to obtain probabilities (38), which Advances in High Energy Physics 7

2 2 are proportional to (𝜙/𝑐 ) . A simple analysis shows that 6.2. Photon Emission and Mass Quantization. Here, we de- 2 2 inclusion in (35) terms of the order of (𝜙/𝑐 ) would change scribe the suggested realistic experiment in more detail. We 2 𝑡=0 electron passive gravitational mass of the order of (𝜙/𝑐 )𝑚𝑒 ∼ considerahydrogenatomtobeinitsgroundstate,at , −9 𝑅 10 𝑚𝑒, which is much smaller than the typical distance and located at distance fromacenteroftheEarth,wherethe 𝑔 2 gravitational potential is small. The wave function of a ground between the quantized values in (34), 𝛿𝑚𝑝 ∼𝛼𝑚𝑒 ∼ −4 state, corresponding to the Hamiltonian (7)and(8), can be 10 𝑚𝑒.)Wepayattentiontothefactthatsmallvalues −18 written as of probabilities (33), (38), 𝑃𝑛 ∼10 ,donotcontradict the existing Eotvostypemeasurements[¨ 13], which have 𝜙 3/2 𝜙 Ψ̃ (𝑟,) 𝑡 =(1− ) Ψ [(1 − )𝑟] confirmed the equivalence principle with the accuracy of 1 2 1 2 −12 −13 𝑐 𝑐 the order of 𝛿𝑚/𝑚 ∼10 –10 .Aswementionedin 2 2 the previous section, for our case, it is crucial that the −𝑖𝑚𝑒𝑐 (1 + 𝜙/𝑐 )𝑡 excited levels of a hydrogen atom spontaneously decay with × [ ] exp ℏ (41) time; therefore, one can detect the quantization law (34)by measuring electromagnetic radiation, emitted by a macro- −𝑖𝐸 (1 + 𝜙/𝑐2)𝑡 scopic ensemble of hydrogen atoms. The abovementioned 1 × exp [ ], optical method is much more sensitive than the Eotvos¨ type ℏ measurements and we, therefore, believe that it will allow us to detect the breakdown of the equivalence between passive where 𝜙=𝜙(𝑅). At arbitrary moment of time, 𝑡>0,elec- gravitational mass and energy, revealed in the paper. tron wave function and time-dependent perturbation for the Hamiltonian (7)and(8) in inertial coordinate system, related 6. Suggested Realistic Experiment to the spacecraft (satellite), can be expressed as

6.1. Hamiltonian. Let us consider a realistic experiment, 𝜙 3/2 ∞ 𝜙 which can be done on the Earth’s orbit to detect photons, Ψ̃ (𝑟,) 𝑡 =(1− ) ∑𝑎̃ (𝑡) Ψ [(1 − )𝑟] 𝑐2 𝑛 𝑛 𝑐2 emitted by a macroscopic ensemble of hydrogen atoms with 𝑛=1 the following frequencies: 2 2 −𝑖𝑚𝑒𝑐 (1 + 𝜙/𝑐 )𝑡 (𝐸𝑛 −𝐸1) × exp [ ] (42) 𝜔 = . (39) ℏ 𝑛,1 ℏ As discussed above, these photons are the consequences of −𝑖𝐸 (1 + 𝜙/𝑐2)𝑡 × [ 𝑛 ], the quantization rule (34), breaking Einstein’s equation for exp ℏ energy and passive gravitational mass. In the experiment we have to use a macroscopic ensemble of hydrogen atoms to 𝜙 (𝑅−𝑢𝑡) −𝜙(𝑅) p̂2 𝑒2 make the number of the emitted photons to be large. More 𝑈̂ (r,𝑅,𝑡) = (3 −2 ) . 𝑐2 2𝑚 𝑟 (43) specifically, a tank of a pressurized hydrogen is located in 𝑒 small spacecraft or satellite and moved from a distant place, where gravitational potential is small, |𝜙(𝑅)| ≪ |𝜙(𝑅0)|, Application of the time-dependent quantum mechanical with constant velocity, 𝑢≪𝛼𝑐, towards the Earth. Note perturbation theory [9] gives the following solutions for 𝑎̃ (𝑡) that the latter inequality allows us to disregard additional functions 𝑛 in (42): velocity-dependent corrections to the Hamiltonian (7)and 𝑉 (8), which can be derived from the Lagrangian of [3]. It is 𝑎̃ (𝑡) =− 𝑛,1 𝑛 2 also important that each hydrogen atom is at rest with respect ℏ𝜔𝑛,1𝑐 to the spacecraft (satellite), which means that gravitational force is compensated by some forces of nongravitational ×{[𝜙(𝑅−𝑢𝑡) −𝜙(𝑅)] exp (𝑖𝜔𝑛,1𝑡) nature. Note that this changes a little the allowed frequencies (39). Nevertheless, the latter effect is out of our current 𝑡 𝑑𝜙 (𝑅+𝑢𝑡) consideration, since it is possible to show [14]thatthechanges 𝑢 + ∫ 𝑑[exp (𝑖𝜔𝑛,1𝑡)]} , of the frequencies (39) are less than the existing accuracies of 𝑖𝜔𝑛,1 0 𝑑𝑅 theirmeasurements.Otherwords,eachhydrogenatomdoes 𝑛 =1,̸ not feel gravitational acceleration, g, but rather feels time- dependent gravitational potential, 𝜙(𝑅 − 𝑢𝑡). Therefore, each (44) hydrogen atom is affected by the following time-dependent Hamiltonian: where 𝑉𝑛,1 and 𝜔𝑛,1 are given by (32)and(39). It is important p̂2 𝑒2 𝜙 (𝑅−𝑢𝑡) −𝜙(𝑅) p̂2 𝑒2 that under the suggested experiment the following inequality 𝐻̂ = − + [𝑚 +3 −2 ] . 2 𝑒 is obviously fulfilled: 2𝑚𝑒 𝑟 𝑐 2𝑚𝑒 𝑟 (40) 𝑅0 13 𝑢≪𝜔𝑛,1𝑅∼𝛼𝑐( )∼10 𝑐; (45) (For more rigorous derivation of (40), see [6].) 𝑟𝐵 8 Advances in High Energy Physics therefore, we can disregard the second term in the amplitude 7. Active Gravitational Mass in (44): Classical Physics

𝑉𝑛,1 Here, we introduce active gravitational mass for a classical 𝑎̃𝑛 (𝑡) =− [𝜙 (𝑅−𝑢𝑡) −𝜙(𝑅)] exp (𝑖𝜔𝑛,1𝑡) , ℏ𝜔 𝑐2 model of a hydrogen atom. Suppose that we have a heavy 𝑛,1 (46) positively charged particle (i.e., proton) with bare mass 𝑚𝑝 𝑛 =1.̸ and light negatively charged particle (i.e., electron) with bare mass 𝑚𝑒,where𝑚𝑝 ≫𝑚𝑒. At large distances, 𝑅≫𝑟𝐵,from Since |𝜙(𝑅)| ≪ |𝜙(𝑅 −𝑢𝑡)|, we can write probabilities, the atom, gravitational potential in the first approximation is corresponding to amplitudes of (46), in the following way: 𝑚𝑝 +𝑚𝑒 𝜙 (𝑅) =−𝐺 , (50) 2 𝑅 𝑉 2 𝜙(𝑅−𝑢𝑡)2 𝑉 2 𝜙(𝑅󸀠) 𝑃̃ (𝑡) =( 𝑛,1 ) =( 𝑛,1 ) [ ] , 𝑛 4 2 where we do not take into account kinetic and Coulomb ℏ𝜔𝑛,1 𝑐 𝐸𝑛 −𝐸1 𝑐 potential energies contributions. Since 𝑚𝑝 ≫𝑚𝑒,we (47) below disregard kinetic energy of proton and consider it as a center of mass of the atom. The next step is to define 𝑅󸀠 =𝑅−𝑢𝑡 where . It is important that the probabilities how kinetic and Coulomb potential energies of electron 𝜙󸀠 =𝜙(𝑅󸀠) (47) depend only on gravitational potential, ,in contributetotheelectronactivegravitationalmass.Tobe the final position of a spacecraft (satellite). Moreover, they more specific, we define active gravitational mass of the atom coincide with the probabilities, obtained in both thought from gravitational potential acting on a small test body at experiments (see (33)and(38)).Thisallowsustoclarifytheir rest at distances much high than the “size” of the atom, 𝑟𝐵. physical meaning. Indeed, since the probabilities (47), (33), For simplicity, we prescribe potential and kinetic energies and (38)areequal,wecanconcludethatallphotons,emitted to electron and, therefore, consider corrections to electron by a macroscopic ensemble of hydrogen atoms during the gravitational mass. It is possible to show from general theory suggested realistic experiment, correspond to the breakdown of a weak gravitational field [1, 13]thatgravitationalpotential of Einstein’s equation for passive gravitation mass due to in our case can be written as quantization of the mass (34). As we discussed above, the 𝑚 +𝑚 Δ𝑇𝑘𝑖𝑛 (𝑡, r) +Δ𝑇𝑝𝑜𝑡 (𝑡, r) excited levels spontaneously decay with time and, therefore, 𝜙 (𝑅,) 𝑡 =−𝐺 𝑝 𝑒 −𝐺∫ 𝛼𝛼 𝛼𝛼 𝑑3r, it is possible to observe the quantization law (34)indirectly 𝑅 𝑐2𝑅 by measuring electromagnetic radiation from a macroscopic (51) ensemble of the atoms. In this case, (47)givesprobabilities 𝑘𝑖𝑛 𝑝𝑜𝑡 that a hydrogen atom emits a photon with frequencies (39) where Δ𝑇𝛼𝛽 (𝑡, r) and Δ𝑇𝛼𝛽 (𝑡, r) are changes of stress-energy 𝑡 during the time interval .(Wenotethatdipolematrix tensor component 𝑇𝛼𝛽(𝑡, r) due to kinetic and Coulomb elements for 𝑛𝑆 → 1𝑆 quantum transitions are zero. Nev- potential energies, respectively. (Note that in (51)andevery- ertheless, the corresponding photons can be emitted due to where below we disregard the so-called retardation effects.) 2 quadrupole effects.) Therefore, in the second approximation in 1/𝑐 ,activeelec- Let us estimate the probabilities (47): tron gravitational mass can be written as

2 2 󸀠 2 𝑔 1 𝑘𝑖𝑛 𝑝𝑜𝑡 3 𝑉 𝜙 (𝑅 ) 𝑉 𝑚 =𝑚𝑒 + ∫ [Δ𝑇 (𝑡, r) +Δ𝑇 (𝑡, r)]𝑑 r. (52) 𝑃̃ = ( 𝑛,1 ) ≃ 0.49 × 10−18( 𝑛,1 ) , 𝑎 𝑐2 𝛼𝛼 𝛼𝛼 𝑛 4 𝐸𝑛 −𝐸1 𝑐 𝐸𝑛 −𝐸1 (48) Let us write the standard expression for stress-energy tensor of a moving point particle without electrical charge where, in (48), we use the following numerical values of the [1, 13]: 24 Earth’s mass, 𝑀≃6×10 kg, and its radius, 𝑅0 ≃ 6.36 × 𝛼 𝛽 6 𝛼𝛽 𝑚V (𝑡) V (𝑡) 3 10 m. It is important that although the probabilities (47) 𝑇 (r,𝑡) = 𝛿 [r − rp (𝑡)], (53) √1−V2/𝑐2 and (48)aresmall,thenumberofphotons,𝑁,emittedby macroscopic ensemble of the atoms, can be large since the 𝛼 3 2 2 where V is a four-velocity, 𝛿 [⋅⋅⋅] is the three-dimensional factor 𝑉 /(𝐸𝑛 −𝐸1) is of the order of unity. For instance, for 𝑛,1 Dirac’s delta function, and rp(𝑡) is a trajectory of the particle 𝑁 1000 moles of hydrogen atoms, is estimated as inthree-dimensionalspace.Itiseasytoshowbymeansof(53) V ≪𝑐 2 that at low enough velocity, , 8 𝑉𝑛,1 𝑁𝑛,1 = 2.95 × 10 ( ) , 2 𝐸 −𝐸 𝑘𝑖𝑛 𝑚V 𝑛 1 (49) Δ𝑇 =3 . (54) 𝛼𝛼 2 8 𝑁2,1 = 0.9 × 10 , The standard expression for stress-energy tensor of electromagnetic field1 [ ]canbewrittenas which can be experimentally detected, where 𝑁𝑛,1 stands for a number of photons, emitted with frequency 𝜔𝑛,1 =(𝐸𝑛−𝐸1)/ℏ 1 1 𝑇𝜇] = [𝐹𝜇𝛼𝐹] − 𝜂𝜇]𝐹 𝐹𝛼𝛽], (see (39)). 𝑒𝑚 4𝜋 𝛼 4 𝛼𝛽 (55) Advances in High Energy Physics 9

𝛼𝛽 where 𝐹 is the so-called electromagnetic field tensor and which can be easily quantized: 𝜂𝛼𝛽 is metric of the Minkowski spacetime. This expression can be significantly simplified in our case, where only electrical ((p̂2/2𝑚 )−(𝑒2/𝑟)) 𝑚̂𝑔 =𝑚 + 𝑒 field is present. As a result, we obtain the following formula 𝑎 𝑒 𝑐2 forchangeofthestress-energytensorinthepresenceofthe (61) Coulomb potential energy: ((2 (p̂2/2𝑚 )) − (𝑒2/𝑟)) + 𝑒 . 𝑐2 𝑒2 Δ𝑇𝑝𝑜𝑡 =−2 . (56) 𝛼𝛼 𝑟 Therefore, in the framework of semiclassical theory of gravity, the expectation value of active gravitational mass, corresponding to macroscopic ensemble of the atoms with each Therefore, the total electron active gravitational mass can be of them being in its ground state, is equal to written in the same way as the electron passive mass: ⟨(p̂2/2𝑚 )−(𝑒2/𝑟)⟩ 2 2 𝑔 𝑒 ((𝑚 k /2) − (𝑒 /𝑟)) ⟨𝑚̂ ⟩=𝑚𝑒 + 𝑚𝑔 =𝑚 + 𝑒 𝑎 𝑐2 𝑎 𝑒 𝑐2 2 2 (57) ⟨(2 (p̂ /2𝑚𝑒)) − (𝑒 /𝑟)⟩ ((2 (𝑚 k2/2)) − (𝑒2/𝑟)) + (62) + 𝑒 , 𝑐2 𝑐2 𝐸 =𝑚 + 1 . 𝑒 𝑐2 wherethelasttermisthevirialone.Sincethevirialterm changes with time, we come to the conclusion that active Thus, we conclude that the expectation values of active gravitational mass of a classical body changes with time. gravitational mass and energy are equivalent for stationary Nevertheless we can introduce averaged over time active quantum states. gravitational mass, which occurs to be equivalent to energy [3, 4], since the averaged over time virial term is zero: 9. Inequivalence between Active Gravitational 2 2 Mass and Energy for Superpositions of ⟨((p̂ /2𝑚𝑒)) − (𝑒 /𝑟)⟩ ⟨𝑚𝑔⟩=𝑚 + 𝑡 𝑎 𝑒 𝑐2 Stationary States 2 2 In the previous section we have established the equivalence ⟨(2 (𝑚𝑒k /2)) − (𝑒 /𝑟)⟩ + 𝑡 (58) for the expectation values of active gravitational mass and 2 𝑐 energy for stationary quantum states. Below, we study if 𝐸 such equivalence survives for superpositions of stationary =𝑚 + . 𝑒 𝑐2 quantum states. As in Section 4, we consider the simplest superpositionofthegroundandfirstexciteds-wavestatesina hydrogen atom, where electron wave function can be written 8. Equivalence of the Expectation Values of as Active Gravitational Mass and Energy for 1 Ψ1,2 (𝑟,) 𝑡 = [Ψ1 (𝑟) exp (−𝑖𝐸1𝑡) +Ψ2 (𝑟) exp (−𝑖𝐸2𝑡)] . Stationary States √2 Here, we use the so-called semiclassical approach to the gen- (63) eral relativity (see, e.g., [16, 17]), where Einstein’s gravitational equationcanbewrittenas As we discussed this before, the expectation value of energy for the wave function (63) does not depend on time. Nev- 1 8𝜋𝐺 ertheless, the expectation value of active gravitational mass 𝑅 − 𝑅𝑔 = ⟨𝑇̂ ⟩, 𝜇] 2 𝜇] 𝑐2 𝜇] (59) operator (61) oscillates with time and has the following form: 𝐸 +𝐸 𝑉 (𝐸 −𝐸 )𝑡 ⟨𝑚̂𝑔⟩=𝑚 + 1 2 + 1,2 [ 1 2 ], where the last term represents the expectation value of 𝑎 𝑒 2𝑐2 𝑐2 cos ℏ (64) quantum stress-energy operator of the matter. In our case, expression for active gravitational mass (57)canberepre- which coincides with (24), describing time-dependent oscil- sented as the following Hamiltonian: lations of passive gravitational mass, where matrix element of the virial operator, 𝑉1,2,isgivenby(25). As we discussed 2 2 ((p /2𝑚𝑒)−(𝑒 /𝑟)) before such oscillations are of a pure quantum origin and 𝑚𝑔 =𝑚 + 𝑎 𝑒 𝑐2 do not have classical analogs. They directly demonstrate (60) inequivalence of active gravitational mass and energy at 2 2 ((2 (p /2𝑚𝑒)) − (𝑒 /𝑟)) a macroscopic level. Nevertheless, in the same way as in + , 𝑐2 Section 4, we can introduce modified equivalence principle 10 Advances in High Energy Physics

for the expectation values of active gravitational mass and [15] E.Fischbach,B.S.Freeman,andW.-K.Cheng,“General-relativ- energy by means of averaging of (64)overtime: istic effects in hydrogenic systems,” Physical Review D,vol.23, no.10,pp.2157–2180,1981. 𝑔 (𝐸1 +𝐸2) 𝐸 [16] N. D. Birrell and P.C. W.Davis, QuantumFieldsinCurvedSpace, ⟨⟨𝑚̂ ⟩⟩ =𝑚 + =⟨ ⟩. (65) 𝑎 𝑡 𝑒 2𝑐2 𝑐2 Cambridge University Press, Cambridge, UK, 1982. [17] D. N. Page and C. D. Geilker, “Indirect evidence for quantum As we stressed in Section 4, such averaging procedure is gravity,” Physical Review Letters,vol.47,no.14,pp.979–982, principally different from that in1 ( )and(58) and does not 1981. have classical analogs.

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments The authors are thankful to N. N. Bagmet (Lebed), V.A. Belin- ski, Steven Carlip, Douglas Singleton, and V. E. Zakharov for useful discussions. This work was supported by the NSF under Grant DMR-1104512.

References

[1] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Butterworth-Heineman, Amsterdam, The Netherlands, 4th edition, 2003. [2]C.W.MisnerandP.Putnam,“Activegravitationalmass,”Phys- ical Review,vol.116,no.4,pp.1045–1046,1959. [3] K. Nordtvedt, “Post-Newtonian gravity: its theory–experiment interface,” Classical and Quantum Gravity,vol.11,no.6,article A119, 1994. [4] S. Carlip, “Kinetic energy and the equivalence principle,” Amer- ican Journal of Physics, vol. 66, no. 5, pp. 409–413, 1998. [5]A.G.Lebed,“Breakdownoftheequivalencebetweenpassive gravitational mass and energy for a quantum body,” in Proceed- ings of the Marcel Grossmann Meeting-13, Sweden, Stockholm, July 2012. [6]A.G.Lebed,“Isgravitationalmassofacompositequantum body equivalent to its energy?” Central European Journal of Physics, vol. 11, no. 8, pp. 969–976, 2013. [7] A. G. Lebed, “Breakdown of the equivalence between gravitational mass and energy for a composite quantum body,” Journal of Physics: Conference Series.Inpress. [8] F. Schwabl, Advanced Quantum Mechanics, Springer, Berlin, Germany, 3rd edition, 2005. [9] D. Park, Introduction to the Quantum Theory,Dover,NewYork, NY, USA, 3rd edition, 2005. [10] W. Lucha and F. F. Schoberl, “Relativistic virial theorem,” Physical Review Letters,vol.64,no.23,pp.2733–2735,1990. [11] R. Colella, A. W.Overhauser, and S. A. Werner, “Observation of gravitationally induced quantum interference,” Physical Review Letters,vol.34,no.23,pp.1472–1474,1975. [ 1 2 ] V. V. Ne s v i z h e v s k y, H . G . B orner,¨ A. K. Petukhov et al., “Quan- tum states of neutrons in the Earth’s gravitational field,” Nature, vol.415,no.6869,pp.297–299,2002. [13] C.W.Misner,K.S.Thorne,andJ.A.Wheeler,Gravitation, W.H. Freeman and Co., New York, NY, USA, 1973. [14] A. G. Lebed, in a preparation. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2014, Article ID 597384, 16 pages http://dx.doi.org/10.1155/2014/597384

Research Article Quantum-Spacetime Scenarios and Soft Spectral Lags of the Remarkable GRB130427A

Giovanni Amelino-Camelia,1 Fabrizio Fiore,2 Dafne Guetta,2,3 and Simonetta Puccetti2,4

1 DipartimentodiFisica,SapienzaUniversitadiRomaandINFN,Sez.Roma1,P.leA.Moro2,00185Roma,Italy` 2 INAF-Osservatorio Astronomico di Roma, Via Frascati 33, 00040 Monteporzio Catone, Italy 3 Department of Physics, ORT Braude, Snunit 51 Street P.O. Box 78, 21982 Karmiel, Israel 4 ASI Science Data Center, Via Galileo Galilei, 00044 Frascati, Italy

Correspondence should be addressed to Giovanni Amelino-Camelia; [email protected]

Received 2 October 2013; Accepted 25 November 2013; Published 20 January 2014

Academic Editor: Piero Nicolini

Copyright © 2014 Giovanni Amelino-Camelia et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

We process the Fermi LAT data on GRB130427A using the Fermi Science Tools, and we summarize some of the key facts that render this observation truly remarkable. We then perform a search of spectral lags, of the type that has been of interest for its relevance in quantum-spacetime research. We do find some evidence of systematic soft spectral lags: when confining the analysis to photons of energies greater than 5 GeV there is an early hard development of minibursts within the burst. The effect is well characterized by a linear dependence, within such a miniburst, of the detection time on energy. We also observe that some support for these features is noticeable also in earlier Fermi-LAT GRBs. Some aspects of the comparison of these features for GRBs at different redshifts could be described within a quantum-spacetime picture, but taking into account results previously obtained by other studies we favour the interpretation as intrinsic properties of GRBs. Even if our spectral lags do turn out to have astrophysical origin their understanding will be important for quantum-spacetime research, since any attempt to reveal minute quantum-spacetime-induced spectral lags evidently requires a good understanding of intrinsic mechanisms at the sources that can produce spectral lags.

1. Introduction and Motivation here take the perspective that GRB130427A is an opportunity to look at a long GRB in “high resolution,” allowing us to The study of Gamma-Ray Bursts (GRBs) has been for some notice features which could not be noticed in previous GRBs. time one of the main themes of astrophysics research, and In this study we test the potentialities of using over the last 15 years it became also of interest for research GRB130427A in this way by focusing on a much studied class on quantum gravity. These more recent developments look at of effects, involving spectral lags with a linear dependence GRBsassignalsandhopetodecodepropertiesofquantum- on energy. These possible effects deserve intrinsic interest spacetime from the implications of spacetime quantization for how such signals propagate from the distant source to our and their study is further motivated by quantum-spacetime telescope. Even tiny quantum-spacetime effects modifying research: in some quantum-spacetime pictures one has that the structure of the signal could cumulate along the way, as a group of photons ideally emitted all in exact simultaneity the signal travels over cosmological distances. at some distant source should reach our telescope with As we here contribute to assess, the remarkable very systematic linear spectral lags. recent observation of GRB130427A is bound to teach us a lot In the next section, besides discussing some aspects of about the astrophysics of GRBs and has the potential to also the exceptionality of GRB130427A and the procedures we empower some of the quantum-spacetime studies of GRBs. followed for retrieving Fermi-LAT data on this GRB, we This GRB130427A was extremely powerful, also thanks to briefly describe the quantum-spacetime picture that inspired thefactthatitwasamongthenearestlongGRBsobserved.We this GRB phenomenology. 2 Advances in High Energy Physics

Then in Section 3 we look at GRB130427A Fermi-LAT was determined [8–10]tobe0.3399 ± 0.0002,soweprobably data from the perspective of a search of linear spectral lags. got to see from up close (in “high definition”) a GRB which We do not find any evidence of such spectral lags on the already intrinsically was rather powerful. low side of the LAT energy range, but we notice that if For the purposes of the analysis we are here reporting, we one restricts attention to GRB130427A photons with energy retrieved LAT Extended Data, in a circular region centered on in excess of 5 GeV there is some rather robust evidence of the optical position [11] R.A. = 11:32:32.84, Dec. = +27:41:56 spectral lags, specifically linear soft spectral lags (lags that fromtheFermiLATdataserver. are linear in energy and such that higher-energy photons are We prepared the data files for the analysis using the LAT detected earlier). This feature is even more pronounced if ScienceTools-v9r31p1 package, which is available from the one restricts attention to GRB130427A photons with energy Fermi ScienceSupportCenter.Anoverallcharacterizationof in excess of 15 GeV. In the course of characterizing these the data we retrieved is offered in Figure 1,whichshowsthe spectral lags we also stumble upon a feature which might Fermi-LAT light curve of GRB130427A. be labeled “hard minibursts”: if one restricts attention to Following the photon selection suggested by the Fermi GRB130427A photons with energy in excess of 5 GeV a team for GRB analysis, we selected all the events of significant fraction of them is found to be part of bursts “P7TRANSIENT” class or better, in the energy range that last much shorter than the time scale set by the overall 100 MeV–300 GeV and with the zenith angle ≤ 100 deg. duration of the GRB130427A signal in the LAT. Our analysis “P7TRANSIENT” class contains lower quality photons, the finds that the first photon in such hard minibursts is always “transient” class events. These cuts are optimized [12]for the highest-energy photon in the miniburst. transient sources for which the relevant timescales are suf- Section 4 iswherewetrytouseourfindingsfor ficiently short that the additional residual charged-particle GRB130427A as guidance for noticing features in data avail- backgrounds are much less significant. We selected photons able on previous GRBs. in a circular region centered on the GRB130427A optical In the closing Section 5 we offer a perspective on our position and with a maximum radius of 10 degrees which findings and comment on various scenarios for their inter- ensures that all GRB photons (“FRONT” and “BACK,” as pretation. described in [13]) are considered. Nevertheless, since our study is mainly focused on photons with energy greater than 5GeVandbecausetheLATPointSpreadFunctionisenergy 2. Preliminaries on GRB130427A and dependent, we have decided to use also a more conservative Quantum-Spacetime source extraction region with a maximum radius of 3 degrees, which corresponds to about 60%andabout90%ofthe 2.1. The Remarkable GRB130427A and Fermi-LAT Data. At LAT Point Spread Function at energy greater than 5 GeV for 07:47:06 UT on 27 April, 2013, Fermi-LAT detected [1] the P7TRANSIENT and P7SOURCE class [14], respectively. high-energy emission from GRB 130427A, which was also More on this aspect is in Section 3.3. detected [2] at lower energies by Fermi-GBM and by several The times and energies of detection of GRB130427A other telescopes including Swift [3], Konus-Wind [4], SPI- photonswithenergyinexcessof5GeVareshowninFigure 2. ACS/INTEGRAL [5], and AGILE [6]. From these early (and however sketchy) reports it 2.2. A Scenario for Quantum-Spacetime. A rich phenomeno- emerges that GRB130427A is a record setter in many ways. logical program was developed over the last decade on From the point of view of high-energy astrophysics and fun- the basis of results for models of spacetime quantization damental physics, relevant for the study we are here report- suggesting that (see, e.g., [15–19]) it is possible for the quan- ing, particularly significant is the fact that GRB130427A is tum properties of spacetime to introduce small violations thenewrecordholderasthehighest-fluenceLAT-detected of the special relativistic properties of classical spacetime. GRB. Moreover the GRB130427A signal contains 𝑎≃95GeV A particularly valuable opportunity for phenomenology is photon, which establishes the new record for a GRB, by a provided by pictures of quantum-spacetime in which the very wide margin (all previous Fermi-LAT GRB-photons had time needed for a ultrarelativistic particle (of course the only energy lower than 35 GeV). regime of particle propagation that is relevant for this paper is Indications of an extreme fluence for this GRB is also the ultrarelativistic regime, since photons have no mass and confirmed at lower energies by the SPI-ACS detector onboard the neutrinos we will consider in Section 4.5 have energies INTEGRAL and by Swift, that reported an extremely lumi- such that their mass is negligible) to travel from a given source nous X-ray afterglow. to a given detector is 𝑡=𝑡0 +𝑡𝑄𝐺.Here𝑡0 is the time that Looking back at GRB catalogues one can tentatively wouldbepredictedinclassicalspacetime,while𝑡𝑄𝐺 is the estimate that such an exceptional GRB is only observed not contribution to the travel time due to quantum properties more frequently than once in a quarter century (a good of spacetime. For energies much smaller than 𝑀𝑄𝐺,the benchmark [7], on the basis of fluence, might be GRB881024), characteristic scale of these quantum-spacetime effects, one and the opportunity is exciting since the quality of our expects that at lowest order 𝑡𝑄𝐺 is given by [20] telescopes made so much progress over this last quarter of acentury. The long list of records is surely in part due to the fact 𝐸 𝐷 (𝑧) 𝑡𝑄𝐺 =−𝑠± , (1) that GRB130427A was a rather near long GRB. The redshift 𝑀𝑄𝐺 𝑐 Advances in High Energy Physics 3

30 100 LAT: no selection 20 80

Counts/bin 60 0 0 200 400 600 800 (GeV) E 40 Time since trigger (s) 20 20 LAT: >100 MeV 15 0 0 100 200 300 400 500 600 10 t−t0 (s) 5 Counts/bin 0 Figure 2: Times and energies of detection of GRB130427A photons 0 200 400 600 800 with energy in excess of 5 GeV, selected within 10 degrees of Time since trigger (s) the GRB130427A optical position. Darker points are for photons observed from within 3 degrees of the GRB130427A optical position. 6 LAT: >1 GeV 4 It is important that some quantum-spacetime models 2 allow for laws roughly of the type (1) to apply differently Counts/bin to different types of particles. For example, in the so-called 0 “Moyal noncommutative spacetime,” which is one of the 0 200 400 600 800 most studied quantum-spacetimes, it is remarkably found Time since trigger (s) [23] that the implications of spacetime quantization for particle propagation end up depending on the standard- Figure 1: Fermi-LAT light curves for GRB130427A (bin size of 3 model charges carried by the particle and its associated seconds). Photons selected within 10 degrees of the GRB130427A coupling to other particles. optical position. The contribution from the subset of photons within But let us stress, since it will be relevant later on in 3 degrees of the GRB130427A optical position is shown in blue. our discussion, that, while particle-dependent effects would indeed not be surprising, we know of no model that predicts anabruptonsetofsucheffects:thequantum-spacetime where model building so far available offers candidates for particle- 𝑐 𝑧 (1+𝜁) dependent effects, but in none of the known models the 𝐷 (𝑧) = ∫ 𝑑𝜁 . (2) effects are exactly absent at low energy and then switch onto 𝐻0 0 √ 3 ΩΛ + (1+𝜁) Ω𝑚 full strength above some threshold value of energy. Formula (1) illustrates the type of mechanisms that are known in the Here the information cosmology gives us on spacetime quantum-spacetime literature, with the effects confined to the curvature is coded in the denominator for the integrand in ultraviolet by the behaviour of analytic functions introducing 𝐷(𝑧),with𝑧 being the redshift and ΩΛ, 𝐻0,andΩ𝑚 denoting, energy dependence such that the effects are small at low as usual, respectively, the cosmological constant, the Hubble energies and become gradually stronger at higher energies. parameter, and the matter fraction. The “sign parameter” 𝑠±, This point is important since there is an apparently robust 1 −1 𝑀 𝑀 > 1.2 𝑀 with allowed values of or ,aswellasthescale 𝑄𝐺, limit at 𝑄𝐺 Planck derived in [24] (also see [22, 25– would have to be determined experimentally. We must stress 28]) applying two different techniques to GRB090510 data. however that most theorists favor naturalness arguments One technique, in which photons with energy below 5 GeV suggesting that 𝑀𝑄𝐺 should take a value that is rather close play the primarily role, is based primarily on statistical prop- 5 19 erties of the arrival-time versus energy of photons, whereas to the “Planck scale” 𝑀 = √ℏ𝑐 /𝐺𝑁 ≃ 1.22 ⋅ 10 GeV. planck the other technique exploits mainly the good coincidence The picture of quantum-spacetime effects summarized in of detection times between a single 31 GeV photon and a (1) does not apply to all quantum-spacetime models. One very short duration burst (miniburst) of GRB090510 at lower can envisage quantum-spacetime pictures that do not violate energies. It should also be noticed that at energies higher than Lorentz symmetry at all, and even among the most studied the range of energies we shall consider for GRB130427A the quantum-spacetime pictures that do violate Lorentz symme- implications of (1) have been studied through observations of try one also finds variants producing (see, e.g., [15, 21, 22]) blazars by ground telescopes such as MAGIC and HESS, and 𝐸/𝑀 features analogous to (1) but with the ratio 𝑄𝐺 replaced this also produces [27, 29, 30]significantboundson𝑀𝑄𝐺 at 2 by its square, (𝐸/𝑀𝑄𝐺) , in which case the effects would be a level just below the Planck scale. In light of this the initial much weaker and practically undetectable at present. objective of our study was to establish whether the unusual 4 Advances in High Energy Physics quality of GRB130427A data could be used to study the effects 100 of (1) with sensitivity providing constraints on the energy scaleevenslightlyabovethePlanckMass. 80

3. Soft Spectral Lags for Minibursts within 60

GRB130427A (GeV)

E 40 3.1. Spectral Lags for Minibursts within GRB130427A. As evident from what we already summarized in Section 2.2, 20 part of our interest in GRB130427A originates from it being an opportunity for looking, at higher energies than in previ- 0 ous GRBs, for spectral lags of the type predicted for GRBs 0 100 200 300 400 500 600 on the basis of (1). Let us however postpone commenting t−t0 (s) on the possible quantum-spacetime picture and (1)and instead contemplate first from a more general perspective Figure 3: The three parallel straight lines here shown identify cases where photons on one such worldline could have been even the possibility of spectral lags with linear dependence on emitted simultaneously (up to experimental errors) if (1)holdsat energy. Looking from this perspective at the GRB130427A the relevant energies. data summarized in Figure 1, including only photons with energy greater than 5 GeV, one cannot fail to notice a very clear hint of systematic spectral lags. This is highlighted 120 in Figure 3: for each of the four highest-energy photons in GRB130427A we find at least one more photon which lies 100 on the same “spectral lag line,” for a fixed choice of slope of 80 such lines. The central line of Figure 3 is impressive because it suggests that 3 of the top-9 most energetic photons in 60 GRB130427A lie on one of such parallel spectral-lag lines. (GeV) E Another way to visualize the content of Figure 3 is to 40 consider the 7 points on those 3 lines and fit the values of the time translations needed to superpose the 3 lines. The result 20 of this fit is shown in Figure 4,wherewealsoshowthesingle 0 line which then best fits the 7 points. −35 −30 −25 −20 −15 −10 −5 0 For the slope one finds −2.81 ± 0.27 GeV/s. We observe t∗ (s) that from the perspective of (1) this slope of about −3GeV/s couldbedescribedasthecasewith𝑠± =1and 𝑀𝑄𝐺 ≃4⋅ Figure 4: Here we show the results of a fit that superposes the three 10−2 𝑀 planck. parallel lines in Figure 3. The seven points here shown are the ones on those parallel lines of Figure 3. 3.2. Supporting Evidence from a Correlation Study. The observation visualized in Figures 3 and 4 is evidently noteworthy butitisnoteasytoquantifyfromitastatisticalsignificance of photon energy versus time realizations assuming the same of the feature we are contemplating. Therefore we have photon energy distribution as the data. We run the simu- performed a correlation study. We essentially studied the lationsbyeitherrandomizingthetimearrivalofthehigh- frequency of occurrence of linear spectral lags in our sample, energy photons or scrambling them, obtaining similar results that is, the number of times a line passing from a high-energy (Figure 5 shows the case with randomization of arrival times). photon intercepts (within errors) a lower energy photon (in Figure 5 shows that the probability of chance occurrence of anycasewithenergygreaterthan5or15GeV).Thisultimately spectral lags of about 3 GeV/second is ≾ 0.23% for an energy gives us an alternative way for determining the slope of threshold of 15 GeV and just below the 1%levelforanenergy lines such as those in Figure 3 that fit well the spectral lags threshold of 5 GeV. present in the data, allowing for an easy estimate of statistical This comparison between our correlation study with a significance. 5GeVlowercutoffandourcorrelationstudywitha 15GeV Our result for this correlation study is summarized in lower cutoff also suggests that the feature we are exposing Figure 5, which gives a result in perfect agreement with emerges at high energies. This after all is also the message of the message conveyed in Figures 3 and 4 of the previous Figure 3 where one immediately sees a pattern of spectral-lag subsection, including the determination of the slope 𝑑𝐸/𝑡 correlations among the highest-energy photons and between of about −3 in GeV per second. Light dashed face, bold the highest energy photons and some lower-energy photons, dashed, and dotted lines in Figure 5 represent, respectively, whereas such correlations among pairs of photons both with the 10%, 1%, and 0.23% probability that the given frequency energy smaller than 15 GeV are not clearly visible (if at all of occurrence of spectral lags is a chance occurrence. The lines present). We confirm this message with the study visualized have been calculated using 10000 Monte Carlo simulations in Figure 6 which is of the same type reported in Figure 5 but Advances in High Energy Physics 5

focuses on GRB130427A photons with energy between 1 and E =5GeV 5GeV:contrarytowhatwasshowninFigure 5 we find no 0.3 min significant signal in the study shown in Figure 6. E =15 min GeV

3.3.AsideontheStructureatLateTimes. It might be noteworthy that the statistical significance of our analysis increases if 0.2 we focus on late times in the development of GRB130427A. To seethatthisisthecasewelookatthedatapointsinFigure 3 focusing on a 450-second interval, from 190 seconds to 640 Frequency seconds after the trigger. What we find is shown in Figure 7. Comparing Figure 7 to Figure 5 one sees clearly that after the 0.1 first 190 seconds the evidence of spectral lags with a slope of about −3 GeV/s is significantly stronger. In particular, when focusing on times after the first 190 seconds also the study of correlations taking into account all photons with energy greater than 5 GeV produces an outcome whose probability 0 0.23 −6 −4 −2 of chance occurrence is significantly less than %. dE/dt Considering the potential significance of the findings (GeV/s) reported in Figure 7 it is appropriate here to pause for an assessment of the possibility that background events might Figure 5: Here shown is a study of the frequency of occurrence be affecting the rather striking evidence we are gathering. of spectral lags in our sample as a function of a minimum photon In this respect it is noteworthy that two events among those energy and without any prior assumption on high energy photons. The red (resp., blue) solid line describes the frequency of occurrence that contribute to the result of Figure 7 were observed at of a certain spectral-lag slope within the sample of GRB130427A more than 3 degrees (but less than 10 degrees; see Section 1) events with energy grater than 5 GeV (resp., 15 GeV). Dotted/dashed off the optical position of GRB130427A. We must therefore lines represent the 10%, 1%, and 0.23% probability that the given attribute to them a somewhat higher risk (higher than for frequency of occurrence of spectral lags is a chance occurrence. The those observed within 3 degrees of GRB130427A) of being peak of the 𝑑𝐸/𝑑𝑡 distribution is consistent with the estimate of background events. These two events are the 19 GeV event −2.8 GeV/s obtained from the best-fit analysis described in Figure 4. visible in the bottom central part of Figure 3 at about 250 seconds after trigger and the 5.2 GeV event visible in the bottomrightpartofFigure 3 at about 610 seconds after trigger. In Figures 8 and 9 we offer a quantification of how different the outlook of our analysis would be when considering 0.3 E=1 5 – GeV the possibility of excluding from it one of these two events. Comparing Figure 8 to Figure 7 one sees that renouncing to the 5.2 GeV event has tangible but not very significant implications for our analysis. Comparing Figure 9 to Figure 7 0.2 one sees that renouncing to the 19 GeV event does have rather significant implications for our analysis, but our findings wouldbenoteworthyevenwithoutthe19GeVevent.

Frequency Inlightofthisspecialroleplayedbythe19GeVevent, we have estimated the probability of a background event 0.1 of energy between 15 and 30 GeV within the 450-second window considered for Figure 7. We find that this probability is of about 15% by both modeling the background and using data available in a temporal and directional neighborhood of GRB130427A.Itappearslegitimatetoalsonoticethatifsucha 0 19 GeV event was truly background and therefore unrelated to −6 −4 −2 GRB130427A it could have come at any time within that 450- dE/dt (GeV/s) second window, but it turned out to contribute so strongly to the significance of our analysis only because it happened to be an excellent match for the central straight line in Figure 3.In Figure6:Hereshownaretheresultsofastudyofthetypealready described in Figure 5, but now restricting the analysis to photons orderforittobesuchastrongcontributortothesignificance with energy between 1 GeV and 5 GeV. Again the dotted line and of our analysis it had to be detected within a temporal window the dashed line estimate, respectively, the 1%and10%probability around that straight line of about 5 seconds, so we are dealing that the given frequency of occurrence of spectral lags is a chance with a 15%probabilityoverallbutonly𝑎 ∼ 0.17%probability occurrence. of a background event contributing that strongly to our case. 6 Advances in High Energy Physics

0.8 0.8

E =5 min GeV E =5 min GeV E =15 min GeV E =15 min GeV 0.6 0.6

0.4 0.4 Frequency Frequency

0.2 0.2

0 0 −6 −4 −2 −6 −4 −2 dE/dt (GeV/s) dE/dt (GeV/s)

Figure 7: Here shown are the results of a study of the type Figure 9: Here shown are the results of a study exactly of the type already described in Figure 5, but now restricting the analysis already described in the previous Figure 7 (including the restriction to the time interval from 190 seconds to 640 seconds after the to the time interval from 190 seconds to 640 seconds after the trigger. Dotted/dashed lines represent again the 10%, 1%, and 0.23% trigger), but now excluding from the analysis a 19 GeV event from probability that the given frequency of occurrence of spectral lags is about 250 seconds after trigger. Dotted/dashed lines represent again a chance occurrence. the 10%, 1%and0.23% probability that the given frequency of occurrence of spectral lags is a chance occurrence.

0.8 (Note however that we are assuming “a posteriori” that the E =5 19 GeV photon is correlated with the 95 and 50 GeV events.) min GeV As stressed above, if we were to exclude the 19 GeV E =15GeV min event we would still be left with the noteworthy outcome 0.6 shown in Figure 9. And evidently the blue line in Figure 9 is independent of the contribution by the 5.2 GeV event. It would require a significant “conspiracy” for it to be the case that both the 19 GeV event and the 5.2 event were background 0.4 events which happened to be detected just at the right times to improve the outlook of our analysis. Frequency

4. Seeking Further Evidence and 0.2 Redshift Dependence WeareassumingthatlookingatGRB130427Aislikelooking at GRBs in high definition. This allowed us to notice some 0 unexpectedfeaturesintheprevioussection,andweshall −6 −4 −2 nowtrytousethisguidancetogobacktopreviousbright dE/dt (GeV/s) high-energy GRBs and try to recognize the same features. As we do this we shall also monitor any indication of redshift Figure 8: Here shown are the results of a study exactly of the type dependence of these features. It would be of particular already described in Figure 7 (including the restriction to the time interest from a quantum-spacetime perspective if we could interval from 190 seconds to 640 seconds after the trigger), but now find encouragement for redshift dependence governed by the excluding from the analysis a 5.2 GeV event from about 610 seconds 𝐷(𝑧) of (1), but we shall remain open to other options (and after trigger. Dotted/dashed lines represent again the 10%, 1%, and we shall end up favouring the hypothesis of description of 0.23% probability that the given frequency of occurrence of spectral the features presented in this work not based on quantum- lags is a chance occurrence. spacetime pictures). Advances in High Energy Physics 7

4.1. The Case of GRB080916C. GRB080916C is our first attempt to recognize in another GRB the features here GRB080916C uncovered for GRB130427A. This is a very interesting GRB 15 forourpurposes,particularlyfortheissueofpossibleredshift dependence, since GRB080916C was [28, 31]atredshift of 4.35, much higher than the 0.34 of GRB130427A. And 10 GRB080916Cmustbeviewedasaverypowerfulhigh-energy

GRB, if one indeed reassesses its LAT fluence in light of its (GeV) high redshift. E There is at least one more criterion which would char- 5 acterize GRB080916C as a GRB in the Fermi-LAT catalogue whose comparison to GRB130427A is particularly significant from the viewpoint of the analysis we are here reporting: 0 020406080 GRB080916C contains a 13.3 GeV photon, a 7 GeV photon, t−t and a 6.2 GeV photon, and when converting these energies 0 (s) at time of detection into energies at time of emission, taking into account the large redshift of 4.35, one concludes that Figure 10: Here shown is the Fermi-LAT sequence of photons with allthese3photonshadenergyattimeofemissionofmore energy greater than 1 GeV for GRB080916C. We visualize error bars than 30 GeV. This compares well with one aspect which we at the 15% level for the energies of the most energetic events, which stressed about GRB130427A, which has 4 photons above play a special role from our perspective. The slope of the straight 30 GeV (actually above 40 GeV). line here shown is not fit on this data but rather is obtained from the It is clear that the presence, in a GRB, of photons of such slope fit in Figure 4 (for GRB130427A) by rigidly rescaling it by the factor of 𝐷(0.34)/𝐷(4.35) dictated by (1). high inferred energies at emission time does not mean that we are assured to see one of our “hard minibursts with spectral lags.”Onthecontrarytheprobabilityofsuchsuccessremains rather low, since surely often when we see 3 or 4 such photons the GRB080916C slope and the GRB130427A slope is fixed to they belong to different hard minibursts within the GRB. To be 𝐷(0.34)/𝐷(4.35). The slope of straight lines characteristic see how this is the case imagine that from GRB130427A we of high-energy spectral lags should be then for GRB080916C only observed 3 rather than 9 photons with energy greater much lower than for GRB130427A. A straight line with such than 19 GeV: if we take off randomly 6 photons among the a slope is shown in Figure 10 anditgivesaratherintriguing top-9 most energetic ones from Figure 3 no hard miniburst result:thatstraightlineobtainedfromtheonesofSection 3 wouldbenoticeable.Butinsomecasesweareboundtoget by correcting for the redshift, assuming the validity of (1), lucky, and this is what at least appears to be the case with is consistent with as many as 3 other GRB080916C events. GRB080916C, as shown in Figure 10,whichreportstheevents And one should notice that among the 4 events that would for GRB080916C with energy higher than 1 GeV. reasonably fit this straight line one finds the 3 highest- Our readers should easily notice from Figure 10 that some energy photons which we highlighted at the beginning of this features we exposed earlier in this paper for GRB130427A subsection. are present also in this GRB080916C case. In particular, As mentioned, if the features we are uncovering do not the highest energy photon (13.3 GeV) is detected as first have quantum-spacetime origin (and therefore are pure- photon after the longest time interval of “high-energy silence” astrophysics effects) then one would not expect redshift during the first 25 seconds. And then some other high- dependence governed by the 𝐷(𝑧) of (1). Rather than specu- energy photons are detected soon after this highest-energy lating here about the possible forms of redshift dependence photon. So actually GRB080916C provides us with a very that astrophysics could produce, we choose to explore the clear example of the feature we are proposing, describable as issue by simply assuming redshift (and context) indepen- early hard development of minibursts within a long burst. dence of the slope of spectral lags within hard minibursts. Next we should use GRB080916C as an opportunity for Thissimplehypothesisisprobablyunrealistic,butatleastit finding further evidence of linear spectral lags. A challenge gives us a hypothesis alternative to the redshift dependence forthisofcoursecomesfromthefactthattheslopeof of (1) and this will help us to assess the significance of what these spectral lags could depend on redshift. In the quantum- we find assuming the redshift dependence1 of( ). In Figure 11 spacetime picture of (1) the redshift dependence would be we let go through the point (with error bar) corresponding governed by the 𝐷(𝑧) of (1), but if (as we shall end up favour- to the highest-energy photon a straight line with the slope we ing) the linear spectral lags hve astrophysical origin then determined in the previous section. various forms of redshift dependence could be conjectured Figure 11 shows that by not rescaling the slope for red- (a promising hypothesis will be discussed in Section 5,but shift one establishes no connection between the highest- several alternatives could be considered). energy photon and the second and third most energetic In order to test the hypothesis of redshift dependence photons in the sample. This is disappointing for the redshift- governed by the 𝐷(𝑧) of (1)oneshoulddoasinFigure 10, independence hypothesis. But of course it may well be that that is, rescale the slope taking into account the differ- theremarkablefeaturehighlightedinFigure 10,for(1)- ence of redshifts. According to1 ( )therelationshipbetween governed-redshift case, is merely accidental and instead atrue 8 Advances in High Energy Physics

GRB080916C GRB090926A 25 15 20

10 15 (GeV) (GeV) E E 10 5 5

0 0 0 20406080 20 40 60 80 100

t−t0 (s) t−t0 (s) (a) Figure 11: Here shown are the same data points already shown in Figure 10, but now the straight line added on top of the data point GRB090926A has exactly the slope we derived in Section 3 (Figure 4), without any 25 rescaling. 20 connection is the one visible in Figure 11 for the redshift- 15

independence hypothesis, with a linear soft spectral lag (GeV) between the highest-energy photon and the photon detected E 10 immediately after it. Stillitisnoteworthythatifonereliesonjustthe 5 comparison of Figures 10 and 11 the conclusion must be 0 that GRB080916C scores a point for the redshift dependence 20 40 60 80 100 governed by the 𝐷(𝑧) of (1). t−t0 (s)

(b) 4.2. The Case of GRB090926A. GRB090926A is another long burst of the pre-GRB130427A Fermi-LAT catalogue that Figure 12: Here shown is the Fermi-LAT sequence of photons with compares well in terms of the brightness of the high-energy energy greater than 1.5 GeV for GRB090926A. We visualize an error component with GRB080916C. And GRB090926A was at a bar at the 15% level for the energy of the two most energetic events. redshift [28, 31]of2.106,sothatitprovidesacasewithavalue (b) A straight line with the slope we derived in Section 3.(a)A of redshift somewhere in between the ones for GRB130427A straight line with slope obtained from the slope derived in Section 3 𝐷(0.34)/𝐷(2.106) and GRB080916C. by rigidly rescaling it by the factor of (2.106 being the redshift of GRB090926A) dictated by our1 ( ). In considering GRB090926A we restrict our focus on the photons shown in Figure 12,withenergygreaterthan 1.5 GeV. Again in Figure 12 we see that for GRB090926A (just as observed above for GRB130427A and GRB080916C) Figure 12(a) which apparently contains the most intriguing there is a clearly noticeable “hard miniburst” with early hard message: both in GRB130427A and in GRB080916C the slope development whose onset coincides with the highest-energy predicted by (1)forthesamevalueof𝑀𝑄𝐺 (and taking into photon in the whole signal: the figure shows several photons account the redshift) gave a straight line compatible with the rather densely distributed between 6 and 21 seconds after highest-energy photon and two other of the most energetic trigger; then “hard-photon silence” for 5 seconds, and at 26 photons in the bursts, and here again, as shown indeed in seconds after trigger the highest-energy photon in the burst Figure 12, the same strategy of analysis places on the same is observed, followed by other high-energy photons. spectral-lag straight line the highest-energy photon and the Concerning the other feature of our interest, involving second-highest-energy photon of GRB090926A. Again the thepresenceofsoftspectrallagswhicharelinearinenergy slope of the straight line shown in Figure 12(a) was obtained by rigidly rescaling the GRB130427A slope found in Figure 4 and the possible redshift dependence of this linear law, in the 𝐷(0.34)/𝐷(2.106) spirit of what we did in the preceding subsection we show in by the factor as dictated by the redshift Figure 12 straight lines on which the highest-energy photon dependence of (1). lies (within errors). In Figure 12(b) thelinehasexactlythe same slope derived in Section 3 (our redshift-independent 4.3. The Case of GRB090902B. GRB090902B, which was at scenario) and it is notable that the first photon in our sample aredshift[28, 31] of 1.822, is the other long burst in the after the highest-energy photon lies on that line. But it is pre-GRB130427A Fermi-LAT catalogue that compares well in Advances in High Energy Physics 9

terms of the brightness of the high-energy component with GRB090902B GRB080916 and GRB090926A. 40 As stressed already in the previous subsection, even in presence of a few high-energy photons one has no guarantee 30 toseeinthedataasinglehardminiburstcontainingmore than one of the highest-energy photons. More often than not the few highest-energy photons Fermi does catch will have 20 (GeV)

originated from different hard minibursts within the burst. E Inlightofthisitmayappearthatwewereunreasonably lucky in Figure 10 for GRB080916C and in Figure 12(a) for 10 GRB090926A. This unreasonably high level of success of our ansatz does not apply to the case of GRB090902B which we 0 are now considering: as we show in Figure 13(a),focusingon 0 50 100 150 GRB090902B photons with energy greater than 1 GeV, our t−t0 (s) spectral-lag straight lines, with slope computed from the one (a) derived in Section 3 according to the redshift dependence codedin(1) (the correction factor for the slope in this case 090902 40 GRB B being 𝐷(0.34)/𝐷(1.822)), do not connect the highest-energy photon with any other one of the most energetic photons in GRB090902B. Figure 13(a) does show that the highest- 30 energy photon of GRB090902B is reasonably compatible within our ansatz with a group of five photons with energy 1<𝐸≲5GeV detected between 100 and 125 seconds after 20 (GeV)

trigger, but this is far less impressive than what we found for E GRB130427A, GRB080916C, and GRB090926A. 10 As shown in Figure 13(b),eveninthecaseof GRB090902B the redshift-independence hypothesis does not provide a better match to the data than the hypothesis of 0 redshift dependence governed by1 ( ). 0 50 100 150 Particularly noteworthy is the fact that even in the case t−t0 (s) of GRB090902B we can observe again the structure of our (b) “hard minibursts”: the sequence of high-energy photons has a relatively long silent time interval between the highest-energy Figure 13: Here shown is the Fermi-LAT sequence of photons with photon and the photon that precedes it (long compared to the energy greater than 1 GeV for GRB090902B. We visualize an error typical time interval between subsequent detections of these bar at the 15% level for the energy of the most energetic event. (b) A GRB090902B high-energy photons). So the highest-energy straight line with the slope we derived in Section 3.(a)Somestraight photon is detected after a sizable time interval of high-energy lines with slope obtained from the slope derived in Section 3 by rigidly rescaling it by the factor of 𝐷(0.34)/𝐷(1.822) dictated by our silence and then a few high-energy photons are detected soon (1) for a GRB at redshift of 1.822. after.

4.4. The Case of a Short Burst: GRB090510. Our analysis did not add any particularly decisive evidence but offered an of the four Fermi-LAT bursts with most significant high- overall consistent picture. As stressed above (particularly for energy signal, GRB130427A, GRB080916C, GRB090902B, GRB080916C and GRB090926A) this additional supporting and GRB090926A, gave rather consistent indications. At least evidence for the linear quantification of the soft spectral lags one miniburst with early hard development was found in all was consistent with the possibility of redshift dependence of them in correspondence of the highest-energy photon in governed by (1), as required by the quantum-spacetime the GRB. And in all four cases the onset of this miniburst hypothesis. However, the next burst we now want to analyze containing the highest-energy photon was preceded by a will lead us to favour overall the interpretation as intrinsic relatively large time interval of high-energy silence: one finds properties of the long GRBs. This burst that we want to in all these four GRBs a suitable lower cutoff on the energies of analyze next is the only short bursts with strong high-energy the photons to be included in the analysis such that there are signal in the Fermi catalogue5: GRB090510 whose sequence no detections of photons in such a sample for a certain time of detection times for photons with energy greater than 1 GeV interval before the detection of the highest-energy photons, is shown in Figure 14. while going at still earlier times one finds a denser rate of In this case let us start by discussing our test with detections of photons in the sample. the spectral-lag straight lines derived in Section 3 from Concerning the quantification of the soft spectral lags GRB130427A data. As shown in Figure 14 one does find within such minibursts in terms of a linear law of dependence some potential candidates as high-energy photons lagging on energy GRB080916C, GRB090902B, and GRB090926A the highest energy photon according to the quantification we 10 Advances in High Energy Physics first derived in Section 3,butwithrespecttotheoutcome GRB090510 40 of the analogous tests we did previously on long bursts in this case the outlook is less promising: both for the redshift-independent hypothesis and for1 ( )basedredshift- 30 dependent hypothesis the test does not fail badly but it also gives only marginally plausible candidates. This is because 20

in Figure 14 one notices that the highest-energy photon lies (GeV) in time-of-detection quasi-coincidence with several other E high-energy photons. It is hard to believe that this time 10 coincidence is accidental (not a manifestation of an inter- relation between the highest-energy photons an those other nearly coincident high-energy photons). Figures 14(a) and 0 14(b) would suggest that the highest-energy photon happens 010203040 to have been detected in reasonably good time coincidence t−t0 (s) with those other high-energy photons but actually it is part of (a) a miniburst involving photons detected at much later times. This is certainly possible, but we feel it is somewhat hard to GRB090510 40 believe. WeobservethatintheFermi-LATdataonGRB090510 there is only one photon with energy greater than 30 GeV 30 (indeed the 31-GeV event we discussed), and, considering that the redshift of GRB090510 was28 [ , 31] 0.897, there was no 20

other photon in GRB090510 data set with energy greater than (GeV)

30GeVevenattimeofemission. E 10 4.5. An Observation about GRB Neutrinos. Having considered the brightest pre-GRB130427A Fermi-LAT GRBs we 0 010203040 shall now offer also an observation about GRB neutrinos. This involves data of more uncertain status, but it highlights t−t0 (s) another striking quantitative match for one of the features (b) we noticed in GRB130427A data. Actually this will also render manifest why we were in good position to notice such Figure 14: Here shown is the Fermi-LAT sequence of photons with features in GRB130427A data, since this observation about energy greater than 1 GeV for GRB090510. We visualize an error bar GRB neutrinos was already reported by some of us in [32], at the 15% level for the energy of the most energetic event. (b) A before GRB130427A (the arXiv preprint version of [32]is straight line with the slope we derived in Section 3.(a)Astraight from March 2013). line with slope obtained from the slope derived in Section 3 by Reference [32] also contemplates the type of energy rigidly rescaling it by the factor of 𝐷(0.34)/𝐷(0.897) (0.897 being dependence of times of arrival that is produced by (1), exactly the redshift of GRB090510), as dictated1 by( ). aswearedoinginthepresentstudy.But[32]wasaimedatthe context of GRB-neutrino searches by IceCube. Recently IceCube reported [33] no detection of any GRB- neutrinos: no current GRB model suggests that neutrinos associated neutrino in a data set taken from April 2008 to May could be emitted thousands of seconds before a GRB. But a 2010. Reference [32] focuses however on 3 Icecube neutrinos collection of GRB-neutrino candidates all sizably in advance that are described in publicly available studies [33, 34]and of corresponding GRB triggers is just what one would expect are plausible (though weak) GRB-neutrino candidates. These from (1)(for𝑠± =1). Using this observation in [32]it ∘ are [34] a 1.3 TeV neutrino 1.95 off GRB090417B, with was shown that possibly 2 of the 3 mentioned GRB-neutrino ∘ localization uncertainty of 1.61 , and detection time 2249 candidates could be actual GRB neutrinos, governed by (1). seconds before the trigger of GRB090417B, a 3.3 TeV neutrino The most appealing possibility discussed in[32]isforthe ∘ ∘ 6.11 off GRB090219, with a localization uncertainty of 6.12 , 3.3 TeV and the 109 TeV neutrinos to be actual GRB neutrinos and detection time 3594 seconds before the GRB090219 and requires 𝑀𝑄𝐺 ≃𝑀 /25.Theotherpossibilityis ∘ planck trigger, and a 109 TeV neutrino, within 0.2 of GRB091230A, for the 1.3 TeV and the 3.3 TeV neutrinos to be actual GRB ∘ with a localization uncertainty of 0.2 , and detected some 14 neutrinos and requires an even smaller value of 𝑀𝑄𝐺,ofabout 𝑀 ≃𝑀 /100 hours before the GRB091230A trigger. 𝑄𝐺 planck . The fact that all 3 of these GRB-neutrino candidates Evidently what we found in the present analysis inspired were detected sizably in advance of the triggers of the GRBs by and based in part on GRB130427A is at least to some theycouldbeassociatedwithisnotparticularlysignificant extent consistent with what was found in [32] for the 3.3 TeV from the standard perspective of this sort of analysis, and and the 109 TeV neutrinos. In order to be more precise actually obstructs any such attempt to view them as GRB in this assessment we must take into account the fact Advances in High Energy Physics 11 that the redshifts of GRB090219 and of GRB091230A were identify two more of such hard minibursts, with exactly the not determined. We shall deal with this in the next (closing) same structure (see Figure 3). section by handling the uncertainty in the redshift as another We have some evidence for a specific behaviour of these source of uncertainty for testing (1)withthesetwoneutrinos: soft spectral lags, with linear dependence on energy. This since GRB090219 was a short burst we can reasonably [35] found rather sizable support in our analysis of GRB130427, assume that its redshift was between 0.2 and 1, whereas for as shown in Section 3.1 by exhibiting some rather compelling GRB091230A, a long burst, we can reasonably [36] assume candidate photons for 3 hard minibursts and shown in that its redshift was between 0.3 and 6. Sections 3.2 and 3.3 by statistical analyses taking the shape of studies of the frequency of occurrence of a given type of spectral lags in our GRB130427A sample. 5. Perspective on Results and Closing Remarks GRB130427A on its own provides a rather compelling case for the presence of these linear-in-energy soft spectral We here took as starting point the remarkable Fermi-LAT lags. And when we considered available data on other observation of GRB130427A. We looked at this GRB assum- ingthatitwasgivingusachancetoseealongbrightburstin bright Fermi-LAT GRBs we did find some additional sup- high resolution. Using as guidance the quantum-spacetime- porting evidence, though significantly weaker than in the inspired ansatz of (1) we stumbled upon a few related features GRB130427A case. present in the highest-energy portion of the Fermi-LAT data GRB130427A. We then used this insight gained from 5.2. Strength and Weakness of the Quantum-Spacetime- the high-resolution GRB130427A to recover some evidence Inspired Hypothesis. When we looked to other bright Fermi- ofthesamefeaturesinthebrightestpreviousFermi-LAT LAT GRBs as opportunities for additional evidence support- observations of GRBs and even for some candidate GRB ing our linear spectral lags it was necessary to contemplate neutrinos. a redshift dependence of the linear law. We had a candi- Since we considered a few features in several contexts it date ready for the task, taking the shape of the redshift is useful to close this paper by offering a perspective on our dependence codified by 𝐷(𝑧) for the linear spectral lags findings. We articulate this section into subsections, each one of (1). We explored the efficaciousness of this hypothesis devotedtooneofthefeaturesandaspectswehaveconsidered by taking the linear energy-versus-time dependence derived and ordered according to the strength of the supporting on GRB130427A data and assuming that it should apply to evidence which we found. other bursts upon rigidly rescaling the slope by 𝐷(0.34)/𝐷(𝑧), (where 𝐷(0.34) is for the redshift of 0.34 of GRB130427A). A dramatic confirmation would have been to find for 5.1. Structure of Hard Minibursts. The feature that emerged each burst several photons lining up (in energy versus most robustly from our analysis concerns the presence of time of detection) just according to the linear relation- hard minibursts, whose onset is marked by the highest- ship, with slope rigidly rescaled by 𝐷(0.34)/𝐷(𝑧).Butthis energy photon observed in a given GRB. We found this fea- was clearly unlikely since, even for the bright long bursts, ture (here illustrated in Figure 15)inallthetopfourbrightest GRB080916C, GRB090902B, and GRB090926A the Fermi- high-energy GRBs in the Fermi-LAT catalogue GRB130427A, LAT only detected a rather small overall number of pho- GRB080916C, GRB090902B, and GRB090926A. tons with energy greater than a few GeV. Nonetheless The minibursts become noticeable only when focusing at least in two cases our investigation of this possibility the analysis on the few highest-energy photons in the signal: appearstohavebeensuccessfulbeyondexpectations.For for GRB130427A it was clearly visible for photons with energy GRB080916C by rigidly rescaling the slope of our lin- higher than 5 GeV, while for GRB080916C, GRB090902B, ear law by 𝐷(0.34)/𝐷(4.35) (with 4.35 for the redshift of and GRB090926A it manifested itself when restricting the GRB080916C) and insisting that the relevant straight line be analysis to photons of energy higher than 1 GeV or 2 GeV. consistent with the highest-energy photon, we found a rather The structure of these hard minibursts was in all cases noteworthy group of four photons with energy and detection such that the high-energy signal (composed of photons timesallconsistentwithourlinearlawforsoftspectrallags. selected with energy above a certain high value, as we just These are the four photons on the straight line in Figure 10, stressed) has a long silent time interval between the highest- which include in particular the highest-energy photon and energy photon and the high-energy photon that precedes thesecondandthirdmostenergeticphotonsofGRB080916C. it (long compared to the typical time interval between AndintheanalogousanalysisofGRB090926Awefound subsequent detections of these high-energy photons). So the that our quantum-spacetime ansatz for the dependence on highest-energy photon is detected after a sizable time interval redshift of the linear spectral lags remarkably also placed of high-energy silence of the signal, and then a few high- in connection the two highest energy photons. While all energy photons are detected soon after. this may well be accidental, the cases of GRB080916C and Besides being identifiable in connection with the GRB090926A clearly can seen as providing encouragement highest-energy photon of all the four brightest long Fermi- for the law of redshift dependence of1 ( ) and with it the LAT GRBs, GRB130427A, GRB080916C, GRB090902B, and possible quantum-spacetime interpretation of the features we GRB090926A, in the case of GRB130427A one can tentatively here exposed. 12 Advances in High Energy Physics

GRB090902B GRB090926A 40 25

30 20

15 20 (GeV) (GeV) E E 10

10 5

0 0 40 60 80 100 10 20 30 40 50 60

t−t0 (s) t−t0 (s)

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10 60 (GeV) (GeV) E E 40 5 20

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t−t0 (s) t−t0 (s)

Figure 15: Here (d) shows data points with energy greater than 15 GeV, which illustrate the structure of our hard minibursts: a miniburst occurring between 230 and 260 seconds after trigger is preceded by a relatively long time interval (≃ 100 seconds) of “high-energy silence,” and farther back in time one finds rather frequent detections of high-energy photons. Even for GRB130427A (observed in “high resolution”) the hard minibursts are composed of only a few photons, characterized by soft spectral lags, with in particular the highest-energy photon in the miniburst observed as the first photon in the miniburst. When we looked for supporting evidence for these features within the three brightest pre-GRB130427A Fermi-LAT long GRBs (other three panels in this figure (a), (b), and (c)) we did not find conclusive evidence of small bunches of photons governed by soft spectral lags. This also suggests that the size of the spectral lags within such hard minibursts gets bigger at higher redshift. We did find that in all three of these pre-GRB130427A Fermi-LAT long GRBs, when selecting only events abovea suitably-high energy, the highest-energy photon was observed after a relatively large time interval of “high-energy silence,” as first photon of a phase of rather frequent detections of high-energy photons.

Let us therefore summarize the most striking elements in plots. The change of perspective we are now adopting is support of this spacetime interpretation which were encoun- to compare data not by rescaling the slope on the basis tered in our analysis. For this purpose, since we are going to of redshift but instead equivalently rescaling the measured compare data obtained from GRBs at different redshifts, it is times 𝑡𝑄𝐺 in such a way as to englobe the redshift depen- useful to look at data from the perspective of the following dence of 𝐷(𝑧) in the new time observable 𝜏𝑄𝐺(𝑧) = way to rewrite (1) (restricting of course our focus to 𝑠± =1): −𝐷(0.34)𝑡𝑄𝐺/𝐷(𝑧). 𝑐𝑀 ThisallowsustoproduceFigure 16 in which we collect the 𝐸= 𝑄𝐺 𝜏 (𝑧) , 𝐷 (0.34) 𝑄𝐺 (3) 7 events from GRB130427A already considered in Figure 4, the mentioned 4 events from GRB080916C that are consistent with with our (1) based straight line of soft spectral lags, and we 𝐷 (0.34) tentatively also show the two candidate GRB neutrinos of 𝜏 (𝑧) =− 𝑡 . 𝑄𝐺 𝐷 (𝑧) 𝑄𝐺 (4) 3.3 TeV and 109 TeV discussed in Section 4.5. The horizontal orange segments reflect a 30% energy uncertainty for the By rewriting (1)inthiswaywecanplacedatafromdifferent neutrinos, while the vertical segments reflect an uncertainty GRBs (at different redshifts) on the same 𝐸-versus-𝜏𝑄𝐺(𝑧) in the computation of 𝜏𝑄𝐺(𝑧) for those two neutrinos, Advances in High Energy Physics 13 which, as mentioned, is due to the fact that the redshifts of the GRBs to which they are tentatively associated is not 104 known. Specifically the two neutrino points are for a 3.3 TeV 1000 (with 30% uncertainty) neutrino with 𝑡𝑄𝐺 of −3594 seconds assumedtohaveoriginatedfromasourceatredshiftbetween 0.2 and 1 (the short GRB090219) and for a 109 TeV (with 30% (s) 100 QG uncertainty) neutrino with 𝑡𝑄𝐺 of −14 hours assumed to have t 10 originated from a source at redshift between 0.3 and 6 (the long GRB091230A). 1 Figure 16 contains information from 13 data points, much of which was unconstrained by the setup of the analysis. We 0.1 already discussed in Section 3 how the information contained 0.1 1 10 100 1000 104 105 in the seven points from GRB130427A is only in part used E (GeV) to the determine the slope of our linear relation between energy and time of detection and the parameters of rigid time Figure 16: We here show in a logarithmic plot a straight line with translation needed to align the 3 straight lines from Figure 3. the slope we determined in Section 3.Alsoshown(asthickpoints) Since the slope of the straight line in Figure 16 is still the arethe7datapointsforGRB130427AalreadyshowninFigure 4.The one determined using that procedure, the information in the red triangles describe data on four events found on a single spectral- remaining 6 points in Figure 16 was largely unconstrained lag straight line in Figure 10 for GRB080916C. The orange point within our analysis. The two neutrino points are located in with sizable error bars are for the 3.3 TeV neutrino and the 109 TeV thefigurejustwheretheynaturallyshouldbe:forspectral neutrino discussed from our perspective in [32], with 30 percent lags of thousands of seconds with respect to a GRB trigger uncertainty on neutrino energies reflected in horizontal segments there is nothing to be gained by leaving some freedom for the and vertical segments reflecting the uncertainty in redshift of the GRB possibly associated to the two neutrinos. actual emission time of the neutrino within the GRB event. The information in the 4 points from GRB080916C was nearly all unconstrained: the points were only used to determine thesingleparameterofrigidtimetranslationneededtoreset of photons with energy between 1 and 2 GeV. Figure 14 the onset of that candidate hard miniburst of GRB080916C suggests that one could interpret that 31 GeV event consis- conveniently at 𝜏𝑄𝐺 =0. tently with (1) and consistently with the slope determined The message of Figure 16 is rather intriguing and may in Figure 4 from GRB130427A data by arguing that the encourage further investigation (in future studies and par- coincidence in timing is accidental: the 31 GeV event should ticularly with future observations) of the quantum-spacetime have happened to occur at some point during those special interpretation of the features we here exposed. 0.2 seconds just by accident, without having any connection But as much as the message contained in Figure 16 is between the mechanism and time of emission of the 1-to- intriguing, we feel that on balance our findings disfavour the 2 GeV photons being detected in that 0.2 seconds interval and quantum-spacetime interpretation. A first aspect of this intu- the mechanism and time of emission of that 31 GeV photon. ition merely originates from theory prejudice: as mentioned Thisisnotimpossiblebutitishardtobelieve. there are solid models of spacetime quantization that predict abehaviourofthetypegivenin(1) but none of these known 5.3. The Possibility of an Astrophysical Interpretation. Our models predicts that the onset of the validity of (1)should study was set in motion by interest in possible quantum- be abrupt at some energy scale. We get the intriguing picture spacetime signatures, but we stumbled upon features for of Figure 16 focusing on a few high-energy photons from which, as stressed in the previous subsection, we remain GRBs, but the possibility that the spectral lags predicted by open to a quantum-spacetime interpretation (which however thestraightlineinFigure 16 apply also to the abundant obser- would require a novel formalism, not exactly producing (1), vations of lower-energy GRB photons is already excluded at least not at all scales) but ended up actually favouring experimentally [22, 24–28]. And, as mentioned, the available an astrophysical interpretation, as intrinsic properties of the literature suggests that at energies above the range which mechanisms producing the GRBs. was here considered for Fermi-LAT GRBs observations one Theonlyothercasewecontemplatedintheprevious shouldtakeintoaccountdatafromobservationsofsome sections as an alternative to the case of redshift dependence Blazars which also exclude the applicability of (1)with𝑀𝑄𝐺 of the slope governed by the 𝐷(𝑧) of (1)isthecaseofa 𝑀 /25 as low as planck . slope that is redshift independent and context independent. Of course we could (and eventually would) set aside For us this served exclusively the purpose of placing in this theory prejudice in the face of a clear experimental some perspective the outcome of our investigation of the situation showing the validity of (1) only in a rather specific hypothesis of redshift dependence governed by (1). Of course energy range. But even then there is a severe interpretational if one was to exclude the interpretation as a quantum- challenge in the data which resides in a single but very spacetime/propagation effect, it would be most natural not to well observed photon: the 31 GeV event in GRB090510. That think of some redshift independent and context independent 31 GeV event occurred during a time of about 0.2 seconds features, but rather think of intrinsic features of astrophysical forwhichFermi’sLATobservedaratherintenseburst origin, and with some context dependence. 14 Advances in High Energy Physics

An example of avenue for a description based on intrinsic GRB080916C properties of the sources can be inspired by the fact that the 15 features here discussed all are relevant at very high energies: this may suggest an inverse-compton origin, which in turn may allow a description of the soft spectral lags in terms of the corresponding cooling time being faster at progressively 10

high energies. (GeV) We should stress that, independently of the actual nature E of the mechanism involved, even within an astrophysical 5 interpretation of the timing features we here highlighted one could expect some redshift dependence. To see this one 0 can think of the simple-minded hypothesis of an energy- 020406080 dependent timing features which at the emission are always t−t exactly the same: then because of redshift and time dilation 0 (s) one would have a picture of such energy-dependent timing features at detection which does depend on redshift. Such Figure17:Thebluelineandthedatapointsinthisfigureexactly aninterpretationwouldleadonetocontemplateforour match the ones for Figure 10, concerning GRB08091C. The only picture the possibility that the slope of the relevant linear difference is that now we add the red line whose slope is obtained −2 from the slope fit in Figure 4 (for GRB130427A) by rigidly rescaling dependence might scale with redshift according to (1 + 𝑧) . 2 2 2 2 it by the factor of 1.34 /5.35 ,thatis,(1 + 𝑧1) /(1 + 𝑧2) for 𝑧1 = 0.34 So in comparing this slope of linear spectral lags between a (GRB130427A) and 𝑧2 = 4.35 (GRB080916C). GRB at redshift of 𝑧1 and a redshift of 𝑧2 one would expect a 2 behaviour at least roughly rescaling the slope by (1+𝑧1) /(1+ 𝑧 )2 2 (given a certain value for the slope found for a GRB at featureandontheothersidethefactthattheseareafterall redshift 𝑧1 then for a GRB at higher redshift 𝑧2 one would 2 just 4 GRBs. While the “lower resolution” of observed high- expect the slope to be less steep by a factor of (1 + 𝑧1) /(1 + 2 energy images of more distant GRBs did not allow us to 𝑧2) ). conclusively identify bunches of photons governed by soft It is interesting to observe that this natural estimate of spectral lags, we did interpret as encouraging the consistent 2 2 (1 + 𝑧1) /(1 + 𝑧2) for the redshift dependence in case of an pattern highlighted in Figure 15 forthestructureofatime astrophysical interpretation leads to a quantification which is interval around the time of detection of the highest-energy not wildly different from the quantum-spacetime prediction photon in a GRB. Also considering that this hard-miniburst 𝐷(𝑧1)/𝐷(𝑧2). We give an intuitive characterization of this in feature is something that was not expected by us and was Figure 17, where we compare the two predictions for the case recognized on available data (rather than being predicted of GRB080916C, which is the highest-redshift GRB among in advance) we feel it should at this point be considered as the bright GRBs we here considered. a promising hypothesis to be tested on future data. In this Confirmingwhatwejuststressed,theblueandthered respect we should acknowledge that a limitation of the notion lines in Figure 17 reflect slopes which are not wildly different of hard miniburst which we here introduced is its rather and as a result they lead to a comparably compelling picture qualitative nature, indeed best summarized in Figure 15.We based on the hypothesis of linear soft spectral lags for the data notice however that a quantitative requirement could be available for GRB080916C. This also shows that (assuming based on the studies here reported in Section 3.3.Therewe that linear soft spectral lags within hard miniburst are even- focused on “late times,” essentially defined as times starting tually fully established) it might require more than just a few withthedetectionofthehighest-energyphotonintheburst, GRBs with suitable properties in order to distinguish between and we used Monte Carlo-randomization criteria for showing the hypothesis of slope with astrophysics-inspired redshift that in that late-time window the data on GRB130427A 2 2 manifest a strong bias toward soft spectral lags that is rather dependence of (1 + 𝑧1) /(1 + 𝑧2) and the hypothesis of slope with the quantum-spacetime-inspired redshift dependence of unlikely as just a chance occurrence. This is a quantitative aspect of our hypothesis that the highest-energy photons in 𝐷(𝑧1)/𝐷(𝑧2). a long GRB might all be part of hard miniburst involving soft spectral lags (as here observed for GRB130427A). 5.4. Outlook. We here provided evidence both for the more For what concerns the robustness of the hypothesis that at general structure of hard minibursts, as perhaps best sum- least in some bright high-energy GRBs such hard minibursts marized in Figure 15, and for the specific feature of linear- involve soft spectral lags with linear dependence on energy in-energy soft spectral lags within the development of such the most impressive indication of significance of the evidence hard minibursts, as perhaps best summarized in Figures 5, 7, we here found is contained in Figure 5 and in Figure 7. and 16. Especially Figure 7 indicates that this feature is present with In gaining an intuition for the robustness of the more gen- a rather high statistical significance in GRB130427A. Still eral hard-miniburst structure one should take into account some prudence may be invited by the observation that on one side the fact that Figure 15 involves all the Fermi- this statistical significance emerges from a remarkable sharp LAT long GRBs bright enough for one to search for such a timing feature for just very few events (think in particular Advances in High Energy Physics 15 ofthebluelineinFigure 7, which focuses only on the few induced spectral lags evidently requires as a prerequisite our events with energy greater than 15 GeV at times greater than best possible understanding of intrinsic mechanisms at the 190 seconds after the GRB130427A trigger). This of course sources that can produce spectral lags. As the understanding is automatically taken into account in the assessment of the ofourhardminiburstsimprovestheymaywellturninto statistical significance shown in Figure 7,butstillsomehow, “standard candles,” with distinctive standard features, ideally becauseofthefactthatonlyfeweventsareinvolved,we suited for the search of possible additional contributions to do feel it would be important to gather additional evidence. the spectral lags with characteristic dependence on redshift This aspect however illustrates very clearly the significance of the type 𝐷(𝑧) and therefore attributable to quantum- of the observation of GRB130427A for the overall effort spacetime effects. of understanding GRBs: one could perhaps have noted the implications for linear soft spectral lags of Figure 10 for Conflict of Interests GRB080916C, but that observation, even when combined with a few similar observations for other bright GRBs, could The authors declare that there is no conflict of interests never lead to a robust indication. The indication in favour regarding the publication of this paper. of linear soft spectral lags contained in the “high-resolution” image for GRB130427A contained in Figure 3 brings us already so close to conclusive evidence that we can already Acknowledgment describe Figure 10 for GRB080916C as supporting evidence. ThisresearchwassupportedinpartbytheJohnTempleton In order for the evidence to be assessed as conclusive, it would Foundation (GAC). now take only a few more GRBs, even with “low-resolution” images like Figure 10 for GRB080916C. Even taking as working assumption that this linear soft References spectral lags are a true feature a lot remains to be worked [1] S. Zhu, J. Racusin, D. Kocevski et al., “Fermi-LAT detection of a out about their interpretation. Evidently the most fascinating burst,” GCN Circular, no. 14471, GRB 130427A, 2013. interpretationwouldbetheonebasedonquantum-spacetime [2]A.vonKienli,“Fermi GBM observation,” GCN Circular,no. effects. This quantum-spacetime interpretation requires that 14473, GRB 130427A, 2013. the effects are present in exactly the same way in all bursts, with the only differences allowed being governed by redshift, [3] A. Maselli, A. P. Beardmore, A. Y. Lien et al., “Swift detection 𝐷(𝑧) of a very bright burst with a likely bright optical counterpart,” through a function of redshift of the type of the of GCN Circular, no. 14448, GRB 130427A, 2013. (1). One is allowed (in spite of this being something that [4]S.Golenetskii,R.Aptekar,D.Frederiksetal.,“Konus-Wind lacks any support in the quantum-spacetime formalisms so observation of GRB 130427A,” GCN Circulars,no.14487,2013. far explored) to speculate about possible scales of abrupt [5] A. Pozanenko, P. Minaev, and A. Volnova, “SPI- onset or offset of the spacetime effects, but a dependence 𝐷(𝑧) ACS/INTEGRAL observations,” GCN Circulars,no.14484, on redshift of the type of the is nonnegotiable for the GRB 130427A, 2013. quantum-spacetime interpretation. At least on the basis of the [6] F.Verrecchia, C. Pittori, A. Giuliani et al., “High energy gamma- consistency between Figure 3 for GRB130427A and Figure 10 ray detection by AGILE,” GCN Circulars, no. 14515, GRB for GRB080916C further exploration of this interpretation is 130427A, 2013. to some extent encouraged. But the specific model based on [7] A. Chernenko and I. Mitrofanov, “Decomposition of a cosmic (1), extending over all values of energy, is clearly inadequate gamma-ray burst into two non-correlated radiation compo- for such explorations because of the limits established for nents,” Monthly Notices of the Royal Astronomical Society,vol. effects of this magnitude in previous studies of GRB photons 274, no. 2, pp. 361–368, 1995. at energies below the ones that played a key role here and [8]H.Flores,S.Covino,D.Xuetal.,“VLT/X-shooterredshift in previous studies of blazars at energies above the ones confirmation,” GCN Circulars, no. 14491, GRB 130427A, 2013. that played a key role here. Moreover, as we also stressed, [9] A. J. Levan, S. B. Cenko, D. A. Perley, and N. R. Tanvir, “Gemini- the quantum-spacetime interpretation of the features here North redshift,” GCN Circulars, no. 14455, GRB 130427A, 2013. exposed is significantly challenged by the applicability to [10] D. Xu, C. Cao, S.-M. Hu, and C.-M. Zhang, “Weihai optical GRB090510, with its 31 GeV photon detected in sharp time observations,” GCN Circulars, no. 14458, GRB 130427A, 2013. coincidence with several photons of energy between 1 GeV [11] L. Elenin, A. Volnova, V. Savanevych, A. Bryukhovetskiy, I. and 2 GeV. Molotov, and A. Pozanenko, “Early optical observations,” GCN Considering the outmost importance of the scientific Circulars, no. 14450, GRB 130427A, 2013. issues at stake the quantum-spacetime avenue should be [12] M. Ackermann, M. Ajello, L. Baldini et al., “Constraints on nonetheless pursued vigorously, but our findings evidently the cosmic-ray density gradient beyond the solar circle from favour a description of the features here highlighted based Fermi 𝛾-ray observations of the third galactic quadrant,” The on intrinsic properties of the sources, in terms of a pure Astrophysical Journal, vol. 726, no. 2, article 81, 2011. astrophysics interpretation. [13] W.B. Atwood, A. A. Abdo, M. Ackermann et al., “The large area It is intriguing that even if spectral lags within our hard telescope on the Fermi gamma-ray space telescope mission ,” The minibursts do not have a quantum-spacetime origin their Astrophysical Journal, vol. 697, no. 2, Article ID 1071, 2009. understanding will be important for quantum-spacetime [14] M. Ackermann, M. Ajello, A. Albert et al., “The Fermi large area research: the search of possibly minute quantum-spacetime- telescope on orbit: event classification, instrument response 16 Advances in High Energy Physics

functions, and calibration,” The Astrophysical Journal Supple- [34] N. Whitehorne, A search for high-energy neutrino emission ment Series,vol.203,no.1,article4,2012. from gamma-ray bursts [Ph.D. thesis], University of Wisconsin- [15] G.Amelino-Camelia,J.Ellis,N.E.Mavromatos,D.V.Nanopou- Madison, Madison, Wis, USA, 2012, https://docushare los, and S. Sarkar, “Tests of quantum gravity from observations .icecube.wisc.edu/dsweb/Get/Document-60879/thesis.pdf. of 𝛾-ray bursts,” Nature,vol.393,pp.763–765,1998. [35] D. Wanderman and T. Piran, “The luminosity function and the [16] R. Gambini and J. Pullin, “Nonstandard optics from quantum rate of Swift’s gamma-ray bursts,” Monthly Notices of the Royal space-time,” Physical Review D,vol.59,no.12,ArticleID124021, Astronomical Society,vol.406,no.3,pp.1944–1958,2010. 4 pages, 1999. [36] D.Coward,E.J.Howell,T.Piranetal.,“TheSwift short gamma- [17]J.Alfaro,H.A.Morales-Tecotl,andL.F.Urrutia,“Quantum ray burst rate density: implications for binary neutron star gravity corrections to neutrino propagation,” Physical Review merger rates,” Monthly Notices of the Royal Astronomical Society, Letters,vol.84,no.11,pp.2318–2321,2000. vol. 425, no. 4, pp. 2668–2673, 2012. [18] G. Amelino-Camelia and S. Majid, “ Waves on noncommutative space-time and gamma-ray bursts,” International Journal of Modern Physics A,vol.15,no.27,article4301,2000. [19] R. C. Myers and M. Pospelov, “Ultraviolet modifications of dispersion relations in effective field theory,” Physical Review Letters, vol. 90, no. 21, Article ID 211601, 4 pages, 2003. [20] U. Jacob and T. Piran, “Lorentz-violation-induced arrival delays of cosmological particles,” Journal of Cosmology and Astroparti- cle Physics,vol.2008,no.1,article031,2008. [21] D. Mattingly, “Modern tests of Lorentz invariance,” Living Reviews in Relativity,vol.8,article5,2005. [22] G. Amelino-Camelia and L. Smolin, “Prospects for constraining quantum gravity dispersion with near term observations,” Physical Review D,vol.80,no.8,ArticleID084017,14pages, 2009. [23] R. J. Szabo, “Quantum field theory on noncommutative spaces,” Physics Reports,vol.378,no.4,pp.207–299,2003. [24] A. A. Abdo, M. Ackermann, M. Ajello et al., “A limit on the variationofthespeedoflightarisingfromquantumgravity effects,” Nature,vol.462,pp.331–334,2009. [25] A. Abdo, M. Ackermann, M. Ajello et al., “Detection of high-energy gamma-ray emission from the globular cluster 47 Tucanae with Fermi,” Science, vol. 323, no. 5942, pp. 845–848, 2009. [26] J. Ellis, N. E. Mavromatos, and D. V. Nanopoulos, “Probing a possible vacuum refractive index with 𝛾-ray telescopes,” Physics Letters B,vol.674,no.2,pp.83–86,2009. [27] J. Bolmont and A. Jacholkowska, “Lorentz symmetry breaking studies with photons from astrophysical observations,” Advances in Space Research, vol. 47, no. 2, pp. 380–391, 2011. [28]R.J.Nemiroff,R.Connolly,J.Holmes,andA.B.Kostinski, “Bounds on spectral dispersion from Fermi-detected gamma ray bursts,” Physical Review Letters,vol.108,no.23,ArticleID 231103, 4 pages, 2012. [29] J. Albert, E. Aliu, H. Anderhub et al., “Probing quantum gravity using photons from a flare of the active galactic nucleus Markarian 501 observed by the MAGIC telescope,” Physics Letters B,vol.668,no.4,pp.253–257,2008. [30] F. Aharonian, A. G. Akhperjanian, U. B. de Almeida et al., “Limits on an energy dependence of the speed of light from a flare of the active galaxy PKS 2155-304,” Physical Review Letters, vol. 101, no. 17, Article ID 170402, 5 pages, 2008. [31] M. Ackermann, M. Ajello, K. Asano et al., “The first Fermi-LAT gamma-ray burst catalog,” The Astrophysical Journal Supplement Series, vol. 209, no. 1, article 11, 2013. [32] G. Amelino-Camelia, D. Guetta, and T. Piran, “Possible relevance of quantum spacetime for neutrino-telescope data analyses,” http://arxiv.org/abs/1303.1826. [33]R.Abbasi,Y.Abdou,T.Abu-Zayyadetal.,“Anabsenceof neutrinos associated with cosmic-ray acceleration in 𝛾-ray bursts,” Nature,vol.484,pp.351–354,2012. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2013, Article ID 106135, 6 pages http://dx.doi.org/10.1155/2013/106135

Research Article Kaluza-Klein Multidimensional Models with Ricci-Flat Internal Spaces: The Absence of the KK Particles

Alexey Chopovsky,1,2 Maxim Eingorn,3 and Alexander Zhuk2

1 Department of Theoretical Physics, Odessa National University, Dvoryanskaya Street 2, Odessa 65082, Ukraine 2 Astronomical Observatory, Odessa National University, Dvoryanskaya Street 2, Odessa 65082, Ukraine 3 North Carolina Central University, CREST and NASA Research Centers, Fayetteville Street 1801, Durham, NC 27707, USA

Correspondence should be addressed to Maxim Eingorn; [email protected]

Received 31 October 2013; Accepted 10 December 2013

Academic Editor: Douglas Singleton

Copyright © 2013 Alexey Chopovsky et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider a multidimensional Kaluza-Klein (KK) model with a Ricci-flat internal space, for example, a Calabi-Yau manifold. We perturb this background metrics by a system of gravitating masses, for example, astrophysical objects such as our Sun. We suppose that these masses are pressureless in the external space but they have relativistic pressure in the internal space. We show that metric perturbations do not depend on coordinates of the internal space and gravitating masses should be uniformly smeared over the internal space. This means, first, that KK modes corresponding to the metric fluctuations are absent and, second, particles should be only in the ground quantum state with respect to the internal space. In our opinion, these results look very unnatural. According to statistical physics, any nonzero temperature should result in fluctuations, that is, in KK modes. We also get formulae for the metric correction terms which enable us to calculate the gravitational tests: the deflection of light, the time-delay of the radar echoes, and the perihelion advance.

1. Introduction The internal compact space can have different topology. The Ricci-flat space is of special interest because Calabi- It may turn out that our Universe is multidimensional and has Yau manifolds, which are widely used in the superstring more than three spatial dimensions. This fantastic idea was theory [10], belong to this class. A particular case of toroidal first proposed in the twenties of the last century in the papers compactification was considered in our previous paper [11]. by Kaluza and Klein [1, 2], and since then it has consistently We demonstrate in this paper that metric perturbations attracted attention of researchers. Obviously, to be compat- depend only on coordinates of the external spacetime; that is, ible with our observable four-dimensional spacetime, the the corresponding KK particles are absent. In our approach, extra dimensions should be compact and sufficiently small. the metric perturbations are caused by a system of gravitating According to the recent accelerator experiments, their size masses, for example, by astrophysical objects. This is the −17 should be of the order or less than the Fermi length 10 cm. main difference between our investigation and studies in (In the brane-world models, the extra dimensions can be [8, 9] where the metric and form-field perturbations are significantly larger3 [ , 4]. We do not consider such models considered without taking into account the reason of such in our paper.) If the internal space is very small, the hope to fluctuations. Our analysis in11 [ ]clearlyshowsthatthe detect them is connected with Kaluza-Klein (KK) particles inclusion of the matter sources, being responsible for the which correspond to appropriate eigenfunctions of the inter- perturbations, imposes strong restrictions on the model. In nal manifold, that is, excitations of fields in the given internal the present paper, we investigate this important phenomenon space. Such excitations were investigated in many articles in the case of Ricci-flat internal spaces and arrive again at the (see, e.g., the classical papers [5–7]). Quite recently, KK same conclusions. That is, KK modes corresponding to the particles were considered, for example, in the papers [8, 9]. metric fluctuations are absent. We also show that gravitating 2 Advances in High Energy Physics masses should be uniformly smeared over the internal space. where 𝜔 is the parameter of EoS (we should note that the From the point of quantum mechanics, this means that pressure (4) is the intrinsic pressure of a gravitating mass in particles must be only in the ground state with respect to the the extra dimensions and is not connected with its motion; internal space. Of course, it looks very unnatural. According in other words, 𝜔 is the parameter of a particle) and the rest- to statistical physics, any nonzero temperature should result mass density is in fluctuations, that is, in KK states. 𝑁 󵄨 󵄨 −1/2 The paper is structured as follows. In Section 2,we 󵄨 (D)󵄨 ̃ ̃ 𝜌≡∑[󵄨𝑔 󵄨] 𝑚𝑝𝛿(X − X𝑝), (5) consider the multidimensional model with the Ricci-flat 𝑝=1 internal space. This background spacetime is perturbed by ̃ the system of gravitating masses. We demonstrate that the where X𝑝 is a 𝐷-dimensional “radius vector” of the 𝑝th metric perturbations do not depend on the coordinates of the 0 particle. In EoS, we include the factor 𝑑𝑠/𝑑𝑥 just for internal space and gravitating masses are uniformly smeared convenience.WecanalsoeliminatethisfactorfromEoS over the internal space. That is, KK modes are absent. In and include it in the definition of the rest-mass density5 ( ) Section 3, we provide the exact expressions for the metric (see, e.g., [11]). For our purpose, it is sufficient to consider a correction terms. The main results are briefly summarized in steady-state model. (For example, we can consider just one concluding Section 4. gravitating mass and place the origin of a reference frame on it.) That is, we disregard the spatial velocities of the gravitating 2. Smeared Extra Dimensions masses. In the weak-field approximation, the perturbed metric We consider a block-diagonal background metrics, componentscanbewritteninthefollowingform: (D) (4) (𝑑) [𝑔̂𝑀𝑁 (𝑦)] = [𝜂𝜇] ]⊕[𝑔̂𝑚𝑛 (𝑦)] , (1) 𝑀𝑁 𝑀𝑁 𝑀𝑁 𝑔𝑀𝑁 ≈ 𝑔̂𝑀𝑁 +ℎ𝑀𝑁,𝑔≈ 𝑔̂ −ℎ , (6) which is determined on (D =1+𝐷=4+𝑑)-dimensional 2 𝑁 𝑁𝑇 (4) where ℎ𝑀𝑁 ∼𝑂(1/𝑐) and ℎ𝑀 ≡ 𝑔̂ ℎ𝑀𝑇.Hereafter,the product manifold MD = M4 × M𝑑.Here,𝜂 (𝜇, ] = 𝜇] “hats” denote the unperturbed quantities. For example, the 0, 1, 2, 3) is the metrics of the four-dimensional Minkowski 𝑔̂(𝑑)(𝑦)(𝑚,𝑛=4,...,𝐷) metric coefficients in3 ( )–(5) are already perturbed quantities. spacetime and 𝑚𝑛 is the metrics of To get the metric correction terms, we should solve some 𝑑-dimensional internal (usually compact) Ricci-flat 𝑦≡𝑦𝑚 (in the corresponding order) the multidimensional Einstein space with coordinates : equation 𝑅̂(𝑑) [𝑔̂(𝑑)] =0. 𝑚𝑛 𝑚𝑛 (2) 2𝑆 𝐺̃ 1 𝑅 = 𝐷 D [𝑇 − 𝑇𝑔 ], (7) 𝑀 𝑀𝑁 𝑐4 𝑀𝑁 𝐷−1 𝑀𝑁 In what follows, 𝑋 (𝑀=0,1,2,...,𝐷)are coordinates 𝜇 𝜇 on MD, 𝑥 =𝑋 (𝜇= 0,1,2,3)are coordinates on M4,and 𝐷/2 𝑚 𝑚 where 𝑆𝐷 =2𝜋 /Γ(𝐷/2) is the total solid angle (the surface 𝑦 =𝑋 (𝑚 = 4,5,...,3+𝑑)are coordinates on M𝑑.We (𝐷 − 1) ̃ area of the -dimensional sphere of the unit radius) will also use the definitions for the spatial coordinates 𝑋: 𝐺̃ (D =𝐷+1) ̃ and D is the gravitational constant in the - 𝑋𝑀 (𝑀=1,2,...,𝐷)̃ 𝑥:𝑥̃ 𝜇̃ =𝑋𝜇̃ (𝜇̃ = 1, 2, 3) 2 and . dimensional spacetime. For our model and up to the 𝑂(1/𝑐 ) 𝑁 Now,weperturbthemetrics(1)byasystemof discrete terms,theEinsteinequation(7) is reduced to the following 𝑚 𝑝=1,...,𝑁 massive (with rest masses 𝑝, )bodies.Wesup- system of equations (see [11] for details): pose that the pressure of these bodies in the external three- 1 2𝑆 𝐺̃ 𝐷−2+𝜔𝑑 dimensional space is much less than their energy density. 𝑅 [𝑔(D)]≈− ∇̂ ∇̂𝐿̃ℎ = 𝐷 D ( )𝜌, This is a natural approximation for ordinary astrophysical 00 2 𝐿̃ 00 𝑐2 𝐷−1 (8) objects such as our Sun. For example, in general relativity, this approach works well for calculating the gravitational 𝐿=1,2,...,𝐷,̃ experiments in the solar system [12]. In other words, the 1 2𝑆 𝐺̃ 1−𝜔𝑑 gravitating bodies are pressureless in the external/our space. 𝑅 [𝑔(D)]≈− ∇̂ ∇̂𝐿̃ℎ = 𝐷 D ( )𝜌𝛿 , 𝜇̃̃] 𝐿̃ 𝜇̃̃] 2 𝜇̃̃] On the other hand, we suppose that they may have pressure in 2 𝑐 𝐷−1 (9) the extra dimensions. Therefore, nonzero components of the 𝜇,̃ ̃] =1,2,3, energy-momentum tensor of the system can be written in the following form for a perfect fluid [11, 12]: 1 𝑅 [𝑔(D)]≈− ∇̂ ∇̂𝐿̃ℎ 𝑀 𝑚𝑛 𝐿̃ 𝑚𝑛 𝑀] 2 𝑑𝑠 𝑀 ] 𝑀 𝑑𝑋 2 𝑇 =𝜌𝑐 𝑢 𝑢 ,𝑢= , (10) 𝑑𝑥0 𝑑𝑠 2𝑆 𝐺̃ 𝜔 (𝐷−1) +1−𝜔𝑑 (3) =− 𝐷 D ( )𝜌𝑔̂ , 𝑑𝑠 𝑐2 𝐷−1 𝑚𝑛 𝑇𝑚𝑛 =−𝑝 𝑔𝑚𝑛 +𝜌𝑐2 𝑢𝑚𝑢𝑛, 𝑚 0 𝑑𝑥 (D) where the Ricci-tensor components 𝑅𝑀𝑁[𝑔 ] are deter- with the equation of state (EoS) in the internal space, mined by (A.11). 2 𝑑𝑠 Obviously, the solution of this system is determined by 𝑝 =𝜔𝜌𝑐 , 𝑚=4,5,...,𝐷, (4) ̃ 𝑚 𝑑𝑥0 the rest-mass density 𝜌(𝑋). Let us suppose, for example, Advances in High Energy Physics 3

̃ that the physically reasonable solution ℎ00(𝑋) of (8) exists. be uniformly smeared over the internal space. In short, we (The exact form of such solution depends on the topology usually call such internal spaces “smeared” extra dimensions. of the internal space. For example, in the case of toroidal This conclusion has the following important effect. Sup- internal spaces with the periodic boundary conditions in pose that we have solved for the considered particle a multidi- the directions of the extra dimensions, the corresponding mensional quantum Schrodinger¨ equation and found its wave solutions can be found in [13, 14]. However, as we will see later, function Ψ(𝑋)̃ . In general, this function depends on all 𝐷 the exact form of -dimensional solutions is not important spatial coordinates 𝑋=(̃ 𝑥,̃ 𝑦), and we can expand it in for our purpose. A similar situation took place in our paper appropriate eigenfunctions of the compact internal space, [15] where this point was discussed in detail after Equation that is, in the Kaluza-Klein (KK) modes. The ground state (2.56).) Then, we can write it in the form corresponds to the absence of these particles. In this state the 𝑥̃ 2𝜑 (𝑋)̃ wave function may depend only on the coordinates of the ℎ (𝑋)̃ ≡ . (11) external space. The classical rest-mass density is proportional 00 2 2 𝑐 to the probability density |Ψ| . Therefore, the demand that 𝜑(𝑋)̃ the rest-mass density depends only on the coordinates of the It is clear that the function describes the nonrela- externalspacemeansthattheparticlecanbeonlyinthe tivistic gravitational potential created by the system of masses ground quantum state, and KK excitations are absent. This with the rest-mass density (5). From (9)and(10)weget looks very unnatural from the viewpoint of quantum and 1−𝜔𝑑 statistical physics. ℎ = ℎ 𝛿 , 𝜇̃̃] 𝐷−2+𝜔𝑑 00 𝜇̃̃] (12) 𝜔 (𝐷−1) +1−𝜔𝑑 3. Metric Components ℎ𝑚𝑛 =− ℎ00𝑔̂𝑚𝑛. 2 𝐷−2+𝜔𝑑 Now, we want to get the expression for the metrics up to 1/𝑐 0 terms. Since 𝑥 =𝑐𝑡,thecomponent𝑔00 should be calculated It is worth noting that now we can easily determine (via 4 ̃ up to 𝑂(1/𝑐 ): the gravitational potential 𝜑(𝑋))thefunctions𝜉1, 𝜉2,and𝜉3 introduced in (A.10). 𝑔00 ≈𝜂00 +ℎ00 +𝑓00, (17) It is well known (see, e.g., [16])that,tobeinagreement with the gravitational tests (the deflection of light and the 2 4 where ℎ00 ∼𝑂(1/𝑐) and 𝑓00 ∼𝑂(1/𝑐). For the rest metric Shapiro time-delay experiment) at the same level of accuracy 𝑔 ≈ 𝑔̂ +ℎ ℎ ∼ as general relativity, the metric correction terms should coefficients, we still have (6): 𝑀̃𝑁̃ 𝑀̃𝑁̃ 𝑀̃𝑁̃, 𝑀̃𝑁̃ 𝑂(1/𝑐2) satisfy the equalities .Werecallthatweconsiderthesteady-statecase: 𝑔0𝑀̃ =0. ℎ00 =ℎ11 =ℎ22 =ℎ33. (13) According to the results of the previous section, the gravitating masses are uniformly smeared over the internal This results in the black string/brane condition17 [ , 18]: space. Therefore, the rest-mass density (5) now reads 1 𝜔=− . (14) 1 𝑁 𝛿(̃x −̃x ) 1 󰜚 2 𝜌= ∑𝑚 𝑝 = 3 , 𝑉 𝑝 𝑉 (18) 𝑑 𝑝=1 √−𝑔(4) 𝑑 √−𝑔(4) Thus, to be in agreement with observations, this parameter should not be negligibly small; that is, 𝜔∼𝑂(1). The solutions (11)-(12) should satisfy the gauge condition 𝑉 =∫ 𝑑𝑑𝑦√|𝑔(𝑑)| where 𝑑 M is the volume of the internal ℎ=𝑔̂𝑀𝑁ℎ =[2(𝜔𝑑− 𝑑 (A.7). Taking into account that 𝑀𝑁 space and 1)/(𝐷 − 2 + 𝜔𝑑)]ℎ00,wecaneasilygetthatfor𝑁=] (] = 0, 1, 2, 3) this condition is automatically satisfied. However, in 𝑁 𝑁=𝑛 (𝑛=4,5,...,𝐷) the case we arrive at the condition 󰜚3 (𝑥̃) ≡ ∑𝑚𝑝𝛿(̃x −̃x𝑝). (19) 𝑝=1 ̂ 𝐿𝑇 1 𝜔 (𝐷−1) ∇𝐿𝑔̂ ℎ𝑇𝑛 − 𝜕𝑛ℎ=− 𝜕𝑛ℎ00 =0. (15) 2 𝐷−2+𝜔𝑑 Here, x̃ is a three-dimensional radius vector in the external 𝜉 𝜉 𝜉 Because 𝜔 =0̸, to satisfy this equation, we must demand that space. Additionally, it is clear that all prefactors 1, 2,and 3 in (A.10) are also functions of the external spatial coordinates 𝑥̃ 𝜉 :𝜉(𝑥)̃ ≡ 2 ̃ . For convenience, we redetermine the function 3 3 𝜕𝑛ℎ00 = 𝜕𝑛𝜑 (𝑋) =0. (16) 2 𝑐2 𝛼(𝑥)̃ .Then,ℎ𝑚𝑛(𝑥, 𝑦) = 𝛼(𝑥)𝑔̂𝑚𝑛(𝑦),where𝛼(𝑥) ∼ 𝑂(1/𝑐 ). To get the metric correction terms, we should solve the Hence, the gravitational potential is the function of the 2 Einstein equation (7). First, we consider the 𝑂(1/𝑐 ) terms spatial coordinates of the external/our space: 𝜑(𝑋)̃ = 𝜑(𝑥)̃ . ℎ𝑀𝑁. These terms satisfy the system (8)–(10), where we must Obviously, it takes place only if the rest-mass density also ̂ ̂ perform the following replacements: 𝜌→󰜚3/𝑉𝑑,where𝑉𝑑 = depends only on the spatial coordinates of the external space: 𝑑 𝜌(𝑋)̃ = 𝜌(𝑥)̃ ∫ 𝑑 𝑦√|𝑔̂(𝑑)| is the volume of the unperturbed internal . In other words, the gravitating masses should M𝑑 4 Advances in High Energy Physics

̂ ̂𝐿̃ 𝜇̃̃] 2 𝜇̃ ̃] x space and −∇𝐿̃∇ →Δ=𝛿𝜕 /𝜕𝑥 𝜕𝑥 .Introducingthe at a point with the radius vector 𝑝 produced by all particles, 2 𝑝 definition ℎ00 ≡2𝜑/𝑐, we can easily get from this system except the th: 󸀠 𝑁 𝜑 (̃x )= ∑ 𝜑 (̃x −̃x ), 𝑚𝑝 𝑝 𝑞 𝑝 𝑞 𝜑=−𝐺 ∑ 󵄨 󵄨, 𝑞 =𝑝̸ 𝑁 󵄨 󵄨 (20) (26) 𝑝=1 󵄨x̃ −̃x𝑝󵄨 Δ𝜑𝑝 (̃x −̃x𝑝)=4𝜋𝐺𝑁𝑚𝑝𝛿(̃x −̃x𝑝). where Therefore, the solution of (25)is 𝑆𝐷 (𝐷−2+𝜔𝑑) ̃ 𝐺𝑁 = 𝐺D (21) 𝑁 2𝜋𝑉̂ (𝐷−1) 2 2 2 󸀠 𝑑 𝑓 (x̃) = 𝜑 (x̃) + ∑𝜑 (̃x −̃x )𝜑 (̃x ). 00 4 4 𝑝 𝑝 𝑝 (27) 𝑐 𝑐 𝑝=1 is the Newtonian gravitational constant and 𝜑(𝑥)̃ is the nonre- 𝑁 lativistic gravitational potential created by masses. At the Obviously, in the case of the toroidal internal space, same time this section reproduces the results of our paper [11]. The 1−𝜔𝑑 found metric correction terms enable us to calculate the ℎ = ℎ 𝛿 , 𝜇̃̃] 𝐷−2+𝜔𝑑 00 𝜇̃̃] gravitational tests (the deflection of light, the time-delay of the radar echoes, and the perihelion advance) in full analogy ℎ𝑚𝑛 (𝑥, 𝑦) =𝛼 (𝑥) 𝑔̂𝑚𝑛 (𝑦) , (22) with the paper [18]. 𝜔 (𝐷−1) +1−𝜔𝑑 𝛼=− ℎ . 𝐷−2+𝜔𝑑 00 4. Conclusion In our paper, we have studied the multidimensional KK 𝑓 ∼𝑂(1/𝑐4) Now, we intend to determine the 00 correc- model with the Ricci-flat internal space. Such models are tionterm.Obviously,todoit,weneedtosolveonlythe00- of special interest because Calabi-Yau manifolds, which are component of the Einstein equation (7). Keeping in mind the widely used in the superstring theory [10], belong to this class. 1/𝑐4 prefactor in the right hand side of (7), it is clear that In the absence of matter sources, the background manifold is 𝑇 the energy-momentum tensor component 00 and the trace a direct product of the four-dimensional Minkowski space- 𝑇 𝑂(1) must be determined up to . Therefore, similar to the time and the Ricci-flat internal space. Then, we perturbed paper [11], we get this background metrics by a system of gravitating masses, forexample,astrophysicalobjectssuchasourSun.Itiswell 󰜚 𝑐2 3𝐷−4+𝜔𝑑 𝑇 ≈ 3 [1 + ℎ ], known that pressure inside these objects is much less than 00 ̂ 2 𝐷−2+𝜔𝑑 00 𝑉𝑑 ( ) the energy density. Therefore, we supposed that gravitating (23) masses are pressureless in the external space. However, 󰜚 𝑐2 𝐷−𝜔𝑑 𝑇≈ 3 [1 + ℎ ] (1−𝜔𝑑) . they may have relativistic pressure in the internal space. ̂ 2 𝐷−2+𝜔𝑑 00 𝑉𝑑 ( ) Moreover, the presence of such pressure is the necessary condition for agreement with the gravitational tests (the 4 To get the Ricci-tensor component 𝑅00 up to 𝑂(1/𝑐 ),we deflection of light, the time-delay of the radar echoes, and the can use (B.2) and Equations (2.29), (2.38) and (2.39) in [15] perihelion advance) [16–18]. We clearly demonstrated that and arrive at the following formula: metric perturbations do not depend on the coordinates of the internal space and gravitating masses are uniformly smeared 1 2 1 𝑅 ≈ Δ(𝑓 − 𝜑2)+ Δ𝜑 over the internal space. This means, first, that KK modes 00 2 00 𝑐4 𝑐2 corresponding to the metric fluctuations are absent and, (24) 𝐷−1 2 second, the particles corresponding to the gravitating masses + 𝜑Δ𝜑, should be only in the ground quantum state with respect 4 𝐷−2+𝜔𝑑𝑐 to the internal space. Of course, it looks very unnatural. According to statistical physics, any nonzero temperature Δ=𝛿𝜇̃̃]𝜕2/𝜕𝑥𝜇̃𝜕𝑥̃] where is the 3-dimensional Laplace oper- shouldresultinfluctuations,thatis,inKKmodes.Onthe 00 ator. Therefore, the -component of the Einstein equation other hand, the conclusion about the absence of KK particles reads is consistent with the results of [19, 20], where a mechanism 1 2 1 for the symmetries formation of the internal spaces was Δ(𝑓 − 𝜑2)= 𝜑Δ𝜑 2 00 𝑐4 𝑐4 proposed. According to this mechanism, due to the entropy decreasing in the compact subspace, its metric undergoes 4𝜋𝐺 𝑁 (25) the process of symmetrization during some time after its = 𝑁 ∑𝑚 𝜑󸀠 (̃x )𝛿(̃x −̃x ), 𝑐4 𝑝 𝑝 𝑝 quantum nucleation. A high symmetry of internal spaces 𝑝=1 actually means suppression of the extra degrees of freedom. We also obtained the formulae for the metric correction Δ𝜑 = 4𝜋𝐺 ∑𝑁 𝑚 𝛿(̃x −̃x ) 4 whereweusedtheequation 𝑁 𝑝=1 𝑝 𝑝 . terms. For the 00-component, we got it up to 𝑂(1/𝑐 ).We 󸀠 2 The function 𝜑 (̃x𝑝) is the potential of the gravitational field found other components up to 𝑂(1/𝑐 ). This enables us to Advances in High Energy Physics 5 calculate the gravitational tests: the deflection of light, the simplify considerably this expression: time-delay of the radar echoes, and the perihelion advance. ̂𝐿 ̂𝐿 𝑃 ̂ 𝑃 ̂ 𝑃 It also gives a possibility to construct the Lagrange function 𝑄𝑀𝑁 =−(𝑅𝑁𝑃𝑀 + 𝑅𝑀𝑃𝑁)ℎ𝐿 + 𝑅𝑃𝑀ℎ𝑁 + 𝑅𝑃𝑁ℎ𝑀. (A.8) for the considered many-body system (see, e.g., [11]). Now, we suppose that the manifold MD is a product M = M ×M Appendices manifold D 4 𝑑 with a block-diagonal background metrics of the form (1). However, in this appendix we do not M A. Ricci Tensor in the Weak-Field assume that the internal manifold 𝑑 is a Ricci-flat one. For the metrics (1) and an arbitrary internal space M𝑑,theonly Approximation ̂ (D) nonzero components of the Riemann tensor 𝑅𝐿𝑁𝑀𝑃[𝑔̂ ] (+,−,−, ̂ (D) In our paper we use the signature of the metrics and the Ricci tensor 𝑅𝑁𝑀[𝑔̂ ] are −,...)and determine the Ricci-tensor components as follows: ̂ (D) ̂ (𝑑) 𝑆 𝐿 𝐿 𝐿 𝑆 𝑆 𝐿 𝑅𝑙𝑛𝑚𝑝 [𝑔̂ ]=𝑅𝑙𝑛𝑚𝑝 [𝑔̂ ], 𝑅𝑀𝑁 ≡𝑅 =𝜕𝐿Γ −𝜕𝑁Γ +Γ Γ −Γ Γ , 𝑀𝑆𝑁 𝑀𝑁 𝑀𝐿 𝑀𝑁 𝐿𝑆 𝑀𝐿 𝑁𝑆 (A.9) (A.1) ̂ (D) ̂ (𝑑) 𝑅𝑚𝑛 [𝑔̂ ]=𝑅𝑚𝑛 [𝑔̂ ]. where the Christoffel symbols are 1 Typically, the matter, which perturbs the background Γ𝐿 = 𝑔𝐿𝑇 (−𝜕 𝑔 +𝜕 𝑔 +𝜕 𝑔 ), metrics,istakenintheformofaperfectfluidwithisotropic 𝑀𝑁 2 𝑇 𝑀𝑁 𝑀 𝑁𝑇 𝑁 𝑀𝑇 (A.2) pressure. In the case of the product manifold, the pressure 𝑀,𝑁,𝐿,...=0,1,2,...,𝐷. is isotropic in each factor manifold (see, e.g., the energy- momentum tensor (3)). Obviously, such perturbation does (D) The metrics 𝑔𝑀𝑁 ≡𝑔𝑀𝑁 is determined on (D =1+𝐷)- not change the topology of the model, for example, the dimensional manifold MD. topologies of the factor manifolds in the case of the product Now, we suppose that the metric coefficients 𝑔𝑀𝑁 cor- manifold. Additionally, it preserves also the block-diagonal respondtotheperturbedmetrics,andintheweak-field structure of the metric tensor. In the case of a steady-state ̃ approximation they can be decomposed as model (our case) the nondiagonal perturbations ℎ0𝑀̃ (𝑀= 1, 2, 3, . . . , 𝐷) are also absent. Therefore, the metric correction 𝑔 ≈ 𝑔̂ +ℎ ,𝑔𝑀𝑁 ≈ 𝑔̂𝑀𝑁 −ℎ𝑀𝑁, 𝑀𝑁 𝑀𝑁 𝑀𝑁 (A.3) terms are conformal to the background metrics: 𝑔̂ where 𝑀𝑁 is the background unperturbed metrics and the (D) 2 ℎ00 =𝜉1𝑔̂00 =𝜉1𝜂00, metric perturbations ℎ𝑀𝑁 ∼𝑂(1/𝑐) (𝑐 is the speed of 𝑁 𝑁𝑇 light). As usual, ℎ𝑀 ≡ 𝑔̂ ℎ𝑇𝑀. Here and after, the “hats” (D) ℎ𝜇̃̃] =𝜉2𝑔̂ =−𝜉2𝛿𝜇̃̃], 𝜇,̃ ̃] =1,2,3, (A.10) denote unperturbed quantities. Then, up to linear terms ℎ, 𝜇̃̃] the Christoffel symbols read (D) (𝑑) ℎ𝑚𝑛 =𝜉3𝑔̂𝑚𝑛 =𝜉3𝑔̂𝑚𝑛, 𝐿 ̂𝐿 1 𝐿𝑇 ̂𝑃 Γ𝑀𝑁 ≈ Γ𝑀𝑁 + 𝑔̂ [−𝜕𝑇ℎ𝑀𝑁 +(𝜕𝑀ℎ𝑁𝑇 − Γ𝑀𝑁ℎ𝑃𝑇) 2 where 𝜉1,2,3 are some scalar functions of all spatial coordinates ̃ ̂𝑃 𝑋. It can be easily verified that under these conditions (it is +(𝜕𝑁ℎ𝑀𝑇 − Γ𝑁𝑀ℎ𝑃𝑇)] (A.4) worth noting that, as it follows from (A.8), 𝑄𝑀𝑁 is also equal 𝑅̂𝐿 =−𝑅̂𝐿 ̂𝐿 1 𝐿𝑇 ̂ ̂ ̂ to zero when the following condition holds: 𝑁𝑃𝑀 𝑀𝑃𝑁, = Γ𝑀𝑁 + 𝑔̂ [−∇𝑇ℎ𝑀𝑁 + ∇𝑀ℎ𝑁𝑇 + ∇𝑁ℎ𝑀𝑇], ̂ 2 resulting in the Ricci-flat space 𝑅𝑀𝑁 =0(but not vice versa)) the tensor 𝑄𝑀𝑁 in (A.8)isequaltozero:𝑄𝑀𝑁 =0. Therefore, 2 and for the Ricci tensor we get up to 𝑂(1/𝑐 ) terms, we get 1 𝐿 𝑅 ≈ 𝑅̂ + (−∇̂ ∇̂ ℎ +𝑄 ) , (A.5) 1 𝑀𝑁 𝑀𝑁 2 𝐿 𝑀𝑁 𝑀𝑁 𝑅 [𝑔(D)]≈𝑅̂ [𝑔̂(D)]− ∇̂ ∇̂𝐿ℎ 𝑀𝑁 𝑀𝑁 2 𝐿 𝑀𝑁 where we introduced a new tensor: (A.11) 1 ̃ 1 𝐿 𝐿 𝐿 ̂ (D) ̂ ̂𝐿 𝑄 ≡ ∇̂ ∇̂ ℎ + ∇̂ ∇̂ ℎ − ∇̂ ∇̂ ℎ = 𝑅𝑀𝑁 [𝑔̂ ]− ∇𝐿̃∇ ℎ𝑀𝑁 +𝑜( ), 𝑀𝑁 𝐿 𝑀 𝑁 𝐿 𝑁 𝑀 𝑁 𝑀 𝐿 2 𝑐2 1 1 =[∇̂ (∇̂ ℎ𝐿 − 𝜕 ℎ𝐿)+∇̂ (∇̂ ℎ𝐿 − 𝜕 ℎ𝐿)] 𝐿=1,2,...,𝐷̃ 𝑋0 =𝑐𝑡 𝑀 𝐿 𝑁 2 𝑁 𝐿 𝑁 𝐿 𝑀 2 𝑀 𝐿 where and we took into account that . −(𝑅̂𝐿 + 𝑅̂𝐿 )ℎ𝑃 + 𝑅̂ ℎ𝑃 + 𝑅̂ ℎ𝑃 . 𝑁𝑃𝑀 𝑀𝑃𝑁 𝐿 𝑃𝑀 𝑁 𝑃𝑁 𝑀 B. Ricci Tensor for a Block-Diagonal Metrics (A.6) In this appendix we consider a block-diagonal metrics: To get the last expression, we used the relation (91.8) in (D) (𝑑1) (𝑑2) [12]. The gauge conditions (see, e.g., the paragraph 105 in[12]) [𝑔𝑀𝑁 (𝑥, 𝑦)] =𝜇 [𝑔 ] (𝑥)]⊕[𝑔𝑚𝑛 (𝑥, 𝑦)] 1 (B.1) ̂ 𝐿 𝐿 (𝑑 ) 𝛼(𝑥) (𝑑 ) ∇𝐿ℎ − 𝜕𝑁ℎ =0 (A.7) =[𝑔 1 (𝑥)]⊕[𝑒 𝑔̂ 2 (𝑦)] . 𝑁 2 𝐿 𝜇] 𝑚𝑛 6 Advances in High Energy Physics

Then, for the Ricci-tensor components we get the follow- [13] M. Eingorn and A. Zhuk, “Multidimensional gravity in non- ing expressions: relativistic limit,” Physical Review D,vol.80,no.12,ArticleID 124037, 5 pages, 2009. (𝑑 ) 𝑑 (𝑑 ) 1 𝑅 [𝑔 ]=𝑅 [𝑔 1 ]− 2 [∇ 1 (𝜕 𝛼) + (𝜕 𝛼) (𝜕 𝛼)], [14] M. Eingorn and A. Zhuk, “Non-relativistic limit of multidimen- 𝜇] 𝑀𝑁 𝜇] 𝜆𝜏 2 𝜇 ] 2 𝜇 ] sional gravity: exact solutions and applications,” Classical and (B.2) Quantum Gravity,vol.27,no.5,ArticleID055002,p.17,2010. [15] M. Eingorn and A. Zhuk, “Classical tests of multidimensional 𝑅𝑚] [𝑔𝑀𝑁] =0, (B.3) gravity: negative result,” Classical and Quantum Gravity,vol.27, no. 20, Article ID 205014, p. 18, 2010. (𝑑 ) 1 𝛼(𝑥) (𝑑 ) (𝑑 )𝜆𝜏 2 ̂ 2 1 𝑅𝑚𝑛 [𝑔𝑀𝑁]=𝑅𝑚𝑛 [𝑔𝑘𝑙 ]− 𝑒 𝑔𝑚𝑛 𝑔 [16] M. Eingorn and A. Zhuk, “Remarks on gravitational interaction 2 in Kaluza-Klein models,” Physics Letters B,vol.713,no.3,pp. (B.4) (𝑑 ) 𝑑 154–159, 2012. ×[∇ 1 (𝜕 𝛼) + 2 (𝜕 𝛼) (𝜕 𝛼)] . 𝜆 𝜏 2 𝜆 𝜏 [17] M. Eingorn and A. Zhuk, “Kaluza-Klein models: can we construct a viable example?” Physical Review D,vol.83,no.4,Arti- cle ID 044005, 11 pages, 2011. Acknowledgments [18] M. Eingorn, O. R. De Medeiros, L. C. B. Crispino, and A. Zhuk, “Latent solitons, black strings, black branes, and equations of This work was supported by the “Cosmomicrophysics-2” state in Kaluza-Klein models,” Physical Review D,vol.84,no.2, programme of the Physics and Astronomy Division of the Article ID 024031, 8 pages, 2011. National Academy of Sciences of Ukraine. The work of [19] A. A. Kirillov, A. A. Korotkevich, and S. G. Rubin, “Emergence Maxim Eingorn was supported by NSF CREST Award HRD- of symmetries,” Physics Letters B,vol.718,no.2,pp.237–240, 0833184 and NASA Grant NNX09AV07A. 2012. [20] K. A. Bronnikov and S. G. Rubin, Black Holes, Cosmology and References Extra Dimensions, World Scientific, Singapore, 2012.

[1] T. Kaluza, “On the problem of unity in physics,” Sitzungsberichte der Koniglich¨ Preussischen Akademie der Wissenschaften zu Berlin,vol.1921,pp.966–972,1921. [2] O. Klein, “Quantum theory and five-dimensional theory of relativity,” Zeitschriftur f¨ Physik,vol.37,no.12,pp.895–906,1926. [3] N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, “The hierarchy problem and new dimensions at a millimeter,” Physics Letters B, vol. 429, no. 3-4, pp. 263–272, 1998. [4] I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, “New dimensions at a millimeter to a fermi and superstrings at aTeV,”Physics Letters B, vol. 436, no. 3-4, pp. 257–263, 1998. [5]A.SalamandJ.Strathdee,“OnKaluza-Kleintheory,”Annals of Physics,vol.141,no.2,pp.316–352,1982. [6] P.van Nieuwenhuizen, “The complete mass spectrum of 𝑑=11 supergravity compactified on S4 andageneralmassformulafor arbitrary cosets M4,” ClassicalandQuantumGravity,vol.2,no. 1, pp. 1–20, 1985. [7] H. J. Kim, L. J. Romans, and P. van Nieuwenhuizen, “The mass 5 spectrum of chiral 𝑁=2𝐷=10supergravity on 𝑆 ,” Physical Review D,vol.32,no.2,pp.389–399,1985. [8] K. Hinterbichler, J. Levin, and C. Zukowski, “Kaluza-Klein towers on generalmanifolds,” http://arxiv.org/abs/1310.6353. [9]A.R.BrownandA.Dahlen,“Stabilityandspectrumofcom- pactifications on product manifolds,” http://arxiv.org/abs/1310 .6360. [10] M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory: Loop Amplitudes, Anomalies and Phenomenology,vol.2of Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, UK, 1987. [11] A. Chopovsky, M. Eingorn, and A. Zhuk, “Many-body problem in Kaluza-Klein models with toroidal compactification,” accepted in European Physical Journal C. [12] L. D. Landau and E. M. Lifshitz, “The classical theory of fields,” vol. 2 of Course of Theoretical Physics,PergamonPress,Oxford, UK, 4th edition, 2000. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2013, Article ID 967805, 5 pages http://dx.doi.org/10.1155/2013/967805

Research Article Closing a Window for Massive Photons

Sergio A. Hojman1,2,3 and Benjamin Koch4

1 Departamento de Ciencias, Facultad de Artes Liberales, Facultad de Ingenier´ıa y Ciencias, Universidad Adolfo Iba´nez,˜ Santiago, Chile 2 Departamento de F´ısica, Facultad de Ciencias, Universidad de Chile, Santiago, Chile 3 Centro de Recursos Educativos Avanzados (CREA), Santiago, Chile 4 Instituto de F´ısica, Pontificia Universidad CatolicadeChile,AvenidaVicu´ na˜ Mackenna 4860, 7820436 Santiago, Chile

Correspondence should be addressed to Benjamin Koch; [email protected]

Received 23 August 2013; Accepted 17 November 2013

Academic Editor: Douglas Singleton

Copyright © 2013 S. A. Hojman and B. Koch. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Working with the assumption of nonzero photon mass and a trajectory that is described by the nongeodesic world line of a spinning top we find, by deriving new astrophysical bounds, that this assumption is in contradiction with current experimental results. This yields the conclusion that such photons have to be exactly massless.

1. Introduction also involves the invariant functions of the corresponding additional degrees of freedom: Although there are good theoretical reasons to believe that the photon mass should be exactly zero, there is no exper- 𝛼𝜇 ]𝛽 2 𝑔𝜇]𝑔𝛼𝛽𝜎 𝜎 = tr 𝜎 , imental proof of this belief. A long series of very different experiments lead to the current experimental upper bound 𝜇 ]𝛼 𝛽𝛾 𝛿 −18 𝑔𝜇]𝑔𝛼𝛽𝑔𝛾𝛿𝑢 𝜎 𝜎 𝑢 = 𝑢𝜎𝜎𝑢, (2) on the photon mass 𝑚𝛾 <10 eV.However,evenwith further improvement of the experimental technology and 𝑔 𝑔 𝑔 𝑔 𝜎𝛿𝜇𝜎]𝜌𝜎𝜏𝛼𝜎𝛽𝛾 = 𝜎4. precision a complete exclusion of a nonzero photon mass by 𝜇] 𝜌𝜏 𝛼𝛽 𝛾𝛿 tr those techniques will never be possible. In order to improve on this situation we will work with the assumption that This generic approach to the geometric action which includes the photon mass is different from zero 𝑚𝛾 =0̸ .Wewillnot spin and mass was developed and discussed in the context of speculate on the origin of this mass and its underlying theory. a spinning top, a point on a world line to which a rotating In the simplest case, such a photon will be described by a wave frame has been attached [1–10]. The resulting equations of equation of the Proca type. For waves describing massless motion do not depend on the particular form in which the particles with spin, it is well understood, in the eikonal Lagrangian terms (1, 2) are combined. This formulation and approximation, how the propagating wave can be treated as the corresponding equations of motion can also be derived geometric path that minimizes a Lagrangian described by the as consistent eikonal limit as quantum field theory in curved length space-time. This has been shown independently for spin one half [11, 12], for spin one [13, 14],andforspinthreehalf[15, 16]. 𝜇 ] 2 𝑔𝜇]𝑢 𝑢 =𝑢, (1) Exact solutions to those equations coupled to gravity 𝑔𝜇] 𝜇 𝜇 via the metric show that spinning tops without mass where 𝑢 = 𝑥̇. As soon as mass and rotational degrees of follow geodesics, while massive spinning tops have different 2 𝜇] free-dom (𝑚 ,𝜎 ) are involved, it is natural to assume that, trajectories. Further implications and discussions of the in the eikonal approximation, the resulting geometric action spinning top approach can be found in [17–22]. 2 Advances in High Energy Physics

2. Astrophysical Bound The orientation of this problem is chosen such that the angle 𝜃=𝜋/2is constant along the trajectories, which implies that As master equation for the trajectory of massive particles with spin we use the solution of the spinning top equations in a Schwarzschild background [5, 6, 9, 10]: 𝑃𝜃 =0. (6) 𝑑𝜙 2𝜂 + 1 𝑃 =( )( 𝜙 ), 𝑑𝑟 𝜂−1 𝑟2𝑃𝑟 (3) where one has to insert the following definitions: 𝜇 2 2 The solution is such that ±𝜙 =±𝑡 and 𝑃𝜇𝑃 =𝑚𝛾𝑐 are ful- 𝐽2𝑟 filled. 𝜂= 0 , For small spin corrections and large radii one can approx- 2𝑚2𝑐2𝑟3 (4) 𝛾 imate where 𝐽=ℎis the photon spin. For a given angular momentum 𝑗, the momenta are 2 𝑑𝑟 4 1 2ℎ 2 (±𝑡)ℎ 2 ( ) =𝑟 ( + (± ) ) −𝑟 +𝑟𝑟 (1− ) . −𝑗 + (±𝜙) 𝐸𝐽/𝛾 (𝑚 𝑐 ) 2 𝜙 3 0 𝑑𝜙 𝑗 𝑐𝑗 𝑚𝛾 𝑐𝑗𝑚𝛾 𝑃𝜙 = , 1−𝜂 (7) 𝐸−(±)𝑗𝐽𝑟 /(2𝑚 𝑟3) 𝑃 = 𝑡 0 𝛾 , 𝑡 1−𝜂 (5) In this equation one can easily verify that for spinless case 2 1/2 𝑃2 𝑃 𝑟 (ℎ→0), the geodesic trajectory is recovered. One can 𝑃𝑟 =±[ 𝑡 −( 𝜙 +𝑚2𝑐2)(1− 0 )] . 𝑗 𝑐2 𝑟2 𝑟 express the angular momentum in terms of the minimal radial distance 𝑟𝑚:

2𝑐𝐸ℎ𝑚 ± (3𝑟 −2𝑟 )𝑟4 𝑗= 0 𝑚 𝑚 +2√𝑟 (𝑟 −𝑟) 2 2 3 2 4 𝑚 𝑚 0 𝑐ℎ 𝑟0 𝑟𝑚 +4𝑐 𝑚 (𝑟0 −𝑟𝑚)𝑟𝑚

2 4 2 4 8 6 9 6 2 4 6 2 2 4 3 2 2 7 4 2 3 2 2 7 4 2 √𝐸 ℎ 𝑟0 𝑟𝑚 +4𝑐 𝑚 (𝑟0 −𝑟𝑚)𝑟𝑚 +𝑐 ℎ 𝑚 𝑟0𝑟𝑚 (4𝑟𝑚 −3𝑟0)+𝑐 ℎ 𝑟0 (ℎ 𝑟0 /4 − 4𝐸 𝑚 𝑟𝑚)+𝑐 𝑚 𝑟𝑚 (4𝐸 𝑚 𝑟𝑚 −ℎ 𝑟0 𝑟𝑚) ⋅ . 2 2 3 2 4 𝑐ℎ 𝑟0 𝑟𝑚 +4𝑐 𝑚 (𝑟0 −𝑟𝑚)𝑟𝑚 (8)

Due to the steep behavior of 𝑑𝜙/𝑑𝑟 at the minimal radius This modification is inversely proportional to the photon 𝑟𝑚 one has to use (8), since one cannot rely a priori on the mass 𝑚𝛾. This inverse proportionality implies that practically 3/2 approximate value 𝑗≈𝑟𝑚 /√𝑟𝑚 −𝑟0. no deviations from the usual trajectories are observable Due to the involved form of (8), the complete angular for massive standard model particles. Please note that this deviation has to be computed essentially numerically, as it was nonperturbative feature in 𝑚𝛾 is also known from theories done here. In order to gain a better intuition of the numerical of massive gravity where the limit 𝑚𝑔 →0does not give results one can however do approximations and expansions standard gravity where the graviton was massless right from 2 𝑚 =0 for small ℎ, 𝑚𝑐 /𝐸,and𝑟0/𝑟𝑚 as follows: the start 𝑔 . Further similarities to massive gravity have been recently discussed in [23]. It is further interesting to ∞ note that Δ 𝐽 actually increases the usual angular deflection. 𝑑𝜙 𝑟0 ℎ Δ𝜙 = 2 (∫ ( ) 𝑑𝑟) −𝜋≈2 (1+ ) , (9) However, a deviation from the geodesic bending of light has 𝑟 𝑑𝑟 𝑟 𝑟 𝑐𝑚 𝑚 𝑚 𝑚 𝛾 been excluded to high precision [24]. Thus, relation10 ( )can beinterpretedasanestimateforthenumericallowerphoton where ±𝑡 =−1and ±𝜙 =1were used. One observes that the mass limit: standard result 2𝑟0/𝑟𝑚 is modified by a spin-dependent cor- ℎ 𝑚 ≫ ≈3×10−16 /𝑐2. rection. In order to be in agreement with the experimentally 𝛾 𝑟 𝑐 eV (11) well confirmed gravitational lensing effect, this dimensionless 𝑚 modification has to be much smaller than one: In Figure 1 we compare this estimate to the observed value [25, 26] and to the precise numerical results by using the solar ℎ 8 radius 𝑟𝑚 = 6.96 × 10 m, the solar Schwarzschild radius 𝑟0 = 1≫Δ𝐽 ≈ . (10) 𝑟𝑚𝑐𝑚𝛾 2964 m,andphotonenergyof𝐸=1eV. Advances in High Energy Physics 3

thephotonmass.However,ifonerefersonlytothemore stringent upper limits on 𝑚𝛾 [28–30]onefinds

3.0 −18 2 −17 2 𝑚𝛾 < {1.2 × 10 eV/𝑐 ,5.6×10 eV/𝑐 }, (13) ) 󳰀󳰀 ( 2.0 which is in clear contradiction to (12). Therefore, one can con-

Δ𝜙 clude that confronting the assumption of a nonzero photon 1.5 mass with an astrophysical lower bound and the established upper bounds excludes the existence of any photon mass. Thus, the photon, if it is actually correctly described by the model that is discussed throughout this paper, has to be 1×10−16 5×10−161×10−15 5×10−151×10−14 5×10−14 massless right from the start: m (eV) 𝑚𝛾 =0. (14) Figure 1: Numerical results for the angular deviation Δ𝜙 as a function of 𝑚𝛾. The horizontal lines represent a conservative range At this point, another word of caution is at place: when for the measured gravitational lensing by the sun [25]. Please note combining bounds that arise from different descriptions, as that this range can be largely improved by measurements using radio itwasdoneherewhencombiningtheboundsfromageo- astronomy [24, 26]. The red line is the numerical value for both metrical description (12)withboundsthatarisefromawave configurations of ±𝜙 =±𝑡.Theblackdottedlineistheanalytic 𝑚 description (13), one might end up comparing parameters estimate (9). The vertical line is the deduced minimal value for 𝛾 of different models. Even within the upper bounds obtained given in (12). from wave descriptions of the massive photons there exist problems of generality, which imply that bounds that were observed for one model of 𝑚𝛾 do not apply to other models Onefindsthattheestimateddeviationisinverygood of 𝑚𝛾 [27]. agreement with the lower numerical result. By using conser- Thus, despite of the generality of the geometric equations vative exclusion ranges for Δ𝜙 one can read from the figure a of motion, the criterion of applicability that was used here more precise limit: can be made more precise by explicitly deriving specific geometrical equations as eikonal limit of a particular wave −15 2 model of photon mass, as it was done in [14, 53]. A useful 𝑚𝛾 > 1.1 × 10 eV/𝑐 . (12) guiding tool for anticipating those results might be the peculiar noncontinuous behavior in the 𝑚𝛾 →0limit of The lower limit (12) can be combined with the upper limit the geometric description, as it appears in (10). For example for a photon mass from the Particle Data Group [27]; 𝑚𝛾 < for the wave descriptions of 𝑚𝛾 that use the Proca equations 10−18 /𝑐2 eV . Some of the limits in the PDG tables [27]are [54], it has been shown that the limit 𝑚𝛾 →0converges 𝜇 however derived by assuming a spinless coupling of the toMaxwell’stheoryonlyifonedemands𝜕𝜇𝐴 =0[55– photon to gravity and to matter. But this might not be true. 60]. On the other hand, the same limit for Proca equations 𝜇 When deriving the limit (12), we found that the gravitational with 𝜕𝜇𝐴 =0̸ does not lead continuously to Maxwell’s theory coupling of the massive particles with spin is different from a [61, 62]. The corresponding geometrical descriptions in the spinless or massless coupling. Thus, out of the limits in [27] eikonal limit are expected to have an analogous behavior in only those can be applied straightforwardly, where the grav- the 𝑚𝛾 →0limit as their counterpart in the wave de- itational coupling is not relevant. For example, results from scription. However, the eikonal limit of Maxwell’s theory is laboratory experiments that do not involve astrophysical or described by the usual geodesics. Therefore, one might expect gravitational components [28–34]canbeuseddirectly.Also that the limit (12) could only apply to Proca mass models 𝜇 𝜇 results from other experiments that do involve astrophysical with 𝜕𝜇𝐴 =0̸ andnottoProcamassmodelswith𝜕𝜇𝐴 =0. components but where the actual trajectory of the photon However, this argument does not disqualify the applicability does not play any role [35–45]shouldbeapplicable,buta of the geometric approximation (2) a priori. For example, in 𝑚 sound revision is in order. Nevertheless, the bounds on 𝛾 [61] it has been shown that although the longitudinal modes 𝜇 from some experiments are not directly applicable because of the Proca equation (even with 𝜕𝜇𝐴 =0)decouplefrom they are not sufficiently general [46–48]orbecausethey matter for 𝑚𝛾 →0in flat space-times, the coupling to gravity make explicit use of the photon trajectory in a gravitational of those modes does not vanish in this limit. field [34, 49–52] (note that those experimental references are ordered by the strength of their respective bounds on 𝑚𝛾). The experimental bounds that are directly applicable are 3. Discussion and Conclusion 𝑚 < {1.2 × 10−18 /𝑐2 5.6 × 10−17 /𝑐2 𝛾 eV [28, 29], eV [30], Under the assumption that the trajectory of a photon can −14 2 −10 2 −4 2 1×10 eV/𝑐 [31], 4.5×10 eV/𝑐 [32, 33], 1.6×10 eV/𝑐 be described by the world line of a top with one spin, it [34]}. was shown that such a photon has to have either zero mass −15 2 Thus, three of the applicable experimental limits [31– 𝑚𝛾 =0or a minimal nonzero mass 𝑚𝛾 > 1.1 × 10 eV/𝑐 . 34], combined with the limit (12), leave only a window for This new phenomenological bound is the main result of this 4 Advances in High Energy Physics

paper. If one further imposes an agreement with the observed [19] A. Barducci, R. Casalbuoni, and L. Lusanna, “Classical scalar gravitational light bending by the sun [24–26]onefindsthat and spinning particles interacting with external Yang-Mills when combining those results with the experimental bounds fields,” Nuclear Physics B,vol.124,no.1,pp.93–108,1977. −17 2 from laboratory experiments such as 𝑚𝛾 < 5.6 × 10 eV/𝑐 [20]F.W.Hehl,P.VonDerHeyde,G.D.Kerlick,andJ.M.Nester, [28–30], in this theoretical framework, there is no room for a “General relativity with spin and torsion: foundations and photon mass different from zero. prospects,” ReviewsofModernPhysics,vol.48,no.3,pp.393– 416, 1976. [21] D. Piriz, M. Roy, and J. Wudka, “Neutrino oscillations in strong Acknowledgments gravitational fields,” Physical Review D,vol.54,no.2,pp.1587– 1599, 1996. The work of Benjamin Koch was supported by project Fonde- [22] Q. G. Bailey, “Lorentz-violating gravitoelectromagnetism,” cyt 1120360 and Anillo Atlas Andino 10201. Many thanks to S. Physical Review D,vol.82,no.6,ArticleID065012,11pages, Deser for his helpful remarks. 2010. [23] S. Deser and A. Waldron, “Acausality of massive gravity,” References Physical Review Letters, vol. 110, Article ID 111101, 4 pages, 2013. [24] E. Fomalont, S. Kopeikin, G. Lanyi, and J. Benson, “Progress in [1] M. Mathisson, “Neue Mechanik materieller Systeme,” Acta measurements of the gravitational bending of radio waves using Physica Polonica,vol.6,pp.163–200,1937. the vlba,” Astrophysical Journal Letters,vol.699,no.2,pp.1395– [2] M. Mathisson, “Das Zitternde Elektron und seine Dynamik,” 1402, 2009. Acta Physica Polonica,vol.6,pp.218–227,1937. [25] S. Weinberg, Gravitation and Cosmology,JohnWiley&Sons, [3] A. Papapetrou, “Spinning test-particles in general relativity. I,” New York, NY, USA, 1972. Proceedings of the Royal Society A,vol.209,pp.248–258,1951. [26] C. M. Will, Theory and Experiment in Gravitational Physics, [4] E. Corinaldesi and A. Papapetrou, “Spinning test-particles in Cambridge University Press, Cambridge, UK, 1993. general relativity. II,” Proceedings of the Royal Society A,vol.209, [27] K. Nakamura, “Review of particle physics,” Journal of Physics G, pp.259–268,1951. vol. 37, no. 7A, Article ID 075021. [5] A. J. Hanson and T. Regge, “The relativistic spherical top,” [28]G.Spavieri,J.Quintero,G.T.Gillies,andM.Rodr´ıguez, “Asur- Annals of Physics, vol. 87, no. 2, pp. 498–566, 1974. vey of existing and proposed classical and quantum approaches [6] S. Hojman, M. Rosenbaum, M. P. Ryan, and L. C. Shepley, to the photon mass,” European Physical Journal D,vol.61,no.3, “Gauge invariance, minimal coupling, and torsion,” Physical pp. 531–550, 2011. Review D,vol.17,no.12,pp.3141–3146,1978. [29] G. Spavieri and M. Rodriguez, “Photon mass and quantum ef- [7] S. A. Hojman, Electromagnetic and gravitational interactions of fects of the Aharonov-Bohm type,” Physical Review A,vol.75, a spherical relativistic top [Ph.D. thesis], Princeton University, no. 5, Article ID 052113, 2007. 1975. [30] P. A. Franken and G. W. Ampulski, “Photon rest mass,” Physical [8] B. Mashhoon, “Massless spinning test particles in a gravitational Review Letters,vol.26,no.2,pp.115–117,1971. field,” Annals of Physics,vol.89,no.1,pp.254–257,1975. [31]E.R.Williams,J.E.Faller,andH.A.Hill,“Newexperimental [9] S. Hojman, “Lagrangian theory of the motion of spinning test of Coulomb’s Law: a laboratory upper limit on the photon particles in torsion gravitational theories,” Physical Review D, rest mass,” Physical Review Letters,vol.26,no.12,pp.721–724, vol. 18, no. 8, pp. 2741–2744, 1978. 1971. [10] S. A. Hojman and F. A. Asenjo, “Can gravitation accelerate neu- [32] M.A.Chernikov,C.J.Gerber,H.R.Ott,andH.-J.Gerber,“Low- trinos?” http://arxiv.org/abs/1203.5008. temperature upper limit of the photon mass: experimental null test of Amperes` law,” Physical Review Letters,vol.68,no.23,pp. [11] J. Audretsch, “Trajectories and spin motion of massive spin-1/2 3383–3386, 1992. particles in gravitational fields,” Journal of Physics D,vol.14,no. 2, pp. 411–422, 1981. [33] M. A. Chernikov, C. J. Gerber, H. R. Ott, and H.-J. Gerber, “Erratum: Low-temperature upper limit of the photon mass: [12] J. Audretsch, “Dirac electron in space-times with torsion: spinor experimental null test of Ampere’s` law,” Physical Review Letters, propagation, spin precession, and nongeodesic orbits,” Physical vol.69,no.20,p.2999,1992. Review D,vol.24,no.6,pp.1470–1477,1981. [34] A. Accioly, J. Helayel-Neto,¨ and E. Scatena, “Upper bounds on [13] M. Seitz, “Proca field in a spacetime with curvature and torsion,” the photon mass,” Physical Review D,vol.82,no.6,ArticleID ClassicalandQuantumGravity,vol.3,no.6,pp.1265–1273,1986. 065026, 2010. [14] R. Spinosa, “Proca test field in a spacetime with torsion,” [35]J.Luo,L.J.Tu,Z.K.Hu,andE.J.Luan,“Newexperimental Classical and Quantum Gravity,vol.4,no.2,pp.473–484,1987. limit on the photon rest mass with a rotating torsion balance,” [15] R. Spinosa, “Spin-3/2 test field in a spacetime with torsion,” Physical Review Letters,vol.90,ArticleID081801,4pages,2003. ClassicalandQuantumGravity,vol.4,no.6,pp.1799–1808,1987. [36] J.Luo,L.J.Tu,Z.K.Hu,andE.J.Luan,“Reply:,”Physical Review [16] R. Spinosa, “Uber¨ spin-3/2-felder in Raum-Zeiten mit torsion,” Letters,vol.91,ArticleID149102,1page,2003. Annalen der Physik,vol.501,no.3,pp.179–188,1989. [37] D. D. Ryutov, “Relating the proca photon mass and cosmic [17] R. Wald, “Gravitational spin interaction,” Physical Review D,vol. vector potential via solar wind,” Physical Review Letters,vol.103, 6, no. 2, pp. 406–413, 1972. no. 20, Article ID 201803, 2009. [18] B. M. Barker and R. F. O’Connell, “Gravitational two-body [38] R. Lakes, “Experimental limits on the photon mass and cosmic problem with arbitrary masses, spins, and quadrupole mo- magnetic vector potential,” Physical Review Letters,vol.80,no. ments,” Physical Review D,vol.12,no.2,pp.329–335,1975. 9, pp. 1826–1829, 1998. Advances in High Energy Physics 5

[39] L. Davis Jr., A. S. Goldhaber, and M. M. Nieto, “Limit on the [60] D. G. Boulware and W. Gilbert, “Connection between gauge photon mass deduced from Pioneer-10 observations of Jupiter’s invariance and mass,” Physical Review,vol.126,no.4,pp.1563– magnetic field,” Physical Review Letters,vol.35,no.21,pp.1402– 1567, 1962. 1405, 1975. [61] S. Deser, Annales de L’Institut Henri Poincare A,vol.16,p.79, [40]J.V.Hollweg,“Improvedlimitonphotonrestmass,”Physical 1972. Review Letters,vol.32,no.17,pp.961–962,1974. [62] D. G. Boulware and S. Deser, “Aharonov-bohm effect and the [41] E. Fischbach, H. Kloor, R. A. Langel, A. T. Y.Lui, and M. Peredo, mass of the photon,” Physical Review Letters,vol.63,no.21,pp. “New geomagnetic limits on the photon mass and on long- 2319–2321, 1989. range forces coexisting with electromagnetism,” Physical Review Letters,vol.73,no.4,pp.514–517,1994. [42] M. A. Gintsburg, “Structure of the equations of cosmic electrodynamics,” Soviet Astronomy,vol.7,p.536,1964. [43] M. A. Gintsburg, “Structure of the equations of cosmic electrodynamics,” Astronomicheskii Zhurnal,vol.40,p.703,1963. [44] V.L. Patel, “Structure of the equations of cosmic electrodynamics and the photon rest mass,” Physics Letters,vol.14,no.2,pp. 105–106, 1965. [45] A. S. Goldhaber and M. M. Nieto, “Photon and graviton mass limits,” Reviews of Modern Physics,vol.82,no.1,pp.939–979, 2010. [46] G. V. Chibisov, “Astrophysical upper limits on the photon rest mass,” Soviet Physics Uspekhi,vol.19,no.7,pp.624–626,1976. [47] G. V.Chibisov, “Astrophysical upper limits on photon rest mass,” Uspekhi Fizicheskikh Nauk, vol. 119, pp. 551–555, 2002. [48]E.Adelberger,G.Dvali,andA.Gruzinov,“Photon-massbound destroyed by vortices,” Physical Review Letters, vol. 98, no. 1, Article ID 010402, 2007. [49] M. Fullekrug,¨ “Probing the speed of light with radio waves at extremely low frequencies,” Physical Review Letters,vol.93,no. 4, Article ID 043901, 2004. [50] D. S. Robertson, W. E. Carter, and W. H. Dillinger, “New measurement of solar gravitational deflection of radio signals using VLBI,” Nature,vol.349,no.6312,pp.768–770,1991. [51] S. S. Shapiro, J. L. Davis, D. E. Lebach, and J. S. Gregory, “Measurement of the solar gravitational deflection of radio waves using geodetic very-long-baseline interferometry data, 1979–1999,” Physical Review Letters, vol. 92, no. 12, Article ID 121101, 2004. [52] S. B. Lambert and C. Le Poncin-Lafitte, “Determination of the relativistic parameter gammausing very long baseline interferometry,” http://arxiv.org/abs/0903.1615. [53] S. Hojman and B. Koch, Work in progress; we thank S. Deser for stimulating this line of research. [54] A. l. Proca, “Sur la theorie´ ondulatoire des electrons´ positifs et negatifs,”´ JournaldePhysiqueetleRadium,vol.7,no.8,pp.347– 353, 1936. [55] L. Bass and E. Schrodinger,¨ “Must the photon mass be zero?” Proceedings of the Royal Society A,vol.232,pp.1–6,1955. [56] F. Coester, “Quantum electrodynamics with nonvanishing photon mass,” Physical Review, vol. 83, no. 4, pp. 798–800, 1951. [57] H. Umezawa, “On the structure of the interactions of the elementary particles. II. Does the interaction of the second kind exist in the nature?” Progress of Theoretical Physics,vol.7,pp. 551–562, 1952. [58] R. J. Glauber, “On the gauge invariance of the neutral vector meson theory,” Progress of Theoretical Physics,vol.9,pp.295– 298, 1953. [59] E. C. G. Stueckelberg, “Theorie´ de la radiation de photons de masse arbitrairement petite,” Helvetica Physica Acta,vol.30,pp. 209–215, 1957. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2013, Article ID 812084, 8 pages http://dx.doi.org/10.1155/2013/812084

Research Article The Final Stage of Gravitationally Collapsed Thick Matter Layers

Piero Nicolini,1 Alessio Orlandi,2 and Euro Spallucci3

1 Frankfurt Institute for Advanced Studies (FIAS) and Institute for Theoretical Physics, Johann Wolfgang Goethe University, D-60438 Frankfurt am Main, Germany 2 Dipartimento di Fisica, Universita` di Bologna and INFN, I-40126 Bologna, Italy 3 Dipartimento di Fisica, Universita` di Trieste and INFN, I-34151 Trieste, Italy

Correspondence should be addressed to Alessio Orlandi; [email protected]

Received 17 September 2013; Accepted 8 November 2013

Academic Editor: Emil Akhmedov

Copyright © 2013 Piero Nicolini et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In the presence of a minimal length, physical objects cannot collapse to an infinite density, singular, matter point. In this paper, we consider the possible final stage of the gravitational collapse of “thick” matter layers. The energy momentum tensor we choose to model these shell-like objects is a proper modification of the source for “noncommutative geometry inspired,” regular black holes. By using higher momenta of Gaussian distribution to localize matter at finite distance from the origin, we obtain new solutions of the Einstein equation which smoothly interpolates between Minkowski’s geometry near the center of the shell and Schwarzschild’s spacetime far away from the matter layer. The metric is curvature singularity free. Black hole type solutions exist only for “heavy” shells; that is, 𝑀≥𝑀𝑒,where𝑀𝑒 is the mass of the extremal configuration. We determine the Hawking temperature and a modified area law taking into account the extended nature of the source.

1. Introduction In this framework, self-gravitating quantum shells open a window over the still “murky” quantum features of evaporat- Relativistic, self-gravitating matter shells have been thor- ing mini black holes [7–11]. oughly investigated in different sectors of theoretical physics: In this paper, we are going to investigate the static, “...from cosmic inflation to hadronic bags” [1]. Remarkable final stage of collapsed matter shell in the presence ofa applications of gravitational shell models can be found in fundamental minimal length forbidding the shell to contract the framework of inflationary cosmology, where both the into a singular matter point. The emergence of a minimal “birth” and the evolution of vacuum bubbles can be effectively length, as a new fundamental constant of nature on the same described in terms of the dynamics of the boundary surface ground as 𝑐 and ℎ, is a general feature of different approaches engulfing a false vacuum domain [1–4]. Matter shells in to quantum gravity [12–14]. general relativity are modeled as “zero-thickness” membranes In recent years, we showed that the very concept of endowed with some characteristic tension determined by “point particle” is meaningless if there exists a lower bound the underlying classical, or quantum, physics. Neglecting the to physically measurable lengths. For instance, in a series realwidthofthemass-energydistributionaffectsEinstein’s of papers the repercussions of a natural ultraviolet cutoff equations by introducing a surface of discontinuity in the have been analyzed in the context of quantum field theory background spacetime. This approximation allows to encode [15–20].Thisbasicnotioncanbealsoencodedintothe all the dynamics in the matching condition between the inner Einstein equations through a proper choice of the energy and outer geometries [5, 6]. Furthermore, contracting matter momentum tensor. The most remarkable outcomes of this shells provide useful analytic toy models of collapsing massive procedurearethedisappearanceofcurvaturesingularitiesin bodies leading to black hole formation. In the spherically thesolutionsoftheEinsteinequations[21], a regular behavior symmetric case, the only dynamical degree of freedom is of the Hawking temperature which allows to determine the givenbytheshellradiusandthesystemcanbequantized physical character of the evaporation remnant [22–26], and according to the standard principle of quantum mechanics. a different form of the relation between entropy and area of 2 Advances in High Energy Physics

the event horizon reproducing the celebrated area law for where 𝑇𝜇] = diag(𝜌,𝑟 𝑝 ,𝑝⊥,𝑝⊥). The energy density is a min- large, semiclassical black holes [27, 28]. imal width Gaussian distribution Theprocedurehasbeenappliedinthewholearrayof 𝑀 −𝑟2/4𝜃 physically meaningful black hole solutions like the neutral, 𝜌=𝜌0 (𝑟) ≡ 𝑒 , (4𝜋𝜃)3/2 (2) nonrotating case [29–32], the charged, nonrotating case [33, 34], the “dirty,” neutral, and nonrotating case [35], and the representing a “blob-like” object, centred around the origin, spinning, neutral [36]andchargedcases[37]. In addition, with a characteristic extension given by 𝑙∝√𝜃. A simple way the regularity of the above metrics has been exploited in toderivetheaboveenergyprofileisbasedonthefollowing several complementary contexts like the decay of the de Sitter considerations. At classical level point-like objects in spheri- universe by quantum black hole nucleation [38]andthecase cal coordinates are described by a profile of dimensionally reduced spacetimes [39]. The procedure we 𝑀 followed has been also recognized as a special result [40]of 𝜌 (𝑟) = 𝛿 (𝑟) , recently formulated nonlocal gravity proposals [41, 42]with cl 4𝜋𝑟2 (3) important cross-fertilization in the AdS/CFT paradigm [43]. 𝛿(𝑟) The paper is organized as follows. In Section 2,wewill where is the Dirac delta. We recall that a Dirac delta function can be represented as the derivative of a Heaviside derive new solutions of Einstein’s equations describing spher- Θ(𝑟) ically symmetric, static, self-gravitating, thick matter layers in step-function the presence of a fundamental minimal length. These types 𝑑 𝛿 (𝑟) = Θ (𝑟) . (4) of objects are modeled by means of a proper modification of 𝑑𝑟 thesourcefor“noncommutativegeometryinspired,”regular black holes. We replace the Gaussian profile of mass-energy, However, in the presence of a minimal length the very con- peaked around the origin, with higher moments of Gaussian cept of sharp step is no longer meaningful. Rather we expect distribution with maxima shifted at finite distance from the the local loss of resolution to sweeten the step. Accordingly, origin. The finite width of the distribution is determined it can be shown that in the framework of a minimal length by the minimal length. In the case of lump-type objects, a modified step function can be defined through an integral the minimal length measures the spread of mass-energy representation of the Heaviside function without taking the around the origin and removes the curvature singularity. For limit √𝜃→0[44]: finite width matter layers, with energy density vanishing at 𝑟 1 2 short distance, not only the central curvature singularity is Θ (𝑟) 󳨀→ 𝑃 (𝑟) = ∫ 𝑥2𝑒−𝑥 /4𝜃𝑑𝑥. 0 √ 3/2 (5) removed, but the extrinsic curvature discontinuity between 4𝜋𝜃 0 “inner” and “outer” geometry is cured as well. Spacetime The Gaussian profile2 ( ) is therefore obtained as geometry is continuous and differentiable everywhere and smoothly interpolates between Minkowski’s metric near the 𝑀 𝑑 𝜌 (𝑟) = 𝑃 (𝑟) . (6) center and Schwarzschild’s spacetime far away from the 0 4𝜋𝑟2 𝑑𝑟 0 matter layer. As in the cases previously discussed, black hole type In order to solve Einstein’s equations, we need to determine the remaining components of the energy-momentum tensor. solutions exist only for “heavy” shells; that is, 𝑀≥𝑀𝑒,where The radial pressure 𝑝𝑟 is fixed by the equation of state 𝑝𝑟 = 𝑀𝑒 is the mass of the extremal configuration. “Light” matter −𝜌 reproducing the de Sitter “vacuum” equation of state at layers with 𝑀<𝑀𝑒 will settle down in a smooth solitonic type configuration with no horizons or curvature singularity. short distance. This is a key feature to build up a regular, In Section 3, we study the thermodynamic properties stable configuration, where the negative pressure balances the of black hole solutions and determine both the Hawking gravitational pull. In other words, the singularity theorem is evaded by a violation of null energy condition triggered by temperature, 𝑇𝐻, and the relation between entropy and area of the horizon. We recover the celebrated area law in the the short distance vacuum fluctuations. Finally, the tangential pressure 𝑝⊥ is obtained in terms of limit of large, semiclassical black holes. On a general ground, 𝜌 ∇ 𝑇𝜇] =0 we find the leading term is one-fourth of the area butin by the divergence-free condition 𝜇 . Asthesourceisstaticandsphericallysymmetric,theline units of an “effective” gravitational coupling constant, 𝐺𝑁(𝑟+), element can be cast in the form depending on the radius of black hole. For “large” 𝑟+,the Newton constant is recovered. Finally, in Section 4,wewill 𝑑𝑟2 𝑑𝑠2 =−𝑓(𝑟) 𝑑𝑡2 + +𝑟2Ω2 draw the conclusions. 𝑓 (𝑟) (7)

2. Thick Shells with 2𝐺 𝑚 (𝑟) 𝑓 (𝑟) =1− 𝑁 . (8) InapreviousseriesofpaperswesolvedtheEinsteinequations 𝑟 including the effects of a minimal length on a proper energy- momentum tensor The cumulative mass distribution 𝑚(𝑟) is given by 1 𝑟 2 𝑚 𝑟 =4𝜋∫ 𝑑𝑟󸀠(𝑟󸀠) 𝜌 (𝑟󸀠). 𝑅𝜇] − 𝑔𝜇]𝑅=8𝜋𝐺𝑁𝑇𝜇], (1) ( ) 0 (9) 2 0 Advances in High Energy Physics 3

1 0.005

0.004 0.8 0.003

0.002 0.6 0.001 r) ( k P

0.4 2 4 6 8 10

Figure 2: Plot of the radial density profile 𝜌(𝑟) as a function of 𝑟 in √𝜃 𝑘=1 𝑘=2 𝑘=7 𝑘=8 0.2 -units for (cyan), (blue), (red), (pink), 𝑘=20(green), and 𝑘=21(yellow). The function 𝜌(𝑟) has been normalized so that its integration over a spherical volume gives 1.

0 0 5 10 15 20 25 function. By replacing 𝑃0 with a generic 𝑃𝑘 in (6), we derive r the following energy density profile: √𝜃 2𝑘 −𝑟2/4𝜃 √ 𝑟 𝑒 Figure 1: Plot of the function 𝑃𝑘(𝑟) as a function of 𝑟/ 𝜃 for 𝑘=0 𝜌 (𝑟) ≡𝑀 , 𝑘 𝑘+2 𝑘+3/2 (13) (purple), 𝑘=1(cyan), 𝑘=2(blue), 𝑘=7(red), 𝑘=8(pink), 4 𝜋𝜃 Γ (𝑘+3/2) 𝑘=20(green), and 𝑘=21(yellow). For comparison, the Heaviside function is plotted in black. which corresponds to higher moments of the Gaussian. For 𝑘=0,thefunction𝜌𝑘 turns into the Gaussian distribution, centred around the origin, while for 𝑘≥1the Wenotice that the parameter 𝑀 corresponds to the total mass matter distribution is more and more diluted near the origin, √ energy of the system; namely, being peaked at 𝑟𝑀 =2 𝑘𝜃. As a result, the density function (13) describes a whole family of “mass-degenerate” shells, 𝑀= 𝑚 (𝑟) . 𝑟→∞lim (10) with the same 𝑀, but concentrated at a distance given by 𝑟𝑀 (see Figure 2). Equation (13) discloses further properties The above profile cures the usual Dirac delta (singular) dis- of the minimal length, which assumes a new intriguing tributionassociatedtoapointparticle,andleadstoafamily meaning. In the case of point particle, √𝜃 represents the of regular black hole solutions [29–39]. spread of the object around the origin. For matter shells, we A crucial point at the basis of the above line of reasoning see that √𝜃 relatestotheshellthicknessor,inotherwords,a is the modification of the step function in the presence ofa measure of the intrinsic fuzziness of the layer. Just as a particle minimal length. We notice that the profile5 ( )isjustoneof cannot be exactly localized at a single point, we cannot have thepossiblechoicesonecanhavetoaccountforthelossof zero-width layers as well. Thus, as the matter distribution is resolution of the edge of the step [44]. In other words, there smooth everywhere, there is no discontinuity in the extrinsic exist alternative representations of the Heaviside function for √𝜃→0 curvature between the “inner” and “outer” geometries. No which one can deliberately avoid the limit .For matching condition is required and we can look for a single, instance, the family of distributions smooth, metric inside, across, and outside the matter layer. 𝑟 1 (𝑘+1)! 2 We can define the distance between two shells corre- Θ (𝑟) 󳨀→ 𝑃 (𝑟) = ∫ 𝑥2𝑘+2𝑒−𝑥 /4𝜃𝑑𝑥 𝑘 𝑘+3/2 sponding to different moments as the distance between the √𝜋𝜃 [2 (𝑘+1)]! 0 peaks: (11) Δ𝑟 account for minimal length effects growing with the index 𝑘, 𝑀 = √𝑘+1−√𝑘. √ (14) where 𝑘 = 0, 1, 2, . . is a natural number (see Figure 1). This 2 𝜃 canbeseenbythelimit𝑟/√𝜃≫1 We see that for higher moments Δ𝑟𝑀 vanishes as we are at 𝑘+1 2𝑘+1 √𝜃 4 (𝑘+1)! 𝑟 2 length scale much larger than and the relative distance 𝑃 (𝑟) ≈1− ( ) 𝑒−𝑟 /4𝜃 (12) 𝑘 √𝜋 [2 (𝑘+1)]! 2√𝜃 cannot be resolved anymore. A stable solution of the Einstein equations can be which shows that the distributions 𝑃𝑘(𝑟) reach the plateau obtained by sourcing the gravitational field by the energy 𝑝 = (i.e., 𝑃𝑘 ≈1)moreslowlyas𝑘 increases. momentum tensor of an anisotropic fluid; the choice 𝑟 −1 Here, we want to explore the nature of the solutions −𝜌𝑘 for the matter equation of state allows to have 𝑔00 =−𝑔𝑟𝑟 . emerging for the above admissible profiles for a modified step Furthermore, the hydrodynamic equilibrium equation will 4 Advances in High Energy Physics

1.5 40

1.0 30 0.5 ) H 20 r) ( U(r

00 2 4 6 8 10 g −0.5 r 10

−1.0 2 4 6 8 10

−1.5 rH 𝑔 (𝑟) 𝑟 √𝜃 √ Figure 3: Plot of 00 as a function of in -units for fixed value Figure 4: Plot of 𝑈(𝑟𝐻) as a function of 𝑟𝐻 in 𝜃-units for 𝑘=0 of 𝑘=1and different values of the mass 𝑀;thatis,𝑀=1(blue), (blue), 𝑘=1(magenta), and 𝑘=4(brown). 𝑀=𝑀𝑒 (magenta), and 𝑀=5(brown). Here, 𝑀𝑒 ≃ 2.36976 is the extremal mass for the case 𝑘=1. points for the motion of a “test particle” of energy 𝑀 subject to the “potential” 𝑈(𝑟𝐻).Theintersection(s)betweenaline give 𝑝⊥ in terms of 𝜌𝑘, while the metric itself results in being 𝑀=const.andtheplotofthefunction𝑈(𝑟𝐻),inthe independent from 𝑝⊥. The solution of the Einstein equations plane 𝑀-𝑟𝐻, represent the allowed radii of inner/outer event reads in geometric units, 𝑐=1, 𝐺𝑁 =1: horizons (see Figure 4). The asymptotic behavior of 𝑈(𝑟𝐻) is 2𝑚 (𝑟) 2𝑚 (𝑟) −1 obtained from the corresponding approximate forms of 𝛾: 𝑑𝑠2 =−(1− )𝑑𝑡2 +(1− ) 𝑑𝑟2 +𝑟2𝑑Ω2, 𝑟 𝑟 𝑟 𝑈(𝑟 )≈ 𝐻 ,𝑟≫ √𝜃, (15) 𝐻 2 𝐻 2 −(𝑘+1) (20) 𝑟 𝛾(𝑘+3/2;𝑟 /4𝜃) 𝑟2 󸀠 󸀠2 󸀠 √ 5 𝐻 √ 𝑚 (𝑟) ≡4𝜋∫ 𝑑𝑟 𝑟 𝜌(𝑟 )=𝑀 , (16) 𝑈(𝑟𝐻)≈ 𝜃Γ (𝑘 + )( ) ,𝑟𝐻 ≪ 𝜃. 0 Γ (𝑘+3/2) 2 4𝜃

2 where 𝛾(𝑘+3/2;𝑟 /4𝜃) is the lower, incomplete Euler Gamma Thus, 𝑈(𝑟𝐻) is a convex function with a single minimum function. By taking into account the asymptotic form of 𝛾(𝑘+ determined by the condition 3/2; 𝑟2/4𝜃) for small argument, one finds that the metric (15) 2𝑘+3 2 𝑑𝑈 𝑟𝑒 3 𝑟𝑒 𝑟2/4𝜃 is essentially flat near the origin, the contrary to what happens =0󳨀→ =𝛾(𝑘+ ; ) 𝑒 𝑒 . 2𝑘+2 𝑘+3/2 (21) for the case 𝑘=0in which a regular de Sitter core forms. 𝑑𝑟𝐻 2 𝜃 2 4𝜃 Indeed, one finds that The radius 𝑟𝑒 represents the size of an extremal configuration 2 𝑘+3/2 1 4𝑀 𝑟 with a couple of degenerate horizons: 𝑟− =𝑟+ ≡𝑟𝑒. 𝑓 (𝑟) ≃1− [ ]( ) , (17) 2𝑘 + 3 𝑟Γ (𝑘+3/2) 4𝜃 Once 𝑟𝑒 is numerically determined from (21), one gets the corresponding mass 𝑀𝑒 from the potential: 𝑑𝑠2 =−(1−𝑂(𝑟2𝑘+2)) 𝑑𝑡2 𝑀𝑒 =𝑈(𝑟𝑒)=𝑈min . (22) (18) 2𝑘+2 −1 2 2 2 +(1−𝑂(𝑟 )) 𝑑𝑟 +𝑟 𝑑Ω . An order of magnitude estimate for 𝑟𝑒 and 𝑀𝑒 can be obtained by keeping only the leading theta term: The asymptotic behavior (18) can be seen as a generalization ofthe“GaussTheorem”:insideanempty,classical,thinshell √𝜃 𝑟 ∝ √𝜃󳨀→𝑀 ∝ 𝑀 , (23) of matter, the Newtonian gravitational field is zero; that is, 𝑒 𝑒 𝐿 Pl. Pl. spacetime is flat. Introducing a finite width density, smoothly decreasing both toward the origin and space-like infinity, 𝐿 𝐿 =𝑀−1 = √𝐺 where Pl. is the Planck length; that is, Pl. Pl. 𝑁. causes small deviations from perfect flatness. This estimate suggests that (near/)extremal configurations Going back to (15), the presence of event horizons is read are close to a full quantum gravity regime. As such, they −1 from the zeroes of 𝑔𝑟𝑟 (see Figure 3). Due to the nontrivial are appropriate candidates to describe the end-point of structureofthemetric,itismoreconvenienttolookatthe the Hawking evaporation process where the semiclassical “horizon equation” in the form description breaks down. In summary 𝑟𝐻 Γ (𝑘+3/2) 𝑀≡𝑈(𝑟𝐻), 𝑈(𝑟𝐻)≡ . 2 (19) 𝑀>𝑀𝑒 2 𝛾(𝑘+3/2;𝑟𝐻/4𝜃) (i) for , we find a geometry with noncoincident inner and outer horizons 𝑟− <𝑟+.Itisworthto The problem of finding the horizons in the metric (15)is remark that there is no curvature singularity in 𝑟=0. mapped in the equivalent problem of determining the turning Spacetime is flat near the origin; Advances in High Energy Physics 5

(ii) for 𝑀=𝑀𝑒,wehaveanextremal configuration with 0.05 a pair of degenerate horizons, 𝑟− =𝑟+ ≡𝑟𝑒. 0.04 (iii) for 𝑀<𝑀𝑒, there are neither horizons nor curvature singularities. The metric is smooth and regular every- 0.03

where. The shell is too light and diluted to produce ) any relevant alteration of the spacetime fabric. After + collapse, it will settle into a sort of solitonic object with T(r 0.02 no horizon of curvature singularity. 0.01 Massive layers will produce horizons, but no curvature singularities. This is a crucial point which marks a departure with respect to all the existing literatures. Our approach must not 0 2 4 6 8 10 12 14 be confused with previous contributions in which generalized r+ matter shells containing polytropic and Chaplygin gas cannot √ ultimately resolve the emergence of singularities [45]. Figure 5: Plot of 𝑇(𝑟+) as a function of 𝑟+ in 𝜃-units for 𝑘=0 We can estimate the size of these objects by solving iter- (blue), 𝑘=1(magenta), 𝑘=4(brown), and 𝑘=15(green). The atively (19) and truncating the procedure at the first order in dashed line represents the classical case; that is, 𝜃=0. 2 the expansion parameter exp(−𝑀 /𝜃):

𝑘+1/2 (𝑀2/𝜃) Table 1: Some values of parameters of the matter shells. [ −𝑀2/𝜃] 𝑟+ ≃2𝑀 1− 𝑒 . (24) 𝑟 𝑀 𝑇 𝑟 Γ (𝑘+3/2) 𝑒 𝑒 max max [ ] 𝑘=0 3.0224 1.9041 0.014937 4.76421 𝑘=1 This quantity can be compared with the shell mean radius 4.0431 2.3698 0.012783 5.72632 𝑘=4 5.9269 3.2647 0.009960 7.56069 ∞ 4𝜋 2 Γ (𝑘+2) 𝑘=15 9.7347 5.1245 0.006799 11.3419 ⟨𝑟⟩ ≡ ∫ 𝑑𝑟𝑟 𝑟𝜌 (𝑟) =2√𝜃 . (25) 𝑀 0 Γ (𝑘+3/2)

We notice that for 𝑘≫1the leading contribution to the 𝑟 /⟨𝑟⟩ temperature corresponds to an infinite discontinuity of the ratio + is given by heat capacity which is usually interpreted as the signal of a “change of state” for the system. For 𝑟+ >𝑟max, the black hole 𝑟+ 𝑀 1 ≈ . (26) is thermodynamically unstable and increases its temperature ⟨𝑟⟩ √𝜃 √𝑘+1 by radiating away its own mass. After crossing 𝑟max,that 𝑟 <𝑟 Thus, for an assigned 𝑀 the ratio decreases with 𝑘.The is, for + max,theblackholeentersastabilityphase horizon radius is determined by the total mass energy 𝑀 and asymptotically ending into a degenerate (zero-temperature) extremal configuration. By increasing 𝑘, we just lower down is weakly 𝑘 dependent. On the contrary, ⟨𝑟⟩ ∝ √𝑘+1√𝜃 the maximum temperature. As a result, the solution is and grows with 𝑘, which means that for higher moments the unaffected by any relevant quantum back reaction as already horizon is surrounded by a cloud of matter. proved for the case 𝑘=0in [31]. A further important consequence of 𝑇𝐻 ≤𝑇max is that (in 3. Thermodynamics extradimensional models) the black hole mainly radiates on the brane [46],itneverbecomeshotenoughtowarmupthe Massive shells will collapse into black holes described by bulk in a significant way. the line element (7), (8). This is not the end of story as The area-entropy law can be recovered from the relation these objects are semiclassically unstable under Hawking’s emission. We are now ready to study the thermodynamic 𝑑𝑀𝐻 =𝑇 𝑑𝑆, (28) properties of these solutions starting from the Hawking temperature: where 𝑑𝑆 is the horizon entropy variation triggered by a 2𝑘+3 −𝑟2 /4𝜃 𝑑𝑀 𝑀 1 𝑟 𝑒 + variation in the total mass energy . In order to translate 𝑇 = [1 − + ]. 𝐻 4𝜋𝑟 22𝑘+2𝜃𝑘+3/2 𝛾(𝑘+3/2;𝑟2 /4𝜃) (27) the first law of black hole thermodynamics28 ( )intoarelation + + involving the area of the event horizon, we need to write mass energy variation as Itcanbeeasilyverifiedthat𝑇𝐻 is vanishing for the extremal 𝑇 (𝑟 =𝑟)=0 𝑘=0 configuration 𝐻 + 𝑒 .For ,weobtain 𝜕𝑈 the temperature of the black hole as in [31]. The behavior 𝑑𝑀 = 𝑑𝑟+ (29) ofthetemperaturecanbefoundinFigure 5 and Table 1.We 𝜕𝑟+ notice that the regularity of the manifold for any 𝑘 leads to a cooling down of the horizon in the terminal phase of the andtotakeintoaccountthattheminimumof𝑈(𝑟+) is the Hawking process. At 𝑟+ =𝑟max, the presence of a maximum mass of the extremal black hole. Thus, when integrating 6 Advances in High Energy Physics

(29), the lower integration limit is the radius of the extremal 4. Conclusions configuration In this paper, we derived new static, spherically symmetric 𝑟 + 1 𝜕𝑈 𝑆= ∫ 𝑑𝑥 regular solutions of Einstein’s equations, describing the final 𝑇 𝜕𝑥 state of collapsing matter shells in the presence of an effective 𝑟𝑒 𝐻 minimal length. This derivation is the result of a long path 𝑟 (30) 3 + 𝑟 =2𝜋Γ(𝑘+ ) ∫ 𝑑𝑟 , starting with the quest of quantum gravity regularized black 2 2 𝑟𝑒 𝛾 (𝑘 + 3/2; 𝑟 /4𝜃) hole solutions [21, 29–40]. We showed that our previous discoveries of noncommutative geometry inspired solutions wherewehaveinserted(27)and(29)into(30). By performing correspond to the simplest case within a larger class of regular the integral, one gets gravitational objects. This class, the family of gravitational shells here presented, has peculiar properties which descend 3 𝑆=𝜋Γ(𝑘+ ) from the common key feature of the absence of curvature 2 singularities. Such properties are the existence of extremal 𝑟2 𝑟2 configurations, characterized by a minimal mass below which ×( + − 𝑒 ) horizon cannot form, even in the case of neutral, nonrotating 𝛾 (𝑘 + 3/2; 𝑟2 /4𝜃) 𝛾(𝑘+3/2;𝑟2/4𝜃) (31) + 𝑒 black holes; the occurrence of a phase transition from a 𝑟 󸀠 thermodynamically unstable classical phase to a thermody- 3 + 𝛾 +𝜋Γ(𝑘+ ) ∫ 𝑑𝑟𝑟2 . 2 namically stable cooling down in the final stage of the horizon 2 𝑟 𝛾 𝑒 evaporation. As for the case of regular black holes, these matter shells correspond to final configurations of dynamical The first term can be written in terms of the area of the event processes of gravitational collapse in which matter cannot horizon as be compressed below a fundamental length scale. The above 1 3 properties are also common to other models of quantum 𝑆= Γ(𝑘+ ) geometry that have been derived by means of different 4𝐺𝑁 2 formalisms. Specifically the shell profile of the energy density 𝐴 𝐴 resembles what one has in the case of loop quantum black ×( 𝐻 − 𝑒 )+⋅⋅⋅ 2 2 holes, with consequent thermodynamic similarities [47–50]. 𝛾(𝑘+3/2;𝑟+/4𝜃) 𝛾 (𝑘 + 3/2;𝑒 𝑟 /4𝜃) (32) Thestudyoftheseobjectsisfarfrombeingcomplete.We believe that these new solutions can have repercussions in andrepresentsthe“arealaw”inourcase.Wehavereinserted a variety of fields. Here we just mention that these regular the Newton constant into (32) for reasons to become clear in shells could affect the stability of the de Sitter space, at a while. The standard form, which is one-fourth of the area, least during inflationary epochs [38]andtheycouldlead √ is recovered in the large black hole limit, 𝑟𝐻 ≫ 𝜃. tonewinsightsintothephysicsofnuclearmatterviathe Once (32) is written in natural units, we can define an gauge/gravity duality paradigm as well [43]. effective Newton constant as Acknowledgments 𝛾(𝑘+3/2;𝑟2 /4𝜃) 𝐺 󳨀→ 𝐺 (𝑟 )≡𝐺 + (33) 𝑁 𝑁 + 𝑁 Γ (𝑘+3/2) This work has been supported by the project “Evaporation of microscopic black holes” under the Grant NI 1282/2-1 of and introduce a “modified” area law as the German Research Foundation (DFG), by the Helmholtz International Center for FAIR within the framework of 𝜋𝑟2 𝑆(𝑟 )= 𝐻 . the LOEWE program (Landesoffensive zur Entwicklung 𝐻 (34) ¨ 4𝐺𝑁 (𝑟𝐻) Wissenschaftlich-Okonomischer Exzellenz) launched by the State of Hesse and in part by the European Cooperation The interesting feature of 𝐺𝑁(𝑟𝐻) is to be “asymptotically free” in Science and Technology (COST) action MP0905 “Black in the sense that it is smaller than 𝐺𝑁, which represents the Holes in a Violent Universe.” Alessio Orlandi would like to asymptotic value of the gravitational coupling for large black thank the Frankfurt Institute for Advanced Studies (FIAS), holes only (it is interesting to remark that if we apply the Frankfurt am Main, Germany, for the initial period of work- same reasoning to the Newton constant in (8), and replace ing on this project. Piero Nicolini and Alessio Orlandi would 𝑟𝐻 with the radial coordinate 𝑟, we find a radial distance- like to thank the Perimeter Institute for Theoretical Physics, dependent gravitational coupling, which is quickly vanishing Waterloo, ON, Canada, for the kind hospitality during the in the limit 𝑟→0. This asymptotically free behavior of our final period of working on this project. model, at short distance, provides an alternative explanation 𝑟=0 for the absence of curvature singularity in ). References Finally,thesecondtermin(31) gives exponentially small corrections to the leading area term. It can be analytically [1] A. Aurilia, R. S. Kissack, R. Mann, and E. Spallucci, “Relativistic computed for 𝑘≫1, but it does not introduce any relevant bubble dynamics: from cosmic inflation to hadronic bags,” new effect. Physical Review D,vol.35,no.10,pp.2961–2975,1987. Advances in High Energy Physics 7

[2] A. Aurilia, G. Denardo, F. Legovini, and E. Spallucci, “Vacuum [22] R. Banerjee, B. R. Majhi, and S. Samanta, “Noncommutative tensioneffectsontheevolutionofdomainwallsintheearly black hole thermodynamics,” Physical Review D,vol.77,no.12, universe,” Nuclear Physics B,vol.252,pp.523–537,1985. Article ID 124035, 2008. [3]S.K.Blau,E.I.Guendelman,andA.H.Guth,“Dynamicsof [23] R. Casadio and P. Nicolini, “The decay-time of non-commu- false-vacuum bubbles,” Physical Review D,vol.35,no.6,pp. tative micro-black holes,” JournalofHighEnergyPhysics,vol. 1747–1766, 1987. 2008,no.11,article072,2008. [4]E.Farhi,A.H.Guth,andJ.Guven,“Isitpossibletocreate [24] Y. S. Myung, Y.-W. Kim, and Y.-J. Park, “Thermodynamics and a universe in the laboratory by quantum tunneling?” Nuclear evaporation of the noncommutative black hole,” Journal of High Physics B,vol.339,no.2,pp.417–490,1990. Energy Physics,vol.2007,no.2,article012,2007. [5] W. Israel, “Singular hypersurfaces and thin shells in general [25] Y. S. Myung, Y.-W. Kim, and Y.-J. Park, “Quantum cooling relativity,” Il Nuovo Cimento B,vol.44,no.1,pp.1–14,1966. evaporation process in regular black holes,” Physics Letters B, vol. 656, no. 4-5, pp. 221–225, 2007. [6] W. Israel, “Singular hypersurfaces and thinshells in general relativity,” Il Nuovo Cimento B,vol.48,no.2,p.463,1967. [26] Y. S. Myung, Y. W. Kim, and Y. J. Park, “Thermodynamics of regular black hole,” General Relativity and Gravitation,vol.41, [7]S.Ansoldi,A.Aurilia,R.Balbinot,andE.Spallucci,“Classical no.5,pp.1051–1067,2009. and quantum shell dynamics, and vacuum decay,” Classical and [27] R. Brustein, D. Gorbonos, and M. Hadad, “Wald’s entropy is Quantum Gravity,vol.14,no.10,pp.2727–2755,1997. equaltoaquarterofthehorizonareainunitsoftheeffective [8] G. L. Alberghi, R. Casadio, and G. Venturi, “Effective action gravitational coupling,” Physical Review D,vol.79,no.4,Article and thermodynamics of radiating shells in general relativity,” ID 044025, 9 pages, 2009. Physical Review D,vol.60,no.12,ArticleID124018,11pages, [28] P. Nicolini, “Entropic force, noncommutative gravity, and 1999. ungravity,” Physical Review D,vol.82,no.4,ArticleID044030, [9] S. Ansoldi, “WKB metastable quantum states of a de Sitter- 2010. Reissner-Nordstrom dust shell,” Classical and Quantum Grav- [29] P.Nicolini, “Amodel of radiating black hole in noncommutative ity,vol.19,no.24,pp.6321–6344,2002. geometry,” Journal of Physics A,vol.38,no.39,pp.L631–L638, [10] G. L. Alberghi, R. Casadio, and G. Venturi, “Thermodynamics 2005. for radiating shells in anti-de Sitter space-time,” Physics Letters [30] P.Nicolini, A. Smailagic, and E. Spallucci, “The fate of radiating B, vol. 557, no. 1-2, pp. 7–11, 2003. black holes in noncommutative geometry,” in Proceedings of the [11] G. L. Alberghi, R. Casadio, and D. Fazi, “Classical dynamics 13thGeneralConferenceofEuropeanPhysicalSociety;Beyond and stability of collapsing thick shells of matter,” Classical and Einstein, European Space Agency, November 2005. Quantum Gravity,vol.23,no.5,pp.1493–1506,2006. [31] P. Nicolini, A. Smailagic, and E. Spallucci, “Noncommutative [12] L. J. Garay, “Quantum gravity and minimum length,” Interna- geometry inspired Schwarzschild black hole,” Physics Letters B, tionalJournalofModernPhysicsA,vol.10,no.2,pp.145–165, vol. 632, no. 4, pp. 547–551, 2006. 1995. [32]E.SpallucciandS.Ansoldi,“RegularblackholesinUVself- [13] X. Calmet, M. Graesser, and S. D. H. Hsu, “Minimum length complete quantum gravity,” Physics Letters B,vol.701,no.4,pp. from quantum mechanics and classical general relativity,” Phys- 471–474, 2011. ical Review Letters, vol. 93, no. 21, Article ID 211101, 2004. [33] S. Ansoldi, P. Nicolini, A. Smailagic, and E. Spallucci, “Non- [14] M. Fontanini, E. Spallucci, and T. Padmanabhan, “Zero-point commutative geometry inspired charged black holes,” Physics length from string fluctuations,” Physics Letters B,vol.633,no. Letters B,vol.645,no.2-3,pp.261–266,2007. 4-5, pp. 627–630, 2006. [34] E. Spallucci, A. Smailagic, and P. Nicolini, “Non-commutative geometry inspired higher-dimensional charged black holes,” [15] A. Smailagic and E. Spallucci, “UV divergence-free QFT on Physics Letters B,vol.670,no.4-5,pp.449–454,2009. noncommutative plane,” Journal of Physics A,vol.36,no.39,pp. L517–L521, 2003. [35] P. Nicolini and E. Spallucci, “Noncommutative geometry- inspired dirty black holes,” Classical and Quantum Gravity,vol. [16] A. Smailagic and E. Spallucci, “Feynman path integral on the 27, no. 1, Article ID 015010, 2010. non-commutative plane,” Journal of Physics A,vol.36,no.33, [36] A. Smailagic and E. Spallucci, “‘Kerrr’ black hole: the lord of the pp.L467–L471,2003. string,” Physics Letters B,vol.688,no.1,pp.82–87,2010. [17] A. Smailagic, E. Spallucci, and Lorentz invariance, “Lorentz [37] L. Modesto and P.Nicolini, “Charged rotating noncommutative invariance and unitarity in UV finite NCQFT,” Journal of Physics black holes,” Physical Review D,vol.82,no.10,ArticleID104035, A,vol.37,pp.1–10,2004. 2010. [18] A. Smailagic and E. Spallucci, “Lorentz invariance, unitarity and [38] R. B. Mann and P. Nicolini, “Cosmological production of non- UV-finiteness of QFT on noncommutative spacetime,” Journal commutative black holes,” Physical Review D,vol.84,no.6, of Physics A,vol.37,no.28,pp.7169–7178,2004. Article ID 064014, 2011. [19]E.Spallucci,A.Smailagic,andP.Nicolini,“Traceanomalyona [39] J. R. Mureika and P. Nicolini, “Aspects of noncommutative (1 + quantum spacetime manifold,” Physical Review D,vol.73,no.8, 1)-dimensional black holes,” Physical Review D,vol.84,no.4, Article ID 084004, 2006. Article ID 044020, 2011. [20] M. Kober and P. Nicolini, “Minimal scales from an extended [40] L. Modesto, J. W. Moffat, and P. Nicolini, “Black holes in an hilbert space,” Classical and Quantum Gravity,vol.27,no.24, ultraviolet complete quantum gravity,” Physics Letters B,vol. Article ID 245024, 2010. 695, no. 1–4, pp. 397–400, 2011. [21] P. Nicolini, “Noncommutative black holes, the final appeal to [41] P. Gaete, J. A. Helayel-Neto,andE.Spallucci,“Un-graviton¨ quantum gravity: a review,” International Journal of Modern corrections to the Schwarzschild black hole,” Physics Letters B, Physics A,vol.24,no.7,pp.1229–1308,2009. vol. 693, no. 2, pp. 155–158, 2010. 8 Advances in High Energy Physics

[42] J. R. Mureika and E. Spallucci, “Vector unparticle enhanced black holes: exact solutions and thermodynamics,” Physics Letters B,vol.693,no.2,pp.129–133,2010. [43] P. Nicolini and G. Torrieri, “The Hawking-Page crossover in noncommutative anti-deSitter space,” Journal of High Energy Physics, vol. 2011, article 97, 2011. [44] J. Mureika, P. Nicolini, and E. Spallucci, “Could any black holes be produced at the LHC?” Physical Review D,vol.85,no.10, Article ID 106007, 8 pages, 2012. [45] J. J. Oha and C. Park, “Gravitational collapse of the shells with the smeared gravitational source in noncommutative geometry,” JournalofHighEnergyPhysics, vol. 2010, article 86, 2010. [46] P. Nicolini and E. Winstanley, “Hawking emission from quantum gravity black holes,” Journal of High Energy Physics,vol. 2011, article 75, 2011. [47] L. Modesto, “Loop quantum black hole,” Classical and Quantum Gravity,vol.23,no.18,pp.5587–5601,2006. [48] L. Modesto and I. Premont-Schwarz, “Self-dual black holes in LQG: theory and phenomenology,” Physical Review D,vol.80, no.6,ArticleID064041,17pages,2009. [49] S. Hossenfelder, L. Modesto, and I. Premont-Schwarz,´ “Model for nonsingular black hole collapse and evaporation,” Physical Review D,vol.81,no.4,ArticleID044036,7pages,2010. [50] L. Modesto, “Semiclassical loop quantum black hole,” Interna- tional Journal of Theoretical Physics,vol.49,no.8,pp.1649–1683, 2010. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2013, Article ID 873686, 6 pages http://dx.doi.org/10.1155/2013/873686

Research Article The 5D Standing Wave Braneworld with Real Scalar Field

Merab Gogberashvili1,2 and Pavle Midodashvili3

1 Andronikashvili Institute of Physics, 6 Tamarashvili Street, 0177 Tbilisi, Georgia 2 Javakhishvili State University, 3 Chavchavadze Avenue, 0128 Tbilisi, Georgia 3 Ilia State University, 3/5 Cholokashvili Avenue, 0162 Tbilisi, Georgia

Correspondence should be addressed to Merab Gogberashvili; [email protected]

Received 22 October 2013; Accepted 1 December 2013

Academic Editor: Douglas Singleton

Copyright © 2013 M. Gogberashvili and P. Midodashvili. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We introduce the new 5D braneworld with the real scalar field in the bulk. The model represents the brane which bounds collective oscillations of gravitational and scalar field standing waves. These waves are out of phase; that is, the energy of oscillations passes back and forth between the scalar and gravitational waves. When the amplitude of the standing waves is small, the brane width and the size of the horizon in extra space are of a same order of magnitude, and matter fields are localized in extra dimension due to the presence of the horizon. When oscillations are large, trapping of matter fields on the brane is provided mainly by the pressure of bulk waves. It is shown that in this case the mass of the lightest KK mode is determined by the smaller energy scale corresponding to the horizon size; that is, these modes can be created in accelerators at relatively low energies, which gives a chance to check the present model.

1. Introduction 2. The Model Braneworld models involving large extra dimensions [1–6] We consider 5D space-time without bulk cosmological con- have been very useful in addressing several open questions stant containing a brane and a non-self-interacting real scalar in high energy physics (e.g., the hierarchy problem and field coupled to gravity the nature of flavor). Most of the braneworlds are realized 3 5 𝑀 1 𝑀𝑁 as time-independent field configurations. However, mostly 𝑆=∫ 𝑑 𝑥√𝑔( 𝑅+ 𝑔 𝜕𝑀𝜑𝜕𝑁𝜑+𝐿𝐵). (1) within the framework of cosmological studies, there were 2 2 proposed models with time-dependent metrics and matter Here 𝐿𝐵 is the brane Lagrangian and 𝑀 is the 5D fundamental fields, as well as branes with tensions varying in time7 [ – scale, which relates to the 5D Newton constant, 𝐺= 10]. One of the 5D braneworld models with nonstationary 3 1/(8𝜋𝑀 ). Capital Latin indexes numerate the coordinates of metric coefficients was proposed by one of us11 in[ –14](for 5D space-time with the signature (+,−,−,−,−),andweuse the generalization to 6D case, see [15, 16]). In this scenario the units where 𝑐=ℏ=1. Variation of the action (1)with the braneworld was generated by standing gravitational waves respect to 𝑔𝐴𝐵 leads to the 5D Einstein equations coupled to a phantom-like bulk scalar field, rapid oscillations of these waves provide universal gravitational trapping of zero 1 1 𝑅 − 𝑔 𝑅= (𝜎 +𝑇 ). (2) modes of all kinds of matter fields on the brane [17–21]. 𝐴𝐵 2 𝐴𝐵 𝑀3 𝐴𝐵 𝐴𝐵 In this paper we introduce the new nonstationary 5D Here the source term is the combination of the energy- braneworld generated by standing waves of the gravitational momentum tensors of the brane, 𝜎𝐴𝐵,andofthebulkscalar and real scalar fields, instead of the phantom-like scalar field, fields of [11–14]. The model also has new features: it does not require a bulk cosmological constant and the oscillation 1 𝐶 𝑇𝐴𝐵 =𝜕𝐴𝜑𝜕𝐵𝜑− 𝑔𝐴𝐵𝜕 𝜑𝜕𝐶𝜑. (3) frequency of the standing waves can be small. 2 2 Advances in High Energy Physics

Using (3),theEinsteinequations(2)canberewritteninthe andforthebranetensions form 𝑆 3 2 󸀠 1 𝑒 𝑀 𝛿 (𝑟) ( 𝑎+𝑆)=𝜎𝑡𝑡 − 𝜎, 1 1 3 3 (1−𝑎|𝑟|)2/3 𝑅 = (𝜎 − 𝑔 𝜎+𝜕 𝜑𝜕 𝜑) , 𝐴𝐵 𝑀3 𝐴𝐵 3 𝐴𝐵 𝐴 𝐵 (4) 3 2 󸀠 −𝑆+𝑢 1 2/3 𝑢 𝑀 𝛿 (𝑟) ( 𝑎−𝑢)𝑒 =𝜎𝑥𝑥 + (1−𝑎|𝑟|) 𝑒 𝜎, 𝐴𝐵 3 3 where 𝜎=𝑔 𝜎𝐴𝐵. 2 1 To solve (4), we take the metric ansatz 𝑀3𝛿 (𝑟) ( 𝑎−𝑢󸀠)𝑒−𝑆+𝑢 =𝜎 + (1−𝑎|𝑟|)2/3𝑒𝑢𝜎, 3 𝑦𝑦 3 𝑒𝑆 2 1 𝑑𝑠2 = (𝑑𝑡2 −𝑑𝑟2) 𝑀3𝛿 (𝑟) ( 𝑎+2𝑢󸀠)𝑒−𝑆−2𝑢 =𝜎 + (1−𝑎|𝑟|)2/3𝑒−2𝑢𝜎, (1−𝑎|𝑟|)2/3 3 𝑧𝑧 3 (5) 2/3 𝑢 2 𝑢 2 −2𝑢 2 4 1 𝑒𝑆 − (1−𝑎|𝑟|) (𝑒 𝑑𝑥 +𝑒 𝑑𝑦 +𝑒 𝑑𝑧 ), 𝑀3𝛿 (𝑟) ( 𝑎−𝑆󸀠)=𝜎 + 𝜎. 3 𝑟𝑟 3 (1−𝑎|𝑟|)2/3 where 𝑎 is a positive constant and 𝑆 = 𝑆(|𝑟|) and 𝑢 = 𝑢(𝑡, |𝑟|) (11) are some functions. The solution to (9)and(10)is The metric (5) is some combination of the 5D general- 𝜔 izations of the known domain wall solution [22–24](when 𝑢 (𝑡, |𝑟|) =𝐴 (𝜔𝑡) 𝐽 ( −𝜔|𝑟|), sin 0 𝑎 𝑆=𝑢=0)andofthecollidingplanewavesolutions[25– 27](when𝑆=𝑎=0). Similar ansatz in the 4D case was 3 √ 3𝑀 𝜔 considered by one of us in [28]. 𝜑 (𝑡, |𝑟|) = 𝐴 cos (𝜔𝑡) 𝐽0 ( −𝜔|𝑟|), To find a solution to the system of Einstein and back- 2 𝑎 ground real scalar field equations, let us assume that the 5D 3𝜔2(1−𝑎|𝑟|)2 𝑆 (|𝑟|) = scalar field, 2 2𝑎 (12) 𝜑≡𝜑(𝑡, |𝑟|) , 𝜔 𝜔 (6) ×𝐴2 [𝐽2 ( −𝜔|𝑟|)+𝐽2 ( −𝜔|𝑟|) 0 𝑎 1 𝑎 depends only on time, 𝑡,andontheabsolutevalueoftheextra 𝑎 𝜔 |𝑟| − 𝐽 ( −𝜔|𝑟|) coordinate, . Then its equation of motion 𝜔 (1−𝑎|𝑟|) 0 𝑎

1 𝑀𝑁 𝜔 𝜕 (√𝑔𝑔 𝜕 𝜑) = 0, ×𝐽1 ( −𝜔|𝑟|)] − 𝐵, √𝑔 𝑀 𝑁 (7) 𝑎 where 𝐴 and 𝐵 are some dimensionless constants, 𝐽0 and where 𝐽1 are Bessel functions of the first kind, and the integration constant 𝜔 corresponds to the frequency of standing waves. 1/3 𝑆 √𝑔=(1−𝑎|𝑟|) 𝑒 (8) Using (12)from(11), one can easily find the brane energy- momentum tensor 𝐵 3 𝑡 𝑥 𝑦 𝑧 is the determinant of the background metric (5), reduces to 𝜎𝐴 =𝑀𝛿 (𝑟) diag [𝜏𝑡 ,𝜏𝑥 ,𝜏𝑦 ,𝜏𝑧 ,0], (13) wherethebranetensionsare 󸀠󸀠 𝑎 󸀠 𝜑 +[2𝛿(𝑟) − ]𝜑 − 𝜑=0,̈ (9) 𝑡 1−𝑎|𝑟| 𝜏𝑡 =2𝑎, 2 𝜔 𝜏𝑥 =𝜏𝑦 = 𝑎+𝑎𝐵+𝐴𝜔 (𝜔𝑡) 𝐽 ( ), where overdots and primes denote derivatives with respect to 𝑥 𝑦 3 sin 1 𝑎 (14) 𝑡 and |𝑟|,respectively. For the ansatz (5), the Einstein equations (4)splitintothe 𝑧 2 𝜔 𝜏 = 𝑎+𝑎𝐵−2𝐴𝜔sin (𝜔𝑡) 𝐽1 ( ). system of the equations for the metric functions 𝑧 3 𝑎 To interpret solution (12) as describing the brane at 𝑟= 3 1 1 𝑎 1 − 𝑢̇2 + 𝑆󸀠󸀠 − 𝑆󸀠 = 𝜑̇2, 0, which bounds the scalar and gravitational bulk standing 3 2 2 2 (1−𝑎|𝑟|) 𝑀 waves, one needs to assume that the oscillatory metric 3 1 functions and 5D scalar field vanish at the origin; that is, − 𝑢󸀠𝑢=̇ 𝜑󸀠𝜑,̇ 󵄨 2 𝑀3 𝑢| =0, 𝜑󵄨 =0, 𝑆| =0. (10) 𝑟=0 󵄨𝑟=0 𝑟=0 (15) (1−𝑎|𝑟|) (𝑢󸀠󸀠 − 𝑢)̈ − 𝑎𝑢󸀠 =0, These conditions can be achieved assuming the relation between 𝜔 and 𝑎 as follows: 2 3 󸀠 1 󸀠󸀠 1 𝑎 󸀠 1 󸀠2 𝜔 (𝐽 ) − 𝑢 − 𝑆 − 𝑆 = 𝜑 =𝑍 0 , 2 2 2 (1−𝑎|𝑟|) 𝑀3 𝑎 𝑛 (16) Advances in High Energy Physics 3

(𝐽0) where 𝑍𝑛 is the 𝑛th zero of the function 𝐽0,andsimultane- are finite there. It resembles the situation with the Schwarz- ously fixing the constant 𝐵 as follows: schild Black Hole; however, in contrast the determinant of 3𝜔2𝐴2 𝜔 our metric (20)becomeszeroat|𝑟| = 1/𝑎.Astheresult, 𝐵= 𝐽2 ( ). (17) 2𝑎2 1 𝑎 nothing can cross the horizon of (20), and matter fields are confinedinsideofthe3-braneofthewidth∼1/𝑎 in the extra In fact, the conditions (15) lead to the quantization, (16), of space. To provide experimentally acceptable localization of 𝜔 the ratio of the standing wave frequency, ,tothecurvature matter fields, the actual size of the extra dimension must be 𝑎 scale, . sufficiently small, ≤1/𝑀𝐻,where𝑀𝐻 denotes the Higgs scale. From (12), one can also see that the metric function, So in the limiting case 𝐴≪1,thecurvaturescale𝑎 must be 𝑢(𝑡, |𝑟|) 𝜑(𝑡, |𝑟|) ,andthescalarfield, , having similar depen- large, 𝑀𝐻 ≤𝑎≤𝑀,where𝑀 is the 5D fundamental scale. As 𝜋/2 dence on spatial coordinates, are oscillating out of phase regards the brane, its width, in fact, is defined by the horizon in time; that is, the energy of the oscillations is passing back size. and forth between the gravitational and scalar field standing waves bounded by the brane. 4. The Large Extra Space There are three free parameters in our model: the constant 𝐴 (defining the amplitude of oscillations), the curvature scale Inthesecondlargeamplitudelimitingcase,itisobviousthat 𝑎 2 (giving the size of extra space for observers on the brane), 𝐴 ∼𝐵≫1. (22) (𝐽0) and the ratio 𝑍𝑛 =𝜔/𝑎(the 𝑛th zero of Bessel function Now, assuming that curvature scale 𝑎 is relatively small, the 𝐽0). Below, we consider two limiting cases corresponding to 𝐴≪1 𝐴≫1 width of the brane, located at the origin of the large (of the the small, ,andthelarge, , amplitudes of bulk ∼1/𝑎 standing waves. size ) but finite extra space, is determined by the metric function 𝑆(|𝑟|) in (5). Indeed, for small 𝑎∼𝜔,thetime- dependent terms in the brane tensions (14) are negligible, 3. The Small Extra Space 𝐵 3 𝜎𝐴 ≈𝑀𝛿 (𝑟) diag [2𝑎, 𝑎𝐵, 𝑎𝐵,] 𝑎𝐵,0 , (23) In the first limiting case, the amplitude is 𝐴≪1,and andthebranewidthisoftheorderof∼1/(𝑎𝐵). So trapping of consequently the energy of the oscillations is small. Then, matter fields is caused by the pressure of the bulk oscillations neglecting the terms containing the constants 𝐴 and 𝐵∼ 𝐴2 and not by the existence of the horizon in the extra space. in the brane tensions (14),onegetsthebraneenergy- As an illustrative example of this trapping mechanism, let momentum tensor, Φ(𝑥𝐴) 2 2 2 us consider localization of a real massless scalar field, , 𝜎𝐵 ≈𝑀3𝛿 (𝑟) [2𝑎, 𝑎, 𝑎, 𝑎, 0] , 𝐴 diag 3 3 3 (18) on the brane in the background metric (5). From the 5D action, E =3P E P 1 which obeys the equation of state ,with and 𝑆 = ∫ 𝑑𝑥5√𝑔𝑔𝑀𝑁𝜕 Φ𝜕 Φ, being the brane energy density and pressure, respectively. Φ 2 𝑀 𝑁 (24) Moreover, from (12),itisclearthatinthiscasethe 𝐴 𝑢 𝑆 𝜑 we obtain the Klein-Gordon equation for Φ(𝑥 ) as follows: functions , ,and do not play significant role and from 1 the very beginning one can assume the following: 𝜕 (√𝑔𝑔𝑀𝑁𝜕 Φ) = 0. √𝑔 𝑀 𝑁 (25) 𝑆 (|𝑟|) =𝑢(𝑡, |𝑟|) =𝜑(𝑡, |𝑟|) =0 (19) Using (8), equation (25)canbewrittenas and consider the metric ansatz without oscillatory metric 𝑆 functions as follows: 2 𝑒 −𝑢 2 −𝑢 2 2𝑢 2 1 [𝜕 − (𝑒 𝜕 +𝑒 𝜕 +𝑒 𝜕 )] Φ 𝑑𝑠2 = (𝑑𝑡2 −𝑑𝑟2) 𝑡 (1−𝑎|𝑟|)4/3 𝑥 𝑦 𝑧 (1−𝑎|𝑟|)2/3 (26) (20) 1 = 𝜕 [(1−𝑎|𝑟|) 𝜕 Φ] . − (1−𝑎|𝑟|)2/3 (𝑑𝑥2 +𝑑𝑦2 +𝑑𝑧2). (1−𝑎|𝑟|) 𝑟 𝑟 This metric is 5D generalizations of the 4D domain wall In addition, it is easy to find that the assumption22 ( )leadsto solution of [22–24]. Due to the presence of the absolute value the following relation between the metric functions: 󵄨 󵄨 of the extra coordinate, |𝑟|, the Ricci tensor at 𝑟=0has 󵄨𝑢 (𝑡, |𝑟|)󵄨 󵄨 󵄨 ≪1. (27) 𝛿-function-like singularity which corresponds to the brane 󵄨 𝑆 (|𝑟|) 󵄨 tension; see (11). Solution (20) has also new features, since, 𝑎 Then, in the scalar field equation26 ( ), we can drop the in contrast to 4D domain walls [22–24], the parameter has 𝑢(𝑡, 𝑟) theoppositesign.Duetothisfact,themetric(20)hasthe function in the exponents and rewrite it as 𝑆 horizon at |𝑟| = 1/𝑎 in the bulk. At these points 𝑅𝑡𝑡 and 𝑅𝑟𝑟 𝑒 [𝜕2 − (𝜕2 +𝜕2 +𝜕2)] Φ components of the Ricci tensor get infinite values, while all 𝑡 (1−𝑎|𝑟|)4/3 𝑥 𝑦 𝑧 gravitational invariants, for example, Ricci scalar, (28) 1 −𝑆 2/3 3 󸀠2 2 󸀠󸀠 = 𝜕 [(1−𝑎|𝑟|) 𝜕 Φ] . 𝑅=𝑒 (1−𝑎|𝑟|) [ (𝑢 − 𝑢̇)+𝑆 ] 𝑟 𝑟 2 (1−𝑎|𝑟|) 8 (21) We look for the solution in the form −𝑎(2𝐵+ )𝛿(𝑟) , 𝜇 3 Φ(𝑡,𝑥,𝑦,𝑧,𝑟)=𝜙(𝑥)𝜌(𝑟) , (29) 4 Advances in High Energy Physics

where Greek indexes numerate 4D coordinates, and the 4D where 𝐶1 and 𝐶2 are integration constants, and 𝐴𝑖 and 𝐵𝑖 are 𝜇 factor of the scalar field wavefunction 𝜙(𝑥 ) obeys the equa- Airy functions. To fulfill the conditions (33), the constants 𝐶1 tion and 𝐶2 must obey the relation ]𝛽 𝜇 2 𝜇 𝜂 𝜕 𝜕 𝜙(𝑥 )=−𝑚𝜙(𝑥 ). 2 4 4 ] 𝛽 (30) 𝐵𝐸 1 3 𝑎 1 3 𝑎 𝐶 = −(2√3 𝐴𝑖󸀠 ( √ )−𝐴𝑖( √ )) 2 𝑎2 4 𝐵2𝐸4 4 𝐵2𝐸4 In what follows, we assume

𝜇 −𝑖(𝐸𝑡−𝑝𝑥𝑥−𝑝𝑦𝑦−𝑝𝑧𝑧) 2 4 𝜙 (𝑥 ) =𝑒 . (31) 𝐵𝐸 1 3 𝑎 ×(2√3 𝐵𝑖󸀠 ( √ ) 𝑎2 4 𝐵2𝐸4 Then extra dimension factor 𝜌(|𝑟|) ofthescalarfieldwave- function obeys the equation −1 4 1 3 𝑎 𝑎 (𝑟) 𝑒𝑆 −𝐵𝑖( √ )) 𝐶 , 𝜌󸀠󸀠 − sgn 𝜌󸀠 −𝐸2 [ −1]𝜌 4 𝐵2𝐸4 1 1−𝑎|𝑟| (1−𝑎|𝑟|)4/3 (32) (39) 𝑒𝑆 =−𝑚2 𝜌, 𝐴𝑖󸀠 𝐵𝑖󸀠 4/3 where and denote the first derivatives of Airy func- (1−𝑎|𝑟|) tions. Then, at the origin of the extra space, that is, on 𝜌 (|𝑟|) with the boundary conditions the brane, the function 0 will have the following series expansion: 󸀠󵄨 𝜌 󵄨 =0, 1 󵄨|𝑟|=0 𝜌 (𝑟) | =𝐶(1− 𝐵𝐸2𝑎|𝑟|3)+𝑂(𝑎4|𝑟|4), (33) 0 |𝑟| → 0 6 (40) 𝜌||𝑟| → 1/𝑎 =0. where we introduced the constant For the scalar field zero mode wavefunction, 𝜌0(𝑟),withthe 2 4 4 𝐵𝐸 1 3 𝑎 1 3 𝑎 dispersion relation 𝐶=2𝐶√3 (𝐴𝑖 ( √ )𝐵𝑖󸀠 ( √ ) 1 𝑎2 4 𝐵2𝐸4 4 𝐵2𝐸4 2 2 2 2 𝐸 =𝑝𝑥 +𝑝𝑦 +𝑝𝑧, (34) 4 4 1 3 𝑎 1 3 𝑎 equation (32)reducesto −𝐴𝑖󸀠 ( √ )𝐵𝑖( √ )) 4 𝐵2𝐸4 4 𝐵2𝐸4 𝑆 󸀠󸀠 𝑎 sgn 󸀠 2 𝑒 𝜌0 − 𝜌0 −𝐸 [ −1] 𝜌0 =0. (35) (1−𝑎|𝑟|) (1−𝑎|𝑟|)4/3 2 4 𝐵𝐸 1 3 𝑎 ×(2√3 𝐵𝑖󸀠 ( √ ) 2 2 4 To show that (35) has the solution localized on the brane, 𝑎 4 𝐵 𝐸 we investigate the equation in two limiting regions: close to −1 the brane (|𝑟| →) 0 and close to the horizon in the extra 4 1 3 𝑎 space (|𝑟| → 1/𝑎). −𝐵𝑖 ( √ )) . 4 𝐵2𝐸4 In the first limiting region, |𝑟| →, 0 (41) 2 𝑢| = (𝜔𝑡) [√ 𝐵𝑎 |𝑟| +𝑂(𝑎2|𝑟|2)] , |𝑟| → 0 sin 3 In the second limiting region, |𝑟| → 1/𝑎, 1 𝜔2 𝑆| =−𝐵𝑎|𝑟| +𝑂(𝑎2|𝑟|2), (36) 𝑢| = (𝜔𝑡) [𝐴 − 𝐴 (1−𝑎|𝑟|)2 |𝑟| → 0 |𝑟| → 1/𝑎 sin 4 𝑎2 󵄨 1 √𝑔󵄨 =1−(𝐵+ )𝑎|𝑟| +𝑂(𝑎2|𝑟|2), 󵄨|𝑟| → 0 3 +𝑂 ( 1−𝑎|𝑟|)3)], equation (35) has the following asymptotic form: 3 𝜔2 󸀠󸀠 󸀠 2 2 2 4 𝜌 −𝑎 (𝑟) 𝜌 +𝐵𝐸 𝑎 |𝑟| 𝜌 =0. 𝑆||𝑟| → 1/𝑎 =−𝐵+ 𝐴 (1−𝑎|𝑟|) +𝑂((1−𝑎|𝑟|) ), 0 sgn 0 0 (37) 4 𝑎2 This equation has the general solution 󵄨 −𝐵 1/3 7/3 √𝑔󵄨|𝑟| → 1/𝑎 =𝑒 (1−𝑎|𝑟|) +𝑂((1−𝑎|𝑟|) ), 4 2 (42) 1 3 𝑎 𝐵𝐸 𝜌 (𝑟) =𝑒𝑎|𝑟|/2 [𝐶 𝐴𝑖 ( √ − √3 𝑎 |𝑟|) 0 1 4 𝐵2𝐸4 𝑎2 equation (35) has the following asymptotic form: [ 𝑎 (𝑟) 𝑐2 𝜌󸀠󸀠 − sgn 𝜌󸀠 − 𝜌 =0, 4 2 0 0 4/3 0 (43) 1 3 𝑎 3 𝐵𝐸 1−𝑎|𝑟| (1−𝑎|𝑟|) +𝐶 𝐵𝑖 ( √ − √ 𝑎 |𝑟|)] , 2 4 𝐵2𝐸4 𝑎2 ] where 2 2 −𝐵 (38) 𝑐 =𝐸𝑒 . (44) Advances in High Energy Physics 5

The general solution to this equation is from which we get the KK mass spectrum of scalar field on 3𝑐 3𝑐 the brane √3 √3 𝜌0 (𝑟) =𝐶3𝐼0 ( 1−𝑎|𝑟|)+𝐶4𝐾0 ( 1−𝑎|𝑟|), (45) (𝐽1) 𝑎 𝑎 𝑚𝑛 =𝑎𝑍𝑛 , (52)

𝐶 𝐶 𝐼 (𝐽1) where 3 and 4 aresomeintegrationconstantsand 0 and where 𝑍𝑛 is the 𝑛th zero of the Bessel function 𝐽1.Sothe 𝐾 0 are zero-order modified Bessel functions of the first and mass gap between the zero and the first massive modes will second kind, respectively. To fulfill the conditions33 ( ), we be 𝐶 =0 𝐾 must choose 4 (the function 0 is unbounded function (𝐽 ) Δ =𝑚 =𝑎𝑍 1 ≈3.8𝑎. at |𝑟| → 1/𝑎). Then, for the series expansion of 𝜌0(𝑟) at 𝑚 1 1 (53) |𝑟| → 1/𝑎 ,weget In the limiting case of this section, the curvature scale 𝑎, 󵄨 9𝑐2 which determines the size of extra dimension, is smaller than 𝜌 (𝑟)󵄨 =𝐶 [1 + (1−𝑎|𝑟|)2/3] ∼𝑎𝐵 0 󵄨|𝑟|󳨀→1/𝑎 3 4𝑎2 the scale associated with the width of the brane, ,where (46) 𝐵≫1. So KK modes, having the masses ≈3.8 𝑎,canbecreated +𝑂((1−𝑎|𝑟|)4/3). in accelerators at relatively low energies, what gives a chance to check this model. According to (27), the action (24)ofthe5Dscalarfiled zero mode reduces to 5. Conclusion 1 𝑆 = ∫ 𝑑𝑥4𝑑𝑟 [𝑄 (𝑟) 𝜕 𝜙2 −𝑄 (𝑟) 0 2 1 𝑡 2 In this paper, we have introduced the new nonstationary 5D braneworld model with the real scalar field in the bulk. 2 2 2 2 ×(𝜕𝑥𝜙 +𝜕𝑦𝜙 +𝜕𝑧𝜙 )−𝑄3 (𝑟) 𝜙 ], The model represents single brane which bounds collective (47) bulk oscillations of gravitational and scalar field standing waves. These waves are out of phase; that is, the energy of where integration over 𝑟 isperformedonthefiniteintegration oscillations passes back and forth between the scalar and region [−1/𝑎, +1/𝑎],and𝑄𝑖 functions are defined as gravitational waves. The metric of the model has the horizon in the extra space, and, consequently, the extra space is finite 𝑄 (𝑟) = (1−𝑎|𝑟|) 𝜌2, 1 0 for the observer on the brane. We have investigated limiting cases of large and small extra space, depending on the two 𝑄 (𝑟) = (1−𝑎|𝑟|)−1/3𝑒𝑆𝜌2, 2 0 (48) dimensional parameters of the model, the curvature scale and 󸀠2 the energy scale associated with the brane. 𝑄3 (𝑟) = (1−𝑎|𝑟|) 𝜌 . 0 In the limiting case of small extra space, when the ampli- It is easy to see that, for the extra dimension factor of the tude of the standing waves is small, these two parameters are scalar field zero mode, 𝜌0(𝑟), having asymptotes (40)and of the same order of magnitude, and matter fields are localized (46), the integral over extra coordinate 𝑟 in (47)isfinite. on the brane due to the presence of the metric horizon. This means that the scalar field zero mode wavefunction is Inthecaseoflargeoscillations,thedistancetothehorizon localized on the brane. Also note that on the brane, 𝑟=0, is relatively large as compared with the brane width, and due to the boundary conditions (33),theLagrangianinthe trapping of matter fields on the brane is caused mainly by action (47) gets the standard 4D form for the massless scalar the pressure of bulk oscillations. The mass of the lightest KK field. mode in this case is determined by the smaller energy scale To estimate the masses of KK excitations on the brane, we associated with the horizon size, and therefore such particles 𝑝2 =0 consider the scalar particles with zero 3-momentum, . can be created in future accelerators at relatively low energies. Then (32) for the massive modes reduces to 𝑎 (𝑟) 𝜌󸀠󸀠 − sgn 𝜌󸀠 +𝑚2𝜌=0. Acknowledgments 1−𝑎|𝑟| (49) Merab Gogberashvili was partially supported by the Grant The exact general solution to this equation is of Shota Rustaveli National Science Foundation no. DI/8/6- 𝑚 𝑚 100/12. The research of Pavle Midodashvili was supported by 𝜌 (𝑟) =𝐷𝐽 ( (1−𝑎|𝑟|))+𝐷 𝑌 ( (1−𝑎|𝑟|)), 1 0 𝑎 2 0 𝑎 Ilia State University. (50) References where 𝐷1 and 𝐷2 are some constants. Imposing the boundary conditions (33), we get [1] N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, “The hierarchy problem and new dimensions at a millimeter,” Physics Letters B, 𝐷2 =0, vol. 429, no. 3-4, pp. 263–272, 1998. 𝑚 (51) [2] I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, 𝐽1 ( )=0, “New dimensions at a millimeter to a fermi and superstrings at 𝑎 aTeV,”Physics Letters B,vol.436,no.3-4,pp.257–263,1998. 6 Advances in High Energy Physics

[3] M. Gogberashvili, “Gravitational trapping for extended extra Modern Physics Letters A,vol.28,no.20,ArticleID1350092,12 dimension,” International Journal of Modern Physics D, vol. 11, pages, 2013. no. 10, pp. 1639–1642, 2002. [22] A. H. Taub, “Isentropic hydrodynamics in plane symmetric [4] M. Gogberashvili, “Four dimensionality in non-compact Kalu- space-times,” vol. 103, pp. 454–467, 1956. za-Klein model,” Modern Physics Letters A,vol.14,no.29,pp. [23] A. Vilenkin, “Gravitational field of vacuum domain walls and 2025–2031, 1999. strings,” Physical Review D,vol.23,no.4,pp.852–857,1981. [5] L. Randall and R. Sundrum, “Large mass hierarchy from a small [24] J. Ipser and P. Sikivie, “Gravitationally repulsive domain wall,” extra dimension,” Physical Review Letters,vol.83,no.17,pp. Physical Review D,vol.30,no.4,pp.712–719,1984. 3370–3373, 1999. [25] U. Yurtsever, “Structure of the singularities produced by collid- [6] L. Randall and R. Sundrum, “An alternative to compactifica- ing plane waves,” Physical Review D,vol.38,no.6,pp.1706–1730, tion,” Physical Review Letters,vol.83,no.23,pp.4690–4693, 1988. 1999. [26] A. Feinstein and J. Ibanez,˜ “Curvature-singularity-free solutions [7] M. Gutperle and A. Strominger, “Spacelike branes,” Journal of for colliding plane gravitational waves with broken 𝑢-V symme- High Energy Physics,vol.2002,no.4,article18,19pages,2002. try,” Physical Review D,vol.39,no.2,pp.470–473,1989. [8] M. Kruczenski, R. C. Myers, and A. W. Peet, “Supergravity S- [27] J. B. Griffiths, Colliding Plane Waves in General Relativity,chap- branes,” JournalofHighEnergyPhysics,vol.2002,no.5,article ter 10, Oxford University Press, Oxford, UK, 1991. 39,23pages,2002. [28] M. Gogberashvili, S. Myrzakul, and D. Singleton, “Standing [9]V.D.IvashchukandD.Singleton,“CompositeelectricS-brane gravitational waves from domain walls,” Physical Review D,vol. solutions with maximal number of branes,” Journal of High 80,no.2,ArticleID024040,5pages,2009. Energy Physics,vol.2004,no.10,article61,18pages,2004. [10] C. P. Burgess, F. Quevedo, R. Rabadan,G.Tasinato,andI.´ Zavala, “On bouncing brane worlds, S-branes and branonium cosmology,” Journal of Cosmology and Astroparticle Physics,vol. 2004,no.2,article8,30pages,2004. [11] M. Gogberashvili and D. Singleton, “Anti-de Sitter island- universes from 5D standing waves,” Modern Physics Letters A, vol. 25, no. 25, pp. 2131–2143, 2010. [12] M. Gogberashvili, A. Herrera-Aguilar, and D. Malagon-More-´ jon,´ “An anisotropic standing wave braneworld and associated Sturm-Liouville problem,” Classical and Quantum Gravity,vol. 29, no. 2, Article ID 025007, 14 pages, 2012. [13]M.Gogberashvili,A.Herrera-Aguilar,D.Malagon-Morej´ on,´ R. R. Mora-Luna, and U. Nucamendi, “Thick brane isotropization in a generalized 5D anisotropic standing wave braneworld model,” Physical Review D,vol.87,no.8,ArticleID084059,8 pages, 2013. [14] M. Gogberashvili, A. Herrera-Aguilar, D. Malagon-Morej´ on,´ and R. R. Mora-Luna, “Anisotropic inflation in a 5D standing wave braneworld and effective dimensional reduction,” Physics Letters B, vol. 725, no. 4-5, pp. 208–211, 2013. [15] L.J.S.Sousa,J.E.G.Silva,andC.A.S.Almeida,“A6Dstanding waveBraneworld,” http://arxiv.org/abs/1209.2727 . [16] P. Midodashvili, “Localization of matter fields in the 6D standing wave braneworld,” International Journal of Theoretical Physics,2013. [17] M. Gogberashvili, P.Midodashvili, and L. Midodashvili, “Local- ization of scalar and tensor fields in the standing wave braneworld with increasing warp factor,” Physics Letters B,vol. 702, no. 4, pp. 276–280, 2011. [18] M. Gogberashvili, P.Midodashvili, and L. Midodashvili, “Local- ization of gauge bosons in the 5D standing wave braneworld,” Physics Letters B,vol.707,no.1,pp.169–172,2012. [19] M. Gogberashvili, P.Midodashvili, and L. Midodashvili, “Local- ization problem in the 5d standing wave braneworld,” Interna- tional Journal of Modern Physics D,vol.21,no.10,ArticleID 1250081, 21 pages, 2012. [20] M. Gogberashvili, “Localization of matter fields in the 5D standing wave braneworld,” Journal of High Energy Physics,vol. 2012, article 56, 2012. [21] M. Gogberashvili, O. Sakhelashvili, and G. Tukhashvili, “Numerical solutions in 5D standing wave braneworld,” Hindawi Publishing Corporation Advances in High Energy Physics Volume 2013, Article ID 531696, 7 pages http://dx.doi.org/10.1155/2013/531696

Research Article Why the Length of a Quantum String Cannot Be Lorentz Contracted

Antonio Aurilia1 and Euro Spallucci2

1 Department of Physics and Astronomy, California State Polytechnic University, Pomona, CA 91768, USA 2 Dipartimento di Fisica, Sezione Teorica, Universita` di Trieste and INFN, 34100 Trieste, Italy

Correspondence should be addressed to Euro Spallucci; [email protected]

Received 6 September 2013; Accepted 18 September 2013

Academic Editor: Piero Nicolini

Copyright © 2013 A. Aurilia and E. Spallucci. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We propose a quantum gravity-extended form of the classical length contraction law obtained in special relativity. More specifically, the framework of our discussion is the UV self-complete theory of quantum gravity. We show how our results are consistent with (i) the generalized form of the uncertainty principle (GUP), (ii) the so-called hoop-conjecture, and (iii) the intriguing notion of “classicalization” of trans-Planckian physics. We argue that there is a physical limit to the Lorentz contraction rule in the form of some minimal universal length determined by quantum gravity, say the Planck Length, or any of its current embodiments such as the string length, or the TeV quantum gravity length scale. In the latter case, we determine the critical boost that separates the ordinary “particle phase,” characterized by the Compton wavelength, from the “black hole phase,” characterized by the effective Schwarzschild radius of the colliding system.

1. Introduction and Background that is, 14 TeV [1–10]. In this new physics, the distinction between “point-like” elementary particles and “extended” High energy particle physics is based on the notion that quantum gravity excitations, whatever they are, that is, black smaller and smaller distance scales can be investigated by holes, 𝐷-branes, string balls, and so forth, turns out to be increasing the energy of the probe particle. Elementary fuzzy,sothatstandardnotions,suchastheLorentz-Fitzgerald projectiles colliding with a target can resolve distances length contraction, require a substantial revision, at least comparable with their quantum mechanical wavelength. The insofar as its domain of validity is concerned. For instance, more is the energy, the shorter is the wavelength in agreement a fundamental, quantum string is, presumably, the smallest with the relativistic rule of length contraction. Quantum √ 󸀠 object in nature with a linear size given by 𝑙𝑠 = 𝛼 .Thus,in mechanics and special relativity work together to open a this string perspective,nodistanceshorterthan𝑙𝑠 canbegiven window on the microscopic world. a physical meaning. Furthermore, by supplying more and This simple picture becomes less clear when we begin to more energy, higher and higher vibration modes are excited approach the Planck scale of distance or energy and consider making the string longer and longer, in conflict with the the concomitant quantum gravity effects.Thisproblemhas length-contraction rule but not unlike the increasing size of a long been ignored on the basis that the Planck energy, roughly 19 Schwarzschild black hole. To our mind, this signals the end of 10 GeV, is so huge that no particle accelerator will ever be the validity of special relativity and the onset of gravitational able to approach it. However, the picture is completely different when we effects. consider the string-inspired unified models with large extra- How can we account for that? dimensions, where the unification scale can be as low as some Before trying to answer this question, it is useful to TeV. In this kind of scenario, quantum gravity effects, includ- recall the derivation of the fundamental units that define the ing microblack hole production in partonic hard scattering, domain of quantum gravity, as the answer to our question lies have been suggested to occur near the LHC peak energy, in the very definition of those units. 2 Advances in High Energy Physics

The appropriate standards of length, mass, and time were the minimal length which is physically meaningful since, for a originally introduced by Max Planck on a purely dimensional shorter one, the uncertainty is larger than the length itself. In basis by combining the speed of light 𝑐,thegravitational contrast to this, it seems worth observing that the Planck mass coupling constant 𝐺∗ (𝐺∗ can be either the Newton constant is neither an absolute minimum nor an absolute maximum. or the higher-dimension gravitational coupling of TeV quan- It is, rather, an extremal value or turning point, in the sense tum gravity.), and the Planck constant ℎ.Inotherwords, that, as implied by definition2 ( ), it represents the largest mass Planck recognized that it is possible to combine special that an elementary particle may possess or the smallest mass relativity, quantum mechanics, and gravity in the following attributable to a microblack hole. Interestingly enough, we dimensional package: will argue in Section 2 as well as in Section 3 of this paper that there exists in nature a universal, unsurpassable linear ℎ𝐺 ℎ𝐺 ℎ𝑐 energy density or tension that lies at the core of every black 𝐿 ∝ √ ∗ ,𝑇∝ √ ∗ ,𝑀∝ √ . 𝑃 3 𝑃 5 𝑃 (1) 𝑐 𝑐 𝐺∗ hole, regardless of its mass or size.

Clearly, this dimensional approach defines the Planck units 2. Critical Boost and Minimal Length up to a numerical factor providing only an “orders of magnitude” estimate. In the old days, the Planck world was We have remarked earlier that the existence of a “quantum envisaged as the arena of violent quantum gravity fluctua- of length”[16, 17] is in conflict with the conventional rule of tions disrupting the very fabric of space and time [11, 12]. “length contraction” derived in special relativity. In a nutshell, Eventually, the notion of “space-time foam” evolved into the quantum of length 𝐿∗ is a new universal constant on the a “Planckian phase” with a different description according same footing as 𝑐 and ℎ, and as such it must be an observer to string/M-theory, loop quantum gravity, noncommutative independent. It follows that 𝐿∗ must act as an unbreakable geometry, fractal space-time, and so forth. barrier to the Lorentz-Fitzgerald contraction. In order to determine the numerical constants in (1), We propose to get around this problem by redefining some extra argument is due. the Lorentz-Fitzgerald contraction law in the presence of a Forinstance,onemaydeclarethatthePlanckmassis short-distance Planck barrier. This is the crux of the following defined by the equality between the quantum mechanical discussion. wavelength of a particle and its gravitational critical radius In special relativity, a rod of length 𝐿0 in its rest frame is ℎ 2𝑀 𝐺 seen to be contracted in the direction of motion according to = ∗ ∗ . 2 (2) the rule 𝑀∗𝑐 𝑐 2 V 𝐿(𝛽)=𝐿0 √1−𝛽 ,𝛽≡. (7) Thus, 𝑐 𝐿 2ℎ𝐺 ℎ𝑐 An immediate consequence of (7)isthat can contract to 𝐿 = √ ∗ ,𝑀= √ . (3) an arbitrarily small length as 𝛽→1. There are, however, at ∗ 𝑐3 ∗ 2𝐺 ∗ least two types of objections to this conclusion that require a redefinition of the contraction rule. An alternative, but consistent, definition of (3), which to our knowledgehasneverbeennotedbefore,isthefollowing.𝐿∗ (i) “Quantum” objection or the absence of ℎ:eventhough is the geometric mean of the quantum mechanical wavelength (7) is routinely applied to the world of particle 𝜆𝐶 =ℎ/𝑚𝑐of the particle and its critical gravitational radius physics, it was conceived with macroscopic,thatis, 2 𝑅𝑠 =2𝑚𝐺∗/𝑐 : “classical” objects in mind. Stated otherwise, the quantum of action ℎ seemingly has no effect in the 2ℎ𝐺 length-contraction rule, but we expect this to change 𝐿 ≡ √𝜆 𝑅 = √ 𝑁 . (4) ∗ 𝐶 𝑠 𝑐3 in the ultrarelativistic regime when one approaches distances of the order of the Planck length. 𝐿 Further insight into the physical meaning of ∗ can be (ii) “Gravitational” objection or the absence of 𝐺𝑁:(7) obtained from the generalized uncertainty principle (GUP) refers to “abstract” lengths ignoring the fact that 𝐿 [13–15], where ∗ is often identified with the string length, any physical object produces its own gravitational √ 󸀠 that is, 𝐿∗ = 𝛼 : field and thus introduces a “critical” gravitational length scale, that is, the Schwarzschild radius 𝑅𝑠 = 𝐿2 ℎ ∗ Δ𝑝 2𝑀𝐺 /𝑐2 𝐿≤𝑅 Δ𝑥 ≥ + . (5) 𝑁 .If 𝑠,therodisnotarodanymore, Δ𝑝 4 ℎ rather,itwilllooklikeablack hole! This is the so-called “hoop-conjecture”: any physical object extending By minimizing the uncertainties, one finds along a certain direction less than its Schwarzschild 2ℎ radius collapses into a black hole [18]. How a black hole Δ𝑝 = ,Δ𝑥=𝐿 . ∗ ∗ ∗ (6) appearsinaboostedframeisanoverlookedproblem 𝐿∗ except in the somewhat ambiguous “shock wave limit” From (6)weseethat𝐿∗ represents the minimal uncertainty in where 𝛾→∞, 𝑀→0while the product is kept the particle/string localization. From this point of view, 𝐿∗ is finite, that is, 0<𝛾𝑀<∞[19]. Advances in High Energy Physics 3

By considering both arguments at the same time, one expects (i) Take for 𝐿0 the Compton wavelength 𝜆𝐶 =1/𝑚of a that quantum gravity imposes intrinsic limits to the rela- particle. In analogy with the string improved GUP,we tivistic contraction of physical objects. Presently, the most obtain the following modified de Broglie formula: promising candidate for a self-consistent theory of quantum 𝛼󸀠𝜃 (𝛽) gravitational phenomena is super-string theory. From its ̃ 2 𝐻 𝜆(𝛽)=𝜆𝐶√1−𝛽 + . vantage point, string theory “solves” the problem from the 2 (11) 4𝜆𝐶√1−𝛽 very beginning by assuming that the building blocks of everything are finite length, vibrating strings. Nothing can be 󸀠 As 𝜆(𝛽) cannot be smaller than √𝛼 it follows that the “smaller,”in the sense that any distance (length) smaller than √ 󸀠 mass spectrum of an “elementary” particle is bounded the string length 𝛼 does not have physical meaning. As 󸀠 from above by the limiting mass 1/√𝛼 string theory is a quantum theory of gravity, the string length −33 1 maybeidentifiedwiththePlanckLength𝐿𝑃 ≈10 cm. √ 󸀠 𝜆≥ 𝛼 󳨀→ 𝑚 ≤ . (12) Unfortunately, to our knowledge string theory says nothing √𝛼󸀠 about the Lorentz-Fitzgerald contraction and how to modify it. (ii) Next, consider the case of a Schwarzschild black hole, Our foregoing discussion, on the other hand, requires 𝐿0 =𝑅𝑠.Equation(8) tells us how the horizon radius that any quantum gravity-inspired extension ought to contain will appear from a Lorentz boosted frame as follows: both Newton and Planck constant, 𝐺𝑁 and ℎ,andreproduces 𝛼󸀠𝜃 (𝛽) (7)when𝐺𝑁 or ℎ are “switched off ”. 2 𝐻 𝑅𝑠 (𝛽) = 2𝑀𝐺𝑁√1−𝛽 + . According to the hoop-conjecture there must be a critical 2 (13) 8𝑀𝐺𝑁√1−𝛽 2 boost factor 𝛾∗ ≡1/√1−𝛽∗ that characterizes the transition from a gravitationally interacting two-particle system into a The first term shows how the Schwarzschild radius of a moving mass appears contracted as any other black hole. In order to determine 𝛾∗,letustentativelychange the contraction formula into the following expression: physical length. The second term in (13)takesinto account the existence of a “hard core” characterized 𝐿2 𝜃 (𝛽) by a universal, unsurpassable linear energy density or 𝐿(𝛽)=𝐿̃ √1−𝛽2 + ∗ 𝐻 , 0 (8) tension that prevents further contraction or collapses 4𝐿 √1−𝛽2 0 into a point singularity. We will come back to this essential point in the concluding section of this paper. where 𝜃𝐻 is the Heaviside step function which guarantees that the extra term does not affect the measure of 𝐿 at rest. (We The critical boost is define 𝜃𝐻(𝑥) as 𝜃𝐻(𝑥)=1←𝑥>0and 𝜃𝐻(𝑥)=0←𝑥≤0. 4𝑀𝐺𝑁 Sometimes, it is conventionally chosen as 𝜃𝐻(0) ≡ 1/2.Inthis 𝛾∗ = . (14) 2 √𝛼󸀠 case, a 𝛽-independent quantity 𝐿∗/8𝐿0 must be subtracted in (8).) Moreover, since any macroscopic length is tens of orders For 𝛾=𝛾∗,theSchwarzschildradiusreachesitsminimal 𝐿 ∗ √ 󸀠 of magnitude larger than ∗,thesecondtermin(8)givesa value 𝑅𝐻(𝛽 )= 𝛼 . A snapshot of a black hole at this 𝛽≈ relevant contribution only in the ultrarelativistic regime minimal size will show an object with an effective mass 𝑀∗ ̃ 1. The minimum of the function 𝐿(𝛽) is defined as 󸀠 𝑑𝐿̃ 2𝐿 ∗ √𝛼 =0󳨀→𝛾 = 0 , 𝐿(̃ 𝛽 ) =𝐿 . 𝑅 (𝛽 ) ≡2𝑀 𝐺 󳨀→ 𝑀 = . (15) ∗ ∗ ∗ (9) 𝐻 ∗ 𝑁 ∗ 2𝐺 𝑑𝛽 𝐿∗ 𝑁 ̃ For 𝛾>𝛾∗,thefunction𝐿(𝛽) “bounces back” and increases as In a string theoretical formulation of quantum gravity, the stipulated in our earlier discussion on the basis that a similar ReggeslopecanberelatedtotheNewtonconstantthrough 𝛼󸀠 =2𝐺 𝑀 1/√2𝐺 𝑀 𝑅 (𝛽∗)= behavior is shown by a fundamental string which cannot 𝑁.Thus,wefind ∗ = 𝑁 = 𝑃 and 𝐻 √ 󸀠 𝐿𝑃. shrink below its minimal length 𝑙𝑠 = 𝛼 , while increasing its energy excites higher and higher vibration modes forcing If we formally assign to the black hole a Compton wavelength 𝜆𝐵𝐻 ≡1/𝑀,wecanwrite(13) as follows: thestringtoelongate.Thus,anaturalchoicefor𝐿∗ is 𝐿∗ = √ 󸀠 2 𝑙𝑠 = 𝛼 . For later convenience, it seems also worth recalling 𝐿 𝜆 𝜃 (𝛽) 𝑅 (𝛽) = 𝑃 √1−𝛽2 + 𝐵𝐻 𝐻 . again that a highly excited string looks rather like a black hole. 𝑠 (16) 𝜆𝐵𝐻 √ 2 With this identification,9 ( ) tells that in any inertial reference 4 1−𝛽 frame no physical length can be smaller than the string length Comparison with (11)showsthat𝜆𝐵𝐻 enters the modified 𝐿(𝛽)≥̃ √𝛼󸀠, (10) contraction law in the inverse way with respect to 𝜆𝐶,thus suggesting that the de Broglie wavelength of a black hole can and the critical boost representing the turning point between be written as √ 󸀠 𝛾∗ =2𝐿0/ 𝛼 contraction and dilatation turns out to be . 𝑑𝐵 𝜆𝐵𝐻 Now, let us take a closer look at (8)bywayofsome 𝜆𝐵𝐻 ≡ =𝛾𝜆𝐵𝐻. 2 (17) illustrative examples. √1−𝛽 4 Advances in High Energy Physics

√ 󸀠 Once the critical boost 𝛾∗ =2𝑀 𝛼 =𝑀/𝑀𝑃 is passed, which is unphysical. Only by moving at a speed greater than thefirsttermin(16) is negligible and the Schwarzschild radius the speed of light can an object turn into a black hole. Thus, expands as follows: even in the presence of quantum corrections, there is no inertial framewhereaclassicalobjectwithlinearsize𝐿0 >𝑅𝑠 may 𝜆𝑑𝐵 appear as a black hole. 𝑅 (𝛽) ≈ 𝐵𝐻 . (18) 𝑠 4 Let us consider now a quantum particle rather than a classical object as follows: It is worth mentioning that, recently, a new family of 𝛼󸀠𝜃 (𝛽) singularity-free black hole-metrics was reported in [20–25], 2 𝐻 𝜆𝐶√1−𝛽 + where the existence of a minimal length is assumed at the 4𝜆 √1−𝛽2 outset in the Einstein equations. A remarkable property of 𝐶 (21) these black holes is to admit extremal configurations even in 𝛼󸀠𝜃 (𝛽) 2 𝐻 the neutral nonspinning case. Extremality corresponds to the ≤𝑅𝑠√1−𝛽 + . 2 lowest mass state of the system and to a minimal radius of the 4𝑅𝑠√1−𝛽 event horizon which equals few times the minimal length. In some simple models, it is possible to choose the free length Once more, the “terminal boost” is unphysical, that is, 𝛽>1, scale that regularizes the short distance behavior in such a and the only acceptable solution is way that the radius of the extremal configuration is exactly the Planck length [9, 26, 27]. Without introducing the improved 1 𝜆𝐶 =𝑅𝑠 󳨀→ 𝑚 = =𝑀𝑃. (22) Lorentz law, the very idea of a minimal size object would 𝛼󸀠 become observer dependent. To sum up, at this point we have Thus, as before, there does not exist an inertial frame where an isolated elementary particle with (invariant) mass 𝑚< 𝑀 (1) equation (8)fora(semi)classicallength𝐿0 with string 𝑃 looks like a black hole. However, this conclusion does corrections; not prevent the production of a black hole in the final state ofatwo-bodyhighenergyscatteringwherethehoopcon- (2) equation (11) for the de Broglie wave length with jecture has been validated using numerical/computational string corrections; techniques [28]. This different case will be discussed in the (3) equation (13) for the Schwarzschild radius with string next section using a more analytical approach. corrections. 2.1. High Energy Collisions and Black Hole Production. We are Now, it is time to consider the hoop-conjecture and check the now ready to extend (11)tothecaseofatwo-bodysystem self-consistency of our formulae. (1) of colliding partons in the framework of higher dimensional Let us start with case and address the central question quantum gravity. In this case, the gravitational coupling as follows. Canaboostedobjectbeseencontractedbelowits 𝐺 [𝐺 ]=𝑙𝑑−1 Schwarzschild radius? constant is ∗ with dimensions (in natural units) ∗ , much below the Planck energy, and 𝑑 isthenumberofspace- If so, the hoop-conjecture would imply the original object ≥ is seen as a black hole: like dimensions ( 3). If the two partons have four-momenta 𝑝1 and 𝑝2, it is useful to introduce the Mandelstam variable 𝛼󸀠𝜃 (𝛽) 𝑠=−(𝑝+𝑝 )2 𝑠 2 𝐻 1 2 .Intermsof , we can define the “effective 𝐿0√1−𝛽 + 2 Schwarzschild radius”ofthesystemas 4𝐿0√1−𝛽 1/(𝑑−2) 1/(𝑑−2) (19) 𝑟 (𝑠) = (2√𝑠𝐺 ) ≡ (√𝑠𝐿𝑑−1) , (23) 𝛼󸀠𝜃 (𝛽) 𝐻 ∗ ∗ 2 𝐻 ≤𝑅𝑠√1−𝛽 + . 2 𝐿 4𝑅𝑠√1−𝛽 where ∗ is the higher dimensional minimal length. The hoop-conjecture states that whenever the two partons collide 𝑏≤𝑟(𝑠) We may regard this relation either as an equation for the with an impact parameter 𝐻 ,amicroblackholeis produced. In our approach, we can rephrase this statement radius 𝐿0 below which the object is shielded by horizon, or 𝛽̃ as follows: the two-parton system will collapse into a black as an equation for a hypothetical “terminal speed” that, hole if the de Broglie wavelength (11)issmallerorequalto once surpassed, will make the object appear inside its own the Schwarzschild radius (13) Schwarzschild hoop. In the first case, it is immediate to rec- 𝐿 ≤𝑅 𝛽 2 2 ognize that 0 𝑠 is the independent, somewhat “trivial” 1 𝐿 𝑑−1 1/(𝑑−2) 𝐿 𝑑−1 −1/(𝑑−2) + ∗ √𝑠 ≤ (√𝑠𝐿 ) + ∗ (√𝑠𝐿 ) , (24) solution. In order to be seen as a black object, the maximal √𝑠 4 ∗ 4 ∗ length, at rest, must be smaller than the Schwarzschild radius 𝑅 𝐿 >𝑅 𝑠. On the other hand, if one assumes that 0 𝑠 and tries wherewehaveswitchedtonaturalunitsℎ=𝑐=1and no ̃ to determine 𝛽 from (19), then one finds step-function is needed as the two particles are by definition in a relative state of motion. Solving for 𝑠,wefindthethreshold 󸀠 ̃2 𝛼 invariant energy for the creation of a microblack hole. This is a 𝛽 =1+ >1 (20) 4𝑅𝑠𝐿0 necessary, but not a sufficient condition for this event to occur. Advances in High Energy Physics 5

As it can be expected, the production channel opens up once that marks the sharp transition from special relativity to the the quantum gravity energy scale is reached: “quantum gravity” regime has been related to the threshold 1 energy where gravitationally interacting point particles col- √𝑠 ≥ =𝑀 . lapse into an extended microblack hole. This threshold energy 𝐿 ∗ (25) ∗ is determined by the final unification scale where quantum gravity becomes as strong as the other interactions. Equation (25) tells us that in a high energy scattering Assuming that the “superunification” scale is the Planck experiment we can probe distances down to 𝐿∗ but not scale, is there any clue as to what the expression of the “supe- beyond. The would be trans-Planckian region is shielded by runified force” might be? This question leads us to confront the creation of a black hole with linear dimension increasing the notion of maximal tension introduced by Gibbons in the with 𝑠. This argument is the essence of a recent proposal by framework of general relativity [34]. Dvali and coworkers [29, 30] to explain how quantum gravity Gibbons’ conjecture is that there exists in nature a limiting can self-regularize in the deep ultraviolet region [26](The force; let us call it a superforce,whoseexact expression is given scenario of UV self-complete quantum gravity is especially by attractive when realized in the more general framework of TeV quantum gravity. In this case, the Planck scale is lowered 𝑐4 𝐹 = . down to the TeV scale, and it opens the exciting possibility 𝑠 4𝐺 (26) to detect quantum gravity signals at LHC.). Thus, the trans- 𝑁 Planckian regime is actually inaccessible, and the deep UV With the above expression in hand, we are in a position to region is dominated by large, “classical” field configurations. add some final remarks that may shed a different light on This mechanism has been dubbed “classicalization” [31, 32]. the whole sequence of arguments presented in this paper. A There is a second important consequence of the relation more comprehensive account of the following points will be (24) regarding the final stage of black hole evaporation. presented in a forthcoming article [35]. Microblackholesareknowntobesemiclassicallyunsta- ble because of Hawking radiation. However, the standard (1)Note,first,thatourdefinitionsofPlanckunitsare description of thermal decay breaks down when the black consistent with Gibbons’ expression of the superforce. hole approaches the full quantum gravity regime. Even In other words, on dimensional grounds alone, the worse, no semiclassical model can foresee the endpoint of superforce is the “Planck force.” Having established the process which remains open to largely unsubstantiated that, it takes an elementary calculation to verify that 4 speculations. the Gibbons-Planck force 𝐹𝑠 = 𝑐 /4𝐺𝑁 is, indeed, Equation (24), on the other hand, shows not only the the Planckian limit of both the electrostatic Coulomb transition of a two-particle system into black hole, but the force and the static gravitational Newton’s force! inverse process as well. Start from the black hole region and Whilethisisadefinitecluethat(26)istheunification decrease the invariant mass of the object. Effective models point of the electrogravitational force, it remains an of “quantum gravity-improved” black holes suggest that for open question whether it also represents the supe- 𝑀≫𝑀∗ the semiclassical model is correct and the particle runified value of all fundamental forces. emission is to a good degree of approximation a grey-body (2) With hindsight, the conspicuous absence of ℎ from thermal radiation. However, as the mass of the black hole the Gibbons-Planck expression seems to support the approaches the value 𝑀∗,andthesizebecomescomparable classicalization idea as well as the idea of a transi- with 𝐿∗,themassoftheobjectrevealsadiscretespectrum tion from “contraction” to “dilation” in the modified and the decay process goes on through the emission of a expression of the Lorentz-Fitzgerald formula. Again, few quanta while jumping quantum mechanically towards with hindsight, both ideas are inherent in the funda- the ground state. In this late stage of decay, the black hole mental relationship (2) behaves like a hadronic resonance or an unstable nucleus, rather than a hot body. Thus, it is not surprising that after ℎ 2𝑀 𝐺 = ∗ ∗ . crossing the critical point 𝑀=𝑀∗ one is left with an 2 (27) 𝑀∗𝑐 𝑐 “ordinary” elementary particle system [33]. As a matter of fact, inspection of the above equation 3. Final Remarks: A New ‘‘Black Hole shows that, on the one hand, it defines the Planck Universal Constant’’ and the Existence of scale of mass-energy, but, on the other hand, it signals a trade-off, at the Planck scale of energy, a Maximal Force in Nature between a quantum length (Compton) and a classical In this note we have proposed a consistent framework to one (Schwarzschild) with the concomitant transition reconcile the existence of a new fundamental constant of from “Lorentz contraction” in the particle phase to nature,withlengthdimension,withtheLorentz-Fitzgerald “Schwarzschild expansion” in the black hole phase. contraction expected from special relativity. The presence of (3)Theappearanceof𝐺𝑁 in 𝐹𝑠 makes one wonder an ultimate length barrier has been related to the presence of about the specific role of gravity in the unification a black hole barrier that shields the trans-Planckian regime of fundamental forces. Here we offer an alternative, from a direct investigation. The critical boost factor 𝛾∗ gravity inspired, interpretation of the superforce: it 6 Advances in High Energy Physics

represents the ultimate linear energy density of a [2] M. Cavaglia, “Black hole and brane production in TeV gravity: black hole. In order to underscore this point, consider a review,” JournalofModernPhysicsA,vol.18,p.1843,2003. the conventional volume density of a body. In the [3] T. G. Rizzo, “Non-commutative inspired black holes in extra case of a black hole, this leads to the somewhat dimensions,” JournalofHighEnergyPhysics,vol.2006,no.9, counterintuitive result that the density is inversely article 21, 2006. proportional to the square of the mass [4] A. Casanova and E. Spallucci, “TeV mini black hole decay at future colliders,” Classical Quantum Gravity,vol.23,no.3,pp. 𝑀 3𝑐6 1 R45–R62, 2006. 𝜌 = = , 𝐵𝐻 3 3 2 (28) [5] R. Casadio and P. Nicolini, “The decay-time of non- 4𝜋𝑅𝑠 /3 32𝜋𝐺 𝑀 𝑁 commutative micro-black holes,” Journal of High Energy so that, while mini black holes may possess a nuclear Physics,vol.2008,no.11,article72,2008. density, galactic black holes can be less dense than [6] D. M. Gingrich, “Non-commutative geometry inspired black holes in higher dimensions at the LHC,” JournalofHighEnergy water. On the other hand, by considering the linear Physics,vol.2010,no.5,article22,2010. energy density, one obtains the universal constant [7] P. Nicolini and E. Winstanley, “Hawking emission from quan- 𝑀𝑐2 𝑐4 tum gravity black holes,” JournalofHighEnergyPhysics,vol. 𝜌 = = . 2011, no. 11, article 75, 2011. ∗ 2𝑅 4𝐺 (29) 𝑠 𝑁 [8] M. Bleicher and P. Nicolini, “Large extra dimensions and small black holes at the LHC,” Journal of Physics,vol.237,no.1,Article In other words, there exists in nature a limiting ID 012008, 2010. linear density that is a universal characteristic of all [9] E. Spallucci and A. Smailagic, “Black holes production in self- (Schwarzschild) black holes regardless of their mass complete quantum gravity,” Physics Letters B,vol.709,no.3,pp. or size.Atfirstsight,thisresultmayseemsurprising 266–269, 2012. and hard to understand. In actual fact, the physical [10] J. Mureika, P. Nicolini, and E. Spallucci, “Could any black holes explanation rests on the duality between deep UV be produced at the LHC?” Physical Review D,vol.85,no.10, and far IR domains in quantum gravity. The unique ArticleID106007,8pages,2012. properties of black holes bridge the gap between [11] J. Wheeler, “On the nature of quantum geometrodynamics,” trans-Planckian and classical physics [36]! Annals of Physics,vol.2,no.6,pp.604–614,1957. (4) Last but not least, given the background of ideas [12] J. A. Wheeler and K. Ford, Geons, Black Holes, and Quantum advanced in this paper, it seems natural to identify Foam: A Life in Physics, Norton, New York, NY, USA, 1998. [13]A.Kempf,G.Mangano,andR.B.Mann,“Hilbertspace 𝜌∗ with the energy density of a relativistic string,and, therefore, we identify the superforce equation (29) representation of the minimal length uncertainty relation,” Physical Review D,vol.52,no.2,pp.1108–1118,1995. with the universal string tension [14] A. Kempf and G. Mangano, “Minimal length uncertainty ℎ𝑐 relation and ultraviolet regularization,” Physical Review D,vol. 𝜌 ≡ ≡𝑇. (30) ∗ 2𝜋𝛼󸀠 𝑠 55,no.12,ArticleID7909,pp.7909–7920,1997. [15] E. Witten, “Reections on the fate of spacetime,” Physics Today, vol. 96, p. 24, 1996. Therefore, there are two equivalent ways of writing 𝜌∗: [16] L. J. Garay, “Quantum gravity and minimum length,” Interna- (i) classical, macroscopic form given by Gibbons’ maxi- tional Journal of Modern Physics A,vol.10,pp.145–166,1995. mal tension (29); [17] M. Sprenger, P.Nicolini, and M. Bleicher, “Physics on the smallest scales: an introduction to minimal length phenomenology ,” (ii) quantum, microscopic form which is the string tension European Journal of Physics,vol.33,no.4,p.853,2012. (30). [18] K. S. Thorne, “Nonspherical gravitational collapse, a short review,” in Magic Without Magic,J.R.Klauder,Ed.,pp.231–258, The two definitions are linked through Freeman, San Francisco, Calif, USA, 1972. (i) the existence of a universal linear energy density for [19]P.C.AichelburgandR.U.Sexl,“Onthegravitationalfieldofa massless particle,” General Relativity and Gravitation,vol.2,no. black holes exposing their “stringy nature.” [37]; 4, pp. 303–312, 1971. (ii) the “classicalization mechanism” of quantum gravity [20] P. Nicolini, A. Smailagic, and E. Spallucci, “Non-commutative that identifies trans-Planckian black holes with clas- geometry inspired Schwarzschild black hole,” Physics Letters B, sical, macroscopic objects. vol. 632, no. 4, pp. 547–551, 2006. [21] S. Ansoldi, P. Nicolini, A. Smailagic, and E. Spallucci, “Non- It seems to be a unique property of gravity to bridge the gap commutative geometry inspired charged black holes,” Physics between micro- and macroworlds. Letters B,vol.645,no.2-3,pp.261–266,2007. [22] E. Spallucci, A. Smailagic, and P. Nicolini, “Non-commutative References geometry inspired higher-dimensional charged black holes,” Physics Letters B,vol.670,no.4-5,pp.449–454,2009. [1] S. B. Giddings and S. D. Thomas, “High energy colliders as [23] P. Nicolini, “Non-commutative black holes, the final appeal to black hole factories: the end of short distance physics,” Physical quantum gravity: a review,” International Journal of Modern Review D,vol.65,no.5,ArticleID056010,12pages,2002. Physics A,vol.24,no.7,p.1229,2009. Advances in High Energy Physics 7

[24] A. Smailagic and E. Spallucci, “‘Kerrr’ black hole: the lord of the string,” Physics Letters B,vol.688,no.1,pp.82–87,2010. [25] P. Nicolini and E. Spallucci, “Non-commutative geometry- inspired dirty black holes,” Classical and Quantum Gravity,vol. 27, no. 1, Article ID 015010, 2010. [26] E. Spallucci and S. Ansoldi, “Regular black holes in UV self- complete quantum gravity,” Physics Letters B,vol.701,no.4,pp. 471–474, 2011. [27] P. Nicolini and E. Spallucci, “Holographic screens in ultraviolet self-complete quantum gravity,” http://arxiv.org/abs/1210.0015 . [28] M. W. Choptuik and F. Pretorius, “Ultrarelativistic particle collisions,” Physical Review Letters, vol. 104, no. 11, Article ID 111101, 4 pages, 2010. [29] G. Dvali and C. Gomez, “Self-completeness of einstein gravity,” http://arxiv.org/abs/1005.3497v1 . [30] G. Dvali, S. Folkerts, and C. Germani, “Physics of trans- Planckian gravity,” Physical Review D,vol.84,no.2,ArticleID 024039, 14 pages, 2011. [31] G. Dvali, G. F. Giudice, C. Gomez, and A. Kehagias, “V- completion by classicalization,” Journal of High Energy Physics, vol. 2011, no. 8, article 108, 2011. [32] G. Dvali and D. Pirtskhalava, “Dynamics of unitarization by classicalization,” Physics Letters B,vol.699,no.1-2,pp.78–86, 2011. [33] P. Meade and L. Randall, “Black holes and quantum gravity at the LHC,” JournalofHighEnergyPhysics,vol.2008,no.5,article 3, 2008. [34] G. W. Gibbons, “The maximum tension principle in general relativity,” Foundations of Physics,vol.32,no.12,pp.1891–1901, 2002. [35] A. Aurilia and E. Spallucci, “Planck’s uncertainty principle and the saturation of Lorentz boosts by Planckian black holes,” http://arxiv.org/abs/1309.7186. [36] B. P. Kosyakov, “Black holes: interfacing the classical and the quantum,” Foundations of Physics, vol. 38, no. 7, pp. 678–694, 2008. [37] G. T. Horowitz and J. Polchinski, “Correspondence principle for black holes and strings,” Physical Review D,vol.55,no.10,pp. 6189–6197, 1997.