DOCSLIB.ORG

Gravity in Quantum Mechanics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Gravity in Quantum Mechanics COMMENTARY | INSIGHT Gravity in quantum mechanics Giovanni Amelino-Camelia Gravity and quantum mechanics tend to stay out of each other’s way, but this might change as we devise new experiments to test the applicability of quantum theory to macroscopic systems and larger length scales. he remarkable accomplishments of Among the numerous approaches1 on phenomena of a mainly relativistic twentieth-century physics revolve used to define the interface between and gravitational nature that are studied Taround the success of two theoretical relativity and quantum mechanics, I find with experimental sensitivities for which paradigms. On the one hand, we have it easiest to focus on the one centred one might expect tiny effects originating phenomena described by quantum on modified uncertainty relations and/ from the interface between gravity and mechanics and involving interactions or modified commutator relations. A quantum mechanics. Some of the most that are governed by the standard model well-studied scenario3,4 assumes that the interesting opportunities for such tests of particle physics. On the other hand, uncertainty relations for measuring the concern the description of the propagation we have (general) relativity and its position coordinates xj and momenta pk of particles over astrophysical distances. description of gravitational phenomena. are produced by non-commutativity of the Relativity makes firm predictions for These two very different theories manage relevant observables of the form: these laws of propagation — assuming, to share quarters by keeping clear of however, that spacetime coordinates are 1 2 2 each other . Gravity is negligible in [xj, pk] = iδjkh(1 + α p ) (1) unaffected by uncertainty principles. the typical applications of quantum But new uncertainty principles, such mechanics, which involve microscopic where δjk is 1 only when j = k (0 otherwise) as the one encoded through equation particles and relatively short distances. and α is a length-scale characteristic (2), would affect the structure of the Analogously, quantum mechanical of the modification to be determined signal in photons and neutrinos seen by effects are usually inconsequential experimentally. The standard Heisenberg telescopes monitoring distant explosions in when we study macroscopic bodies and commutator is recovered in the limiting astrophysical bodies. large distance scales, where gravity is case where the effects of α can be neglected. Such imprints are now being sought in charge. In addition, there may also be new with the Fermi (see Fig. 1), HESS and Still, we do not expect gravity to be uncertainty principles and non-trivial MAGIC telescopes for photons, and truly absent at microscopic scales or that commutators involving pairs of position with IceCube for neutrinos. There is a quantum mechanics should somehow coordinates of the form5,6: determined effort to find evidence of switch off at macroscopic distances: it is spacetime fuzziness effects. The data m just that the effects they produce in those [xj, xk] = iΘjk + iΦjkxm (2) analysis would be very simple if we could regimes are very small and we have not yet assume that the astrophysical source m managed to develop the technologies and where the matrices Θjk and Фjk would emits a burst of high-energy photons devise the experiments capable of seeing have to be characterized by small length and neutrinos all in exact simultaneity: such small effects. But this frustratingly scales, small enough to explain why if such a short-duration burst propagates leaves us without a clue about how quantum mechanics was so far successful in a classical spacetime, then all particles these very different theories manage to ignoring them. in the burst must reach our telescope cooperate when they both must be taken For measurements involving large (nearly) simultaneously. One of the m into account. distance scales, the term Фjk in equation (2) possible implications of the modified The early Universe is the prototypical could be important in two very different uncertainty relations is that the signal example of where we expect both theories ways. There will be situations that we would propagate with some fuzziness to produce large effects1,2. However, with usually describe only in terms of relativity and the particles would not reach our no direct experimental access to the and gravitational effects and in these cases telescope simultaneously — the arrival conditions in the early Universe we have to the analysis of the spacetime properties times might therefore exhibit some sort of look elsewhere, and our best chances are will be affected by the new properties statistical spread. in regimes where one of the two theories of spacetime coordinates governed by However, we know that the duration of m dominates the description of the dynamics Фjk. And there will be situations that particle bursts from astrophysical sources and yet the smaller effects of the other we usually describe using quantum is not ideally small: in the best cases the theory might come within the reach of theory alone — here the analysis of the bursts last a few seconds. This decreases some high-sensitivity experiments. More quantum uncertainties might receive small the sensitivity of the studies, but we are simply put, we should then be looking corrections of quantum-gravitational origin learning how to compensate for these m either for (i) modifications of gravity by governed by Фjk. aspects of the emission mechanisms. quantum mechanics or, conversely, for The first of these two possibilities has The results so far have been negative, (ii) modifications of quantum mechanics already been studied intensely, particularly but the expected pace of improvement by gravity. over the past decade2. These efforts focused in sensitivities for the next decade or so 254 NATURE PHYSICS | VOL 10 | APRIL 2014 | www.nature.com/naturephysics © 2014 Macmillan Publishers Limited. All rights reserved INSIGHT | COMMENTARY provides hope that a discovery might be just around the next corner. Much less has been done for the Ursa Major second possibility, the case of phenomena primarily governed by standard quantum Leo mechanics, but affected by small corrections originating from the interface between gravity and quantum mechanics. Nevertheless, over the past couple of years GRB 130427A there have been some studies that I believe might set the stage for quick progress in this line of research. In this respect I feel it is very significant that techniques are being developed for testing some of the most striking features of quantum mechanics — such as entanglement — in experiments involving the exchange of particles over truly macroscopic distances, including the COLLABORATION LAT NASA/DOE/FERMI possibility of exchanging particles between Figure 1 | Searching the skies for the tiny effects that originate from the interface between gravity and a ground laboratory and a satellite7 (see also quantum mechanics. Gamma-ray bursts are short-lived and very bright. The highest-energy light ever the Review by Shadbolt et al. in this issue8). detected from such an event (GRB130427A) was observed in 2013. The image in the left panel taken by On the theory side, we will have to catch NASA’s Fermi Gamma-ray telescope shows how the northern galactic hemisphere of the gamma-ray sky up with these experimental opportunities looked just before the GRB130427A burst depicted in the right panel. and we have just recently started to make progress9,10 in understanding how the non-commutativity of coordinates exploited in the search of manifestations of I used here. More realistic theoretical (equation (2)) could affect entanglement the interface between gravity and quantum descriptions of macroscopic bodies could and other striking aspects of quantum mechanics. provide guidance for these experiments. theory. For instance, we expect that the It is useful to contemplate an idealized I, for one, am not at all frustrated by the presence of further contributions to the description12,14,15 of a macroscopic body fact that theory might have to catch up with uncertainties that grow with distance composed of N identical particles in experiments. For a long time — a time that would produce a loss of coherence in the terms of its centre-of-mass coordinates might be eventually viewed as the dark ages 1 N n 1 N n quantum states that becomes increasingly Xj =-N ∑n=1 xj and Pj =-N ∑n=1 pj where of quantum-gravity research — it seemed n n significant over large distances. This x j and p j denote the coordinates and that the study of the interface between coherence loss would lead to a gradual loss momentum of the nth particle. In the gravity and quantum mechanics should of entanglement. So with an entangled state current formulation of quantum mechanics, be a unique case of ‘pure-theory science’. shared between Earth and a distant satellite, the validity of standard commutators It was not expected that experiments m n we could find an otherwise unexpected for the constituent particles [x j, x k] = 0 would ever reach the level of the theory. m n loss of coherence — possibly signalling a and [x j, pk] = iδnmδjkh implies that also But things are now changing, and it would quantum-gravity effect. [Xj, Xk] = 0 and [Xj, Pk] = iδjkh, meaning be extremely exciting if experiments It would be very interesting to look that the same commutation relations apply took the lead in some areas of quantum for such an effect, even though the to the centre-of-mass degree of freedom gravity research. ❐ theoretical efforts aimed at modelling such of the macroscopic body. This striking phenomena cannot give us much guidance property of standard quantum mechanics Giovanni Amelino-Camelia is in the Physics yet. We understand qualitatively the turns out to be fragile, and as soon as any Department, Sapienza University of Rome, m mechanism that should produce the loss of the parameters α, Θjk, Фjk in equations Rome 00185, Italy.
Load more

Recommended publications
  • Kaluza-Klein Gravity, Concentrating on the General Rel- Ativity, Rather Than Particle Physics Side of the Subject
    Kaluza-Klein Gravity J. M. Overduin Department of Physics and Astronomy, University of Victoria, P.O. Box 3055, Victoria, British Columbia, Canada, V8W 3P6 and P. S. Wesson Department of Physics, University of Waterloo, Ontario, Canada N2L 3G1 and Gravity Probe-B, Hansen Physics Laboratories, Stanford University, Stanford, California, U.S.A. 94305 Abstract We review higher-dimensional unified theories from the general relativity, rather than the particle physics side. Three distinct approaches to the subject are identi- fied and contrasted: compactified, projective and noncompactified. We discuss the cosmological and astrophysical implications of extra dimensions, and conclude that none of the three approaches can be ruled out on observational grounds at the present time. arXiv:gr-qc/9805018v1 7 May 1998 Preprint submitted to Elsevier Preprint 3 February 2008 1 Introduction Kaluza’s [1] achievement was to show that five-dimensional general relativity contains both Einstein’s four-dimensional theory of gravity and Maxwell’s the- ory of electromagnetism. He however imposed a somewhat artificial restriction (the cylinder condition) on the coordinates, essentially barring the fifth one a priori from making a direct appearance in the laws of physics. Klein’s [2] con- tribution was to make this restriction less artificial by suggesting a plausible physical basis for it in compactification of the fifth dimension. This idea was enthusiastically received by unified-field theorists, and when the time came to include the strong and weak forces by extending Kaluza’s mechanism to higher dimensions, it was assumed that these too would be compact. This line of thinking has led through eleven-dimensional supergravity theories in the 1980s to the current favorite contenders for a possible “theory of everything,” ten-dimensional superstrings.
    [Show full text]
  • Conformal Symmetry in Field Theory and in Quantum Gravity
    universe Review Conformal Symmetry in Field Theory and in Quantum Gravity Lesław Rachwał Instituto de Física, Universidade de Brasília, Brasília DF 70910-900, Brazil; [email protected] Received: 29 August 2018; Accepted: 9 November 2018; Published: 15 November 2018 Abstract: Conformal symmetry always played an important role in field theory (both quantum and classical) and in gravity. We present construction of quantum conformal gravity and discuss its features regarding scattering amplitudes and quantum effective action. First, the long and complicated story of UV-divergences is recalled. With the development of UV-finite higher derivative (or non-local) gravitational theory, all problems with infinities and spacetime singularities might be completely solved. Moreover, the non-local quantum conformal theory reveals itself to be ghost-free, so the unitarity of the theory should be safe. After the construction of UV-finite theory, we focused on making it manifestly conformally invariant using the dilaton trick. We also argue that in this class of theories conformal anomaly can be taken to vanish by fine-tuning the couplings. As applications of this theory, the constraints of the conformal symmetry on the form of the effective action and on the scattering amplitudes are shown. We also remark about the preservation of the unitarity bound for scattering. Finally, the old model of conformal supergravity by Fradkin and Tseytlin is briefly presented. Keywords: quantum gravity; conformal gravity; quantum field theory; non-local gravity; super- renormalizable gravity; UV-finite gravity; conformal anomaly; scattering amplitudes; conformal symmetry; conformal supergravity 1. Introduction From the beginning of research on theories enjoying invariance under local spacetime-dependent transformations, conformal symmetry played a pivotal role—first introduced by Weyl related changes of meters to measure distances (and also due to relativity changes of periods of clocks to measure time intervals).
    [Show full text]
  • Unification of Gravity and Quantum Theory Adam Daniels Old Dominion University, [email protected]
    Old Dominion University ODU Digital Commons Faculty-Sponsored Student Research Electrical & Computer Engineering 2017 Unification of Gravity and Quantum Theory Adam Daniels Old Dominion University, [email protected] Follow this and additional works at: https://digitalcommons.odu.edu/engineering_students Part of the Elementary Particles and Fields and String Theory Commons, Engineering Physics Commons, and the Quantum Physics Commons Repository Citation Daniels, Adam, "Unification of Gravity and Quantum Theory" (2017). Faculty-Sponsored Student Research. 1. https://digitalcommons.odu.edu/engineering_students/1 This Report is brought to you for free and open access by the Electrical & Computer Engineering at ODU Digital Commons. It has been accepted for inclusion in Faculty-Sponsored Student Research by an authorized administrator of ODU Digital Commons. For more information, please contact [email protected]. Unification of Gravity and Quantum Theory Adam D. Daniels [email protected] Electrical and Computer Engineering Department, Old Dominion University Norfolk, Virginia, United States Abstract- An overview of the four fundamental forces of objects falling on earth. Newton’s insight was that the force that physics as described by the Standard Model (SM) and prevalent governs matter here on Earth was the same force governing the unifying theories beyond it is provided. Background knowledge matter in space. Another critical step forward in unification was of the particles governing the fundamental forces is provided, accomplished in the 1860s when James C. Maxwell wrote down as it will be useful in understanding the way in which the his famous Maxwell’s Equations, showing that electricity and unification efforts of particle physics has evolved, either from magnetism were just two facets of a more fundamental the SM, or apart from it.
    [Show full text]
  • Post-Newtonian Approximation
    Post-Newtonian gravity and gravitational-wave astronomy Polarization waveforms in the SSB reference frame Relativistic binary systems Effective one-body formalism Post-Newtonian Approximation Piotr Jaranowski Faculty of Physcis, University of Bia lystok,Poland 01.07.2013 P. Jaranowski School of Gravitational Waves, 01{05.07.2013, Warsaw Post-Newtonian gravity and gravitational-wave astronomy Polarization waveforms in the SSB reference frame Relativistic binary systems Effective one-body formalism 1 Post-Newtonian gravity and gravitational-wave astronomy 2 Polarization waveforms in the SSB reference frame 3 Relativistic binary systems Leading-order waveforms (Newtonian binary dynamics) Leading-order waveforms without radiation-reaction effects Leading-order waveforms with radiation-reaction effects Post-Newtonian corrections Post-Newtonian spin-dependent effects 4 Effective one-body formalism EOB-improved 3PN-accurate Hamiltonian Usage of Pad´eapproximants EOB flexibility parameters P. Jaranowski School of Gravitational Waves, 01{05.07.2013, Warsaw Post-Newtonian gravity and gravitational-wave astronomy Polarization waveforms in the SSB reference frame Relativistic binary systems Effective one-body formalism 1 Post-Newtonian gravity and gravitational-wave astronomy 2 Polarization waveforms in the SSB reference frame 3 Relativistic binary systems Leading-order waveforms (Newtonian binary dynamics) Leading-order waveforms without radiation-reaction effects Leading-order waveforms with radiation-reaction effects Post-Newtonian corrections Post-Newtonian spin-dependent effects 4 Effective one-body formalism EOB-improved 3PN-accurate Hamiltonian Usage of Pad´eapproximants EOB flexibility parameters P. Jaranowski School of Gravitational Waves, 01{05.07.2013, Warsaw Relativistic binary systems exist in nature, they comprise compact objects: neutron stars or black holes. These systems emit gravitational waves, which experimenters try to detect within the LIGO/VIRGO/GEO600 projects.
    [Show full text]
  • Gravity, Inertia, and Quantum Vacuum Zero Point Fields
    Foundations of Physics, Vol.31,No. 5, 2001 Gravity, Inertia, and Quantum Vacuum Zero Point Fields James F. Woodward1 Received July 27, 2000 Over the past several years Haisch, Rueda, and others have made the claim that the origin of inertial reaction forces can be explained as the interaction of electri- cally charged elementary particles with the vacuum electromagnetic zero-point field expected on the basis of quantum field theory. After pointing out that this claim, in light of the fact that the inertial masses of the hadrons reside in the electrically chargeless, photon-like gluons that bind their constituent quarks, is untenable, the question of the role of quantum zero-point fields generally in the origin of inertia is explored. It is shown that, although non-gravitational zero-point fields might be the cause of the gravitational properties of normal matter, the action of non-gravitational zero-point fields cannot be the cause of inertial reac- tion forces. The gravitational origin of inertial reaction forces is then briefly revisited. Recent claims critical of the gravitational origin of inertial reaction forces by Haisch and his collaborators are then shown to be without merit. 1. INTRODUCTION Several years ago Haisch, Rueda, and Puthoff (1) (hereafter, HRP) pub- lished a lengthy paper in which they claimed that a substantial part, indeed perhaps all of normal inertial reaction forces could be understood as the action of the electromagnetic zero-point field (EZPF), expected on the basis of quantum field theory, on electric charges of normal matter. In several subsequent papers Haisch and Rueda particularly have pressed this claim, making various modifications to the fundamental argument to try to deflect several criticisms.
    [Show full text]
  • Loop Quantum Cosmology, Modified Gravity and Extra Dimensions
    universe Review Loop Quantum Cosmology, Modified Gravity and Extra Dimensions Xiangdong Zhang Department of Physics, South China University of Technology, Guangzhou 510641, China; [email protected] Academic Editor: Jaume Haro Received: 24 May 2016; Accepted: 2 August 2016; Published: 10 August 2016 Abstract: Loop quantum cosmology (LQC) is a framework of quantum cosmology based on the quantization of symmetry reduced models following the quantization techniques of loop quantum gravity (LQG). This paper is devoted to reviewing LQC as well as its various extensions including modified gravity and higher dimensions. For simplicity considerations, we mainly focus on the effective theory, which captures main quantum corrections at the cosmological level. We set up the basic structure of Brans–Dicke (BD) and higher dimensional LQC. The effective dynamical equations of these theories are also obtained, which lay a foundation for the future phenomenological investigations to probe possible quantum gravity effects in cosmology. Some outlooks and future extensions are also discussed. Keywords: loop quantum cosmology; singularity resolution; effective equation 1. Introduction Loop quantum gravity (LQG) is a quantum gravity scheme that tries to quantize general relativity (GR) with the nonperturbative techniques consistently [1–4]. Many issues of LQG have been carried out in the past thirty years. In particular, among these issues, loop quantum cosmology (LQC), which is the cosmological sector of LQG has received increasing interest and has become one of the most thriving and fruitful directions of LQG [5–9]. It is well known that GR suffers singularity problems and this, in turn, implies that our universe also has an infinitely dense singularity point that is highly unphysical.
    [Show full text]
  • Gravitational Interaction to One Loop in Effective Quantum Gravity A
    IT9700281 LABORATORI NAZIONALI Dl FRASCATI SIS - Pubblicazioni LNF-96/0S8 (P) ITHf 00 Z%i 31 ottobre 1996 gr-qc/9611018 Gravitational Interaction to one Loop in Effective Quantum Gravity A. Akhundov" S. Bellucci6 A. Shiekhcl "Universitat-Gesamthochschule Siegen, D-57076 Siegen, Germany, and Institute of Physics, Azerbaijan Academy of Sciences, pr. Azizbekova 33, AZ-370143 Baku, Azerbaijan 6INFN-Laboratori Nazionali di Frascati, P.O. Box 13, 00044 Frascati, Italy ^International Centre for Theoretical Physics, Strada Costiera 11, P.O. Box 586, 34014 Trieste, Italy Abstract We carry out the first step of a program conceived, in order to build a realistic model, having the particle spectrum of the standard model and renormalized masses, interaction terms and couplings, etc. which include the class of quantum gravity corrections, obtained by handling gravity as an effective theory. This provides an adequate picture at low energies, i.e. much less than the scale of strong gravity (the Planck mass). Hence our results are valid, irrespectively of any proposal for the full quantum gravity as a fundamental theory. We consider only non-analytic contributions to the one-loop scattering matrix elements, which provide the dominant quantum effect at long distance. These contributions are finite and independent from the finite value of the renormalization counter terms of the effective lagrangian. We calculate the interaction of two heavy scalar particles, i.e. close to rest, due to the effective quantum gravity to the one loop order and compare with similar results in the literature. PACS.: 04.60.+n Submitted to Physics Letters B 1 E-mail addresses: [email protected], bellucciQlnf.infn.it, [email protected] — 2 1 Introduction A longstanding puzzle in quantum physics is how to marry the description of gravity with the field theory of elementary particles.
    [Show full text]
  • Einstein's Gravitational Field
    Einstein’s gravitational field Abstract: There exists some confusion, as evidenced in the literature, regarding the nature the gravitational field in Einstein’s General Theory of Relativity. It is argued here that this confusion is a result of a change in interpretation of the gravitational field. Einstein identified the existence of gravity with the inertial motion of accelerating bodies (i.e. bodies in free-fall) whereas contemporary physicists identify the existence of gravity with space-time curvature (i.e. tidal forces). The interpretation of gravity as a curvature in space-time is an interpretation Einstein did not agree with. 1 Author: Peter M. Brown e-mail: [email protected] 2 INTRODUCTION Einstein’s General Theory of Relativity (EGR) has been credited as the greatest intellectual achievement of the 20th Century. This accomplishment is reflected in Time Magazine’s December 31, 1999 issue 1, which declares Einstein the Person of the Century. Indeed, Einstein is often taken as the model of genius for his work in relativity. It is widely assumed that, according to Einstein’s general theory of relativity, gravitation is a curvature in space-time. There is a well- accepted definition of space-time curvature. As stated by Thorne 2 space-time curvature and tidal gravity are the same thing expressed in different languages, the former in the language of relativity, the later in the language of Newtonian gravity. However one of the main tenants of general relativity is the Principle of Equivalence: A uniform gravitational field is equivalent to a uniformly accelerating frame of reference. This implies that one can create a uniform gravitational field simply by changing one’s frame of reference from an inertial frame of reference to an accelerating frame, which is rather difficult idea to accept.
    [Show full text]
  • Principle of Relativity and Inertial Dragging
    ThePrincipleofRelativityandInertialDragging By ØyvindG.Grøn OsloUniversityCollege,Departmentofengineering,St.OlavsPl.4,0130Oslo, Norway Email: [email protected] Mach’s principle and the principle of relativity have been discussed by H. I. HartmanandC.Nissim-Sabatinthisjournal.Severalphenomenaweresaidtoviolate the principle of relativity as applied to rotating motion. These claims have recently been contested. However, in neither of these articles have the general relativistic phenomenonofinertialdraggingbeeninvoked,andnocalculationhavebeenoffered byeithersidetosubstantiatetheirclaims.HereIdiscussthepossiblevalidityofthe principleofrelativityforrotatingmotionwithinthecontextofthegeneraltheoryof relativity, and point out the significance of inertial dragging in this connection. Although my main points are of a qualitative nature, I also provide the necessary calculationstodemonstratehowthesepointscomeoutasconsequencesofthegeneral theoryofrelativity. 1 1.Introduction H. I. Hartman and C. Nissim-Sabat 1 have argued that “one cannot ascribe all pertinentobservationssolelytorelativemotionbetweenasystemandtheuniverse”. They consider an UR-scenario in which a bucket with water is atrest in a rotating universe,andaBR-scenariowherethebucketrotatesinanon-rotatinguniverseand givefiveexamplestoshowthatthesesituationsarenotphysicallyequivalent,i.e.that theprincipleofrelativityisnotvalidforrotationalmotion. When Einstein 2 presented the general theory of relativity he emphasized the importanceofthegeneralprincipleofrelativity.Inasectiontitled“TheNeedforan
    [Show full text]
  • Vacuum Energy
    Vacuum Energy Mark D. Roberts, 117 Queen’s Road, Wimbledon, London SW19 8NS, Email:[email protected] http://cosmology.mth.uct.ac.za/ roberts ∼ February 1, 2008 Eprint: hep-th/0012062 Comments: A comprehensive review of Vacuum Energy, which is an extended version of a poster presented at L¨uderitz (2000). This is not a review of the cosmolog- ical constant per se, but rather vacuum energy in general, my approach to the cosmological constant is not standard. Lots of very small changes and several additions for the second and third versions: constructive feedback still welcome, but the next version will be sometime in coming due to my sporadiac internet access. First Version 153 pages, 368 references. Second Version 161 pages, 399 references. arXiv:hep-th/0012062v3 22 Jul 2001 Third Version 167 pages, 412 references. The 1999 PACS Physics and Astronomy Classification Scheme: http://publish.aps.org/eprint/gateway/pacslist 11.10.+x, 04.62.+v, 98.80.-k, 03.70.+k; The 2000 Mathematical Classification Scheme: http://www.ams.org/msc 81T20, 83E99, 81Q99, 83F05. 3 KEYPHRASES: Vacuum Energy, Inertial Mass, Principle of Equivalence. 1 Abstract There appears to be three, perhaps related, ways of approaching the nature of vacuum energy. The first is to say that it is just the lowest energy state of a given, usually quantum, system. The second is to equate vacuum energy with the Casimir energy. The third is to note that an energy difference from a complete vacuum might have some long range effect, typically this energy difference is interpreted as the cosmological constant.
    [Show full text]
  • 5D Kaluza-Klein Theories - a Brief Review
    5D Kaluza-Klein theories - a brief review Andr´eMorgado Patr´ıcio, no 67898 Departamento de F´ısica, Instituto Superior T´ecnico, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal (Dated: 21 de Novembro de 2013) We review Kaluza-Klein theory in five dimensions from the General Relativity side. Kaluza's original idea is examined and two distinct approaches to the subject are presented and contrasted: compactified and noncompactified theories. We also discuss some cosmological implications of the noncompactified theory at the end of this paper. ^ ^ 1 ^ I. INTRODUCTION where GAB ≡ RAB − 2 g^ABR is the Einstein tensor. Inspired by the ties between Minkowki's 4D space- For the metric, we identify the αβ-part ofg ^AB with time and Maxwell's EM unification, Nordstr¨om[3] in 1914 gαβ, the α4-part as the electromagnetic potential Aα and and Kaluza[4] in 1921 showed that 5D general relativ- g^44 with a scalar field φ, parametrizing it as follows: ity contains both Einstein's 4D gravity and Maxwell's g + κ2φ2A A κφ2A EM. However, they imposed an artificial restriction of no g^ (x; y) = αβ α β α ; (4) AB κφ2A φ2 dependence on the fifth coordinate (cylinder condition). β Klein[5], in 1926, suggested a physical basis to avoid this where κ is a multiplicative factor. If we identifyp it in problem in the compactification of the fifth dimension, terms of the 4D gravitational constant by κ = 4 πG, idea now used in higher-dimensional generalisations to then, using the metric4 and applying the cylinder con- include weak and strong interactions.
    [Show full text]
  • Gravity, Orbital Motion, and Relativity
    Gravity, Orbital Motion,& Relativity Early Astronomy Early Times • As far as we know, humans have always been interested in the motions of objects in the sky. • Not only did early humans navigate by means of the sky, but the motions of objects in the sky predicted the changing of the seasons, etc. • There were many early attempts both to describe and explain the motions of stars and planets in the sky. • All were unsatisfactory, for one reason or another. The Earth-Centered Universe • A geocentric (Earth-centered) solar system is often credited to Ptolemy, an Alexandrian Greek, although the idea is very old. • Ptolemy’s solar system could be made to fit the observational data pretty well, but only by becoming very complicated. Copernicus’ Solar System • The Polish cleric Copernicus proposed a heliocentric (Sun centered) solar system in the 1500’s. Objections to Copernicus How could Earth be moving at enormous speeds when we don’t feel it? . (Copernicus didn’t know about inertia.) Why can’t we detect Earth’s motion against the background stars (stellar parallax)? Copernicus’ model did not fit the observational data very well. Galileo • Galileo Galilei - February15,1564 – January 8, 1642 • Galileo became convinced that Copernicus was correct by observations of the Sun, Venus, and the moons of Jupiter using the newly-invented telescope. • Perhaps Galileo was motivated to understand inertia by his desire to understand and defend Copernicus’ ideas. Orbital Motion Tycho and Kepler • In the late 1500’s, a Danish nobleman named Tycho Brahe set out to make the most accurate measurements of planetary motions to date, in order to validate his own ideas of planetary motion.
    [Show full text]