New Developments in Cosmology and Gravitation from Extended Theories of General Relativity

Advances in High Energy Physics

Guest Editors: Mauricio Bellini, Kishor Adhav, José Edgar Madriz Aguilar, and Dandala R. K. Reddy New Developments in Cosmology and Gravitation from Extended Theories of General Relativity Advances in High Energy Physics

New Developments in Cosmology and Gravitation from Extended Theories of General Relativity

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 G. Scholtz, South Africa Gongnan Xie, China Chao-Qiang Geng, Taiwan George Siopsis, USA Hong-Jian He, China Terry Sloan, UK Contents

New Developments in Cosmology and Gravitation from Extended Theories of General Relativity, Mauricio Bellini, Kishor Adhav, Jose´ Edgar Madriz Aguilar, and Dandala R. K. Reddy Volume 2014, Article ID 563125, 1 page

Necessity of Dark Energy from Thermodynamic Arguments,H.Moradpour,A.Sheykhi,N.Riazi, and B. Wang Volume2014,ArticleID718583,9pages

Primordial Dark Energy from a Condensate of Spinors in a 5D Vacuum, Pablo Alejandro SanchezandMauricioBellini´ Volume 2013, Article ID 789476, 7 pages

On Higher Dimensional Kaluza-Klein Theories,AurelBejancu Volume 2013, Article ID 148417, 12 pages

Anisotropic Bulk Viscous String Cosmological Model in a Scalar-Tensor Theory of Gravitation, D. R. K. Reddy, Ch. Purnachandra Rao, T. Vidyasagar, and R. Bhuvana Vijaya Volume 2013, Article ID 609807, 5 pages

Scaling Relations for the Cosmological “Constant” in Five-Dimensional Relativity, Paul S. Wesson and James M. Overduin Volume 2013, Article ID 214172, 6 pages

Gauss-Bonnet Braneworld Cosmology with Modified Induced Gravity on the Brane,KouroshNozari, Faeze Kiani, and Narges Rashidi Volume 2013, Article ID 968016, 12 pages

Noether Current of the Surface Term of Einstein-Hilbert Action, Virasoro Algebra, and Entropy, Bibhas Ranjan Majhi Volume 2013, Article ID 386342, 10 pages Hindawi Publishing Corporation Advances in High Energy Physics Volume 2014, Article ID 563125, 1 page http://dx.doi.org/10.1155/2014/563125

Editorial New Developments in Cosmology and Gravitation from Extended Theories of General Relativity

Mauricio Bellini,1 Kishor Adhav,2 José Edgar Madriz Aguilar,3 and Dandala R. K. Reddy4

1 IFIMAR, University of Mar del Plata and CONICET, Funes 3350, 7600 Mar del Plata, Argentina 2 Department of Mathematics, Sant Gadge Baba Amravati University, Amaravati, Mahasastra 444602, India 3 Department of Mathematics, CUCEI, University of Guadalajara, Avenue Revolucion´ 1500 S. R, 44430 Guadalajara, Mexico 4 Department of Science and Humanities, MVGR College of Engineering, Vizianagaram, Andhra Pradesh 530017, India

Correspondence should be addressed to Mauricio Bellini; [email protected]

Received 29 January 2014; Accepted 29 January 2014; Published 3 March 2014

Copyright © 2014 Mauricio Bellini 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.

Extensions and modifications to the standard 4D theory of interesting to study the motion on a 4D space induced from general relativity are topics that have an increasing impact a(4+n)D general gauge Kaluza-Klein space. in top original research on gravitation and cosmology. This Anisotropic cosmological models are really an interesting issue compiles exciting papers that can be very interesting topic for cosmologists and researchers, especially when this for researchers who could be interested in the study of topic is studied in the framework of a scalar-tensor theory extended theories of general relativity focused on cosmology of gravitation with a source such that the energy momentum and gravitation. tensor is a bulk viscous fluid containing one-dimensional An important topic for the present day cosmology con- cosmic strings. sists in explaining why the universe is accelerated and how Braneworld cosmology from modified induced gravity on thisaccelerationisrelatedtoanunknownkindofenergy the brane is a very close approach to modern Kaluza-Klein (or called “dark energy.” In this issue it is considered a cosmic Induced Matter theory of gravity). These kinds of approaches fluid as a quasi-static thermodynamic system. The status of can predict very plausible cosmological models that describe thegeneralizedsecondlawofthermodynamicsisinvestigated the history of the universe. and the range of validity for the equation of state parameter There are very interesting approaches related to extended is derived for a few important cosmological models. The theories of gravity applied to topological defects such as black physical origin of dark energy can be explored using a holes. In this context some thermodynamical considerations condensate of spinors which are free of interactions in a 5D can be explored when the horizon structure is invariant. relativistic vacuum defined in an extended de Sitter spacetime Bycompilingthesepapers,wehopetoenrichreaders which is Riemann flat. This condensate of spinors could bean with new investigations on this exciting research area and interesting candidate to explain the presence of dark energy stimulate new investigations which could be fundamental for in the early universe, during the inflationary stage. the future of the physics. A new method for the study of general higher dimensional Kaluza-Klein theories based on the Riemannian Mauricio Bellini adapted connection and on a theory of adapted tensor fields Kishor Adhav in the ambient space is also interesting to be studied. This JoseEdgarMadrizAguilar´ topiccanbeexploredinacovariantformanditisvery DandalaR.K.Reddy Hindawi Publishing Corporation Advances in High Energy Physics Volume 2014, Article ID 718583, 9 pages http://dx.doi.org/10.1155/2014/718583

Research Article Necessity of Dark Energy from Thermodynamic Arguments

H. Moradpour,1 A. Sheykhi,1,2 N. Riazi,3 and B. Wang4

1 Physics Department and Biruni Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran 2 Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O. Box 55134-441, Maragha, Iran 3 Physics Department, Shahid Beheshti University, Evin, Tehran 19839, Iran 4 INPAC and Department of Physics, Shanghai Jiao Tong University, Shanghai 200240, China

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

Received 17 August 2013; Accepted 26 December 2013; Published 20 January 2014

Academic Editor: Kishor Adhav

Copyright © 2014 H. Moradpour 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.

Considering the cosmic fluid as a quasi-static thermodynamic system, the status of the generalized second law of thermodynamics is investigated and the valid range of the equation of state parameter is derived for a few important cosmological models. Our study shows that the satisfaction of the laws of thermodynamics in these cosmological models requires the existence of some kind of energy in our universe with 𝜔<−1/3. In other words, the existence of a dark energy component, or equivalently modified gravity theory, is unavoidable if the cosmological model is to approach thermal equilibrium in late times.

1. Introduction [11–34] and references therein). The studies have also been generalized to higher order gravity theories such as Gauss- Thermodynamical nature of Einstein’s theory of general rela- Bonnet and Lovelock gravity [35]. All these attempts indicate tivity was first disclosed by Jacobson1 [ ] who showed that the ahopetoachieveadeeperconnectionbetweengravityand hyperbolic second order partial differential field equations of thermodynamics. gravity can be derived by applying the first law of thermodynamics on any local Rindler horizon. Generalization of this From the second law of thermodynamics, we know that method to 𝑓(𝑅) gravity, by introducing the entropy gener- every closed system moves towards its maximum entropy ation term due to nonequilibrium nature of spacetime, was state which is an equilibrium state. This leads to the con- investigated in [2]. More attempts to reveal the connection clusion that the second derivative of the entropy should be between thermodynamics and various theories of gravity can negative [36]. The assumption that the second derivative be found in [3–8]. An elegant example is the derivation of the of entropy is negative comes from the fact that while the Friedmann equations as a consequence of the validity of the entropy is increasing as the system approaches equilibrium, first law of thermodynamics on the apparent horizon of the it should tend to a maximum; hence the first derivative Friedmann-Robertson-Walker (FRW) universe [9]. which is already positive tends to zero, leading to a negative Recently, an entropic origin for gravity was proposed by second derivative. Indeed, our discussion in this paper is Verlinde [10]. He argued that the laws of gravity are not based on that the natural tendency of systems to evolve fundamental and in particular they emerge as an entropic toward thermodynamical equilibrium is characterized by force caused by the changes in the information associated two properties of its entropy function, 𝑆(𝑥);namely,itis with the positions of material bodies. Verlinde’s derivation always a nondecreasing function, 𝑑𝑆(𝑥)/𝑑𝑥 ≥0,andisa 2 2 of Newton’s law of gravitation at the very least offers a convex function, 𝑑 𝑆(𝑥)/𝑑𝑥 ≤0[37]. In the context of strong analogy with the well-understood statistical approach. cosmology, this implies that the total entropy including the Therefore, this derivation opens a new window to under- entropy associated with the apparent horizon together with standing gravity from first principles. The entropic approach the matter field entropy inside the apparent horizon must be 󸀠 to gravity has arisen a lot of enthusiasm, recently (see, e.g., a nondecreasing function of scale factor, 𝑆 (𝑎) ≥,wherethe 0 2 Advances in High Energy Physics prime stands for the derivative with respect to scale factor Also, we suppose the equation of state as 𝑎. This statement is usually called the generalized second law (GSL) of thermodynamics. In addition as 𝑎→∞we 𝑝=𝜔𝜌, (5) 󸀠󸀠 must have 𝑆 (𝑎) ≤. 0 Applying these requirements to the cosmological setup leads to interesting constraints on the where, for radiation and byronic matter, we have 𝜔≥0.From equation of state parameter of the cosmic fluid filling the the continuity (4), one finds universe. −3(1+𝜔) We consider a background which is filled by a homoge- 𝜌=𝜌0𝑎 . (6) neous fluid with the equation of state 𝑝=𝜔𝜌and endorsing situations in that the background component of the total In the long run where 𝑎→∞, the energy density goes to entropy plays the role of dominate entropy in fulfilling the zero provided that 𝜔>−1.For𝜔<−1,thatis,ghost GSLofthermodynamicsinthelongrunlimit.In[38, 39], matter,wehavetheunusualsituationwheredensitygrows authors have divided the entropy of the universe into three with expansion. Throughout this paper, we assume a spatially parts arising from the matter field, geometry (horizon), and flat FRW spacetime. In the next section, we study thermo- 𝑝=𝜔𝜌 an unknown fluid with the equation of state . dynamic nature of ghost dark energy model. Cyclic model, Then, they studied situations in which the background Horava-Liftshitz deformed model, DGP model, and Gauss- componentofentropyhasamajorcontributiontothetotal Bonnet model will be discussed in the subsequent sections. entropy of the universe and that it satisfies GSL. In this way, The last section is devoted to our concluding remarks. the authors of [38, 39] could find the proper range of 𝜔 which is consistent with dark energy. In the present work, we do not separate matter component from the dominant fluid and try 2. Ghost Dark Energy to reach the corresponding implications for the equation of A new dark energy model called “ghost dark energy” state in various theories of gravity. Further calculations using was recently proposed to explain the observed accelerating this approach were presented in [38, 39]. expansion of the Universe [44–51]. In this model, dark Let us have a glimpse at general properties of a homoge- energy originates from the Veneziano ghost of QCD [52, 53] neous and isotropic FRW universe which is described by the with an energy density which is proportional to the Hubble line element 𝜌 =𝛼𝐻 𝛼 Λ3 parameter, 𝐷 ,where isaconstantoforder QCD or 𝑑𝑠2 =ℎ 𝑑𝑥𝜇𝑑𝑥] + 𝑟̃2 [𝑑𝜃2 + 2𝜃𝑑𝜙2], Λ ∼ 𝜇] sin (1) QCD mass scale. Taking into account the fact that QCD −33 2 2 100 MeV and 𝐻∼10 eV for the present time, this gives the where 𝑟=𝑎(𝑡)𝑟̃ and ℎ𝜇] = diag(−1, 𝑎 (𝑡)/(1 − 𝑘𝑟 )) with −3 4 right order of magnitude 𝜌𝐷 ∼(3×10 eV) for the ghost 𝑥0 =𝑡 𝑥1 =𝑟 𝑘 and and is spatial curvature taking the val- energy density [44]. In this section we would like to con- 0, −1 1 ues ,and for a flat, open, and closed universe, resp- straint the equation of state parameter of ghost dark energy 𝑎(𝑡) ectively [40]. Here is the scale factor which carries the using the entropy argument. Let us rewrite the ghost energy effect of expansion on the spatial degrees of freedom. Recent density as [44] observations have confirmed an accelerating universe (i.e., 𝑎≥0̇ 𝑎≥0̈ −1 and )[41–43]. The validity of the first condition 𝐻=𝛼 𝜌. (7) (𝑎≥0̇ ) is clear. Using the fact that 𝐻=𝑎/𝑎̇ ,wehave 󸀠 𝑎=𝑎𝐻(𝐻+𝑎𝐻̈ ), (2) Substituting (6)and(7)in(2), after simple calculations, we find 𝜔≤−1/3in order to have an accelerating universe; where the prime stands for derivative with respect to the scale namely, 𝑎≥0̈ . Assuming the background is filled by a factor 𝑎. For every model of cosmology this relation makes typical fluid with energy density7 ( ), we want to see under anupperboundontheequationofstateparameter𝜔 and so which circumstances the GSL is preserved for the universe must be evaluated for each model separately. The apparent filled with ghost dark energy. The entropy associated with horizon which is defined as a marginally trapped surface the apparent horizon in a flat FRW universe obeys the well- with vanishing expansion can be determined by relation known area law [54]: 𝜇 ] ℎ𝜇]𝜕 𝑟𝜕̃ 𝑟=0̃ . A simple calculation yields 𝐴 −2 2 −2 6(1+𝜔) 1 𝑆 = =𝜋𝐻 =𝜋𝛼𝜌 𝑎 , (8) 𝑟̃ = . ℎ 4 0 𝐴 √𝐻2 +𝑘/𝑎2 (3) −2 where 𝐴=4𝜋𝐻 is the apparent horizon area. Throughout Suppose that the energy-momentum tensor of the total 𝐺=𝑐=ℎ=𝑘 = matter and energy in the universe has the form of a perfect this paper, we employ units in which 𝐵 𝑇 =𝑝𝑔 +(𝜌+𝑝)𝑈 𝑈 𝑈] 1. Taking the first and the second derivative of the entropy fluid, 𝜇] 𝜇] 𝜇 ],where denotes the four- 𝑆 velocity of the fluid and 𝜌 and 𝑝 are the total energy density function ℎ withrespecttothescalefactor,weget and pressure of the fluid, respectively. The energy conserva- 𝜇] 𝑆󸀠 =6𝜋𝛼2𝜌−2 (𝜔+1) 𝑎6𝜔+5, tion law ∇𝜇𝑇 =0leads to the continuity equation in the ℎ 0 form 5 (9) 𝑆󸀠󸀠 = 36𝜋𝛼2𝜌−2 (𝜔+1) (𝜔 + )𝑎6𝜔+4. 𝜌̇ + 3𝐻 (𝜌 + 𝑝) =0. (4) ℎ 0 6 Advances in High Energy Physics 3

Hence, we arrive at 3. Cyclic Universe Model 󸀠 The modified Friedman equation of cyclic universe, which 𝑆ℎ ≥0󳨐⇒𝜔≥−1, can be obtained from the effective theory of loop quantum (10) 󸀠󸀠 5 cosmology, can be written as [57, 58] 𝑆ℎ ≤0󳨐⇒−1≤𝜔≤− . 6 8𝜋𝐺 𝜌 𝐻2 = 𝜌(1− ), 3 𝜌 (17) Thus, for both conditions to be satisfied simultaneously, we 𝑐 −1≤𝜔≤−5/6 must have which is in agreement with the 76 4 accelerating condition 𝜔≤−1/3. Next, we consider the entr- where 𝜌𝑐 ≈10 (GeV) is the critical density, set by quantum opy of the perfect fluid inside the horizon. The entropy of the gravity [59] and disparate from the usual critical density 2 2 universe inside the horizon 𝑆𝑓 can be related to its energy and 3𝑀𝑝𝐻 .Using(2), (6), and (17) in the late time (low density), pressure in the horizon by the Gibbs equation [55]: the upper bound for 𝜔 can be obtained as 𝜔≤−1/3,similar to what we found for ghost dark energy model. Using area 𝑇 𝑑𝑆 =𝑑𝐸 +𝑝 𝑑𝑉, −2 𝑓 𝑓 𝑓 𝑓 (11) law for the horizon entropy, 𝑆ℎ =𝐴/4=𝜋𝐻 ,aswellas Friedmann (17), the first derivative of the entropy with respect where 𝐸𝑓 is the energy of the fluid within the Hubble horizon to scalar factor reads 𝐿=𝐻−1 with radius , 9 (𝜔+1) 𝜌−2𝜌2/𝜌 𝑆󸀠 = 𝑐 . ℎ 2 (18) −3 3 8𝐺 2 4𝜋𝐻 𝜌 4𝜋𝛼 𝑎(𝜌 −𝜌 /𝜌𝑐) 𝐸 =𝜌 𝑉= 𝑓 = , 𝑓 𝑓 3 2 (12) 3𝜌𝑓 󸀠 In two cases 𝑆ℎ can be positive. First for 𝜌>𝜌𝑐/2 and 𝜔<−1, andsecondfor𝜌<𝜌𝑐/2 and 𝜔>−1.Forthelatetimecosm- wherewehavealsoused(7). Therefore, we have ology where 𝑎→∞(𝜌→0), (18)reducesto

3 𝑑𝑆𝑓 𝑑 4𝜋𝛼 𝑑𝑉 󸀠 3(𝜔+(2/3)) 𝑇 = ( )+𝜔𝜌 . 𝑆ℎ ≈ (𝜔+1) 𝑎 . (19) 𝑓 𝑑𝑎 𝑑𝑎 2 𝑓 𝑑𝑎 (13) 3𝜌𝑓 󸀠 Therefore, in order to have 𝑆ℎ ≥0the equation of state para- Theevolutionofthetemperatureofthematterfieldcanbe meter should satisfy 𝜔≥−1. Now, we consider the second de- determined by (𝑑 ln 𝑇𝑓/𝑑 ln 𝑎) = −3𝜔, which leads to 𝑇𝑓 = rivative of the entropy: −3𝜔 𝑇0𝑎 [56]. From (13)onefinds 󸀠󸀠 9 (𝜔+1) 𝑆ℎ = 3 8𝐺𝑎2(𝜌 −2 𝜌 /𝜌 )2 󸀠 4𝜋𝜌0𝛼 9𝜔+5 9 (𝜔−1) 12𝜔+8 𝑐 𝑆𝑓 = (𝜔+1) (6𝑎 + 𝑎 ), (14) 3𝑇0 2 4 𝜌2 2𝜌2 × [ −3(𝜔+ )𝜌 +12(𝜔+1) + 𝜌=𝜌 3 𝜌 𝜌 (20) wherewehaveusedthefactthat 𝑓.Thesecond [ 𝑐 𝑐 derivative of the fluid entropy can be obtained as 2 (𝜌 − 2𝜌2/𝜌 ) 4𝜌 𝜋𝛼3 +6(𝜔+1) 𝑐 ] . 󸀠󸀠 0 2 𝑆𝑓 = (𝜔+1) 𝜌−𝜌 /𝜌𝑐 3𝑇0 ] (15) 9 (𝜔−1)(12𝜔 + 8) In the late time where 𝜌→0,theaboverelationcanbesim- ×[6(9𝜔 + 5) 𝑎9𝜔+4 + 𝑎12𝜔+7]. 2 plified as

󸀠󸀠 3𝜔+1 The ratio between derivatives of different components, the 𝑆ℎ ≈ (𝜔+1)(3𝜔 + 2) 𝑎 , (21) entropy of the fluid, and the entropy associated with the 󸀠󸀠 Hubble horizon can be obtained in the long run limit as which indicates that −1≤𝜔≤−2/3for which we have 𝑆ℎ ≤ 0. Next, we turn to the calculation of the entropy of a fluid 󸀠 󸀠󸀠 𝑆 𝑆 1 which fills the background spacetime. Following the method 𝑓 ≈ 𝑓 ∼𝑎3𝜔 +𝑎6𝜔+3 󳨐⇒ 𝜔 <− , 󸀠 󸀠󸀠 (16) of the previous section, we arrive at 𝑆ℎ 𝑆ℎ 2 𝜌 (𝜔+1) 2𝜋 𝑆󸀠 = 0 which implies that −3/2<𝜔<−1/2.Bycomparingwith(10), 𝑓 3/2 (8𝜋𝐺/3)3/2𝑇 𝑎4(𝜌 − 𝜌2/𝜌 ) it becomes obvious that the domain −1≤𝜔≤−5/6is an 0 𝑐 appropriate domain for the equation of state parameter of (22) 𝜌−2𝜌2/𝜌 3 ghost dark energy which satisfies all thermodynamic condi- ×[3𝜔 𝑐 + +1]. 2 tions simultaneously. 𝜌−𝜌 /𝜌𝑐 𝜌𝑐 (1 − 𝜌/𝜌𝑐) 4 Advances in High Energy Physics

Onecaneasilycheckthatatthelatetimetheaboverelation the above action L𝑚 stands for the lagrangian of the matter reduces to field, 𝑅 and 𝑅𝑖𝑗 are three-dimensional spatial Ricci scalar and 𝐶 󸀠 9𝜔/2+1/2 Ricci tensor, and 𝑖𝑗 is the Cotton tensor defined as 𝑆𝑓 󳨀→ (𝜔+1)(3𝜔 + 1) 𝑎 , (23) 𝑖𝑗 𝑖𝑘𝑙 𝑗 1 𝑗 𝑖𝑘𝑙 𝑗 1 𝑖𝑘𝑗 𝐶 =𝜖 ∇ (𝑅 − 𝑅𝛿 )=𝜖 ∇ 𝑅 − 𝜖 𝜕 𝑅, (28) andthusforthesecondderivativeinthislimitweobtain 𝑘 𝑙 4 𝑙 𝑘 𝑙 4 𝑘 𝑖𝑘𝑙 󸀠󸀠 (9𝜔 + 1)(3𝜔 + 1)(𝜔+1) 9𝜔−1/2 𝜖 𝑆 ≈ 𝑎 . (24) where is the totally antisymmetric unit tensor. It is worth 𝑓 2 mentioning that the IR vacuum of this theory is an anti de 𝜔<1 Sitter (AdS) spacetime. Hence, it is interesting to take a limit This implies for the dominating background compo- of the theory, which may lead to a Minkowskian spacetime in nent. This is due to the fact that theIRsector.Forthispurpose,onemaymodifythetheoryby 󸀠 󸀠󸀠 𝜇4𝑅 Λ →0 𝑆𝑓 𝑆𝑓 introducing andthentakethe 𝑊 limit [70]. This ∼ 󳨀→ 𝑎 3(𝜔−1)/2 󳨐⇒ 𝜔 <1. (25) 𝑆󸀠 𝑆󸀠󸀠 does not alter the UV properties of the theory, but it changes ℎ ℎ the IR properties. That is, there exists a Minkowski vacuum, −1 ≤ 𝜔 ≤ −2/3 insteadofanAdSvacuum.Wewillnowconsiderthelimitof Now,itisclearthatinthelimitof ,the Λ →0 background piece of entropy dominates and is responsible for this theory such that 𝑊 . The deformed action of the satisfying the GSL of thermodynamics. From (18)and(20), nonrelativistic renormalizable gravitational theory is given by in the high density limit, one finds condition 𝜔≤−3/2for [70]: 2 satisfying the GSL, which is compatible with phantom regime 3 2 𝑖𝑗 2 𝜅 𝑖𝑗 𝜔<−1) 𝐼 = ∫ 𝑑𝑡𝑑 𝑥√𝑔𝑁 { (𝐾 𝐾 −𝜆𝐾 )− 𝐶 𝐶 ( . SH 𝜅2 𝑖𝑗 2𝜉4 𝑖𝑗

4. IR Deformed Horava-Liftshitz Cosmology 𝜇2𝜅2 𝜅2𝜇 − 𝑅 𝑅𝑖𝑗 + 𝜖𝑖𝑗𝑘 𝑅 ∇ 𝑅𝑙 8 𝑖𝑗 2𝜉2 𝑖𝑙 𝑗 𝑘 Inspired by Lifshitz theory in solid state physics, Horava proposed a field theory model for a UV complete theory 𝜅2𝜇2 (1−4𝜆) of gravity [60–62].Thistheoryisanonrelativisticreno- + 𝑅2 +𝜇4𝑅} . rmalizable theory of gravity and reduces to Einstein’s gen- 32 (1−3𝜆) eral relativity at large scales. The theory is usually referred (29) to as the Horava-Lifshitz (HL) theory. It has also manifested In the IR limit, action (29) can be written as the standard three-dimensional spatial general covariance and time rep- Einstein-Hilbert action in the ADM formalism provided [70] arametrization invariance. Various aspects of HL gravity have been investigated in the literature [63–69]. In HL theory, that 𝜅2𝜇4 𝜅2 𝜆=1,2 𝑐 = ,𝐺= . (30) reduces to Einstein’s general relativity at large scales, quantum 2 32𝜋𝑐 field theory definitions of time and space come over Einstein’s proposalandso,inthelargescales,Lorentziansymmetryis The constant 𝜉 is given by [71] achievable. The action of HL gravity is given by60 [ –62] 8𝜇2 (3𝜆 − 1) 𝜉≡ . 2 (31) 𝐼 = ∫ 𝑑𝑡𝑑3𝑥(L + L̃ + L ), 𝜅 SH 0 1 𝑚 𝜉→∞ 𝜅2 →0 2 2 2 Besides, for (equivalently, ), action (29)red- 2 𝜅 𝜇 (Λ 𝑊𝑅−3Λ ) L = √𝑔𝑁 { (𝐾 𝐾𝑖𝑗 −𝜆𝐾2)+ 𝑊 }, uces to the action of Einstein gravity. 0 𝜅2 𝑖𝑗 8 (1−3𝜆) The Friedmann equation, resulting from variation of

2 2 2 2 action (29)withrespecttoFRWmetric,turnsouttobe[70] ̃ 𝜅 𝜇 (1−4𝜆) 2 𝜅 𝜇𝜉 L1 = √𝑔𝑁 { 𝑅 − (𝐶𝑖𝑗 − 𝑅𝑖𝑗 ) 𝜅2 6𝑘𝜇4 3𝑘2𝜅2𝜇2 32 (1−3𝜆) 2𝜉4 2 𝐻2 = (𝜌 − − ). 6 (3𝜆 − 1) 𝑎2 8 (3𝜆 − 1) 𝑎4 (32) 𝜇𝜉2 ×(𝐶𝑖𝑗 − 𝑅𝑖𝑗 )} , 2 For 𝑘=0, there is no contribution from the higher order 𝑘 =0̸ (26) derivative terms in the action. However, for ,thehigher derivative terms are significant for small volume, that is, for 2 𝑎 𝑎 where 𝜅 , 𝜆,and𝜉 are dimensionless constant parameters small , and become insignificant for large , where it agrees while 𝜇 and Λ 𝑊 are constant parameters with mass dimen- with general relativity. The standard Friedmann equation is 𝑐=1 sions. Here 𝐾𝑖𝑗 is the extrinsic curvature which takes the form recovered, in units where , provided we define [68, 69] 𝜅2 1 𝐺 = , 𝐾 = (𝑔̇−∇𝑁 −∇𝑁 ), cosm (33) 𝑖𝑗 2𝑁 𝑖𝑗 𝑖 𝑗 𝑗 𝑖 (27) 16𝜋 (3𝜆 − 1) 2 4 𝑡 𝜅 𝜇 and a dot denotes a derivative with respect to and covariant =1, (34) derivatives defined with respect to the spatial metric 𝑔𝑖𝑗 .In 3𝜆 − 1 Advances in High Energy Physics 5 where condition (34)alsoagreesfor𝜆=1=𝑐with second The ratio between the fluid and the horizon entropy is given 𝐺 relation in (30). Here cosm is the “cosmological” Newton’s by 𝜆=1 constant. In the IR limit where the Lorentz invariance is 󸀠󸀠 󸀠 𝐺 =𝐺 𝑆𝑓 𝑆𝑓 restored, and hence cosm . Using the above identifica- ≃ ∼𝑎6𝜔+3, 󸀠󸀠 󸀠 (44) tions, as well as definition (31), the Friedmann equation (32) 𝑆ℎ 𝑆ℎ canberewrittenas 2 which implies 𝜔<−1/2, for dominating spacetime entropy. 2 𝑘 8𝜋𝐺 𝑘 𝐻 + = cosm 𝜌+ . (35) By comparing this condition with what we saw from (42), we 𝑎2 3 2𝜉𝑎4 get Onecaneasilyseethatinthelimit𝜉→∞the dark radiation 2 term vanishes and the standard Friedmann equation is −1 ≤ 𝜔 ≤ − , (45) 𝜆=1 𝐺 =𝐺 3 restored for ( cosm ), as expected. For a flat universe (𝑘=0), this equation reduces to as a proper domain. Another condition arises from our 3𝜔 + 3 ≥ 0 8𝜋𝐺 second assumption which implies .Ifweconsider 𝐻2 = cosm 𝜌. 3𝜔 + 3 < 0 3 (36) opposite statement ( )thenforsatisfyingtheGSL by background fluid we find 𝜔<−1, which is consistent with In the deformed HL gravity, the entropy associated with the (44). event horizon of a static spherically symmetric black hole takes the form [72]: 5. DGP Braneworld Model 𝐴 𝜋 𝐴 𝑆 = + , ℎ 4𝐺 𝜉 ln 𝐺 (37) ForspatiallyflatFRWbackground,theFriedmannequation in DGP braneworld can be written as [73, 74] 𝐴=4𝜋𝑟2 𝑘 = where + is the area of the black hole horizon and 𝐵 2 𝑐=ℎ=1set for simplicity. Replacing 𝑟+ with the Hubble 𝜇2𝜌 1 𝜖 −1 𝐻2 =(√ + + ) , radius 𝐻 in the flat FRW Universe, we have for the entropy 2 (46) 3 4𝑟 2𝑟𝑐 of IR modified HL cosmology: 𝑐 −2 −2 2 2 2 𝜋𝐻 𝜋 4𝜋𝐻 where 𝜇 =8𝜋𝐺4, 𝑟𝑐 =𝑀𝑝/2𝑀5 is the crossover scale which 𝑆 = + . (38) ℎ 𝐺 𝜉 ln 𝐺 determines the transition from 4𝐷 to 5𝐷 behavior, and 𝜖=±1 corresponds to the two branches of the DGP braneworld [75]. Taking the first derivative of the entropy with respect to scale Equation (46)with𝜖=1and 𝜌=0has an interesting self- factor, one gets accelerating solution with a Hubble parameter given by the 󸀠 3 (𝜔+1) 𝜋 3 3(𝜔+1) inverse of the crossover scale 𝑟𝑐 [76]. It was shown in [77] 𝑆 = [ + 𝑎 ], (39) ℎ 𝑎 𝜉 8𝐺2 that there are some cosmological constraints that confine this model beside its prediction about cross over scale which is wherewehaveused inconsistent with reality (because its scale is of the order of the 𝑑 ln 𝐻 3 (𝜔+1) =− (40) Solar system). However, it has attracted some investigations. 𝑑 ln 𝑎 2 The reason of these attempts comes from its view about and relation (36). For the second derivative of entropy, we acceleration which have been argued in ample details in the have literatures [78–80]. Using (2)and(46) it is clear that in the 3 (𝜔+1) 𝜋 3 low densities the condition 𝑎≥0̈ is satisfied, independent of 𝑆󸀠󸀠 = [− + (3𝜔 + 2) 𝑎3(𝜔+1)]. 𝜔 𝜔≤−1/3 ℎ 𝑎2 𝜉 8𝐺2 (41) . For high densities, emerges as an upper bound for 𝜔. The entropy associated with the apparent horizon in In the long run limit where 𝑎→∞,wehave3𝜔 + 3 ≥ 0,and this model is given by [81] so we arrive at 𝐴 𝜖𝑟̃ 󸀠 3𝜔+2 𝑆 = (1 − 𝐴 ). 𝑆ℎ ≈ (𝜔+1) 𝑎 , ℎ (47) 4𝐺4 3𝑟𝑐 (42) 󸀠󸀠 3𝜔+1 𝜖=−1 𝑆ℎ ≈ (𝜔+1)(3𝜔 + 2) 𝑎 . It is obvious that, for entropy is positive, whereas for 𝜖=+1, the positivity of entropy implies that 𝑟̃𝐴 <3𝑟𝑐,which −1≤𝜔≤−2/3 These results imply the range ( )forsatisfying is consistent with numerical simulations [77]. By evaluating 𝑆󸀠 ≥0 𝑆󸀠󸀠 ≤0 ℎ and ℎ . For the fluid which is enveloped by the the first derivative of the entropy, we find Hubble Horizon, we get 2𝜋𝜇2 (𝜔+1) 𝜌 9 (𝜔+1)(3𝜔 + 1) 𝑎9𝜔+5𝜌 𝑆󸀠 = 𝑆󸀠 = 0 , ℎ 3 𝑓 2 √ 2 2 √ 2 2 32𝜋𝐺 𝑇0 𝑎𝐺4 𝜇 𝜌/3 +1/(2𝑟𝑐) ( 𝜇 𝜌/3 + 1/(2𝑟𝑐) +𝜖(1/2𝑟𝑐)) (43) 9𝜔+4 󸀠󸀠 9 (𝜔+1)(3𝜔 + 1)(9𝜔 + 5) 𝑎 𝜌0 𝑆 = . ×(1−(𝜖𝑟̃𝐴/2𝑟𝑐)) . 𝑓 32𝜋𝐺2𝑇 0 (48) 6 Advances in High Energy Physics

For 𝜖=−1theaboveexpressionwillbenonnegativeprovided term in four dimensions, and its expansion around flat 󸀠 that 𝜔≥−1.Forpositivevalueof𝜖,thecondition𝑆ℎ ≥0 spacetime is ghost-free. Although the action includes higher yields a similar result for 𝑟̃𝐴 <2𝑟𝑐,andphantomregimefor order derivatives of curvature terms, there are no more than 2𝑟𝑐 < 𝑟̃𝐴 <3𝑟𝑐.Inthelimitof𝑎→∞,onegets second-order derivatives of metric in the equation of motion. The entropy of the static spherically symmetric black hole in 3 (𝜔+1) 𝜖√3 𝑆󸀠 ≈ 𝑎3𝜔+2 (1 − 𝑎−3((𝜔+1)/2)). (𝑛 + 1)-dimensional Gauss-Bonnet theory has the following ℎ 2 (49) 𝜇 2𝑟𝑐𝜇 form [82–86]:

𝜖 𝐴 𝑛−12𝛼̃ For nonphantom regime and independent of the value of , 𝑆ℎ = (1 + ), (55) 󸀠 (3𝜔+2) 󸀠󸀠 3𝜔+1 4𝐺 𝑛−3𝑟2 we get 𝑆ℎ ≈(𝜔+1)𝑎 and 𝑆ℎ ≈ (𝜔+1)(3𝜔+2)𝑎 ,which + −1≤𝜔≤−2/3 yields as a proper domain for satisfying GSL 𝛼̃ = (𝑛−2)(𝑛−3)𝛼 𝐴=𝑛Ω𝑟𝑛−1 by background. For the entropy of fluid, from (11)wefind where and 𝑛 + .Takinginto account the entropy formula (55) for the apparent horizon, using the apparent horizon radius 𝑟̃𝐴 instead of the black 3𝑉𝜌 (𝜔+1) 𝜇2𝜌 󸀠 hole radius 𝑟+, and applying the first law, one obtains the 𝑆𝑓 = ( −1), (50) 𝑎𝑇 2 2 𝐻√𝜇 𝜌/3 + 1/4𝑟𝑐 Friedmann equation in the Gauss-Bonnet cosmology as [9]

2 4 16𝜋𝐺 𝐻 + 𝛼𝐻̃ = 𝜌. (56) whereinthelongrunlimityields 𝑛 (𝑛−1) 󸀠 (9𝜔+1)/2 Taking the derivative with respect to scale factor 𝑎 and using 𝑆𝑓 ≈𝑎 , (51) relation (6), one finds 𝑆󸀠󸀠 ≈𝑎(9𝜔−1)/2. 𝑓 𝑑𝐻 𝑛 (𝜔+1) 𝛽𝜌 =− , 𝑑𝑎 2𝑎[𝐻+2𝛼𝐻̃ 3] (57) The ratio between fluid and horizon entropy is givenby

󸀠󸀠 󸀠 where 𝛽 = 16𝜋𝐺/𝑛(𝑛 −1). After inserting (57)into(2)and 𝑆𝑓 𝑆𝑓 ≃ ≈𝑎(3𝜔−3)/2 󳨐⇒ 𝜔 <1. (52) doing simple calculations, one arrives at 𝑆󸀠󸀠 𝑆󸀠 ℎ ℎ 2𝛼𝐻̃ 4 2−𝑛 𝜔≤ + , 𝑛𝛽𝜌 𝑛 (58) Comparing this condition with one previously obtained for −1 ≤ the satisfaction of the GSL, it becomes clear that, for 𝐻 𝜔≤−2/3 where obviously decreases by expansion [87–89]; there- , the DGP braneworld model approaches thermal fore, for small values of 𝐻 (late time limit), this reduces to equilibrium in the late time, which is consistent with observations at the present time [41, 42]. In the phantom regime 2 𝑛 𝜔≤ (𝛼𝛽𝜌̃ +1− ). (59) where 2𝑟𝑐 < 𝑟̃𝐴 <3𝑟𝑐, 𝜔<−1, the ratio between fluid and 𝑛 2 horizon entropy is given by As a proper upper bound for density, one can consider 2 󸀠󸀠 󸀠 (2𝛼𝛽𝜌/𝑛̃ + (2 − 𝑛)/𝑛) in which we have used 𝐻 ∼𝛽𝜌.Taking 𝑆𝑓 𝑆𝑓 ≃ ≈𝑎3𝜔 󳨐⇒ 𝜔 <0. (53) the first derivative of the entropy of the apparent horizon with 𝑆󸀠󸀠 𝑆󸀠 ℎ ℎ respect to the scale factor, we find 𝑛2 (𝑛−1) Ω (𝜔+1) 𝛽𝜌𝑟̃𝑛 By calculating the asymptotic behavior of the entropy deriva- 𝑆󸀠 = 𝑛 𝐴 , (60) 󸀠 (3𝜔+1)/2 󸀠󸀠 ℎ 8𝐺𝑎𝐻 tivesinthisregime,namely,𝑆ℎ ≃ −(𝜔 + 1)𝑎 , 𝑆ℎ ≃ (3𝜔−1)/2 −(𝜔+1)(3𝜔+1)𝑎 , and considering the results of (53)we wherewehaveused get 𝜔<−1. 𝑑𝑟̃ 𝑑𝑟̃ 𝑑𝐻 𝑛 (𝜔+1) 𝛽𝜌 𝑟̃󸀠 = 𝐴 = 𝐴 = . ℎ 𝑑𝑎 𝑑𝐻 𝑑𝑎 2𝑎𝐻 [𝐻2 +2𝛼𝐻̃ 4] (61) 6. Gauss Bonnet Gravity Employing the second law of thermodynamics, we get 𝑤≥ The action of Gauss-Bonnet gravity in (𝑛+1) dimensions can −𝑛(𝜔+1) −1, which implies a decreasing density, 𝜌=𝜌0𝑎 ,inthe be written as long run limit for negative values of 𝜔. Therefore, we can neg- 1 𝑛+1 lect the effects of higher orders of 𝐻 in the left hand side of 𝐼= ∫ 𝑑 𝑥√−𝑔 (𝑅 +𝛼L )+𝐼𝑚, (54) 2 16𝜋𝐺 GB (56) and justify relation 𝐻 ≃𝛽𝜌.Byevaluatingthesecond derivative we get L =𝑅2 −4𝑅 𝑅𝜇] +𝑅 𝑅𝜇]𝜆𝜌 where GB 𝜇] 𝜇]𝜆𝜌 is the lagrangian 𝑛2 (𝑛−1) Ω (𝜔+1) 𝛽𝜌 of the Gauss-Bonnet correction term, 𝛼 is the Gauss-Bonnet 𝑆󸀠󸀠 = 𝑛 ℎ 8𝐺𝑎2𝐻𝑛+1 coefficient which is a positive constant, and 𝐼𝑚 denotes (62) 󸀠 the matter action. The Gauss-Bonnet term is a topological ×[−(1+𝑛𝜔+𝑛) +𝑎𝐻𝑟̃𝐴 (𝑛+1)]. Advances in High Energy Physics 7

󸀠 In the long run limit 𝑟̃𝐴 ≃ 𝑛(𝜔 + 1)/2𝑎𝐻 and we have Acknowledgments 𝜌 𝑆󸀠󸀠 ≈ (𝜔+1) [(𝜔+1) (𝑛2 −𝑛)−2] . A. Sheykhi thanks the Research Council of Shiraz University. ℎ 𝑎2𝐻𝑛+1 (63) The work of A. Sheykhi has been supported financially Using the GSL of thermodynamics we find −1≤𝜔≤−1+ by Research Institute for Astronomy and Astrophysics of 2 2/(𝑛 −𝑛), which shows that the GSL can be satisfied by the Maragha (RIAAM), Iran. 2 background fluid. Note that 𝜔=−1and 𝜔=−1+2/(𝑛 −𝑛) 𝑆󸀠󸀠 =0 𝑆󸀠󸀠 ≤0 are roots of equation ℎ , and therefore condition ℎ is References satisfied. Also, by using the result of applying the second law to (60) which includes 𝜔≥−1,itisobviousthatGSLissati- [1] T. Jacobson, “Thermodynamics of spacetime: the Einstein equ- sfied. For 𝑛=3,theresultisthesameasEinstein’sgravity[39]. ationofstate,”Physical Review Letters,vol.75,no.7,pp.1260– By increasing 𝑛, 𝜔 approaches −1 as a limiting value. Using the 1263, 1995. Gibbs’ law for the fluid, we obtain [2] C. Eling, R. Guedens, and T. Jacobson, “Nonequilibrium thermodynamics of spacetime,” Physical Review Letters,vol.96,no. 𝑛 (𝜔+1) 𝑉 𝜌 𝑛 (𝜔+1) 𝛽𝜌 𝑆󸀠 = 𝑛 0 [ −1]. 12, Article ID 121301, 4 pages, 2006. 𝑓 𝑛+1 2 4 (64) 𝑎 𝑇0 2(𝐻 +2𝛼𝐻̃ ) [3] T. Padmanabhan, “Classical and quantum thermodynamics of horizons in spherically symmetric spacetimes,” Classical and 𝑆󸀠 ∼ ((𝑛(𝜔 + 1) − 2)/ Thus, in the long run limit we have 𝑓 Quantum Gravity,vol.19,no.21,pp.5387–5408,2002. (𝑛2(𝜔+1)−2𝑛−2)/2 󸀠󸀠 (𝑛2(𝜔+1)−2𝑛−4)/2 2)𝑎 and 𝑆𝑓 ∼𝑎 . The relations bet- [4] T. Padmanabhan, “Gravity and the thermodynamics of hori- ween various components of entropy are as follows: zons,” Physics Reports,vol.406,no.2,pp.49–125,2005. 󸀠󸀠 󸀠 [5] T. Padmanabhan, “Gravity: a new holographic perspective,” 𝑆𝑓 𝑆𝑓 International Journal of Modern Physics D,vol.15,no.10,p.1659, ≃ ∼𝑎−(𝑛/2)(𝜔+3) 󳨐⇒ 𝑤 > −3. 󸀠󸀠 󸀠 (65) 2006. 𝑆ℎ 𝑆ℎ [6] M. Akbar and R. G. 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Research Article Primordial Dark Energy from a Condensate of Spinors in a 5D Vacuum

Pablo Alejandro Sánchez1,2 and Mauricio Bellini1,2

1 Departamento de F´ısica, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, Funes 3350, 7600 Mar del Plata, Argentina 2 Instituto de Investigaciones F´ısicas de Mar del Plata (IFIMAR), Consejo Nacional de Investigaciones Cient´ıficas y Tecnicas´ (CONICET), 7600 Mar del Plata, Argentina

Correspondence should be addressed to Mauricio Bellini; [email protected]

Received 28 October 2013; Accepted 9 December 2013

Academic Editor: Kishor Adhav

Copyright © 2013 P.A. Sanchez´ and M. Bellini. 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 explore the possibility that the expansion of the universe can be driven by a condensate of spinors which are free of interactions in a 5D relativistic vacuum defined in an extended de Sitter spacetime which is Riemann flat. The extra coordinate is considered as noncompact. After making a static foliation on the extra coordinate, we obtain an effective 4D (inflationary) de Sitter expansion which describes an inflationary universe. We found that the condensate of spinors studied here could be an interesting candidate to explain the presence of dark energy in the early universe. The dark energy density which we are talking about is poured into smaller subhorizon scales with the evolution of the inflationary expansion.

1. Introduction Sitter spacetime (where the 3D Euclidean space is in Carte- sian coordinates), small letters 𝑎 and 𝑏 run from 0 to 5 Modern versions of 5D general relativity abandon the cylin- in a 5D Minkowsky spacetime (in cartasian coordinates), der and compactification conditions used in original Kalu- Greek letters 𝛼 and 𝛽 run from 0 to 3, and latin letters 𝑖 za-Klein (KK) theories, which caused problems with the and 𝑗 run from 1 to 3.). Due to this fact the stress-energy cosmological constant and the masses of particles, and may be a 4D manifestation of the embedding geometry and, consider a large extra dimension. The main question that therefore, by making a static foliation on the space-like extra these approaches address is whether the four-dimensional coordinate of an extended 5D de Sitter spacetime, it is possible properties of matter can be viewed as being purely geo- to obtain an effective 4D universe that suffered an exponential metrical in origin. In particular, the induced matter theory accelerated expansion driven by an effective scalar field with (IMT) [1, 2] is based on the assumption that ordinary matter an equation of state typically dominated by vacuum [4– and physical fields that we can observe in our 4D universe 7]. An interesting problem in modern cosmology relies on can be geometrically induced from a 5D Ricci-flat metric explaining the physical origin of the cosmological constant, with a space-like noncompact extra dimension in which we which is responsible for the exponential expansion of the define a physical vacuum. The Campbell-Magaard theorem early inflationary universe. The standard explanation for the (CMT) [3] serves as a ladder to go between manifolds whose early universe expansion is that it is driven by the inflaton dimensionality differs by one. Due to this theorem one can field [8]. Many cosmologists mean that such acceleration say that every solution of the 4D Einstein equations with (as well as the present day accelerated expansion of the arbitrary energy-momentum tensor can be embedded, at universe) could be driven by some exotic energy called dark least locally, in a solution of the 5D Einstein field equations energy. Most versions of inflationary cosmology require one in a relativistic vacuum: 𝐺𝐴𝐵 =0. (We shall consider that scalar inflaton field which drives the accelerated expansion capital letters 𝐴 and 𝐵 run from 0 to 4 in a 5D extended de of the early universe with an equation of state governed by 2 Advances in High Energy Physics the vacuum [9]. The parameters of this scalar field must and we obtain that the equation for the spinor Ψ takes the be rather finely tuned in order to allow adequate inflation form and an acceptable magnitude for density perturbations. The 𝑔𝐴𝐵∇ ∇ Ψ=0. need for this field is one of the less satisfactory features of 𝐴 𝐵 (9) inflationary models. Consequently, we believe that it is of The same procedure yields an identical equation for the field interest to explore variations of inflation in which the role Ψ. On the other hand, the 5D Einstein equation for the of the scalar field is played by some other field10 [ , 11]. Riemann-flat metric1 ( )is Recently the possibility that such expansion can be explained by a condensate of dark spinors has been explored [12]. This ⟨0| 𝑇𝐴𝐵 |0⟩ =0, (10) interesting idea was recently revived in the framework of the induced matter theory (IMT) [13]. In this work we shall where ⟨0|𝑇𝐴𝐵|0⟩ denotes the expectation value of 𝑇𝐴𝐵 in the extend this idea. vacuum state |0⟩.

2. The Effective Lagrangian in 2.1. Tensorial Formulation of the Equation of Motion. Using a 5D Riemann-Flat Spacetime the formalism previously introduced, the double Nabla can be expressed explicitly: Weareconcernedwitha5DRiemann-flatspacetimewitha ∇ ∇ Ψ=𝜕 ∇ Ψ+Γ ∇ Ψ−𝜔 𝐶∇ Ψ line element given by 𝐴 𝐵 𝐴 𝐵 𝐴 𝐵 𝐴𝐵 𝐶 2 1 1 2 𝜓 2 2𝑡/𝜓 2 2 2 2 =𝜕 𝜕 Ψ− 𝜕 𝜔𝑎𝑏𝛾 𝛾 Ψ− 𝜔𝑎𝑏𝛾 𝛾 𝜕 Ψ 𝑑𝑆 = ( ) [𝑑𝑡 −𝑒 0 (𝑑𝑥 +𝑑𝑦 +𝑑𝑧 )] −𝑑𝜓 , (1) 𝐴 𝐵 4 𝐴 𝐵 𝑎 𝑏 4 𝐵 𝑎 𝑏 𝐴 𝜓0 1 where 𝑡, 𝑥, 𝑦 and 𝑧 aretheusuallocalspacetimecoordinate − 𝜔𝑎𝑏𝛾 𝛾 𝜕 Ψ 4 𝐴 𝑎 𝑏 𝐵 (11) systems and 𝜓 is the noncompact space-like extra dimension. 1 We start from an effective Lagrangian density for non + 𝜔𝑎𝑏𝜔𝑐𝑑𝛾 𝛾 𝛾 𝛾 Ψ−𝜔𝐶 𝜕 Ψ massive fermions in 5D: 16 𝐴 𝐵 𝑎 𝑏 𝑐 𝑑 𝐴𝐵 𝐶 1 𝐴 L =− (∇ Ψ) (∇ Ψ) . 1 𝐶 𝑎𝑏 eff 𝐴 (2) + 𝜔 𝜔 𝛾 𝛾 Ψ, 2 4 𝐴𝐵 𝐶 𝑎 𝑏

Atthispointitiseasytoobtaintheequationsofmotionfroma 𝑎𝑏 𝑎 𝑏 𝐴𝑁 where the spin connection is 𝜔𝑀 =−𝑒𝑁[𝜕𝑀𝑒𝐴𝑔 + variational principle. The Euler-Lagrange equations for both 𝑏 𝐴𝐵 𝑁 𝑎𝑏 Ψ Ψ 𝑒𝐵𝑔 Γ𝐴𝑀] and Γ𝑀 = −(1/8)𝜔𝑀[𝛾𝑎,𝛾𝑏]. and can be obtained making the following functional 𝐴 derivatives: (Thetensorscanbewrittenusingthevielbein𝑒𝑎 and its 𝑎 𝐴 𝑏 𝑏 𝛿L inverse 𝑒𝐴, such that, if 𝑒𝑎 𝑒𝐴 =𝛿𝑎,then eff =0, (3) 𝛿Ψ 𝐴 𝐵 𝜂𝑎𝑏 =𝑒𝑎 𝑒𝑏 𝑔𝐴𝐵, (12) 𝛿L 1 eff 𝐴 ∇𝐴 = ∇𝐴∇ Ψ, where 𝜂𝑎𝑏 is the 5D Minkowsky tensor metric with signature 𝛿(∇ Ψ) 2 (4) 𝐴 (+,−,−,−,−).) 𝛿L 1 Thus, after replacing the last expression in the equation of eff 𝑀 𝑁 𝑀𝑁𝑃 = ∇ Ψ∇ Ψ+∇𝑃𝐽 , (5) motion (9)weobtain 𝛿𝑔𝑀𝑁 2 𝐴𝐵 1 𝐴𝐵 𝑎𝑏 𝑀𝑁𝑃 𝑔 𝜕 𝜕 Ψ− 𝑔 𝜔 𝜎 𝜕 Ψ where the effective current 𝐽 is symmetric with respect to 𝐴 𝐵 2 (𝐴 𝑎𝑏 𝐵) permutations of 𝑀 and 𝑁, 𝐴𝐵 𝐶 1 𝐴𝐵 𝑎𝑏 1 −𝑔 𝜔𝐴𝐵 𝜕𝐶Ψ+ 𝑔 𝜕𝐴𝜔𝐵 𝜎𝑎𝑏Ψ 𝐽𝑀𝑁𝑃 = (∇𝑀Ψ𝑓𝑁𝑃 Ψ+Ψ∇𝑀Ψ) , 4 8 (6) 1 𝐴𝐵 𝑎𝑏 𝑐𝑑 1 𝐴𝐵 𝐶 𝑎𝑏 𝑁𝑃 𝑁 𝑃 + 𝑔 𝜔 𝜔 𝜎 𝜎 Ψ− 𝑔 𝜔 𝜔 𝜎 Ψ=0. and 𝑓 =[𝛾 ,𝛾 ] is antisymmetric [14]. At this point we 16 𝐴 𝐵 𝑎𝑏 𝑐𝑑 4 𝐴𝐵 𝐶 𝑎𝑏 𝑀𝑁 are in conditions of introducing the stress tensor 𝑇 = (13) 2(𝛿L /𝛿𝑔 )−𝑔𝑀𝑁 L eff 𝑀𝑁 eff: Here,wehavemadeuseofthefactthat𝛾𝑎𝛾𝑏 = (1/2){𝛾𝑎,𝛾𝑏}+ 1 𝑎𝑏 𝑏𝑎 𝑎𝑏 𝑎𝑏 𝑀𝑁 𝑀 𝑁 𝑀𝑁𝑃 𝑀𝑁 𝐴 𝐵 (1/2)[𝛾𝑎,𝛾𝑏]=𝑔𝑎𝑏I +𝜎𝑎𝑏, 𝜔 =−𝜔 and 𝜔 𝛾𝑎𝛾𝑏 =𝜔 𝜎𝑎𝑏. 𝑇 =∇ Ψ∇ Ψ+2∇𝑃𝐽 + 𝑔 (𝑔𝐴𝐵 ∇ Ψ∇ Ψ) . 𝑀 𝑀 𝑀 𝑀 2 Once we simplify some terms, we obtain (7) 1 1 𝑔𝐴𝐵𝜔𝑎𝑏𝜎 𝜕 Ψ= 𝑔𝐴𝐵𝜔𝑎𝑏𝜎 𝜕 Ψ, Applying the compatibility condition on the metric (𝐴 𝑎𝑏 𝐵) 𝐴 𝑎𝑏 𝐵 𝐴𝐵 2 2 ∇𝐶𝑔 =0,weobtain 𝜔 𝐶 =𝜔𝐷𝐶𝑔 =𝜔𝑑𝑐𝑔 𝑒𝐷𝑒𝐶, (14) 𝐴 𝐴𝐵 𝐴𝐵 𝐴𝐵 𝐴𝐵 𝐴 𝐷𝐵 𝐴 𝐷𝐵 𝑑 𝑐 ∇𝐴∇ Ψ=∇𝐴 (𝑔 ∇𝐵Ψ) = (∇𝐴𝑔 )∇𝐵Ψ+𝑔 ∇𝐴∇𝐵Ψ=0, 𝐴𝐵 𝐶 𝐴𝐵 𝑑𝑐 𝐷 𝐶 𝑑𝑐 𝐴 𝐷 𝐶 𝑑𝑐 𝐴 𝐶 (8) 𝑔 𝜔𝐴𝐵 =𝑔 𝜔𝐴 𝑔𝐷𝐵𝑒𝑑 𝑒𝑐 =𝜔𝐴 𝛿𝐷𝑒𝑑 𝑒𝑐 =𝜔𝐴 𝑒𝑑 𝑒𝑐 . Advances in High Energy Physics 3

Finally, the equation of motion for the spinors assumes its the first equation, while the other coupling becomes a source final form for the second equation:

4𝜓0 𝜕Φ+ 4 𝜕Φ+ 5 𝐴𝐵 1 𝐴𝐵 𝑎𝑏 ̂Φ + − − Φ =0, 𝑔 𝜕 𝜕 Ψ− 𝑔 𝜔 𝜎 𝜕 Ψ Ø + 2 2 + (20) 𝐴 𝐵 2 𝐴 𝑎𝑏 𝐵 𝜓 𝜕𝑡 𝜓 𝜕𝜓 4𝜓 2𝜓 𝜕Φ 𝜕Φ 𝑎𝑏 𝐴 𝐶 1 𝐴𝐵 𝑎𝑏 ̂ 0 − 4 − 7 −𝜔 𝑒 𝑒 𝜕 Ψ+ 𝑔 𝜕 𝜔 𝜎 Ψ ØΦ− + − + Φ− 𝐴 𝑎 𝑏 𝐶 4 𝐴 𝐵 𝑎𝑏 𝜓2 𝜕𝑡 𝜓 𝜕𝜓 4𝜓2 (15) 1 (21) 𝐴𝐵 𝑎𝑏 𝑐𝑑 𝑖2𝜓0 −𝑡/𝜓 + 𝑔 𝜔𝐴 𝜔𝐵 𝜎𝑎𝑏𝜎𝑐𝑑Ψ = 𝑒 0 𝜎⋅⃗ ∇Φ⃗ . 16 𝜓2 +

1 𝑎𝑏 𝐴 𝐶 𝑐𝑑 − 𝜔𝐴 𝑒𝑎 𝑒𝑏 𝜔𝐶 𝜎𝑐𝑑Ψ=0, Then, after few calculations, the Lagrangian density written 4 intermsofthenewfieldstakestheform 1 L =− (∇ 𝜑 ∇𝐴𝜑 +∇ 𝜑 ∇𝐴𝜑 ) which is very difficult to be resolved because the fields are eff 2 𝐴 1 1 𝐴 2 2 (22) coupled. or, alternatively, can be written as a function of the pair 𝜑1 = (1/2)(Φ+ +Φ−), 𝜑2 = (1/2𝑖)(Φ+ −Φ−): 2.2. Conformal Mapping Based Solution. In order to simplify the structure of (15), we shall introduce the following trans- 1 𝐴 𝐴 L =− (∇ Φ ∇ Φ +∇ Φ ∇ Φ ). (23) formation on the spinor components: eff 4 𝐴 + + 𝐴 − − On the other hand the 5D energy-momentum (EM) tensor 𝜑1 (5) 𝐴𝐵 Ψ=( ), is represented by 𝑇𝐴𝐵 =2(𝛿L /𝛿𝑔 )−𝑔𝐴𝐵L . 𝜑 (16) eff eff 2 This procedure takes place in a 5D vacuum. Therefore, the effective Lagrangian and the EM tensor are involved directly where components are grouped as with the cosmological observables we wish to evaluate. The observables to which we refer are energy density and pressure. 𝜓1 𝜓3 Both come from the diagonal part of the EM tensor. 𝜑1 =( ), 𝜑2 =( ). (17) 𝜓2 𝜓4 2.3. Extra Dimensional Solution for Φ+. We shall use the With this representation we obtain the equation of motion for variable separation method in the homogeneous PDE (20); we obtain the following set of ODEs: 𝜑1 and 𝜑2: 2 2 ∇ 𝑅+𝜅 𝑅=0, (24) 3𝜓 𝜕𝜑 4 𝜕𝜑 1 𝑖𝜓 −𝑡/𝜓 ̂𝜑 + 0 1 − 1 + 𝜑 − 0 𝑒 0 𝜎⋅⃗ ∇𝜑⃗ Ø 1 2 2 1 2 1 2 (+) (+) 𝜓 𝜕𝑡 𝜓 𝜕𝜓 4𝜓 𝜓 𝜕 𝑇 2 𝜕𝑇 2 −2𝑡/𝜓 2 (+) 𝜅 + 𝜅 +(𝜅 𝑒 0 −𝑀 )𝑇 =0, 2 1 𝜅 (25) 𝜕𝑡 𝜓0 𝜕𝑡 𝑖𝜓 𝜕𝜑 3𝑖 𝜓 −𝑡/𝜓 =− 0 2 + 𝜑 − 0 𝑒 0 𝜎⋅⃗ ∇𝜑⃗ , 𝜓2 𝜕𝑡 2𝜓2 2 𝜓2 2 𝜕2Λ 𝜕Λ 5 𝜓2 +4𝜓 +( −𝑀2𝜓2)Λ=0. (18) 2 1 0 (26) 3𝜓 𝜕𝜑 4 𝜕𝜑 1 𝑖𝜓 𝜕𝜓 𝜕𝜓 4 ̂ 0 2 2 0 −𝑡/𝜓0 ⃗ Ø𝜑2 + − + 𝜑2 + 𝑒 𝜎⋅⃗ ∇𝜑2 𝜓2 𝜕𝑡 𝜓 𝜕𝜓 4𝜓2 𝜓2 Equation (24)hasasolutionthatcanbewrittenintermsof plane wavefront 𝑖𝜓0 𝜕𝜑1 3𝑖 𝜓0 −𝑡/𝜓 = − 𝜑 − 𝑒 0 𝜎⋅⃗ ∇𝜑⃗ . ±𝑖𝜅⋅⃗𝑟⃗ 𝜓2 𝜕𝑡 2𝜓2 1 𝜓2 1 𝑅 (𝑟)⃗∼𝑒 . (27) The second equation (25) has a general solution: Here,wehaveadoptedthefollowingconventions: −((3/2)+√1+𝑀2𝜓2) (+) 𝜓 1 0 2 2 2 2 Λ (𝜓) =1 𝐶 ( ) 𝜓 𝜕 𝜑 𝜓 −2𝑡/𝜓 2 𝜕 𝜑 𝜓 ̂𝜑=( 0 ) −( 0 ) 𝑒 0 ∇ 𝜑− , 0 Ø 𝜓 𝜕𝑡2 𝜓 𝜕𝜓2 (28) −((3/2)−√1+𝑀2𝜓2) 𝜓 1 0 +𝐶 ( ) . 𝜎=𝜎⃗ ̂𝚤+𝜎̂𝚥+𝜎̂𝑘, (19) 2 1 2 3 𝜓0 ⃗ 𝜕𝜑 𝜕𝜑 𝜕𝜑 Since we are interested in “localized” static solutions, that is, 𝜎⋅⃗ ∇𝜑 =1 𝜎 +𝜎2 +𝜎3 . 𝜕𝑥 𝜕𝑦 𝜕𝑧 those that decay to zero when 𝜓 tends to infinity, we must 2 2 choose 𝐶2 =0,sothat𝑛 ≡ ((3/2) + √1+𝑀1 𝜓0 )>0.This Now we can use the conformal mapping defining new com- 2 2 2 choice makes 𝑀1 = (((𝑛 − (3/2)) − 1)/𝜓0 )≥0,with𝑛≥3 plex fields Φ+ =𝜑1 +𝑖𝜑2 and Φ− =𝜑1 −𝑖𝜑2.Rewriting √ 2 2 (18) in terms of these new fields, it is possible to decouple and 𝑛∈R, in order to get 1+𝑀1 𝜓0 ≥0. 4 Advances in High Energy Physics

2.4. Extra Dimensional Solution for Φ−. Now we are able to 4D de Sitter expansion, we shall consider a static foliation calculate the coupling term of the inhomogeneous equation on the 5D metric (1). The resulting 4D hypersurface after (21) for each mode (see (37)): making the static foliation 𝜓=𝜓0 =1/𝐻0 describes an effective 3D spatially flat, isotropic, and homogeneous de 2𝑖𝜓 −𝑡/𝜓 0 𝑒 0 𝜎⋅⃗ ∇Φ⃗ Sitter four-dimensional expanding universe with a constant 2 +,𝜅 𝜓 Hubble parameter 𝐻0, with a line element 2]Γ (]) =− 2 2 2 2𝐻0𝑡 2 ]−1 (1/2)+] (29) 𝑑𝑆 󳨀→ 𝑑 𝑠 =𝑑𝑡 −𝑒 𝑑𝑟 . (33) √𝜋𝜅 𝜓0

−((7/2)+√]2−3) 𝑖𝜅⋅⃗𝑥⃗(]−3)(𝑡/𝜓 ) 𝜓 From the relativistic point of view, an observer moves in ×𝑒 𝑒 0 ( ) . 𝜓 𝜓 a comoving frame with the five-velocity 𝑈 =0on a 0 (4) 2 4D hypersurface with a scalar curvature 𝑅 = 12/𝜓0 = 2 Using the last expression in (21), we obtain a degenerate 12𝐻0 , such that the Hubble parameter 𝐻0 and thus also the Φ 2 two-component system; for the spinor −,𝜅.(Henceforth cosmological constant Λ 0 =3𝐻0 /(8𝜋𝐺) are defined by the we are concerned with asymptotic solutions; that is, only −1 foliation 𝐻0 =𝜓0 . the infrared limit has cosmological significance.) Again, a plane wavefront satisfies the spatial part. By inserting Φ−,𝜅 = 𝑖𝜅⋅⃗𝑟⃗ 2 Φ 𝐺𝜅(𝑡, 𝜓)𝑒 and multiplying by (𝜓/𝜓0) ,weobtain 3.1. Time-Dependent Modes of +. The solution for the time- dependent equation (26) can be expanded in terms of first- 2 (−) (−) 𝜕 𝐺 2 𝜕𝐺 2 −2𝑡/𝜓 7 and second-kind Hankel functions: 𝜅 + 𝜅 +(𝜅 𝑒 0 + )𝐺 2 2 𝜅 𝜕𝑡 𝜓0 𝜕𝑡 4𝜓 0 (+) 𝑇𝜅 (𝑡) 𝜓 2 𝜕2𝐺(−) 4𝜓 𝜕𝐺(−) −[( ) 𝜅 + 𝜅 ] 𝜅 𝜅 2 2 (30) −2𝐻0𝑡 (1) −𝐻0𝑡 (2) −𝐻0𝑡 𝜓0 𝜕𝜓 𝜓0 𝜕𝜓 =𝑒 [𝐶3H] ( 𝑒 )+𝐶4H] ( 𝑒 )] , 𝐻0 𝐻0 ] 1−] −[(3/2)+√]2−3] (34) 2 Γ (]) 𝜅 ((]−3)/𝜓 )𝑡 𝜓 ≃− 𝑒 0 ( ) . (1/2)+] 𝜓 √𝜋𝜓0 0 2 2 where ] = √4+𝑀1 𝜓0 ≥2. After making a Bunch-Davies This inhomogeneous PDE can be converted to one with a normalization of the modes [15]weobtainthesolution constant coupling. In order to make the right side of (30) constant, we shall propose 𝑖 𝜋 𝜅 (+) −2𝐻0𝑡 (2) −𝐻0𝑡 2 𝑇 (𝑡) = √ 𝑒 H ( 𝑒 ). 𝜓 −[(3/2)+√] −3] 𝜅 2 𝐻 ] 𝐻 (35) (−) −2 ((]−3)/𝜓0)𝑡 (−) 0 0 𝐺𝜅 (𝑡, 𝜓) =𝜓0 𝑒 ( ) 𝐾𝜅 (𝑡, 𝜓) . 𝜓0 (31) Since we are interested to describe the universe on super- −𝑡/𝜓 𝜅𝜓 𝑒 0 ≪1 Finally, we must solve the follwing equation: Hubble cosmological scales, we must acquire 0 ; werejectsolutionsthatgotozeroatlatetimes.Theasymptotic 2 (−) (−) 2 (−) (+) 𝜕 𝐾 2 (] −2) 𝜕𝐾 𝜓 𝜕𝐾 behavior of 𝑇𝜅 (𝑡) on cosmological scales will be 𝜅 + 𝜅 −( ) 𝜅 2 2 𝜕𝑡 𝜓0 𝜕𝑡 𝜓0 𝜕𝜓 −] (+) 𝑖 𝜋 −2𝐻 𝑡 𝜅 −𝐻 𝑡 2 1 𝜕𝐾(−) 𝑇 (𝑡) ≃ √ Γ (]) 𝑒 0 ( 𝑒 0 ) . − [ − √]2 −3]𝜓 𝜅 𝜅 2 𝐻 2𝐻 (36) 2 0 0 𝜓0 2 𝜕𝜓 (32) 2 −2𝑡/𝜓 10 − 4] (−) +[𝜅 𝑒 0 + ]𝐾 Finally, the degenerate two-component spinor Φ+ can be 2 𝜅 𝜓0 expanded as a function of the modes 2]Γ (]) ≃− . ]−1 −(3/2)+] 1 √𝜋𝜅 𝜓0 Φ+,𝜅 (𝑡, 𝑟,⃗ 0 𝜓 = ) 𝐻0 (37) ]−1 3. Effective Dynamics on the 4D 2 Γ (]) ]−(1/2) −] 𝑖𝜅⋅⃗𝑟⃗(]−2)𝐻 𝑡 ≃𝑖𝐶 𝐻 𝜅 𝑒 𝑒 0 Hypersurface 𝜓=1/𝐻0 1 √𝜋 0 In order to describe the effective 4D dynamics of the physical system in the early inflationary universe with an effective and their conjugated complex. Advances in High Energy Physics 5

𝜕Φ 1 𝑛 1 3.2. The Time-Dependent Modes for Φ−. The homogeneous + ∇4Φ+ = ( )=−( )Φ+ ( ), 𝐾(−)(𝑡, 𝜓)| 𝜕𝜓 1 𝜓 1 solution 𝜅 hom,of(32), is 󵄨 𝜕Φ 1 𝑚 1 (−) 󵄨 ∇ Φ = − ( )=−( )Φ ( ), 𝐾𝜅 (𝑡, 𝜓)󵄨 4 − − 󵄨hom 𝜕𝜓 1 𝜓 1 −(]−2)𝑡/𝜓 (1) −(𝑡/𝜓 ) 0 0 (41) =𝑒 [𝐶3H𝜇 (𝜅0 𝜓 𝑒 )

(2) −(𝑡/𝜓0) + 𝐶4 H𝜇 (𝜅0 𝜓 𝑒 )] theeffective4Dpotentialresultingis

(√]2−3−√]2−4]+7+𝑀2𝜓2) (38) 󵄨 𝜓 2 0 1 4 4 󵄨 [ 𝑉(Φ+,Φ−)=−( )[∇4Φ+∇ Φ+ +∇4Φ−∇ Φ−]󵄨 × 𝐶1( ) 4 󵄨 𝜓 𝜓=1/𝐻0 [ 0 2 𝐻0 2󵄩 󵄩2 2󵄩 󵄩2 󵄨 √ 2 2 2 2 = (𝑛 󵄩Φ 󵄩 +𝑚 󵄩Φ 󵄩 )󵄨 , ( ] −3+√] −4]+7+𝑀2 𝜓0 ) 󵄩 +󵄩 󵄩 −󵄩 󵄨 𝜓 4 󵄨𝜓=1/𝐻0 + 𝐶 ( ) ] , 2 𝜓 (42) 0 ] which is induced by the static foliation on the fifth coordinate 𝜇=√]2 −4] +4+𝑀2𝜓2 ≥0 where 2 0 and the squared mass 𝜓=𝜓0 =1/𝐻0. This effective 4D potential is responsible 2 2 2 Φ (𝑥𝜇,𝜓 ) of Φ(−) is [𝑀2(𝑚, 𝑛)] = ((𝑚 − 3/2) − (𝑛 − 3/2) +(4 to provide us with the dynamics of the fields ± 0 on theeffective4Dhypersurfaceonwhichtheequationofstate √(𝑛 − 3/2)2 + 3−10))/𝜓2 𝑚≥ 0 , which is definitely positive for is 𝜔=𝑃/𝜌=−1. The energy density and pressure related to 1 𝑛≥3 𝜓 (with ). In order get to the -dependent solution of thesefieldsareobtainedfromthediagonalpartoftheenergy- 𝐾(−)(𝑡, 𝜓)| →0 𝐶 =0 lim𝜓→∞ 𝜅 hom , we shall require that 2 and momentum tensor written in a mixed manner: 2 𝑚>3/2+√3+(√(𝑛 − 3/2)2 +3−2) 𝑛≥3 󵄨 1 ,for ,suchthat 󵄨 󵄩 󵄩2 󵄩 󵄩2 𝜌= ⟨𝐸󵄨 [󵄩∇0Φ+󵄩 + 󵄩∇0Φ−󵄩 ] (𝑚, 𝑛) ∈ Z. After taking the asymptotic limit on cosmological 󵄨 4 ⃗ scales we obtain that the modes Φ−,𝜅(𝑡, 𝑟,0 𝜓 =1/𝐻0),for −2𝐻 𝑡 𝑒 0 𝜇=1,are − [∇Φ⃗ ⋅ ∇⃗Φ + ∇Φ⃗ ⋅ ∇⃗Φ ] 4 − − + + 𝐻1/2 󵄨 󵄨 Φ (𝑡, 𝑟,⃗ 𝜓 )≃𝐴 0 𝑒𝑖𝜅⋅⃗𝑟⃗, (39) +𝑉(Φ ,Φ )+𝐹0 󵄨𝐸⟩󵄨 , −,𝜅 0 2 √ + − 0 󵄨 󵄨 𝜋𝜅 󵄨 󵄨𝜓=1/𝐻0 󵄨 (43) 󵄨 1 󵄩 󵄩2 󵄩 󵄩2 where 𝐴2 = 𝐶4[𝐶]1.Noticethatwehaveneglectedthe 𝑃= ⟨𝐸󵄨 [󵄩∇ Φ 󵄩 + 󵄩∇ Φ 󵄩 ] 󵄨 4 󵄩 0 +󵄩 󵄩 0 −󵄩 inhomogenoues part of its solution because it is negligible on −2𝐻 𝑡 these large super-Hubble scales at the end of inflation. 𝑒 0 (−) ⃗ ⃗ ⃗ ⃗ − [∇Φ− ⋅ ∇Φ− + ∇Φ+ ⋅ ∇Φ+] As can be demonstrated the solution 𝐾𝜅 (𝑡,0 𝜓 )≃ 12 𝐾(−)(𝑡, 𝜓 )| 𝐻 = 𝜅 0 hom oncosmologicalscales,onceweconsider 0 󵄨 󵄨 −9 𝑖 𝑖 󵄨 󵄨 1/𝜓0 =1×10𝑀𝑝. Hence, the homogeneous solution −𝑉(Φ+,Φ−)+𝐹𝑗𝛿𝑗 󵄨𝐸⟩󵄨 , 󵄨 󵄨𝜓=1/𝐻 𝐾(−)(𝑡, 𝜓 )| 0 𝜅 0 hom is a very acceptable solution at the end of inflation for the time-dependent modes of Φ−.Inother 0 7 2 where |𝐸⟩ is some quantum state, 𝐹0 =𝐶3/𝜋[(𝐻0 /8𝜅 )+ words, at the end of inflation the effective 4D bosons Φ± can (𝐻9/𝜅4)] 𝐹𝑖 =𝐴 /𝜋[(15𝐻7/32𝜅2)+(𝐻9/2𝜅4)]𝛿𝑖 be decoupled on cosmological scales. 0 , 𝑗 3 0 0 𝑗,and

4𝐷 1 1 3.3. Einstein Equations. The effective 4D Lagrangian ∇ Φ =[𝜕 ∓ ]Φ ( ), 𝜇 0 ± 0 ± 1 density (23) is expressed in terms of the fields Φ±(𝑥 ,𝜓0), 4𝜓0 which can be thought of as two minimally coupled bosons; 1 ∇ Φ =𝜕Φ ( ), 1 𝑗 + 𝑗 + 1 L =− [∇ Φ ∇𝜇Φ +∇ Φ ∇𝜇Φ ] +𝑉(Φ ,Φ ) . eff 𝜇 + + 𝜇 − − + − (40) 4 𝐻 𝑡 𝐻 𝑒 0 1 ∇ Φ =[𝜕Φ +𝑖 0 Φ ]( ), Since 1 − 1 − 2 − 1

𝐻 𝑡 𝜕Φ+ 𝑛 1 𝐻 𝑒 0 −𝑖 ∇ Φ = (11)=−( ) Φ (11), ∇ Φ =𝜕Φ ( )+𝑖 0 Φ ( ), 4 + 𝜕𝜓 𝜓 + 2 − 2 − 1 2 − 𝑖

𝐻 𝑡 𝜕Φ 𝑚 1 𝐻 𝑒 0 1 ∇ Φ = − (11)=−( ) Φ (11), ∇ Φ =𝜕Φ∗ ( )+𝑖 0 Φ∗ ( ), 4 − 𝜕𝜓 𝜓 − 3 − 3 − 1 2 − −1 6 Advances in High Energy Physics

𝐻0𝑡 𝐻0𝑒 𝑛=5/2and 𝑚=7/2. In order to calculate the coefficients ∇ Φ =[𝜕Φ −𝑖 Φ ] (11) , ±𝐻 𝑡 1 − 1 − 2 − corresponding to the factors 𝑒 0 andifwerequirethat𝜌= −𝑃 = 3𝐻2/(8𝜋𝐺) 𝐻 𝑡 0 ,weobtainthat 𝐻 𝑒 0 ∇ Φ =𝜕Φ∗ (11) −𝑖 0 Φ∗ (𝑖−𝑖) , 2 − 2 − 2 − 6𝑘 𝜋2 (−519 + 16𝜖2) 𝐶2 = ∗ 1 2 2 2 2 2 𝐻0𝑡 𝐻0𝜖 [32𝐻 (−519 + 16𝜖 )−𝑘 (101 + 64𝜖 )] ∗ 𝐻0𝑒 ∗ 0 ∗ ∇3Φ− =𝜕3Φ− (11) −𝑖 Φ− (1−1) . 2 12𝑘 𝜋2 (657 + 16𝜖2) = ∗ , (44) 2 2 2 2 2 𝐻0𝜖 [64𝐻0 (657 + 16𝜖 ) − 15𝑘∗ (−99 + 64𝜖 )] An interesting asymptotic solution can be obtained by (47) considering the expectation values of, for instance, some 2 2 2 2 2 Σ (𝑥,⃗ 𝑡) 3𝐶1 (101 + 64𝜖 ) 45𝐶1 (−99 + 64𝜖 ) quadratic scalar ,as 𝐴2 =− =− , 2 2 2 (48) 󵄨 󵄨 2 (−519 + 16𝜖 ) 4 (657 + 16𝜖 ) 2 󵄨 2 󵄨 Σ (𝑡) =⟨𝐸󵄨 Σ (𝑥,⃗ 𝑡) 󵄨𝐸⟩ such that from (47)weobtainthat𝜖 = 6.68586.Furthermore, 𝜖𝜅±(𝑡) (45) 𝜌=−𝑃= 1 0 3 ∗ due to the fact that the equation of state is = ∫ 𝑑 𝜅Σ𝜅 (𝑥,⃗ ) 𝑡 Σ (𝑡)(𝑥,⃗ ) 𝑡 , 2 3 𝜅 3𝐻0 /(8𝜋𝐺),wemustrequirethat (2𝜋) 𝜅∗ 32𝐻2 (519 − 16𝜖2)+𝑘2 (101 + 64𝜖2) where 𝜅∗ >0is some minimum cut for the wavenumber 0 ∗ + 𝐻 𝑡 − 𝐻 𝑡 𝜅 (𝑡) = 𝐻 𝑒 0 𝜅 (𝑡) = 2𝐻 𝑒 0 2 to be determined and 0 0 and 0 0 𝑘∗ (−519 + 16𝜖 ) are the maximum wavenumbers to the modes of Φ+ and (49) Φ 2 2 2 2 −, respectively. The expectation values for the radiation 64𝐻0 (657 + 16𝜖 ) − 15𝑘∗ (−99 + 64𝜖 ) 𝜌 𝑃 = , energy density and the pressure aregivenbythefollwing 2𝑘 (657 + 16𝜖2) expressions: ∗ from which we obtain that 𝜅∗ = 1.45598𝐻0 = 1.45598 × 4 3 4 8 −9 2 173𝐻 𝜖 𝜖 𝐻 𝐻 𝜖 𝐻 𝑡 10 𝑀 𝑃 𝜌 𝜌= [𝐴 ( 0 − 0 )+𝐶 0 ]𝑒 0 𝑝.Noticethatwehaveneglectedin and terms 2 128𝜋3 24𝜋3 3 16𝜋3 2 whichareverysmallwithrespectto3𝐻0 /(8𝜋𝐺) and decrease −2𝐻 𝑡 𝑒 0 𝜅 𝐻 173𝑘 𝐻3 101𝐻5 as . With the values earlier mentioned for ∗, 0,and −𝐴2 ∗ 0 +𝐶2 0 𝑀 (𝐶 )2 = −2.63042 × 2 3 1 3 𝑝, we arrive at the numerical values 1 128𝜋 32𝑘∗𝜋 32 2 33 70 −4 10 , (𝐴2) = 5.95601 × 10 , 𝐶3 = 4.8693 × 10 𝑀𝑝 ,and 𝑘 𝐻7 𝐻9 𝐴 = 9.081 × 1070𝑀−4 𝜌=−𝑃= +𝐶 (− ∗ 0 + 0 ) 3 𝑝 ,whichcorrespondto 3 16𝜋3 2𝑘 𝜋3 −19 4 ∗ 1.19366 × 10 𝑀𝑝. These values perfectly accord with 4 4 8 that one expects during an inflationary vacuum dominated 2 101𝐻 𝐻 𝜖 𝐻 −𝐻 𝑡 −[𝐶 ( 0 + 0 )+𝐶 0 ]𝑒 0 expansion of the early universe. A very important fact is that 1 3 3 3 3 64𝜋 𝜖 𝜋 4𝜋 𝜖 the dark energy is outside the horizon at the beginning of 3 3 inflation but during the inflationary epoch enters to causally 2 𝑘∗𝐻 2 𝑘 𝐻0 −2𝐻 𝑡 +[𝐶 0 +𝐴 ∗ ]𝑒 0 , connected regions. In other words the dark energy is concen- 1 3 2 3 ± 2𝜋 24𝜋 trated in the range of scales (physical scales) 2𝜋/[𝜖𝜅0 (𝑡)] ≃ −𝐻 𝑡 (46) (𝜋/𝐻 )𝑒 0 <𝜆 <2𝜋/𝜅 8 4 4 3 0 phys ∗. Hence, the effective 4D scalar 15𝐻 𝜖 2 219𝐻 𝜖 𝐻 𝜖 𝐻 𝑡 0 0 0 0 (massive) field Φ− should be an interesting candidate to 𝑃= [𝐴3 −𝐴2 ( + )] 𝑒 64𝜋3 128𝜋3 24𝜋3 explain dark energy in the early inflationary universe. 99𝐻5 219𝑘 𝐻3 −𝐶2 0 +𝐴2 ∗ 0 1 3 2 3 4. Final Remarks 32𝑘∗𝜋 128𝜋 15𝑘 𝐻7 𝐻9 We have explored the possibility that the expansion of the +𝐴 (− ∗ 0 + 0 ) universe during the primordial inflationary phase of the 3 3 3 64𝜋 4𝑘∗𝜋 universe can be driven by a condensate of spinor fields. In our picture 𝜙± are effective fields which from a condensate 99𝐻4 𝐻4𝜖 𝐻8 2 0 0 0 −𝐻0𝑡 of two entangled spinors. The fields 𝜙± are decoupled at the +[𝐶1 ( − )−𝐴3 ]𝑒 64𝜋3𝜖 𝜋3 8𝜋3𝜖 end of inflation. In all our analysis we have neglected the

3 3 role of the inflaton field, which (in a de Sitter expansion) is 2 𝑘 𝐻0 2 𝑘∗𝐻 −2𝐻 𝑡 +[𝐴 ∗ +𝐶 0 ]𝑒 0 . freezed in amplitude and nearly scale invariant but decays at 2 24𝜋3 1 2𝜋3 the end of inflation into other fields. The point here is how we explain the existence of dark energy once the inflaton Since we are interested to find solutions with 𝜇=1and ] =2 field energy density goes to zero. Our proposal is consistent that correspond to 𝜕0Φ± =0,wemustconsiderthevalues to prove that the dark energy could be physically explained Advances in High Energy Physics 7

though the entanglement of spinor fields that behave as [14] G. Ch. Bohmer,J.Bournett,D.F.Mota,andD.J.Shaw,“Dark¨ effective1-spinand0-spinbosonsona4Dhypersurfaceon spinor models in gravitation and cosmology,” Journal of High which the universe suffers a vacuum dominated expansion. Energy Physics,vol.2010,no.7,article53,2010. The equation of state of the universe is determined by the [15] T. S. Bunch and P. C. W. Davies, “Quantum field theory in de static foliation 𝜓=1/𝐻0. Our calculations show that the Sitter space: renormalization by point-splitting,” Proceedings of vector boson 𝜙+ is massless and with spin 1, and therefore the Royal Society A,vol.360,no.1700,pp.117–134,1978. compatible with the properties of a massless vector boson. [16] A. Mazumdar and J. Rocher, “Particle physics models of On the other hand the field 𝜙− is a scalar boson which could inflation and curvaton scenarios,” Physics Reports,vol.497,no. be (jointly with the inflaton) responsible for the expansion 4-5, pp. 85–215, 2011. oftheuniverseandwouldbeagoodcandidatetoexplain the existence of the dark energy. (Other fields such as the curvaton field16 [ ]havebeenproposedintheliteratureto explain it.) A very interesting fact is that the (dark) energy density which we are talking about is poured into smaller subhorizon scales with the evolution of the inflationary expansion.

Acknowledgments The authors acknowledge UNMdP and CONICET, Argenti- na, for financial support.

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Research Article On Higher Dimensional Kaluza-Klein Theories

Aurel Bejancu

Department of Mathematics, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait

Correspondence should be addressed to Aurel Bejancu; [email protected]

Received 2 September 2013; Accepted 8 October 2013

Academic Editor: Kishor Adhav

Copyright © 2013 Aurel Bejancu. 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 present a new method for the study of general higher dimensional Kaluza-Klein theories. Our new approach is based on the Riemannian adapted connection and on a theory of adapted tensor fields in the ambient space. We obtain, in a covariant form, the fully general 4D equations of motion in a (4 + n)D general gauge Kaluza-Klein space. This enables us to classify the geodesics of the (4 + n)D space and to show that the induced motions in the 4D space bring more information than motions from both the 4D general relativity and the 4D Lorentz force equations. Finally, we note that all the previous studies on higher dimensional Kaluza-Klein theories are particular cases of the general case considered in the present paper.

1. Introduction “bulk”) with particles and fields trapped on the brane, while gravity is free to access the bulk (cf. [9]). The other one is As it is well known, by the Kaluza-Klein theory, the unifica- called space-time-matter theory and assumes that matter in tion of Einstein’s theory of general relativity with Maxwell’s the 4D spacetime is a manifestation of the fifth dimension (cf. theory of electromagnetism was achieved. In a modern [10, 11]). terminology, this theory is developed on a trivial principal 4 𝑈(1) Recently, we presented a new point of view on a general bundle over the usual Dspacetime,with as fibre type. Kaluza-Klein theory in a 5D space (cf. [12]).Weremovedboth Thus, a natural generalization of Kaluza-Klein theory consists theaboveconditionsandgaveanewmethodofstudybased in replacing 𝑈(1) by a nonabelian gauge group 𝐺 (cf. [1–5]). on the Riemannian horizontal connection. This enabled us There have been also some other generalizations wherein the to give a new definition of the fifth force in 4D physics (cf. internal space has been considered a homogeneous space of 5 type 𝐺/𝐻 (cf. [6, 7]). [13]) and to obtain a classification of the warped Dspaces Two conditions have been imposed in the classical satisfying Einstein equations with cosmological constant (cf. Kaluza-Klein theory and in most of the above generalizations: [14]). the “cylinder condition” and the “compactification condi- The present paper is the first in a series of papers devoted tion.” The former condition assumes that all the local com- to the study of general Kaluza-Klein theory with arbitrary ponents of the pseudo-Riemannian metric on the ambient gaugegroup.Moreprecisely,ourapproachisdevelopedon space do not depend on the extra dimensions, while the latter a principal bundle 𝑀 over the 4Dspacetime𝑀,withan𝑛- requires that the fibre must be a compact manifold. dimensional Lie group 𝐺 as fibre type. Moreover, both the In 1938, Einstein and Bergmann [8]presentedthefirst cylinder condition and the compactification condition are generalization in this direction. According to it, the local removed. In other words, the theory we develop here contains components of the 4D Lorentz metric in a 5Dspaceare as particular cases all the other generalizations of Kaluza- supposed to be periodic functions of the fifth coordinate. Kleintheorythathavebeenpresentedabove. Later on, two other important generalizations have been ThewholestudyisbasedontheRiemannianadapted intensively studied. One is called brane-world theory and connectionthatweconstructinthispaperandona4Dtensor assumes that the observable universe is a 4-surface (the calculus that we introduce via a natural splitting of the tangent “brane”) embedded in a (4 + 𝑛)-dimensional spacetime (the bundle of the ambient space. We obtain, in a covariant form 2 Advances in High Energy Physics

4 𝛾 and in their full generality, the D equations of motion as 𝛾 𝜕𝑥̃ 𝛼 (4 + 𝑛) 𝑑𝑥̃ = 𝑑𝑥 , (3a) part of equations of motion in a Dspace.Weanalyze 𝜕𝑥𝛼 theseequationsanddeducethattheinducedmotionsonthe 𝑖 𝑖 base manifold bring more information than both the motions 𝑖 𝜕𝑦̃ 𝛼 𝜕𝑦̃ 𝑗 𝑑𝑦̃ = 𝑑𝑥 + 𝑑𝑦 , (3b) from general relativity and the motions from Lorentz force 𝜕𝑥𝛼 𝜕𝑦𝑗 equations. Moreover, these equations show the existence of an extra force, which, in a particular case, is perpendicular to the 4D velocity. The general study of the extra force will be respectively. presented in a forthcoming paper. Throughout the paper we use the ranges of indices: 𝛼,𝛽,𝛾,⋅⋅⋅ ∈ {0,1,2,3} 𝑖,𝑗,𝑘,⋅⋅⋅ ∈ {4,...,𝑛 + 3},⋅⋅⋅𝐴, Now, we outline the content of the paper. In Section 2 we , 𝐵,𝐶,...,∈ {0,...,3+ 𝑛} 𝑇(𝑥, 𝑦) present the general gauge Kaluza-Klein space (𝑀, 𝑔, 𝐻𝑀), .By we denote a function 𝑇 𝑀 where 𝑀 isthetotalspaceofaprincipalbundleovera4D that is locally defined on .Also,foranyvectorbundle 𝐸 𝑀 Γ(𝐸) F(𝑀) space time with a Lie group 𝐺 as fibre type. The pseudo- over , we denote by the -module of smooth Riemannian metric 𝑔 determines the orthogonal splitting sections of 𝐸,whereF(𝑀) is the algebra of smooth functions (5) and enables us to construct the adapted frame field on 𝑀. 𝛼 𝑖 {𝛿/𝛿𝑥 ,𝜕/𝜕𝑦} (see (12)). Our study is based on a 4D Next, from (2b) we see that there exists a vector bundle 𝑖 tensor calculus developed in Section 3.Theelectromagnetic 𝑉𝑀 over 𝑀 of rank 𝑛 which is locally spanned by {𝜕/𝜕𝑦 }. 𝑘 𝑉𝑀 𝑀 tensor field 𝐹=(𝐹𝛼𝛽) given by (41)andtheadapted We call the vertical distribution on .Then,wesuppose tensor fields 𝐻 and 𝑉 given by (44)and(45), respectively, that there exists on 𝑀 a pseudo-Riemannian metric 𝑔 whose ⋆ play an important role in our approach. In Section 4 we restriction to 𝑉𝑀 is a Riemannian metric 𝑔 .Denoteby𝐻𝑀 construct the Riemannian adapted connection, that is, a the complementary orthogonal distribution to 𝑉𝑀 in 𝑇𝑀, metric connection with respect to which both distributions and call it the horizontal distribution on 𝑀.Supposethat𝐻𝑀 𝐻𝑀 𝑉𝑀 and are parallel, and its torsion is given by (58a), is invariant with respect to the action of 𝐺 on 𝑀 on the right; (58b), and (58c). Section 5 is the main section of the paper that is, we have and presents the 4D equations of motions in (𝑀, 𝑔, 𝐻𝑀) (cf. (85a)and(85b)). Also, in a particular case, we show that the (𝑅𝑎) (𝐻𝑀) = 𝐻𝑀, ∀𝑎 ∈𝐺, (4) extra force is orthogonal to the 4D velocity and therefore ⋆ does not contradict the 4D physics. Finally, in Section 6 we show that the set of geodesics in (𝑀, 𝑔, 𝐻𝑀) splits into where 𝑅𝑎⋆ is the differential of the right translation 𝑅𝑎 of 𝐺. three categories: horizontal, vertical, and oblique geodesics. Thus 𝐻𝑀 defines an Ehresmann connection on 𝑀 (cf. [15, Both, the horizontal and oblique geodesics induce some p. 359]). Also, suppose that the restriction of 𝑔 to 𝐻𝑀 is new motions on the 4Dspacetime.Weclosethepaperwith a Lorentz metric 𝑔;thatis,𝑔 is nondegenerate of signature conclusions. (+,+,+,−).Thus𝑇𝑀 is endowed with a Lorentz distribu- ⋆ tion (𝐻𝑀, 𝑔) and a Riemannian distribution (𝑉𝑀, 𝑔 ) and 2. General Gauge Kaluza-Klein Space admits the orthogonal direct decomposition

Let 𝑀 be a 4-dimensional manifold and 𝐺 an 𝑛-dimensional 𝑇𝑀=𝐻𝑀⊕𝑉𝑀. (5) Liegroup.TheKaluza-Kleintheorywepresentinthepaper is developed on a principal bundle 𝑀 with base manifold 𝑀 𝐺 (𝑥𝛼) 𝑀 As we apply the above objects to physics, we need a and structure group . Any coordinate system on will coordinate presentation for them. First, we recall (cf. [15,p. (𝑥𝛼,𝑦𝑖) 𝑀 (𝑦𝑖) define a coordinate system on ,where are the 359], [16,p.64])thattheEhresmannconnectiondefinedby (𝑥𝛼,𝑦𝑖) fibre coordinates. Two such coordinate systems and 𝐻𝑀 is completely determined by a 1-form 𝜔 on 𝑀 with values (𝑥̃𝜇, 𝑦̃𝑗) are related by the following general transformations: in the Lie algebra 𝐿(𝐺) of 𝐺, satisfying the conditions

𝛼 𝛼 𝜇 𝑥̃ = 𝑥̃ (𝑥 ) , (1a) ⋆ 𝜔 (𝐴 ) =𝐴, ∀𝐴∈𝐿(𝐺) , (6a) 𝑖 𝑖 𝛼 𝑗 𝑦̃ = 𝑦̃ (𝑥 ,𝑦 ). (1b) −1 𝜔 ((𝑅𝑎⋆ )𝑋)=𝑎𝑑(𝑎 )𝜔(𝑋) ,∀𝑎∈𝐺,∀𝑋∈Γ(𝑇𝑀) , (6b) Then, the transformations of the natural frame and coframe 𝑀 fields on have the forms ⋆ where 𝐴 is the fundamental vector field corresponding to 𝐴 and 𝑎𝑑 denotes the adjoint representation of 𝐺 in 𝐿(𝐺).Now, 𝛾 𝑖 𝜕 𝜕𝑥̃ 𝜕 𝜕𝑦̃ 𝜕 suppose that {𝐾𝑖} is a basis of left invariant vector fields in = + , (2a) 𝐿(𝐺) 𝜕𝑥𝛼 𝜕𝑥𝛼 𝜕𝑥̃𝛾 𝜕𝑥𝛼 𝜕𝑦̃𝑖 and put

𝑖 𝜕 𝜕𝑦̃ 𝜕 ⋆ 𝑗 𝜕 = , (2b) 𝐾 =𝐾 (𝑦) , 𝜕𝑦𝑗 𝜕𝑦𝑗 𝜕𝑦̃𝑖 𝑖 𝑖 𝜕𝑦𝑗 (7) Advances in High Energy Physics 3

𝑗 where [𝐾𝑖 (𝑦)] is a nonsingular matrix whose inverse we have a great role in the study. First, by direct calculations 𝑗 𝑖 using (12)weobtain denote by [𝐾𝑖 (𝑦)].Thenweput𝜔=𝜔𝐾𝑖,andbyusing(6a), (6b), and (7)wededucethat 𝛿 𝜕 𝜕 [ , ] =𝐿 𝑘 , 𝛿𝑥𝛼 𝜕𝑦𝑖 𝑖𝛼𝜕𝑦𝑘 (17a) 𝜕 𝑖 𝜔𝑖 ( )=𝐾 , 𝜕𝑦𝑗 𝑗 (8a) 𝛿 𝛿 𝜕 [ , ]=𝐹𝑘 , 𝛿𝑥𝛽 𝛿𝑥𝛼 𝛼𝛽 𝜕𝑦𝑘 (17b) 𝑖 𝜕 −1 𝑖 𝜔 ((𝑅𝑎⋆ ) )=𝑎𝑑(𝑎 ) 𝐾𝑗. (8b) 𝜕𝑦𝑗 𝜕 [ ,𝐾⋆]=0, 𝜕𝑥𝛼 𝑖 (17c) As it is well known, 𝐻𝑀 is the kernel of the connection form 𝜔. In order to present two other local characterizations of where we put 𝑘 𝐻𝑀, we consider a local basis {𝐸𝛼} in Γ(𝐻𝑀) and put 𝜕𝐿 𝐿 𝑘 = 𝛼 , (18a) 𝑖𝛼 𝜕𝑦𝑖 𝜕 𝛾 𝑖 𝜕 =𝐿𝛼 (𝑥, 𝑦)𝛾 𝐸 +𝐿𝛼 (𝑥, 𝑦) . (9) 𝜕𝑥𝛼 𝜕𝑦𝑖 𝑘 𝛿𝐿 𝛿𝐿𝑘 𝐹𝑘 = 𝛽 − 𝛼 . (18b) 𝑖 𝛼𝛽 𝛿𝑥𝛼 𝛿𝑥𝛽 As the transition matrix from {𝐸𝛼,𝛿/𝛿𝑦} to the natural 𝛾 𝑖 frame field {𝜕/𝜕𝑥 ,𝜕/𝜕𝑦} has the form Next, we show that 𝛿 𝐿𝛾 0 [ ,𝐾⋆]=0. 𝛼 𝛼 𝑖 (19) [ ] , 𝛿𝑥 𝐿𝑖 𝛿𝑖 (10) [ 𝛼 𝑗] First,accordingtoageneralresultstatedinpage78inthe book of Kobayashi and Nomizu [16], we deduce that the 𝛾 we infer that the 4×4matrix [𝐿𝛼] is nonsingular. Hence the vectorfieldsinthelefthandsideof(19) must be horizontal. vector fields On the other hand, by using (7)and(17a), we obtain 𝛿 ℎ 𝛾 𝛿 ⋆ 𝛿𝐾𝑖 𝑗 ℎ 𝜕 =𝐿𝛼𝐸𝛾,𝛼∈{0, 1, 2, 3} , (11) [ ,𝐾 ]=( +𝐾𝐿 ) . (20) 𝛿𝑥𝛼 𝛿𝑥𝛼 𝑖 𝛿𝑥𝛼 𝑖 𝑗𝛼 𝜕𝑦ℎ form a local basis in Γ(𝐻𝑀),too.Moreover,from(9)we That is, these vectors fields are vertical, too. This proves (19) obtain via (5). As 𝐿(𝐺) is isomorphic to the Lie algebra of vertical vector fields, we have 𝛿 𝜕 𝜕 = −𝐿𝑖 . ⋆ ⋆ 𝑘 ⋆ 𝛿𝑥𝛼 𝜕𝑥𝛼 𝛼 𝜕𝑦𝑖 (12) [𝐾𝑖 ,𝐾𝑗 ] =𝐶𝑖𝑗𝐾𝑘 , (21) 𝑘 𝛼 𝛼 where 𝐶 are the structure constants of the Lie group 𝐺. Note that 𝛿/𝛿𝑥 is just the projection of 𝜕/𝜕𝑥 on 𝐻𝑀.Also, 𝑖𝑗 we define the local 1-forms Then, by using17a ( )–(17c), (15), (19), (21), and (7), we deduce that 𝑖 𝑖 𝑖 𝛼 𝛿𝑦 =𝑑𝑦 +𝐿 𝑑𝑥 , (13) 𝑘 ⋆ℎ 𝑘 𝛼 𝐹 𝛼𝛽 =𝐹 𝛼𝛽𝐾ℎ, (22) and by using (8a)and(13), we deduce that where we put ℎ 𝑖 𝑖 𝑗 𝜕𝐴 ℎ ⋆ℎ 𝛽 𝜕𝐴 𝑖 𝑗 ℎ 𝜔 = 𝐾𝑗𝛿𝑦 . (14) 𝐹 = − 𝛼 +𝐴 𝐴 𝐶 . (23) 𝛼𝛽 𝜕𝑥𝛼 𝜕𝑥𝛽 𝛼 𝛽 𝑖𝑗 Hence, 𝐻𝑀 is locally represented by the kernel of the 1-forms Now, taking into account (17c), from (19), we obtain 𝑖 ⋆ {𝛿𝑦 }. Now, by using the fundamental vector fields {𝐾 } we 𝑖 𝐾⋆ (𝐴ℎ )=−𝐶 ℎ 𝐴𝑗 , put 𝑖 𝛼 𝑖𝑗 𝛼 (24) which together with (23)implies 𝛿 𝜕 𝑖 ⋆ = −𝐴 (𝑥,) 𝑦 𝐾 , (15) 𝛿𝑥𝛼 𝜕𝑥𝛼 𝛼 𝑖 ⋆ ⋆ℎ ℎ ⋆𝑗 𝐾𝑖 (𝐹 𝛼𝛽) =−𝐶𝑖𝑗𝐹 𝛼𝛽. (25) and comparing (12)with(15)weobtain ⋆ℎ By using (24)and(25)weareentitledtocall𝐹 𝛼𝛽 the Yang- 𝑖 𝑗 𝑖 ℎ 𝐿 =𝐴 𝐾 , Mills fields corresponding to gauge potentials 𝐴𝛼.Alsoby 𝛼 𝛼 𝑗 (16) 𝑘 (18b)wemaycall𝐹 𝛼𝛽 the electromagnetic tensor field corre- 𝛼 𝑖 𝛼 ⋆ 𝑘 via (7). The frame fields {𝛿/𝛿𝑥 ,𝜕/𝜕𝑦} and {𝛿/𝛿𝑥 ,𝐾𝑖 } are sponding to the electromagnetic potentials 𝐿𝛼.Itisimportant called adapted frame fields with respect to the decomposition to note that these objects come from different physical (5). The commutation formulas for these vector fields will theories, and they are related by (22)and(16). 4 Advances in High Energy Physics

𝑖 Remark 1. By a different method, the above Yang-Mills fields 𝑖 𝜕𝑦̃ 𝑗 𝛿𝑦̃ = 𝛿𝑦 , (30b) have been first introduced by Cho [4]. On the other hand, we 𝜕𝑦𝑗 𝑘 should stress that we find it more convenient to use 𝐹 and 𝛼𝛽 ̃𝑖 𝛾 ̃𝑖 𝑘 ⋆ℎ ℎ 𝑗 𝜕𝑦 ̃𝑖 𝜕𝑥̃ 𝜕𝑦 𝐿𝛼 instead of 𝐹 𝛼𝛽 and 𝐴𝛼. 𝐿 = 𝐿 + . (30c) 𝛼 𝜕𝑦𝑗 𝛾 𝜕𝑥𝛼 𝜕𝑥𝛼 Next, we express the pseudo-Riemannian metric 𝑔 on 𝑀 𝛼 𝑖 Now, we put with respect to the adapted frame field {𝛿/𝛿𝑥 ,𝜕/𝜕𝑦};that is, we have ⋆ ̃𝑗 𝜕 𝐾𝑖 = 𝐾𝑖 , (31) 𝛿 𝛿 𝛿 𝛿 𝜕𝑦̃𝑗 𝑔 (𝑥, 𝑦) = 𝑔( , )=𝑔( , ), 𝛼𝛽 𝛿𝑥𝛼 𝛿𝑥𝛽 𝛿𝑥𝛼 𝛿𝑥𝛽 (26a) and by using (16)into(30c)wededucethat ⋆ 𝜕 𝜕 𝜕 𝜕 𝑔 (𝑥,) 𝑦 =𝑔 ( , ) = 𝑔( , ) , 𝛾 𝑗 𝑖𝑗 𝜕𝑦𝑖 𝜕𝑦𝑗 𝜕𝑦𝑖 𝜕𝑦𝑗 (26b) 𝑖 𝜕𝑥̃ 𝑖 𝑗 𝜕𝑦̃ (𝐴 − 𝐴̃ ) 𝐾̃ = . (32) 𝛼 𝜕𝑥𝛼 𝛾 𝑖 𝜕𝑥𝛼 𝛿 𝜕 𝑔( , )=0. (26c) 𝛿𝑥𝛼 𝜕𝑦𝑖 The transformations30c ( )and(32)haveagaugecharacter. Apartfromthemwewillmeettransformationswithtensorial Thus the local line element representing 𝑔 has the form character. Here we observe that by using (26a), (26b), (30a), 2 𝛼 𝛽 𝑖 𝑗 and (2b) we obtain the first such transformations 𝑑𝑠 =𝑔𝛼𝛽 (𝑥, 𝑦) 𝑑𝑥 𝑑𝑥 +𝑔𝑖𝑗 (𝑥, 𝑦) 𝛿𝑦 𝛿𝑦 𝜕𝑥̃𝜇 𝜕𝑥̃] 𝛼 𝛽 𝑖 𝑖 𝛼 𝑔𝛼𝛽 = 𝑔̃𝜇] , (33a) =𝑔𝛼𝛽 (𝑥, 𝑦) 𝑑𝑥 𝑑𝑥 +𝑔𝑖𝑗 (𝑥, 𝑦) (𝑑𝑦 +𝐿𝛼𝑑𝑥 ) 𝜕𝑥𝛼 𝜕𝑥𝛽

ℎ 𝑘 ×(𝑑𝑦𝑗 +𝐿𝑗 𝑑𝑥𝛽) 𝜕𝑦̃ 𝜕𝑦̃ 𝛽 𝑔 = 𝑔̃ . (33b) 𝑖𝑗 ℎ𝑘 𝜕𝑦𝑖 𝜕𝑦𝑗 𝛼 𝛽 𝑖 ℎ 𝑖 𝛼 =𝑔𝛼𝛽 (𝑥, 𝑦) 𝑑𝑥 𝑑𝑥 +𝑔𝑖𝑗 (𝑥, 𝑦) (𝑑𝑦 +𝐴𝛼𝐾ℎ𝑑𝑥 )

×(𝑑𝑦𝑗 +𝐴ℎ 𝐾𝑗𝑑𝑥𝛽). 𝛽 ℎ 3. Adapted Tensor Fields on (𝑀, 𝑔, 𝐻𝑀) (27) In the present section we develop a tensor calculus on Hence 𝑔 is locally given by the matrices 𝑀 that is adapted to the decomposition (5). For example,

𝑔𝛼𝛽 0 we construct some adapted tensor fields which have an [ ], (28) important role in the general Kaluza-Klein theory which we 0𝑔𝑖𝑗 develop in a series of papers. In particular, we show that the electromagnetic tensor field is indeed an adapted tensor field. and ⋆ ⋆ First, we consider the dual vector bundles 𝐻𝑀 and 𝑉𝑀 𝑔 +𝑔 𝐿𝑖 𝐿𝑗 𝑔 𝐿𝑖 𝛼𝛽 𝑖𝑗 𝛼 𝛽 𝑖𝑗 𝛼 of 𝐻𝑀 and 𝑉𝑀,respectively.Then,anF(𝑀)−(𝑝+𝑞)-linear [ 𝑗 ] 𝑔𝑖𝑗 𝐿𝛽 𝑔𝑖𝑗 mapping (29) 𝑖 𝑗 ℎ 𝑘 𝑖 ℎ ⋆ 𝑝 𝑞 𝑔𝛼𝛽 +𝑔𝑖𝑗 𝐾ℎ𝐾𝑘𝐴𝛼𝐴𝛽 𝑔𝑖𝑗 𝐾ℎ𝐴𝛼 𝑇:Γ(𝐻𝑀 ) ×Γ(𝐻𝑀) 󳨀→ F (𝑀) (34) =[ 𝑗 𝑘 ], 𝑔𝑖𝑗 𝐾𝑘𝐴𝛽 𝑔𝑖𝑗 is called a horizontal tensor field of type (𝑝, 𝑞).Similarly,an 𝛼 𝑖 withrespecttotheframefields{𝛿/𝛿𝑥 ,𝜕/𝜕𝑦} and F(𝑀) − (𝑟 -linear+ 𝑠) mapping {𝜕/𝜕𝑥𝛼,𝜕/𝜕𝑦𝑖} ,respectively.Formally,(29) is identical ⋆ 𝑟 𝑠 to(13.31) from [17], but in the latter the local components are 𝑇:Γ(𝑉𝑀 ) ×Γ(𝑉𝑀) 󳨀→ F (𝑀) (35) 𝛼 supposed to be functions of (𝑥 ) alone. So 𝑔 given by (27) is the most general Kaluza-Klein metric considered in any is called a vertical tensor field of type (𝑟, 𝑠). For example, 𝑔 ⋆ Kaluza-Klein theory. The principal bundle 𝑀,togetherwith (resp., 𝑔 ) is a horizontal (resp., vertical) tensor field of type 𝛼 𝑖 the metric 𝑔 and the Ehresmann connection defined by the (0, 2). Also, 𝑑𝑥 (resp., 𝛿𝑦 ) are horizontal (resp., vertical) 𝛼 𝑖 horizontal distribution 𝐻𝑀, is denoted by (𝑀, 𝑔, 𝐻𝑀),and covector fields, while 𝛿/𝛿𝑥 (resp., 𝜕/𝜕𝑦 )arehorizontal it is called a general gauge Kaluza-Klein space. (resp., vertical) vector fields, locally defined on 𝑀.More 𝛼 𝑗 Finally, we consider two coordinate systems (𝑥 ,𝑦 ) and generally, an F(𝑀) − (𝑝 + 𝑞 + -linear𝑟 +𝑠) mapping 𝛾 𝑖 (𝑥̃ , 𝑦̃ ) and by using (12), (13), (2a), (2b), (3a), and (3b), we ⋆ 𝑝 𝑞 ⋆ 𝑟 obtain 𝑇:Γ(𝐻𝑀 ) ×Γ(𝐻𝑀) ×Γ(𝑉𝑀 ) 𝛿 𝜕𝑥̃𝛾 𝛿 (36) = , (30a) 𝑠 𝛿𝑥𝛼 𝜕𝑥𝛼 𝛿𝑥̃𝛾 ×Γ(𝑉𝑀) 󳨀→ F (𝑀) Advances in High Energy Physics 5 is an adapted tensor field of type (𝑝,𝑞;𝑟,𝑠)on 𝑀.Locally,𝑇 is given by given by the functions 1 𝐻 (ℎ𝑋, ℎ𝑌, V𝑍) = {V𝑍(𝑔(ℎ𝑋, ℎ𝑌))−𝑔(ℎ [V𝑍, ℎ𝑋] ,ℎ𝑌) 𝛼1⋅⋅⋅𝛼𝑝𝑖1⋅⋅⋅𝑖𝑟 2 𝑇𝛽 ⋅⋅⋅𝛽 𝑗 ⋅⋅⋅𝑗 (𝑥, 𝑦) 1 𝑞 1 𝑠 −𝑔(ℎ [V𝑍, ℎ𝑌] ,ℎ𝑋)}, 𝛼 𝛼 𝛿 𝛿 =𝑇(𝑑𝑥 1 ,...,𝑑𝑥 𝑝 , ,..., , (44) 𝛽 𝛽 𝛿𝑥 1 𝛿𝑥 𝑞 (37) 1 𝑉 (V𝑋, V𝑌, ℎ𝑍) = {ℎ𝑍 (𝑔⋆ (V𝑋, V𝑌))−𝑔⋆ (V [ℎ𝑍, V𝑋] , V𝑌) 𝑖 𝑖 𝜕 𝜕 𝛿𝑦 1 ,...,𝛿𝑦𝑟 , ,..., ). 2 𝜕𝑦𝑗1 𝜕𝑦𝑗𝑠 −𝑔⋆ (V [ℎ𝑍, V𝑌] , V𝑋)}, Then by using (2b), (3a), (30a), and (30b)wededucethatthere (45) exists an adapted tensor field of type (𝑝,𝑞;𝑟,𝑠)on 𝑀,ifand for all 𝑋, 𝑌, 𝑍 ∈ Γ(𝑇𝑀).Itiseasytoverifythatboth𝐻 only if, on the domain of each coordinate system, there exist 𝑉 F(𝑀) 𝑝+𝑞 𝑟+𝑠 𝛼1⋅⋅⋅𝛼𝑝𝑖1⋅⋅⋅𝑖𝑟 and are -3-linear mappings and therefore define 4 ⋅𝑛 functions 𝑇 satisfying 𝛽1⋅⋅⋅𝛽𝑞𝑗1⋅⋅⋅𝑗𝑠 the adapted tensor fields of types (0,2;0,1) and (0,1;0,2), 𝐻 𝑉 𝐻𝑀 𝛾 𝛾 𝑘 𝑘 respectively. By using and and the metrics on and 𝛼 ⋅⋅⋅𝛼 𝑖 ⋅⋅⋅𝑖 𝜕𝑥̃ 1 𝜕𝑥̃ 𝑝 𝜕𝑦̃ 1 𝜕𝑦̃ 𝑟 𝑇 1 𝑝 1 𝑟 ⋅⋅⋅ ⋅⋅⋅ 𝑉𝑀, we define two adapted tensor fields denoted by the same 𝛽 ⋅⋅⋅𝛽 𝑗 ⋅⋅⋅𝑗 𝛼 𝛼 𝑖 𝑖 1 𝑞 1 𝑠 𝜕𝑥 1 𝜕𝑥 𝑝 𝜕𝑦 1 𝜕𝑦 𝑟 symbols and given by (38) 𝜇 𝜇 ℎ ℎ 𝛾 ⋅⋅⋅𝛾 𝑘 ⋅⋅⋅𝑘 𝜕𝑥̃ 1 𝜕𝑥̃ 𝑞 𝜕𝑦̃ 1 𝜕𝑦̃ 𝑠 𝐻:Γ(𝐻𝑀) × Γ 𝑀)(𝑉 󳨀→ Γ (𝐻𝑀) , = 𝑇̃ 1 𝑝 1 𝑟 ⋅⋅⋅ ⋅⋅⋅ , 𝜇 ⋅⋅⋅𝜇 ℎ ⋅⋅⋅ℎ 𝛽 𝛽 𝑗 𝑗 1 𝑞 1 𝑠 𝜕𝑥 1 𝜕𝑥 𝑞 𝜕𝑦 1 𝜕𝑦 𝑠 (46) 𝑉:Γ(𝑉𝑀) × Γ 𝑀)(𝐻 󳨀→ Γ (𝑉𝑀) , with respect to the transformations (1a)and(1b). Also, we note that any F(𝑀) − (𝑞 + 𝑟-linear +𝑠) mapping 𝑔 (ℎ𝑋, 𝐻 (ℎ𝑌, V𝑍)) =𝐻(ℎ𝑋, ℎ𝑌, V𝑍) , (47a) 𝑞 ⋆ 𝑟 𝑠 𝑇:Γ(𝐻𝑀) ×Γ(𝑉𝑀 ) ×Γ(𝑉𝑀) 󳨀→ Γ (𝐻𝑀) ⋆ (39) 𝑔 (V𝑋, 𝑉 (V𝑌, ℎ𝑍)) =𝑉(V𝑋, V𝑌, ℎ𝑍) , (47b) defines an adapted tensor field of type (1,𝑞;𝑟,𝑠).Similarly, for all 𝑋, 𝑌, 𝑍 ∈ Γ(𝑇𝑀). any F(𝑀) − (𝑝 + 𝑞-linear +𝑠) mapping We close this section with a local presentation of the 𝐻 𝑉 𝑝 𝑞 𝑠 adapted tensor fields and .First,from(33a)and(33b)we ⋆ 𝛼𝛽 𝑖𝑗 𝑇:Γ(𝐻𝑀 ) ×Γ(𝐻𝑀) ×Γ(𝑉𝑀) 󳨀→ Γ ( 𝑉 𝑀) (40) deduce that entries 𝑔 (resp., 𝑔 ) of the inverse of the matrix [𝑔𝛼𝛽] (resp., [𝑔𝑖𝑗 ]) define a horizontal (resp., vertical) tensor (𝑝,𝑞;1,𝑠) defines an adapted tensor field of type .Moreabout field of type (2, 0). Then, weput adaptedtensorfieldscanbefoundinthebookofBejancuand 𝛿 𝛿 𝜕 Farran [18]. 𝐻( , , )=𝐻 , 𝛽 𝛼 𝑖 𝑖𝛼𝛽 (48a) Next, we will construct some adapted tensor fields which 𝛿𝑥 𝛿𝑥 𝜕𝑦 ℎ are deeply involved in our study. First, we denote by 𝛿 𝜕 𝛿 V 𝑇𝑀 𝐻𝑀 𝑉𝑀 𝐻( , )=𝐻 𝛾 , and the projection morphisms of on and , 𝛿𝑥𝛼 𝜕𝑦𝑖 𝑖𝛼 𝛿𝑥𝛾 (48b) respectively. Then, we consider the mapping 𝜕 𝜕 𝛿 2 𝑉 ( , , ) =𝑉 , 𝐹:Γ(𝐻𝑀) 󳨀→ Γ ( 𝑉 𝑀) , 𝜕𝑦𝑗 𝜕𝑦𝑖 𝛿𝑥𝛼 𝛼𝑖𝑗 (48c) (41) 𝐹 (ℎ𝑋, ℎ𝑌) =−V [ℎ𝑋, ℎ𝑌] ,∀𝑋,𝑌∈Γ(𝑇𝑀) . 𝜕 𝛿 𝜕 𝑉 ( , ) =𝑉 𝑘 , 𝜕𝑦𝑖 𝛿𝑥𝛼 𝛼𝑖 𝜕𝑦𝑘 (48d) It is easy to check that 𝐹 is F(𝑀)-bilinear mapping. Thus 𝐹 is anadaptedtensorfieldoftype(0,2;1,0).Byusing(17b)and and by using (44), (45), (47a), (47b), (48a)–(48d), (26a), (41)weobtain (26b), and (17a)–(17c), we obtain

𝛿 𝛿 𝑘 𝜕 𝐹 ( , ) =𝐹 , 1 𝜕𝑔𝛼𝛽 𝛿𝑥𝛼 𝛽 𝛼𝛽 𝑘 (42) 𝐻 = , 𝛿𝑥 𝜕𝑦 𝑖𝛼𝛽 2 𝜕𝑦𝑖 (49a) 𝑘 𝛾 where 𝐹 𝛼𝛽 are given by (18b). Hence the electromagnetic 𝛾𝛽 𝐻𝑖𝛼 =𝑔 𝐻𝑖𝛼𝛽, (49b) tensor field is indeed an adapted tensor field. Next, we define the mappings: 1 𝛿𝑔 2 𝑉 = { 𝑖𝑗 −𝑔 𝐿 𝑘 ,−𝑔 𝐿 𝑘 }, 𝐻:Γ(𝐻𝑀) ×Γ(𝑉𝑀) 󳨀→ F (𝑀) , 𝛼𝑖𝑗 2 𝛿𝑥𝛼 𝑘𝑗 𝑖𝛼 𝑖𝑘 𝑗𝛼 (50a) (43) 2 𝑘 𝑘𝑗 𝑉: Γ(𝑉𝑀) ×Γ(𝐻𝑀) 󳨀→ F (𝑀) , 𝑉𝛼𝑖 =𝑔 𝑉𝛼𝑖𝑗. (50b) 6 Advances in High Energy Physics

𝐺=𝑈(1) 𝐻 ℎ V Remark 2. If in particular ,then 1𝛼𝛽 represent the Also,itiseasytoshowthatanadaptedconnection∇=(∇, ∇) local components of the extrinsic curvature used in brane- is metric; that is, world theory (cf. [9]) and in space-time-matter theory (cf. [12, 19]). For this reason we call 𝐻 given by (44)theextrinsic (∇𝑋𝑔) (𝑌, 𝑍) =0, ∀𝑋,𝑌,𝑍∈Γ(𝑇𝑀) , (55) curvature of the horizontal distribution. ℎ V Remark 3. In all the papers published so far on Kaluza-Klein if and only if both ∇ and ∇ are metric connections; that is, theories with nonabelian gauge group, the local components 𝑔𝛼𝛽 of the Lorentz metric 𝑔 are supposed to be independent ℎ 𝑖 (∇𝑋𝑔) (ℎ𝑌,ℎ𝑍) =0, (56a) of 𝑦 (cf. [2, 4–7]). From (49a)and(49b) we see that this particular case occurs if and only if the extrinsic curvature V ⋆ of 𝐻𝑀 vanishes identically on 𝑀. (∇𝑋𝑔 ) (V𝑌, V𝑍) =0, (56b)

4. A Remarkable Linear for all 𝑋, 𝑌, 𝑍 ∈ Γ(𝑇𝑀). The torsion tensor field of ∇ is given Connection on (𝑀, 𝑔, 𝐻𝑀) by

In a previous paper (cf. [12]), we constructed the Riemannian 𝑇 (𝑋, 𝑌) =∇𝑋𝑌−∇𝑌𝑋−[𝑋, 𝑌] . (57) horizontal connection on the horizontal distribution of a 5 4 D general Kaluza-Klein theory and obtain both the D Now, we can prove the following important result. equations of motion and 4D Einstein equations. As in that case the vertical bundle was of rank 1, it was not necessary Theorem 4. Let (𝑀, 𝑔, 𝐻𝑀) be a general gauge Kaluza- to consider a linear connection on it. On the contrary, the Klein space. Then there exists a unique metric adapted linear geometric configuration of (𝑀, 𝑔, 𝐻𝑀) from the present ℎ V connection ∇=(∇, ∇) whose torsion tensor field 𝑇 is given by paper requires such connections on both 𝐻𝑀 and 𝑉𝑀. The construction of these connections is the purpose of this section. ∇ First, we denote by the Levi-Civita connection on 𝑇 (ℎ𝑋, ℎ𝑌) =𝐹(ℎ𝑋, ℎ𝑌) , (58a) (𝑀, 𝑔, 𝐻𝑀) given by (cf. [20,p.61]) 𝑇 (V𝑋, V𝑌) =0, (58b) 2𝑔(∇𝑋𝑌,𝑍) =𝑔 𝑋( (𝑌, 𝑍))+𝑌(𝑔 (𝑍, 𝑋))−𝑍(𝑔 (𝑋, 𝑌)) 𝑇 (ℎ𝑋, V𝑌) =𝑉(V𝑌, ℎ𝑋) −𝐻(ℎ𝑋, V𝑌) , + 𝑔 ([𝑋, 𝑌] ,𝑍) − 𝑔 ([𝑌, 𝑍] ,𝑋) (58c)

+ 𝑔 ([𝑍, 𝑋] ,𝑌) , for all 𝑋, 𝑌 ∈ Γ(𝑇𝑀). (51) ℎ V for all 𝑋, 𝑌, 𝑍 ∈ Γ(𝑇𝑀).Recallthat∇ is the unique linear Proof. First, define ∇ and ∇ as follows: connection on 𝑀 which is metric and torsion free. ∇ ℎ Next, we say that is an adapted linear connection on ∇ℎ𝑋ℎ𝑌 = ℎ∇ℎ𝑋ℎ𝑌, (59a) (𝑀, 𝑔, 𝐻𝑀) if both distributions 𝐻𝑀 and 𝑉𝑀 are parallel with respect to ∇; that is, we have ℎ ∇V𝑋ℎ𝑌 = ℎ [V𝑋, ℎ𝑌] +𝐻(ℎ𝑌, V𝑋) , (59b)

∇𝑋ℎ𝑌 ∈ Γ (𝐻𝑀) , (52a) V ∇V𝑋V𝑌=V∇V𝑋V𝑌, (59c) ∇ V𝑌∈Γ(𝑉𝑀) , 𝑋 (52b) V ∇ℎ𝑋V𝑌=V [ℎ𝑋, V𝑌] +𝑉(V𝑌, ℎ𝑋) , (59d) for all 𝑋, 𝑌 ∈ Γ(𝑇𝑀). Then there exist two linear connections ℎ V ∇ and ∇ on 𝐻𝑀 and 𝑉𝑀,respectively,givenby for all 𝑋, 𝑌 ∈ Γ(𝑇𝑀).Then,itiseasytocheckthat ℎ V ℎ ∇=(∇, ∇) ∇ ℎ𝑌 = ∇ ℎ𝑌, (53a) given by (59a)–(59d) is a metric adapted linear 𝑋 𝑋 connection whose torsion tensor field satisfies58a ( )–(58c). ℎ V V 󸀠 󸀠 󸀠 ∇𝑋V𝑌=∇𝑋V𝑌. (53b) Next, suppose that ∇ =(∇ , ∇ ) is an another metric adapted

ℎ V linear connection satisfying (58a)–(58c). Then, from58c ( )we Conversely, given two linear connections ∇ and ∇ on 𝐻𝑀 and deduce that 𝑉𝑀, respectively, there exists an adapted linear connection ∇ V V𝑌−V [ℎ𝑋, V𝑌] −𝑉(V𝑌, ℎ𝑋) on 𝑀 given by ∇ℎ𝑋 (60) ℎ V ℎ ∇𝑋𝑌=∇𝑋ℎ𝑌 + ∇𝑋V𝑌. (54) = ∇V𝑌ℎ𝑋 − ℎ [V𝑌, ℎ𝑋] −𝐻(ℎ𝑋, V𝑌) , Advances in High Energy Physics 7

∇󸀠 V 𝜕 𝑘 𝜕 which implies both (59b)and(59d)for ,via(5). Now, we 𝑗 =Γ , ∇𝜕/𝜕𝑦 𝑖 𝑖𝑗 (63c) note that (58a)isequivalentto 𝜕𝑦 𝜕𝑦𝑘

V ℎ ℎ 𝜕 𝑘 𝜕 ∇𝛿/𝛿𝑥𝛼 =Γ . ∇ℎ𝑋ℎ𝑌 − ∇ℎ𝑌ℎ𝑋−ℎ[ℎ𝑋, ℎ𝑌] =0, ∀𝑋,𝑌∈Γ(𝑇𝑀) . 𝜕𝑦𝑖 𝑖𝛼𝜕𝑦𝑘 (63d) (61) 𝛽 𝛼 𝜇 Then, we take 𝑋=𝛿/𝛿𝑥,𝑌 = 𝛿/𝛿𝑥 , and 𝑍=𝛿/𝛿𝑥 into ℎ 󸀠 (51), and using (59a), (63a)(26a), and (17b), we obtain Then by using (56a)and(61)for∇ and taking into account (51), we obtain 1 𝛿𝑔 𝛿𝑔 𝛿𝑔 Γ 𝛾 = 𝑔𝛾𝜇 { 𝜇𝛼 + 𝜇𝛽 − 𝛼𝛽 }. 𝛼𝛽 𝛼 𝜇 (64) ℎ ℎ 2 𝛿𝑥𝛽 𝛿𝑥 𝛿𝑥 󸀠 󸀠 0= (∇ ℎ𝑋𝑔) (ℎ𝑌,ℎ𝑍) +(∇ ℎ𝑌𝑔) (ℎ𝑍, ℎ𝑋) 𝑗 𝑖 𝑘 Similarly, we take 𝑋=𝜕/𝜕𝑦,𝑌 = 𝜕/𝜕𝑦, and 𝑍=𝜕/𝜕𝑦 in ℎ (51), and by using (59c), (63c), and (26b), we infer that 󸀠 −(∇ ℎ𝑍𝑔) (ℎ𝑋, ℎ𝑌) 1 𝜕𝑔 𝜕𝑔 𝜕𝑔 Γ 𝑘 = 𝑔𝑘ℎ { ℎ𝑖 + ℎ𝑗 − 𝑖𝑗 }. 𝑖𝑗 2 𝜕𝑦𝑗 𝜕𝑦𝑖 ℎ (65) =ℎ𝑋(𝑔 (ℎ𝑌,ℎ𝑍))+ℎ𝑌(𝑔 (ℎ𝑍, ℎ𝑋)) 𝜕𝑦

−ℎ𝑍(𝑔 (ℎ𝑋, ℎ𝑌)) Also, by direct calculations using (59b), (59d), (63b), (63d), (62) (17a), (48b), and (48d), we deduce that + 𝑔 ([ℎ𝑋, ℎ𝑌] ,ℎ𝑍) − 𝑔 ([ℎ𝑌,ℎ𝑍] ,ℎ𝑋) 𝛾 𝛾 Γ =𝐻 , (66a) + 𝑔 ([ℎ𝑍, ℎ𝑋] ,ℎ𝑌) 𝛼𝑖 𝑖𝛼 Γ 𝑘 =𝐿 𝑘 +𝑉 𝑘. ℎ 𝑖𝛼 𝑖𝛼 𝛼𝑖 (66b) 󸀠 −2𝑔(∇ ℎ𝑋ℎ𝑌,ℎ𝑍) According to the splitting in (5),theRiemannianadapted ℎ V ℎ connection ∇=(∇, ∇) definestwotypesofcovariant 󸀠 𝛾𝑘 =2𝑔(ℎ∇ℎ𝑋ℎ𝑌 − ∇ ℎ𝑋ℎ𝑌,ℎ𝑍) , derivatives. More precisely, if 𝑇𝛽𝑗 are the local components of an adapted tensor field of type (1,1;1,1),thenwehave

ℎ 𝛾𝑘 ∇󸀠 𝛿𝑇 which proves (59a)for .Inasimilarway(59c)isprovedfor 𝑇𝛾𝑘 = 𝛽𝑗 +𝑇𝜇𝑘Γ 𝛾 +𝑇𝛾ℎΓ 𝑘 −𝑇𝛾𝑘Γ 𝜇 −𝑇𝛾𝑘Γ ℎ , V 𝛽𝑗| 𝛼 𝛽𝑗 𝜇𝛼 𝛽𝑗 ℎ𝛼 𝜇𝑗 𝛽𝛼 𝛽ℎ 𝑗𝛼 ∇󸀠 ∇󸀠 =∇ 𝛼 𝛿𝑥 .Thus ,andtheproofiscomplete. (67a)

ℎ V 𝜕𝑇𝛾𝑘 As ∇ and ∇ satisfy (56a)and(56b), we call them 𝛾𝑘 𝛽𝑗 𝜇𝑘 𝛾 𝛾ℎ 𝑘 𝛾ℎ 𝜇 𝛾𝑘 ℎ 𝑇𝛽𝑗| = +𝑇𝛽𝑗 Γ𝜇𝑖+𝑇𝛽𝑗 Γℎ𝑖−𝑇𝜇𝑗 Γ𝛽𝑖−𝑇𝛽ℎΓ𝑗𝑖. the Riemannian horizontal connection and the Riemannian 𝑖 𝜕𝑦𝑖 ℎ V vertical connection,respectively.Also,∇=(∇, ∇) given by (67b) (59a), (59b), (59c), and (59d)iscalledRiemannian adapted connection on (𝑀, 𝑔, 𝐻𝑀). In particular, from (56a)and(56b)wededucethat

Remark 5. It is important to note that both ∇ℎ𝑋𝑇 and ∇V𝑋𝑇 𝑇 𝑔 =0, are adapted tensor fields, where is an adapted tensor field 𝛼𝛽|𝛾 (68a) and ∇ is given by (59a)–(59d). 𝛼𝛽 𝑔 | =0, (68b) Remark 6. Throughout the paper, all local components for 𝛾 linear connections and adapted tensor fields are defined with 𝑔 =0, 𝛼 𝑖 𝑖𝑗|𝛼 (68c) respect to the adapted frame field {𝛿/𝛿𝑥 ,𝜕/𝜕𝑦} and the 𝛼 𝑖 𝑖𝑗 adapted coframe field {𝑑𝑥 ,𝛿𝑦}. 𝑔 =0, |𝛼 (68d) ℎ V ∇=(∇, ∇) 𝑔 =0, Next, we consider given by (59a)–(59d)and 𝛼𝛽|𝑖 (69a) put 𝑔𝛼𝛽 =0, |𝑖 (69b) ℎ 𝛿 𝛾 𝛿 ∇ 𝛽 =Γ , 𝛿/𝛿𝑥 𝛼 𝛼𝛽 𝛾 (63a) 𝑔 =0, 𝛿𝑥 𝛿𝑥 𝑖𝑗|𝑘 (69c) ℎ 𝛿 𝛾 𝛿 𝑔𝑖𝑗 =0. ∇ 𝑖 =Γ , (63b) | (69d) 𝜕/𝜕𝑦 𝛿𝑥𝛼 𝛼𝑖𝛿𝑥𝛾 𝑘 8 Advances in High Energy Physics

𝛼𝛽 𝑖𝑗 Throughout the paper we use 𝑔𝛼𝛽,𝑔 ,𝑔𝑖𝑗 ,and𝑔 for raising 5. 4D Equations of Motion in (𝑀, 𝑔, 𝐻𝑀) and lowering indices of adapted tensor fields as follows: In this section we present the first achievement of the new 𝑘 𝑘𝑖 𝐻 𝛼𝛽 =𝑔 𝐻𝑖𝛼𝛽, (70a) method which we develop on general (4 + 𝑛)DKaluza- 4 𝛾 𝛾𝛼 Klein theories. We obtain, in a covariant form, the D 𝑉 𝑖𝑗 =𝑔 𝑉𝛼𝑖𝑗, (70b) equations of motion induced by the equations of motion in (𝑀, 𝑔, 𝐻𝑀) 𝛾 𝛾𝛽 𝑘 . This enables us to study the geodesics of the 𝐹𝑖𝛼 =𝑔𝑖𝑘𝑔 𝐹 𝛼𝛽. (70c) ambient space according to their positions with respect to horizontal distribution. It is noteworthy that the geodesics Now, we state the following. which are tangent to 𝐻𝑀 must be autoparallel curve for the ℎ Theorem 7. The Levi-Civita connection ∇ on the general gauge Riemannian horizontal connection ∇.Themotionsonthe Kaluza-Klein space (𝑀, 𝑔, 𝐻𝑀) is expressed as follows: base manifold are defined as projections of the motions in (𝑀, 𝑔, 𝐻𝑀). Let 𝐶 be a smooth curve in 𝑀 given by parametric 𝛿 𝛿 1 𝜕 equations ∇ =Γ 𝛾 +( 𝐹𝑘 −𝐻𝑘 ) , 𝛿/𝛿𝑥𝛽 𝛼 𝛼𝛽 𝛾 𝛼𝛽 𝛼𝛽 𝑘 (71a) 𝛿𝑥 𝛿𝑥 2 𝜕𝑦 𝛼 𝛼 𝑥 =𝑥 (𝑡) , (78a) 𝛿 𝛾 1 𝛾 𝛿 𝑘 𝜕 ∇ 𝑖 =(𝐻 + 𝐹 ) +𝑉 , 𝑖 𝑖 𝜕/𝜕𝑦 𝛿𝑥𝛼 𝑖𝛼 2 𝑖𝛼 𝛿𝑥𝛾 𝛼𝑖 𝜕𝑦𝑘 (71b) 𝑦 =𝑦 (𝑡) ,𝑡∈[𝑎,] 𝑏 ,𝛼∈{0, 1, 2, 3} ,𝑖∈{4,...,3+𝑛} .

𝜕 𝛾 1 𝛾 𝛿 𝑘 𝜕 ∇ 𝛼 =(𝐻 + 𝐹 ) +Γ , (78b) 𝛿/𝛿𝑥 𝜕𝑦𝑖 𝑖𝛼 2 𝑖𝛼 𝛿𝑥𝛾 𝑖𝛼𝜕𝑦𝑘 (71c) 𝑑/𝑑𝑡 𝐶 𝜕 𝛾 𝛿 𝑘 𝜕 Then, we express the tangent vector field to with ∇ 𝑗 =−𝑉 +Γ . 𝜕/𝜕𝑦 𝜕𝑦𝑖 𝑖𝑗 𝛿𝑥𝛾 𝑖𝑗𝜕𝑦𝑘 (71d) respect to the natural frame field as follows: 𝛼 𝑖 Proof. According to decomposition (5)weput 𝑑 𝑑𝑥 𝜕 𝑑𝑦 𝜕 = + . (79) 𝑑𝑡 𝑑𝑡 𝜕𝑥𝛼 𝑑𝑡 𝜕𝑦𝑖 𝛿 𝛾 𝛿 𝑘 𝜕 ∇ = Γ + Γ . 𝛿/𝛿𝑥𝛽 𝛿𝑥𝛼 𝛼𝛽𝛿𝑥𝛾 𝛼𝛽𝜕𝑦𝑘 (72) Taking into account decomposition (5)andusing(12)into Then by using (59a)and(63a), we deduce that (79), we obtain 𝛾 Γ =Γ 𝛾 . 𝑑 𝑑𝑥𝛼 𝛿 𝛿𝑦𝑖 𝜕 𝛼𝛽 𝛼𝛽 (73) = + , (80) 𝑑𝑡 𝑑𝑡 𝛿𝑥𝛼 𝛿𝑡 𝜕𝑦𝑖 𝛽 𝛼 𝑖 Next, take 𝑋=𝛿/𝛿𝑥 ,𝑌 = 𝛿/𝛿𝑥 ,and𝑍=𝜕/𝜕𝑦 in (51)and by using (26a), (26c), (17a), (17b), (48a), and (70a), we obtain where we put

𝑘 1 Γ = 𝐹𝑘 −𝐻𝑘 . 𝛿𝑦𝑖 𝑑𝑦𝑖 𝑑𝑥𝛼 𝛼𝛽 2 𝛼𝛽 𝛼𝛽 (74) = +𝐿𝑖 . (81) 𝛿𝑡 𝑑𝑡 𝛼 𝑑𝑡 Thus (71a)isobtainedfrom(72). Similarly, we put Next, by direct calculations using (71a)–(71d)and(80), we 𝛿 𝛾 𝛿 𝑘 𝜕 ∇ 𝑖 = Γ + Γ . deduce that 𝜕/𝜕𝑦 𝛿𝑥𝛼 𝛼𝑖𝛿𝑥𝛾 𝛼𝑖𝜕𝑦𝑘 (75) 𝛽 𝑖 𝛿 𝛾 𝑑𝑥 𝛾 1 𝛾 𝛿𝑦 𝛿 𝑖 𝛼 𝜇 ∇ ={Γ +(𝐻 + 𝐹 ) } Then, take 𝑋=𝜕/𝜕𝑦,𝑌=𝛿/𝛿𝑥 ,and𝑍=𝛿/𝛿𝑥 in (51)and 𝑑/𝑑𝑡 𝛿𝑥𝛼 𝛼𝛽𝑑𝑡 𝑖𝛼 2 𝑖𝛼 𝛿𝑡 𝛿𝑥𝛾 by using (26a), (26c), (17a), (17b), (49a), and (70c), we infer 𝛽 𝑖 that 1 𝑘 𝑘 𝑑𝑥 𝑘 𝛿𝑦 𝜕 +{( 𝐹 𝛼𝛽 −𝐻 𝛼𝛽) +𝑉𝛼𝑖 } , 𝛾 𝛾 1 𝛾 2 𝑑𝑡 𝛿𝑡 𝜕𝑦𝑘 Γ𝛼𝑖=𝐻 + 𝐹 . (76) 𝑖𝛼 2 𝑖𝛼 (82a) 𝑖 𝛼 𝑖 Also, take 𝑋=𝜕/𝜕𝑦,𝑌 = 𝛿/𝛿𝑥 ,and𝑍=𝜕/𝜕𝑦 in (51)and 𝜕 1 𝑑𝑥𝛼 𝛿𝑦𝑗 𝛿 ∇ ={(𝐻𝛾 + 𝐹 𝛾) −𝑉𝛾 } by using (26b), (26c), (17a), and (50b), we deduce that 𝑑/𝑑𝑡 𝜕𝑦𝑖 𝑖𝛼 2 𝑖𝛼 𝑑𝑡 𝑖𝑗 𝛿𝑡 𝛿𝑥𝛾 𝑘 𝑘 (82b) Γ𝛼𝑖=𝑉𝛼𝑖 . (77) 𝛼 𝑗 𝑘 𝑑𝑥 𝑘 𝛿𝑦 𝜕 +{Γ𝑖𝛼 +Γ𝑖𝑗 } , Thus (71b)isobtainedfrom(75). Now, taking into account 𝑑𝑡 𝛿𝑡 𝜕𝑦𝑘 that ∇ is a torsion-free connection and using (17a), (71b), and (66b)weobtain(71c). Finally, (71d) is deduced in a similar where ∇ is the Levi-Civita connection on (𝑀, 𝑔, 𝐻𝑀).Then, way as (71a). by using (80), (82a), and (82b) and taking into account that Advances in High Energy Physics 9

𝑘 𝛾 𝐹 𝛼𝛽 are skew symmetric with respect to Greek indices, we 𝐹𝑖𝛼 =0, (86b) obtain 𝑉𝛾 =0, 𝑑 𝑖𝑗 (86c) ∇𝑑/𝑑𝑡 𝑑𝑡 for all 𝛼,𝛾 ∈ {0,1,2,3} and 𝑖,𝑗 ∈ {4,...,3+.Notethat 𝑛} 𝑑𝑥𝛼 𝛿 𝛿𝑦𝑖 𝜕 all these conditions have geometrical (physical) meaning, = ∇ { + } 𝑑/𝑑𝑡 𝛼 𝑖 because they are invariant with respect to the transformations 𝑑𝑡 𝛿𝑥 𝛿𝑡 𝜕𝑦 (1a)and(1b). Taking into account (86a), (49a), (49b), and 𝑔 𝐻𝑀 𝑑2𝑥𝛾 𝛿 𝑑 𝛿𝑦𝑘 𝜕 𝑑𝑥𝛼 𝛿 (26a), we deduce that the Lorentz metric on can be = + ( ) + ∇ considered as a Lorentz metric on the base manifold 𝑀.Thus, 𝑑𝑡2 𝛿𝑥𝛾 𝑑𝑡 𝛿𝑡 𝜕𝑦𝑘 𝑑𝑡 𝑑/𝑑𝑡 𝛿𝑥𝛼 𝛾 𝛼 in this particular case, Γ𝛼𝛽given by (64)arefunctionsof(𝑥 ) 𝛿𝑦𝑖 𝜕 alone and they are given by + ∇𝑑/𝑑𝑡 𝛿𝑡 𝜕𝑦𝑖 1 𝜕𝑔 𝜕𝑔 𝜕𝑔 Γ 𝛾 (𝑥𝜇)= 𝑔𝛾𝜇 { 𝜇𝛼 + 𝜇𝛽 − 𝛼𝛽 }. (83) 𝛼𝛽 𝛽 𝛼 𝜇 (87) 𝑑2𝑥𝛾 𝑑𝑥𝛼 𝑑𝑥𝛽 2 𝜕𝑥 𝜕𝑥 𝜕𝑥 ={ +Γ 𝛾 𝑑𝑡2 𝛼𝛽𝑑𝑡 𝑑𝑡 Moreover, (85a)becomes 𝑑𝑥𝛼 𝛿𝑦𝑖 𝛿𝑦𝑖 𝛿𝑦𝑗 𝛿 2 𝛾 𝛼 𝛽 𝛾 𝛾 𝛾 𝑑 𝑥 𝛾 𝜇 𝑑𝑥 𝑑𝑥 +(2𝐻𝑖𝛼 +𝐹𝑖𝛼 ) −𝑉 𝑖𝑗 } +Γ (𝑥 ) =0. (88) 𝑑𝑡 𝑑𝑡 𝛿𝑡 𝛿𝑡 𝛿𝑥𝛾 𝑑𝑡2 𝛼𝛽 𝑑𝑡 𝑑𝑡

𝑘 𝛼 𝛽 𝑑 𝛿𝑦 𝑘 𝑑𝑥 𝑑𝑥 That is, we obtain the equations of motion in the 4Dspace- +{ ( )−𝐻 𝜇 𝑑𝑡 𝛿𝑡 𝛼𝛽 𝑑𝑡 𝑑𝑡 time (𝑀, 𝑔𝛼𝛽 =𝑔 (𝑥 )).Hence,the projections of geodesics of (𝑀, 𝑔, 𝐻𝑀) on 𝑀 coincide with the geodesics of the spacetime 𝑑𝑥𝛼 𝛿𝑦𝑖 𝛿𝑦𝑖 𝛿𝑦𝑗 𝜕 (𝑀, 𝑔) 4 +(Γ 𝑘 +𝑉 𝑘) +Γ 𝑘 } . .Thisjustifiesthename D equations of motion for 𝑖𝛼 𝛼𝑖 𝑑𝑡 𝛿𝑡 𝑖𝑗𝛿𝑡 𝛿𝑡 𝜕𝑦𝑘 (85a). Next, we suppose that only (86a)and(86c) are satisfied. Now, we recall that 𝐶 is a geodesic of (𝑀, 𝑔, 𝐻𝑀) if and only Then (85a)becomes if it is a curve of acceleration zero; that is, we have (cf. [20,p. 𝑑2𝑥𝛾 𝑑𝑥𝛼 𝑑𝑥𝛽 𝑑𝑥𝛼 𝛿𝑦𝑖 67]) +Γ 𝛾 (𝑥𝜇) +𝐹 𝛾 (𝑥, 𝑦) =0. (89) 𝑑𝑡2 𝛼𝛽 𝑑𝑡 𝑑𝑡 𝑖𝛼 𝑑𝑡 𝛿𝑡 𝑑 ∇ =0. (84) 𝑑/𝑑𝑡 𝑑𝑡 In this case, we show that there exists an extra force which does not contradict the 4D physics. First, we define the Thus, using (84), (83), and decomposition (5), we can state 4 𝐶 the main result of this section. D velocity along a geodesic as the horizontal vector field 𝑈(𝑡) given by Theorem 8. Theequationsofmotioninageneralgauge 𝑑𝑥𝛼 𝛿 (𝑀, 𝑔, 𝐻𝑀) 𝑈 (𝑡) = . (90) Kaluza-Klein space are expressed as follows: 𝑑𝑡 𝛿𝑥𝛼

Then, define the extra force induced by extra dimensions as 𝑑2𝑥𝛾 𝑑𝑥𝛼 𝑑𝑥𝛽 the horizontal vector field 𝐹 given by +Γ 𝛾 2 𝛼𝛽 𝑑𝑡 𝑑𝑡 𝑑𝑡 ℎ 𝐹 (𝑡) = ∇ 𝑈 (𝑡) , (91) 𝑑𝑥𝛼 𝛿𝑦𝑖 𝛿𝑦𝑖 𝛿𝑦𝑗 𝑑/𝑑𝑡 +(2𝐻𝛾 +𝐹 𝛾) −𝑉𝛾 =0, 𝑖𝛼 𝑖𝛼 𝑑𝑡 𝑑𝑡 𝑖𝑗 𝛿𝑡 𝛿𝑡 ℎ 𝑑/𝑑𝑡 ∇ (85a) where is given by (80)and is the Riemannian horizontal connection. Now, we put 𝑘 𝛼 𝛽 𝑑 𝛿𝑦 𝑘 𝑑𝑥 𝑑𝑥 ( )− 𝐻 𝛾 𝛿 𝛼𝛽 𝐹 (𝑡) =𝐹 (𝑡) , (92) 𝑑𝑡 𝛿𝑡 𝑑𝑡 𝑑𝑡 𝛿𝑥𝛾 𝑑𝑥𝛼 𝛿𝑦𝑖 𝛿𝑦𝑖 𝛿𝑦𝑗 +(Γ𝑘 +𝑉 𝑘) +Γ 𝑘 =0. and by using (92), (90), (80), and (91), we deduce that 𝑖𝛼 𝛼𝑖 𝑑𝑡 𝛿𝑡 𝑖𝑗𝛿𝑡 𝛿𝑡 2 𝛾 𝛼 𝛽 (85b) 𝑑 𝑥 𝛾 𝑑𝑥 𝑑𝑥 𝐹𝛾 (𝑡) = +Γ (𝑥𝜇) . (93) 𝑑𝑡2 𝛼𝛽 𝑑𝑡 𝑑𝑡 We call (85a)the4D equations of motion in (𝑀, 𝑔, 𝐻𝑀). Wejustifythisnameasfollows.Supposethatthefollowing Thus, from (89)weobtain conditions are satisfied: 𝑑𝑥𝛼 𝛿𝑦𝑖 𝛾 𝐹𝛾 (𝑡) =−𝐹 𝛾 (𝑥,) 𝑦 . (94) 𝐻𝑖𝛼 =0, (86a) 𝑖𝛼 𝑑𝑡 𝛿𝑡 10 Advances in High Energy Physics

Then, by using90 ( ), (92), and (94) and taking into account Theorem 9. (i) A curve 𝐶 is a horizontal geodesic in that 𝐹𝑖𝛼𝛽 are skew symmetric with respect to Greek indices, (𝑀, 𝑔, 𝐻𝑀) if and only if (97b) and the following equations we infer that are satisfied:

𝑑𝑥𝛼 𝑑𝑥𝛽 𝛿𝑦𝑖 2 𝛾 𝛼 𝛽 𝛾 𝑑 𝑥 𝛾 𝑑𝑥 𝑑𝑥 𝑔 (𝐹 (𝑡) ,𝑈(𝑡)) =−𝑔𝛾𝛽𝐹𝑖𝛼 (𝑥, 𝑦) +Γ (𝑥, 𝑦) =0, 𝑑𝑡 𝑑𝑡 𝛿𝑡 𝑑𝑡2 𝛼𝛽 𝑑𝑡 𝑑𝑡 (95) (99a) 𝑑𝑥𝛼 𝑑𝑥𝛽 𝛿𝑦𝑖 ∀𝛾 ∈ {0, 1, 2, 3} , =−𝐹 (𝑥, 𝑦) =0. 𝑖𝛼𝛽 𝑑𝑡 𝑑𝑡 𝛿𝑡 𝑑𝑥𝛼 𝑑𝑥𝛽 𝐻𝑘 =0. ∀𝑘∈{4,...,3+𝑛} . (99b) Thus the extra force is perpendicular to the 4Dvelocity,which 𝛼𝛽 𝑑𝑡 𝑑𝑡 is a well-known property of the extra force in classical Kaluza- Klein theory. The above result on the extra force enables us (ii) A curve 𝐶 is a vertical geodesic in (𝑀, 𝑔, 𝐻𝑀) if and to call (89) the Lorentz force equations induced in the space only if (98b) and the following equations are satisfied: time (𝑀, 𝑔).Finally,inthisparticularcase,weseethatour equations (89)coincidewith(44) obtained by Kerner [2]. 𝑑 𝛿𝑦𝑘 𝛿𝑦𝑖 𝛿𝑦𝑗 (𝑀, 𝑔, 𝐻𝑀) ( )+Γ 𝑘 (𝑥, 𝑦) =0, 6. Motions in and Induced 𝑑𝑡 𝛿𝑡 𝑖𝑗 𝛿𝑡 𝛿𝑡 Motions on the Base Manifold 𝑀 (100a) ∀𝑘 ∈ {4,...,3+𝑛} , In this section we show that the set of geodesics in (𝑀, 𝑔, 𝐻𝑀) splits into three categories and state charac- 𝛿𝑦𝑖 𝛿𝑦𝑗 𝑉𝛾 =0, ∀𝛾∈{0, 1, 2, 3} . (100b) terizations of each category. Also, we define and study the 𝑖𝑗 𝛿𝑡 𝛿𝑡 induced motions on the base manifold. The study of geodesics of (𝑀, 𝑔, 𝐻𝑀) is based on their It is noteworthy that the equations in (99a)and(99b) positions with respect to the distributions 𝐻𝑀 and 𝑉𝑀. are related to the geometry of the horizontal distribution. To First, we see from (80)that,apartfromthe4Dvelocity𝑈(𝑡) emphasize this, we give some definitions. First, we say that given by (90), there exists an 𝑛D velocity 𝑊(𝑡) given by acurve𝐶 in 𝑀 is an autoparallel curve with respect to the ℎ ∇ 𝛿𝑦𝑖 𝜕 Riemannian horizontal connection if it is a horizontal curve 𝑊 (𝑡) = . (96) satisfying 𝛿𝑡 𝜕𝑦𝑖 ℎ 𝑑 ∇ =0, (101) Thewholestudyisdevelopedinacoordinateneighbourhood 𝑑/𝑑𝑡 𝑑𝑡 U around a point 𝑃0 ∈ 𝑀.Wesaythatacurve𝐶 passing 𝑑/𝑑𝑡 through 𝑃0 is horizontal (resp.,vertical)ifits𝑛Dvelocity where is given by (97a). Then, by direct calculations (resp., 4Dvelocity)vanishesonU.By(80)and(81)wesee using (97a)and(63a), we deduce that (101)isequivalentto (99a). Now, according to (71a)wemaysaythat that 𝐶 is a horizontal curve if and only if one of the following conditions is satisfied: 𝑘 1 𝑘 𝑘 𝐾 𝛼𝛽 = 𝐹 𝛼𝛽 −𝐻 𝛼𝛽 (102) 𝑑 𝑑𝑥𝛼 𝛿 2 = 𝛼 (97a) 𝑑𝑡 𝑑𝑡 𝛿𝑥 are local components of the second fundamental form of the 𝐻𝑀 𝐾𝑘 or distribution .Notethat 𝛼𝛽 are symmetric with respect to Greek indices if and only if 𝑀 is an integrable distribution. 𝛿𝑦𝑖 𝑑𝑦𝑖 𝑑𝑥𝛼 1 𝑖 If this is the case and 𝑛=1,then−𝐾 is just the extrinsic = +𝐿𝛼 =0. (97b) 𝛼𝛽 𝛿𝑡 𝑑𝑡 𝑑𝑡 curvature which has been intensively used in both the braneworld theory (cf. [9]) and space-time-matter theory (cf. [19]). 𝐶 Similarly, isaverticalcurveifandonlyifwehave Coming back to the general case, we say that a curve 𝐶 𝑀 𝐻𝑀 𝑑 𝛿𝑦𝑖 𝜕 in is an asymptotic line for if it is a horizontal curve = , (98a) satisfying 𝑑𝑡 𝛿𝑡 𝜕𝑦𝑖 𝑑𝑥𝛼 𝑑𝑥𝛽 or 𝐾𝑘 =0. ∀𝑘∈{4,...,3+𝑛} . (103) 𝛼𝛽 𝑑𝑡 𝑑𝑡 𝑑𝑥𝛼 =0. (98b) 𝐹𝑘 𝑑𝑡 Then taking into account the skew symmetry of 𝛼𝛽,we deduce that (103)isequivalentto(99b). Summing up this Then, by using85a ( ), (85b), (97b), and (98b)wecanstatethe discussion and using assertion (i) in Theorem 9,wecanstate following. the following characterization of horizontal geodesics. Advances in High Energy Physics 11

Corollary 10. Acurve𝐶 is a horizontal geodesic of (𝑀, 𝑔, Corollary 12. An oblique geodesic of (𝑀, 𝑔, 𝐻𝑀) is given by 𝐻𝑀) if and only if the following conditions are satisfied: the system of equations

(a) 𝐶 is an autoparallel curve with respect to the Rieman- ℎ 𝛾 𝑖 𝑗 𝛼 𝑖 nian horizontal connection ∇ on 𝐻𝑀; ℎ 𝛾 𝛿𝑦 𝛿𝑦 𝛾 𝛾 𝑑𝑥 𝛿𝑦 (∇𝑑/𝑑𝑡𝑈(𝑡)) =𝑉𝑖𝑗 −(𝐻𝑖𝛼 +𝐹𝑖𝛼 ) , (b) 𝐶 is an asymptotic line for 𝐻𝑀. 𝛿𝑡 𝛿𝑡 𝑑𝑡 𝛿𝑡

Remark 11. A similar characterization can be given for (107a) vertical geodesics in (𝑀, 𝑔, 𝐻𝑀).However,weomitithere becauseaswewillseeinthelastpartofthepaperthevertical V 𝑘 𝑑𝑥𝛼 𝑑𝑥𝛽 𝑑𝑥𝛼 𝛿𝑦𝑖 (∇ 𝑊(𝑡)) =𝐻𝑘 −𝑉 𝑘 . geodesics do not induce any motion on the base manifold. 𝑑/𝑑𝑡 𝛼𝛽 𝑑𝑡 𝛿 𝛼𝑖 𝑑𝑡 𝛿𝑡 (107b) Next, we consider the case of the integrable horizontal distribution; that is, (86b) is satisfied. Then, any leaf of 𝐻𝑀 Next,wesaythat𝐶 passing through 𝑃0 ∈ 𝑈 is a is locally given by the equations projectable curve around 𝑃0,ifits4Dvelocityisnonzero 𝑖 𝑖 around 𝑃0. Taking into account (90), we deduce that through 𝑦 =𝑐,𝑖∈{4,...,3+𝑛} , (104) the projection point 𝑄0 of 𝑃0 on 𝑀 is passing a smooth curve 𝐶 in 𝑀 givenbytheequations(see(78a)and(78b)) and it is denoted by 𝑀(𝑐). In this case, any horizontal geodesic 𝛼 𝛼 must lie in only one leaf of 𝐻𝑀,andbyTheorem 9 it is given 𝑥 =𝑥 (𝑡) ,𝑡∈[𝑎,] 𝑏 ,𝛼∈{0, 1, 2, 3} . (108) by the following system of equations: In case 𝐶 is a geodesic in (𝑀, 𝑔, 𝐻𝑀),wecall𝐶 the induced 𝑑2𝑥𝛾 𝑑𝑥𝛼 𝑑𝑥𝛽 +Γ 𝛾 (𝑥,) 𝑐 =0, motion on 𝑀 by the motion 𝐶 in 𝑀. Taking into account 2 𝛼𝛽 (105a) 𝑑𝑡 𝑑𝑡 𝑑𝑡 the definitions of the above three categories of geodesics 𝑑𝑥𝛼 𝑑𝑥𝛽 in (𝑀, 𝑔, 𝐻𝑀) we conclude that horizontal geodesics and 𝐻𝑘 (𝑥,) 𝑐 =0, (105b) 𝛼𝛽 𝑑𝑡 𝑑𝑡 oblique geodesics are projectable curves, and therefore they will induce some motions in the base manifold 𝑀.Hence, 𝛼 𝑘 𝑑𝑥 the vertical geodesics have no influence on the 4Ddynamics 𝐿 (𝑥,) 𝑐 =0, (105c) 𝛼 𝑑𝑡 in 𝑀. According to the two particular cases considered at the end of Section 5 (see (88)and(89)) we conclude for all 𝛾 ∈ {0,1,2,3} and 𝑘 ∈ {4,...,3 + 𝑛}.By(105a) that, in general, the induced motions on 𝑀 bring more we see that horizontal geodesics in (𝑀, 𝑔, 𝐻𝑀) are in fact information than both the motions from general relativity some particular geodesics of the 4D Lorentz manifolds and the solutions of the Lorentz force equations. This is (𝑀(𝑐),𝛼𝛽 𝑔 (𝑥, 𝑐)). due to the existence of extra dimensions and to the action 𝐺 𝑀 Now, we say that 𝐶 is an oblique geodesic through a point of the Lie group on . Something interesting can be 𝐻𝑀 𝑃0 if both the 4Dvelocityand𝑛Dvelocityarenonzeroat𝑃0. observed from the particular case, where is integrable By continuity, we deduce that 𝐶 is an oblique geodesic if and (see (105a), (105b), and (105c)). Let 𝐶1 and 𝐶2 be two hori- only if both 𝑈(𝑡) and 𝑊(𝑡) are nonzero for any 𝑡∈[𝑎,𝑏].It zontal geodesics in 𝑀(𝑐1) and 𝑀(𝑐2), with initial conditions 𝛼 𝑖 𝛼 𝑖 𝛼 𝑖 𝛼 𝑖 isimportanttonotethatbothvelocities𝑈(𝑡) and 𝑊(𝑡) are {(𝑥0 ,𝑐1), (𝑢 , V1)} and {(𝑥0 ,𝑐2), (𝑢 , V2)},respectively.Then involved in the equations of motion in (𝑀, 𝑔, 𝐻𝑀).First,by the induced motions 𝐶1 and 𝐶2 on 𝑀 have the same initial (𝑥𝛼,𝑢𝛼) using (90), (96), and the Riemannian adapted connection ∇= conditions 0 , but they come from different systems of ℎ V equations, and therefore they do not necessarily coincide. (∇, ∇) given by (63a), (63b), (63c), and (63d), we obtain This might be used to detect extra dimensions experimentally. ℎ 𝑈 (𝑡) ∇𝑑/𝑑𝑡 7. Conclusions 𝑑2𝑥𝛾 𝑑𝑥𝛼 𝑑𝑥𝛽 𝑑𝑥𝛼 𝛿𝑦𝑖 𝛿 (106a) ={ +Γ 𝛾 +𝐻 𝛾 } , In the present paper we obtain, for the first time in the 𝑑𝑡2 𝛼𝛽𝑑𝑡 𝑑𝑡 𝑖𝛼 𝑑𝑡 𝛿𝑡 𝛿𝑥𝛾 literature, the fully general equations of motion in a general gauge Kaluza-Klein space (cf. (85a)and(85b)). We pay V 4 ∇𝑑/𝑑𝑡𝑊 (𝑡) attention to the D equations of motion, which of course modify the well-known motions in 4D Einstein gravity. 𝑑 𝛿𝑦𝑘 𝑑𝑥𝛼 𝛿𝑦𝑖 𝛿𝑦𝑖 𝛿𝑦𝑗 𝜕 Comparing (85a)withusual4D equations of motion (88), ={ ( )+Γ 𝑘 +Γ 𝑘 } . 𝑑𝑡 𝛿𝑡 𝑖𝛼𝑑𝑡 𝛿𝑡 𝑖𝑗𝛿𝑡 𝛿𝑡 𝜕𝑦𝑘 we note two important differences. First, the local coefficients (Γ 𝛾 ,𝐻 𝛾) (106b) of the Riemannian horizontal connection 𝛼𝛽 𝑖𝛼 do depend on the extra dimensions. Then, there are some extra 4 (𝐹 𝛾,𝑉𝛾 ) Then, taking into account106a ( )and(106b)in(85a)and(85b) terms given by the Dtensorfields 𝑖𝛼 𝑖𝑗 ,which,in we can state the following. principle,canbeusedtotestthetheory.Suchtermsin 12 Advances in High Energy Physics

astrophysics might appear for usual velocities of galaxies or [15] M. Spivak, A Comprehensive Introduction to Differential Geom- clusters of galaxies. etry,vol.2,PublishorPerish,Houston,Tex,USA,1979. The method developed in the present paper opens new [16] S. Kobayashi and K. Nomizu, Foundations of Differential Geom- perspectives in the study of some other important concepts etry, vol. 1, Interscience Publishers, New York, NY, USA, 1963. from higher dimensional physical theories. Here we have in [17] P.D. B. Collins, A. D. Martin, and E. J. Squires, Particles, Physics mind an approach of the dynamics in such spaces under and Cosmology, John Wiley & Sons, New York, NY, USA, 1989. the effect of an extra force whose existence is guaranteed by [18] A. Bejancu and H. R. Farran, Foliations and Geometric Struc- the extra dimensions. In a particular case (see Section 5)we tures, Springer, Dordrecht, The Netherlands, 2006. have seen that such force does not contradict the 4D physics. [19] J. P. de Leon, “Equivalence between space-time-matter and It is an open question whether this result is still valid in brane-world theories,” Modern Physics Letters A,vol.16,no.35, case of a general gauge Kaluza-Klein space. Also, we should pp. 2291–2303, 2001. stress that the Riemannian adapted connection constructed [20] B. O’Neill, Semi-Riemannian Geometry and Applications to in Section 4 plays in this general theory the same role as the Relativity, Academic Press, New York, NY, USA, 1983. Levi-Civita connection on the 4D spacetime in the classical Kaluza-Klein theory. This connection together with theory of adapted tensor fields (see Section 3)enablesustothinkof some 4D Einstein equations induced by the (4 + 𝑛)DEinstein equations on the ambient space. All these problems deserve further studies which might show how far the concepts induced by the extra dimensions canberelatedtotherealmatter.

References

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Research Article Anisotropic Bulk Viscous String Cosmological Model in a Scalar-Tensor Theory of Gravitation

D. R. K. Reddy,1 Ch. Purnachandra Rao,1 T. Vidyasagar,2 and R. Bhuvana Vijaya3

1 Department of Engineering Mathematics, MVGR College of Engineering, Vizianagaram 535001, India 2 Miracle Educational Society Group of Institutions, Vizianagaram 535001, India 3 Department of Mathematics, JNTU College of Engineering, Anantapur 515002, India

Correspondence should be addressed to D. R. K. Reddy; reddy [email protected]

Received 28 August 2013; Accepted 15 October 2013

Academic Editor: Jose Edgar Madriz Aguilar

Copyright © 2013 D. R. K. Reddy 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.

Spatially homogeneous, anisotropic, and tilted Bianchi type-VI0 model is investigated in a new scalar-tensor theory of gravitation proposed by Saez and Ballester (1986) when the source for energy momentum tensor is a bulk viscous fluid containing one- dimensional cosmic strings. Exact solution of the highly nonlinear field equations is obtained using the following plausible physical conditions: (i) scalar expansion of the space-time which is proportional to the shear scalar, (ii) the barotropic equations of state for pressure and energy density, and (iii) a special law of variation for Hubble’s parameter proposed by Berman (1983). Some physical and kinematical properties of the model are also discussed.

1. Introduction regime appears in the theory. Also, this theory suggests a possible way to solve the “missing matter problem” in nonflat It is well known that Einstein’s general theory of relativity FRW cosmologies. The field equations given byaez S´ and has been successful in finding different models for the uni- Ballester for the combined scalar and tensor fields are verse. Friedmann-Robertson-Walker (FRW) models describe 1 1 spatially homogenous and isotropic universes. But, they have 𝑅 − 𝑔 𝑅−𝜔0𝑛 (0 0 − 𝑔 0 0,𝑘)=−8𝜋𝑇 , 𝑖𝑗 2 𝑖𝑗 ,𝑖 ,𝑗 2 𝑖𝑗 ,𝑘 𝑖𝑗 (1) higher symmetries than the real universe and therefore they are probably poor approximations for very early universe. and the scalar field 0 satisfies the following equation: The measurements of the cosmic microwave background 𝑛 ,𝑖 𝑛−1 ,𝑘 (CMB) anisotropy support the existence of anisotropies at the 20 0;𝑖 +𝑛0 0,𝑘0 =0. (2) early universe [1–4]. Hence, in order to understand the early stages of evolution of the universe, spatially homogeneous Also, we have 𝑖𝑗 anisotropic and tilted Bianchi type-Vl0 cosmological models, 𝑇 =0, (3) are studied. In the tilted cosmological models the matter does ;𝑗 not move orthogonally to the hyper surface of homogeneity. which is a consequence of the field equations1 ( )and(2). The general behavior of tilted cosmological models has been Here, 𝜔 and 𝑛 are constants. 𝑇𝑖𝑗 is the energy tensor of studied by King and Ellis [5],CollinsandEllis[6], and Bali the matter, 𝑅𝑖𝑗 is the Ricci tensor, 𝑅 is the Ricci scalar, and Sharma [7]. and comma and semicolon denote partial and covariant Saez´ and Ballester [8]havedevelopedanewscalar-tensor derivatives, respectively. Singh and Agrawal [9], Reddy and theory of gravitation in which the metric is coupled with a Venkateswara Rao [10], Reddy et al. [11], Mohanty and Sahu dimensionless scalar field in a simple manner. This coupling [12, 13], Adhav et al. [14], and Tripathy et al. [15]aresome gives a satisfactory description of the weak fields. In spite of of the authors who have studied several aspects of the Saez-´ the dimensionless character of the scalar field, an antigravity Ballester scalar-tensor theory. 2 Advances in High Energy Physics

In recent years, there has been a considerable interest The energy momentum tensor for a bulk viscous fluid in the investigation of Bianchi-type cosmological models containing one-dimensional cosmic strings is given by when the source for energy momentum tensor is a bulk 𝑇 =(𝜌+𝑝)𝑢𝑢 +𝑝𝑔 −𝜆𝑥𝑥 , viscous fluid containing one-dimensional cosmic strings. 𝑖𝑗 𝑖 𝑗 𝑖𝑗 𝑖 𝑗 (5) Bulk viscosity plays a significant role in the early evolution 𝑝=𝑝−3𝜁𝐻, of the universe and contributes to the accelerated expansion phase of the universe popularly known as the inflationary where 𝜌 is the rest energy density of the system, 𝜁(𝑡) is the phase. A review of the universe models with viscosity is given coefficient of bulk vissocity, 3𝜁𝐻 is usually known as bulk by Grøn [16]. Strings arise as a random network of stable viscous pressure, 𝐻 is Hubble’s parameter, and 𝜆 is string line-like topological defects during the phase transition in the tension density. 𝑖 𝑖 early universe. Massive closed loops of strings serve as seeds Also, 𝑢 =𝛿 is a four-velocity vector which satisfies for the formation of large structures like galaxies and cluster 4 𝑖 𝑖 𝑖 of galaxies at the early stages of evolution of the universe. A 𝑔𝑖𝑗 𝑢 𝑢𝑗 =−𝑥𝑥𝑗 =−1, 𝑢𝑥𝑖 =0. (6) good many authors have investigated about different aspects of string cosmological models either in the frame work of Here, we, also consider 𝜌, 𝑝, and 𝜆 as functions of time 𝑡 only. Einstein’s theory or in the modified theories gravity17 [ –25]. Using comoving coordinates and (5)-(6), Saez-Ballester´ Veryrecently,Reddyetal.[26] have investigated Kaluza- field equations1 ( )–(3)forthemetric(4)takethefollowing Klein bulk viscous cosmic string universe in Saez-Ballester´ form: theory while Naidu et al. [27]havediscussedthesameuni- 𝐵 𝐶 𝐵 𝐶 𝛼2 𝜔 verse in Brans-Dicke [28] scalar-tensor theory of gravitation. 44 + 44 + 4 4 + − 0𝑛02 =−8𝜋(𝑝 − 𝜆) , (7) 𝐵 𝐶 𝐵 𝐶 𝐴2 2 4 Reddy et al. [29] presented LRS Bianchi type-II universe with cosmic strings and bulk viscosity in the 𝑓(𝑅, 𝑇) theory of 𝐴 𝐶 𝐴 𝐶 𝛼2 𝜔 44 + 44 + 4 4 − − 0𝑛02 =−8𝜋𝑝, (8) gravity proposed by Harko et al. [30] while Reddy et al. [31] 𝐴 𝐶 𝐴 𝐶 𝐴2 2 4 have studied Kaluza-Klein bulk viscous cosmic string model 2 in 𝑓(𝑅, 𝑇) gravity. Naidu et al. [32] have obtained a Bianchi 𝐴 𝐵 𝐴 𝐵 𝛼 𝜔 𝑛 2 44 + 44 + 4 4 − − 0 0 =−8𝜋𝑝, (9) type-V bulk viscous string model in 𝑓(𝑅, 𝑇) gravity. Reddy 𝐴 𝐵 𝐴 𝐵 𝐴2 2 4 et al. [33] have discussed LRS Bianchi type-II bulk viscous 𝐴 𝐵 𝐵 𝐶 𝐴 𝐶 2 cosmic string cosmological model in the scale covariant 4 4 4 4 4 4 𝛼 𝜔 𝑛 2 + + − + 0 04 =8𝜋𝜌, (10) theory of gravitation formulated by Canuto et al. [34]. Also, 𝐴 𝐵 𝐵 𝐶 𝐴 𝐶 𝐴2 2 Kiran and Reddy [35] have established the nonexistence of 𝐵 𝐶 4 − 4 =0, Bianchi type-III bulk viscous string cosmological models in 𝐵 𝐶 (11) 𝑓(𝑅, 𝑇) gravity. Motivated by the above investigations of bulk viscous cos- 𝐴4 𝐵4 𝐶4 𝐴4 𝜌 +(𝜌+𝑝) ( + + )−𝜆 =0, (12) mic string Bianchi type models in modified theories of grav- 4 𝐴 𝐵 𝐶 𝐴 itation, we, in this paper, investigate spatially homogeneous, 𝐴 𝐵 𝐶 𝑛 02 anisotropic, and tilted Bianchi type-VI0 cosmological model 4 4 4 4 044 +04 ( + + )+ =0. (13) in the presence of bulk viscous fluid with one-dimensional 𝐴 𝐵 𝐶 2 0 cosmic strings. The paper is organized as follows. Section 2 Here, and in what follows, a subscript 4 after an unknown deals with the derivation of the field equations inaez- S´ function indicates differentiation with respect to 𝑡. Ballester theory in Bianchi type-VI0 space-time when the The spatial volume is given by source for energy momentum tensor is bulk viscous fluid 3 with one dimensional cosmic strings. Section 3 is devoted 𝑉=𝐴𝐵𝐶=𝑎, (14) to the solutions of the nonlinear field equations under some specific physical conditions. In Section 4,wediscusssome where 𝑎(𝑡) is the scale factor of the universe. physical and kinematical properties of the cosmological The expressions for scalar of expansion 𝜃 and shear scalar 2 model. Section 5 contains some conclusions. 𝜎 are (kinematical parameters)

𝑖 𝐴 𝐵 𝐶 𝜃=𝑢 = 4 + 4 + 4 . (15) 2. Metric and Field Equations ;𝑗 𝐴 𝐵 𝐶 Hubble parameter 𝐻 and the mean anisotropy parameter are We consider the spatially homogenous, anisotropic, and defined as tilted Bianchi type-VI0 space-time described by the following 𝐴 𝐵 𝐶 metric: 3𝐻=𝜃=3( 4 + 4 + 4 ), 𝐴 𝐵 𝐶

2 2 2 2 2 −2𝛼𝑥 2 2 2𝛼𝑥 2 3 2 𝑑𝑠 =−𝑑𝑡 +𝐴 𝑑𝑥 +𝐵 𝑒 𝑑𝑦 +𝐶 𝑒 𝑑𝑧 , (4) Δ𝐻𝑖 3𝐴ℎ = ∑( ) ,Δ𝐻𝑖 =𝐻𝑖 −𝐻,𝑖=1,2,3, (16) 𝐼=1 𝐻 𝐴 𝐵 𝐶 𝑡 𝛼 where , ,and are functions of cosmic time and is a 2𝜎2 =𝜎𝑖𝑗 𝜎 =3𝐴2 −𝐻2. constant. 𝑖𝑗 ℎ Advances in High Energy Physics 3

3. Solutions and the Model Now from (14), (18), (24), and (26), we obtain metric potentials as Equation (12) gives, on integration, 𝐴= (𝑐𝑡 +𝑑)3𝑚/(2𝑚+1)(1+𝑞) , 𝐵=𝑘𝐶, (17) (27) 𝐵=𝐶=(𝑐𝑡 +𝑑)3/(2𝑚+1)(1+𝑞). where 𝑘 is a constant of integration which can be chosen as unity without any loss of generality, so that we have Using (27)andbyasuitablechoiceofcoordinatesand 𝑑=0 𝑐=1 𝐵=𝐶. constants (i.e., taking and ), the metric (4)can (18) be written as Using (18),thefieldequations(7)–(13)reducetothefollowing 𝑑𝑠2 =−𝑑𝑡2 +𝑡6𝑚/(2𝑚+1)(1+𝑞) 𝑑𝑥2 system of independent equations: (28) +𝑡6/(2𝑚+1)(1+𝑞) [𝑒−𝛼𝑥𝑑𝑦2 +𝑒2𝛼𝑥𝑑𝑧2]. 𝐵 𝐵 2 𝛼2 𝜔 2 44 + ( 4 ) + − 0𝑛02 =−8𝜋(𝑝 −𝜆) , (19) 𝐵 𝐵 𝐴2 2 4 4. Physical Discussion of the Model 𝐴 𝐵 𝐴 𝐵 𝛼2 𝜔 44 + 44 + 4 4 − − 0𝑛02 =−8𝜋𝑝, (20) 𝐴 𝐵 𝐴 𝐵 𝐴2 2 4 Equation (28) represents the anisotropic Bianchi type-VI0 bulk viscous string cosmological model in Saez-Ballester´ 𝐴 𝐵 𝐵 2 𝛼2 𝜔 scalar-tensor theory of gravitation with the following expres- 2 4 4 + ( 4 ) − + 0𝑛02 =8𝜋𝜌, (21) 𝐴 𝐵 𝐵 𝐴2 2 4 sions for physical and kinematical parameters which are significant in the physical discussion of the cosmological 𝐴 𝐵 𝑛 02 0 +0 ( 4 +2 4 )+ 4 =0. (22) model. 44 4 𝐴 𝐵 2 0 Spatial volume is 3 3/(1+𝑞) Now, (19)–(22) are a system of four independent equa- 𝑉 =𝑡 . (29) tions in six unknowns 𝐴, 𝐵, 𝑝, 0,𝜌,and𝜆.Also,theequations are highly nonlinear. Hence, to find a determinate solution, Scalar expansion is we use the following plausible physical conditions. 3 𝜃= . (30) (i) Variation of Hubble’s parameter proposed by Berman (1 + 𝑞) 𝑡 [36] that yields constant deceleration parameter models of the universe is defined by The mean Hubble parameter is 𝑎𝑎 1 𝑞=− 44 = . 𝐻= . (31) 2 constant (23) (1 + 𝑞) 𝑡 𝑎4

2 The mean anisotropy parameter is (ii) The shear scalar 𝜎 is proportional to scalar expansion 𝜃 (𝑚−1)2 of the space-time (4)sothatwecantake 𝐴 =2 . ℎ 2 (32) 𝑚 (𝑚+2) 𝐴=𝐵 , (24) Theshearscalaris where 𝑚 =0̸ is a constant (Collins et al. [37]). 2 2 (𝑚−1) 𝜎 =3 . (33) (iii) For a barotropic fluid, the combined effects of the (𝑚+2)2(1 + 𝑞)2𝑡2 proper pressure and the bulk viscous pressure can be expressed as Energy density is 𝜔 𝑝=𝑝−3𝜁𝐻=𝜀𝜌, 8𝜋𝜌 =𝛼2𝑡−6𝑚/(1+𝑞)(𝑚+2) − 02𝑡−6/(1+𝑞) (25) 2 0 𝑝=𝜀𝜌, 𝑜 9 (2𝑚 + 1) (34) − . (1 + 𝑞)2(𝑚+2)2𝑡2 where 𝜀=𝜀𝑜 −𝛽(0≤𝑜 𝜀 ≤1)and 𝜀,𝑜 𝜀 ,and𝛽 are constants Isotropic pressure is Now, (23) admits the following solution: 2 −6𝑚/(1+𝑞)(𝑚+2) 𝜔 2 −6/(1+𝑞) 1/(1+𝑞) 8𝜋𝑝0 =𝜀 [𝛼 𝑡 − 0 𝑡 𝑎 (𝑡) = (𝑐𝑡 +𝑑) , (26) 2 0 (35) where 𝑐 =0̸ and 𝑑 are constants of integration. This equation 9 (2𝑚 + 1) implies that the condition for expansion of the universe is − ]. (1 + 𝑞)2(𝑚+2)2𝑡2 1+𝑞>0. 4 Advances in High Energy Physics

Coefficient of bulk viscosity is the scalar field vanishes at the initial epoch. Bulk viscosity in the model decreases with time leading to inflationary (𝜀 −𝜀)(1+𝑞) 8𝜋𝜁 = 0 model. The model will be useful in the discussion of structure 9 formationintheearlyuniverseinscalar-tensorcosmology. 𝜔 ×𝑡[𝛼2𝑡−6𝑚/(1+𝑞)(𝑚+2) − 02𝑡−6/(1+𝑞) 2 0 (36) References [1] D. N. Spergel, L. Verde, H. V. Peiris et al., “First-year Wilkinson 9 (2𝑚 + 1) microwave anisotropy probe (WMAP) observations: determi- − ]. 2 nation of cosmological parameters,” The Astrophysical Journal (1 + 𝑞) (𝑚+2)2𝑡2 Supplement Series,vol.148,no.1,p.175,2003. [2]D.N.Spergel,R.Bean,O.Dore´ et al., “Three-year Wilkinson The scalar field in the model is microwave anisotropy probe ( WMAP) observations: impli- 2/(𝑛+2) cations for cosmology,” The Astrophysical Journal Supplement 0 (𝑛+2) (1 + 𝑞) 0=[ 0 𝑡(1+𝑞)/(𝑞−2)] . (37) Series,vol.170,no.2,p.377,2007. 2 (𝑞 − 2) [3]J.Dunkley,E.Komatsu,M.R.Noltaetal.,“Five-yearWilkinson microwave anisotropy probe observations: likelihoods and String tension density is parameters from the WMAP data,” The Astrophysical Journal Supplement Series,vol.180,no.2,p.306,2009. 3 (𝑚−1) (𝑞 − 2) 2 −6𝑚/(1+𝑞)(𝑚+2) [4]C.L.Bennett,M.Halpern,G.Hinshawetal.,“First-year 8𝜋𝜆 = −2𝛼 𝑡 . (38) (1+𝑞)2 (𝑚+2) 𝑡2 Wilkinson microwave anisotropy probe (WMAP) observations: preliminary maps and basic results,” The Astrophysical Journal Supplement Series,vol.148,no.1,2003. Using the above results, we now discuss the behavior of the cosmological model given by (30). The result (29)shows [5] A. R. King and G. F.R. Ellis, “Tilted homogeneous cosmological models,” Communications in Mathematical Physics,vol.31,no. that the model is expanding with time since 1+𝑞>0.Itcan 3,pp.209–242,1973. be observed that the space-time given by (28)hasnoinitial 𝑡=0 𝜃 𝐻 𝜌 [6]C.B.CollinsandG.F.R.Ellis,“SingularitiesinBianchi singularity, that is, at . It can also be observed that , , , cosmologies,” Physics Reports,vol.56,no.2,pp.65–105,1979. 𝑝, 𝜆,and𝜁 decrease with time and approach zero as 𝑡→∞ [7] R. Bali and K. Sharma, “Tilted bianchi type I models with heat and they all diverge at 𝑡=0. The scalar field increases with 𝑡=0 conduction filled with disordered radiations of perfect fluid in time and at ,itvanishes. general relativity,” Astrophysics and Space Science,vol.271,no.3, 𝐴 =0̸ 𝜎2/𝜃2 =0̸ Also, since ℎ and ,theuniverseremains pp. 227–235, 2000. anisotropic throughout the evolution of the universe. It is [8] D. Saez´ and V.J. Ballester, “Asimple coupling with cosmological also interesting to note from (32)and(33)thatwhen𝑚= 2 implications,” Physics Letters A, vol. 113, no. 9, pp. 467–470, 1986. 1, 𝐴ℎ =0and 𝜎 =0andhencetheuniversebecomes [9] T. Singh and A. K. Agrawal, “Some Bianchi-type cosmological isotropic and shear free. Also, when 𝛼=0and 𝑚=1,we models in a new scalar-tensor theory,” Astrophysics and Space observe, from (38), that 𝜆=0which shows that strings do not Science,vol.182,no.2,pp.289–312,1991. surviveinthisparticularcase.Bulkviscosity,inthemodel, [10] D. R. K. Reddy and N. Venkateswara Rao, “Some cosmological decreases as 𝑡 increases which is in accordance with the well- models in scalar-tensor theory of gravitation,” Astrophysics and known fact that bulk viscosity decreases with time and leads Space Science,vol.277,no.3,pp.461–472,2001. to inflationary model [38]. [11] D. R. K. Reddy, R. L. Naidu, and V.U. M. 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P.Bhende, “Bianchi in the framework of a scalar-tensor theory of gravitation type VI string cosmological model in Saez-Ballester’s scalar- proposed by Saez´ and Ballester [8] in the presence of bulk tensor theory of gravitation,” International Journal of Theoretical viscous fluid containing one-dimensional cosmic strings. Physics,vol.46,no.12,pp.3122–3127,2007. The model is obtained using the special law of variation [15] S. K. Tripathy, S. K. Sahu, and T. R. Routray, “String cloud for Hubble’s parameter proposed by Berman [36], scalar cosmologies for Bianchi type-III models with electromagnetic expansion of the space-time which is proportional to shear field,” Astrophysics and Space Science,vol.315,no.1–4,pp.105– scalar (Collins [37]), and the barotropic equation of state for 110, 2008. pressure and energy density. It is observed that the model is [16] Ø. Grøn, “Viscous inflationary universe models,” Astrophysics expanding, nonsingular, and nonrotating. 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Research Article Scaling Relations for the Cosmological ‘‘Constant’’ in Five-Dimensional Relativity

Paul S. Wesson1 and James M. Overduin2

1 Department of Physics and Astronomy, University of Waterloo, Waterloo, ON, Canada N2L 3G1 2 Department of Physics, Astronomy and Geosciences, Towson University, Towson, MD 21252, USA

Correspondence should be addressed to James M. Overduin; [email protected]

Received 6 September 2013; Accepted 8 October 2013

Academic Editor: Jose Edgar Madriz Aguilar

Copyright © 2013 P. S. Wesson and J. M. Overduin. 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.

When the cosmological “constant” is derived from modern five-dimensional relativity, exact solutions imply that for small systems it scales in proportion to the square of the mass. However, a duality transformation implies that for large systems it scales as the inverse square of the mass.

1. Introduction [21, 22]; the vacuum and gauge fields associated with elementary particles [23, 24]; and the wave-particle duality The cosmological “constant” as it appears in Einstein’s general connected with certain Λ-dominated 5D metrics [25–27]. relativity has several puzzling aspects, and it is a serious prob- Most of our results are in Section 2. There we will reexamine lem to understand why its value as inferred from cosmology the meaning of Λ, reinterpret two classes of known solutions, is much smaller than its magnitude as implied by particle and present a new class with interesting properties. Section 3 physics. However, it has been known for a long time that is a conclusion. the cosmological “constant” appears more naturally when To streamline the work, we will often absorb the speed the world is taken to be five-dimensional1 [ ], and recently of light 𝑐,thegravitationalconstant𝐺,andthequantumof there has been intense work on the modern versions of 5D action ℎ, except in places where they are made explicit to aid relativity where the extra dimension is not compactified2 [ – in understanding. As usual, uppercase Latin letters run 𝐴, 𝐵 = 4]. The purpose of the present paper is to draw together 0, 1, 2, 3, 4 for time, space and the extra dimension. We label 4 various results in the literature which indicate that there the last 𝑥 =𝑙to avoid confusion. Lowercase Greek letters may be simple scaling relations between the values of the run 𝛼, 𝛽 = 0, 1,. 2,3 Other notation is standard. cosmological “constant” Λ and the mass 𝑚 of the system 2 concerned. Tentatively, we identify Λ∼𝑚 for small systems 2 and Λ∼1/𝑚 for large, gravitationally-dominated systems. 2. The Cosmological ‘‘Constant’’ and Possible While these relations cannot be rigorously established with Scaling Relations our present level of understanding, we believe that it is useful to point them out as guides for future research. In this section, we will examine certain subjects which involve The subjects which indicate possible relations are diverse the cosmological “constant” Λ of a spacetime and the mass and include the embedding of Λ-dominated solutions of 4D 𝑚 of a test particle moving in it. That these parameters may general relativity in the so-called 5D canonical metric [5–8]; be linked can be appreciated by noting that 5D relativity is the embeddings which lead to variable values of Λ [9–13]; the broader than Einstein’s4D theory, being in general an account equations of motion for canonical and related metrics [14– of gravity, electromagnetism, and a scalar field, where the last 20]; conformal transformations which affect Λ and possibly 𝑚 is widely believed to be concerned with how particles acquire 2 Advances in High Energy Physics

mass [2–4]. However, in 5D neither Λ nor 𝑚 are in general Compton wavelength, 𝜆𝑐 = ℎ/𝑚𝑐. Then, the average density constants. Rather, they depend on the field equations and is approximately solutions of them. It is common to take the field equations 𝑚 ℎ𝑘4 tobegivenintermsoftheRiccitensorby 𝜌 ∼ = . V 3 (3) 𝜆𝑐 𝑐 𝑅 =0 (𝐴, 𝐵 = 0, 1, 2, 3,4) . 𝐴𝐵 (1) This expression is formally identical to the one above. But the high-density vacuum is now confined to the particle, as These apparently empty 5D equations actually contain Ein- expectedifitistheproductofascalarfieldwhichcouplesto stein’s 4D equations with a finite energy-momentum tensor, matter (see below). There is no conflict between3 ( )andthe a result guaranteed by Campbell’s embedding theorem [5–7]. all-pervasive cosmological vacuum discussed above, so the This means that the 4D theory is smoothly contained in the cosmological-constant problem is avoided. 5D one and that the latter can be brought into agreement with The best way to incorporate a scalar field into physics observations at some level. is to take its potential Φ to be the extra, diagonal element In Einstein’s theory, the cosmological “constant” is usually of an extended 5D metric tensor. Then, following Kaluza introduced by adding a term Λ𝑔𝛼𝛽 to the field equations: the extra, nondiagonal elements can be identified with the potentials of electromagnetism, while the 4D block remains as a description of the 4D Einsteinian gravity. Since we are 8𝜋𝐺 here mainly interested in the scalar field, we can eliminate the 𝐺𝛼𝛽 +Λ𝑔𝛼𝛽 =( )𝑇𝛼𝛽 (𝛼,𝛽=0,1,2,3). (2) 𝑐4 electromagnetic potentials by a suitable use of the coordinate degrees of freedom of the metric, so the interval for the gravitational and scalar fields is Here, 𝑔𝛼𝛽 is the metric tensor, whose covariant derivative is zero, hence the acceptability of the noted term. We recognize 𝑑𝑆2 =𝑔 𝑑𝑥𝛼𝑑𝑥𝛽 +𝜀Φ2𝑑𝑙2. that the Λ term is a kind of a gauge term. It is sometimes 𝛼𝛽 (4) moved to the right-hand side of Einstein’s equations, where it 2 Here 𝑔𝛼𝛽 and Φ depend in general on both the coordinates canbeviewedasavacuumfluidwithdensity𝜌V =Λ𝑐/(8𝜋𝐺) 𝛾 𝑝=−𝜌𝑐2 of spacetime (𝑥 ) and the extra dimension (𝑙). The symbol and equation of state V .However,itshouldbe 𝜀=±1 recalled that the coupling constant between the left-hand (or indicates whether the extra dimension is spacelike or geometrical) side of the Einstein equations and the right- timelike, both being allowed in modern 5D theory (the extra 4 dimension does not have the physical nature of an extra time, hand (or matter) side is 8𝜋𝐺/𝑐 . This, therefore, cancels the so for 𝜀=+1there is no problem with closed timelike paths). similar coefficient of the vacuum density, leading us back Many solutions are known of the field equations1 ( )forthe to the realization that Λ is really a stand-alone parameter metric (4)[2–4]. It transpires that the easiest way to approach insofar as general relativity is concerned (this is in line with 𝐿−2 thefieldequationsisbysplittingthe4Dpartofthemetricinto the fact that its physical dimensions or units are ,matching two functions; thus, thoseoftherestofthefieldequations,whichinvolvethe second derivatives of the dimensionless metric coefficients 2 𝛾 𝛾 𝛼 𝛽 2 2 𝑑𝑆 =𝑓(𝑥 ,𝑙)𝑔𝛼𝛽 (𝑥 )𝑑𝑥 𝑑𝑥 +𝜀Φ 𝑑𝑙 . (5) with respect to the coordinates.) An implication of this is that Λ when is derived from a 5D as opposed to a 4D theory, it Here, 𝑓 is a gauge function which determines the behavior maybeconnectednotwithgravitybutwiththescalarfield,a 4 in 𝑥 ,while𝑔𝛼𝛽 depends only on the spacetime coordinates possibility we will return to later. 𝛾 𝑥 . While the form (5) provides a mathematical advantage, it The quantum vacuum, as opposed to the classical one, is involves a physical quandary: does an observer experience the frequently attributed an energy density which is calculated 𝑑𝑠2 =𝑓𝑔 𝑑𝑥𝛼𝑑𝑥𝛽 in terms of many simple harmonic oscillators and expressed whole 4D space 𝛼𝛽 or only the spacetime- 2 𝛼 𝛽 in terms of an effective value of Λ [23]. This energy density dependent subspace 𝑑𝑠 = 𝑔𝛼𝛽𝑑𝑥 𝑑𝑥 ? This question is is formally divergent, unless it is cut off by introducing akin to the argument for the so-called Jordan frame versus a minimum wavelength or equivalently a maximum wave the Einstein frame in old 4D scalar-tensor theory, where a number 𝑘. With this being understood, there results Λ∼ scalar function was applied to the 4D metric with no fifth 4 𝜌V ∼ℎ𝑘/𝑐.Ifthecutoffin𝑘 is chosen to be the inverse of dimension. It did not find a definitive answer then and has not 2 112 −3 the Planck length, this has the size of 𝜌V𝑐 ∼10 erg cm . done so today. There is a difference in the physics between the For comparison, the cosmologically determined value of Λ twoframes,butsolongasthefunction𝑓 is slowly varying, −56 −2 (∼10 cm ) corresponds to an energy density of order this will be minor. Cosmological observations may one day −8 −3 120 10 erg cm . The discrepancy, of order 10 ,isthecruxof reveal the difference between the two frames, but for now we proceed with the view that they yield complementary physics. the cosmological-constant problem. 2 An alternative interpretation of the result in the preceding An instructive case of the metric (5)has𝑓 = (𝑙/𝐿) and Φ=1 𝑔 paragraph is to imagine that the quantum vacuum does not ,where 𝛼𝛽 is any solution of the Einstein equations spread through ordinary 3D space but is concentrated in without ordinary matter but with a vacuum fluid whose particles of mass 𝑚. It is reasonable to suppose that the stuff density is measured by Λ. This is known as the (pure) canon- of each particle occupies a volume whose size is given by the ical metric. There is a large literature on this case (see8 [ ] Advances in High Energy Physics 3 for a review). It includes the Schwarzschild-de Sitter metric The second line here requires lengthy calculations for Λ and 2 for the sun and the solar system and the de Sitter metric 𝑚 [9, 10, 16–20], so the fact that we again find |Λ| ∼ 𝑚 is for the universe in its inflationary stage. It turns out that the significant. equations of motion for a test particle in the 5D metric (5)are Thethirdcasewepresentismorecomplicatedthanthe the same as those in the 4D theory, a result which enforces canonical metrics studied in the two preceding paragraphs. 𝛾 agreement with the classical tests of relativity [28, 29]. The In (5), we put 𝑓=exp(𝑙Φ/𝐿), where Φ=Φ(𝑥).Thismay dynamics may be obtained either by using the 5D geodesic beshowntosatisfythefieldequations(1), which break 2 equation or by putting 𝑑𝑆 =0in (5). The latter is based on down into sets: ten relations which determine the energy- 2 the fact that null paths in 5D with 𝑑𝑆 =0reproduce the momentum tensor 𝑇𝛼𝛽 necessary to balance Einstein’s equa- 2 timelike paths of massive particles in 4D with 𝑑𝑠 >0,as tions; four conservation-type relations which fix a 4-tensor 2 well as the paths of photons with 𝑑𝑠 =0. The definition of 𝑃𝛼𝛽 that has an associated scalar 𝑃;andonewaveequation 2 dynamics and causality by 𝑑𝑆 =0matches the null nature for the scalar field Φ. The work is tedious (see2 [ –4]; indices of the field equations (1). It turns out that the nature of the are raised and lowered using 𝑔𝛼𝛽 =𝑓𝑔𝛼𝛽 of (5)). The metric motion in the extra dimension 𝑙=𝑙(𝑠)depends on the choice andfinalresultsofthefieldequationsreadasfollows: of 𝜀 in the metric (5), as does the sign of Λ. Thus introducing 2 𝑙Φ 𝛾 𝛼 𝛽 2 𝛾 2 aconstant𝑙∗,wefind 𝑑𝑆 = ( ) 𝑔 (𝑥 )𝑑𝑥 𝑑𝑥 +𝜀Φ (𝑥 )𝑑𝑙 , exp 𝐿 𝛼𝛽 (9a) 𝑠 3 Φ 𝜀𝑔 𝑙=𝑙∗ exp (± ), Λ = + ,𝜀=−1,(6a) ,𝛼;𝛽 𝛼𝛽 𝐿 𝐿2 8𝜋𝑇 = − , (9b) 𝛼𝛽 Φ 2𝐿2 𝑖𝑠 3 𝑙=𝑙 (± ), Λ =− ,𝜀=+1. 𝛽 ∗ exp 2 (6b) 3𝛿 𝐿 𝐿 𝑃𝛽 =− 𝛼 , (9c) 𝛼 2𝐿 The second of these equations is of particular interest, because 𝜀Φ Ψ= ◻Φ + =0. it is the same as the expression for the wave function 𝐿2 (9d) Ψ∗ exp(±𝑖𝑚𝑐𝑠/ℎ) in old wave mechanics. In fact, it may be shown that the 5D geodesic equation for the (pure) canonical Here, a comma denotes the partial derivative, a semicolon 𝑙 𝛼𝛽 metric reproduces the Klein-Gordon equation with in place denotes the (4D) covariant derivative, and ◻Φ ≡ 𝑔 Φ,𝛼;𝛽 of Ψ and 1/𝐿 in place of 𝑚 [25–27]. We will meet the Klein- 𝛼𝛽 𝛼𝛽 𝛾 where 𝑔 = exp(−𝑙Φ/𝐿)𝑔 (𝑥 ). Gordon equation again below. Here, we note that the (pure) There are scalar quantities associated with the above canonical metric suggests the possibility that which are of physical interest. For example, 𝑇 can be obtained 𝑃 3 𝑚𝑐 2 by contracting (9b)andusing(9d) to simplify it; as given by |Λ| = =3( ) . 2 (7) the contraction of (9c) is a conserved quantity; and the (4D) 𝐿 ℎ Ricci or curvature scalar 𝑅 can be expressed in its general form and in the special form it takes for the metric (9a). Thus, Here, 𝑚 has been written in terms of the Compton wave- ◻Φ 2𝜀 3𝜀 length. This identification presupposes that the observer 8𝜋𝑇 = − =− , 2 2 (10a) experiences the 4D spacetime 𝑔𝛼𝛽 in (5) rather than the Φ 𝐿 𝐿 𝑓𝑔 composite spacetime defined by 𝛼𝛽. This is a subtle issue, 6 𝑃=− , as noted above, and we will return to it below. 𝐿 (10b) The next most simple case of(5)iswhenashift𝑙→ 𝜀 𝛼𝛽 2 3𝜀 (𝑙 −0 𝑙 ) is applied to the extra coordinate in the canonical 𝛼𝛽 𝑅= [𝑔,4 𝑔𝛼𝛽,4 +(𝑔 𝑔𝛼𝛽,4) ]= . (10c) metric.Thismayappeartobeclosetotrivial,butitisnot 4Φ2 𝐿2 because of the way in which the 4D Ricci scalar transforms Λ These relations and (9a), (9b), (9c), and (9d)canbegiven and with it [9, 10, 21, 22]. The equations of motion and the physical interpretations along the lines of what has been mass of a test particle for the shifted canonical metric were done for other solutions in the literature [2–4]. The energy- worked out by Ponce de Leon [16–20]. He used the principle momentum tensor (9b) shows that the source consists of the of the least action and the eikonal equation for massive and scalar field plus a term which, because of its proportionality massless particles, as opposed to the geodesic equation used 𝑔 Λ>0 to 𝛼𝛽,wouldusuallybeattributedtoavacuumfluidwith by Mashhoon et al. [14, 15]. As before, it turns out that Λ=−𝜀/(2𝐿2) for a spacelike extra dimension (𝜀=−1)andΛ<0for a cosmological constant . The conserved tensor 𝑃𝛽 =0 timelike one (𝜀=+1). The metric and the expressions for Λ of (9c)obeys 𝛼;𝛽 bythefieldequations,anditsscalar and 𝑚 are 𝑃 has in other works been linked to the rest mass of a test particle, which here is 𝑚=1/𝐿[25–27]. This is confirmed by 2 (𝑙 − 𝑙 ) the wave equation (9d), which deserves some discussion. 𝑑𝑆2 = 0 𝑔 (𝑥𝛾)𝑑𝑥𝛼𝑑𝑥𝛽 +𝜀𝑑𝑙2, (8a) 𝐿2 𝛼𝛽 Relation (9d), depending on the choice for 𝜀=±1,is known either as the Helmholtz equation or as the Klein- 3 𝑙 2 𝑚𝑐 2 |Λ| = ( ) =3( ) . Gordon equation. Many solutions to it are known with 2 (8b) 𝐿 𝑙−𝑙0 ℎ applications to problems in atomic physics (like diffusion) 4 Advances in High Energy Physics and elementary particle physics (like wave mechanics). There interactions of particles with gravity. What is, however, of are different modes of behavior, depending on whether 𝜀= the latter interaction? It is natural to wonder if there is not −1 or 𝜀=+1, which correspond to the monotonic and a complementary relation to what we have found above, but oscillatory modes (6a)and(6b) of the canonical metric for macroscopic gravity-dominated systems. discussed before. For the present metric (9a), the scalar field This subject will require detailed analysis, but some Φ may be real or complex, and in the latter case for 𝜀=+1the comments of a preliminary type may be made. It is useful, in wave equation (9d) is identical to the Klein-Gordon equation, this context, to reconsider the traditional distinction between with 𝐿=ℎ/𝑚𝑐beingtheComptonwavelengthofthetest inertial mass (𝑚𝑖)andgravitationalmass(𝑚𝑔). The Kaluza- particle. This is similar to a previous interpretation based on Klein equation involves the former, so our previous consider- 2 the shifted-canonical metric25 [ –27]. (In (9d), the oscillation ations have concerned 𝐿=ℎ/𝑚𝑖𝑐 and Λ∼𝑚𝑖 as the scaling is in Φ, whereas in the corresponding equation of [25–27]itis relation for the cosmological “constant”. It is clear that this in 𝑙, because in the canonical metric it is presumed that Φ=1, scaling rule cannot persist to arbitrarily large masses without so the physical behavior is moved from one parameter to the leading to excessive curvature of empty spacetime (𝑅=4Λ). other. In (9a), the problem can be made explicitly complex by We expect, therefore, that it might pass over to some other writing exp(𝑖𝑙Φ/𝐿), if so desired.) scaling relation Λ=Λ(𝑚𝑔) for large gravitational masses. It may seem strange that a classical field theory yields an Such a relation is actually implicit in certain works on equation typical of (old) quantum theory, but it should be the canonical metric [2–4, 8–22]. We recall that the 4D recalled that the wave equation (9d) comes from the field part of the 5D canonical metric involves the combination (𝑙/𝐿)𝑑𝑠 equation 𝑅44 =0, which does not exist in standard general .Thiscanbecomparedtotheelementofactionfor relativity. In fact, the present interpretation of the metric classical mechanics, 𝑚𝑑𝑠. Two obvious identifications are (9a) is fully consistent with the approach to noncompactified possible: 𝐿∼1/𝑚and 𝑙∼𝑚. We have already explored 5D relativity known as Space-Time-Matter theory, where the former, so attention is focused on the latter. In fact the 𝑥4 ∼𝑙∼𝑚 matter on the macroscopic and microscopic scales is taken possibility 𝑔 has been considered, mainly in tobetheresultofhigher-dimensionalgeometry[2–4]. By relation to cosmology, and cannot be ruled out [2–4, 11– contrast, while the metric (9a)mayresemblethewarpmetric 13]. As regards Λ, we note that its behavior depends on the of the alternative approach to 5D relativity known as the coordinate frame experienced by an observer (see above). Membrane theory, in that approach, the “Φ” in the exponent To illustrate this, consider a vacuum spacetime with the ofthe4Dpartofthemetricisabsent,whichmeansthat (pure) canonical metric, where the 4D part of the interval is 𝛼 𝛽 2 𝛾 𝛼 𝛽 the metric does not satisfy the field equations in the simple 𝑔𝛼𝛽𝑑𝑥 𝑑𝑥 = (𝑙/𝐿) 𝑔𝛼𝛽(𝑥 )𝑑𝑥 𝑑𝑥 .TheeffectivevalueofΛ form (1). Our view is that (9a), (9b), (9c), and (9d)show can be obtained from either the Ricci scalar or the Einstein the wave-mechanical properties of matter. The scalars (10a), tensor and depends on whether the observer experiences only 𝛾 𝛾 (10b), and (10c)associatedwiththesolutionbearthisout. 𝑔𝛼𝛽(𝑥 ) or the full 𝑔𝛼𝛽(𝑥 ,𝑙).Theresultsare,respectively, With conventional units restored, the conserved quantity 𝑃 is 3/𝐿2 3/𝑙2 𝐿=𝜆 = and , and both appear in the literature. Let us inversely proportional to the Compton wavelength 𝑐 take the second alternative and combine it with the physical ℎ/𝑚𝑐 of a test particle moving in the spacetime. Viewed as a identification 𝑙∼𝑚𝑔 noted above. The obvious parameter wave which couples to matter, we expect that the Compton 𝐺𝑚 /𝑐2 wavelength should be consistent with the radius of curvature with which to geometrize the gravitational mass is 𝑔 , 𝑅 theSchwarzschildradius.Thenwefindthatintotal,Λ= of the spacetime, and this is confirmed by the relation for . 2 −2 2 3(𝐺𝑚 /𝑐 ) Lastly,wenotethattheaforementionedrelationΛ=−𝜀/(2𝐿) 𝑔 . That is, for large gravitationally dominated 2 systems we expect Λ to scale as the inverse square of the mass. shows once again that Λ∼𝑚. The argument of the preceding paragraph is tentative, This relation is common to the three classes of solutions but can be checked by combining it with the more detailed examined above, which come from the different choices of the 𝑓(𝑥𝛾,𝑙) 𝑓 = (𝑙/𝐿)2 work concerning the inertial mass which went before. For gauge function in (5). They involve which simplicity, we take the numerical factors to be those of the 𝑓=[(𝑙−𝑙)/𝐿]2 gives (6a), (6b), 0 which gives (8a), (8b), and canonical case and consider a proton (inertial mass 𝑚𝑝)and 𝑓= (𝑙Φ/𝐿) exp which gives (9a), (9b), (9c), and (9d). By com- theobservablepartoftheuniverse(gravitationalmass𝑀𝑢). parison with known physics, we infer that the constant length Then, the scaling relations for the cosmological “constant” 𝐿 is inversely proportional to the particle mass 𝑚,whichwe 2 2 2 read Λ 𝑝 =3(𝑚𝑝𝑐/ℎ) and Λ 𝑢 =3(𝑐/𝐺𝑀𝑢) .Thesecanbe can write in terms of the Compton wavelength as 𝐿∼𝜆𝑐 = combinedtogivethenumberofbaryonsintheobservable ℎ/𝑚𝑐. The exponential gauge, in particular, leads from the universe as field equation 𝑅44 =0to the Klein-Gordon equation, which is the basic relation in wave mechanics (its low-energy limit 𝑀 3𝑐3/𝐺ℎ 𝑁= 𝑢 = . is the Schrodinger¨ equation which underlies the physics of 1/2 (11) 𝑚𝑝 (Λ Λ ) the hydrogen atom). The implication is that the scalar field of 𝑝 𝑢 5D relativity is connected to the mass of a particle, and with the phenomenon of wave-particle duality ([25–27]; the Klein- In this, we substitute the quantum field theoretical value of Λ =2×1026 −2 Λ = Gordon equation can have real or complex forms). These 𝑝 cm and the cosmological value of 𝑢 −56 −2 2 comments are in accordance with the longstanding view that 3×10 cm (obtained from Λ 𝑢 =8𝜋𝐺𝜌V/𝑐 ,where 𝜌 =Ω𝜌 𝜌 3𝐻2/8𝜋𝐺 theories of Kaluza-Klein type provide a way of unifying the V Λ crit and crit= 0 ,togetherwithcurrent Advances in High Energy Physics 5

−1 −1 observational data giving 𝐻0 =74±2km s Mpc and References 80 ΩΛ = 0.73 ± 0.04). The result is 𝑁=10,whichisin agreement with conventional estimates. [1] V. A. Rubakov and M. E. Shaposhnikov, “Extra space-time dimensions: towards a solution to the cosmological constant The two scaling relations considered in this section problem,” Physics Letters B,vol.125,no.2-3,pp.139–143,1983. should be regarded as complementary. The first is better based [2]J.M.OverduinandP.S.Wesson,“Kaluza-Kleingravity,”Physics on theory than the second, since it can be examined in three Report,vol.283,no.5-6,pp.303–378,1997. gauges rather than one. However, there is in principle no con- [3] P. S. Wesson, Five-Dimensional Relativity, World Scientific, flict between them, and in practice we expect the first to grade 2 Singapore, 2006. into the second. The Λ∼𝑚 rule should be dominant on −13 2 [4] P. S. Wesson, “The geometrical unification of gravity with its the particle scale (∼10 cm), and the Λ∼1/𝑚 rule should 28 source,” General Relativity and Gravitation,vol.40,no.6,pp. be dominant on the cosmological scale (∼10 cm). Theoret- 1353–1365, 2008. ically, they should be comparable on scales of order 100 km, [5] S. Rippl, C. Romero, and R. Tavakol, “D-dimensional gravity which in practice is rough where quantum interactions and from (D + 1) dimensions,” Classical and Quantum Gravity,vol. solid-state forces are superseded by the effects of gravity. 12, no. 10, pp. 2411–2421, 1995. [6] C. Romero, R. Tavakol, and R. Zalaletdinov, “The embedding of general relativity in five dimensions,” General Relativity and 3. Conclusion Gravitation,vol.28,no.3,pp.365–376,1996. [7]J.E.Lidsey,C.Romero,R.Tavakol,andS.Rippl,“On We have seen in the preceding section that the cosmological applications of Campbell’s embedding theorem,” Classical and constant is open to reinterpretation, particularly as a measure Quantum Gravity,vol.14,no.4,pp.865–879,1997. of the energy density of the vacuum fields of particles. It is [8] P. S. Wesson, “The embedding of general relativity in five- somewhat better understood in cosmology, where its theoret- dimensional canonical space: a short history and a review of ical status is relatively clear in Einstein’s equations, and where recent physical progress,” http://arxiv.org/abs/1011.0214. observations establish its approximate value. Unfortunately, [9] B. Mashhoon and P. S. Wesson, “Gauge-dependent cosmologi- there is a very large mismatch between the microscopic and cal ‘constant’,” Classical and Quantum Gravity,vol.21,no.14,pp. the macroscopic domains. This can in principle be alleviated 3611–3620, 2004. by using a five-dimensional theory, of the kind indicated by [10] B. Mashhoon and P.Wesson, “An embedding for general relativ- unification, where in general Λ isnotauniversalconstantbut ity and its implications for new physics,” General Relativity and a variable. This is shown most clearly by the 5D canonical Gravitation,vol.39,no.9,pp.1403–1412,2007. gauge, where Λ scales according to the size of the potential [11] J. M. Overduin, “Nonsingular models with a variable cosmolog- well (𝐿) or the value of the extra coordinate (𝑙). Since the ical term,” Astrophysical Journal Letters,vol.517,no.1,pp.L1–L4, mass (𝑚) of a test particle also depends on these parameters, 1999. we are tentatively led to suggest scaling relations of the form [12] J. M. Overduin, P. S. Wesson, and B. Mashhoon, “Decaying Λ = Λ(𝑚). For the canonical gauge in its pure and shifted dark energy in higher-dimensional gravity,” Astronomy and forms, the scaling relation is for small 𝑚 andhastheform Astrophysics,vol.473,no.3,pp.727–731,2007. 2 Λ∼𝑚. This is also the form derived from the exponential [13] P. S. Wesson, B. Mashhoon, and J. M. Overduin, “Cosmology gauge, which has the advantage of showing that the extra with decaying dark energy and cosmological quot ‘constant’,” International Journal of Modern Physics D,vol.17,no.13-14,pp. field equation resembles the Klein-Gordon equation of wave 2527–2533, 2008. mechanics, implying that the scalar field is connected with [14] B. Mashhoon, H. Liu, and P. S. Wesson, “Particle masses and the particle mass. cosmological constant in Kaluza-Klein theory,” Physics Letters Thereis,however,analternativeinterpretationofthe B,vol.331,pp.305–312,1994. canonical gauge and others like it. The 4D part of this involves [15] B. Mashhoon, P. Wesson, and H. 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Ponce de Leon, “Invariant definition of rest mass and universe. This result (11) agrees with conventional estimates, dynamics of particles in 4d from bulk geodesics in brane-world which may be seen as provisional support for the idea that the and non-compact kaluza-klein theories,” International Journal cosmological “constant” varies with scale. of Modern Physics D,vol.12,p.757,2003. [19] J. Ponce de Leon, “ The principle of least action for test particles in a four-dimensional spacetime embedded in 5D,” Modern Acknowledgment Physics Letters A,vol.23,p.249,2008. [20] J. Ponce de Leon, “Embeddings for 4D Einstein equations with a Thanks for comments are due to members of the Space-Time- cosmological constant,” Gravitation and Cosmology,vol.14,pp. Matter group (5Dstm.org). 241–247, 2008. 6 Advances in High Energy Physics

[21] R. M. Wald, General Relativity, University of Chicago Press, Chicago, Ill, USA, 1984. [22] P. S. Wesson, “Particle masses and the cosmological “Constant” in five dimensions,” http://arxiv.org/abs/1111.4698. [23] S. M. Carroll, Spacetime and Geometry, Addison-Wesley, San Francisco, Calif, USA, 2004. [24] Y. Hosotani, “Gauge-higgs unification: stable higgs bosons as cold dark matter,” International Journal of Modern Physics A,vol. 25,p.5068,2010. [25] P.S. Wesson, “General relativity and quantum mechanics in five dimensions,” Physics Letters B,vol.701,no.4,pp.379–383,2011. [26] P. S. Wesson, “The cosmological “constant” and quantization in five dimensions,” Physics Letters B,vol.706,no.1,pp.1–5,2011. [27] P. S. Wesson, “Vacuum waves,” Physics Letters B,vol.722,pp. 1–4, 2013. [28] H. Liu and P. S. Wesson, “The motion of a spinning object in a higher-dimensional spacetime,” Classical and Quantum Gravity,vol.13,no.8,p.2311,1996. [29] J. M. Overduin, R. D. Everett, and P.S. Wesson, “Constraints on Kaluza-Klein gravity from Gravity Probe B,” General Relativity and Gravitation,vol.45,no.9,pp.1723–1731,2013. Hindawi Publishing Corporation Advances in High Energy Physics Volume 2013, Article ID 968016, 12 pages http://dx.doi.org/10.1155/2013/968016

Research Article Gauss-Bonnet Braneworld Cosmology with Modified Induced Gravity on the Brane

Kourosh Nozari,1 Faeze Kiani,2 and Narges Rashidi2

1 Center for Excellence in Astronomy and Astrophysics (CEAAI-RIAAM), P.O. Box 55134-441, Maragha, Iran 2 Department of Physics, Faculty of Basic Sciences, University of Mazandaran, P.O. Box 47416-95447, Babolsar, Iran

Correspondence should be addressed to Kourosh Nozari; [email protected]

Received 21 May 2013; Accepted 19 August 2013

Academic Editor: Jose Edgar Madriz Aguilar

Copyright © 2013 Kourosh Nozari 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 analyze the background cosmology for an extension of the DGP gravity with Gauss-Bonnet term in the bulk and 𝑓(𝑅) gravity on the brane. We investigate implications of this setup on the late-time cosmic history. Within a dynamical system approach, we study cosmological dynamics of this setup focusing on the role played by curvature effects. Finally, we constrain the parameters of the model by confrontation with recent observational data.

1. Introduction theinducedgravityonthebrane(Wecallthe𝑓(𝑅) term on the brane the modified induced gravity since this braneworld One of the most significant astronomical observations in scenario is an extension of the DGP model. In the DGP the last decade is the accelerated expansion of the universe model, the gravity is induced on the brane through inter- [1–14]. One way to explain this accelerating phase of the action of the bulk graviton with loops of matter on the universe expansion is invoking a dark energy component brane. So the phrase “induced gravity” is coming from the in the matter sector of the Einstein field equations [15–31]. DGP character of our model.) on the brane. This can be However, it is possible also to modify the geometric part of achieved by treating the induced gravity in the framework of the field equations to achieve this goal32 [ –45]. In the spirit 𝑓(𝑅) gravity [61–66]. This extension can be considered as a of the second viewpoint, the braneworld model proposed manifestation of the scalar-tensor gravity on the brane since by Dvali, Gabadadze, and Porrati (DGP) provides a natural 𝑓(𝑅) gravity can be reconstructed as a general relativity plus explanation of late-time accelerated expansion in its self- ascalarfield[32–45]. Some features of this extension such accelerating branch of the solutions [46–51]. Unfortunately, as self-acceleration in the normal branch of the scenario are the self-accelerating branch of this scenario suffers from studied recently [61–68]. Here, we generalize this viewpoint ghost instabilities [52, 53], and, therefore, it is desirable to the case that the Gauss-Bonnet curvature effect is also to invoke other possibilities in this braneworld setup. An taken into account. We consider a DGP-inspired braneworld amazing feature of the DGP setup is that the normal branch of model where the induced gravity on the brane is modified this scenario, which is not self-accelerating, has the capability in the spirit of 𝑓(𝑅) gravity, and the bulk action contains the to realize phantom-like behavior without introducing any Gauss-Bonnet term to incorporate higher-order curvature phantom field neither in the bulk nor on the brane54 [ –60]. effects. Our motivation is to study possible influences of the By the phantom-like behavior one means that an effective curvature corrections on the cosmological dynamics on the energy density, which is positive, grows with time and its normal branch of the DGP setup. We analyze the background 𝜔 =𝑝/𝜌 equation of state parameter ( eff eff eff) stays always cosmology and possible realization of the phantom-like less than −1. The phantom-like prescription breaks down if behavior in this setup. By introducing a curvature fluid that this effective energy density becomes negative. An interesting plays the role of dark energy component, we show that this extension of the DGP setup is possible modification of model realizes phantom-like behavior on the normal branch 2 Advances in High Energy Physics of the scenario in some subspaces of the model parameter where space, without appealing to phantom fields neither in the bulk 1 1 Λ(5) 2E 𝐺(5) nor on the brane and by respecting the null energy condition 𝑈=− ± √1+4𝛼( + 0 ), in the phantom-like phase of expansion. We show also that in 4𝛼 4𝛼 6 𝑎4 this setup there is smooth crossing of the phantom divide line (6) Λ(4) by the equation of state parameter and the universe transits 𝜆≡ . smoothly from a quintessence-like phase to a phantom-like 8𝜋𝐺(4) phase. We present a detailed analysis of cosmological dynam- E ics in this setup within a dynamical system approach in order 0 is referred to hypothetically as the mass of the bulk black to reveal some yet unknown features of these kinds of models hole, and the corresponding term is called the bulk radiation in their phase space. Finally, confrontation of our model term. Note that, when one adopts the positive sign, the above with recent observational data leads us to some constraint on equation can be reduced to the generalized DGP model as 𝛼→0 model parameters. , but the branch with negative sign cannot be reduced to the generalized DGP model in this regime. Therefore, we just consider the plus sign of the above equation [81–84]. 2. The Setup We note that, depending on the choice of the bulk space, 2.1. Gauss-Bonnet Braneworld with Induced Gravity on the the brane FRW equations are different (see [85] for details). Brane. The action of a GBIG (the Gauss-Bonnet term in the The bulk space in the present model is a 5-dimensional AdS bulk and induced gravity on the brane) braneworld scenario black hole. In which follows, we assume that there is no cosmological constant on the brane or in the bulk; that is, canbewrittenasfollows[69–80]: (4) (5) Λ =Λ =0.Also,weignorethebulkradiationterm 𝑆=𝑆 +𝑆 , bulk brane (1) sinceitdecaysveryfastintheearlystagesoftheevolution (note, however, that this term is important when one treats where, by definition, cosmological perturbations on the brane). So the Friedmann 1 5 (5) (5) equation in this case reduces to the following form: 𝑆 = ∫ 𝑑 𝑥√−𝑔R [ −2Λ +𝛼L ], (2) bulk 16𝜋𝐺(5) GB 2 8𝜋𝐺(4) 4 8𝛼 2 [𝐻2 − 𝜌] = [1 + 𝐻2] 𝐻2. with 2 (7) 3 𝑟𝑐 3 2 L =(R(5)) −4R(5) R(5)AB + R(5) R(5)ABCD, GB AB ABCD It has been shown that it is possible to realize the phantom- 1 (3) like behavior in this setup without introducing any phantom 𝑆 = ∫ 𝑑4𝑥√−𝑞R [ −2Λ(4)]. brane 16𝜋𝐺(4) matter on the brane [86–91]. In which follows, we generalize this setup to the case that induced gravity on the brane is (5) (4) 𝑓(𝑅) 𝐺 is the 5D Newton constant in the bulk, and 𝐺 is modified in the spirit of gravity, and we explore the L cosmological dynamics of this extended braneworld scenario. the corresponding 4D quantity on the brane. GB is the Gauss-Bonnet term, and 𝛼 is a constant with dimension of 2 [length] . 𝑞 is the induced metric on the brane. We choose the 2.2. Modified GBIG Gravity. In this subsection, we firstly coordinate of the extra dimension to be 𝑦,sothatthebrane formulate a GBIG scenario that induced gravity on the brane is located at 𝑦=0.TheDGPcrossoverdistance,whichis acquires a modification in the spirit of 𝑓(𝑅) gravity. To defined as obtain the generalized Friedmann equation of this model, we proceed as follows. Firstly, the Friedmann equation for pure 𝐺(5) 𝜅2 𝑟 = = 5 , DGP scenario is given as follows [49–51, 92, 93]: 𝑐 (4) 2 (4) 𝐺 2𝜅4 E Λ 𝑘 [ ] 𝜖√𝐻2 − 0 − 5 + has the dimension of length and will appear in our forth- 4 2 coming equations. We note that this scenario is UV/IR 𝑎 6 𝑎 (8) complete in some sense, since it contains both the Gauss- 𝑘 8𝜋𝐺(4) Bonnet term as a string-inspired modification of the UV =𝑟 [(𝐻2 + )− (𝜌 + 𝜆)], 𝑐 2 (ultraviolet) sector and the induced gravity as IR (infra- 𝑎 3 red) modification to the general relativity. The cosmological 𝜖=±1 dynamics of this GBIG scenario is described fully by the where is corresponding to two possible embeddings following Friedmann equation [81–84]: of the brane in the bulk. Considering a Minkowski bulk with Λ 5 =0and by setting E0 =0with a tensionless brane (𝜆=0), 2 (𝑘=0) 𝑘 8𝜋𝐺(4) (𝜌 + 𝜆) for a flat brane ,wefindthat [𝐻2 + − ] 𝑎2 3 (4) 2 8𝜋𝐺 𝐻 𝐻 = 𝜌± . (9) 3 𝑟 4 8𝛼 𝑘 𝑈 2 𝑘 𝑐 = [1 + (𝐻2 + + )] (𝐻2 + −𝑈), 2 2 2 𝑟𝑐 3 𝑎 2 𝑎 The normal branch of the scenario is corresponding to (5) the minus sign on the right-hand side of this equation. Advances in High Energy Physics 3

The second term on the right is the source of the phantom- scale 𝑟𝑐 which is the horizon for the self-accelerated branch, like behavior on the normal branch: the key feature of this the theory reduces to a scalar-tensor model, with the scalar phase is that the brane is extrinsically curved in such a way sector (brane bending mode) being described by a simple that shortcuts through the bulk allow gravity to screen the Galileon self-interaction [94]as effects of the brane energy-momentum contents at Hubble 𝐻∼𝑟−1 (𝜕𝜋)2 ◻𝜋 parameters 𝑐 , and this is not the case for the self- L =𝜋◻𝜋− , (13) accelerating phase [54–60]. 𝜋 Λ3 In the next step, we incorporate possible modification of the induced gravity by inclusion of a 𝑓(𝑅) term on the brane. which, in spite of the presence of higher derivatives, propa- This extension can be considered as a manifestation of the gatesasinglehealthydegreeoffreedom.Theghostontheself- scalar-tensor gravity on the brane. In this case, we find the accelerated branch arises merely due to the fact that 𝜋 gets following generalized Friedmann equation (see, e.g., [61–68, a nontrivial profile and the kinetic term for its perturbation 92, 93]): flips the sign on the background. A similar argument can be applied in the present work to overcome the ghost instabilities 𝑀𝐺(5) Λ 𝑘 in this extended braneworld setup. 𝜖√𝐻2 − − 5 + 𝜌 𝑎4 6 𝑎2 In which follows, we assume that the energy density onthebraneisduetocolddarkmatter(CDM)with𝜌𝑚 = (4) 3 𝑘 8𝜋𝐺 𝜌0𝑚(1 + 𝑧) . We can rewrite the Friedmann equation in =𝑟 [(𝐻2 + )𝑓󸀠 (𝑅) − 𝑐 𝑎2 3 terms of observational parameters such as the redshift 𝑧 and dimensionless energy densities Ω𝑖 as follows: 1 ×[𝜌+𝜆+( [𝑅𝑓󸀠 (𝑅) −𝑓󸀠 (𝑅)]−3𝐻𝑅𝑓̇ 󸀠󸀠 (𝑅))] ], 2 Ω𝑚 3 3(1+𝑤 ) 2 𝐸 = (1+𝑧) +Ω (1+𝑧) curv 𝑓󸀠 (𝑅) curv (10) (14) √Ω where a prime marks derivative with respect to 𝑅.Inthe 𝑟𝑐 −2 [1 + Ω 𝐸2 (𝑧)]𝐸(𝑧) , third step, we need the GBIG Friedmann equation in the 𝑓󸀠 (𝑅) 𝛼 absence of any modification of the induced gravity on the 𝑓(𝑅) brane, that is, without term on the brane. This has been where obtained in the previous subsection; see (5). Now, we have all 𝐻 prerequisites to obtain the Friedmann equation of our GBIG- 𝐸 (𝑧) ≡ , modified gravity scenario. The comparison between previous 𝐻0 equations gives this Friedmann equation of cosmological 𝜅2 8 dynamics as follows (we note that this equation can be Ω ≡ 4 𝜌 ,Ω≡ 𝛼𝐻2, 𝑚 2 0𝑚 𝛼 0 derived using the generalized junction conditions on the 3𝐻0 3 brane straightforwardly; see [81–84]): (15) 1 𝜅2 2 2 4 𝜅 𝜅 1 8 Ω𝑟 ≡ ,Ω≡ 𝜌0 , 2 4 4 2 𝑐 4𝑟2𝐻2 curv 3𝐻2 curv 𝐻 = 𝜌+ 𝜌 ± (1 + 𝛼𝐻 )𝐻, (11) 𝑐 0 0 3𝑓󸀠 (𝑅) 3 curv 𝑟 𝑓󸀠 (𝑅) 3 𝑐 𝑝 𝑤 = curv . where we have defined hypothetically the following energy curv 𝜌 curv density corresponding to curvature effect: 𝑝 1 1 curv, the hypothetical pressure of the curvature effect, can be 𝜌 = ( [𝑅𝑓󸀠 (𝑅) −𝑓(𝑅)]−3𝐻𝑅𝑓̇ 󸀠󸀠 (𝑅)). obtained by the following equation of continuity: curv 𝑓󸀠 (𝑅) 2 (12) 𝑅𝑓̇ 󸀠󸀠 (𝑅) Note that, to obtain this relation, E0, Λ 5, 𝜆,and𝑘 are set equal 𝜌̇ +3𝐻(𝜌 +𝑝 + ) curv curv curv 󸀠 2 to zero. From now on, we restrict our attention to the normal 𝑟𝑐[𝑓 (𝑅)] branch of the scenario, that is, the minus sign in (11)because (16) 3𝐻2Ω 𝑅𝑓̇ 󸀠󸀠 (𝑅) there are no ghost instabilities in this branch only if the DGP = 0 𝑚 𝑎−3. character of the model is considered. [𝑓󸀠 (𝑅)]2 Note, however, that although we refer to the normal (ghost-free) branch of the DGP model (in the sense that for 𝑓(𝑅) =𝑅 One can obtain a constraint on the cosmological parameters theobtainedsolutionsreducetothisbranch)as of the model at 𝑧=0as follows: an indication of the ghost-free property of the considered solutions, it is not a priori guaranteed that, on the obtained Ω𝑚 =1−Ω +2√Ω𝑟 (1+Ω𝛼) . (17) de Sitter backgrounds which generalize the normal DGP curv 𝑐 branch,theghostdoesnotreappear.Infact,theghoston 󸀠 the self-accelerated branch of the DGP model is entirely the Note that we have used the normalization 𝑓 (𝑅)|𝑧=0 =1in problemofthedeSitterbackground.Withinthecrossover this relation which is observationally a viable assumption. 4 Advances in High Energy Physics

𝑤 >−1 3. Cosmological Dynamics in (1) curv : in this case, curvature fluid plays the the Modified GBIG Scenario role of quintessence component; then, the minimum value happens asymptotically at 𝑧=−1,andwewill Now, we study cosmological dynamics in this setup. To solve obtain 𝜌min =0.Inthissituation,wecandefinethree the Friedmann equation for the normal branch of this sce- possible regimes: a high-energy regime with 𝜌1 < 𝜌,a nario, it is convenient (following the papers by Bohamdi- limiting regime with 𝜌1 = 𝜌, and a low-energy regime Lopez in [86–90]) to introduce the dimensionless variables with 𝜌1 > 𝜌. In each of these cases, depending on the as follows: sign of 𝑁, there are different solutions86 [ –91]. 8 𝛼 𝑤 <−1 𝐻≡ 𝐻=2Ω √Ω 𝐸 (𝑧) , (2) curv :inthiscase,thecurvaturefluidplaysthe 󸀠 𝛼 𝑟 3 𝑟𝑐𝑓 (𝑅) role of a phantom component (we will investigate its phantom-like behavior in the next section), and the 32 𝜅2𝛼2 𝜌 𝑧 = 0.18 𝜌≡ 5 (𝜌 +𝑓󸀠 (𝑅) 𝜌 ) minimum value of happens at .Sowefind 3 𝑚 curv 𝜌 27 [𝑟 𝑓󸀠 (𝑅)] the min as follows: 𝑐 (18) 2 3(1+𝑤 ) 2 3 3(1+𝑤 ) 𝜌 =4Ω Ω [0.43 + Ω (1 + 0.18) curv ] , =4Ω Ω [Ω (1+𝑧) +Ω (1+𝑧) curv ], min 𝛼 𝑟𝑐 curv (25) 𝛼 𝑟 𝑚 curv 8 𝛼 Ω = 0.26 𝑤 wherewehaveset 𝑚 .Wenotethat curv is not 𝑏≡ =4Ω𝛼Ω𝑟. 3 󸀠 2 [𝑟𝑐𝑓 (𝑅)] constant, and, as we will show, it evolves from quintessence to phantom phase. We note also that the value of redshift An effective crossover distance which appeared on the right- when 𝜌min occurs (i.e., 𝑧 = 0.18) has no dependence on the 𝑤 𝜌 < 𝜌 hand side of these relations can be defined as follows, values of curv. Here, we treat only the case 1 min with details. When this condition is satisfied, the function N is 𝑟≡𝑟𝑓󸀠 (𝑅) , 𝑐 (19) positive, and there is a unique solution for expansion of the 2 2 brane described by and, by definition, Ω𝑟 ≡1/4𝑟𝐻0 . Then, the Friedmann equationcanberewrittenas 1 √ 𝜂 𝐻1 = [2 1−3𝑏cosh ( )] , (26) 3 2 3 3 𝐻 + 𝐻 +𝑏𝐻 − 𝜌 =0. (20) where 𝜂 is defined as The number of real roots of this equation is determined by N 𝑆 thesignofthediscriminantfunction defined as cosh (𝜂) = . √−𝑄3 (27) 3 2 N =𝑄 +𝑆 , (21) We note that this condition provides a constraint on the 𝑄 𝑆 where and are defined as dimensionless parameters Ω𝑖 as follows: 1 1 1 2 𝑄= (𝑏 − ), − [𝑏 − [1 + √(1−3𝑏)3]] 3 3 3𝑏 9 (22) 1 1 1 (28) 3(1+𝑤 ) 𝑆= 𝑏+ 𝜌− , <Ω [0.43 + Ω (1 + 0.18) curv ]. 6 2 27 𝛼 curv respectively. We can rewrite N as Figure 1 shows the phase space of the above relation. In 3(1+𝑤 ) Ψ ≡ 0.43+Ω (1+0.18) curv 1 this figure, we have defined curv . N = (𝜌−𝜌 )(𝜌−𝜌 ), 4 1 2 (23) The relation (28) is fulfilled for upper region (region II) of this figure. In this case, there are three possible regimes as was where mentioned above. A point that should be emphasized here is 1 2 the fact that, in the presence of modified induced gravity on 𝜌 =− {𝑏 − [1 + √(1−3𝑏)3 ]} , 1 3 9 the brane, the solution of the generalized Friedmann equation (11) is actually rather involved due to simultaneous presence (24) ̇ ̈ 1 2 of 𝐻, 𝐻,and𝐻. A thorough analysis of this problem is 𝜌 =− {𝑏 − [1 − √(1−3𝑏)3 ]} . 2 3 9 outofthescopeofthisstudy,buttherearesomeattempts (suchascosmography)inthisdirectiontoconstructan In which follows, we consider just the real and positive roots operational framework to treat this problem; see, for instance of the Friedmann equation (20). For 0<𝑏<1/4, 𝜌1 >0and [41].Here,wehavetriedtofindasolutionof(11)byusing 𝜌2 <0.Then,thenumberofrealrootsofthecubicFriedmann the discriminant function N,theresultofwhichisgivenby equation depends on the minimum energy density of the (26). However, we note that a complete analysis is needed, braneandthesituationof𝜌1 relative to this minimum. Since, for instance, in the framework of cosmography of brane 𝑓(𝑅) in our setup, curvature effect plays the role of the dark energy gravity [95]. component on the brane, we can consider two different To investigate cosmology described by solution (26), we regimes to determine the minimum value of 𝜌 as follows. rewrite the original Friedmann equation in the following Advances in High Energy Physics 5

1.0 4

0.8 3

0.6 Ψ 2 II eff 𝜌 0.4 1 I 0.2

0 0.05 0.10 0.15 0.20 0.25 0 b=4Ω𝛼Ωr 2 46810 z Figure 1: The upper region (region II) of this figure is corresponding (Ω ,Ω ,Ω ,𝑤 ) −0.2 to the set 𝑟 𝛼 curv curv that fulfils inequality28 ( ). Figure 2: Variation of effective energy density versus the redshift for a specific 𝑓(𝑅) model described as (33)with𝑛 = 0.25. form in order to create a general relativistic description of our model:

2 (𝜅4) 𝐻 8 The effective energy density shows a phantom-like behav- 𝐻2 = eff 𝜌 +𝜌 − (1 + 𝛼𝐻2), (29) 𝑚 curv 󸀠 ior; that is, it increases with cosmic time. This is a nec- 3 𝑟𝑐𝑓 (𝑅) 3 essary condition to have phantom-like behavior, but it is 𝜌 notsufficient:weshouldcheckstatusofthedeceleration where curv isdefinedin(12). Comparing this relation with the Friedmann equation in GR parameter and also equation of state parameter. In a general relativistic description of our model, one can rewrite the 𝜅2 energy conservation equation as follows: 𝐻2 = 4 (𝜌 +𝜌 ), (30) 3 𝑚 eff 𝜌̇ +3𝐻(1+𝜔 ) 𝜌 =0 eff eff eff (34) we obtain an effective energy density 2 which leads to the following relation for 1+𝜔 : 𝜅 𝐻 8 2 eff 4 𝜌 =𝜌 − (1 + 𝛼𝐻 ), (31) 3 eff curv 𝑟 𝑓󸀠 (𝑅) 3 𝑐 𝜌̇ 1+𝜔 =− eff . eff 3𝐻𝜌 (35) which can be rewritten as follows: eff

2√Ω𝑟 3(1+𝑤 ) 𝑐 2 Figure 3 shows variation of the effective equation of state 𝜌 =Ω (1+𝑧) curv − (1 + Ω 𝐸 )𝐸(𝑧) . eff curv 𝑓󸀠 (𝑅) 𝛼 parameter versus the redshift for a specific 𝑓(𝑅) model 𝑛 = 0.25 (32) described as (33)with .Theeffectiveequationof 1+𝜔 <0 state parameter transits to the phantom phase eff , 𝜌 but there is no smooth crossing of the phantom divide line in The dependence of eff on the redshift depends itself on the regimes introduced above and the form of 𝑓(𝑅) function. this setup. We note that adopting other general ansatz, such 𝜌 as the Hu-Sawicki model [96] Figure 2 shows the variation of eff versus the redshift for

𝑛 𝑛 2 𝑅 𝛼 (𝑅/𝑅 ) 𝑓 (𝑅) =𝑅−(𝑛−1) 𝜁 ( ) (33) 0 𝑐 𝜁2 𝑓 (𝑅) =𝑅−𝑅𝑐 𝑛 (36) 1+𝛽0(𝑅/𝑅𝑐) with 𝑛 = 0.25.Thisvalueof𝑛 lies well in the range of observationally acceptable values of 𝑛 from solar-system tests (where both 𝛼 and 𝑅𝑐 are free positive parameters), does [32–45]. (We note, however, that the key issue with regards notchangethisresultinourframework.Thedeceleration to passing solar-system tests is not the value of 𝑛 but the parameter defined as 󸀠 value of 𝑓 (𝑅) today. In fact, experimental data tell us that 󸀠 −6 󸀠 𝑓 (𝑅) − 1 < 10 ,when𝑓 (𝑅) is parameterized to be exactly 𝐻̇ 𝑞≡−1− (37) 1 in the far past.) 𝐻2 6 Advances in High Energy Physics

0.10 5

0.05 3

eff 0.00 2 5.41 5.43 5.45 5.47 5.49 1+𝜔 z 1 −0.05

−1 0 12345 z −0.10 −1

Figure 3: Variation of effective equation of state parameter versus Figure 4: Variation of the deceleration parameter 𝑞 versus 𝑧 for the redshift for a specific 𝑓(𝑅) model described as (33)with𝑛 = 0.25. 𝑛 = 0.25. The transition from deceleration to the acceleration phase occurs at 𝑧≈2.

0.1 takes the following form in our setup:

3 󸀠̇ 󸀠 −1 0 12345 Ω𝑚(1+𝑧) ((3/2) −(𝑓 (𝑅) /𝑓 (𝑅))(𝐻0/𝐸 (𝑧))) 𝑞=−1− z 𝑓󸀠 (𝑅) 𝐸2 (𝑧) + √Ω (1 + 3Ω 𝐸2 (𝑧))𝐸(𝑧) −0.1 𝑟𝑐 𝛼 Ḣ 𝑓󸀠̇(𝑅) 1 −(√Ω (1 + Ω 𝐸2 (𝑧)) −0.2 𝑟𝑐 󸀠 𝛼 𝑓 (𝑅) 𝐻0 −0.3 3 󸀠 3(1+𝜔 ) − Ω 𝑓 (𝑅)(1+𝑧) 𝑐 (1 + 𝜔 )) 2 curv 𝑐 −0.4 −1 ×(𝑓󸀠 (𝑅) 𝐸2 (𝑧) + √Ω (1 + 3Ω 𝐸2 (𝑧))𝐸(𝑧)) . 𝑟𝑐 𝛼 (38) −0.5

Figure 5: In this model, 𝐻̇is always negative, and, therefore, there 𝐻 𝑓(𝑅) In this relation, and are defined as (26)and(33). is no superacceleration or big-rip singularity. Figure 4 shows variation of 𝑞 versus the redshift for 𝑛 = 0.25. The universe enters the accelerated phase of expansion at 𝑧≃2.Anotherimportantissuetobeinvestigatedinthis setup is the big-rip singularity. To avoid super acceleration on 4. A Dynamical System Viewpoint the brane, it is necessary to show that Hubble rate decreases Up to this point, we have shown that there are effective quan- as the brane expands and that there is no big-rip singularity tities that create an effective phantom-like behavior on the in the future. Figure 5 shows variation of 𝐻̇versus 𝑧.We brane. In this respect, one can define a potential related to see that in this model 𝐻<0̇ always, and, therefore, there theeffectivephantomscalarfield𝜙 as follows [97]: is no super-acceleration or future big-rip singularity in this eff setup. All of the previous considerations show that this model 3 2 3 󸀠 𝑉 (𝑧) Ω (1+𝑧) 1 𝑑(𝐸 −(Ω𝑚(1+𝑧) /𝑓 (𝑅))) accounts for realization of the phantom-like behavior without eff =𝐸2 − 𝑚 + , 2 󸀠 3 introducing a phantom field neither on the brane nor in the 3𝐻0 𝑓 (𝑅) 2 𝑑 ln (1+𝑧) bulk. Nevertheless, we have to check the status of the null energy condition in this setup. Figure 6 shows the variation 2 3 󸀠 𝜙 (𝑧) 𝑑𝑧 𝑑(𝐸 −(Ω𝑚(1+𝑧) /𝑓 (𝑅))) of (𝜌 + 𝑝) versus the redshift. We see that this condition is eff =−∫ √ . tot 3 fulfilled at least in some subspaces of the phantom-like region √3 (1+𝑧) 𝐸 𝑑 ln (1+𝑧) of the model parameter space. (39) Advances in High Energy Physics 7

4 This constraint means that the phase space can be defined by 2 2 󸀠 −6 the relation 𝑥 +𝑦 −2𝜐≥1,since𝑓 (𝑅) − 1 < 10 by solar- system constraints and 𝜐 is a positive quantity. Introducing the new time variable 𝜏=ln 𝑎 and eliminating 𝜒 and 𝐸,we 3 obtain the following autonomous system: 1 𝑥󸀠 =𝑥(𝑞− ), tot 2 2 󸀠 𝑦 3 2

(𝜌 + p) 𝑦 = [ (1 + 𝑤 )(1+𝑥 )−(𝑞+1) 𝑦2 −1 2 curv (42) 1 1 ×[1+𝑥2 −2𝜐+ (2+𝜐) (𝑥2 +𝑦2 −1)]], 2 𝜐󸀠 =𝜐(𝑞+1). 𝜏 𝑞= −1 −0.5 0 0.5 1 Here, primes denote differentiation with respect to ,and −𝑎𝑎/̈ 𝑎2̇ z stands for the deceleration parameter 3 Figure 6: The status of the null energy condition in this model. 𝑞+1= (𝑥2 +[1+𝑤 (1 − 𝑥2)] 𝑦2) 2 curv 3 1 9 ×(4𝑥2 +8𝑦2 −𝑥2𝑦2 − 𝑦4 + 𝜐𝑦4 − 𝜐𝑦2 𝑉 = (43) We note that in principle these equations can lead to eff 2 4 4 𝑉 (𝜙 ) eff eff , but in practice the inversion cannot be performed 1 19 −1 analytically. Now, we define the following normalized expan- − 𝜐𝑥2 − 𝜐−5) . sion variables [98–101]: 2 2 To study cosmological evolution in the dynamical system √Ω 𝑓󸀠 (𝑅) √Ω𝑚 curv approach, it is necessary to find fixed (or critical) points of 𝑥= ,𝑦= , ∗ ∗ ∗ 3/2 (3/2)(1+𝑤 ) (𝑥 ,𝑦 ,𝜐 ) 𝑎 𝐸 𝑎 curv 𝐸 the model that are denoted by .Thesepointsare (40) achieved by fulfillment of the following condition: √Ω 𝑟𝑐 𝜐= ,𝜒=√Ω 𝐸. 𝑔𝑖 (𝑥∗,𝑦∗,𝜐∗) =0, 𝐸 𝛼 (44)

With these definitions, the Friedmann equation14 ( )takesthe where following form: 󸀠𝑖 𝑖 ∗ ∗ ∗ 𝑥 =𝑔 (𝑥 ,𝑦 ,𝜐 ) , (45) 2 2 2 󸀠 𝑥 +𝑦 −2𝜐(1+𝜒 )=𝑓 (𝑅) . (41) where in Table 1 𝑥𝐵 and 𝑦𝐵 are as follows:

𝑦 = √𝑥2 (𝑤 −1)+(1+𝑤 ), 𝐵 𝐵 curv curv (46) −17𝑤 +4+6𝑤2 + √297𝑤2 − 120𝑤 − 180𝑤3 + 16 + 36𝑤4 𝑥 = √ curv curv curv curv curv curv . 𝐵 2(𝑤 −1) curv

A part of dynamical system analysis of this model is summa- in this model. Existence of a stable de Sitter point and an rized in Table 1. unstable matter-dominated phase (in addition to radiation- The critical points 𝐴 and 𝐵 demonstrate the early- dominated era) in the universe expansion history is required time, matter-dominated epoch which leads to a positive for cosmological viability of any cosmological model. In order deceleration parameter. Points 𝐶 and 𝐷 which are phases to investigate the stability of these points, one can obtain the with vanishing matter character, that is, Ω𝑚 =0, can explain eigenvalues of these points separately. Based on Table 2,in the positively accelerated phase of the universe expansion order for point 𝐴 to be an unstable point, it is necessary to 𝑤 ≥ −6.88 𝐸 𝑤 <1/4 𝐴 for curv .Criticalcurve also demonstrates a have curv . Therefore, the point as a saddle point positively accelerated phase for all values of the equation of agrees with what we have shown in Figure 7.Now,thestability state parameter of the curvature fluid. Critical curve 𝐹,which of the positively accelerated phases of the model depends on 𝑤 =−1 exists only for the case curv , is a de Sitter phase whether the curvature fluid plays the role of a quintessence 8 Advances in High Energy Physics

Table 1: Location and deceleration parameter of the critical points. The location of point 𝐵 and the deceleration factor of points 𝐶 and 𝐷 are 𝑤 𝐹 𝑤 =−1 dependent on the equation of state parameter of the perfect fluid, curv.Thecriticalcurve exists just for curv . Name 𝑥𝑦𝜐 𝑞 𝐴 √15/3 0 0 1/2

𝐵𝑥𝐵 𝑦𝐵 01/2 𝐶 0 1/2 0 −(1.17 + 0.17𝑤 ) curv 𝐷 0 7/2 0 −(1.13 + 0.13𝑤 ) curv 𝐸00𝜐∗ −1 𝐹0𝑦∗ 𝜐∗ −1

𝜍(𝑤 ) 𝜉(𝑤 ) 𝜁(𝑤 ) 𝑤 Table 2: Eigenvalues and the stability regions of the critical points. curv , curv ,and curv are complicated functions of curv.The 𝐹 𝑤 =−1 critical curve for which curv isastabledeSitterphase. Name Eigenvalue Stable region 𝐴−9,−1/2,1−4𝑤 𝑤 >1/4 curv curv 𝐵𝜍(𝑤), 𝜉(𝑤 ), 𝜁(𝑤 ) curv curv curv — 𝐶 −(1 + 0.1𝑤 ), −(1 + 𝑤 ), −(2 + 0.17𝑤 )𝑤>−1 curv curv curv curv 𝐷 −(1 + 0.1𝑤 ), −(1 + 𝑤 ), −(2 + 0.13𝑤 )𝑤>−1 curv curv curv curv 𝐸 −3/2, −3/2(1 +𝑤 ), −2 𝑤 >−1 curv curv 𝐹 −3/2, 0, −2 𝑤 =−1 curv

1 the curve can be determined by the nonzero eigenvalues, because near this critical point there are essentially no dynamics along the critical curve [102]. We have plotted the 𝑥−𝑦 𝑤 =−1 phase space of this model in subspace with curv . 0.5 As we see, point 𝐴 is a saddle point, and curve 𝐹 is a stable de Sitter curve.

y 0 5. Confrontation with Recent Curve F Point A Observational Data In this section, we use the combined data from Planck + WMAP+highL+lensing+BAO[103] to confront our −0.5 model with recent observation. In this way, we obtain some constraints on the model parameters, especially, the Gauss- Bonnet curvature contribution. For this purpose, we consider Ω Ω the relation between curv and 𝑚 in the background of the −1 Ω mentioned observational data. We suppose that curv plays −1 012theroleofdarkenergyinthissetup.Figure 8 shows the result Ω ∼10−4 x of our numerical study. In this model, with 𝑟𝑐 (see [104], e.g.,), Ω𝛼 is constrained as follows: Figure 7: The phase subspace 𝑥−𝑦that the curvature fluid plays the role of a cosmological constant. This model is cosmologically viable 𝑤 = −0.5, 0.008 < Ω < 0.011, sincethereareanunstablematter-dominatedpointandastablede curv 𝛼 Sitter phase. 𝑤 = −0.92, 0.01 < Ω < 0.055, curv 𝛼 (47) 𝑤 = −1.05, 0.012 < Ω < 0.073. curv 𝛼 scalar field or not. Points 𝐶, 𝐷,and𝐸 of Table 2 are stable 𝑤 >−1 Ω phases of this model if curv , whereas if the curvature On the other hand, if we consider the eff defined as 𝐹 Ω =(𝜅2/3𝐻2)𝜌 fluid plays the role of a cosmological constant, the point eff 4 0 eff as our main parameter, the result will will be a stable de Sitter phase. We note that, generally, if a be as shown in Figure 9. In this case, we have the following nonlinear system has a critical curve, the Jacobian matrix of constraint on Ω𝛼: the linearized system at a critical point on the curve (line in our 2-dimensional subspace) has a zero eigenvalue with Ω = 0.7, 0.0043 < Ω < 0.036, curv 𝛼 an associated eigenvector tangent to the critical curve at (48) Ω = 0.5, 0.0048 < Ω < 0.0081. the chosen point. The stability of a specific critical point on curv 𝛼 Advances in High Energy Physics 9

0.72 curvature effects are taken into account by incorporation of theGauss-Bonnetterminthebulkaction.Itiswellknown that the normal branch of the DGP braneworld scenario, 0.71 which is not self-accelerating, has the potential to realize phantom-like behavior without introducing any phantom fields neither on the brane nor in the bulk. Our motivation 0.70 here to study this extension of the DGP setup is to explore possible influences of the curvature corrections, especially,

curv the modified induced gravity, on the cosmological dynamics Ω of the normal branch of the DGP setup. In this regard, 0.69 cosmological dynamics of this scenario as an alternative Ω𝛼 for dark energy proposal is studied, and the effects of the Ω𝛼 curvature corrections on the phantom-like dynamics of the 0.68 modelareinvestigated.Thecompleteanalysisofthegener- alized Friedmann equation needs a cosmographic viewpoint Ω𝛼 to 𝑓(𝑅) gravity,butherewehavetriedtofindaspecial 0.67 solution of this generalized equation via the discriminant function method. In our framework, effective energy density 0.28 0.29 0.30 0.31 0.32 0.33 attributed to the curvature plays the role of effective dark Ω m energy density. In other words, we defined a curvature fluid 𝜔c = −1.05 with varying equation of state parameter that incorporates in 𝜔 = −0.92 c the definition of effective dark energy density. The equation 𝜔 = −0.5 c of state parameter of this curvature fluid is evolving, and Ω Ω the effective dark energy equation of state parameter has Figure 8: curv versus 𝑚 in the background of Planck + WMAP + highL+lensing+BAOjointdata. transition from quintessence to the phantom phase in a nonsmooth manner. We have considered a cosmologically viable (Hu-Sawicki) ansatz for 𝑓(𝑅) gravityonthebraneto 0.72 havemorepracticalresults.Wehaveshownthatthismodel mimics the phantom-like behavior on the normal branch of the scenario in some subspaces of the model parameter 0.71 space without introduction of any phantom matter neither in the bulk nor on the brane. At the same time, the null energy condition is respected in the phantom-like phase of 0.70 the model parameter space. There is no super-acceleration or big-rip singularity in this setup. Incorporation of the

eff curvature effects both in the bulk (via the Gauss-Bonnet Ω 0.69 term) and on the brane (via modified induced gravity) results in the facility that curvature fluid plays the role of dark energy component. On the other hand, this extension 0.68 allows the model to mimic the phantom-like prescription in relatively wider range of redshifts in comparison to the case

Ω𝛼 that induced gravity is not modified. This effective phantom- Ω like behavior permits us to study cosmological dynamics of 0.67 𝛼 this setup from a dynamical system viewpoint. This analysis has been performed with details, and its consequences are 0.28 0.29 0.30 0.31 0.32 0.33 explained. The detailed dynamical system analysis of this Ω m setupismoreinvolvedrelativetothecasethatthereareno Ωcurv = 0.7 curvature effects. Wehave shown that, with suitable condition Ωcurv = 0.5 on equation of state parameter of curvature fluid, there are an unstable matter era and a stable de Sitter phase in this scenario Ω Ω Figure 9: eff versus 𝑚 in the background of Planck + WMAP + leading to the conclusion that this model is cosmologically highL+lensing+BAOjointdata. viable.Wehaveconstrainedourmodelbasedontherecent observational data from joint Planck + WMAP + high L+ lensing + BAO data sets. In this way, some constraints on 6. Summary and Conclusion Gauss-Bonnet coupling contribution are presented. We note that no big-rip singularity is present in this model since In this paper, we have constructed a DGP-inspired bra- the Gauss-Bonnet contribution to this model is essentially neworld scenario where induced gravity on the brane is a stringy, quantum gravity effect that prevents the big-rip modified in the spirit of 𝑓(𝑅) gravity, and higher-order singularity (see, e.g., [105] for details). 10 Advances in High Energy Physics

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Research Article Noether Current of the Surface Term of Einstein-Hilbert Action, Virasoro Algebra, and Entropy

Bibhas Ranjan Majhi

IUCAA, Ganeshkhind, Pune University Campus, Post Bag 4, Pune 411 007, India

Correspondence should be addressed to Bibhas Ranjan Majhi; [email protected]

Received 30 April 2013; Accepted 28 June 2013

Academic Editor: Mauricio Bellini

Copyright © 2013 Bibhas Ranjan Majhi. 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.

A derivation of Noether current from the surface term of Einstein-Hilbert action is given. We show that the corresponding charge, calculatedonthehorizon,isrelatedtotheBekenstein-Hawkingentropy.Alsousingthecharge,thesameentropyisfoundbasedon the Virasoro algebra and Cardy formula approach. In this approach, the relevant diffeomorphisms are found by imposing a very simple physical argument: diffeomorphisms keep the horizon structure invariant. This complements similar earlier results (Majhi and Padmanabhan (2012)) (arXiv:1204.1422) obtained from York-Gibbons-Hawking surface term. Finally we discuss the technical simplicities and improvements over the earlier attempts and also various important physical implications.

1. Introduction (EH) action, and the analysis was on shell; that is, equation of motion has been used explicitly. Later an off shell analysis The thermodynamic properties of horizon arise from the and a generalization to Lanczos-Lovelock gravity have been combination of the general theory of relativity and the quan- presented in [54]. tum field theory. This was first observed in the case of black Earlier [55], based on the Virasoro algebra approach, holes [1, 2]. Now it is evident that it is much more general, we showed that the entropy can also be obtained from the and a local Rindler observer can attribute temperature and Noether current corresponding to the York-Gibbons-Hawk- entropy to the null surfaces in the context of the emergent ing surface term. But it is not clear if the same can be achieved paradigm of gravity [3–6]. Such a generality might provide us from the surface term of the Einstein-Hilbert action since a deeper insight into the quantum nature of the spacetime. So they are not exactly identical. So it is necessary to investigate far several attempts have been made to know the microscopic thisissueinthelightofVirasoroalgebracontext,particullary origin of the entropy, but every method has its own merits becausebothsurfacetermsleadtothesameentropyonthe and demerits. Among others, Carlip made an attempt [7, 8] horizon. This will complement our earlier work55 [ ]. in the context of Virasoro algebra to illuminate this aspect In this paper we will use the Noether current associated which is basically the generalisation of the method by Brown with the surface term of EH action. Before going into the andHenneaux[9]. In brief, in this method one first defines motivations for taking the surface term only, let us first a bracket among the Noether charges and calculate it for highlight some peculiar facts of EH action which are essential certain diffeomorphisms, chosen by some physical consider- for the present purpose. ations. It turns out that the algebra is identical to the Virasoro algebra. The central charge and the zero mode eigenvalue (i) It is an unavoidable fact that to obtain the equation of the Fourier modes of the charge are then automatically of motion in the Lagrangian formalism one has to identified in which after substituting in the Cardy formula impose some extra prescription, like adding extra [10–12] one finds the Bekenstein-Hawking entropy. (For a boundary term (in this case York-Gibbons-Hawking complete list of works which lead to further development of term). This is because the action contains second- this method, see [13–53].) In all the previous attempts, the order derivative of metric tensor 𝑔𝑎𝑏. But unfortu- Noether current was taken in relation to the Einstein-Hilbert nately the choice of the surface term is not unique. 2 Advances in High Energy Physics

Thisisquitedifferentfromotherwell-knownfield description which was originally obtained only for the black theories. hole horizon. Moreover, all the quantities will be observer (ii) The EH action can be separated into two terms: one dependent. In this paper we will proceed as follows. First a detailed contains the squires of the Christoffel connections 𝑎 ΓΓ ΓΓ derivation of the Noether current for a diffeomorphism 𝑥 → (i.e., it is in - structure) and the other one 𝑥𝑎 +𝜉𝑎 𝐿 containsthetotalderivationofΓ (𝜕Γ-𝜕Γ structure). , corresponding to sur, will be given. This is important 𝐿 𝐿 because it has not been done earlier, and therefore the We will call them quad and sur,respectively.Inter- estingly, Einstein’sequation of motion can be obtained properties of the current have not been explored. Here we will 𝐿 show that the corresponding charge 𝑄[𝜉],calculatedonthe solely from quad by using the usual variation princi- 𝜉𝑎 ple where no additional prescription is required [56]. null surface for to be Killing, yields exactly one-quarter of the horizon area after multiplying it by 2𝜋/𝜅,where𝜅 is the (iii) The most important one is that these two terms are acceleration of the observer or the surface gravity in the case related by an algebraic relation, usually known as of a black hole. Next, a definition of the bracket among the holographic relation [57, 58]. chargeswillbegiven.Thiswillbedonebytakingvariation 𝑎 𝑎 of the charge 𝑄[𝜉1] for another transformation 𝑥 →𝑥+ Interestingly, all the previous features happened to be com- 𝑎 𝜉 .Finally,weneedtocalculateallthesequantitiesfora monevenfortheLanczos-Lovelocktheory[59]. For a recent 2 particular diffeomorphism. To identify the relevant diffeo- review in this direction, see [60]. morphisms from which the algebra has to be constructed, Although an extensive study on the Noether current of following our earlier work [55], we use the criterion that the gravity has been done starting from Wald et al. [61–63], dis- diffeomorphism should leave the near horizon form of the cussion on the current derived from 𝐿 is still lacking. To sur metric invariant in some nonsingular coordinate system. This motivate why one should be interested, let us summarise the will lead to a set of diffeomorphism vectors for which the already observed facts as follows. Fourier components of the bracket among the charges will (i) It is expected that the entropy is associated with the be exactly similar to Virasoro algebra. It is then very easy to degrees of freedom around or on the relevant null identify the zero mode eigenvalue and the central extension. surfaceratherthanthebulkgeometryofspacetime. Substitution of all these values in the Cardy formula [10– 12] will yield exactly the Bekenstein-Hawking entropy [1, 2]. (ii) This surface term calculated on the Rindler horizon A similar analysis was done in [53] based on the Noether gives exactly the Bekenstein-Hawking entropy [56]. 𝐿 current corresponding to bulk [64–66]. In this calculation, to (iii) Extremization of the surface term with respect to the obtain the correct value of the entropy, a particular boundary diffeomorphism parameter whose norm is a constant condition(DirichletorNeumann)wasused.Butitsphysical leads to Einstein’s equation [57]. significance is not well understood. (iv) Another interesting fact is that, in a small region Before going into the main calculation, let us summarize around an event, EH action reduces to a pure surface themainfeaturesofthepresentanalysis. term when evaluated in the Riemann normal coordinates. (i) The first is the technical aspect. To obtain the correct entropy, in most of the earlier works, one had to All these indicate that either the bulk and the surface terms either shift the zero mode eigenvalue8 [ ]orchoosea 𝑎 are duplicating all the information or the actual dynamics is parameter contained in the Fourier modes of 𝜉 as the stored in surface term rather than in bulk term. To illuminate surface gravity 𝜅 [53]orboth[54]. Here we will show more on this issue, one needs to study every aspect of the that none of the ad hoc prescriptions will be required. surface term. In this paper, we will discuss the Noether realization of the (ii) The important one is the simplicity of the criterion surface term of the EH action; particularly we will examine (near horizon structure of the metric remains invariant if the Noether current represents the Virasoro algebra for a in some nonsingular coordinate system)tofindthe certain class of diffeomorphisms. This is necessary to have a relevant diffeomorphisms for which we obtain the deeper understanding of the role of the surface term in the Virasoro algebra. This was first introduced by55 us[ ] gravity. Also it will give a further insight into the earlier in this context. The significance of this choice is claim: the actual information of the gravity is stored in the that the full set of diffeomorphism symmetry of the surface. To do this explicitly, we will consider the form of theory is now reduced to a subset which respects the the metric close to the null surface in the local Rindler frame existence of horizon in a given coordinate system. around some event. This is given by the Rindler metric. The Hence it may happen that some of the original reasons for choosing such metric are as follows. According gauge degrees of freedom (which could have been to equivalence principle, gravity can be mimicked by an eliminated by certain diffeomorphisms which are accelerated observer, and an uniformly accelerated frame will now disallowed) now being effectively upgraded to have Rindler metric. Apart from that, it is a relevant frame physical degrees of freedom as far as a particular class for an observer sitting very near to the black hole horizon. of observers are concerned. So all the thermodynamic Hence any thermodynamic feature of the null surface can quantities, attributed to the horizon, become observer be attributed by this metric, and it provides a general dependent. Advances in High Energy Physics 3

(iii) In the present analysis we will not need any use of Therefore, boundary condition like Dirichlet or Neumann to 𝛿𝐿 =𝜕 [𝑆𝑎𝜕 (√−𝑔𝜉𝑏)+√−𝑔𝛿𝑆𝑎]. obtain the exact form of the entropy. sur 𝑎 𝑏 (6) 𝐿 (iv) Since our analysis will be completely based on sur, 𝑎 𝐿 On the other hand, since 𝑆 is not a tensor, its variation cannot where no information about is needed, it will 𝑎 bulk be expressed by simple Lie derivative. To find 𝛿𝑆 we will use definitely illuminate the emergent paradigm of grav- 󸀠𝑎 󸀠 the general definition3 ( ). Let us first calculate 𝑆 (𝑥 ). Under ity, particularly the holographic aspects in the action. 󸀠𝑎 𝑎 𝑎 the change 𝑥 =𝑥 +𝜉 we have We will discuss later more on different aspects and significance of our results. 𝜕𝑥󸀠𝑎 =𝛿𝑎 +𝜕𝜉𝑎, The organization of the paper is as follows. In Section 2, 𝜕𝑥𝑏 𝑏 𝑏 𝐿 the derivation of the Noether current for the sur will be (7) 𝜕𝑥𝑏 presented explicitly. Next we will give the definition of the =𝛿𝑏 −𝜕 𝜉𝑏. bracketamongthechargesandtherelevantdiffeomorphisms 𝜕𝑥󸀠𝑎 𝑎 𝑎 based on the invariance of horizon structure criterion. Section 4 willbedevotedtoshowthattheFouriermodeofthe Here we considered infinitesimal change, and so the terms bracket is exactly like the Virasoro algebra which by the Cardy from 𝜕𝜉𝜕𝜉 have been ignored. This will be followed in later formula will lead to Bekenstein-Hawking entropy. Finally, we analysis. Hence, will conclude. 󸀠𝑎 󸀠 𝑎 𝑎 𝑑 𝑎 𝑑 𝑑 𝑎 𝑎 Γ𝑏𝑐 (𝑥 )=Γ𝑏𝑐 −Γ𝑏𝑑𝜕𝑐𝜉 −Γ𝑐𝑑𝜕𝑏𝜉 +Γ𝑏𝑐𝜕𝑑𝜉 −𝜕𝑏𝜕𝑐𝜉 , 2. Derivation of Noether Current from the 𝑔󸀠𝑏𝑘 (𝑥󸀠)=𝑔𝑏𝑘 +𝑔𝑏𝑓 𝜕 𝜉𝑘 +𝑔𝑘𝑓 𝜕 𝜉𝑏, Surface Term of Einstein-Hilbert Action 𝑓 𝑓 󸀠𝑎𝑑 󸀠 𝑎𝑑 In this section, a detailed derivation of the Noether current 𝑄𝑐𝑘 (𝑥 )=𝑄𝑐𝑘 . and the potential corresponding to the surface term of EH (8) action will be presented. Then we will calculate the charge on the Rindler horizon. 󸀠𝑎 󸀠 󸀠𝑎𝑑 󸀠 󸀠𝑏𝑘 󸀠 󸀠𝑐 󸀠 Substitution of these in 𝑆 (𝑥 )=2𝑄𝑐𝑘 (𝑥 )𝑔 (𝑥 )Γ𝑏𝑑(𝑥 ) The Lagrangian corresponding to the surface term is giv- leads to en by [56] 𝑆󸀠𝑎 (𝑥󸀠)=𝑆𝑎 +𝑆𝑏𝜕 𝜉𝑎 −𝑔𝑏𝑑𝜕 𝜕 𝜉𝑎 +𝑔𝑎𝑏𝜕 𝜕 𝜉𝑐. 𝐿 =𝜕 (√−𝑔𝑆𝑎) , 𝑏 𝑏 𝑑 𝑏 𝑐 (9) sur 𝑎 (1) where Another one is given by

𝑎 𝑎𝑑 𝑏𝑘 𝑐 𝑎𝑑 1 𝑎 𝑑 𝑎 𝑑 𝑎 󸀠 𝑎 𝑏 𝑏 𝑎 𝑏 𝑎 𝑆 =2𝑄 𝑔 Γ ,𝑄= (𝛿 𝛿 −𝛿 𝛿 ). 𝑆 (𝑥 ) =𝑆 (𝑥 +𝜉 ) =𝑆 +𝜉 𝜕𝑏𝑆 . (10) 𝑐𝑘 𝑏𝑑 𝑐𝑘 2 𝑐 𝑘 𝑘 𝑐 (2) 𝑎 Here the normalization 1/16𝜋𝐺 is omitted and it will be Therefore, according to (3), the Lie variation of 𝑆 due to the inserted where necessary. Now our task is to find the varia- diffeomorphism is 󸀠𝑎 𝑎 𝑎 tions of both sides of (1)foradiffeomorphism𝑥 =𝑥 +𝜉 𝑎 𝑎 󸀠 󸀠𝑎 󸀠 𝑏 𝑎 𝑏 𝑎 𝑎 and then equate them. The variation we will consider here is 𝛿𝑆 =𝑆 (𝑥 ) −𝑆 (𝑥 ) =𝜉𝜕𝑏𝑆 −𝑆 𝜕𝑏𝜉 +𝑀 , (11) the Lie variation which is defined, in general, as

󸀠 󸀠 󸀠 where 𝛿𝐴 = 𝐴 (𝑥 )−𝐴 (𝑥 ), (3) 𝑎 𝑏𝑑 𝑎 𝑎𝑏 𝑐 𝑀 =𝑔 𝜕 𝜕 𝜉 −𝑔 𝜕 𝜕 𝜉 . (12) 󸀠 𝑎 𝑏 𝑑 𝑏 𝑐 where 𝐴(𝑥 ) = 𝐴(𝑥+𝜉)= 𝐴(𝑥)+𝜉𝜕𝑎𝐴(𝑥) and 𝐴(𝑥) and 󸀠 󸀠 𝐴 (𝑥 ) are evaluated in two different coordinate systems 𝑥 and Substituting this in (6) we obtain the variation of right-hand 󸀠 𝑥 ,respectively.Inthefollowing,forthenotationalsimplicity, side of (1)as we will denote 𝐴(𝑥) as 𝐴. 𝛿𝐿 =𝜕 [𝜕 (√−𝑔𝑆𝑎𝜉𝑏)−√−𝑔𝑆𝑏𝜕 𝜉𝑎 + √−𝑔𝑀𝑎]. The variation of the right-hand side of1 ( )isgivenby sur 𝑎 𝑏 𝑏 (13) 𝛿𝐿 =𝜕 [𝛿√ ( −𝑔𝑆𝑎)]=𝜕 [𝑆𝑎𝛿(√−𝑔) + √−𝑔𝛿𝑆𝑎] sur 𝑎 𝑎 𝑎 Next we find the variation of left-hand side1 of( ), that is, 𝑆 𝑏𝑐 𝑎 =𝜕 [ √−𝑔𝑔 𝛿𝑔 + √−𝑔𝛿𝑆 ]. 𝐿 .Forthiswewillstartfromthefollowingrelation: 𝑎 2 𝑏𝑐 sur (4) 𝐿 = √−𝑔 (𝐿 −𝐿 ), sur 𝑔 quad (14) Since 𝑔𝑎𝑏 is a tensor, for the Lie variation, 𝛿𝑔𝑎𝑏 is expressed by the Lie derivative and is given by where 𝛿𝑔 =∇𝜉 +∇𝜉 . 𝐿 =𝑅; 𝐿 =2𝑄𝑏𝑐𝑑Γ𝑎 Γ𝑘 , 𝑎𝑏 𝑎 𝑏 𝑏 𝑎 (5) 𝑔 quad 𝑎 𝑑𝑘 𝑏𝑐 (15) 4 Advances in High Energy Physics

𝑄𝑏𝑐𝑑 = (1/2)(𝛿𝑐 𝑔𝑏𝑑 −𝛿𝑑𝑔𝑏𝑐) 𝐿 with 𝑎 𝑎 a .Since 𝑔 is a scalar, by the Hence, 𝑎 definition of Lie derivative, 𝛿𝐿 𝑔 =𝜉𝜕𝑎𝐿𝑔. Therefore using √−𝑔𝛿𝐿 = √−𝑔𝐿 (𝑥󸀠)−√−𝑔𝐿󸀠 (𝑥󸀠) (5)wefind quad quad quad (23) 𝑎 𝑎 𝛿𝐿 =𝛿(√−𝑔𝐿𝑔)−𝛿(√−𝑔𝐿 ) = √−𝑔𝜉 𝜕 𝐿 −𝜕 (√−𝑔𝑀 ). sur quad 𝑎 quad 𝑎 =𝜕 (√−𝑔𝜉𝑎𝐿 )−𝜕 (√−𝑔𝜉𝑎)𝐿 − √−𝑔𝛿𝐿 𝑎 𝑔 𝑎 quad quad Substituting this in (16)weobtain 𝑎 𝑎 =𝜕 [√−𝑔𝜉 (𝐿 −𝐿 )] + √−𝑔𝜉 𝜕 𝐿 𝑎 𝑎 𝑎 𝑔 quad 𝑎 quad 𝛿𝐿 =𝜕 (𝜉 𝐿 + √−𝑔𝑀 ) . sur 𝑎 sur (24)

− √−𝑔𝛿𝐿 𝑎 quad Now equating (13)and(24)weobtain𝜕𝑎𝐽 [𝜉] =,wherethe 0 𝐽𝑎[𝜉] =𝜕 (𝜉𝑎𝐿 )+√−𝑔𝜉𝑎𝜕 𝐿 − √−𝑔𝛿𝐿 . conserved Noether current is given by 𝑎 sur 𝑎 quad quad (16) 𝐽𝑎 [𝜉] =−𝜕 (√−𝑔𝑆𝑎𝜉𝑏) + √−𝑔𝑆𝑏𝜕 𝜉𝑎 +𝜉𝑎𝐿 . 𝑏 𝑏 sur (25) 𝛿𝐿 To find quad, we will proceed as earlier. Under the change 𝑥󸀠𝑎 =𝑥𝑎 +𝜉𝑎 𝐿󸀠 (𝑥󸀠) 𝐿 =𝜕(√−𝑔𝑆𝑎) , quad is calculated as Finally, using sur 𝑎 in the above, we can express the current as the divergence of an antisymmetric two-index 𝐿󸀠 (𝑥󸀠) quad quantity: =2𝑄󸀠𝑏𝑐𝑑 (𝑥󸀠)Γ󸀠𝑎 (𝑥󸀠)Γ󸀠𝑘 (𝑥󸀠) 𝑎 𝑎 𝑏 𝑏 𝑎 𝑎𝑏 𝑎 𝑑𝑘 𝑏𝑐 𝐽 [𝜉]=𝜕𝑏 [√−𝑔 (𝜉 𝑆 −𝜉 𝑆 )] = 𝜕𝑏 [√−𝑔𝐽 [𝜉]] . =𝐿 +𝑔𝑏𝑐Γ𝑘 𝜕 𝜕 𝜉𝑑 +𝑔𝑏𝑐Γ𝑑 𝜕 𝜕 𝜉𝑘 −𝑔𝑏𝑑Γ𝑐 𝜕 𝜕 𝜉𝑘, (26) quad 𝑏𝑐 𝑑 𝑘 𝑑𝑘 𝑏 𝑐 𝑑𝑘 𝑏 𝑐 (17) 𝑎𝑏 It is evident that the anti-symmetric object 𝐽 [𝜉] is not a ten- where (8) has been used. This can be expressed in terms of sor and it is usually called the Noether potential. Therefore, 𝑎 𝑀 in the following way. The second term on the right-hand inserting the proper normalization, the charge is given by side can be expressed in the following form: 1 √ 𝑎𝑏 𝑏𝑐 𝑘 𝑑 𝑄 [𝜉] = ∫ 𝑑Σ𝑎𝑏 ℎ𝐽 [𝜉] , (27) √−𝑔𝑔 Γ𝑏𝑐𝜕𝑑𝜕𝑘𝜉 32𝜋𝐺 H

=[√−𝑔𝑔𝑏𝑐𝑔𝑎𝑘𝜕 𝑔 −𝑔𝑎𝑘𝜕 (√−𝑔)] 𝜕 𝜕 𝜉𝑑 2 𝑏 𝑎𝑐 𝑎 𝑑 𝑘 (18) where 𝑑Σ𝑎𝑏 =−𝑑𝑥(𝑁𝑎𝑀𝑏 −𝑁𝑏𝑀𝑎) is the surface element of the 2-dimensional surface H and ℎ is the determinant of =−𝜕 (√−𝑔𝑔𝑎𝑘)𝜕 𝜕 𝜉𝑑, 𝑎 𝑑 𝑘 the corresponding metric. Since our present discussion will be near the horizon, we choose the unit normals 𝑁𝑎 and 𝑀𝑎 𝑔𝑏𝑐𝑔𝑎𝑘𝜕 𝑔 =−𝜕𝑔𝑎𝑘 whereintheaboveweused 𝑏 𝑎𝑐 𝑎 .Thethird as spacelike and timelike, respectively. term of (17)reducesto Now we will calculate charge (27) explicitly on the hori- 𝑏𝑐 𝑑 𝑘 𝑏𝑐 𝑘 zon. This will be done by considering the form of the metric √−𝑔𝑔 Γ𝑑𝑘𝜕𝑏𝜕𝑐𝜉 =𝜕𝑘 (√−𝑔) 𝑔 𝜕𝑏𝜕𝑐𝜉 . (19) near the horizon: Similarly, the last term can be expressed as 2 2 1 2 2 𝑏𝑑 𝑐 𝑘 𝑏𝑑 𝑐𝑎 𝑘 𝑑𝑠 =−2𝜅𝑥𝑑𝑡 + 𝑑𝑥 +𝑑𝑥⊥, (28) 2√−𝑔𝑔 Γ𝑑𝑘𝜕𝑏𝜕𝑐𝜉 = √−𝑔𝑔 𝑔 𝜕𝑘𝑔𝑎𝑑𝜕𝑏𝜕𝑐𝜉 2𝜅𝑥 (20) 𝑏𝑐 𝑘 =−√−𝑔𝜕𝑘 (𝑔 )𝜕𝑏𝜕𝑐𝜉 , where 𝑥⊥ represents the transverse coordinates. The metric 𝑎 has a timelike Killing vector 𝜒 = (1, 0, 0, 0) and the Killing 𝑔𝑏𝑑𝑔𝑐𝑎𝜕 𝑔 =−𝜕𝑔𝑏𝑐 2 whereinthelastline 𝑘 𝑎𝑑 𝑘 has been used. horizon is given by 𝜒 =0;thatis,𝑥=0.Thenonzero Substituting all these in (17)weobtain Christoffel connections are 𝐿󸀠 (𝑥󸀠) quad 𝑡 1 𝑥 2 𝑥 1 Γ𝑡𝑥 = ,Γ𝑡𝑡 =2𝜅𝑥,𝑥𝑥 Γ =− . (29) 1 2𝑥 2𝑥 =𝐿 − [𝜕 (√−𝑔𝑔𝑎𝑘)𝜕 𝜕 𝜉𝑑 +𝜕 (√−𝑔𝑔𝑏𝑐)𝜕 𝜕 𝜉𝑘] quad 𝑎 𝑑 𝑘 𝑘 𝑏 𝑐 √−𝑔 For metric (28)wefind

1 𝑎 =𝐿 + 𝜕 (√−𝑔𝑀 ). 𝑎 𝑎 1 quad √−𝑔 𝑎 𝑁 = (0, √2𝜅𝑥, 0, 0) ,𝑀 =( ,0,0,0), √2𝜅𝑥 (21) (30) On the other hand, 𝑑Σ =−𝑑2𝑥 𝐿 (𝑥󸀠)=𝐿 (𝑥𝑎 +𝜉𝑎)=𝐿 +𝜉𝑎𝜕 𝐿 . and hence 𝑡𝑥 .Also,(2)yields quad quad quad 𝑎 quad 𝑡 𝑥 (22) 𝑆 =0, 𝑆 =−2𝜅. (31) Advances in High Energy Physics 5

𝑎 Therefore, and in addition the expression for 𝑆 ,givenby(2), we obtain

𝑡𝑥 𝑡 𝑥 𝑥 𝑡 𝑡 𝑎𝑏 𝛿𝜉 (√−𝑔𝐽 [𝜉2]) 𝐽 =(𝜉𝑆 −𝜉 𝑆 )=−2𝜅𝜉. (32) 1 = √−𝑔 [∇ 𝜉𝑚𝐽𝑎𝑏 [𝜉 ] 𝑎 𝑡 𝑡 𝑚 1 2 Now if 𝜉 is a Killing vector, then 𝜉 =𝜒 =1,andsocalcu- 𝑚 𝑎 𝑚 𝑎 𝑏 lating charge (27) explicitly we find +{(𝜉1 ∇𝑚𝜉2 −𝜉2 ∇𝑚𝜉1 )𝑆

𝜅𝐴 +𝜉𝑎 (−2Γ𝑏 ∇𝑚𝜉𝑛 +∇ ∇𝑚𝜉𝑏 +2𝑅𝑏 𝜉𝑚 𝑄[𝜉=𝜒]= ⊥ , 2 𝑚𝑛 1 𝑚 1 𝑚 1 8𝜋𝐺 (33) 𝑛 𝑏 𝑚 𝑚 𝑏 𝑏 𝑚 −Γ𝑛𝑚 (∇ 𝜉1 +∇ 𝜉1)−∇𝑚∇ 𝜉1 ) 2 where 𝐴⊥ =∫ 𝑑 𝑥 is the horizon cross-section area. Multi- H − (𝑎←→𝑏) }]. plying it by the periodicity of time coordinate 2𝜋/𝜅 we obtain exactly the entropy: one-quarter of horizon area. Moreover, (36) theabovecanbeexpressedas𝑄[𝜉 = 𝜒],where =𝑇𝑆 𝑇=𝜅/2𝜋 𝑎 For the present metric (28), 𝑔=−1, 𝑅 =0, and hence is the temperature of the horizon and 𝑆=𝐴⊥/4𝐺 is the 𝑏 √−𝑔Γ𝑛 =𝜕 (√−𝑔) = 0 entropy. Therefore one can call it the Noether energy. Such 𝑛𝑚 𝑚 . Therefore interpretaion was done earlier in [67, 68]. 𝛿 (√−𝑔𝐽𝑎𝑏 [𝜉 ]) So far we found that the Noether charge corresponding 𝜉1 2 to the surface term of EH action alone led to the entropy of =[∇ 𝜉𝑚𝐽𝑎𝑏 [𝜉 ] the Rindler horizon. This was shown earlier for the charge 𝑚 1 2 coming from the total EH action [61–63]. Therefore, the 1 presentanalysisrevealedthatitmaybepossiblethatthe + {(𝜉𝑚∇ 𝜉𝑎 −𝜉𝑚∇ 𝜉𝑎)𝑆𝑏 16𝜋𝐺 1 𝑚 2 2 𝑚 1 information is actually encoded in the surface term rather than the bulk term. Then the natural question arises: what 𝑎 𝑏 𝑚 𝑛 𝑚 𝑏 𝑏 𝑚 +𝜉2 (−2Γ𝑚𝑛∇ 𝜉1 +∇𝑚∇ 𝜉1 −∇𝑚∇ 𝜉1 ) are the degrees of freedom responsible for this entropy? So far they are not known. In the next couple of sections we will − (𝑎←→𝑏) }] give an idea on the nature of the possible degrees of freedom in the context of Virasoro algebra and Cardy formula. 𝑎𝑏 ≡𝐾12 . (37) 3. Bracket among the Charges and the Diffeomorphism Generators Finally we define a bracket as

In the previous section, we have given the expression for the 1 √ 𝑎𝑏 [𝑄 1[𝜉 ],𝑄[𝜉2]] := ∫ 𝑑Σ𝑎𝑏 ℎ[𝐾12 − (1←→2)], charge (see (27)) for an arbitrary diffeomorphism. Here we 2 H will define the bracket among the charges. The relevant diffeo- (38) morphismswillbechosenbyimposingaminimumcondition on the spacetime metric. The charge and the bracket will be which for the present metric (28)reducesto then expressed in terms of these generators. 2 𝑡𝑥 We will find the bracket following our earlier works54 [ , [𝑄 1[𝜉 ],𝑄[𝜉2]] := − ∫ 𝑑 𝑥[𝐾12 − (1←→2)]. (39) 55]. For this let us first calculate the following: H To calculate the above bracket we need to know about the 𝛿 (√−𝑔𝐽𝑎𝑏 [𝜉 ]) 𝜉𝑎 𝜉1 2 generators . We will determine them by using the condition that the horizon structure remains invariant in some nonsin- =𝛿 (√−𝑔)𝑎𝑏 𝐽 [𝜉 ]+√−𝑔𝛿 (𝐽𝑎𝑏 [𝜉 ]) 𝜉1 2 𝜉1 2 gular coordinate system. For that let us first express metric (28) in Gaussian null (or Bondi like) coordinates: 1 𝑚𝑛 𝑎𝑏 =− √−𝑔𝑔𝑚𝑛𝛿𝜉 𝑔 𝐽 [𝜉2] 2 1 𝑑𝑥 𝑑𝑢 = 𝑑𝑡 − ,𝑑𝑋=𝑑𝑥. (40) 2𝜅𝑥 + √−𝑔 [(𝛿 𝜉𝑎)𝑆𝑏 +𝜉𝑎 (𝛿 𝑆𝑏)−(𝑎←→𝑏)]. 𝜉1 2 2 𝜉1 (34) In these coordinates the metric reduces to the following form: 2 2 2 𝑑𝑠 =−2𝜅𝑋𝑑𝑢 − 2𝑑𝑢 𝑑𝑋 +𝑑𝑥 . (41) Using ⊥

Now impose the condition that the metric coefficients 𝑔𝑋𝑋 𝑎𝑏 𝑎𝑏 𝑎 𝑏 𝑏 𝑎 𝑔 𝛿𝜉𝑔 = £𝜉𝑔 =−∇ 𝜉 −∇ 𝜉 , and 𝑢𝑋 do not change under the diffeomorphism; that is, (35) 𝑎 𝑎 𝑎 𝑚 𝑔 =0, 𝑔 =0, 𝛿𝜉Γ𝑏𝑐 =∇𝑏∇𝑐𝜉 +𝑅𝑐𝑚𝑏𝜉 , £𝜉̃ 𝑋𝑋 £𝜉̃ 𝑢𝑋 (42) 6 Advances in High Energy Physics

̃ where £𝜉̃ istheLiederivativealongthevector𝜉.Theseleadto So near the horizon (46)reducesto 𝑢 ̃ 𝑡𝑥 2 1 2 2 £𝜉̃𝑔𝑋𝑋 =−2𝜕𝑋𝜉 =0, 𝐾12 =−2𝜅𝑇1𝜕𝑡𝑇2 +𝑇1𝜕𝑡 𝑇2 − (𝜕𝑡𝑇1𝜕𝑡 𝑇2 +𝜕𝑡 𝑇1𝜕𝑡𝑇2) (43) 2𝜅 𝑔 =−𝜕 𝜉̃𝑢 −2𝜅𝑋𝜕 𝜉̃𝑢 −𝜕 𝜉̃𝑋 =0. 1 1 £𝜉̃ 𝑢𝑋 𝑢 𝑋 𝑋 + 𝑇 𝜕3𝑇 +𝜕𝑇 𝜕 𝑇 − 𝜕3𝑇 𝜕 𝑇 . 2𝜅 2 𝑡 1 𝑡 1 𝑡 2 4𝜅2 1 𝑡 2 The solutions are ̃𝑢 (48) 𝜉 =𝐹(𝑢,𝑥⊥), (44) ̃𝑋 𝜉 =−𝑋𝜕𝑢𝐹(𝑢,𝑥⊥). Substituting this in (39) and inserting the normalization factor, we obtain the expression for the bracket The condition £𝜉̃𝑔𝑢𝑢 =0is automatically satisfied near the horizon, because use of the above solutions leads to £𝜉̃𝑔𝑢𝑢 = [𝑄 1[𝜉 ],𝑄[𝜉2]] O(𝑋). These conditions appeared earlier in [69] in the context 1 2 of late time symmetry near the black hole horizon. Finally := ∫ 𝑑 𝑥[2𝜅(𝑇1𝜕𝑡𝑇2 −𝑇2𝜕𝑡𝑇1) expressing (44)intheoldcoordinates(𝑡, 𝑥)wefind 16𝜋𝐺 H 2 2 𝑡 1 𝑥 −(𝑇1𝜕𝑡 𝑇2 −𝑇2𝜕𝑡 𝑇1) 𝜉 =𝑇− 𝜕𝑡𝑇, 𝜉 = −𝑥𝜕𝑡𝑇, (45) (49) 2𝜅 1 + (𝑇 𝜕3𝑇 −𝑇𝜕3𝑇 ) where 𝑇(𝑡, 𝑥,⊥ 𝑥 )=𝐹(𝑢,𝑥⊥). 1 𝑡 2 2 𝑡 1 𝑡𝑥 2𝜅 Next we calculate 𝐾12 from (37) for the present case. Since 1 𝑡 + (𝜕3𝑇 𝜕 𝑇 −𝜕3𝑇 𝜕 𝑇 )] . 𝑆 =0,wefind 4𝜅2 𝑡 1 𝑡 2 𝑡 2 𝑡 1 𝑡𝑥 𝑚 𝑡 𝑥 𝑚 𝑡 𝑚 𝑡 𝑥 𝐾12 =∇𝑚𝜉1 𝜉2𝑆 +(𝜉1 ∇𝑚𝜉2 −𝜉2 ∇𝑚𝜉1)𝑆

𝑡 𝑥 𝑚 𝑛 𝑚 𝑥 𝑥 𝑚 Similarly, (27) yields +𝜉2 (−2Γ𝑚𝑛∇ 𝜉1 +∇𝑚∇ 𝜉1 −∇𝑚∇ 𝜉1 ) (46)

𝑥 𝑡 𝑚 𝑛 𝑚 𝑡 𝑡 𝑚 1 2 1 −𝜉 (−2Γ ∇ 𝜉 +∇𝑚∇ 𝜉 −∇𝑚∇ 𝜉 ). 𝑄[𝜉]= ∫ 𝑑 𝑥(𝜅𝑇− 𝜕 𝑇) . 2 𝑚𝑛 1 1 1 8𝜋𝐺 2 𝑡 (50) Now since the integration (39) will ultimately be evaluated on the horizon, we will find the value of each term of the above A couple of comments are in order. It must be noted very near to the horizon. Therefore, using (29), (31), and the that, in finding the expression for bracket49 ( ), no use of form of the generators (45)weobtainthevaluesofeachterm boundary conditions (Dirichlet or Neumann) has been used. of the above expression near the horizon 𝑥=0as Earlier this was used for the case of 𝐿 to throw away the 1 bulk ∇ 𝜉𝑚𝜉𝑡 𝑆𝑥 =𝑇𝜕2𝑇 − 𝜕2𝑇 𝜕 𝑇 , noncovariant terms in the bracket without giving any physical 𝑚 1 2 2 𝑡 1 2𝜅 𝑡 1 𝑡 2 𝛿 𝜉𝑎 =0 meaning [53]. Also, we did not use the condition 𝜉1 2 (see (34)) which was adopted in earlier works. For instance, 𝑚 𝑡 𝑥 2 1 2 𝑎 𝜉 ∇ 𝜉 𝑆 =−𝜅𝑇𝜕 𝑇 +𝜅𝑇𝜕 𝑇 +𝑇𝜕 𝑇 − 𝜕 𝑇 𝜕 𝑇 , see [8, 53]. This is logically correct since 𝛿𝜉 𝜉 =0contradicts 1 𝑚 2 1 𝑡 2 2 𝑡 1 1 𝑡 2 2𝜅 𝑡 1 𝑡 2 1 2 the algebra among the Fourier modes of the diffeomorphisms 1 −𝜉𝑚∇ 𝜉𝑡 𝑆𝑥 =𝜅𝑇𝜕 𝑇 −𝜅𝑇𝜕 𝑇 −𝑇𝜕2𝑇 + 𝜕 𝑇 𝜕2𝑇 , (see (52), in next section). 2 𝑚 1 2 𝑡 1 1 𝑡 2 2 𝑡 1 2𝜅 𝑡 2 𝑡 1 4. Virasoro Algebra and Entropy 𝑡 𝑥 𝑚 𝑛 2 1 2 −2𝜉2Γ𝑚𝑛∇ 𝜉1 =−𝑇2𝜕𝑡 𝑇1 + 𝜕 𝑇1𝜕𝑡𝑇2, 2𝜅 In this section, the Fourier modes of the bracket and the 1 1 charge will be found out. We will show that for a particular 𝜉𝑡 ∇ ∇𝑚𝜉𝑥 = 𝑇 𝜕3𝑇 − 𝜕3𝑇 𝜕 𝑇 −2𝜅𝑇𝜕 𝑇 2 𝑚 1 2𝜅 2 𝑡 1 4𝜅2 𝑡 1 𝑡 2 2 𝑡 1 ansatz for the Fourier modes of the generators will lead to the Virasoro algebra. Finally using the Cardy formula, the 2 1 2 entropywillbecalculated. +𝜕𝑡𝑇1𝜕𝑡𝑇2 +𝑇2𝜕𝑡 𝑇1 − 𝜕𝑡 𝑇1𝜕𝑡𝑇2, 2𝜅 Consider the Fourier decompositions of 𝑇1 and 𝑇2: 𝑡 𝑥 𝑚 −𝜉2∇𝑚∇ 𝜉1 =0, 𝑇 = ∑𝐴 𝑇 ,𝑇= ∑𝐵 𝑇 , 1 1 𝑚 𝑚 2 𝑛 𝑛 (51) 2𝜉𝑥Γ𝑡 ∇𝑚𝜉𝑛 =− 𝜕2𝑇 𝜕 𝑇 −2𝜅𝑥𝜕 𝑇 𝜕 𝑇 , 𝑚 𝑛 2 𝑚𝑛 1 2𝜅 𝑡 1 𝑡 2 𝑥 1 𝑡 2 1 𝐴∗ =𝐴 𝐵∗ =𝐵 𝑇 −𝜉𝑥∇ ∇𝑚𝜉𝑡 = 𝜕3𝑇 𝜕 𝑇 , where 𝑚 −𝑚 and 𝑛 −𝑛. The Fourier modes 𝑚 2 𝑚 1 4𝜅2 𝑡 1 𝑡 2 will be chosen such that the Fourier modes of the diffeomor- 𝑆1 1 phisms (44)obeyonesubalgebraisomorphictoDiff. : 𝜉𝑥∇ ∇𝑡𝜉𝑚 =− 𝜕3𝑇 𝜕 𝑇 . 2 𝑚 1 4𝜅2 𝑡 1 𝑡 2 𝑎 𝑎 (47) 𝑖{𝜉𝑚,𝜉𝑛} = (𝑚−𝑛) 𝜉𝑚+𝑛, (52) Advances in High Energy Physics 7 where {, } is the Lie bracket. Now with the use of (51), let us be noted that the transverse directions are noncompact due first find the Fourier modes of bracket (49)andcharge(50). to our Rindler approximations and so we will assume that 𝑇𝑚 Substitution of (51)in(49)yields is periodic in the transverse coordinates with the periodicities 𝐿𝑦 and 𝐿𝑧 on 𝑦 and 𝑧,respectively.Nowsubstituting(57)in [𝑄 1[𝜉 ],𝑄[𝜉2]] (55)and(56) and then integrating over the cross-sectional 𝐴 =𝐿 𝐿 𝐶𝑚,𝑛 2 area ⊥ 𝑦 𝑧,weobtain := ∑ ∫ 𝑑 𝑥[2𝜅(𝑇𝑚𝜕𝑡𝑇𝑛 −𝑇𝑛𝜕𝑡𝑇𝑚) 𝑚,𝑛 16𝜋𝐺 H 𝐴⊥ 𝜅 2 2 𝑄𝑚 = 𝛿𝑚,0, (58) −(𝑇𝑚𝜕𝑡 𝑇𝑛 −𝑇𝑛𝜕𝑡 𝑇𝑚) 8𝜋𝐺 𝛼 1 𝐴 𝜅 𝐴 𝛼 + (𝑇 𝜕3𝑇 −𝑇𝜕3𝑇 ) 𝑖[𝑄 ,𝑄 ]:= ⊥ (𝑚−𝑛) 𝛿 +𝑛3 ⊥ 𝛿 . 2𝜅 𝑚 𝑡 𝑛 𝑛 𝑡 𝑚 𝑚 𝑛 8𝜋𝐺 𝛼 𝑚+𝑛,0 16𝜋𝐺 𝜅 𝑚+𝑛,0 1 (59) + (𝜕3𝑇 𝜕 𝑇 −𝜕3𝑇 𝜕 𝑇 )] , 4𝜅2 𝑡 𝑚 𝑡 𝑛 𝑡 𝑛 𝑡 𝑚 (53) Using (58), (59) can be reexpressed as 𝐴 𝛼 𝐶 =𝐴 𝐵 𝐶∗ =𝐶 𝑖[𝑄 ,𝑄 ]:=(𝑚−𝑛) 𝑄 +𝑛3 ⊥ 𝛿 . where 𝑚,𝑛 𝑚 𝑛 and so 𝑚,𝑛 −𝑚,−𝑛. Next defining the 𝑚 𝑛 𝑚+𝑛 16𝜋𝐺 𝜅 𝑚+𝑛,0 (60) Fourier modes of [𝑄[𝜉1], 𝑄[𝜉2]] as This is exactly identical to Virasoro algebra with the central [𝑄 [𝜉 ],𝑄[𝜉 ]] = ∑𝐶 [𝑄 ,𝑄 ], 1 2 𝑚,𝑛 𝑚 𝑛 charge 𝐶 being identified as 𝑚,𝑛 (54) 𝐶 𝐴 𝛼 = ⊥ . we find 12 16𝜋𝐺 𝜅 (61) [𝑄 ,𝑄 ] 𝑚 𝑛 The zero mode eigenvalue is evaluated from (58)for𝑚=0: 1 := ∫ 𝑑2𝑥[2𝜅(𝑇 𝜕 𝑇 −𝑇𝜕 𝑇 ) 𝐴 𝜅 16𝜋𝐺 𝑚 𝑡 𝑛 𝑛 𝑡 𝑚 𝑄 = ⊥ . H 0 8𝜋𝐺 𝛼 (62) −(𝑇 𝜕2𝑇 −𝑇𝜕2𝑇 ) 𝑚 𝑡 𝑛 𝑛 𝑡 𝑚 Finally using the Cardy formula [10–12], we obtain the entropy as 1 3 3 + (𝑇𝑚𝜕𝑡 𝑇𝑛 −𝑇𝑛𝜕𝑡 𝑇𝑚) 2𝜅 𝐶𝑄 𝐴 𝑆=2𝜋√ 0 = ⊥ , 1 (63) + (𝜕3𝑇 𝜕 𝑇 −𝜕3𝑇 𝜕 𝑇 )] . 6 4𝐺 4𝜅2 𝑡 𝑚 𝑡 𝑛 𝑡 𝑛 𝑡 𝑚 (55) which is exactly the Bekenstein-Hawking entropy.

Similarly from (50), the Fourier modes of the charge are given 5. Conclusions by It has already been observed that several interesting features 1 1 and pieces of information can be obtained from the surface 𝑄 = ∫ 𝑑2𝑥(𝜅𝑇 − 𝜕 𝑇 ), 𝑚 8𝜋𝐺 𝑚 2 𝑡 𝑚 (56) term without incorporating the bulk term of the gravity action.Inthispaperwestudiedthesurfacetermofthe where 𝑄[𝜉]𝑚 =∑ 𝐴𝑚𝑄𝑚.Itmustbenotedthatthepresent Einstein-Hilbert (EH) action in the context of Noether cur- expression (56) is exactly identical to that obtained in [55]for rent. So far we know that this has not been attempted before. the York-Gibbons-Hawking surface term, whereas the other First the current was derived for an arbitrary diffeomorphism expression (55) is different by a total derivative term. This may by using Noether prescription. Then we showed that the be because these two surface terms are not exactly the same. charge evaluated on the horizon for a Killing vector led to But we will show that final result for the bracket is identical to the Bekenstein-Hawking entropy after multiplying it by 2𝜋/𝜅. the earlier analysis, because the total derivative term will not But till now, it is not known about the degrees of freedom contribute to an ansatz for 𝑇𝑚. responsible for the entropy. Here we addressed the issue and To calculate the previous expressions (55)and(56) explic- tried to shed some light. This has been discussed in the itly we need to have 𝑇𝑚’s. Following the earlier arguments, we context of Virasoro algebra and Cardy formula. choose In this paper, we defined the bracket among the charges. It was done, in the sprite of our earlier works [54, 55], by taking 1 𝑖𝑚(𝛼𝑡+𝑔(𝑥)+𝑝⋅𝑥 ) 𝑇 = 𝑒 ⊥ 𝑎𝑏 𝑚 (57) the variation of the Noether potential 𝐽 [𝜉1] for a different 𝛼 𝑎 𝑎 𝑎 diffeomorphism 𝑥 →𝑥 +𝜉2 andthenananti-symmetric such that they satisfy algebra (52). Here 𝛼 is a constant, 𝑝 combination between the indices 1 and 2 and integrating over is an integer, and 𝑔(𝑥) is a function which is regular at the the horizon surface. To achieve the final form, we did not use horizon. This is a standard choice in these computations and Einstein’s equation of motion or any ambiguous prescription, has been used several times in the literature [7, 8, 53]. It must like vanishing of the variation of diffeomorphism parameter 8 Advances in High Energy Physics

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