Abstract
We consider a modified version of Brans-Dicke theory (MBDT) in four dimensions (4D) obtained by applying the induced matter method of Wesson to a 5D generalized Brans-Dicke theory. In 5D the model consists of pure vacuum, with no self-interacting potential, except for a scalar field. Following Wesson’s protocol, we group geometric terms in the 5D Einstein tensor arising from the extra dimension, move them to the other side of the generalized field equations, and identify them as the energy-momentum of the induced matter in 4D. Thus the extra dimension in 5D leads naturally to an effective matter field in 4D. Constraining the 5D geometry to be a generalization of the anisotropic Bianchi type I universe model first studied by Kasner, we derive the induced energy-momentum in MBDT and apply it to the investigation of energy conditions. The specified induced energy-momentum of that MBDT model consists of the energy density and directional pressure which indicate the anisotropy of the universe. We discuss the energy conditions and their bounds in the MBDT with such an induced imperfect fluid, with an eye toward a realistic model of the present-day universe, and consider the large-scale behavior of that spatially homogeneous and anisotropic model. We discuss how the energy conditions would be satisfied or violated in the context of MBDT, with the aim of providing a feasible description of the universe in the current era.
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Notes
Wald theorem [23] implies that if one includes a positive cosmological constant (\(\Lambda > 0\)) in the Bianchi type I spacetime, consequently, as time passes, the dynamics tends to assume a form similar to de Sitter spacetime. Specifically, each of the scale factors in a Bianchi type I spacetime with a positive cosmological constant becomes, over time, \(\mathrm {exp}(\sqrt{\Lambda /3}t)\). Therefore, all of the scale factors become identical and turn the homogeneous and anisotropic spacetime asymptotically into a vacuum de Sitter state. Such a process is equivalent to the cosmic no-hair theorem [24]. In other words, according to the cosmic no-hair theorem, adding a positive cosmological constant to an anisotropic 5D Bianchi type I spacetime leads asymptotically to an isotropic state. For that reason, we did not include a cosmological constant in our model, as that would remove its interesting anisotropic dynamics.
One could consider it as being an inflationary phase which follows the power law at the present time.
References
Brans, C., Dicke, R.H.: Mach’s principle and a relativistic theory of gravitation. Phys. Rev. 124(3), 925 (1961). https://doi.org/10.1103/PhysRev.124.925
Kaluza, T.: Zum unitätsproblem der physik. Sitzungsber. Preuss. Akad. Wiss. Berlin.(Math. Phys.) 1921(966972), 45–46 (1921). https://doi.org/10.1142/S0218271818700017
Chodos, A., Detweiler, S.: Where has the fifth dimension gone? Phys. Rev. D 21(8), 2167 (1980). https://doi.org/10.1103/PhysRevD.21.2167
Einstein, A.: Physics and reality. J. Franklin Inst. 221(3), 349–382 (1936). https://doi.org/10.1016/S0016-0032(36)91047-5
Wesson, P.S., Ponce de Leon, J.: Kaluza–Klein equations, Einstein’s equations, and an effective energy-momentum tensor. J. Math. Phys. 33(11), 3883–3887 (1992). https://doi.org/10.1063/1.529834
Halpern, P.: Exact solutions of five dimensional anisotropic cosmologies. Phys. Rev. D 66(2), 027503 (2002). https://doi.org/10.1103/PhysRevD.66.027503
Wesson, P.S., Ponce de Leon, J.: The equation of motion in Kaluza-Klein cosmology and its implications for astrophysics. Astron. Astrophys. 294, 1–7 (1995)
Rasouli, S., Farhoudi, M., Moniz, P.V.: Modified Brans-Dicke theory in arbitrary dimensions. Classical Quantum Gravity 31(11), 115002 (2014). https://doi.org/10.1088/0264-9381/31/11/115002
Halpern, P.: Behavior of Kasner cosmologies with induced matter. Phys. Rev. D 63(2), 024009 (2000). https://doi.org/10.1103/PhysRevD.63.024009
Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space-time, vol. 1. Cambridge University Press, Cambridge, UK (1975)
Visser, M.: Energy conditions in the epoch of galaxy formation. Science 276(5309), 88–90 (1997). https://doi.org/10.1126/science.276.5309.88
Atazadeh, K., Darabi, F.: Energy conditions in \(f({R},{G})\) gravity. Gen. Relativ. Gravit. 46(2), 1664 (2014). https://doi.org/10.1007/s10714-014-1664-8
Yousaf, Z., Sharif, M., Ilyas, M., Zaeem-ul-Haq Bhatti, M.: Energy conditions in higher derivative \(f({R},\square {R},{T})\) gravity. Int. J. Geom. Methods Mod. Phys. 15(09), 1850146 (2018). https://doi.org/10.1142/S0219887818501463
Capozziello, S., Nojiri, S., Odintsov, S.D.: The role of energy conditions in \(f({R})\) cosmology. Phys. Lett. B 781, 99–106 (2018). https://doi.org/10.1016/j.physletb.2018.03.064
Atazadeh, K., Khaleghi, A., Sepangi, H., Tavakoli, Y.: Energy conditions in \(f({R})\) gravity and Brans-Dicke theories. Int. J. Mod. Phys. D 18(07), 1101–1111 (2009). https://doi.org/10.1142/S0218271809014972
Capozziello, S., Lobo, F.S., Mimoso, J.P.: Energy conditions in modified gravity. Phys. Lett. B 730, 280–283 (2014). https://doi.org/10.1016/j.physletb.2014.01.066
Qiang, L.-E., Ma, Y., Han, M., Yu, D.: Five-dimensional Brans-Dicke theory and cosmic acceleration. Phys. Rev. D 71(6), 061501 (2005). https://doi.org/10.1103/PhysRevD.71.061501
Sen, S., Sen, A.A.: Late time acceleration in Brans-Dicke cosmology. Phys. Rev. D 63, 124006 (2001). https://doi.org/10.1103/PhysRevD.63.124006
De Leon, J.P.: Brans-Dicke cosmology in 4D from scalar-vacuum in 5D. J. Cosmol. Astropart. Phys. 10(03), 030 (2010). https://doi.org/10.1088/1475-7516/2010/03/030
Rasouli, S.M.M., Farhoudi, M., Sepangi, H.R.: An anisotropic cosmological model in a modified Brans-Dicke theory. Classical Quantum Gravity 28(15), 155004 (2011). https://doi.org/10.1088/0264-9381/28/15/155004
De Leon, J.P.: Late time cosmic acceleration from vacuum Brans-Dicke theory in 5D. Classical Quantum Gravity 27(9), 095002 (2010). https://doi.org/10.1088/0264-9381/27/9/095002
Aguilar, J.E.M., Romero, C., Barros, A.: Modified Brans-Dicke theory of gravity from five-dimensional vacuum. Gen. Relativ. Gravit 40(1), 117–130 (2008). https://doi.org/10.1007/s10714-007-0517-0
Wald, R.M.: Asymptotic behavior of homogeneous cosmological models in the presence of a positive cosmological constant. Phys. Rev. D 28, 2118–2120 (1983). https://doi.org/10.1103/PhysRevD.28.2118
Carroll, S.M., Chatwin-Davies, A.: Cosmic equilibration: A holographic no-hair theorem from the generalized second law. Phys. Rev. D 97, 046012 (2018). https://doi.org/10.1103/PhysRevD.97.046012
Fradkin, E.S., Tseytlin, A.A.: Quantum string theory effective action. Nucl. Phys. B 261, 1–27 (1985). https://doi.org/10.1016/0550-3213(85)90559-0
Raychaudhuri, A.: Relativistic cosmology. I. Phys. Rev. 98(4), 1123 (1955). https://doi.org/10.1103/PhysRev.98.1123
Sharif, M., Fatima, H.I.: Energy conditions for Bianchi type I universe in \(f({G})\) gravity. Astrophys. Space Sci. 353(1), 259–265 (2014). https://doi.org/10.1007/s10509-014-2000-1
Pimentel, O.M., Lora-Clavijo, F., González, G.A.: The energy-momentum tensor for a dissipative fluid in general relativity. Gen. Relativ. Gravit 48(10), 17 (2016). https://doi.org/10.1007/s10714-016-2121-7
Curiel, E.: In: Lehmkuhl, D., Schiemann, G., Scholz, E. (eds.) A primer on energy conditions, vol. 13, pp. 43–104. Birkhäuser, New York, NY, (2017). https://doi.org/10.1007/978-1-4939-3210-8_3
Abbott, B.P., Abbott, R., Abbott, T., Acernese, F., Ackley, K., Adams, C., Adams, T., Addesso, P., Adhikari, R., Adya, V.B., et al.: GW170817: observation of gravitational waves from a binary neutron star inspiral. Phys. Rev. Lett. 119(16), 161101 (2017). https://doi.org/10.1103/PhysRevLett.119.161101
Ezquiaga, J.M., Zumalacárregui, M.: Dark energy after GW170817: dead ends and the road ahead. Phys. Rev. Lett. 119(25), 251304 (2017). https://doi.org/10.1103/PhysRevLett.119.251304
Vijaykumar, A., Kapadia, S.J., Ajith, P.: Constraints on the time variation of the gravitational constant using gravitational wave observations of binary neutron stars. Phys. Rev. Lett. 126(14), 141104 (2021). https://doi.org/10.1103/PhysRevLett.126.141104
Verma, P.: Probing gravitational waves from pulsars in Brans-Dicke theory. Universe 7(7), 235 (2021). https://doi.org/10.3390/universe7070235
Heydari-Fard, M., Sepangi, H.: Anisotropic brane gravity with a confining potential. Phys. Lett. B 649(1), 1–11 (2007). https://doi.org/10.1016/j.physletb.2007.04.008
Sharif, M., Shamir, M.F.: Non-vacuum Bianchi types I and V in \(f({R})\) gravity. Gen. Relativ. Gravit 42(11), 2643–2655 (2010). https://doi.org/10.1007/s10714-010-1005-5
Sharif, M., Waheed, S.: Anisotropic universe models in Brans-Dicke theory. Eur. Phys. J. C 72(2), 1–12 (2012). https://doi.org/10.1140/epjc/s10052-012-1876-6
de Araujo, J.C.N.: The dark energy-dominated universe. Astropart. Phys. 23(2), 279–286 (2005). https://doi.org/10.1016/j.astropartphys.2004.12.004
Bennett, C.L., Larson, D., Weiland, J.L., Jarosik, N., Hinshaw, G., Odegard, N., Smith, K., Hill, R., Gold, B.: Nine-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: final maps and results. ApJS 208(2), 20 (2013). https://doi.org/10.1088/0067-0049/208/2/20
Aghanim, N., Akrami, Y., Ashdown, M., Aumont, J., Baccigalupi, C., Ballardini, M., Banday, A., Barreiro, R., Bartolo, N., Basak, S.: Planck 2018 results-VI. Cosmological parameters. Astron. Astrophys. 641, 6 (2020). https://doi.org/10.1051/0004-6361/201833910
Kolassis, C.A., Santos, N.O., Tsoubelis, D.: Energy conditions for an imperfect fluid. Classical Quantum Gravity 5(10), 1329 (1988). https://doi.org/10.1088/0264-9381/5/10/011
Acknowledgements
The first author (H. Amani) is grateful to Dr. Mehrdad Farhoudi for his guidance and valuable discussions at Shahid Beheshti university in 2019. The second author (P. Halpern) is grateful to the late Paul S. Wesson for his kindness and support.
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Amani, H., Halpern, P. Energy conditions in a modified Brans-Dicke theory. Gen Relativ Gravit 54, 64 (2022). https://doi.org/10.1007/s10714-022-02950-3
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DOI: https://doi.org/10.1007/s10714-022-02950-3