The idea of an oscillating Universe has remained a topic of interest even after the discovery of dark energy. This paper confirms this idea by means of another well-established theory in general relativity, the embedding of curved spacetimes in higher-dimensional flat spacetimes: an n-dimensional Riemannian space is said to be of embedding class m if m？+n is the lowest dimension of the flat space in which the given space can be embedded; here . So a four-dimensional Riemannian space is of class two and can therefore be embedded in a six-dimensional flat space. A line element of class two can be reduced to a line element of class one by a suitable coordinate transformation. The extra dimension can be either spacelike or timelike, leading to accelerating and decelerating expansions, respectively. Accordingly, it is proposed in this paper that the free parameter occurring in the transformation be a periodic function of time. The result is a mathematical model that can be interpreted as a periodic change in the signature of the embedding space. This signature change may be the best model for an oscillating Universe and complements various models proposed in the literature.
References
[1]
White, A., Weinfurtner, S. and Visser, M. (2010) Signature Change Events: A Challenge for Quantum Gravity? Classical and Quantum Gravity, 27, Article ID: 045007. https://doi.org/10.1088/0264-9381/27/4/045007
[2]
Sahoo, P.K., Tripathy, S.K. and Sahoo, P. (2018) A Periodic Varying Deceleration Parameter in f(R, T) Gravity. Modern Physics Letters A, 33, Article ID: 1850193. https://doi.org/10.1142/S0217732318501936
[3]
Pati, Laxmipriya, Mishra, B. and Tripathy, S.K. (2021) Model Parameters in the Context of Late Time Cosmic Acceleration in f(Q, T) Gravity. Physica Scripta, 96, Article ID: 105003. https://doi.org/10.1088/1402-4896/ac0f92
[4]
Usmani, A.A., Ghosh, P.P., Mukhopadhyay, U., Ray, P.C. and Ray, S. (2008) The Dark Energy Equation of State. Monthly Notices of the Royal Astronomical Society, 386, L92-L95. https://doi.org/10.1111/j.1745-3933.2008.00468.x
[5]
Ghosh, P.P., Ray, S., Usmani, A.A. and Mukhopadhyay, U. (2013) Oscillatory Univerese, Dark Energy and General Relativity. Astrophysics and Space Science, 345, 367-371. https://doi.org/10.1007/s10509-013-1390-9
[6]
Mars, M., Senovilla, J.M.M. and Vera, R. (2008) Accelerating Expansion and Change of Signature. EAS Publications Series, 30, 307-310. https://doi.org/10.1051/eas:0830050
[7]
Bojowald, M. and Brahma, S. (2020) Loop Quantum Gravity, Signature Change, and the No-boundary Proposal. Physical Review D, 102, Article ID: 106023. https://doi.org/10.1103/PhysRevD.102.106023
[8]
Campbell, J. (1926) A Course of Differential Geometry. Clarendon, Oxford.
[9]
Wesson, P.S. (2015) The Status of Modern Five-Dimensional Gravity. International Journal of Modern Physics D, 24, Article ID: 1530001. https://doi.org/10.1142/S0218271815300013
[10]
Wesson P.S. and Ponce de León, J. (1992) Kaluza-Klein Equations, Einstein’s Equations, and an Effective Energy-Momentum Tensor. Journal of Mathematical Physics, 33, 3883-3887. https://doi.org/10.1063/1.529834
[11]
Seahra S.S. and Wesson, P.S. (2003) Application of the Campbell-Magaard Theorem to Higher-Dimensional Physics. Classical and Quantum Gravity, 20, Article ID: 1321. https://doi.org/10.1088/0264-9381/20/7/306
[12]
Wesson, P.S. (2011) The Cosmological “Constant” and Quantization in Five Dimensions. Physics Letters B, 706, 1-5. https://doi.org/10.1016/j.physletb.2011.10.027
[13]
Maurya, S.K., Deb, D., Ray, S. and Kuhfittig, P.K.F. (2019) A Study of Anisotropic Compact Stars Based on Embedding Class 1 Condition. International Journal of Modern Physics D, 28, Article ID: 1950116. https://doi.org/10.1142/S0218271819501165
[14]
Maurya, S.K., Ratanpal, B.S. and Govender, M. (2017) Anisotropic Stars for Spherically Symmetric Space Times Satisfying the Karmarkar Condition. Annals of Physics, 382, 36-49. https://doi.org/10.1016/j.aop.2017.04.008
[15]
Maurya, S.K. and Maharaj, S.D. (2017) Anisotropic Fluid Spheres of Embedding Class One Using Karmarkar Condition. European Physical Journal C, 77, Article No. 328. https://doi.org/10.1140/epjc/s10052-017-4905-7
[16]
Maurya, S.K. and Govender, M. (2017) Generating Physically Realizable Stellar Structures via Embedding. European Physical Journal C, 77, Article No. 347. https://doi.org/10.1140/epjc/s10052-017-4916-4
[17]
Karmarkar, K.R. (1948) Gravitational Metrics of Spherical Symmetry and Class One. Proceedings of the Indian Academy of Sciences, 27, Article No. 56. https://doi.org/10.1007/BF03173443
[18]
Kuhfittig, P.K.F. and Gladney, V.D. (2018) A Model for Dark Energy Based on the Theory of Embedding. Advanced Studies in Theoretical Physics, 12, 233-241. https://doi.org/10.12988/astp.2018.8627
[19]
Feng, B., Li, M.Z., Piao, Y.-S. and Zhang, X.M. (2006) Oscillating Quintom and the Recurrent Universe. Physics Letters B, 634, 101-105. https://doi.org/10.1016/j.physletb.2006.01.066
[20]
Wald, R.M. (1984) General Relativity. Chicago University Press, Chicago, Chapter 5. https://doi.org/10.7208/chicago/9780226870373.001.0001
[21]
Nandi, K.K., Filippov, A.I., Rahaman, F., Ray, S., Usmani, A.A., Kalam, M. and DeBenedictis, A. (2009) Features of Galactic Halo in a Brane World Model and Observational Constraints. Monththly Notices of the Royal Astronomical Society, 399, 2079-2087. https://doi.org/10.1111/j.1365-2966.2009.15399.x
[22]
Poplawski, N. (2016) Universe in a Black Hole in Einstein-Cartan Gravity. The Astronomical Journal, 832, Article No. 96. https://doi.org/10.3847/0004-637X/832/2/96
[23]
Hartle, J.B. and Hawking, S.W. (1983) Wave Function of the Universe. Physical Review D, 28, 2960-2975. https://doi.org/10.1103/PhysRevD.28.2960
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. So a four-dimensional Riemannian space is of class two and can therefore be embedded in a six-dimensional flat space. A line element of class two can be reduced to a line element of class one by a suitable coordinate transformation. The extra dimension can be either spacelike or timelike, leading to accelerating and decelerating expansions, respectively. Accordingly, it is proposed in this paper that the free parameter occurring in the transformation be a periodic function of time. The result is a mathematical model that can be interpreted as a periodic change in the signature of the embedding space. This signature change may be the best model for an oscillating Universe and complements various models proposed in the literature.
