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Energy conditions in a modified Brans-Dicke theory

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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

  1. 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.

  2. One could consider it as being an inflationary phase which follows the power law at the present time.

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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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