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Recovering semiclassical Einstein’s equation using generalized entropy

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Abstract

In this letter, we show that the Semiclassical Einstein’s Field Equation can be recovered using the generalized entropy \(S_{gen}\). This approach is reminiscent of non-equilibrium thermodynamics. Furthermore, contrary to the entanglement equilibrium approach of deriving the semiclassical Einstein’s equation, this approach does not require any such assumptions and still recovers its correct form. Therefore, in a sense, we also show the validity of the semiclassical approximation, a crucial approach for establishing a number of important ideas such as the Hawking effect.

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Notes

  1. In this paper we set \(k_B=c=1\).

  2. The metric for a static patch in de-Sitter space is given by \(ds^2=-[1-(r/L)^2]dt^2+[1-(r/L)^2]^{-1}dr^2+r^2d\Omega ^2_2\).

  3. As can be noted here. In contrast with the classical case, the semiclassical case requires the vanishing of quantum expansion \(\Theta \) to zeroth order and not the classical expansion \(\theta \) for the semiclassical Einstein’s equation to hold. This can be understood as follows: The quantum fields violate the classical focusing while even in this the quantum focusing (\(d\Theta /d\lambda \le 0\)) holds. Therefore, focusing to the past of p must bring the quantum expansion to zero so that the increase in generalized entropy \(S_{gen}\) is proportional to the killing energy across it. This imposes a condition on the spacetime curvature that is governed by the evolution Eq. (17). The same cannot be said for the classical focusing since it does not hold in the semiclassical case.

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Acknowledgements

We would like to thank the anonymous referee who provided detailed comments and useful suggestions that greatly improved the quality of the manuscript.

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  1. School of Basic Sciences, Indian Institute of Technology, Bhubaneswar, Odisha, 752050, India

    Naman Kumar

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Kumar, N. Recovering semiclassical Einstein’s equation using generalized entropy. Gen Relativ Gravit 55, 127 (2023). https://doi.org/10.1007/s10714-023-03172-x

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