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Black Hole Entropy from Non-dirichlet Sectors, and a Bounce Solution

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Abstract

The relevance of gravitational boundary degrees of freedom and their dynamics in gravity quantization and black hole information has been explored in a series of recent works. In this work we further progress by focusing keenly on the genuine gravitational boundary degrees of freedom as the origin of black hole entropy. Wald’s entropy formula is scrutinized, and the reason that Wald’s formula correctly captures the entropy of a black hole examined. Afterwards, limitations of Wald’s method are discussed; a coherent view of entropy based on boundary dynamics is presented. The discrepancy observed in the literature between holographic and Wald’s entropies is addressed. We generalize the entropy definition so as to handle a time-dependent black hole. Large gauge symmetry plays a pivotal role. Non-Dirichlet boundary conditions and gravitational analogues of Coleman-De Luccia bounce solutions are central in identifying the microstates and differentiating the origins of entropies associated with different classes of solutions. The result in the present work leads to a view that black hole entropy is entanglement entropy in a thermodynamic setup.

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Notes

  1. The same name,“bounce,” is used for the “Lorentzian bounce solutions” in the gravitational context [10,11,12,13,14,15,16]. The main conceptual difference of our view is the instontonic physics aspect of a CdL bounce solution.

  2. For how the worldvolume theory manifests, see sect. 3.3 of [5].

  3. In this work we use the setup only at a conceptual level. See [25] for a recent analysis in which BHI was studied by employing the setup.

  4. In some of the comments, replacing LGS by asymptotic symmetry would be more appropriate; we will not be concerned with this distinction.

  5. In [8] a simplistic gravitational bounce solution based on a shock wave was constructed, postponing construction of a more realistic solution. In fact, the construction should be possible by following [35] of which we have become aware recently.

  6. Related discussions were presented in the earlier works, such as [6, 8]. It is, however, the discussion around Eqs. (5 and 6) below that most clearly and explicitly “pins the issue down.”

  7. In a Hartle-Hawking vacuum, for instance, the boundary degrees of freedom are excited. Since a Dirichlet boundary condition was always assumed, this raises a question on internal consistency of the analysis.

  8. Indeed it is well known that it is Unruh vacuum that describes a decaying BH. The Unruh vacuum was obtained by carefully conducting perturbative analysis around a collapsing solution. Although it shuold be possible to consider a bounce solution associated with decay of Unruh vacuum, the analysis will enevitably be more complicated. To some degree, the simplification involved is analogous to the one achieved by examining an Unruh effect, instead of Unruh vacuum, for BH decay.

  9. This is so in the component notation. In the coordinate-free notation, tensor types are preserved both by Lie and covariant derivatives (but not by a covariant differential) [47].

  10. Another way to see this is through a path integral: the measure should ultimately be over the physical states, which have support on the boundary [40].

  11. Strictly speaking, we believe that the degrees of freedom here should include those within the stretched horizon.

  12. The singularity at the EH is usually referred to as a ‘coordinate singularity.’ All this means is that it is not a curvature singularity. To our view the term coordinate singularity is misleading at the quantum level in that it gives an impression that the singularity is not physical. If, for instance, a non-smooth EH turns out to be real, which we anticipate, the singularity will be very physical, though not a curvature singularity.

  13. The relevance of the EH in entropy computation is long known in other methods, such as those of [55, 56]. In the present framework both the EH and asymptotic boundary play roles: LGS is a symmetry associated with the boundary, whereas the nonzero contributions come from the EH.

  14. In general, this is not strictly true since \(\sqrt{-g}\) is a tensor density. However, with \(\nabla _\mu \xi ^\mu =0\), the quantities under consideration do transform as scalars.

  15. Although the original Wald’s definition requires the presence of a Killing vector, the charge was also given in subsequent works by an expression based on a functional derivative with respect to the Riemann tensor. The present result implies that a formal entropy calculation based on such a definition will not reproduce the entropy based on an Euclidean action.

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  1. Department of Applied Mathematics, Philander Smith College, Little Rock, AR, 72202, USA

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Park, I.Y. Black Hole Entropy from Non-dirichlet Sectors, and a Bounce Solution. Found Phys 53, 74 (2023). https://doi.org/10.1007/s10701-023-00719-5

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