Thermodynamics of Regular Cosmological Black Holes with the de Sitter Interior

Irina Dymnikova,Michał Korpusik

doi:10.3390/e13121967

Irina Dymnikova, Michał Korpusik

Open Access

https://doi.org/10.3390/e13121967

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Abstract

We address the question of thermodynamics of regular cosmological spherically symmetric black holes with the de Sitter center. Space-time is asymptotically de Sitter as r → 0 and as r → ∞. A source term in the Einstein equations connects smoothly two de Sitter vacua with different values of cosmological constant: 8πGTμν = Λδμν as r → 0, 8πGTμν = λδμν as r → ∞ with λ < Λ. It represents an anisotropic vacuum dark fluid defined by symmetry of its stress-energy tensor which is invariant under the radial boosts. In the range of the mass parameter Mcr1 ≤ M ≤ Mcr2 it describes a regular cosmological black hole. Space-time in this case has three horizons: a cosmological horizon rc, a black hole horizon rb < rc, and an internal horizon ra < rb, which is the cosmological horizon for an observer in the internal R-region asymptotically de Sitter as r → 0. We present the basicfeatures of space-time geometry and the detailed analysis of thermodynamics of horizons using the Padmanabhan approach relevant for a multi-horizon space-time with a non-zero pressure. We find that in a certain range of parameters M and q =√Λ/λ there exist a global temperature for an observer in the R-region between the black hole horizon rb and cosmological horizon rc. We show that a second-order phase transition occurs in the course of evaporation, where a specific heat is broken and a temperature achieves its maximal value. Thermodynamical preference for a final point of evaporation is thermodynamically stable double-horizon (ra = rb) remnant with the positive specific heat and zero temperature.

Highlights

In 1973 Four laws of black hole mechanics were formulated [1], and Bekenstein introduced a black hole entropy [2]
Thermodynamics of a regular black hole with de Sitter interior is dictated by the typical behavior of the metric function g(r), generic for the considered class of space-times specified by symmetry (9) of a stress-energy tensor satisfying the weak energy condition
Let us note that this conclusion follows unambiguously from thermodynamics of horizons, so that a regular cosmological black hole, like the asymptotically flat regular black hole [65,80], leaves behind the remnant free of the existential problems

Summary

Introduction

In 1973 Four laws of black hole mechanics were formulated [1], and Bekenstein introduced a black hole entropy [2]. Thermodynamics of a regular black hole with de Sitter interior is dictated by the typical behavior of the metric function g(r), generic for the considered class of space-times specified by symmetry (9) of a stress-energy tensor satisfying the weak energy condition. Relevant physical mechanism can be self-regulation of geometry noted by Poisson and Israel [52]: An asymptotically flat space-time generated during spherically symmetric gravitational collapse can be described by the Schwarzschild vacuum solution down to the quantum barrier This barrier there may exist a layer of an uncertain depth in which geometry could be self-regulatory and describable semiclassically by the Einstein field equations with a source term representing vacuum polarization effects [52]. Detailed investigation for the case of renormalization group improved black hole (without a de Sitter core) has shown that quantum gravity effects weaken the strength of the Cauchy singularity and suggest that presence of some self-regulation mechanism could prevent the local curvature from divergence on the Cauchy horizon [88].

Thermodynamics of the Black Hole Horizon

The Case of Global Temperature for an Observer between rb and rc

Findings

Summary and Discussion