Abstract
We present a systematic review of thermodynamics of horizons in regular spherically symmetric spacetimes of the Kerr-Schild class, d s 2 = g ( r ) d t 2 − g − 1 ( r ) d r 2 − r 2 d Ω 2 , both asymptotically flat and with a positive background cosmological constant λ . Regular solutions of this class have obligatory de Sitter center. A source term in the Einstein equations satisfies T t t = T r r and represents an anisotropic vacuum dark fluid ( p r = − ρ ), defined by the algebraic structure of its stress-energy tensor, which describes a time-evolving and spatially inhomogeneous, distributed or clustering, vacuum dark energy intrinsically related to space-time symmetry. In the case of two vacuum scales it connects smoothly two de Sitter vacua, 8 π G T ν μ = Λ δ ν μ as r → 0 , 8 π G T ν μ = λ δ ν μ as r → ∞ with λ < Λ . In the range of the mass parameter M c r 1 ≤ M ≤ M c r 2 it describes a regular cosmological black hole directly related to a vacuum dark energy. Space-time has at most three horizons: a cosmological horizon r c , a black hole horizon r b < r c , and an internal horizon r a < r b , which is the cosmological horizon for an observer in the internal R-region asymptotically de Sitter as r → 0 . Asymptotically flat regular black holes ( λ = 0 ) can have at most two horizons, r b and r a . We present the basic generic features of thermodynamics of horizons revealed with using the Padmanabhan approach relevant for a multi-horizon space-time with a non-zero pressure. Quantum evaporation of a regular black hole involves a phase transition in which the specific heat capacity is broken and changes sign while a temperature achieves its maximal value, and leaves behind the thermodynamically stable double-horizon ( r a = r b ) remnant with zero temperature and positive specific heat. The mass of objects with the de Sitter center is generically related to vacuum dark energy and to breaking of space-time symmetry. In the cosmological context space-time symmetry provides a mechanism for relaxing cosmological constant to a certain non-zero value. We discuss also observational applications of the presented results.
Highlights
In 1973 Bekenstein introduced a black hole entropy related to the Hawking area theorem [1].A year later Hawking found that an observer at r = const outside a black hole will detect a stationary flux of particles from a black hole with the thermal spectrum [2,3]
This gave the birth to thermodynamics of black holes [4,5,6]
Later generic evolution was studied for evaporation of three horizons in a space-time with two vacuum scales and the phase transition in the course of evaporation was found for a regular cosmological black hole [39]
Summary
In 1973 Bekenstein introduced a black hole entropy related to the Hawking area theorem [1]. Later generic evolution was studied for evaporation of three horizons in a space-time with two vacuum scales and the phase transition in the course of evaporation was found for a regular cosmological black hole [39] In both cases quantum evaporation of horizons leaves behind a thermodynamically stable double-horizon remnant (an extremal black hole) with zero temperature and positive specific heat ([34,39], for a review [40]).
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