Generic Features of Thermodynamics of Horizons in Regular Spherical Space-Times of the Kerr-Schild Class

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.