Gravitational tension, spacetime pressure and black hole volume

Abstract

Abstract:We study the first law of black hole thermodynamics in the presence of surrounding gravitational fields and argue that variations of these fields are naturally incorporated in the first law by defining gravitational tension or gravitational binding energy. We demonstrate that this notion can also be applied in Anti-de Sitter spacetime, in which the surrounding gravitational field is sourced by a cosmological fluid, therefore showing that spacetime volume and gravitational tension encode the same physics as spacetime pressure and black hole volume. We furthermore show that it is possible to introduce a definition of spacetime pressure and black hole volume for any spacetime with characteristic length scales which does not necessarily require a cosmological constant sourcing Einstein equations. However, we show that black hole volume is non-universal in the flat spacetime limit, questioning its significance. We illustrate these ideas by studying the resulting black hole volume of Kaluza-Klein black holes and of a toy model for a black hole binary system in five spacetime dimensions (the black saturn solution) as well as of several novel perturbative black hole solutions. These include the higher-dimensional Kerr-Newman solution in Anti-de Sitter spacetime as well as other black holes in plane wave and Lifshitz spacetimes.

Extremal black hole horizons

“…In addition, the topology of all horizons in D = 5 encountered in this work is either S 3 or S 1 × S 2 , thus agreeing with the extremal near-horizon geometry classification of [43,44] and the uniqueness theorems of [45,46]. It would be interesting to obtain extremal versions of the black holes found in [20,36,47] in (Anti)-de Sitter spacetime and to check whether they also agree with [42,48].…”

Black hole chemistry: thermodynamics with Lambda

“…Amusingly, the result is the same as if the black hole were a ball of radius r + in Euclidean space. A recent contrasting viewpoint [95] is that thermodynamic volume should be replaced with a more general notion of gravitational tension that describes the extra energy associated with the presence of gravitational fields surrounding a black hole. Gravitational tension vanishes in the flat-space limit and is proportional r 3 + /l 3 for a Schwarzschild-AdS black hole.…”

Thermodynamic volume of cosmological solitons

“…However a class of black holes has recently been found that violates this conjecture [45,46], necessitating further investigation of the role and meaning of the volume [47]. The relationship of V to other proposed notions of volume [48,49] is an ongoing subject of investigation.…”