Gravitational Tension, Spacetime Pressure and Black Hole Volume

TL;DR

This paper explores the relationship between gravitational tension, spacetime pressure, and black hole volume, extending thermodynamic laws to various spacetime backgrounds and questioning the universality of black hole volume.

Contribution

It introduces a unified framework connecting gravitational tension and spacetime pressure, applicable beyond AdS spacetime, and examines the non-universality of black hole volume in flat spacetime.

Findings

Gravitational tension can be incorporated into the first law of black hole thermodynamics.

Spacetime volume and gravitational tension encode the same physics as pressure and volume.

Black hole volume is non-universal in flat spacetime, challenging its fundamental significance.

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.