Where is the PdV term in the first law of black hole thermodynamics?

TL;DR

This paper explores the inclusion of pressure and volume in the first law of black hole thermodynamics, proposing a new volume definition and analyzing phase transitions in rotating AdS black holes.

Contribution

It introduces a definition of volume for black holes to extend the first law and analyzes the Van der Waals type phase transition in rotating AdS black holes.

Findings

A new volume definition for black holes is proposed.

The phase transition in rotating AdS black holes is of Van der Waals type.

Mean field exponents characterize the second order phase transition.

Abstract

Traditional treatments of the first law of black hole thermodynamics do not include a discussion of pressure and volume. We give an overview of recent developments proposing a definition of volume that can be used to extend the first law to include these appropriately. New results are also presenting relating to the critical point and the associated second order phase transition for a rotating black-hole in four-dimensional space-time which is asymptotically anti-de Sitter. In line with known results for a non-rotating charged black-hole, this phase transition is shown to be of Van der Waals type with mean field exponents.