Geometric and Thermodynamic Volume of Hairy Black Branes

TL;DR

This paper extends the thermodynamic analysis of hairy black branes in anti-de Sitter space, deriving a generalized Smarr relation, defining a thermodynamic volume proportional to geometric volume, and establishing an extended first law.

Contribution

It introduces a general method to compute the thermodynamic volume and enthalpy for hairy black branes, applicable to a wide class of solutions, using Hamiltonian and Komar techniques.

Findings

Thermodynamic volume is proportional to geometric volume.

A generalized Smarr relation is established for hairy black branes.

An extended first law of thermodynamics is derived for these solutions.

Abstract

With the objective to generalize previous results found for a handful of explicit solutions, we study the extended thermodynamics of a black brane with minimally coupled scalar hair in D-dimensional asymptotically anti-de Sitter spacetimes. Using Komar integration and the Hamiltonian formalism to calculate the conserved charges, we obtain a Smarr relation that is applicable to a wide variety of solutions and suggests a more general definition of the thermodynamic volume. This volume is found to be proportional to the geometric volume, and a simple prescription is given to calculate the constant of proportionality. Moreover, the method of Hamiltonian perturbations yields an extended first law of thermodynamics for hairy black branes, thus giving a definition for their enthalpy. These results are then verified by applying them to some of the explicit solutions that exist in the literature.