Scalar charges and the first law of black hole thermodynamics

TL;DR

This paper develops a variational framework for Einstein-Maxwell-dilaton theory, revealing how scalar charges influence black hole thermodynamics and extending the analysis to AdS spacetime with hairy black holes.

Contribution

It introduces a boundary term formulation for scalar fields in Einstein-Maxwell-dilaton theory and explores their impact on black hole thermodynamics, including in AdS space.

Findings

Scalar charges modify the first law of black hole thermodynamics.

Finite gravitational action and stress tensor are computed with scalar contributions.

Analysis extended to hairy black holes in AdS spacetime.

Abstract

We present a variational formulation of Einstein-Maxwell-dilaton theory in flat spacetime, when the asymptotic value of the scalar field is not fixed. We obtain the boundary terms that make the variational principle well posed and then compute the finite gravitational action and corresponding Brown-York stress tensor. We show that the total energy has a new contribution that depends of the asymptotic value of the scalar field and discuss the role of scalar charges for the first law of thermodynamics. We also extend our analysis to hairy black holes in Anti-de Sitter spacetime and investigate the thermodynamics of an exact solution that breaks the conformal symmetry of the boundary.