Einstein-Katz action, variational principle, Noether charges and the thermodynamics of AdS-black holes

TL;DR

This paper develops a covariant approach to defining conserved charges and thermodynamics for AdS black holes using the Einstein-Katz action, broadening boundary conditions and linking Noether charges to thermodynamic relations.

Contribution

It introduces a variational principle based on the Einstein-Katz action that naturally yields conserved charges and thermodynamic relations for AdS black holes with scalar hair.

Findings

The KBL superpotential provides a straightforward way to compute black hole mass.

Solutions satisfying the variational principle obey the Gibbs relation.

Dyonic black holes follow the first law of thermodynamics with this approach.

Abstract

In this paper we describe 4-dimensional gravity coupled to scalar and Maxwell fields by the Einstein-Katz action, that is, the covariant version of the "Gamma-Gamma $-$ Gamma-Gamma" part of the Hilbert action supplemented by the divergence of a generalized "Katz vector". We consider static solutions of Einstein's equations, parametrized by some integration constants, which describe an ensemble of asymptotically AdS black holes. Instead of the usual Dirichlet boundary conditions, which aim at singling out a specific solution within the ensemble, we impose that the variation of the action vanishes on shell for the broadest possible class of solutions. We will see that, when a long-range scalar "hair" is present, only sub-families of the solutions can obey that criterion. The Katz-Bicak-Lynden-Bell ("KBL") superpotential built on this (generalized) vector will then give straightforwardly the Noether charges associated with the spacetime symmetries (that is, in the static case, the mass). Computing the action on shell, we will see next that the solutions which obey the imposed variational principle, and with Noether charges given by the KBL superpotential, satisfy the Gibbs relation, the Katz vectors playing the role of "counterterms". Finally, we show on the specific example of dyonic black holes that the sub-class selected by our variational principle satisfies the first law of thermodynamics when their mass is defined by the KBL superpotential.