Black holes in vector-tensor theories and their thermodynamics

TL;DR

This paper explores black hole solutions in vector-tensor theories of gravity, analyzing their thermodynamics and uncovering new solutions with unique properties, while highlighting challenges in defining their thermodynamic laws.

Contribution

It provides new exact black hole solutions in Einstein-vector theories, including minimal and non-minimal couplings, and discusses the complexities in their thermodynamic descriptions.

Findings

Black hole solutions resemble Reissner-Nordström black holes without a global charge.

Many solutions are static and maximally symmetric, depending on theory parameters.

Thermodynamics involves subtleties due to vector degrees of freedom, affecting the Wald entropy formula.

Abstract

In this paper, we study Einstein gravity either minimally or non-minimally coupled to a vector field which breaks the gauge symmetry explicitly in general dimensions. We first consider a minimal theory which is simply the Einstein-Proca theory extended with a quartic self-interaction term for the vector field. We obtain its general static maximally symmetric black hole solution and study the thermodynamics using Wald formalism. The aspects of the solution are much like a Reissner-Nordstr{\o}m black hole in spite of that a global charge cannot be defined for the vector. For non-minimal theories, we obtain a lot of exact black hole solutions, depending on the parameters of the theories. In particular, many of the solutions are general static and have maximal symmetry. However, there are some subtleties and ambiguities in the derivation of the first laws because the existence of an algebraic degree of freedom of the vector in general invalids the Wald entropy formula. The thermodynamics of these solutions deserves further studies.