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-Nordstrøm 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.
Highlights
We will adopt the Wald formalism to derive the first law of thermodynamics for all the solutions we obtain
We study generalized Einstein-Proca theories in general dimensions by introducing either a quartic selfinteraction term for the vector or non-minimally coupled terms between the curvature and the vector
We find that there are two distinct class solutions, both of which are general static and have maximal symmetry, depending on whether Ar vanishes
Summary
To govern the validity of Wald entropy formula, one should impose proper boundary conditions on the horizon for both At and Ar , which in general results to a degenerate solution characterized by only one parameter, in contrast to the general two-parameter family solutions. This does not make sense in the derivation of the first law. 4–6, we study non-minimally coupled vector-tensor theories and obtain a lot of exact black hole solutions depending on the parameters of the theories.
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