Black holes with vector hair

TL;DR

This paper explores black hole solutions in Einstein gravity coupled to vector fields, revealing how vector fields, both minimally and non-minimally coupled, modify the thermodynamic laws and produce novel extremal black holes with unique properties.

Contribution

It presents new extremal and non-extremal black hole solutions with vector hair, demonstrating how vector fields alter the first law of thermodynamics in various gravity theories.

Findings

Black hole solutions with vector hair in Einstein gravity.

Modification of the first law of thermodynamics by vector fields.

Existence of asymptotically flat and Lifshitz black holes with vector hair.

Abstract

In this paper, we consider Einstein gravity coupled to a vector field, either minimally or non-minimally, together with a vector potential of the type $V=2\Lambda_0+\ft 12 m^2 A^2+\gamma_4 A^4$. For a simpler non-minimally coupled theory with $\Lambda_0=m=\gamma_4=0$, we obtain both extremal and non-extremal black hole solutions that are asymptotic to Minkowski space-times. We study the global properties of the solutions and derive the first law of thermodynamics using Wald formalism. We find that the thermodynamical first laws of the extremal black holes are modified by a one form associated with the vector field. In particular, due to the existence of the non-minimal coupling, the vector forms thermodynamic conjugates with the graviton mode and partly contributes to the one form modifying the first laws. For a minimally coupled theory with $\Lambda_0\neq 0$, we also obtain one class of asymptotically flat extremal black hole solutions in general dimensions. This is possible because the parameters $(m^2,\gamma_4)$ take certain values such that $V=0$. In particular, we find that the vector also forms thermodynamic conjugates with the graviton mode and contributes to the corresponding first laws, although the non-minimal coupling has been turned off. Thus all the extremal black hole solutions that we obtain provide highly non-trivial examples how the first law of thermodynamics can be modified by a either minimally or non-minimally coupled vector field. We also study Gauss-Bonnet gravity non-minimally coupled to a vector and obtain asymptotically flat black holes and Lifshitz black holes.