Black holes with vector hair

Understand

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.