Counterterm method and thermodynamics of hairy black holes in a vector-tensor theory with Abelian gauge symmetry breaking

Ai-Chen Li

doi:10.1103/physrevd.104.044040

Abstract

For a type of nonminimally coupled vector-tensor theories with Abelian gauge symmetry breaking in four-dimensional spacetime and correspondingly asymptotic non-AdS black hole solutions including a cosmological constant, we construct the appropriate boundary terms and derive the associated junction condition. In order to remove the divergences in the stress tensor which is localized on the spacetime boundary, we also involve the suitable surface counterterms into the total action. Using the counterterm method, we calculate the black hole mass. An implicit relation between the black hole carge $Q$ and other parameters is implied by combining the expression of the black hole mass with the first law of black hole thermodynamics. With this implicit relation, we can prove the inequality $Q\ensuremath{\le}M$ which is a general bound for most of charged black holes. Besides, the phase structure of black holes is also investigated in the grand canonical ensemble.

Highlights

In recent decades, the studies about hairy black holes have become a hot topic
A class of interesting black hole solutions in four-dimensional spacetime were obtained by [48] from a type of vector-tensor theory in which the Abelian gauge symmetry is broken by coupling the vector with gravity in a nonminimal way [49,50]
II, we briefly review a type of vector-tensor theory with broken Abelian gauge symmetry in four-dimensional spacetime and correspondingly asymptotic non-AdS black hole solutions including a cosmological constant

Summary

INTRODUCTION

The studies about hairy black holes have become a hot topic. One motivation is the anti–de Sitter space/conformal field theory correspondence [1,2,3], i.e., AdS=CFT. A class of interesting black hole solutions in four-dimensional spacetime were obtained by [48] from a type of vector-tensor theory in which the Abelian gauge symmetry is broken by coupling the vector with gravity in a nonminimal way [49,50]. Our purpose is to find the appropriate boundary terms and construct the effective holographic surface counterterms for a type of nonminimally coupled vector-tensor theory and correspondingly asymptotic nonAdS black hole solutions including a cosmological constant [48]. The right-hand side (rhs) of (24) is derived from the geometry of bulk spacetime, while the Tab in l.h.s corresponds to the stress tensor of the matter field living on the spacetime boundary

Surface counterterms and holographic energy

The first law of black hole thermodynamics

Phase structures analysis

CONCLUSIONS AND DISCUSSION