Thermodynamics of Black Holes in Massive Gravity

TL;DR

This paper investigates the thermodynamics and phase transitions of charged black holes in higher-dimensional massive gravity with a negative cosmological constant, revealing conditions for stability and phase changes depending on parameters like curvature, chemical potential, and graviton mass.

Contribution

It introduces new charged black hole solutions in higher-dimensional massive gravity and analyzes their thermodynamic behavior and phase structure in different ensembles, highlighting the role of graviton mass terms.

Findings

Hawking-Page phase transition occurs when a specific parameter combination is positive.

Black holes are thermodynamically stable with positive capacity when the parameter combination is negative.

Phase transitions between small and large black holes can occur in higher dimensions even without charge.

Abstract

We present a class of charged black hole solutions in an ($n+2)$-dimensional massive gravity with a negative cosmological constant, and study thermodynamics and phase structure of the black hole solutions both in grand canonical ensemble and canonical ensemble. The black hole horizon can have a positive, zero or negative constant curvature characterized by constant $k$. By using Hamiltonian approach, we obtain conserved charges of the solutions and find black hole entropy still obeys the area formula and the gravitational field equation at the black hole horizon can be cast into the first law form of black hole thermodynamics. In grand canonical ensemble, we find that thermodynamics and phase structure depends on the combination $k -\mu^2/4 +c_2 m^2$ in the four dimensional case, where $\mu$ is the chemical potential and $c_2m^2$ is the coefficient of the second term in the potential associated with graviton mass. When it is positive, the Hawking-Page phase transition can happen, while as it is negative, the black hole is always thermodynamically stable with a positive capacity. In canonical ensemble, the combination turns out to be $k+c_2m^2$ in the four dimensional case. When it is positive, a first order phase transition can happen between small and large black holes if the charge is less than its critical one. In higher dimensional ($n+2 \ge 5$) case, even when the charge is absent, the small/large black hole phase transition can also appear, the coefficients for the third ($c_3m^2$) and/or the fourth ($c_4m^2$) terms in the potential associated with graviton mass in the massive gravity can play the same role as the charge does in the four dimensional case.