Extended thermodynamics of the bumblebee black holes

Abstract

As a vector-tensor theory including nonminimal coupling between the Ricci tensor and a vector field, the bumblebee gravity is a potential theory to test Lorentz symmetry violation. Recently, a new class of numerical spherical black holes in the bumblebee theory was constructed. In this paper, we investigate the associated local thermodynamic properties. By introducing a pair of conjugated thermodynamic quantities $X$ and $Y$, which can be interpreted as an extension of electric potential and charge of the Reissner-Nordstr\"om black holes, we numerically construct a new first law of thermodynamics for bumblebee black holes. We then study the constant-$Y$ processes in the entropy-charge parameter space. For the constant-$Y$ processes, we also calculate the heat capacity to study the local thermodynamic stability of the bumblebee black holes. For a negative nonminimal coupling coefficient $\ensuremath{\xi}$, we find both divergent and smooth phase transitions. For a positive but small $\ensuremath{\xi}$, only a divergent phase transition is found. It turns out that there is a critical value $0.4\ensuremath{\kappa}<{\ensuremath{\xi}}_{c}<0.5\ensuremath{\kappa}$ such that when ${\ensuremath{\xi}}_{c}<\ensuremath{\xi}<2\ensuremath{\kappa}$, even the divergent phase transition disappears and the bumblebee black holes thus become locally thermodynamically unstable regardless of the bumblebee charge. As for $\ensuremath{\xi}>2\ensuremath{\kappa}$, the smooth phase transition arises again but there no longer exists any discontinuous phase transition for the bumblebee black holes.

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