Violation of the first law of black hole thermodynamics in $f(T)$ gravity

TL;DR

This paper demonstrates that the first law of black hole thermodynamics is generally violated in $f(T)$ gravity, leading to possible non-equilibrium thermodynamics and entropy production, unlike in other gravity theories.

Contribution

It reveals the violation of the first law in $f(T)$ gravity due to lack of local Lorentz invariance and explores conditions for avoiding singularities and superluminal speeds.

Findings

Violation of the first law in $f(T)$ gravity.

Entropy production indicates non-equilibrium thermodynamics.

Conditions on $f''(T)$ to avoid singularities and superluminal motion.

Abstract

We prove that, in general, the first law of black hole thermodynamics, $\delta Q=T\delta S$, is violated in $f(T)$ gravity. As a result, it is possible that there exists entropy production, which implies that the black hole thermodynamics can be in non-equilibrium even in the static spacetime. This feature is very different from that of $f(R)$ or that of other higher derivative gravity theories. We find that the violation of first law results from the lack of local Lorentz invariance in $f(T)$ gravity. By investigating two examples, we note that $f"(0)$ should be negative in order to avoid the naked singularities and superluminal motion of light. When $f"(T)$ is small, the entropy of black holes in $f(T)$ gravity is approximatively equal to $\frac{f'(T)}{4}A$.