Black Hole Zeroth Law in Horndeski Gravity

TL;DR

This paper proves the zeroth law of black hole mechanics within a specific class of Horndeski gravity, demonstrating that the surface gravity remains constant on the event horizon under certain conditions.

Contribution

It provides the first proof of the zeroth law in a special Horndeski gravity without assuming spacetime symmetries.

Findings

Surface gravity is constant on the event horizon in the studied Horndeski gravity.

The proof does not require assumptions about spacetime symmetries.

The theory smoothly limits to Einstein gravity as the coupling constant approaches zero.

Abstract

The four laws of black hole mechanics have been put forward for a long time. However, the zeroth law, which states that the surface gravity of a stationary black hole is a constant on the event horizon, still lacks universal proof in various modified gravitational theories. In this paper, we study the zeroth law {in a special Horndeski gravity}, which is an interesting gravitational theory with a nonminimally coupled scalar field. After assuming that {the nonminimally coupled scalar field has the same symmetries with the spacetime,} the minimally coupled matter fields satisfy the dominant energy condition and the Horndeski gravity has a smooth limit to Einstein gravity when the coupling constant approaches zero, we prove the zeroth law based on the gravitational equation in Horndeski gravity without any assumption to the spacetime symmetries.