The zeroth law of black hole thermodynamics in arbitrary higher derivative theories of gravity

TL;DR

This paper proves the zeroth law of black hole thermodynamics, showing surface gravity remains constant on the horizon in arbitrary higher derivative gravity theories, extending classical results to more complex gravitational models.

Contribution

It provides a perturbative proof of the zeroth law in higher derivative gravity theories, connecting surface gravity constancy to equations of motion and near-horizon symmetries.

Findings

Surface gravity remains constant on the horizon in higher derivative theories.

The proof is valid to all orders in perturbation theory.

Near-horizon boost symmetry constrains equations of motion.

Abstract

We consider diffeomorphism invariant theories of gravity with arbitrary higher derivative terms in the Lagrangian as corrections to the leading two derivative theory of Einstein's general relativity. We construct a proof of the zeroth law of black hole thermodynamics in such theories. We assume that a stationary black hole solution in an arbitrary higher derivative theory can be obtained by starting with the corresponding stationary solution in general relativity and correcting it order by order in a perturbative expansion in the coupling constants of the higher derivative Lagrangian. We prove that surface gravity remains constant on its horizon when computed for such stationary black holes, which is the zeroth law. We argue that the constancy of surface gravity on the horizon is related to specific components of the equations of motion in such theories. We further use a specific boost symmetry of the near horizon space-time of the stationary black hole to constrain the off-shell structure of the equations of motion. Our proof for the zeroth law is valid up to arbitrary order in the expansion in the higher derivative couplings.