First law of black hole mechanics in variable background fields

TL;DR

This paper extends the understanding of black hole thermodynamics by analyzing the first law in theories with background fields that break diffeomorphism symmetry, demonstrating the applicability of Noether charge entropy even when symmetry is explicitly broken.

Contribution

It introduces a formalism for the covariant first law of black hole mechanics with background fields, including non-dynamical ones, and derives Smarr formulas in massive gravity.

Findings

Noether entropy remains applicable despite explicit symmetry breaking.

Derived two Smarr formulas for static black branes in massive gravity.

Validated the formalism without relying solely on the first law.

Abstract

It is well known that in general theories of gravity with the diffeomorphism symmetry, the black hole entropy is a Noether charge. But what will happen if the symmetry is explicitly broken? By investigating the covariant first law of black hole mechanics with background fields, we show that the Noether entropy is still applicable due to the local nature of the black hole entropy. Moreover, motivated by the proposal that the cosmological constant behaves as a thermodynamic variable, we allow the non-dynamical background fields to be varied. To illustrate this general formalism, we study a generic static black brane in the massive gravity. Using the first law and the scaling argument, we obtain two Smarr formulas. We show that both of them can be retrieved without relying on the first law, hence providing a self-consistent check of the theory.