Black hole thermodynamics is extensive with variable Newton constant

TL;DR

This paper develops a thermodynamic formalism for non-AdS black holes with a variable Newton constant, showing that the mass becomes extensive and consistent with standard thermodynamics when using new variables.

Contribution

It introduces a novel formalism for black hole thermodynamics with variable Newton constant, ensuring extensivity and standard thermodynamic relations.

Findings

Mass is a homogeneous first-order function of extensive variables.

Chemical potential relates to Euclidean action at the black hole.

Thermodynamics formalism aligns with extensive thermodynamics principles.

Abstract

Inspired by the recent studies on the thermodynamics of AdS black holes in the restricted phase space formalism, we propose a similar formalism for the thermodynamics of non-AdS black holes with variable Newton constant. It is shown that, by introducing the new variables $N,\mu$, where $N$ is proportional to the inverse Newton constant and $\mu$ its conjugate variable, referred to as the chemical potential, the black hole thermodynamics can be formulated in a form which is consistent with the standard extensive thermodynamics for open macroscopic systems, with the first law and the Euler relation hold simultaneously. This formalism has profound implications, in particular, the mass is a homogeneous function of the first order in the extensive variables and the intensive variables are zeroth order homogeneous functions. The chemical potential is shown to be closely related to the Euclidean action evaluated at the black hole configuration.