Wald's entropy in Coincident General Relativity

TL;DR

This paper extends Wald's entropy formula to spacetimes with non-metricity in Coincident General Relativity, showing it yields the same black hole entropy as standard General Relativity and confirming the first law of thermodynamics.

Contribution

It introduces a Wald's Noether charge method for non-metric spacetimes in Coincident GR, establishing entropy equivalence with standard GR.

Findings

Wald's entropy formula applies to non-metric spacetimes.

Black hole entropy in Coincident GR matches standard GR results.

The first law of black hole thermodynamics is validated in this framework.

Abstract

The equivalence principle and its universality enables the geometrical formulation of gravity. In the standard formulation of General Relativity \'a la Einstein, the gravitational interaction is geometrized in terms of the spacetime curvature. However, if we embrace the geometrical character of gravity, two alternative, though equivalent, formulations of General Relativity emerge in flat spacetimes, in which gravity is fully ascribed either to torsion or to non-metricity. The latter allows a much simpler formulation of General Relativity oblivious to the affine spacetime structure, the Coincident General Relativity. The entropy of a black hole can be computed using the Euclidean path integral approach, which strongly relies on the addition of boundary or regulating terms in the standard formulation of General Relativity. A more fundamental derivation can be performed using Wald's formula, in which the entropy is directly related to Noether charges and is applicable to general theories with diffeomorphism invariance. In this work we extend Wald's Noether charge method for calculating black hole entropy to spacetimes endowed with non-metricity. Using this method, we show that Coincident General Relativity with an improved action principle gives the same entropy as the well-known entropy in standard General Relativity. Furthermore the first law of black hole thermodynamics holds and an explicit expression for the energy appearing in the first law is obtained.