Black Hole Mass Formula Is a Vanishing Noether Charge

TL;DR

This paper demonstrates that the black hole mass formula in Einstein's gravity can be derived from a Noether current that vanishes, revealing a fundamental connection between black hole thermodynamics and Noether charges.

Contribution

It shows that the black hole mass formula is a vanishing Noether charge, providing new insights into the thermodynamics of black holes within gravitational theories.

Findings

Black hole mass formula is a vanishing Noether charge.

The first law of black hole thermodynamics can be derived from this vanishing charge.

Provides a new perspective on black hole thermodynamics and conserved charges.

Abstract

The Noether current and its variation relation with respect to diffeomorphism invariance of gravitational theories have been derived from the horizontal variation and vertical-horizontal bi-variation of the Lagrangian, respectively. For Einstein's GR in the stationary, axisymmetric black holes, the mass formula in vacuum can be derived from this Noether current although it definitely vanishes. This indicates that the mass formula of black holes is a vanishing Noether charge in this case. The first law of black hole thermodynamics can also be derived from the variation relation of this vanishing Noether current.