Hawking Temperature as the Total Gauss-Bonnet Invariant of the Region Outside a Black Hole

TL;DR

This paper introduces two novel geometric methods to compute black hole surface gravity and Hawking temperature using integrals of the Gauss-Bonnet invariant and Riemann tensor contractions, applicable to stationary black holes.

Contribution

It presents new integral formulas for Hawking temperature based on the Gauss-Bonnet invariant and a geometric identity involving the Riemann tensor and Killing vectors.

Findings

Applied methods to Kerr and extremal Reissner-Nordström black holes

Derived explicit formulas for surface gravity and Hawking temperature

Established geometric identities involving the Bianchi identity and Gauss-Bonnet tensor

Abstract

We provide two novel ways to compute the surface gravity ($\kappa$) and the Hawking temperature $(T_{H})$ of a stationary black hole: in the first method $T_{H}$ is given as the three-volume integral of the Gauss-Bonnet invariant (or the Kretschmann scalar for Ricci-flat metrics) in the total region outside the event horizon; in the second method it is given as the surface integral of the Riemann tensor contracted with the covariant derivative of a Killing vector on the event horizon. To arrive at these new formulas for the black hole temperature (and the related surface gravity), we first construct a new differential geometric identity using the Bianchi identity and an antisymmetric rank-$2$ tensor, valid for spacetimes with at least one Killing vector field. The Gauss-Bonnet tensor and the Gauss-Bonnet scalar play a particular role in this geometric identity. We calculate the surface gravity and the Hawking temperature of the Kerr and the extremal Reissner-Nordstr\"om holes as examples.