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

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-Nordström holes as examples.

9 pages, 2 figure, extended version to appear in THE EUROPEAN PHYSICAL JOURNAL C (Particles and Fields)

References in corpus (4)

• Is there a novel Einstein-Gauss-Bonnet theory in four dimensions?
• Comment on "Einstein-Gauss-Bonnet Gravity in 4-Dimensional Space-Time''
• Invariant characterization of the Kerr spacetime: Locating the horizon and measuring the mass and spin of rotating black holes using curvature invariants
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Cited by in corpus (2)

• Topological Origin of Horizon Temperature via the Chern-Gauss-Bonnet Theorem
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