https://doi.org/10.1140/epjc/s10052-023-11594-9
Letter
Hawking temperature as the total Gauss–Bonnet invariant of the region outside a black hole
1
Department of Physics, Karamanoglu Mehmetbey University, 70100, Karaman, Turkey
2
Department of Physics, Middle East Technical University, 06800, Ankara, Turkey
Received:
25
November
2022
Accepted:
6
May
2023
Published online:
17
May
2023
We provide two novel ways to compute the surface gravity (
) and the Hawking temperature 
of a stationary black hole: in the first method 
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.
© The Author(s) 2023
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