Hawking temperature as the total Gauss–Bonnet invariant of the region outside a black hole

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Altas E., Tekin B.

European Physical Journal C, vol.83, no.5, 2023 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 83 Issue: 5
  • Publication Date: 2023
  • Doi Number: 10.1140/epjc/s10052-023-11594-9
  • Journal Name: European Physical Journal C
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Aerospace Database, Chemical Abstracts Core, Communication Abstracts, INSPEC, zbMATH, Directory of Open Access Journals
  • Middle East Technical University Affiliated: Yes

Abstract

We provide two novel ways to compute the surface gravity (κ) and the Hawking temperature (TH) of a stationary black hole: in the first method TH 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.