Temperature of a spinning black hole via a simple derivation

TL;DR

This paper presents a simplified, pedagogical derivation of the temperature of a Kerr black hole using classical general relativity and thermodynamics, highlighting implications for black hole evaporation and dark matter.

Contribution

It provides a more accessible derivation of spinning black hole temperature, emphasizing pedagogical clarity and potential cosmological implications.

Findings

Derived Kerr black hole temperature using classical methods

Highlighted potential for small black hole evaporation

Discussed primordial black holes as dark matter candidates

Abstract

According to current theory a black hole has a nonzero temperature and thus radiates like any black body. This remarkable result was first shown by Hawking for a non-spinning black hole using general relativity to describe the black hole gravitational field and quantum field theory to describe the radiation. Since then the temperature of a spinning Kerr black hole has been calculated. There have also been many heuristic derivations for the temperature. In this work we derive the temperature of a Kerr spinning black hole using only classical general relativity and thermodynamics. It is very similar to Ref. 11 but is mathematically simpler and more self-contained. Our purpose is mainly pedagogical, to be more accessible to students and non-specialists with a knowledge of general relativity. We also call further attention to the expected explosive evaporation of small black holes, not yet observed, which would be an almost unique window into Planck scale physics. Finally, we discuss the idea that the cosmological dark matter, whose nature is currently unknown, may be composed of small primordial black hole remnants.