Symmetry operators and separability of massive Klein-Gordon and Dirac equations in the general 5-dimensional Kerr-(anti-)de Sitter black hole background

TL;DR

This paper demonstrates the separability of the Dirac equation in a five-dimensional Kerr-(anti-)de Sitter black hole background and constructs a symmetry operator using a Killing-Yano tensor, extending black hole thermodynamics.

Contribution

It introduces a new symmetry operator for the Dirac equation in 5D Kerr-(A)dS spacetime based on a Killing-Yano tensor, enabling variable separation.

Findings

Dirac equation is separable in 5D Kerr-(A)dS spacetime.

Constructed a symmetry operator from a Killing-Yano tensor.

Extended the first law of black hole thermodynamics to include the cosmological constant.

Abstract

It is shown that the Dirac equation is separable by variables in a five-dimensional rotating Kerr-(anti-)de Sitter black hole with two independent angular momenta. A first order symmetry operator that commutes with the Dirac operator is constructed in terms of a rank-three Killing-Yano tensor whose square is a second order symmetric Stackel-Killing tensor admitted by the five-dimensional Kerr-(anti-)de Sitter spacetime. We highlight the construction procedure of such a symmetry operator. In addition, the first law of black hole thermodynamics has been extended to the case that the cosmological constant can be viewed as a thermodynamical variable.