Separability of the massive Dirac's equation in 5-dimensional Myers-Perry black hole geometry and its relation to a rank-three Killing-Yano tensor

TL;DR

This paper demonstrates the separability of the Dirac equation in a five-dimensional Myers-Perry black hole background and links it to a rank-three Killing-Yano tensor, revealing underlying symmetries and conserved quantities.

Contribution

It explicitly constructs symmetry operators for the Dirac and scalar equations using Killing-Yano and Stackel-Killing tensors in five-dimensional black hole geometry.

Findings

Separation of variables for Dirac and Klein-Gordon equations in 5D Myers-Perry spacetime

Explicit construction of symmetry operators from Killing-Yano tensors

Identification of a rank-three Killing-Yano tensor related to the Stackel-Killing tensor

Abstract

The Dirac equation for the electron around a five-dimensional rotating black hole with two different angular momenta is separated into purely radial and purely angular equations. The general solution is expressed as a superposition of solutions derived from these two decoupled ordinary differential equations. By separating variables for the massive Klein-Gordon equation in the same space-time background, I derive a simple and elegant form for the Stackel-Killing tensor, which can be easily written as the square of a rank-three Killing-Yano tensor. I have also explicitly constructed a symmetry operator that commutes with the scalar Laplacian by using the Stackel-Killing tensor, and the one with the Dirac operator by the Killing-Yano tensor admitted by the five-dimensional Myers-Perry metric, respectively.