Solution of Dirac equation in the near horizon geometry of an extreme Kerr black hole

TL;DR

This paper analytically solves the Dirac equation in the near horizon geometry of an extreme Kerr black hole, revealing exact solutions for the axial part and numerical approaches for the angular part, and confirms the absence of superradiance.

Contribution

It provides the first exact solution for the axial Dirac equation in this geometry and analyzes the angular equation, extending understanding of fermionic behavior near extremal Kerr black holes.

Findings

Exact solution for axial Dirac equation independent of mass

Angular equation reduces to confluent Heun for massless case

Confirmed absence of superradiance for Dirac particles

Abstract

Dirac equation is solved in the near horizon limit geometry of an extreme Kerr black hole. We decouple equations first as usual, into an axial and angular part. The axial equation turns out to be independent of the mass and is solved exactly. The angular equation reduces, in the massless case, to a confluent Heun equation. In general for nonzero mass, the angular equation is expressible at best, as a set of coupled first order differential equations apt for numerical investigation. The axial potentials corresponding to the associated Schrodinger-type equations and their conserved currents are found. Finally, based on our solution, we verify in a similar way the absence of superradiance for Dirac particles in the near horizon, a result which is well-known within the context of general Kerr background.