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

Abstract

The 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 a 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 Schr\"odinger-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 a general Kerr background.