Stationary Solutions of the Dirac Equation in the Gravitational Field of a Charged Black Hole

TL;DR

This paper finds stationary solutions to the Dirac equation in the spacetime of a charged black hole, revealing conditions under which these solutions are physically meaningful and their implications for black hole internal structure and evaporation.

Contribution

It demonstrates the existence of stationary Dirac solutions in Reissner-Nordstrom black hole spacetime and analyzes their physical validity, especially in extremal cases.

Findings

Only one regular stationary state outside and inside the horizons.

Normalization diverges for non-extremal black holes, indicating nonstationary nature.

Finite normalization in extremal black holes allows physically consistent solutions.

Abstract

A stationary solution of the Dirac equation in the metric of a Reissner-Nordstrom black hole has been found. Only one stationary regular state outside the black hole event horizon and only one stationary regular state below the Cauchy horizon are shown to exist. The normalization integral of the wave functions diverges on both horizons if the black hole is non-extremal. This means that the solution found can be only the asymptotic limit of a nonstationary solution. In contrast, in the case of an extremal black hole, the normalization integral is finite and the stationary regular solution is physically self-consistent. The existence of quantum levels below the Cauchy horizon can affect the final stage of Hawking black hole evaporation and opens up the fundamental possibility of investigating the internal structure of black holes using quantum tunneling between external and internal states.