Bound states and decay times of fermions in a Schwarzschild black hole background

TL;DR

This paper analyzes fermion bound states around a Schwarzschild black hole, revealing their complex energy spectrum, decay times, and implications for Hawking radiation, with states shifting as the coupling strength varies.

Contribution

It provides the first detailed computation of fermion bound states in a Schwarzschild background, including their complex energies and decay properties, highlighting the decay of the Dirac sea.

Findings

Bound states have hydrogen-like spectra at small coupling.

Decay times depend on proximity to the black hole.

The Dirac sea decay may contribute to Hawking radiation.

Abstract

We compute the spectrum of normalizable fermion bound states in a Schwarzschild black hole background. The eigenstates have complex energies. The real part of the energies, for small couplings, closely follow a hydrogen-like spectrum. The imaginary parts give decay times for the various states, due to the absorption properties of the hole, with states closer to the hole having shorter half-lives. As the coupling increases, the spectrum departs from that of the hydrogen atom, as states close to the horizon become unfavourable. Beyond a certain coupling the 1S1/2 state is no longer the ground state, which shifts to the 2P3/2 state, and then to states of successively greater angular momentum. For each positive energy state a negative energy counterpart exists, with opposite sign of its real energy, and the same decay factor. It follows that the Dirac sea of negative energy states is decaying, which may provide a physical contribution to Hawking radiation.