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

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 1S 1/2 state is no longer the ground state, which shifts to the 2P 3/2 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.

The Problem of Embedded Eigenvalues for the Dirac Equation in the Schwarzschild Black Hole Metric

“…This also means that the aforementioned approximations were not adequate to study bound states in a curved background. For those who arrive at the same conclusion as us (i.e., absence of bound states in the black hole metric) but use the full atlas (including the inside of the black hole), the physical explanation for the absence of bound states is that the central singularity acts as a current sink [3]. Since we are working in the Schwarzschild coordinates, we think that the presence of the horizon is sufficient to explain the phenomenon.…”

“…Both of them end up with an approximated spectrum resembling that of a hydrogen-like atom. On the other hand, [3] proved that such bound states cannot exist and only resonances are admitted. In their approach, the problem is tackled numerically after writing the Dirac equation in the Schwarzschild black hole (BH) metric with respect to a gauge that is well-suited to a numerical solution.…”

Fermion absorption cross section of a Schwarzschild black hole

“…The direction of time implied by this process is not revealed in Schwarzschild coordinates, however, as these are manifestly timereverse symmetric and are invalid at the horizon. Time-asymmetric coordinates, such as Eddington-Finkelstein coordinates, allow the continuation of the metric across the horizon and allow us to correctly study the properties of wavefunctions [8,9,10]. We then find that ingoing states correspond precisely to those that are regular at the horizon.…”

“…The Dirac equation is clearly seperable in time, so has solutions that go as exp(−iEt). The energy E conjugate to time-translation is independent of the chosen coordinate system, and has a physical definition in terms of the Killing time [10]. We can further exploit the spherical symmetry to seperate the spinor into…”

“…The components of the spin connection are found in the standard way [14] and are particularly simple in the Painlevé-Gullstrand gauge [10],…”