Non-Existence of Black Hole Solutions for a Spherically Symmetric, Static Einstein-Dirac-Maxwell System

TL;DR

This paper proves that for a spherically symmetric, static Einstein-Dirac-Maxwell system, the only black hole solutions are the Reissner-Nordstrom solutions, implying Dirac particles must vanish or disappear inside the horizon.

Contribution

It demonstrates the non-existence of non-trivial Dirac particle solutions in static black hole spacetimes within this system.

Findings

Only Reissner-Nordstrom solutions exist under given conditions.

Dirac particles must vanish or be hidden inside the horizon.

No static, spherically symmetric Dirac black hole solutions exist.

Abstract

We consider for j=1/2, 3/2,... a spherically symmetric, static system of (2j+1) Dirac particles, each having total angular momentum j. The Dirac particles interact via a classical gravitational and electromagnetic field. The Einstein-Dirac-Maxwell equations for this system are derived. It is shown that, under weak regularity conditions on the form of the horizon, the only black hole solutions of the EDM equations are the Reissner-Nordstrom solutions. In other words, the spinors must vanish identically. Applied to the gravitational collapse of a "cloud" of spin-1/2-particles to a black hole, our result indicates that the Dirac particles must eventually disappear inside the event horizon.