Charged Dirac stars

Abstract

In this work we solve the coupled Einstein–Dirac–Maxwell (EDM) system for static spherically symmetric configurations of two fermions in a singlet spinor state within the 3+1 formalism of general relativity. We find different families of stationary self-gravitating solutions for the Dirac field through a numerical shooting method for different values of the electric charge parameter q. Furthermore, we investigate the effect of the charge q on the binding energy, mass, radius, and compactness of the solutions. We show that gravitationally bound configurations exist only for \(q<m\), with m the mass of the Dirac field, and that the mass–frequency relation exhibits the characteristic spiral structure previously found for bosonic fields of spin \(s=0\) and \(s=1\). We are able to show that some of these gravitationally bound configurations have a compactness comparable to that of neutron stars. With these results, we conclude that at least at the classical level, self-gravitating fields with different spins \(s=0,1/2,1\) share some common characteristics when they are coupled to gravity. As has been previously shown for the case of bosonic stars, we also find some super-critical solutions with q slightly larger than m. Such super-critical solutions correspond to configurations such that in the Newtonian regime the Coulomb repulsion overcomes the gravitational attraction, and as such they would not be expected to exist. Nevertheless, even if they do exist in the general relativistic case for a limited range of values of \(q>m\), we find that they are always gravitationally unbound.