Black hole with a Dirac field in 3+1 dimensions
We study a complex Dirac field in the chiral representation minimally coupled to gravity in 3+1 dimensions in the context of Einstein-Cartan theory. Generically the matter content gravitates in two different ways: On the one hand, the energy-momentum induces spacetime curvature; on the other hand, the presence of spin acts as a source for the spacetime torsion, which does not propagate. In this setup we consider the most general stationary spherically symmetric ansatz and we find an analytic black hole solution that supports a nontrivial spinor configuration. The spinor field affects the geometry by inducing spacetime torsion, though, remarkably, it does not alter the black hole metric, which retains its Schwarzschild form. We find solutions both in asymptotically flat and asymptotically (Anti) de Sitter spaces. Additionally, we consider how the solution gets deformed when the so-called Holst term is included in the gravity action. We discuss possible observational effects due to the coupling of fermions in the nonvanishing torsion background. Additionally we include a Maxwell field into the problem and find that it forces the spinor field to be zero. Finally, nonminimal couplings of the spinor with spacetime curvature are considered, obtaining any effect on the spherically symmetric solutions.