Spinor fields in spherical symmetry. Einstein-Dirac and other space-times

TL;DR

This paper investigates static spherically symmetric Einstein-spinor field systems, revealing that their energy-momentum tensor structure resembles that of scalar fields, allowing extension of known scalar field results to spinor fields.

Contribution

It demonstrates that Einstein-spinor systems with nonlinearities share EMT structures with scalar fields, enabling new solutions and insights into their geometries.

Findings

EMT structure of spinor fields matches that of scalar fields with self-interaction.

Extended scalar field results to Einstein-spinor systems.

Presented specific solutions, including a non-asymptotically flat Einstein-Dirac solution.

Abstract

We discuss the static spherically symmetric Einstein-spinor field system in the possible presence of various spinor field nonlinearities. We take into account that the spinor field energy-momentum tensor (EMT) has in general some off-diagonal components, whose vanishing due to the Einstein equations substantially affects the form of the spinor field itself and the space-time geometry. In particular, the EMT structure with any spinor field nonlinearities turns out to be the same as that of the EMT of a minimally coupled scalar field with a self-interaction potential. Therefore many results previously obtained for systems with such scalar fields are directly extended to the Einstein-spinor field system. Some special solutions are obtained and discussed, in particular, a solution for the Einstein-Dirac system (which lack asymptotic flatness) and some examples with spinor field nonlinearities.