The Dirac field in LRS space-times: a covariant approach

TL;DR

This paper develops a covariant formalism for the Dirac field in LRS space-times, describing it as an effective fluid and analyzing conditions for self-gravitating solutions.

Contribution

It introduces a $(1+1+2)$ covariant approach avoiding tetrads, extending previous work with new analytical and numerical solutions.

Findings

Formalism describes Dirac fields as effective fluids in LRS space-times.

Conditions for self-gravitating Dirac fields in LRS space-times are established.

Analytical and numerical solutions demonstrate the framework's applicability.

Abstract

We employ the polar decomposition of the Dirac field to describe it as an effective spinorial fluid. We then construct a $(1+1+2)$ covariant formalism for the Dirac field that avoids the introduction of tetrad fields and Clifford matrices. Within this framework, we analyze the conditions under which a self-gravitating Dirac field can be consistently embedded in Locally Rotationally Symmetric (LRS) space-times of types I, II, and III. In accordance with the LRS symmetry requirements, we extend a previous work by assuming that the velocity and spin vector fields of the Dirac field lie in the planes defined pointwise by the generators of the time-like and space-like congruences, which underlie the $(1+1+2)$ decomposition. We present some analytical and numerical solutions to illustrate the applicability of the proposed framework.