A covariant approach to the Dirac field in LRS space-times

TL;DR

This paper introduces a covariant, tetrad-free method to describe the Dirac field in LRS space-times using a spinorial fluid approach, simplifying equations and analyzing compatibility with specific symmetric space-times.

Contribution

Develops a novel covariant approach for the Dirac field in LRS space-times that avoids tetrads and Clifford matrices, enabling new analysis of self-gravitating Dirac fields.

Findings

Compatible with LRS space-times of types I, II, and III

Provides illustrative examples demonstrating the approach

Simplifies the covariant equations for Dirac fields

Abstract

We use the polar decomposition to describe the Dirac field in terms of an effective spinorial fluid. After reformulating all covariant equations in ``spinorial'' signature $(+ -- )$, we develop a $(1+1+2)$ covariant approach for the Dirac field that does not require the use of tetrad fields or Clifford matrices. By identifying the velocity and spin fields as the generators of time-like and space-like congruences, we examine the compatibility of a self-gravitating Dirac field with Locally Rotationally Symmetric space-times of types I, II, and III. We provide illustrative examples to demonstrate the effectiveness of our construction.