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

TL;DR

This work develops a covariant, tetrad-free formulation of the Dirac field in general relativity by recasting the spinor in polar form and exploiting a (1+1+2) decomposition. By identifying the Dirac velocity and spin with the time-like and space-like congruences, the authors derive a hydrodynamic representation of the spinor and apply it to Locally Rotationally Symmetric spacetimes of types I, II, and III, examining both perfect and non-perfect spinorial fluids. They derive the corresponding covariant Dirac equations, energy–momentum projections, and consistency conditions, and illustrate with exact solutions including FLRW spatially flat, Bianchi-I, and Minkowski spacetimes, showing spinor dust behavior and special flat-space configurations. The results clarify when a self-gravitating Dirac field can be consistently embedded in these symmetric spacetimes and pave the way for tetrad-free covariant treatments of fermionic fields in gravitational contexts.

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.