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

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.