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

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mo>+</mml:mo> <mml:mo>−</mml:mo> <mml:mo>−</mml:mo> <mml:mo>−</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> , we develop a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> 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.