Paper detail

Ricci Coefficients in Covariant Dirac Equation, Symmetry Aspects and Newman-Penrose Approach

The paper investigates how the Ricci rotation coefficients act in the Dirac equation in presence of external gravitational fields described in terms of Riemannian space-time geometry. It is shown that only 8 different combinations of the Ricci coefficients γ_{abc}(x) are involved in the Dirac equation. They are combined in two 4-vectors B_{a}(x) and C_{a}(x) under local Lorentz group which has status of the gauge symmetry group. In all orthogonal coordinates one of these vectors, "pseudovector" C_{a}(x), vanishes identically. The gauge transformation laws of the two vectors are found explicitly. Connection of these B_{a}(x) and A_{a}(x) with the known Newman-Penrose coefficients is established. General study of gauge symmetry aspects in Newman-Penrose formalism is performed. Decomposition of the Ricci object, "tensor" γ_{abc}(x), into two "spinors" γ(x) and \barγ(x) is done. At this Ricci rotation coefficients are divided into two groups:12 complex functions γ(x) and 12 conjugated to them \barγ(x). Components of spinor \barγ(x) coincide with 12 spin coefficients by Newman-Penrose. The formulas for gauge transformations of spin coefficients under local Lorentz group are derived. There are given two solutions to the gauge problem: one in the compact form of transformation laws for spinors γ(x) and \barγ(x), and another as detailed elaboration of the latter in terms of 12 spin coefficients.