Lanczos potentials and curvature-free connections aligned to a geodesic shear-free expanding null congruence

Details Lanczos potentials and curvature-free connections aligned to a geodesic shear-free expanding null congruence Lanczos potentials and curvature-free connections aligned to a geodesic shear-free expanding null congruence

2000-02-03

arxiv.org/abs/gr-qc/0002016v2

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

34 pages, LaTeX file; New section added, Slight changes in old sections

Scientific paper

By the method of rho-integration we obtain all Lanczos potentials L_{ABCA'} of the Weyl spinor that, in a certain sense, are aligned to a geodesic shear-free expanding null congruence. We also obtain all spinors H_{ABA'B'}=Q_{AB}o_{A'}o_{B'}, Q_{AB}=Q_{(AB)} satisfying nabla_{(A}{}^{B'}H_{BC)A'B'}=L_{ABCA'}. We go on to prove that H_{ABA'B'} can be chosen so that Gamma_{ABCA'}=nabla_{(A}{}^{B'} H_{B)CA'B'} defines a metric asymmetric curvature-free connection such that L_{ABCA'}=Gamma_{(ABC)A'} is a Lanczos potential that is aligned to the geodesic shear-free expanding congruence. These results are a generalization to a large class of algebraically special spacetimes (including all vacuum ones for which the principal null direction is expanding) of the curvature-free connection of the Kerr spacetime found by Bergqvist and Ludvigsen, which was used in a construction of quasi-local momentum.

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