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

TL;DR

This paper characterizes all Lanczos potentials aligned with a specific null congruence in algebraically special spacetimes, constructing curvature-free connections that generalize previous results for Kerr spacetime.

Contribution

It provides a comprehensive method to obtain aligned Lanczos potentials and associated curvature-free connections in a broad class of algebraically special spacetimes.

Findings

All aligned Lanczos potentials are obtained via rho-integration.

Existence of metric asymmetric curvature-free connections in these spacetimes.

Generalization of Kerr spacetime curvature-free connection to a larger class.

Abstract

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.