On the space-times admitting two shear-free geodesic null congruences

TL;DR

This paper investigates space-times with two shear-free geodesic null congruences, deriving conditions and explicit tensor expressions, and classifying Petrov types, especially type D, with implications for conformal Killing tensors.

Contribution

It provides a tensorial analysis of such space-times, explicit formulas for curvature tensors, and characterizes Petrov types in terms of the volume element U, including conditions for conformal Killing tensors.

Findings

Explicit expressions for Ricci and Weyl tensors in terms of U.

Characterization of Petrov types via conditions on U.

Type D space-times admit conformal Killing-Yano tensors.

Abstract

We analyze the space-times admitting two shear-free geodesic null congruences. The integrability conditions are presented in a plain tensorial way as equations on the volume element $U$ of the time-like 2--plane that these directions define. From these we easily deduce significant consequences. We obtain explicit expressions for the Ricci and Weyl tensors in terms of $U$ and its first and second order covariant derivatives. We study the different compatible Petrov-Bel types and give the necessary and sufficient conditions that characterize every type in terms of $U$. The type D case is analyzed in detail and we show that every type D space-time admitting a 2+2 conformal Killing tensor also admits a conformal Killing-Yano tensor.