Spacetime characterizations of $\Lambda$-vacuum metrics with a null Killing 2-form

TL;DR

This paper classifies four-dimensional $ abla$-vacuum spacetimes with a null Killing 2-form, providing new characterizations for cases with zero or negative cosmological constant and completing previous classifications.

Contribution

It offers a comprehensive classification of $ abla$-vacuum spacetimes with null Killing 2-forms, extending previous results to include cases with non-zero cosmological constant.

Findings

$ abla< 0$ must hold when $ abla$ is non-zero

Characterization of Einstein spacetimes conformal to pp-waves

Spacetime characterizations of vacuum plane waves and Brinkmann spacetimes

Abstract

An exhaustive list of four-dimensional $\Lambda$-vacuum spacetimes admitting a Killing vector whose self-dual Killing two-form ${\cal F}$ is null is obtained assuming that the self-dual Weyl tensor is proportional to the tensor product of ${\cal F}$ by itself. Our analysis complements previous results concerning the case with non-null ${\cal F}$. We analyze both cases with $\Lambda$ vanishing or not. In the latter case we prove that $\Lambda< 0$ must hold necessarily, and we find a characterization of the Einstein spacetimes conformal to pp-waves. In the former case we obtain spacetime characterizations of vacuum plane waves and of the stationary vacuum Brinkmann spacetimes. At the light of the full set of results, old and new, we reformulate the case with non-null ${\cal F}$ and $\Lambda= 0$. We finally present a table collecting the results for both null, and non-null, ${\cal F}$.