A Goldberg-Sachs theorem in dimension three

TL;DR

This paper establishes a Goldberg-Sachs theorem in three-dimensional Lorentzian manifolds, linking Ricci tensor conditions to null line distributions, with applications to topological massive gravity solutions.

Contribution

It provides the first necessary and sufficient conditions for null line distributions in 3D Lorentzian manifolds satisfying topological massive gravity equations.

Findings

Characterization of null line distributions in 3D Lorentzian manifolds.

Conditions for integrability and total geodesicity of orthogonal complements.

Application to Kundt spacetimes solving topological massive gravity.

Abstract

We prove a Goldberg-Sachs theorem in dimension three. To be precise, given a three-dimensional Lorentzian manifold satisfying the topological massive gravity equations, we provide necessary and sufficient conditions on the tracefree Ricci tensor for the existence of a null line distribution whose orthogonal complement is integrable and totally geodetic. This includes, in particular, Kundt spacetimes that are solutions of the topological massive gravity equations.