Algebraic classification of Robinson-Trautman spacetimes

TL;DR

This paper provides a comprehensive algebraic classification of Robinson-Trautman spacetimes based on Weyl tensor properties, applicable to any four-dimensional metric theory of gravity, with explicit examples including scalar fields and Einstein-Weyl black holes.

Contribution

It introduces a general Petrov classification scheme for Robinson-Trautman spacetimes without relying on field equations, applicable across various gravity theories.

Findings

Classified all algebraically special Robinson-Trautman cases

Identified conditions for multiple and single principal null directions

Applied classification to scalar field and Einstein-Weyl black hole spacetimes

Abstract

We consider a general class of four-dimensional geometries admitting a null vector field that has no twist and no shear but has an arbitrary expansion. We explicitly present the Petrov classification of such Robinson-Trautman (and Kundt) gravitational fields, based on the algebraic properties of the Weyl tensor. In particular, we determine all algebraically special subcases when the optically privileged null vector field is a multiple principal null direction (PND), as well as all the cases when it remains a single PND. No field equations are a priori applied, so that our classification scheme can be used in any metric theory of gravity in four dimensions. In the classic Einstein theory this reproduces previous results for vacuum spacetimes, possibly with a cosmological constant, pure radiation and electromagnetic field, but can be applied to an arbitrary matter content. As non-trivial explicit examples we investigate specific algebraic properties of the Robinson-Trautman spacetimes with a free scalar field, and also black hole spacetimes in the pure Einstein-Weyl gravity.