Algebraic structure of Robinson-Trautman and Kundt geometries in arbitrary dimension

TL;DR

This paper classifies the algebraic structure of a broad class of higher-dimensional geometries with non-twisting, shear-free null vectors, revealing their types and conditions, and specifically classifies Robinson-Trautman vacuum spacetimes.

Contribution

It provides a complete algebraic classification of D-dimensional non-twisting, shear-free geometries, including Robinson-Trautman spacetimes, without relying on field equations.

Findings

All such geometries are of type I(b) or more special.

Derived conditions for multiple Weyl aligned null directions.

Classified Robinson-Trautman vacuum spacetimes in higher dimensions.

Abstract

We investigate the Weyl tensor algebraic structure of a fully general family of D-dimensional geometries that admit a non-twisting and shear-free null vector field k. From the coordinate components of the curvature tensor we explicitly derive all Weyl scalars of various boost weights. This enables us to give a complete algebraic classification of the metrics in the case when the optically privileged null direction k is a (multiple) Weyl aligned null direction (WAND). No field equations are applied, so that the results are valid not only in Einstein's gravity, including its extension to higher dimensions, but also in any metric gravitation theory that admits non-twisting and shear-free spacetimes. We prove that all such geometries are of type I(b), or more special, and we derive surprisingly simple necessary and sufficient conditions under which k is a double, triple or quadruple WAND. All possible algebraically special types, including the refinement to subtypes, are thus identified, namely II(a), II(b), II(c), II(d), III(a), III(b), N, O, IIi, IIIi, D, D(a), D(b), D(c), D(d), and their combinations. Some conditions are identically satisfied in four dimensions. We discuss both important subclasses, namely the Kundt family of geometries with the vanishing expansion (Theta=0) and the Robinson-Trautman family (Theta\not=0, and in particular Theta=1/r). Finally, we apply Einstein's field equations and obtain a classification of all Robinson-Trautman vacuum spacetimes. This reveals fundamental algebraic differences in the D>4 and D=4 cases, namely that in higher dimensions there only exist such spacetimes of types D(a)=D(abd), D(c)=D(bcd) and O.