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

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 . 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 is a (multiple) Weyl aligned null direction (WAND). No field equations are applied, so the results are valid not only in Einsteinʼs 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 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 () and the Robinson–Trautman family (, and in particular ). Finally, we apply Einsteinʼs field equations and obtain a classification of all Robinson–Trautman vacuum spacetimes. This reveals fundamental algebraic differences in the and cases, namely that in higher dimensions there only exist such spacetimes of types D(a) ≡ D(abd), D(c) ≡ D(bcd) and O.