Explicit algebraic classification of Kundt geometries in any dimension

Abstract

We present an algebraic classification, based on the null alignment properties of the Weyl tensor, of the general Kundt class of spacetimes in arbitrary dimension D for which the non-expanding, non-twisting, shear-free null direction k is a (multiple) Weyl aligned null direction (WAND). No field equations are used, so that the results apply not only to Einstein’s gravity and its direct extension to higher dimensions, but also to any metric theory of gravity which admits the Kundt spacetimes. By an explicit evaluation of the Weyl tensor in a natural null frame we demonstrate that all Kundt geometries are of type I(b) or more special, and we derive simple necessary and sufficient conditions under which k becomes a double, triple or quadruple WAND. All possible algebraically special types, including the refinement to subtypes, are identified, namely II(a), II(b), II(c), II(d), III(a), III(b), N, O, IIi, IIIi, D(a), D(b), D(c) and D(d). The corresponding conditions are surprisingly clear and expressed in an invariant geometric form. Some of them are always satisfied in four dimensions. To illustrate our classification scheme, we apply it to the most important subfamilies of the Kundt class, namely the pp-waves, the VSI spacetimes, and the generalization of the Bertotti–Robinson, Nariai, and Plebański–Hacyan direct-product spacetimes of any dimension.