Explicit algebraic classification of Kundt geometries in any dimension

TL;DR

This paper provides a comprehensive algebraic classification of Kundt spacetimes in any dimension based on Weyl tensor properties, applicable to various gravity theories without relying on field equations.

Contribution

It introduces a null alignment-based classification scheme for Kundt geometries in arbitrary dimensions, identifying all algebraic types and conditions for multiple Weyl aligned null directions.

Findings

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

Derived simple invariant conditions for multiple WANDs.

Applied classification to key subfamilies like pp-waves and VSI spacetimes.

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 for which the non-expanding, non-twisting, shear-free null direction \boldk 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 \boldk 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, II_i, III_i, 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 generalization of the Bertotti-Robinson, Nariai, and Plebanski-Hacyan direct-product spacetimes of any dimension.