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 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.

26 pages, no figures. Presentation improved. To appear in Classical and Quantum Gravity

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• Robinson-Trautman spacetimes in higher dimensions
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• Asymptotic structure of radiation in higher dimensions
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• Uniqueness of extremal isolated horizons and their identification with horizons of all type D black holes
• Weyl type N solutions with null electromagnetic fields in the Einstein-Maxwell $p$-form theory
• Lovelock vacua with a recurrent null vector field
• Classification of Robinson-Trautmann and Kundt Geometries with Large D Limit
• All non-expanding gravitational waves in $D$-dimensional (anti-)de Sitter space
• Geodesic deviation in Kundt spacetimes of any dimension
• Exact solutions and their interpretation - session A1
• Analyzes of Algebraic Classification of Higher Dimensional Kundt Geometries with Large $D$ Method