Algebraic classification of higher dimensional spacetimes based on null alignment

Abstract

We review recent developments and applications of the classification of the Weyl tensor in higher dimensional Lorentzian geometries. First, we discuss the general setup, i.e. main definitions and methods for the classification, some refinements and the generalized Newman-Penrose and Geroch-Held-Penrose formalisms. Next, we summarize general results, such as a partial extension of the Goldberg-Sachs theorem, characterization of spacetimes with vanishing (or constant) curvature invariants and the peeling behaviour in asymptotically flat spacetimes. Finally, we discuss certain invariantly defined families of metrics and their relation with the Weyl tensor classification, including: Kundt and Robinson-Trautman spacetimes; the Kerr-Schild ansatz in a constant-curvature background; purely electric and purely magnetic spacetimes; direct and (some) warped products; and geometries with certain symmetries. To conclude, some applications to quadratic gravity are also overviewed. IntroductionAlmost a decade has passed since a classification scheme for the Weyl tensor of higher dimensional spaces with Lorentzian signature was put forward [1,2]. This is based on the concept of null alignment (as explained below) and extends to any dimensions n > 4 the well-known Petrov classification [3,4], to which it reduces for n = 4. Over the past few years, a deeper geometric understanding of the null alignment method has been achieved, along with several developments, and a number of applications have been presented. The aim of the present paper is thus to review those new results which are, in our view, most important. Already published proofs and extended discussions are not repeated here, and readers will be referred to related references for more details.Already in 2008, there appeared a review on the classification of the Weyl tensor in higher dimensions [5], where some useful information complementary to the one given here can be found (see also [6] for a recent introductory review). However, several new results have been published since then, in particular on the Geroch-Held-Penrose formalism [7], on spacetimes "characterized by their invariants" [8], on the Goldberg-Sachs theorem [9-11], on alternative approaches to the classification [12-15], on perturbations of near-horizon geometries [16][17][18] and on other aspects. We thus believe that summarizing some of these and other recent developments will be useful.The plan of the paper is already illustrated in detail by the table of contents, and here it will suffice to just comment on the general structure. The first part (sections 2 and 3) is devoted to presenting the formalism. First, a null alignment classification is set up that can be applied to any tensor. Then, this is specialized to the Weyl tensor, for which refinements and alternative approaches are also mentioned. The Newman-Penrose (NP) and Geroch-Held-Penrose (GHP) formalisms are also described since these are extremely useful computational tools, especially for algebraically special spacetimes, and have been already ...

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