Classification of the Weyl Tensor in Higher Dimensions and Applications

TL;DR

This paper reviews the algebraic classification of the Weyl tensor in higher-dimensional Lorentzian geometry, extending four-dimensional concepts and exploring applications in supergravity and spacetime analysis.

Contribution

It introduces a classification scheme for the Weyl tensor in higher dimensions and discusses its applications, including generalizations of classical theorems and spacetime properties.

Findings

Classification reduces to Petrov type in 4D

Generalizations of Goldberg-Sachs and Peeling theorems

Analysis of higher dimensional spacetimes with special curvature invariants

Abstract

We review the theory of alignment in Lorentzian geometry and apply it to the algebraic classification of the Weyl tensor in higher dimensions. This classification reduces to the the well-known Petrov classification of the Weyl tensor in four dimensions. We discuss the algebraic classification of a number of known higher dimensional spacetimes. There are many applications of the Weyl classification scheme, especially in conjunction with the higher dimensional frame formalism that has been developed in order to generalize the four dimensional Newman--Penrose formalism. For example, we discuss higher dimensional generalizations of the Goldberg-Sachs theorem and the Peeling theorem. We also discuss the higher dimensional Lorentzian spacetimes with vanishing scalar curvature invariants and constant scalar curvature invariants, which are of interest since they are solutions of supergravity theory.