On the algebraic classification of pseudo-Riemannian spaces

TL;DR

This paper develops a new algebraic classification scheme for pseudo-Riemannian spaces of arbitrary signature, introducing boost-weight decomposition and properties to classify Weyl tensors, with applications to degenerate types like VSI spaces.

Contribution

It introduces a novel boost-weight decomposition and algebraic properties for classifying Weyl tensors in arbitrary signature pseudo-Riemannian spaces.

Findings

Defined algebraic properties such as ${f S}_i$- and ${f N}$-properties.

Presented a boost-weight decomposition applicable to arbitrary signatures.

Illustrated the classification with a four-dimensional neutral signature example.

Abstract

We consider arbitrary-dimensional pseudo-Riemannian spaces of signature $(k,k+m)$. We introduce a boost-weight decomposition and define a number of algebraic properties (e.g., the ${\bf S}_i$- and ${\bf N}$-properties) and present a boost-weight decomposition to classify the Weyl tensors of arbitrary signature and discuss degenerate algebraic types (e.g., VSI spaces). We consider the four dimensional neutral signature space as an illustration.