Algebraic classification of spacetimes using discriminating scalar curvature invariants

TL;DR

This paper develops an algebraic classification method for higher-dimensional spacetimes using discriminants of curvature operators and polynomial invariants, enabling the identification of specific tensor types like II or D.

Contribution

It introduces an algorithm to determine eigenvalue structures of curvature operators via discriminants, linking algebraic types to scalar polynomial invariants in arbitrary dimensions.

Findings

Derived scalar invariants for type II and D tensors in 5D.

Presented an algorithm for eigenvalue analysis of curvature operators.

Analyzed the 5D rotating black ring example.

Abstract

The Weyl and Ricci tensors can be algebraically classified in a Lorentzian spacetime of arbitrary dimensions using alignment theory. Used in tandem with the boost weight decomposition and curvature operators, the algebraic classification of the Weyl tensor and the Ricci tensor in higher dimensions can then be refined utilizing their eigenbivector and eigenvalue structure, respectively. In particular, for a tensor of a particular algebraic type, the associated operator will have a restricted eigenvector structure, and this can then be used to determine necessary conditions for a particular algebraic type. We shall present an analysis of the discriminants of the associated characteristic equation for the eigenvalues of an operator to determine the conditions on (the associated) curvature tensor for a given algebraic type. We will describe an algorithm which enables us to completely determine the eigenvalue structure of the curvature operator, up to degeneracies, in terms of a set of discriminants. We then express these conditions (discriminants) in terms of these polynomial curvature invariants. In particular, we can use the techniques described to study the necessary conditions in arbitrary dimensions for the Weyl and Ricci curvature operators (and hence the higher dimensional Weyl and Ricci tensors) to be of algebraic type II or D, and create syzygies which are necessary for the special algebraic type to be fulfilled. We are consequently able to determine the necessary conditions in terms of simple scalar polynomial curvature invariants in order for the higher dimensional Weyl and Ricci tensors to be of type II or D. We explicitly determine the scalar polynomial curvature invariants for a Weyl or Ricci tensor to be of type II or D in 5D. A number of simple examples are presented and, in particular, we present a detailed analysis of the important example of a 5D rotating black ring.