Higher dimensional bivectors and classification of the Weyl operator

TL;DR

This paper extends the bivector formalism to higher-dimensional Lorentzian spacetimes, providing a refined algebraic classification of the Weyl tensor that includes new special cases, especially in five dimensions.

Contribution

It introduces a higher-dimensional bivector formalism and a refined classification of the Weyl tensor based on irreducible spin representations, advancing beyond the traditional boost-weight approach.

Findings

Refined classification of the Weyl tensor in higher dimensions.

Identification of new algebraically special cases.

Detailed analysis of the five-dimensional case.

Abstract

We develop the bivector formalism in higher dimensional Lorentzian spacetimes. We define the Weyl bivector operator in a manner consistent with its boost-weight decomposition. We then algebraically classify the Weyl tensor, which gives rise to a refinement in dimensions higher than four of the usual alignment (boost-weight) classification, in terms of the irreducible representations of the spins. We are consequently able to define a number of new algebraically special cases. In particular, the classification in five dimensions is discussed in some detail. In addition, utilizing the (refined) algebraic classification, we are able to prove some interesting results when the Weyl tensor has (additional) symmetries.