Alignment and the classification of Lorentz-signature tensors

TL;DR

This paper introduces the concept of aligned null directions for Lorentz-signature tensors, providing a classification method based on alignment polynomials, with applications to bivectors and Weyl tensors.

Contribution

It defines a new alignment classification framework for Lorentz-signature tensors, extending eigenvector concepts and including complexified alignment for tensor analysis.

Findings

Defined aligned null directions for arbitrary tensors

Developed algebraic classification using alignment polynomials

Applied classification to bivectors and Weyl-type tensors

Abstract

We define the notion of an aligned null direction, a Lorentz-signature analogue of the eigenvector concept that is valid for arbitrary tensor types. The set of aligned null directions is described by a a system of alignment polynomials whose coefficients are derived from the components of the tensor. The algebraic properties of the alignment polynomials can be used to classify the corresponding tensors and to put them into normal form. The alignment classification paradigm is illustrated with a discussion of bivectors and of Weyl-type tensors. Note: an earlier version of this manuscript was published in the proceedings of SPT 2004. The present version has been expanded to include a discussion of complexified alignment. Section 4 also corrects errors contained in the earlier manuscript.