Alignment and algebraically special tensors in Lorentzian geometry

TL;DR

This paper develops a dimension-independent framework for tensor alignment in Lorentzian geometry, extending classical 4D classifications to higher dimensions and analyzing the algebraic types of Weyl and Ricci tensors.

Contribution

It introduces a general alignment theory applicable across dimensions and classifies Weyl and Ricci tensors using algebraic and geometric methods.

Findings

Alignment condition is equivalent to the PND equation.

In higher dimensions, Weyl tensors generally lack aligned directions.

New algebraic types for alignment configurations are described.

Abstract

We develop a dimension-independent theory of alignment in Lorentzian geometry, and apply it to the tensor classification problem for the Weyl and Ricci tensors. First, we show that the alignment condition is equivalent to the PND equation. In 4D, this recovers the usual Petrov types. For higher dimensions, we prove that, in general, a Weyl tensor does not possess aligned directions. We then go on to describe a number of additional algebraic types for the various alignment configurations. For the case of second-order symmetric (Ricci) tensors, we perform the classification by considering the geometric properties of the corresponding alignment variety.