Petrov Types for the Weyl Tensor via the Riemannian-to-Lorentzian Bridge

TL;DR

This paper classifies certain Weyl tensors on 4-manifolds by deforming Riemannian metrics into Lorentzian ones, revealing they correspond to Petrov Types I and D, with classifications based on critical points of associated quadratic forms.

Contribution

It introduces a geometric classification of Weyl tensors satisfying a specific conformally invariant condition by using a Riemannian-to-Lorentzian metric deformation, linking to Petrov Types in relativity.

Findings

Only Petrov Types I and D occur under the given conditions.

The classification is determined by the number of critical points of the Lorentzian quadratic form.

The approach connects Riemannian geometry with Lorentzian Petrov classification.

Abstract

We analyze oriented Riemannian 4-manifolds whose Weyl tensors $W$ satisfy the conformally invariant condition $W(T,\cdot,\cdot,T) = 0$ for some nonzero vector $T$. While this can be algebraically classified via $W$'s normal form, we find a further geometric classification by deforming the metric into a Lorentzian one via $T$. We show that such a $W$ will have the analogue of Petrov Types from general relativity, that only Types I and D can occur, and that each is completely determined by the number of critical points of $W$'s associated Lorentzian quadratic form. A similar result holds for the Lorentzian version of this question, with $T$ timelike.