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

2024/12/30 by Amir Babak Aazami, Aazami, Amir Babak
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Algebraic and Geometric Analysis #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Relativity and Gravitational Theory

paper · pdf · doi:10.48550/arxiv.2412.20915

openalex publication_date 2024/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We analyze oriented Riemannian 4-manifolds whose Weyl tensors W satisfy the conformally invariant condition W(T,⋅,⋅,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.

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