ALIGNMENT AND ALGEBRAICALLY SPECIAL TENSORS IN LORENTZIAN GEOMETRY

2004/01/31 by R. Milson, R. MILSON, A. Coley +5 · 7 citations
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Homotopy and Cohomology in Algebraic Topology #Noncommutative and Quantum Gravity Theories #gr-qc #math-ph #math.MP

paper · pdf · doi:10.1142/s0219887805000491

published as Int.J.Geom.Meth.Mod.Phys.2:41-61,2005 · 19 pages. Revised presentation

arxiv created 2004/07/20 · openalex publication_date 2005/02/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 principal null direction equation. In 4 dimensions this recovers the usual Petrov types are recovered. For higher dimensions we prove that, in general, a Weyl tensor does not possess any 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.

Citations

• Vanishing scalar invariant spacetimes in higher dimensions

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