Alignment and algebraically special tensors in Lorentzian geometry

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.

arXiv comment19 pages. Revised presentation

References in corpus (5)

• Kaluza-Klein Gravity
• Classification of the Weyl Tensor in Higher Dimensions
• All spacetimes with vanishing curvature invariants
• Vanishing Scalar Invariant Spacetimes in Higher Dimensions
• Bianchi identities in higher dimensions

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