Alignment and algebraically special tensors in Lorentzian geometry

Details Alignment and algebraically special tensors in Lorentzian geometry Alignment and algebraically special tensors in Lorentzian geometry

2004-01-04

arxiv.org/abs/gr-qc/0401010v3

Int.J.Geom.Meth.Mod.Phys.2:41-61,2005

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

19 pages. Revised presentation

Scientific paper

10.1142/S0219887805000491

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.

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