Alignment and the classification of Lorentz-signature tensors

We define the notion of an aligned null direction, a Lorentz-signature analogue of the eigenvector concept that is valid for arbitrary tensor types. The set of aligned null directions is described by a a system of alignment polynomials whose coefficients are derived from the components of the tensor. The algebraic properties of the alignment polynomials can be used to classify the corresponding tensors and to put them into normal form. The alignment classification paradigm is illustrated with a discussion of bivectors and of Weyl-type tensors. Note: an earlier version of this manuscript was published in the proceedings of SPT 2004. The present version has been expanded to include a discussion of complexified alignment. Section 4 also corrects errors contained in the earlier manuscript.Comment: 8 pages. To be published in the proceedings of SPT200

Three-dimensional spacetimes of maximal order

We show that the equivalence problem for three-dimensional Lorentzian manifolds requires at most the fifth covariant derivative of the curvature tensor. We prove that this bound is sharp by exhibiting a class of 3D Lorentzian manifolds which realize this bound. The analysis is based on a three-dimensional analogue of the Newman-Pen-rose formalism, and spinorial classification of the three-dimensional Ricci tensor.Comment: final revision