70 research outputs found
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
On Projective Equivalence of Univariate Polynomial Subspaces
We pose and solve the equivalence problem for subspaces of ,
the dimensional vector space of univariate polynomials of degree . The group of interest is acting by projective transformations
on the Grassmannian variety of -dimensional
subspaces. We establish the equivariance of the Wronski map and use this map to
reduce the subspace equivalence problem to the equivalence problem for binary
forms
Invariant classification of vacuum PP-waves
We solve the equivalence problem for vacuum PP-wave spacetimes by employing
the Karlhede algorithm. Our main result is a suite of Cartan invariants that
allows for the complete invariant classification of the vacuum pp-waves. In
particular, we derive the invariant characterization of the G2 and G3
sub-classes in terms of these invariants. It is known [Collins91] that the
invariant classification of vacuum pp-waves requires at most the fourth order
covariant derivative of the curvature tensor, but no specific examples
requiring the fourth order were known. Using our comprehensive classification,
we prove that the q<=4 bound is sharp and explicitly describe all such maximal
order solutions
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