Killing Tensors from Conformal Killing Vectors

TL;DR

This paper extends methods for constructing Killing tensors from conformal Killing vectors, removing orthogonality restrictions, and shows that all conformal Killing tensors in conformally flat spacetimes can be generated from conformal Killing vectors.

Contribution

It generalizes previous constructions by removing orthogonality constraints and demonstrates that all conformal Killing tensors in conformally flat spacetimes can be derived from conformal Killing vectors.

Findings

Removing orthogonality restriction allows more Killing tensors to be constructed.

All conformal Killing tensors in conformally flat spacetimes can be generated from conformal Killing vectors.

Extended and corrected previous results on Killing tensors from a single conformal Killing vector.

Abstract

Some years ago Koutras presented a method of constructing a conformal Killing tensor from a pair of orthogonal conformal Killing vectors. When the vector associated with the conformal Killing tensor is a gradient, a Killing tensor (in general irreducible) can then be constructed. In this paper it is shown that the severe restriction of orthogonality is unnecessary and thus it is possible that many more Killing tensors can be constructed in this way. We also extend, and in one case correct, some results on Killing tensors constructed from a single conformal Killing vector. Weir's result that, for flat space, there are 84 independent conformal Killing tensors, all of which are reducible, is extended to conformally flat spacetimes. In conformally flat spacetimes it is thus possible to construct all the conformal Killing tensors and in particular all the Killing tensors (which in general will not be reducible) from conformal Killing vectors.