Killing Tensors and Conformal Killing Tensors from Conformal Killing Vectors

TL;DR

This paper generalizes methods for constructing conformal and Killing tensors from conformal Killing vectors, removing previous restrictions and demonstrating broader applicability, especially in conformally flat spaces.

Contribution

It shows that the orthogonality restriction is unnecessary for constructing these tensors and extends results to include non-orthogonal cases, broadening the scope of tensor construction methods.

Findings

All conformal Killing tensors are reducible in conformally flat spaces.

A method to construct all conformal Killing tensors in conformally flat spaces.

Extended the class of constructible tensors beyond previous algorithms.

Abstract

Koutras has proposed some methods to construct reducible proper conformal Killing tensors and Killing tensors (which are, in general, irreducible) when a pair of orthogonal conformal Killing vectors exist in a given space. We give the completely general result demonstrating that this severe restriction of orthogonality is unnecessary. In addition we correct and extend some results concerning Killing tensors constructed from a single conformal Killing vector. A number of examples demonstrate how it is possible to construct a much larger class of reducible proper conformal Killing tensors and Killing tensors than permitted by the Koutras algorithms. In particular, by showing that all conformal Killing tensors are reducible in conformally flat spaces, we have a method of constructing all conformal Killing tensors (including all the Killing tensors which will in general be irreducible) of conformally flat spaces using their conformal Killing vectors.