Higher Order Symmetries and the Koutras Algorithm

Abstract

We investigate the form of Killing tensors, constructed from conformal Killing vectors of a given spacetime (M, g), by utilizing the Koutras algorithm. As an example we find irreducible Killing tensors in Robertson–Walker spacetimes. A number of theorems are given for the existence of Killing tensors in the conformally related spacetime [Formula: see text]. The form of the conformally related Killing tensors are explicitly determined. The conditions on the conformal factor Ω relating the two spacetimes (M, g) and [Formula: see text] are determined for the existence of the tensors. Also we briefly consider the role of recurrent vectors, inheriting conformal vectors and gradient conformal vectors in building Killing tensors.

Killing tensors and conformal Killing tensors from conformal Killing vectors

“…The results in Section 3 strengthen, extend, and, in one case, correct results in the earlier papers [19,20]. In Section 4 we extend a result of Weir [14] for flat spaces to conformally flat spaces and obtain the maximum number of conformal Killing tensors, which shows that they are all reducible in conformally flat spaces.…”

“…Also in a paper by O'Connor and Prince [21] there has been an independent related discussion, but in the narrower context of a particular metric. We shall show that the arguments in these papers can be made more general than in the original presentations; in particular, we shall show that our more general approach enables us to obtain more conformal Killing tensors and hence more irreducible Killing tensors than those which can be obtained by the algorithms in [19,20]. In addition we shall take the opportunity to collect together various results and clarify different definitions in the literature.…”

“…Moreover, in [19,20] there was no explicit definition of a reducible conformal Killing tensor (or its trace-free counterpart) and so the trivial and quite widespread occurence of these tensors seems to have been overlooked. The absence of a means of identifying all reducible conformal Killing tensors meant that all associated Killing tensors could not be found.…”

“…The absence of a means of identifying all reducible conformal Killing tensors meant that all associated Killing tensors could not be found. It would seem that the cause for these less than general results in [19,20] has to do with their emphasis on trace-free conformal Killing tensors. They sought trace-free conformal Killing tensors constructed as the symmetrised product of a pair of orthogonal conformal Killing vectors, i.e., A i B i = 0, (including the special case of the product of a null Killing vector with itself) so that P ab = A (a B b) was automatically trace-free.…”

“…(This is a special case of the more general result [9,13,14,4] In this paper we shall consider an indirect method of constructing irreducible Killing tensors via conformal Killing vectors which has been proposed by Koutras [19], and also used recently by Amery and Maharaj [20]. However, in these two papers the underlying principle is not completely transparent nor are the algorithms obtained the most general; this is partly due to a distraction caused by the trace-free requirement in the definitions of conformal Killing tensors which is used in these two papers [19,20]. Also in a paper by O'Connor and Prince [21] there has been an independent related discussion, but in the narrower context of a particular metric.…”