Killing tensors from conformal Killing vectors

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 and hence all the Killing tensors (which will in general be irreducible) of conformally flat spaces using their conformal Killing vectors.

Section: Introductionsupporting

confidence: 83%

Section: Introductionmentioning

confidence: 99%

Killing tensors and conformal Killing tensors from conformal Killing vectors

Rani

Edgar

Barnes

2003

“…The set of all CKV (respectively, SCKV, HKV and KV) form a finite dimensional Lie algebra denoted by C (respectively, S, H and G). Koutras [25] devised an algorithm to find KTs using CKVs and this algorithm was generalized by Rani, Edgar and Barnes [26], [27]: Given a pair of CKVs X, Y satisfying…”

Section: Introductionmentioning

confidence: 99%

Killing tensors in pp-wave spacetimes

Keane¹,

Tupper²

2010

Abstract. The formal solution of the second order Killing tensor equations for the general pp-wave spacetime is given. The Killing tensor equations are integrated fully for some specific pp-wave spacetimes. In particular, the complete solution is given for the conformally flat plane wave spacetimes and we find that irreducible Killing tensors arise for specific classes. The maximum number of independent irreducible Killing tensors admitted by a conformally flat plane wave spacetime is shown to be six. It is shown that every pp-wave spacetime that admits an homothety will admit a Killing tensor of Koutras type and, with the exception of the singular scale-invariant plane wave spacetimes, this Killing tensor is irreducible.

“…The method of the proof is to see that both µ and σ satisfy the same equations and the same initial conditions, so that they must be identical. The vector field µ is a conformal Killing and then it satisfies L µ (g) =Ψg (6) whereΨ is the associated scale factor. On the other hand, it follows from (5) that for conformal Killing vector fields the finite transformation verifies…”

Section: Proofmentioning

confidence: 99%

“…Most of this work has taken into account true isometries only, but not homothetic isometries and/or conformal ones. Fortunately, there has been a renewed interest for these last types of symmetries recently, and some classifications have been achieved for special matter contents in the spacetime or particular structures of the conformal Lie group, (see, for instance, [2], [3], [4] and [5]), and also some theorems relating conformal Killings with Killing tensors have been found [6].…”

Section: Introductionmentioning

confidence: 99%

Axial symmetry and conformal Killing vectors

Mars

Senovilla

1993

138

Axisymmetric spacetimes with a conformal symmetry are studied and it is shown that, if there is no further conformal symmetry, the axial Killing vector and the conformal Killing vector must commute. As a direct consequence, in conformally stationary and axisymmetric spacetimes, no restriction is made by assuming that the axial symmetry and the conformal timelike symmetry commute. Furthermore, we prove that in axisymmetric spacetimes with another symmetry (such as stationary and axisymmetric or cylindrically symmetric spacetimes) and a conformal symmetry, the commutator of the axial Killing vector with the two others mush vanish or else the symmetry is larger than that originally considered. The results are completely general and do not depend on Einstein's equations or any particular matter content.