Killing tensors in Koutras–Mcintosh spacetimes

Abstract

The Koutras-McIntosh family of metrics include conformally flat pp-waves and the Wils metric. It appeared in a paper of 1996 by Koutras-McIntosh as an example of a pure radiation spacetime without scalar curvature invariants or infinitesimal symmetries. Here we demonstrate that these metrics have no "hidden symmetries", by which we mean Killing tensors of low degrees. For the particular case of Wils metrics we show the nonexistence of Killing tensors up to degree 6. The technique we use is the geometric theory of overdetermined PDEs and the Cartan prolongation-projection method. Application of those allows to prove the nonexistence of polynomial in momenta integrals for the equation of geodesics in a mathematical rigorous way. Using the same technique we can completely classify all lower degree Killing tensors and, in particular, prove that for generic pp-waves all Killing tensors of degree 3 and 4 are reducible.

Killing Invariants: An approach to the sub-classification of geometries with symmetry

“…If a given spacetime admits an irreducible rank 2 Killing tensor field, then the line-elements within this class of spacetimes can be expressed in Boyer-Lindquist form [4]. However, determining the existence of this tensor can be difficult [5]. Given a stationary axisymmetric metric in an arbitrary coordinate system and one in Boyer-Lindquist coordinates, it is then possible to determine if they are equivalent.…”