Killing Tensors in Koutras-McIntosh Spacetimes

TL;DR

This paper proves that Koutras-McIntosh spacetimes, including Wils metrics, lack low-degree Killing tensors, indicating no hidden symmetries, using advanced geometric PDE techniques to classify and confirm the nonexistence of such tensors.

Contribution

It demonstrates the nonexistence of low-degree Killing tensors in Koutras-McIntosh spacetimes and classifies all lower degree Killing tensors using geometric PDE methods.

Findings

No low-degree Killing tensors in Wils metrics up to degree 6

All Killing tensors of degree 3 and 4 are reducible in generic pp-waves

Complete classification of lower degree Killing tensors

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.