Some Spacetimes with Higher Rank Killing-Stackel Tensors

TL;DR

This paper constructs new higher-dimensional Lorentzian spacetimes with irreducible higher-rank Killing tensors using the lightlike Eisenhart lift, including a superintegrable supersymmetric pp-wave, and explores their algebraic and quantum properties.

Contribution

It introduces the first known examples of spacetimes with higher-rank Killing tensors derived from integrable systems via the Eisenhart lift.

Findings

Constructed four- and higher-dimensional Lorentzian spacetimes with higher-rank Killing tensors.

Identified a superintegrable supersymmetric pp-wave spacetime.

Killing tensors form a non-trivial Poisson-Schouten-Nijenhuis algebra.

Abstract

By applying the lightlike Eisenhart lift to several known examples of low-dimensional integrable systems admitting integrals of motion of higher-order in momenta, we obtain four- and higher-dimensional Lorentzian spacetimes with irreducible higher-rank Killing tensors. Such metrics, we believe, are first examples of spacetimes admitting higher-rank Killing tensors. Included in our examples is a four-dimensional supersymmetric pp-wave spacetime, whose geodesic flow is superintegrable. The Killing tensors satisfy a non-trivial Poisson-Schouten-Nijenhuis algebra. We discuss the extension to the quantum regime.