Third rank Killing tensors in general relativity. The (1+1)-dimensional case

TL;DR

This paper classifies (1+1)-dimensional geometries admitting third rank Killing tensors, revealing that the existence condition is a quadratic PDE, which aids in finding new exact solutions related to Einstein equations.

Contribution

It provides a classification of third rank Killing tensors in (1+1)-dimensional geometries and formulates the existence condition as a quadratic PDE, unlike lower-rank cases.

Findings

Existence condition is a quadratic PDE in a Kahler potential.

New exact solutions can be derived from geometries with higher rank Killing tensors.

Higher rank Killing tensors are linked to integrability in Einstein equation models.

Abstract

Third rank Killing tensors in (1+1)-dimensional geometries are investigated and classified. It is found that a necessary and sufficient condition for such a geometry to admit a third rank Killing tensor can always be formulated as a quadratic PDE, of order three or lower, in a Kahler type potential for the metric. This is in contrast to the case of first and second rank Killing tensors for which the integrability condition is a linear PDE. The motivation for studying higher rank Killing tensors in (1+1)-geometries, is the fact that exact solutions of the Einstein equations are often associated with a first or second rank Killing tensor symmetry in the geodesic flow formulation of the dynamics. This is in particular true for the many models of interest for which this formulation is (1+1)-dimensional, where just one additional constant of motion suffices for complete integrability. We show that new exact solutions can be found by classifying geometries admitting higher rank Killing tensors.