Rational first integrals of geodesic equations and generalised hidden symmetries

TL;DR

This paper introduces new types of generalized Killing tensors and rational first integrals for geodesic equations, expanding the understanding of hidden symmetries in differential geometry.

Contribution

It defines inconstructible generalized Killing tensors and rational first integrals, providing methods to identify their inconstructibility and applying these concepts to specific solutions.

Findings

The rational first integral of the Collinson-O'Donnell solution is constructible.

Examples of metrics with inconstructible rational first integrals are provided in 2D and 4D.

A generalization of hidden symmetries like Killing-Yano tensors is proposed.

Abstract

We discuss novel generalisations of Killing tensors, which are introduced by considering rational first integrals of geodesic equations. We introduce the notion of inconstructible generalised Killing tensors, which cannot be constructed from ordinary Killing tensors. Moreover, we introduce inconstructible rational first integrals, which are constructed from inconstructible generalised Killing tensors, and provide a method for checking the inconstructibility of a rational first integral. Using the method, we show that the rational first integral of the Collinson-O'Donnell solution is not inconstructible. We also provide several examples of metrics admitting an inconstructible rational first integral in two and four dimensions, by using the Maciejewski-Przybylska system. Furthermore, we attempt to generalise other hidden symmetries such as Killing-Yano tensors.