Lax pair tensors and integrable spacetimes

TL;DR

This paper explores how Lax pair tensors can unify Killing tensors in integrable spacetimes, demonstrating their application to the Toda lattice and resulting in new geometries with integrable properties.

Contribution

It introduces a tensorial Lax pair framework for Killing tensors and geometrizes the Toda lattice, leading to new integrable spacetime models with Killing vectors.

Findings

Lax pair tensors unify Killing tensors of arbitrary rank.

The Toda lattice is geometrized as a geodesic system in Riemannian geometry.

New four-dimensional spacetimes with integrability and Killing vectors are constructed.

Abstract

The use of Lax pair tensors as a unifying framework for Killing tensors of arbitrary rank is discussed. Some properties of the tensorial Lax pair formulation are stated. A mechanical system with a well-known Lax representation -- the three-particle open Toda lattice -- is geometrized by a suitable canonical transformation. In this way the Toda lattice is realized as the geodesic system of a certain Riemannian geometry. By using different canonical transformations we obtain two inequivalent geometries which both represent the original system. Adding a timelike dimension gives four-dimensional spacetimes which admit two Killing vector fields and are completely integrable.