Concircular tensors in Spaces of Constant Curvature: With Applications to Orthogonal Separation of The Hamilton-Jacobi Equation

TL;DR

This paper investigates concircular tensors in spaces of constant curvature and applies these findings to classify separable coordinates for the Hamilton-Jacobi equation, providing a systematic approach to coordinate transformation and separation algorithms.

Contribution

It introduces canonical forms of concircular tensors, characterizes separable coordinates, and develops methods for coordinate transformation and the BEKM separation algorithm in these spaces.

Findings

Canonical forms of concircular tensors are derived.

Separable coordinates induced by irreducible tensors are characterized.

An algorithm for separating Hamilton-Jacobi equations is constructed.

Abstract

We study concircular tensors in spaces of constant curvature and then apply the results obtained to the problem of the orthogonal separation of the Hamilton-Jacobi equation on these spaces. Any coordinates which separate the geodesic Hamilton-Jacobi equation are called separable. Specifically for spaces of constant curvature, we obtain canonical forms of concircular tensors modulo the action of the isometry group, we obtain the separable coordinates induced by irreducible concircular tensors, and we obtain warped products adapted to reducible concircular tensors. Using these results, we show how to enumerate the isometrically inequivalent orthogonal separable coordinates, construct the transformation from separable to Cartesian coordinates, and execute the Benenti-Eisenhart-Kalnins-Miller (BEKM) separation algorithm for separating natural Hamilton-Jacobi equations.