Complete separation of variables in the geodesic Hamilton-Jacobi equation

TL;DR

This paper completely solves the Stäckel problem by classifying all metrics that allow complete separation of variables in the geodesic Hamilton-Jacobi equation across any dimension and signature.

Contribution

It provides a comprehensive classification of separable metrics, including new cases and detailed proofs, for indefinite metrics with zero diagonal elements.

Findings

Classification of separable metrics into equivalence classes

Complete solution to the Stäckel problem for all signatures and dimensions

Listing of canonical separable metrics in low dimensions

Abstract

We consider a (pseudo)Riemannian manifold of arbitrary dimension. The Hamilton-Jacobi equation for geodesic Hamiltonian admits complete separation of variables for some (separable) metrics in some (separable) coordinate systems. Separable metrics are very important in mathematics and physics. The St\"ackel problem is: ``Which metrics admit complete separation of variables in the geodesic Hamilton-Jacobi equation?'' This problem was solved for inverse metrics with nonzero diagonal elements, in particular, for positive definite Riemannian metrics long ago. However the question is open for indefinite metrics having zeroes on diagonals. We propose the solution. Separable metrics are divided into equivalence classes characterised by the number of commuting Killing vector fields, quadratic indecomposable conservation laws for geodesics, and the number of coisotropic coordinates. The paper contains detailed proofs, sometimes new, of previous results as well as new cases. As an example, we list all canonical separable metrics in each equivalence class in two, three, and four dimensions. Thus the St\"ackel problem is completely solved for metrics of any signature in any number of dimensions.