Separable coordinate systems for the Hamilton-Jacobi, Klein-Gordon and wave equations in curved spaces

Abstract

There are exactly two types of separable coordinates for the Hamilton-Jacobi, Klein-Gordon and wave equations. One type can be reduced to separable coordinates adapted to a (conformal) Killing vector, the other type to orthogonal coordinates adapted to eigenvectors of a (conformal) Killing tensor. A canonical form of the metric tensor which is a necessary and sufficient condition for the existence of a separable coordinate system for the Hamilton-Jacobi equation is derived. For the Klein-Gordon equation the metric is further restricted by a condition on the Ricci tensor. Sufficient conditions for the existence of separable coordinates are given in terms of linear or quadratic constants of motion.