Variables separation in gravity

TL;DR

This paper explores how Stackel metrics enable the complete separation of variables in key equations of motion in curved spacetimes, facilitating exact solutions in gravitational physics.

Contribution

It demonstrates that the class of Stackel metrics uniquely allows the separation of variables for multiple fundamental equations in curved spacetimes.

Findings

Separation of variables is possible for Hamilton-Jacobi, Klein-Gordon-Fock, Dirac, and Weyl equations in Stackel metrics.

Stackel metrics are essential for exact integration of equations of motion in curved spacetimes.

The study clarifies the special role of Stackel spaces in gravitational physics.

Abstract

To solve the problem of exact integration of the field equations or equations of motion of matter in curved spacetimes one can use a class of Riemannian metrics for which the simplest equations of motion can be integrated by the complete separation of variables method. Here, we consider the particular case of the class of Stackel metrics. These metrics admit integration of the Hamilton-Jacobi equation for test particle by the complete separation of variables method. It appears that the other important equations of motion (Klein-Gordon-Fock, Dirac, Weyl) in curved spacetimes can be integrated by complete separation of variables method only for the metrics, belonging to the class of Stackel spaces.