On integrability of the Killing equation

TL;DR

This paper investigates the integrability conditions of the Killing equation, which are crucial for understanding hidden symmetries in space-time and for solving geodesic equations in curved geometries.

Contribution

It formulates the integrability conditions of the Killing equation using Young symmetrizers, providing explicit conditions that limit solutions and their forms.

Findings

Derived explicit integrability conditions for the Killing equation.

Established a method using Young symmetrizers for prolongation.

Provided bounds on the number of independent solutions.

Abstract

Killing tensor fields have been thought of as describing hidden symmetry of space(-time) since they are in one-to-one correspondence with polynomial first integrals of geodesic equations. Many problems in classical mechanics can be formulated as geodesic problems in curved spaces and spacetimes, and thus solving the defining equation for Killing tensor fields (the Killing equation) is a powerful way to integrate the equations of motion. In this paper we attempt to formulate the integrability conditions of the Killing equation, which serve to put an upper bound on the number of linearly independent solutions and also to restrict the possible forms of solutions tightly. To this end, we first show the prologation for the Killing equation in a manner that uses Young symmetrizers. Then, using the prolonged equations, we provide the integrability conditions explicitly.