On integrability of the Killing equation
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.
Cite
@article{arxiv.1704.02074,
title = {On integrability of the Killing equation},
author = {Tsuyoshi Houri and Kentaro Tomoda and Yukinori Yasui},
journal= {arXiv preprint arXiv:1704.02074},
year = {2018}
}
Comments
24 pages, single-column; v2: one of the conjectures in ver. 1 is withdrawn because it was already known in the literature; some references to previous work added; a proof in Sec. II added; v3: published version; application added in section 3.2; minor editing to improve the presentation