Killing Symmetry on Finsler Manifold

TL;DR

This paper reformulates Killing symmetry on Finsler manifolds using a new differential condition involving the Killing non-linear 1-form, enabling the analytical discovery of hidden conserved quantities like the Carter constant and Runge-Lenz vectors.

Contribution

It introduces a simple reformulation of Killing symmetry on Finsler manifolds via the spray operator and Killing non-linear 1-form, facilitating the identification of higher derivative conserved quantities.

Findings

Reformulation of Killing symmetry as elta K^lat =0.

Application to Carter constant on Kerr spacetime.

Application to Runge-Lenz vectors in Newtonian gravity.

Abstract

Killing vector fields $K$ are defined on Finsler manifold. The Killing symmetry is reformulated simply as $\delta K^\flat =0$ by using the Killing non-linear 1-form $K^\flat$ and the spray operator $\delta$ with the Finsler non-linear connection. $K^\flat$ is related to the generalization of Killing tensors on Finsler manifold, and the condition $\delta K^\flat =0$ gives an analytical method of finding higher derivative conserved quantities, which may be called hidden conserved quantities. We show two examples: the Carter constant on Kerr spacetime and the Runge-Lentz vectors in Newtonian gravity.