More on Jacobi metric: Randers-Finsler metrics, frame-dragging, and geometrising techniques

What is it about?

In this article, I demonstrate a new method to derive Jacobi metrics from Randers–Finsler metrics by introducing a more generalised approach to Hamiltonian mechanics for such spacetimes and discuss the related applications and properties. I introduce Hamiltonian mechanics with the constraint for relativistic momentum, including a modification for null curves and two applications as exercises: derivation of a relativistic harmonic oscillator and analysis of Schwarzschild Randers–Finsler metric. Then I describe the main application for constraint mechanics in this article: a new derivation of Jacobi metric for time-like and null curves, comparing the latter with optical metrics. After that, I discuss frame dragging with the Jacobi metric and two applications for Randers–Finsler metrics: an alternative to Eisenhart lift, and different metrics that share the same Jacobi metric.

Featured Image

Why is it important?

This article introduces a new and more standardised approach to relativistic Hamiltonian mechanics using a constraint for canonical momenta applicable to more general Randers-Finsler metrics that deals with massive and light-like particles. This allows a new easier method to derive Jacobi metrics for stationary and Randers-Finsler metrics. Furthermore, it shows that Jacobi metrics for null curves differ from optical metrics for stationary spacetimes. This difference requires further exploration in context of application to optical deflection and gravitational lensing. Since the discussion of Jacobi metric has been extended to stationary metrics, the frame-dragging effect must be present. I have shown how the frame dragging effect is encoded into the Jacobi metric for both stationary Riemannian and Randers-Finsler metrics. This should allow the Jacobi metric to be included in discussions of frame-dragging effect and gravito-magnetism. While the Eisenhart lift is a very useful geometrising technique that transform gauge potentials into geometry, it is restricted to regular Hamiltonian systems associated with non-relativistic mechanics. To extend such methods into the relativistic realm, which would allow transformation of Randers-Finsler metrics to Riemannian metrics. With this aim, I discussed two approaches: the relativistic alternative to Eisenhart lift using a reverse of my procedure to derive Jacobi metric, and metrics sharing the same Jacobi metric.