[Paper Review] More on Jacobi metric: Randers-Finsler metrics, frame dragging and geometrisation techniques

Sumanto Chanda|arXiv (Cornell University)|Nov 14, 2019

Advanced Differential Geometry Research4 citations

TL;DR

This paper proposes two alternatives to the Eisenhart lift—Jacobi lift and Symmetry Replacement—for Randers-Finsler (RF) geometry, enabling exact geodesic reformulation of RF gravity without approximations. By replacing the Hamiltonian with a constraint in Maupertuis' principle, it constructs a Jacobi-Maupertuis-Randers-Finsler (JMRF) metric and demonstrates that optical metrics in stationary spacetimes are naturally RF metrics, allowing exact Riemannian lifts via Jacobi lift and symmetry rescaling.

ABSTRACT

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.

Motivation & Objective

• To address the failure of the Eisenhart lift in Randers-Finsler geometry due to the absence of a Hamiltonian generator in Maupertuis-form Lagrangians.
• To develop a consistent geometric formulation for RF metrics using constraints instead of Hamiltonians, enabling geodesic reformulation.
• To introduce Jacobi lift as a viable alternative to Eisenhart lift in RF geometry by leveraging conformal structure and constraint-based metrics.
• To propose Symmetry Replacement as a method to reframe RF metrics by replacing a cyclic coordinate's symmetry with a more favorable one, preserving signature.
• To demonstrate that optical metrics in stationary spacetimes, such as the Kerr metric, are naturally RF metrics and can be lifted via the Jacobi metric framework.

Proposed method

• Adapts Maupertuis' principle to RF Lagrangians, which are already in the form of a square-root action, bypassing the need for Legendre transformation.
• Introduces a momentum constraint as a substitute for the Hamiltonian in systems where the Lagrangian is already in Maupertuis form, enabling the construction of the Jacobi-Maupertuis-Randers-Finsler (JMRF) metric.
• Derives the Jacobi lift by reversing the Jacobi metric construction, transforming the RF metric into a Riemannian metric via conformal rescaling, valid when a conformal factor exists.
• Applies Symmetry Replacement to RF metrics with cyclic coordinates, replacing the original symmetry direction with a new one by absorbing the linear term into the metric's norm under the square root.
• Identifies the Fermat metric as a special case of the JMRF metric in stationary spacetimes, showing that the optical metric for null geodesics in such spacetimes is a Randers-Finsler metric.
• Demonstrates that the resulting lifted metrics avoid higher-order approximations in the Hessian-based metric computation, enabling exact gravity calculations in RF geometry.

Experimental results

Research questions

• RQ1 Why does the Eisenhart lift fail in Randers-Finsler geometry, and what structural limitation prevents its direct application?
• RQ2 How can a constraint on momentum serve as a viable substitute for the Hamiltonian in deriving geodesic equations for RF systems?
• RQ3 Can the Jacobi metric formulation be generalized to Randers-Finsler metrics, and what conditions must be met for the resulting metric to be geodesically equivalent?
• RQ4 In what way does Symmetry Replacement allow for the reparameterization of RF metrics while preserving their geometric signature and physical equivalence?
• RQ5 How do optical metrics in stationary spacetimes, such as the Kerr metric, relate to the Jacobi-Maupertuis-Randers-Finsler framework, and can they be lifted exactly?

Key findings

• The Eisenhart lift cannot be applied directly to Randers-Finsler geometry because the resulting constraint after lift does not conform to the required geometric structure of RF metrics.
• The Jacobi-Maupertuis-Randers-Finsler (JMRF) metric is successfully constructed using a momentum constraint instead of a Hamiltonian, enabling exact geodesic reformulation in RF systems.
• The Jacobi lift is introduced as a valid alternative to the Eisenhart lift in RF geometry, provided a conformal factor exists, allowing exact transformation into a Riemannian metric.
• Symmetry Replacement enables the re-expression of an RF metric with a cyclic coordinate by replacing the original symmetry direction with a new one, effectively absorbing the linear term into the Riemannian part of the metric.
• The optical metric of stationary spacetimes, such as the Kerr metric, is shown to be a Randers-Finsler metric, and its Fermat metric form is derived as a JMRF metric, enabling exact lifting.
• The proposed methods eliminate higher-order approximations in metric computation (e.g., in the Hessian of the Lagrangian), allowing for exact analysis of gravity in Randers-Finsler spacetimes, as demonstrated in models like Schwarzschild-Randers.