[Paper Review] $t$-periodic light rays in conformally stationary spacetimes via Finsler geometry

Leonardo Biliotti, Miguel Ángel Javaloyes|ArXiv.org|Mar 4, 2008

Advanced Differential Geometry Research31 references4 citations

TL;DR

This paper establishes multiplicity results for $t$-periodic light rays in conformally stationary spacetimes by leveraging Finsler geometry through the Fermat metric, which reduces the problem to closed geodesics on a non-reversible Finsler manifold. The key contribution is proving the existence of infinitely many geometrically distinct $t$-periodic light rays under topological conditions (e.g., nontrivial fundamental group or infinite abelian fundamental group), and deriving a lower bound for the period when the Fermat metric’s flag curvature is $η$-pinched.

ABSTRACT

In this paper we prove several multiplicity results of $t$-periodic light rays in conformally stationary spacetimes using the Fermat metric and the extensions of the classical theorems of Gromoll-Meyer and Bangert-Hingston to Finsler manifolds. Moreover, we exhibit some stationary spacetimes with a finite number of $t$-periodic light rays and compute a lower bound for the period of the light rays when the flag curvature of the Fermat metric is $η$-pinched.

Motivation & Objective

• To establish multiplicity results for $t$-periodic light rays in conformally stationary spacetimes using Finsler geometry.
• To extend classical theorems of Gromoll-Meyer and Bangert-Hingston to non-reversible Finsler metrics arising from the Fermat metric.
• To derive a lower bound for the period of $t$-periodic light rays when the Fermat metric has $η$-pinched flag curvature.
• To analyze the role of reversibility in Fermat metrics and its impact on geodesic length estimates.

Proposed method

• Mapping $t$-periodic light rays in spacetimes to closed pregeodesics in the base manifold via the Fermat metric, a Finsler metric constructed from the spacetime’s Lorentzian structure.
• Applying a Finsler extension of the Gromoll-Meyer theorem to guarantee infinitely many closed geodesics under nontrivial fundamental group conditions.
• Extending the Bangert-Hingston theorem to non-reversible Finsler manifolds to ensure infinitely many geometrically distinct closed geodesics when the fundamental group is infinite abelian.
• Computing the reversibility $\lambda$ of the Fermat metric using Lagrange multipliers on the tangent bundle, based on the vector field $\delta$ and conformal factor $\beta$.
• Translating Rademacher’s geodesic length estimate for $\eta$-pinched Finsler manifolds into a period lower bound for $t$-periodic light rays.
• Expressing the curvature pinching condition and period bound in terms of $\varphi = \max_{x\in M} |\delta|_0 / \sqrt{\beta}$, the normalized magnitude of the drift vector field.

Experimental results

Research questions

• RQ1 Under what topological conditions on the Cauchy surface $M$ do conformally stationary spacetimes admit infinitely many geometrically distinct $t$-periodic light rays?
• RQ2 How can the Fermat metric in a stationary spacetime be used to reduce the existence problem of $t$-periodic light rays to closed geodesics on a Finsler manifold?
• RQ3 What is the optimal lower bound for the period of $t$-periodic light rays when the Fermat metric has $\eta$-pinched flag curvature?
• RQ4 How does the non-reversibility of the Fermat metric affect the multiplicity and length estimates of closed geodesics compared to the Riemannian case?

Key findings

• Infinitely many geometrically distinct $t$-periodic light rays exist in conformally stationary spacetimes when the Cauchy surface $M$ satisfies the Gromoll-Meyer condition.
• Infinitely many geometrically distinct $t$-periodic light rays exist when $\pi_1(M)$ is infinite abelian, via an extension of the Bangert-Hingston theorem to non-reversible Finsler metrics.
• A lower bound for the period of $t$-periodic light rays is derived: $ \frac{2\pi\sqrt{1+\varphi^2}}{\varphi + \sqrt{1+\varphi^2}} $, where $\varphi = \max_{x\in M} |\delta|_0 / \sqrt{\beta} $.
• The reversibility $\lambda$ of the Fermat metric is computed as $ \lambda = \frac{\varphi + \sqrt{1+\varphi^2}}{-\varphi + \sqrt{1+\varphi^2}} $, which quantifies the asymmetry of the Finsler structure.
• The flag curvature pinching condition $ \frac{\varphi + \sqrt{1+\varphi^2}}{2\sqrt{1+\varphi^2}} < K(p) < 1 $ ensures the period lower bound holds uniformly across $M$.
• The results generalize prior static spacetime results and extend the applicability of Finsler-geometric methods to non-reversible, conformally stationary Lorentzian spacetimes.