The Jacobi metric approach for dynamical wormholes

TL;DR

This paper introduces the Jacobi metric formalism to analyze dynamical wormholes, enabling the study of geodesic motion and stability without directly solving geodesic equations, and relates curvature to wormhole conditions.

Contribution

It extends the Jacobi metric approach to dynamical wormholes, providing a new method to analyze geodesics and stability in evolving wormhole spacetimes.

Findings

First integral of geodesic equations found using Jacobi metric

Stable circular orbits characterized in the Jacobi framework

Gaussian curvature relates to wormhole flare-out condition

Abstract

We present the Jacobi metric formalism for dynamical wormholes. We show that in isotropic dynamical spacetimes , a first integral of the geodesic equations can be found using the Jacobi metric, and without any use of geodesic equation. This enables us to reduce the geodesic motion in dynamical wormholes to a dynamics defined in a Riemannian manifold. Then, making use of the Jacobi formalism, we study the circular stable orbits in the Jacobi metric framework for the dynamical wormhole background. Finally, we also show that the Gaussian curvature of the family of Jacobi metrics is directly related, as in the static case, to the flare-out condition of the dynamical wormhole, giving a way to characterize a wormhole spacetime by the sign of the Gaussian curvature of its Jacobi metric only.