Geodesy on surfaces of revolution: A wormhole application

TL;DR

This paper derives equations for geodesic orbits on 2D surfaces of revolution, applies them to a wormhole model, and finds analytic solutions revealing two types of geodesics, one confined and one connecting both sides, with potential implications for interstellar travel.

Contribution

It introduces a general method for deriving geodesic equations on surfaces of revolution and applies it analytically to a wormhole model, revealing new orbit types.

Findings

Analytic solutions for geodesics expressed with elliptic functions.

Identification of two geodesic types: confined and throat-passing.

Numerical illustrations of geodesic behavior on the wormhole.

Abstract

We outline a general procedure to derive first-order differential equations obeyed by geodesic orbits over two-dimensional (2D) surfaces of revolution immersed or embedded in ordinary three-dimensional (3D) Euclidean space. We illustrate that procedure with an application to a wormhole model introduced by Morris and Thorne (MT), which provides a prototypical case of a `splittable space-time' geometry. We obtain analytic solutions for geodesic orbits expressed in terms of elliptic integrals and functions, which are qualitatively similar to, but even more fundamental than, those that we previously reported for Flamm's paraboloid of Schwarzschild geometry. Two kinds of geodesics correspondingly emerge. Regular geodesics have turning points larger than the `throat' radius. Thus, they remain confined to one half of the MT wormhole. Singular geodesics funnel through the throat and connect both halves of the MT wormhole, perhaps providing a possibility of `rapid inter-stellar travel.' We provide numerical illustrations of both kinds of geodesic orbits on the MT wormhole.