Geodesics on an ellipsoid in Minkowski space

TL;DR

This paper explores the geometry of geodesics on a Lorentz ellipsoid in Minkowski space, providing explicit formulas and proving a Poncelet-type theorem for null geodesics, advancing understanding of pseudo-Riemannian geometry.

Contribution

It offers explicit formulas for geodesic properties and establishes a Poncelet-type theorem for null geodesics on a Lorentz ellipsoid, a novel result in pseudo-Riemannian geometry.

Findings

Explicit formulas for geodesic integrals and curvature

Invariant area-forms on geodesics

A Poncelet-type closure theorem for null geodesics

Abstract

We describe the geometry of geodesics on a Lorentz ellipsoid: give explicit formulas for the first integrals (pseudo-confocal coordinates), curvature, geodesically equivalent Riemannian metric, the invariant area-forms on the time- and space-like geodesics and invariant 1-form on the space of null geodesics. We prove a Poncelet-type theorem for null geodesics on the ellipsoid: if such a geodesic close up after several oscillations in the "pseudo-Riemannian belt", so do all other null geodesics on this ellipsoid.