Homologically maximizing geodesics in conformally flat tori

TL;DR

This paper investigates homologically maximizing timelike geodesics in conformally flat tori, establishing compactness, Lipschitz continuity of time separation, and exploring the properties of the stable time separation function.

Contribution

It introduces a compactness theorem for such geodesics, proves Lipschitz continuity of the time separation, and analyzes the concavity of the stable time separation function.

Findings

Proved a compactness result for homologically maximizing geodesics.

Established Lipschitz continuity of the time separation in the universal cover.

Linked concavity properties of the stable time separation to geodesic behavior.

Abstract

We study homologically maximizing timelike geodesics in conformally flat tori. A causal geodesic $\gamma$ in such a torus is said to be homologically maximizing if one (hence every) lift of $\gamma$ to the universal cover is arclength maximizing. First we prove a compactness result for homologically maximizing timelike geodesics. This yields the Lipschitz continuity of the time separation of the universal cover on strict sub-cones of the cone of future pointing vectors. Then we introduce the stable time separation $\mathfrak{l}$. As an application we prove relations between the concavity properties of $\mathfrak{l}$ and the qualitative behavior of homologically maximizing geodesics.