Closed Geodesics in Lorentzian Surfaces

TL;DR

This paper proves that every closed Lorentzian surface has at least two closed geodesics, with examples confirming this is optimal, and relates the number of geodesics to the surface's causal structure and topology.

Contribution

It establishes a minimal number of closed geodesics in Lorentzian surfaces and links this to their causal and topological properties.

Findings

Every closed Lorentzian surface has at least two closed geodesics.

Explicit examples demonstrate the optimality of the two-geodesic minimum.

The number of closed geodesics relates to the surface's causal structure and homotopy type.

Abstract

We show that every closed Lorentzian surface contains at least two closed geodesics. Explicit examples show the optimality of this claim. Refining this result we relate the least number of closed geodesics to the causal structure of the surface and the homotopy type of the Lorentzian metric.