On class A Lorentzian 2-tori with poles I: Closed geodesics pass through poles

TL;DR

This paper proves that on class A Lorentzian 2-tori with poles, every timelike pole has a corresponding closed timelike geodesic passing through it, with results on the existence and non-rigidity of such geodesics.

Contribution

It establishes the existence of closed timelike geodesics passing through poles on class A Lorentzian 2-tori and explores non-rigid phenomena related to timelike poles.

Findings

Existence of closed timelike geodesics through poles

Construction of geodesics in any free homotopy class within the stable time cone

Non-rigid behavior when timelike poles are present

Abstract

In this paper, by studying certain isometries on globally hyperbolic planes, we prove that if $p$ is a timelike pole on a class A Lorentzian 2-torus, then there exists a closed timelike geodesic passing through $p$ with any preassigned free homotopy class in the interior of the stable time cone. We also show a non-rigid result when timelike poles appear.