Refocusing spacetimes need not be strongly refocusing

TL;DR

The paper demonstrates that globally hyperbolic spacetimes can be refocusing without being strongly refocusing, providing new insights into spacetime geometry and answering a specific open question.

Contribution

It shows the existence of refocusing spacetimes that are not strongly refocusing and introduces Legendrian refocusing, linking these concepts.

Findings

Existence of refocusing but not strongly refocusing spacetimes

Strongly refocusing spacetimes admit refocusing metrics in higher dimensions

Introduction of Legendrian refocusing concept

Abstract

We prove that there are globally hyperbolic spacetimes $(X,g)$ which are refocusing but not strongly refocusing. In fact, every globally hyperbolic strongly refocusing spacetime of dimension at least $3$ admits globally hyperbolic metrics which are refocusing but not strongly refocusing. This answers a question by Chernov, Kinlaw, and Sadykov. We then prove that globally hyperbolic spacetimes which are Legendrian refocusing (a notion introduced in this paper) admit globally hyperbolic strongly refocusing metrics.