Refocusing of Light Rays in Space-Time

TL;DR

This paper explores the properties of refocusing and strong refocusing of light rays in space-time, providing new examples and establishing relationships between these phenomena and the geometry of space-time.

Contribution

It constructs examples of refocusing space-times that are not strongly refocusing at certain points and proves that certain covering spaces preserve refocusing properties.

Findings

Refocusing points form a closed set in strongly causal space-times.

Constructed examples of space-times refocusing but not strongly refocusing at some points.

A Lorentz covering space of a strongly causal refocusing space-time is also strongly causal and refocusing.

Abstract

We investigate refocusing and strong refocusing of light rays in a space-time. A strongly refocusing space-time is refocusing. The converse is unknown. We construct examples of space-times which are refocusing, but not strongly so, at a particular point. These space-times are strongly refocusing at other points. The geometrization conjecture proved by Perelman implies that a globally hyperbolic refocusing space-time of dimension $\leq 4$ admits a strongly refocusing Lorentz metric. We show that the possibly empty set of points at which a strongly causal space-time is refocusing is closed. We prove that a Lorentz covering space of a strongly causal refocusing space-time is a strongly causal refocusing space-time. This generalizes the result of Chernov and Rudyak for globally hyperbolic space-times.