Refocusing of Light Rays in Space-Time

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 $\le 4$ 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.

23 pages

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Cited by in corpus (4)

• On the space of light rays of a space-time and a reconstruction theorem by Low
• Causality and skies: is non-refocussing necessary?
• The space of light rays: Causality and $L$-boundary
• Topological consequences of null-geodesic refocusing and applications to $Z^x$ manifolds