Refocusing of light rays in space-time

We discuss several definitions of refocusing of null-geodesics in a strongly causal space-time and prove their equivalence. A space-time which is strongly refocusing at a point x is refocusing at x. Examples are provided of space-times which are refocusing at a point x but not strongly refocusing at x. However, they are strongly refocusing at other points. It is unknown if refocusing implies strong refocusing. Furthermore, the set of points at which a strongly causal space-time is refocusing is closed. A space-time covering of a refocusing space-time is also refocusing. This generalizes the result of Chernov and Rudyak [Comm. Math. Phys. 279, 309 (2008)] for globally hyperbolic space-times. As a consequence of the recent results on refocusing and of the Elliptization Conjecture proved by Perelman, any globally hyperbolic refocusing space-time of dimension ⩽4 admits a strongly refocusing metric.