# Alternate geodesic completions of a Schwarzschild black hole

The Kruskal-Szekeres solution extends the exterior Schwarzschild solution maximally, so that every geodesic not contacting a curvature singularity can be extended arbitrarily far in either direction.

Wikipedia says it is the unique maximal extension that is an analytic, simply connected vacuum solution.

So what happens without "simply"? I.e. anyone know a maximal extension that is an analytic, connected vacuum solution, but is not simply connected?

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My guess is that the reason people specify simply connected in the context of analytic extensions is to avoid situations like the following: you can embed Minkowski space as a patch in all sorts of tori - just wrap the various dimensions. So these tori are analytic extensions of Minkowski space, and I guess the "simply connected" criterion is to avoid a plethora of extensions like this. – twistor59 Dec 21 '12 at 14:22
The non-simplyconnected ones should come as quotions of the simplyconnected one by discrete groups. – MBN Dec 22 '12 at 11:53
Oh, is that all it is? It makes sense, thanks. – Retarded Potential Dec 24 '12 at 2:58
@twistor59 I keep seeing this on the unanswered list, someone want to make an "official" answer? – Retarded Potential Mar 13 '13 at 16:58
@MBN (apparently I can't ping two at once... ping) – Retarded Potential Mar 13 '13 at 16:58