# Is a reasonable spacetime geodesically connected?

There are some theorems concerning whether a spacetime is geodesically connected (whether any two points $$p, q \in M$$ admit a geodesic connecting them) or not, ie [1][2], but all of these are concerned with either rather simplistic spacetimes, which would not really apply to our own universe, or have very specific conditions that seem perhaps too narrow.

Is there a theorem that our own universe would be geodesically connected? Throwing around a few "reasonable" conditions, that may imply spacetimes with the following properties :

• Globally hyperbolic
• Maximally extended
• Obeying some reasonable energy condition (ie the null energy condition or the generic condition)
• Bounded stress-energy tensor

It can be shown rather easily that this is false just admitting global hyperbolicity and the NEC (ie. the spacetime constructed by taking the union of two future light cone in Minkowski space), or by taking a Weyl transform of this spacetime so that the Riemann tensor diverges on its boundary, so that it is also maximally extended.

Is there a specific theorem for a plausible spacetime where it is geodesically connected, or is that simply wrong, or unknown?

• IIRC even simple pp-wave type spacetimes are generally not geodesically connected. So the answer to a title question is probably “no”. I'll try to dig up link and write an answer later. – A.V.S. Mar 22 at 13:34
• @A.V.S. That is some of the reasons I would not consider PP-wave space-times generally physically plausible. The point is that since the wave is infinitely extended, in a (not-rigorously-defined) sense of the word, they carry infinite energy. – Void Mar 22 at 14:19
• @Void: It is not the infinite dimensions but the focusing properties that are behind this geodesic disconnectedness: all geodesics emanating from some points are focused and the “shadow” regions thus have no geodesics from that point. If we limit dimensions of the “lens” and even introduce small enough perturbations, shadows would still persist. – A.V.S. Mar 22 at 15:19
• @A.V.S. Well, I would still like to see a source that produces a wave strong enough for a lens. Binary inspirals have a maximum luminosity at equal masses and coalescence. This is associated with a time scale and a length scale. I.e., you will have a limit on the size of the region where the pp-approximation to a real wave will become bad (you need to include curvature of the wavefront and modulation). – Void Mar 22 at 16:04
• @Void Even weak wave could produce lensing, you just have to increase the distance (both focal points). – A.V.S. Mar 22 at 16:32

Whether a “reasonable” spacetime is geodesically connected depends on the definition of “reasonable”.

If we consider globally hyperbolic spacetime as reasonable, then there is a theorem by Avez (1963) & Seifert (1967), that states that globally hyperbolic spacetimes are causally geodesically connected. Note, that the two points here must be causally connected, the theorem says nothing about points that are not causally related. For a proof of the theorem, as well as other sufficient conditions in terms of disprisonment and pseudoconvexity see the book:

• Beem J.K., Ehrlich P.E., Easley K.L. (1996) Global Lorentzian geometry. Marcel Dekker, New York.

or a more recent (and open access) review:

Universal covering of anti-de Sitter is an example of spacetime that is not geodesically connected: all timelike geodesics that are emanating from a given point $$p$$ are focused in an antipodal point $$q$$, so a point $$r$$, with a spacelike separation from $$q$$ has no geodesics connecting it to $$p$$ despite being causally connected to it (see figure):

While arguably anti-de Sitter is not a reasonable spacetime, since it corresponds to negative cosmological constant, there is Bertotti–Robinson solution of Einstein–Maxwell system which is just $$\text{AdS}_2 \times S_2$$ and so it is maximally extended, with a bounded stress–energy tensor obeying reasonable energy conditions, yet not geodesically connected.

Another interesting group of spacetimes without geodesic connectedness consists of various pp-wave solutions. As first noticed for both purely gravitational and EM+gravitational plane waves (a subclass of pp-waves with a higher symmetry) by R. Penrose in 1965, focusing behavior of null geodesics from a specifically chosen points precludes such spacetimes from being globally hyperbolic and geodesically connected. Various generalizations of such spacetimes has been considered see e.g. here and here. Imposition of “reasonable” constraints on such spacetimes such as finiteness or asymptotic flatness of the front often results in global hyperbolicity and geodesic connectedness.

So, overall, conditions for geodesic connectedness of spacetime must be global in character (so just some specific energy condition would not do).