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Consider a Lorentzian manifold $(M,g)$.

Must a maximal, affinely parameterized geodesic $\gamma: [0,b) \to M$ which intersects a point $p \in M$ infinitely many times be complete (i.e., have $b = \infty$)?

It's easy to show that the answer is positive in the Riemannian setting by considering a normal ball around $p$ and noticing that the geodesic must radially traverse its diameter infinitely many times, accumulating infinite length. The same picture applies in the Lorentzian case, up until the critical conclusion of "accumulating infinite length", since the "diameter" of a normal neighborhood is no longer meaningful. Indeed, the radial trajectories may, in principle become closer and closer to null with subsequent passings of $p$ in such a way that the accumulated length/proper time is finite (or zero if the geodesic was null in the first place).

While the Riemannian reasoning breaks down, is there some other reason that intersecting a single point infinitely many times enforces completeness in general, or can one perhaps find a counterexample? Feel free to assume any causality condition you like, so that the question is really about spacelike geodesics.

It is worthwhile to note that the most obvious way to obtain infinitely repeated intersection of a single point is, of course, to consider a closed loop, which is always complete when maximally extended (at least when non-null, which would be the case under even a very lax causality condition).

Also worth noting is that maximal, incomplete geodesics with limit points can exist (e.g. in the Misner spacetime), which demonstrates that something too close to the Riemannian reasoning (which also rules this out in a Riemannian setting) can't work.

Edit: Didier has helpfully provided the counterexample of the Clifton-Pohl torus. This example, being compact, admits closed timelike curves, however, so I am still interested in the question of whether this is possible under a more restrictive causality condition. In particular, since the issue arising in the above normal neighborhood picture is that the geodesic may become too close to null in normal coordinates, it seems intuitive that $M$'s being stably causal (i.e. $M$ would not admit closed causal curves even under small perturbations to the metric) may rule this out. Can this intuition be made precise, or could even a less restrictive causality condition suffice?

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    $\begingroup$ The Clifton-Pohl torus is an example $\endgroup$
    – Didier
    Commented Dec 20, 2021 at 23:20
  • $\begingroup$ @Didier So it is! Thank you. If you transfer that statement to an answer, I would upvote. Since it answers my question as stated, I will accept it if no other answer treating a more restrictive causality condition emerges. While illuminating for my inquiry, this example is causally pathological in that it (being compact) admits closed timelike curves. $\endgroup$
    – jawheele
    Commented Dec 21, 2021 at 18:05
  • $\begingroup$ @Didier What a cool example. Knowing very little about Lorentzian geometry, I've never encountered this before. $\endgroup$ Commented Dec 21, 2021 at 18:21

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The Clifton-Pohl torus is an example of a compact Lorentzian manifold which is not geodesically complete: more than that, there exists a closed geodesic which is not complete!

(Community wiki as it is a link only answer)

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