Is it possible for two geodesics $\gamma_1, \gamma_2$ to be identical within a finite interval without being identical outside the interval? IOW: $\gamma_1(t) = \gamma_2(t)$ for $t \in (A,B)$ but $\gamma_1(t) \neq \gamma_2(t)$ for some $t \notin (A,B)$.
At first glance, I would think that uniqueness (the Cauchy Lipschitz theorem) would say the answer is no, because if $\gamma_1(t) = \gamma_2(t)$ at even a single point $t = t_0$ then they are equal everywhere.
However, the paper [1] below, by the well known physicist Kip Thorne, would seem to indicate this is not always true. This paper considers the general relativistic motion of an object along a timelike geodesic $\gamma$ in a spacetime that admits closed timelike curves, and came to the surprising conclusion that there are instances where there are multiple solutions to the general relativistic equations of motion. Specifically, if I read the paper correctly, two (or more) solutions $\gamma_1, \gamma_2$ can be equivalent up to time $t_0$ but diverge for $t > t_0$. This property is known as "multiplicity."
I am thinking that the uniqueness theorem assumes a simply connected topology that does not allow closed timelike curves, and the answer to my question requires a sophisticated understanding of topology that I do not yet possess. Perhaps $\gamma_1$ is embedded in 4-manifold $M_1$, and $\gamma_2$ is embedded in a distinct 4-manifold $M_2$, where: $M_1$ and $M_2$ are each embedded within some higher dimensional manifold; the intersection of $M_1$ and $M_2$ is a (finite? bounded?) 4-dimensional region $M_3$; $M_1$ and $M_2$ have distinct global topologies; the segment where $\gamma_1(t) = \gamma_2(t)$ is inside $M_3$, and they are unequal once they exit $M_3$. That's as far as I've gotten.
My background: I know a little about differential equations and analytic continuation but it's been a while since I've really used them. If the answer to my question is "yes, under certain conditions" then I am interested in exploring those conditions in detail, but I still have quite a lot to learn about the relevant mathematics - manifolds, tensors, topology, etc. I would appreciate any suggestions you may have on specific mathematical tools or subfields I should study. I apologize in advance if I have butchered any terminology ;) .
[1] Billiard balls in wormhole spacetimes with closed timelike curves: Classical theory. Fernando Echeverria, Gunnar Klinkhammer, and Kip S. Thorne. Phys. Rev. D 44, 1077. Published 15 August 1991. http://dx.doi.org/10.1103/PhysRevD.44.1077