There are examples continuous automata which propagate, like gliders in Conway's Game of Life. Tipler showed that a compact Lorentzian manifold contains a closed timelike geodesic if a technical condition is satisfied (the covering space has a compact Cauchy surface). No closed timelike curve (CTC) on a Lorentzian manifold, including the abovementioned geodesic, can be continuously deformed as a CTC to a point; otherwise any neighborhood of that point would contain a CTC, contradicting the fact that Lorentzian manifolds are locally causally well behaved. Therefore, some topological feature prevents the CTC from being contracted to a point. An observer on the geodesic will free fall toward, through, and away from that topological feature. In that observer's frame of reference, the topological feature will propagate toward the observer. This is a continuous analog of a glider in Conway's Game of Life.
Furthermore, the topology change theorems of Geroch and Tipler do not rule out topology change on a manifold with a CTC through every point. However, because the topological feature mentioned above propagates, conservation of momentum implies they can only be created or destroyed in multiples of two (Hawking and Gibbons have a similar result by a different argument). This is curiously similar to the behavior of photons.
[grqc/0609054] Topology and Closed Timelike Curves I: Dynamics
Authors:  Hunter Monroe 
Abstract:  No CTC on a Lorentzian manifold can be continuously deformed as a CTC to a point, because Lorentzian manifolds are locally causally wellbehaved. Every CTC must pass through some topological feature, to be called a timelike wormhole, which prevents it from being deformed to a point. A test particle free falling along a closed timelike geodesic transits a timelike wormhole; in the test particle's frame, the timelike wormhole propagates toward the test particle. Tipler's theorem that emergence of a CTC implies a singularity does not apply to a spacetime that contains a CTC through every point. 
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