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In the last words of the prepaper*, Mr Alcubierre says:

As a final comment, I will just mention the fact that even though the spacetime described by the metric (8) is globally hyperbolic, and hence contains no closed causal curves, it is probably not very difficult to construct a spacetime that does contain such curves using a similar idea to the one presented here.

Well, considering that have passed almost 20 years since this arxiv prepaper, exists a article that constructs this kind of spacetime?

(assuming only classical general relativity)

*https://arxiv.org/abs/gr-qc/0009013

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Guess I found the article, unfortunately. =(

https://journals.aps.org/prd/abstract/10.1103/PhysRevD.53.7365

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  • $\begingroup$ Krasnikov also produced a paper on the topic $\endgroup$
    – Slereah
    Apr 9 '18 at 6:56

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