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Let $(M, g)$ be a smooth Lorenzian time-oriented manifold.

Is it possible for the Lorenzian metric induced topology to be different from that of the manifold topology, without CTCs?

We know that the two topologies are not necessarily the same and such matching can happen iff the spacetime is strongly causal.

So if the answer is yes and there exists such spacetime, they surely are not strongly causal.

I want to know how far this mismatch of topologies can go while avoiding CTCs. (avoiding also the case of "up to a metric fluctuation that can create a CTC")

Or does any topological mismatch/deviation result in CTCs(directly or up to metric fluctuation)?

Or in general:

Is there any relation at all between topological deviation, and the existence of CTCs(also up to a metric fluctuation) one way or another?

Bastam Tajik
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  • Thanks. Before taking a closer look, I should ask, if in the final and initial states, the manifold topology and the Pseudo-Riemannian metric induced topology, each match separately or not? @A.V.S. – Bastam Tajik Aug 15 '23 at 16:59
  • Which topology would you induce with the metric, if it's pseudo Riemannian you don't really have a metric topology. – Slereah Aug 15 '23 at 19:22
  • @Slereah that's not true. https://mathoverflow.net/questions/266903/on-the-topology-induced-by-a-lorentzian-metric – Bastam Tajik Aug 15 '23 at 20:10
  • @Slereah but I edit the question so that there's no ambiguity. – Bastam Tajik Aug 15 '23 at 20:51
  • @Slereah Nonetheless It's never mentioned in the question that the topology is metric, but that a topology can be defined, using a Lorentzian metric! – Bastam Tajik Aug 15 '23 at 21:06
  • For context, CTC=closed timilike curve. Related: https://physics.stackexchange.com/q/403523/226902 https://physics.stackexchange.com/q/774810/226902 https://physics.stackexchange.com/q/711578/226902 – Quillo Aug 18 '23 at 21:37

1 Answers1

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What do you mean by "how far this mismatch of topologies can go while avoiding CTCs"? You have already noticed that the two topologies coincide if and only if the spacetime is strongly causal. It is therefore sufficient to consider spacetimes without CTCs (causal) but not strongly causal to obtain that the two topologies do not coincide. Moreover in the case where spacetime is not strongly causal the Alexandrov topology ceases to be Hausdorff.

Or does any topological mismatch/deviation result in CTCs(directly or up to metric fluctuation)?

No, when the two topolgies do not coincide, there are not necessarily CTCs.

Pipe
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  • The point of question is if such spacetime remains free of CTC even after fluctuations, something rather stable. – Bastam Tajik Aug 18 '23 at 21:24
  • For example in this paper by Witten arxiv.org/pdf/1901.03928.pdf that you shared somewhere else with me, on page 24 he gives a 2d example of such spacetimes. But I don't know if it's stable against fluctuations. – Bastam Tajik Aug 18 '23 at 21:27
  • Stable causality implies strong causality, but the viceversa is not true. So the two topologies could be the same even if the spacetime is not stable causal. – Pipe Aug 18 '23 at 21:30
  • stability and causality in general are distinct to me. Why do you think stable causal and stable strongly causal are the same? can one prove that any stable causal spacetime is *at least* strongly causal (regardless of dimensionality also) – Bastam Tajik Aug 18 '23 at 21:36
  • I know Hawking showed that strong causality and stability together are nontrivial and there are spacetimes that are strongly causal but not stable. what about stable but not strongly causal? @pipe – Bastam Tajik Aug 18 '23 at 21:39
  • See remark 3.58 in https://arxiv.org/pdf/gr-qc/0609119.pdf – Pipe Aug 18 '23 at 23:48
  • Very interesting. So can I say Witten's example is unstably causal? And this happens because it's not simply connected. (theorem 3.55) – Bastam Tajik Aug 19 '23 at 03:35
  • Honestly I don't know what happens to unstably causal spacetimes after the perturbation. What becomes of them after a perturbation. What are they stable against? Only that they cannot be made to have CTC is the definition of stability? Can youhelp please? @pipe – Bastam Tajik Aug 19 '23 at 03:38
  • If so this stability is not the sort of stability in terms of energy one faces in Physics literature. But its rather concerned with CTC. Right? @pipe – Bastam Tajik Aug 19 '23 at 03:52
  • Then isn't it a bit of misnomer, since the process of appearemce of CTC by fluctuations is not irreversible as one can perturb back amd remove the CTC. @pipe – Bastam Tajik Aug 19 '23 at 04:04