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I have asked this question at physics stackexchange but got no response. I thought I could try my luck here:

I'm trying to understand the concept of asymptotic flatness in general relativity, and came up with the following question:

If the proper time $\tau$ is infinite for a timelike geodesic, does it mean that the spacetime is asymptotically flat? Or am I confusing concepts here?

Would be grateful for any clarifications.

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    $\begingroup$ Counterexample: de Sitter space has timelike geodesics of infinite proper time length, yet is not asymptotically flat. $\endgroup$ Dec 31, 2015 at 22:42
  • $\begingroup$ I see. So then, in general, is there a way to formulate the concept of asymptotic flatness using geodesics? @IgorKhavkine $\endgroup$
    – GregVoit
    Jan 1, 2016 at 11:06
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    $\begingroup$ Not as far as I know. Asymptotic flatness has to do with the existence of an asymptotic region approaching Minkowski space. There might be multiple ways of characterizing, but I don't know of a simple way of doing that using only geodesics. $\endgroup$ Jan 1, 2016 at 15:57
  • $\begingroup$ @IgorKhavkine could it be useful to calculate the proper time of timeline geodesics? (referring to characterization of asymptotic flatness) For a flat space, time like geodesic would maximize proper time… Is my direction correct? $\endgroup$
    – GregVoit
    Jan 4, 2016 at 13:58
  • $\begingroup$ @GregVoit: timelike geodesics are defined as maximizers of proper time (at least locally). Flatness have nothing to do with it. From the equivalence principle there's really no way to probe asymptotic structure "locally". At the very least you will need more than just one geodesic (you will need a congruence). $\endgroup$ Apr 11, 2016 at 18:49

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