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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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  • $\begingroup$ Are you asking "if at least one geodesic has infinite length, then the space is asymptotically flat" or "if all geodesics have infinite length, then the space is asymptotically flat"? (The answer is no in both cases.) $\endgroup$
    – Ryan Unger
    Commented Feb 24, 2016 at 1:49

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Imagine that you follow a timelike geodesic in de-Sitter space. De-Sitter space is not asymptotically flat (since the Ricci scalar is non-zero everywhere) but you can pass an arbitrarily large amount of proper time on your geodesic.

Another example could be a non-flat universe (with curvature) that has a cosmological constant. Again, you can spend an arbitrary amount of time on a timelike geodesic there (for example if you have constant position in comoving coordinates).

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