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There are interesting non-trivial examples of spacetimes which are asymptotically flat at null and spacelike infinities. For example, the Kerr family of black holes satisfies these conditions. However, another definition of asymptotic flatness is asymptotic flatness at future timelike infinity. Minkowski spacetime is an obvious example of this class of spacetimes, but I can't think of non-trivial examples. The Kerr family does not become asymptotically flat at late times, due to the presence of a black hole in the spacetime, so it is not an example. What are some non-trivial examples of spacetimes that are asymptotically flat at future timelike infinity? I'm particularly interested in spacetimes that are also asymptotically flat at future null infinity, if that can be taken into account in your answer.

My interest in this lays in understanding how powerful is the construction given in arXiv: 1706.09666 [math-ph], which (among other things) prescribes a way of constructing Hadamard states using a bulk-to-boundary correspondence in asymptotically flat spacetimes at future null and timelike infinity.

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  • $\begingroup$ There are several inequivalent definitions of “asymptotically flat at $i^{+}$” which assume, in particular, different levels of regularity of asymptotic unphysical metric (relatively recent paper has references to earlier works). Which one do you have in mind? $\endgroup$
    – A.V.S.
    Commented Feb 8 at 7:44
  • $\begingroup$ @A.V.S. I'm assuming the definition given by arXiv: [1706.09666 [math-ph]](arxiv.org/abs/1706.09666), Def. 2.2.1 and Remark 2.2.1. $\endgroup$
    – Níckolas Alves
    Commented Feb 8 at 14:51
  • $\begingroup$ That paper uses the term “asymptotically flat (at future null infinity) with future time infinity” which is different than “asymptotically flat at timelike infinity” in the sense of Chakraborty et al. I think that metric of a star (e.g. with Schw. interior solution) would satisfy the first definition and fail the second. $\endgroup$
    – A.V.S.
    Commented Feb 8 at 17:44

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