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Asymptotic flatness basically means that you can apply a conformal transformation to your spacetime so that it becomes compact, and it admits a boundary having the same causal structure as the boundary of Minkowski space ($\mathscr{I}^-$, $i^0$, and $\mathscr{I}^+$). Different people give different details for the definition, which seem to include additional requirements such as regularity, no CTCs at infinity, and no matter at infinity. For reference, here is a non-paywalled discussion by Frauendiener. Wald also has a treatment of the topic in ch. 11.

If I go through my repertoire of interesting solutions to the Einstein field equations, the primary examples are ones for which it's either obvious that they're asymptotically flat (Schwarzschild) or obvious that they're not asymptotically flat (cosmological spacetimes, which don't have matter-free regions). In none of these cases do I see much motivation for all the fancy machinery involved in the definition. Can anyone suggest a minimalistic example (or more than one) that would help to provide such a motivation?

Related:

What does asymptotically flat solution mean?

What techniques can be used to prove that a spacetime is not asymptotically flat?

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    $\begingroup$ Motivation for the opposite situation: the old-school definition (in terms of the falloff conditions) works for any number of spacetime dimensions. On the other hand, the modern (coordinate-free) definition is best understood for even dimensions only, because in odd dimensions $\mathscr I$ is a subtle concept (cf. gr-qc/0407014). $\endgroup$ – AccidentalFourierTransform Feb 5 '18 at 2:56
  • $\begingroup$ it's either obvious that they're asymptotically flat... It's obvious only if "right" coordinates are used. If this metric is mangled by some general-covariant transformation, or given solution is only applicable to some patch of a whole spacetime, you need "fancy machinery". So presumably, answers could include such analysis performed even for what would then turn out to be "obvious" examples? $\endgroup$ – A.V.S. Feb 5 '18 at 4:36
  • $\begingroup$ Just a note: You can conformally map say the Einstein Universe to Minkowski space (Ok Ok minus a point at infinity) but you wouldn't call it asymptotically flat by any means, so there would be some extra criteria required to distinguish this. In this example you might need the no matter at infinity clause $\endgroup$ – R. Rankin Feb 5 '18 at 7:13
  • $\begingroup$ @A.V.S.: I understand why you can't necessarily appeal to an $r$ coordinate, as if such a thing were automatically and uniquely well defined, so in that sense I understand the motivation for the modern style of definition. However, if the only example people had had in mind was the Schwarzschild spacetime, it seems unlikely that they would have been motivated to come up with the modern definition. Clearly a lot of the details of the regularity conditions are based on studying examples (IIRC Wald says this). A spacetime with a point source of gravitational radiation would seem natural...? $\endgroup$ – user4552 Feb 5 '18 at 16:01
  • $\begingroup$ @BenCrowell: My point is that solutions to GR rarely arrive in a "nice" choice of coordinates, instead coordinates in which it is easier to solve Einstein equations are used. If we want to make a change to say, "Bondi chart" we need to solve relativistic eikonal equations for the coordinates which is a nontrivial thing by itself. $\endgroup$ – A.V.S. Feb 5 '18 at 20:13
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Just to start with, I would say that formalizing the asymptotic structure is important to properly understanding the peeling theorem, so even the boring cases like Kerr are interesting (to me) in that sense.

But to be a bit pickier, it presumably really matters most when things have interesting time dependence, so that there are substantial differences between spacelike and null infinity — and between different slices of the null infinities.

So, as far as exact solutions, the most interesting one I can think of is the Vaidya metric. For approximate solutions, you have Teukolsky waves. That's probably where things start to get most interesting, because the angular dependence can be very nontrivial. So, for example, you may want to measure the radiation from a perturbed black hole — as approximated by the Teukolsky solution. To do so, you probably want to use Bondi mass and momenta, which are defined explicitly using asymptotic flatness. You could probably mess around trying to take limits along outgoing null rays, for example, but in the end you'd just end up reconstructing the basic notions of asymptotic flatness.

And of course, you also have black-hole binary mergers, for which we have post-Newtonian approximations and numerical simulations. The gravitational waves extracted from those are measured on future null infinity, and the behavior of coordinates (on and) in the neighborhood of $\mathscr{I}^+$ are important to extracting them correctly. (Of course, I would say that, since that's one of the main thrusts of my research.)

Taking a step back, I'll just point out that the limits of asymptotically flat spacetimes are important because they are invariant. Note that I didn't say any points or certainly any coordinates on those limit surfaces are invariant (see BMS transformations especially), but the surfaces themselves. So rather than just arbitrarily choosing, for example, coordinate spheres or time slices in the bulk spacetime, you can at least reduce the gauge freedom of the geometric objects you're discussing, which tends to make your results a bit more invariant.

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