The modern coordinate-indepenent definition of asymptotic flatness was introduced by Geroch in 1972. You can find presentations in Wald 1984 and Townsend 1997. The definition is in terms of existence of something -- a certain type of conformal compactification -- and in many cases such as the Schwarzschild metric, it's straightforward to think up a construction that proves existence. But to prove that a spacetime is not asymptotically flat, you have to prove the nonexistence of any such compactification, which seems harder.

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

The only example of such a technique that I've been able to think of is the following. Suppose we want to prove that an FLRW spacetime is not asymptotically flat. Wald gives a theorem by Ashtekar and Hansen that says that if a spacelike submanifold contains $i^0$, it admits a metric that is nearly Euclidean, so that, e.g., the spatial Ricci tensor falls off like $O(1/r^3)$. This implies that there can't be any lower bound on the Ricci tensor, but an FLRW spacetime can have such a lower bound on a constant-time slice, since such a slice has constant spatial curvature.

Townsend, http://arxiv.org/abs/gr-qc/9707012

Wald, General Relativity

  • $\begingroup$ Is there a definition of asymptotically flat spacetime, which is not vacuum or doesn't contain an isolated object? Are you asking two questions, about conformal and asymptotic flatness? I thought the the Weyl tensor gives you the answer for the first. $\endgroup$ – MBN Aug 29 '13 at 11:05
  • $\begingroup$ You seem to be interchanging conformal flatness with asymptotic flatness--they are obviously related concepts, but the pedant in me says that an asymptotically flat spacetime need only be conformally flat at infinity. $\endgroup$ – Jerry Schirmer Aug 29 '13 at 15:27
  • $\begingroup$ @MBN: Wald defines it first for vacuum spacetimes, but then generalizes the definition to include spacetimes that are vacuum in a neighborhood of $\mathscr{I}^+ \cup \mathscr{I}^- \cup i^0$. No, I'm only asking one question. The modern, coordinate-independent definition of asymptotic flatness involves doing a conformal transformation that basically brings infinity in to a finite distance, and adds idealized points at infinity. $\endgroup$ – Ben Crowell Aug 29 '13 at 15:30
  • $\begingroup$ Oops, I see -- I caused massive confusion by writing "conformally flat" in the second paragraph when I meant to say "asymptotically flat." Sorry. The whole question is only about asymptotic flatness. $\endgroup$ – Ben Crowell Aug 29 '13 at 15:31
  • $\begingroup$ I am asking because if asymptotic flatness is defined for vacuum or vacuum near infinity, then homogenous universe with stuff would be excluded. $\endgroup$ – MBN Aug 29 '13 at 20:39

The notion of asymptotic flatness in 4-dimensions was studied way back in 1962 by Bondi, van der Burg, Metzner (here) and Sachs (here) and more recently by Barnich and Troessaert (the first few papers here)

They described asymptotic flatness in terms of the Bondi coordinates, where the metric takes the form $$ ds^2 = \frac{V}{r} e^{2\beta} du^2 - 2 e^{2\beta} du dr + g_{AB} \left( dx^A - U^A du \right) \left( dx^B - U^B du \right) $$ where $A,B = 2,3$, $x^A = (\theta,\phi)$ and $\det g_{AB} = r^4 \sin^2\theta$. Bondi shows that every 4-dimensional metric can be written in the form above. Techniques developed later by Penrose showed that one should really set a more general condition wherein $\det g_{AB} = \frac{1}{4} r^4 e^{2 {\tilde\varphi}}$ (following Barnich's notation). An asymptotically flat spacetime then satisfies these boundary conditions (at large $r$)

  1. $g_{AB} = r^2 \gamma_{AB} + {\cal O}(r)$ where $\gamma_{AB}$ is conformal to the metric of $S^2$, i.e. $$ \gamma_{AB}dx^A dx^B = e^{2\phi(u,\theta,\phi)} \left( d\theta^2 + \sin^2\theta d\phi^2 \right) $$

  2. $\frac{V}{r} = - 2 r \partial_u {\tilde \varphi} + \Delta {\tilde \varphi} + {\cal O}(r^{-1})$

  3. $\beta = {\cal O}(r^{-2})$

  4. $U^A = {\cal O}(r^{-2})$

Given a metric, its asymptotic flatness can be checked by testing if the above boundary conditions hold.

EDIT: This is a discussion of asymptotic flatness at $\mathscr{I}^+$. An analogous discussion exists for $\mathscr{I}^-$ (simply take $u \to v = u + 2 r$. For a complete description of asymptotic flatness, one must also consider the structure of the spacetime at $i^0$. While this has been discussed in Ashtekar, Hansen, I do not know too much about it. I will leave this to other members to discuss.

  • 1
    $\begingroup$ Thanks for the answer. My understanding is that although there were earlier definitions of asymptotic flatness that used coordinates, the modern approach is the coordinate-independent one introduced by Geroch. I don't know if they're strictly equivalent definitions. I'm not clear on how one would carry out the technique you propose, because in the coordinate-based definition, I guess we'd say a spacetime is asymptotically flat if coordinates exist such that conditions 1-4 hold. But then to prove that a spacetime was not asymptotically flat, I'd have to prove nonexistence of such coordinates. $\endgroup$ – Ben Crowell Aug 29 '13 at 15:35
  • $\begingroup$ It was shown by Sachs, that such a coordinate system always exists, whether or not the spacetime is asymptotically flat. The conditions of flatness is then given by the conditions 1-4. $\endgroup$ – Prahar Aug 29 '13 at 15:51
  • $\begingroup$ Hm...but doesn't the problem then reduce to the problem of finding how the $(r,u,\theta,\phi)$ coordinates could be mapped onto your spacetime? Maybe your description of the technique would be clarified if you could give an example of applying it, e.g., to an FLRW spacetime. $\endgroup$ – Ben Crowell Aug 29 '13 at 15:57
  • $\begingroup$ Can such spacetime satisfying conditions 1-4 be factored by some discrete group? Would resulting spacetime still be asymptotically flat? Alternatively, can we glue together several copies of said asymptotically flat spacetime, keeping conditions 1-4 but obtaining regions outside the limits $r\to \infty $? $\endgroup$ – user23660 Aug 29 '13 at 16:25
  • $\begingroup$ So, if you read Sachs, he describes how one can obtain the $(u,r,\theta,\phi)$ coordinates starting from any generalized coordinates $x^\mu$. For example, the coordinate is $u$ is defined by a scalar function $g^{\mu\nu} \partial_\mu u \partial_\nu u = 0$ (any one solution is acceptable as long as it satisfies $\rho \neq 0$ and $|\sigma|^2 \neq \rho^2$, See Sachs for def.). Once you use this to solve for $u$, $\theta$ and $\phi$ are defined as $k^a \partial_a \theta = k^a \partial_a \phi = 0$ where $k_a = \partial_a u$. The coordinate $r$ is the defined by the determinant condition above. $\endgroup$ – Prahar Aug 29 '13 at 16:27

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