In what $precise$ sense is Minkowski space asymptotically flat?

Question

I've brought this question over from the physics stack exchange, where it didn't generate interest.

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We say a manifold $(M,g)$ is conformally compact if it is the interior of some $(\overline M, \overline g)$, such that $$g = r^{-2}\overline g|_M,\quad \mathcal Z(r) = \partial M,\quad \text{and}\quad \mathrm{d}r_p \neq 0\ \text{for any}\ p\in \partial M.$$ Moreover, we say a conformally compact manifold is asymptotically flat if $|\mathrm{d} r|^2_{\overline g}\equiv 0$ on the boundary. (i.e. the first derivative of the defining function is null on the boundary.)

It is "well known" that Minkowski space is an asymptotically flat manifold - Penrose's famous compactification supposedly shows this. However, in his compactification, there are two points on the boundary for which $\mathrm{d}r_p = 0$: namely spacelike and timelike infinity. I've made my own coordinate free compactifications of Minkowski space, but they also have some singularities at conformal infinity.

How is this formalised? When people say "conformally compact" in literature, do they just mean "there is a dense subset of the boundary satisfying the above conditions"?

Answer 1

The definition I would give of asymptotic flatness is the following:

A time-orientable spacetime $(M,g)$ is asymptotically flat at null infinity if there exists a spacetime $(\bar{M},\bar{g})$ such that,

1. there exists a positive function $\Omega:M\rightarrow \mathbb{R}$ such that $(\bar{M},\bar{g})$ is an extension of the spacetime $(M,\Omega^2g)$. Hence if we regard $M\subset \bar{M}$ then $\bar{g}=\Omega^2g$ on $M$.

2. within $\bar{M}$, M can be extended to a manifold with boundary $\partial M$.

3. $\Omega$ can be extended to a function on $\bar{M}$ such that $\Omega|_{\partial M}=0$ and $d\Omega \neq 0$ on $\partial M$.

4. $\partial M=\mathcal{I}^+\sqcup \mathcal{I}^-$ where $\mathcal{I}^{\pm}$ are diffeomorphic to $\mathbb{R}\times \mathbb{S}^2$.

5. No past (future) directed causal curve starting in $M$ intersects $\mathcal{I}^+$ ($\mathcal{I}^-$) respectively.

6. $\mathcal{I}^{\pm}$ are complete: under the gauge condition $(\nabla_a\nabla_b\Omega)=0$ on $\mathcal{I}^{\pm}$, the generators of $\mathcal{I}^{\pm}$ are complete. ($\nabla_{a}$ is the Levi-Civita associated to $\bar{g}$).

It should be relatively easy to show the conformal compactification of Minkowski spacetime satisfies the definition I gave here.

A point of interest, it can be shown that a spacetime which satisfies this definition 'looks like' Minkowski spacetime 'far away'. I.e. if you work out the metric in a neighborhood of null infinity then the leading order contribution looks like the Minkowski spacetime. This is the statement that this definition formalizes.

Please note that this definition is semi-lifted from Harvey Reall's Black Hole lecture notes which in turn are largely based on Wald and Hawking and Ellis. I believe this definition is in Wald.