Asymptotically flat spacetime

Remarks

Clause 4 with the Einstein equations forces $\mathscr{I}$ to be a null hypersurface for $\tilde g$. There is also a formulation at spatial infinity, which is the setting for the ADM energy. It asks for coordinates on a spacelike slice with $g_{ij} = \delta_{ij} + O(1/r)$ and $\partial_k g_{ij} = O(1/r^2)$ as $r \to \infty$, and a suitable fall-off of the extrinsic curvature. Complete definitions in the literature add further technical conditions, for example that every null generator of $\mathscr{I}^\pm$ is complete.

Example: Minkowski space is asymptotically flat at null infinity, using its compactification into the Einstein static universe with $\Omega = 2 \cos U \cos V$, where $U = \arctan(t - r)$ and $V = \arctan(t + r)$. The Schwarzschild metric is also asymptotically flat, and it is a vacuum solution, so clause 4 holds.

Non-example: de Sitter space solves the Einstein equations with a positive cosmological constant $\Lambda$, which acts as $T_{ab} = -\frac{\Lambda}{8\pi} g_{ab}$. Then $\Omega^{-2} T_{ab} = -\frac{\Lambda}{8\pi} \Omega^{-4} \tilde g_{ab}$ blows up at the boundary, so clause 4 fails; indeed its conformal boundary is spacelike, not null.