Conformal completion

A conformal completion embeds a physical spacetime $(\mathcal{M},g)$ into a manifold with boundary carrying a regular metric $\bar{g}={\Omega }^{2}g$. Infinity is represented by $\Omega =0$ with $d\Omega \ne 0$; the causal character of each boundary component is determined by ${\bar{g}}^{ab}{\bar{\nabla }}_a\Omega {\bar{\nabla }}_b\Omega$.

Future null infinity is the boundary reached by future-directed outgoing null geodesics in an asymptotically flat spacetime. In four dimensions it has topology $\mathbb{R}\times S^2$ under the standard asymptotic-flatness assumptions.