Solution (source code)

= Solution

An <asymptotically flat spacetime> at <future null infinity> admits a smooth <conformal completion> $(\overline{\mathcal M},\bar g)$ with the following properties. The physical spacetime $\mathcal M$ is the interior of $\overline{\mathcal M}$, its metric obeys
$$
\bar g_{ab}=\Omega^2g_{ab},
$$
and the boundary component $\mathcal I^+$ satisfies $\Omega=0$ and $d\Omega\ne0$. Every future-directed outgoing null geodesic has an endpoint there, the generators of $\mathcal I^+$ are complete, and in four spacetime dimensions $\mathcal I^+\simeq\mathbb R\times S^2$. The physical <Einstein field equations> are vacuum in a neighborhood of this boundary.

Let $n_a=\bar\nabla_a\Omega$. Multiplying the supplied conformal Ricci relation by $\Omega^2$, using $R_{ab}=0$, and taking the limit to $\mathcal I^+$ gives
$$
0=-3\bar g_{ab}n^cn_c,
\qquad
\boxed{n^an_a=0\quad\hbox{on }\mathcal I^+}.
$$
The boundary normal is therefore null. Since a null normal is also tangent to its hypersurface, $n^a$ generates $\mathcal I^+$.

Smoothness makes $s=\Omega^{-1}n^2$ finite at the boundary. Multiplication of the same vacuum equation by $\Omega$ gives there
$$
2\bar\nabla_an_b+\bar g_{ab}(\bar\Box\Omega-3s)=0.
$$
Taking the trace yields $s=\tfrac12\bar\Box\Omega$, and substitution gives
$$
\bar\nabla_an_b=\frac14\bar g_{ab}\bar\Box\Omega.
$$
The remaining freedom $\Omega\mapsto\omega\Omega$ can be used to impose $\bar\Box\Omega=0$ on $\mathcal I^+$. In this conformal gauge, $\bar\nabla_an_b=0$ there, so the generators are affinely parametrized, expansion-free null geodesics of the unphysical metric.

Choose a generator coordinate $u$, the defining function $\Omega$, and angular coordinates $x^A$ whose leading metric $q_{AB}$ is the round metric on the <unit sphere>. After the conformal and coordinate choices above, the leading unphysical metric is
$$
\bar g=2\,du\,d\Omega+q_{AB}dx^Adx^B+O(\Omega),
$$
with the Minkowski term $-\Omega^2du^2$ entering at the next relevant order. Setting $r=\Omega^{-1}$ recovers the physical asymptotic form
$$
g=-du^2-2\,du\,dr+r^2q_{AB}dx^Adx^B
+\text{terms lower by powers of }r.
$$
The displayed leading metric is <Minkowski spacetime> in outgoing null coordinates. Smooth conformal extendibility controls the lower-order corrections, while the vacuum equations constrain them to the radiative Bondi--Sachs expansion. This is the precise sense in which the permitted spacetimes approach Minkowski spacetime near $\mathcal I^+$ while still allowing outgoing gravitational radiation.