= Solution
<Birkhoff theorem> states that every spherically symmetric vacuum solution of Einstein's equations with zero cosmological constant is locally Schwarzschild, with a constant mass parameter. In particular its exterior vacuum geometry is locally static even if the spherical material source moves. Assuming no exterior matter or outgoing matter radiation, the region outside the collapsing star is vacuum with its conserved total mass $M$, so the given <Schwarzschild metric> is appropriate there. Spherical gravitational radiation does not provide an extra time-dependent vacuum degree of freedom. Matter emission would invalidate the vacuum premise.
Use units $G=c=1$ and signature $(-,+,+,+)$ as in the displayed metric. For a radial <timelike geodesic>, stationarity gives conserved <energy> per unit mass $\mathcal E=F\dot t$. Proper-time normalization gives
$$
-1=-F\dot t^2+\frac{\dot r^2}{F},\qquad
\dot r=-\sqrt{\mathcal E^2-F}
$$
on the ingoing branch. At the horizon $F\to0$, so $\dot r\to-\mathcal E$, with $\mathcal E>0$ for an infaller from the exterior. Hence
$$
\Delta\tau=\int_{2M}^{r_0}\frac{dr}{\sqrt{\mathcal E^2-F(r)}}
$$
is finite near the horizon. If release is from rest at a finite radius, the integrable square-root behavior at the release point also gives finite elapsed <proper time>. Thus \b[the observer crosses the horizon in finite <proper time>]; the Schwarzschild coordinate singularity is not a physical barrier.
By contrast
$$
\frac{dt}{dr}=-\frac{\mathcal E}{F\sqrt{\mathcal E^2-F}}
\sim-\frac1F\sim-\frac{2M}{r-2M},
$$
so $t\sim-2M\log(r-2M)+\text{constant}$ diverges. Schwarzschild time is the limiting <proper time> of stationary observers at infinity. For actual signals, an outward radial null ray adds a propagation delay involving $r_*=r+2M\log(r/(2M)-1)$; signals emitted ever closer to the horizon reach a distant observer ever later and increasingly redshifted. No signal emitted at or inside the <event horizon> reaches that observer. Thus “seeing crossing only after infinite time” means an asymptotic observational limit, not an eventual finite-time sighting.
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