Schwarzschild radial fall from rest at a finite radius ($r=\frac{r_0}{2}(1+\cos \eta ),s=\sqrt{r_0^3/(8M)}(\eta +\sin \eta )$)

For radial timelike geodesic fall released from rest at $r_0>2M$, the Killing energy is $E=\sqrt{1-2M/r_0}$ and the proper time radial equation is $(dr/ds{)}^2=2M/r-2M/r_0$. The displayed parameterization integrates its inward branch, with $s=0$ at release. The Schwarzschild event horizon is reached at ${\eta }_H=\arccos (4M/r_0-1)$ in finite proper time, even though Schwarzschild time diverges.

// Target: general-relativity.bigb

A freely falling source emits optical pulses at proper times $\delta +nP$, for integers $n\ge 0$. If its horizon-crossing proper time is finite, only pulses beginning before $s_H$ can escape. Their number is the displayed ceiling, for $0\le \delta <P<s_H$. Infinite Schwarzschild time at crossing does not give infinitely many emitted pulses. The pulse phase $\delta$ fixes both the count and the redshift of the last pulse. If a finite pulse straddles the horizon, its exterior portion has an arbitrarily late, increasingly redshifted received tail.