= Outgoing pulse redshift from a radial Schwarzschild infaller
{title2=$\nu_\infty/\nu_{\mathrm{em}}=f/(E+\sqrt{E^2-f})$}
A radial infalling <geodesic> in <Schwarzschild spacetime> has $dt/ds=E/f$ and $dr/ds=-\sqrt{E^2-f}$, where $f=1-2M/r$. An outward <radial null geodesic> has fixed retarded time $u=t-r_*$ with $dr_*/dr=1/f$. Consequently $du/ds=(E+\sqrt{E^2-f})/f$. This factor stretches a short emitted pulse's duration, and its reciprocal multiplies the received frequency at infinity. Near horizon crossing $s_H$, it becomes $du/ds\sim4M/(s_H-s)$, including both <gravitational redshift> and the emitter's <Doppler shift>. The local short-pulse approximation requires its emitted duration to be much smaller than $s_H-s$.
// Target: general-relativity.bigb
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