 Horizon redshift of a Schwarzschild infaller

For smooth radial infall through a future Schwarzschild event horizon, outgoing retarded time obeys $u\sim -4M\log ({\tau }_H-\tau )$. A distant observer sees the redshift grow as $e^{t_{\mathrm{rec}}/(4M)}$. Its exponent in emission Schwarzschild time is instead $1/(2M)$, because $t_{\mathrm{rec}}\sim 2t_{\mathrm{em}}$. At finite fixed observer radius the wavelength ratio includes $\sqrt{1-2M/r_B}$.