Gravitational redshift in Schwarzschild spacetime

What you are seeing: a photon emitted at radius $r_{em}$ from a Schwarzschild black hole of mass $M$ (geometric units $G = c = 1$) arrives at an observer at infinity with frequency $f_\text{obs} = f_\text{em} \sqrt{1 - 2M / r_\text{em}}$. The photon loses energy fighting against the gravitational potential. The redshift factor is exactly $\sqrt{1 - 2M / r_\text{em}}$; at the horizon $r = 2M$ it equals zero, corresponding to infinite redshift.

The top panel plots redshift factor vs $r_\text{em} / (2M)$. The bottom panel renders a green source line ($\lambda_\text{em} = 530$ nm) redshifted to the observed wavelength $\lambda_\text{obs} = \lambda_\text{em} / f$. Just outside the horizon the wavelength stretches to infinity (the source appears infinitely red); at large $r_\text{em}$ the shift is tiny. The Pound-Rebka experiment (1959) measured this at $r_\text{em} / r_\text{obs} - 1 \sim 10^{-15}$ in Earth gravity using gamma rays.