Gravitational Redshift in Schwarzschild

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.

Figure 1. Gravitational redshift outside a Schwarzschild horizon. Method: closed-form Schwarzschild metric factor.

r_em / 2M2.00

speed2

WHAT TO TRY

Lower the emission radius toward the horizon at r = 2M: the photon climbs out of a deeper well, redshifts further, and at the horizon the shift diverges, light frozen at infinity.
Read z, the redshift: clocks deep in gravity run slow, so a signal sent up arrives stretched, the effect Pound and Rebka measured down a Harvard tower.
Move the emitter far out and z falls to zero: far from the mass spacetime is flat and there is no shift, the Newtonian limit.