Static-surface Schwarzschild redshift ($1+z=(1-2GM/(R{c}^2){)}^{-1/2}$)

A static emitter at radius $R>2GM/c^2$ and a static receiver at infinity compare photon wavelengths with $1+z=(1-2GM/(R{c}^2){)}^{-1/2}$. Conservation of the time-translation Killing energy and the static clock rate $d\tau =\sqrt{1-2GM/(r{c}^2)}dt$ prove the relation. Motion of the emitter adds a Doppler factor; a rotating stellar exterior may require a different metric.