Gravitational redshift between static Schwarzschild observers

For static observers at radii $r_A$ and $r_B$, equal Schwarzschild-coordinate time intervals correspond to proper times $d\tau =\sqrt{1-2M/r}dt$. Hence

This static-observer relation is a special case of gravitational redshift, not a definition of every gravitational frequency shift.

Two identical stationary devices counting operations at the same rate per proper time act as clocks. If the upper one sends start and finish light signals over a fixed path to the lower one in a stationary metric, both propagation delays are identical in coordinate time and cancel from the interval comparison. The lower elapsed proper time divided by the upper one is $\sqrt{g_{tt}(r_2)/g_{tt}(r_1)}$. In the weak field the relative lag is $g\Delta h/c^2$, giving the displayed operation deficit. A rate of $10^{12}$ operations per second, a week and a height difference of ten metres give a deficit of roughly $660$ operations. This observable tests the radial clock-rate factor; it does not independently determine the radial spatial metric coefficient because the common signal delay cancels.