Gravity's Effect on Time and the Gravitational Red Shift

We may consider how the proper time for a particle $\tau$ relates to the coordinate time $t$ as a function of distance from the mass at the origin. In the rest frame of the particle,

$$-c^2d\tau^2=-\left(1 - \frac{r_s}{r} \right) c^2 dt^2$$

so the proper time interval at a radius $r$ is.

$$d\tau=\sqrt{1-{r_s\over r}}dt$$

Applying this for $r\rightarrow\infty$ we have.

$$d\tau_\infty=dt$$

So for light that escapes to infinity, the time to emit the light in the rest frame of the atom at radius $r$ is shorter than the time seen at infinity.

$${d\tau\over d\tau_\infty}=\sqrt{1-{r_s\over r}}$$

This causes a red shift in the observed frequency.

$${\nu_{obs}\over\nu_{emit}}=\sqrt{1-{r_s\over r}}$$

Since the energy of a photon is $E=h\nu$, the photon loses energy as it moves outward in the gravitational field.