# Relationship between gravitational time dilation and energy?

The rate that a stationary clock slows down near a massive object, relative to one far away, can be read off from the Schwartzschild metric: $$c^2d\tau^2=\left(1-\frac{r_s}{r}\right)c^2dt^2-\left(1-\frac{r_s}{r}\right)^{-1}dr^2-r^2\left(d\theta^2+\sin^2\theta d\phi^2\right)$$ by setting $dr=d\theta=d\phi=0$ to give: $$\frac{d\tau}{dt}=\left(1-\frac{r_s}{r}\right)^{1/2}$$ where the Schwartzschild radius $r_s=2GM/c^2$.

If the clock is running slowly compared to a distant clock is this equivalent to the clock having a lower energy compared to a distant clock?

If the clock was an atomic system then the frequency of its oscillation would be less near the massive object. As energy is proportional to frequency for atomic systems then I would have thought that this would imply that the energy of the atomic system would be less near the massive object than it was far away.

However in the weak field limit there is a sense in which time dilation can be linked to gravitational potential energy. If the difference in the Newtonian gravitational potential energy between two location is $\Delta\Phi$ then the relative time dilation is approximately given by:
$$\frac{\Delta t_1}{\Delta t_2} \approx \sqrt{1 - \frac{2\Delta\Phi}{c^2}}$$