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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.

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You ask:

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?

but you have to very careful what you mean by energy in general relativity. As it stands your question too vague to be usefully answered.

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:

$$ \frac{\Delta t_1}{\Delta t_2} \approx \sqrt{1 - \frac{2\Delta\Phi}{c^2}} $$

Re your comments about atomic clocks: if you are sitting in a gravitational potential well and you observe the hyperfine emission from a caesium atom then you will observe it to have the usual frequency. This is because the time dilation affects both you and the caesium atom in the same way.

When the emitted light reaches an observer far from the gravity well it will have been red shifted and have a lower energy. because it has had to climb out of the potential well. However I don't think it is useful to conclude from this that the caesium atom in the well has a lower energy except in the limited sense that it's potential energy is more negative.

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