Time slows down at lower gravitational potential

[guv]

Homework Statement

According to Schwarzschild solution (see the formula) the time interval is less than the time interval at a greater distance (higher potential).

Relevant Equations

$d\tau^2 = (1 - \frac{R_S}{R}) d t^2$

Common interpretation is that time slows down at lower potential. I wonder if people are simply saying for the time interval between two events at lower potential, it's smaller than what would be measured at greater potential $d \tau < d t$. i.e. Clock at lower potential shows a time interval 10 seconds, while clock at greater potential shows a time interval 20 seconds. This is similar to the special relativistic effect 'moving clock runs slower' where the clock that is moving measures 10 seconds for the interval but the stationary clock measures 20 seconds for the same interval. Is the above interpretation correct?

 

[anuttarasammyak]

Yes. But SR says relativity, i.e. A’s clock is slow for B, B’s clock is slow for A. GR says A’s clock is slower than B’s for the both A and B.

 

[Ibix]

Is the above interpretation correct?

Sort of. In curved spacetime it is very common that a change of time coordinate from $t$ to $t+dt$ does not correspond to an elapsed time $dt$ on a local clock. The formula you quote tells you how to translate a lapse of coordinate time $dt$ into the time $d\tau$ measured by a clock at rest in those coordinates.

Schwarzschild spacetime is a static spacetime, meaning that it is possible to find a definition of "space" that doesn't change with the corresponding notion of "time". Schwarzschild coordinates use this notion, so they provide a meaningful way to compare the rates of separated clocks: calculate $d\tau|_{r=r_1}$ and $d\tau|_{r=r_2}$ and take the ratio. And yes, you'll find that in the time it takes a clock to tick once, a higher altitude clock will tick once and a bit.

If you see someone using that formula directly to calculate time dilation they are abusing it slightly. They're actually comparing $d\tau|_{r=r_1}$ to $d\tau|_{r=\infty}$ and skipping a couple of steps since the latter is equal to $dt$.