Time Dilation equations for Gravity (from General Relativity)

Thanks to Dan M and Colby for finding the Wikipedia link for gravitational time dilation.

We have mentioned in class that Einstein, from his general theory of relativity (which is our modern theory of gravity), figured out that gravity should affect time. Two clocks that start off synchronized should 'run' at different rates in different gravitational fields. This can be deduced from the principle of equivalence, which states that the effects of acceleration are indistinguishable to the effects of gravity. Acceleration is a change in motion, and Einstein showed how motion (i.e. speed) slows time in his special theory of relativity, so accelerating motions should also slow time. If acceleration affects time, then gravity must, too! This is all verified through experiments using atomic clocks.

We can get some sense of how big (or actually, how tiny!) these gravitational effects are for a few different situations.
- As a function of height above the earth, we can use an approximation:
          T = 1 + gh/c^2     This tells us how much time dilation there would be if one clock was on the ground and one was at a height h. This assumes h is very small compared to the size of the earth (or whatever object we are using for g). Because c^2 is very large, 9 x 10^16, you can see that this is a tiny effect for the earth! So the difference between the times being kept by the two clocks is in that term gh/c^2.

- Above a spherical, non-rotating object
       t_surface = (t_far)*sqrt [1 - 2GM/(rc^2)]
This tells us the difference in two clocks, one on the surface of a spherical object and one far away from the object. M is the mass of the object and r is the distance from the center of the object, which is not rotating. Note that the term 2GM/c^2 is the Schwartzshild radius, which is the radius one would need to crush the object to make it a black hole (escape velocity > c).

- A special case of being above a spherical object is if you are in circular orbit around that object. The above expression becomes
       t_surface = (t_satellite)*sqrt[1 - 3GM/(rc^2)]


In any case, this is very cool stuff! Have fun just by playing around with these expressions and discovering how large or small the effect would be for any scenario/object you wish!
Note I have videos on equivalence principle, warped space-time (from Elegant Universe), basics of general relativity, Gauss's law for 1/r^2, Gauss's law inside Earth, gravitational potential energy, and integration to find g-fields for rings and sticks. There is also videos for E = mc^2, and Einstein's energy equation and consequences.