# Gravitational time dilation

1. Jan 6, 2007

### actionintegral

Hello,

I am trying to learn about Gravitational Time Dilation. I came across the following formula for time dilation in a uniform gravitational field:

$$T_d = {1-} \frac{gh}{c^{2}}}$$

but I cannot find any derivation for this. Can someone point me in the right direction?

Thanks

Last edited: Jan 6, 2007
2. Jan 6, 2007

### cristo

Staff Emeritus
Take a look at http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/gratim.html

Now, the time dilation factor you state above is $$\sqrt{1-\frac{2gh}{c^2}$$ (It's equal to T0/T, in the above link).

Using the binomial theorem $$\sqrt{1+x}=1+\frac{1}{2x}+\cdots$$

And hence, we obtain the result above $$T_d=1-\frac{gh}{c^2}$$

3. Jan 6, 2007

### Chris Hillman

cristo should have explained why we are supposed to take $2m/r = 2gh$ (in relativistic units in which G=c=1) in the Schwarzschild vacuum.

This takes some explaining! The idea is obvious enough: in a small box near the surface [itex]r=r_0[/tex] of a spherical massive isolated object, with the exterior gravitational field modeled by the Schwarzschild vacuum, right down to the surface, the "gravitational acceleration" is the acceleration of static observers sitting on the surface, which is radially outward with magnitude [itex]m/r^2[/tex], the same expression as in Newtonian gravitation, where needless to say I am using the exterior Schwarzschild coordinate chart. So the needed explanation concerns the "small box". I'll let readers think about that...

The notion of a "uniform gravitational field" in gtr is actually rather tricky. Since actionintegral didn't cite his source, I don't know if he misread something, or if the author was too careless to mention caveats (or wanted to avoid confusing his students), or if the source is a bad one (very possible for websites by someone who is not an acknowledged expert in gtr).