What is the equation for gravitational time dilation? I have been studying relativity for a while now, but I am still unsure what the equation is to determine a gravitational time dilation. I am more than aware of the speed time dilation equation, but I would like the gravitational one.
 A: In general, the time dilation $\gamma$, is given by $\frac{d\tau}{dt}=\frac{1}{\gamma}$ where $dt=dx^0$ is the time coordinate and $d\tau^2=-\frac{1}{c^2} ds^2=-\frac{1}{c^2}g_{\mu \nu} dx^{\mu} dx^{\nu}$ is the proper time on the clock that you are calculating time dilation for. This expression is very general. It works for any clock motion (at rest, moving inertially, accelerating, etc) and any spacetime (special relativity flat spacetime, black holes, cosmology, etc). It only requires that you use a coordinate system with a time coordinate $dx^0=dt$.
Now, gravitational time dilation, is specifically for a clock at rest. So that means $dx^1=dx^2=dx^3=0$. Substituting that into the above expression we get $d\tau^2 = -\frac{1}{c^2} g_{\mu \nu} dx^{\mu} dx^{\nu} = -\frac{1}{c^2} g_{00} dt^2$. So then $$\gamma_{grav}=\sqrt{-\frac{c^2}{g_{00}}}$$
Let's try a couple of examples:
In flat Minkowski spacetime $ds^2=-c^2 dt^2 + dx^2 + dy^2 + dz^2$. So $g_{00}=-c^2$ and therefore $\gamma_{grav}=\sqrt{-\frac{c^2}{-c^2}}=1$ meaning that in flat spacetime there is no gravitational time dilation. This is what we expect.
In Schwarzschild coordinates: $$ds^2 = -\left(1-\frac{2GM}{c^2r} \right)c^2 dt^2 + \left(1-\frac{2GM}{c^2r} \right)^{-1} dr^2 + r^2 \left(d\theta^2 + \sin(\theta) d\phi^2 \right)$$ So $g_{00}=-\left(1-\frac{2GM}{c^2r} \right)c^2$ and therefore $$\gamma_{grav}=\left(1-\frac{2GM}{c^2r} \right)^{-1/2}$$ which you can confirm is the expected result for the Schwarzschild coordinates.
