Concept of rest-frame in GR How is a rest frame defined in GR? And how can we determine whether two bodies at different points are at rest with respect to each other? I can not find a clear definition of this concept. For example, I want to compute proper time of a "resting" clock at $\boldsymbol{r}$ in a gravitational field: $cd\tau=\sqrt{g_{00}(\boldsymbol{r})}dt$. I am confused about the implication "clock is at rest" $\implies dx^i =0$. Is this actually the definition of "not mooving"?
 A: Unlike special relativity, in general relativity a rest frame doesn't have any special significance. In GR a frame is just a choice of coordinates, and we have complete freedom to choose any coordinate system we want.
The rest frame of an observer could be any system of coordinates in which the spatial components of the four-velocity are zero. For most purposes we'd choose the rest frame to be a frame with the observer at the origin that is locally Minkowski, but this doesn't have to be the case. For an accelerated observer we can choose a rest frame with the observer at the origin, and in this case the geometry is locally the Rindler metric.
As for when two objects are are rest with respect to each other, this is just as ambiguous. For example in the comoving coordinates used to describe the expanding objects all comoving observers are at rest with respect to each other in the sense that their spatial coordinates are time independent. However the proper distance between such observers increases with time. I would guess that most of us would consider objects to be realtively stationary if the proper distance between them is time independent.
However I suspect you are worrying unnecessarily given the context you provide. For example if we choose the Schwarzschild coordinates to describe the geometry around a spherically symmetric body then an object at rest has the simple interpretation that the spatial coordinates do not change with time. In that case $dr=d\theta=d\phi=0$ as you say, so the metric:
$$ d\tau^2 = g_{00}dt^2 - g_{rr}dr^2 - g_{\theta\theta}d\theta^2 - g_{\phi\phi}d\phi^2 $$
simplifies to:
$$ d\tau^2 = g_{00}dt^2 $$
Giving the equation you cite. This gives us the time dilation for a clock stationary in the frame of the Schwarzschild observer, which how most of us would instinctively interpret the word stationary.
