This is more like a conceptual question. Let's say we have two point masses $m_1$ and $m_2$ separated by the distance $r$. According to the Newton's law of universal gravitation, the objects will be accelerated with respect to each other by a force: $$\vec{F}=-G\tfrac{m_1m_2}{r^2}\hat{r}.$$

I was trying to visualize this problem from the General Relativity point of view - in terms of spacetime curvature and the objects following the geodesics.

So my question is: How would General Relativity explain this attraction between the two objects? I know that their mass will warpe the spacetime around them, but how does this curvature of spacetime explain the fact that they are getting closer together - esspecialy when $r$ is very large? Assuming that the object are perfect spheres and the mass distribution is uniform, isn't the spacetime curvature created by each object going to be isotropic?

I started by imagining that the objects will advance through time at 1 sec/sec in their proper frame, but because the spacetime is warped around them - the motion through time implies motion through space. How far off am I?

To put it more technically - How does the spacetime curvature in this particular case lead to geodesics that will eventually make the objects approach each other?