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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?

Qmechanic
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Nemo
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  • Possible duplicates: https://physics.stackexchange.com/q/211930/2451 , https://physics.stackexchange.com/q/68067/2451 and links therein. – Qmechanic Jul 03 '18 at 18:31
  • "because the spacetime is warped around them - the motion through time implies motion through space" [with acceleration] - Yes, this is correct. Time moves slower near heavy bodies. Objects around moving in time tend to also move in space toward the areas of slower time, because timelike geodesics bend towards the areas of slower time: https://en.wikipedia.org/wiki/Schwarzschild_geodesics – safesphere Jul 03 '18 at 21:22
  • Although temporal curvature may be inferred from recurrent phenomena, Einstein did not favor the pop-sci notion of spatial curvature, which, if it would be measurable, would imply a finite universe: Part of this disfavor may've resulted from his having a schizophrenic son, as "ideas of reference" (such as the notion that we are privileged enough to live soon enough after a universal creation event to have been able to find evidence for it) are characteristic of that psychiatric condition. That evidence has been disputed in the formulation of several cosmological models, notably Penrose's. – Edouard Oct 11 '21 at 21:01

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