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When Sun and Earth are moving, at each moment $t$ they are attracted not to the current position of each other, but to the position of each other at $t-\Delta t$, where $\Delta t$ is the time required for gravity to propagate the distance between each other. Laplace computed the effect of this "retardation" on their orbits and concluded, based on Newtonian mechanics, that unless the speed of gravity is $~10^9$ faster than the speed of light Earth would have fallen into the Sun already.

Now we have a more sophisticated theory of gravity by Einstein and others. This theory has two properties of interest for this question: it is well approximated by the Newtonian mechanics, at least on Earth-Sun scale of masses and distances, and it assumes that the speed of gravity is finite, equal to or at least on the order of magnitude of the speed of light.

The above raises the title question: why didn't Earth fell into the Sun, as Laplace has computed?

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Aah, I remember trying to answer this question to someone else, only to see it morph into a giant discussion. Anyways, this gives a very clear explanation: http://math.ucr.edu/home/baez/physics/Relativity/GR/grav_speed.html

Now, we know that the earth and the sun form a two-body system. We also know that GR tells us that gravitational force isn't directed towards the center of the field. Let's combine these two facts together. If the force on object A points towards the retarded position of B, on the other hand, the effect is to add a new component of force in the direction of A's motion, causing instability of the orbit. This means that the momentum of the entire system must now change. But the only way it can is by some spontaneous increase/decrease in the momentum of some individual object in the system, which isn't possible (in the example, A cannot just lose momentum). So it turns out, there must therefore be compensating terms that partially cancel the instability of the orbit caused by retardation.

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  • $\begingroup$ Very interesting read! $\endgroup$ – Prahar Oct 16 '13 at 4:00
  • $\begingroup$ Sam, would you mind including some of the material in the link in your answer as well? Because of the continuous change of internet websites, link-only answers are highly discouraged on this site. $\endgroup$ – Ali Oct 16 '13 at 4:09
  • $\begingroup$ @Ali, sure thing. The only reason I included the link instead of giving my own explanation is that I once did try to explain it to somebody and it didn't go well... $\endgroup$ – Sam29 Oct 16 '13 at 4:12
  • $\begingroup$ @Sam29: this answer boils down to something like "retardation causes orbit instability, but we know that the orbit is stable, therefore there must a compensating term." However, the point of the question is that I don't see where this compensating term could possible come from. Moreover, another possible explanation would be that there's no compensation term, but GR is fundamentally wrong. Could you point out how to derive the compensating terms? $\endgroup$ – Michael Oct 16 '13 at 22:32
  • $\begingroup$ @Michael: The reason I did not write everything in my answer is because I hoped you would read the accompanying link. It explains my point in a far better manner. Another thing: GR is not wrong. Numerous experiments since GR was put forward have verified this. Again, read through the link, and let us know if you still have some issues. $\endgroup$ – Sam29 Oct 16 '13 at 22:49

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