The reason Newtonian gravity is not compatible with relativity is that its influence is instantaneous. If I move a mass over here, the gravitational effect will be felt light years away immediately. But one of the key facts about special relativity is that "instantaneous" is an incomplete sentence. You need to specify instantaneous from whose perspective. And in general there is no answer to whose notion of instantaneous should be used - so there's a problem.

I believe there were some attempts to reconcile Newton and special relativity by adding a propagation speed for gravity (thus removing the "instantaneous" problem), but they were not entirely successful. Ultimately, Einstein settled the question by developing general relaitivity.

SR showed that no signal can travel faster than light. But, Newton's gravity has instantaneous "action at a distance". As I see @Ibix has just posted.

Also, the basic formula for the gravitational force is ##F = \frac{GMm}{r^2}##, where ##r## is the distance between masses ##M## and ##m##. But, SR had shown that distance was relative and the distance measured in one reference frame would not equal the distance measured in another. That was another fundamental problem.

You can make the Poisson eqn. special relativistic; it becomes the massless Klein Gordon eqn. This theory is consistent but e.g. couples to the trace of the energy momentum tensor and as such cannot describe the deflection of light.

Yes, Einstein tried this, since it was the obvious first thing to try, by analogy with electromagnetism. "Not entirely successful" is being charitable. This theory made predictions grossly contradictory to observation, for example that the orbits of the planets around the Sun should be unstable on fairly short time scales.

Thanks to Peter Donis, houshoffer, PeroK, and Ibix for their quick and clear responses. I should have thought a little deeper before posting the question. Many thanks!

Thanks for the reply. But how do we know that "gravity is instantaneous"? Wikipedia says that gravitational waves travel at lightspeed. If the Sun disappeared Earth would stay in orbit for eight minutes (I'm told). What experiment shows that gravitational effect is instantaneous?

Thanks for your response. Wikipedia says that gravitational waves travel at lightspeed. So if the Sun were to disappear, Earth would remain in orbit for eight minutes (I've always been told). So what experiment could show that "gravity is instantaneous"?

Newtonian gravity is instant action at a distance by mathematical construction. Newton didn't like this, but it is what fit observation. It became well known way before special relativity that even very small delays in the effect of gravity in a Newtonian framework would lead to wildly unstable orbits, inconsistent with observation.

GR is what solved this problem, providing a framework for finite propagation speed while reproducing orbital stability over required time scales. (In principle, no orbits in GR are stable due to gravitational radiation causing orbital decay; but the time scale for this in the solar system is many, many orders of magnitude greater than the age of the universe)

That's a prediction of general relativity, using the equations of general relativity. If you take Newton's equations as your starting point, you won't even have any gravitational waves; Newton's equations predict a different behavior than waves travelling outwards at any speed when you change the mass distribution, and predicts that earth would immediately fall out of orbit if the sun were to disappear.

If Newton's theory is correct but "gravity is instantaneous" is not, then planetary orbits would be unstable; thus our centuries of observation of stable planetary orbits is the experiment that shows that Newtonian gravity must be instantaneous. This is the basic problem that puts Newtonian gravity in conflict with SR: You can't have Newtonian gravity without instantaneous propagation, and you can't have instantaneous propagation and SR, so you can't have Newtonian gravity and SR.

Gravitational waves don't exist in Newtonian gravity. They're a solution of Einstein's Field Equations, and they do indeed travel at the speed of light.

That's a complicated question. GR won't let you just make the Sun disappear, so asking what would happen if it did isn't likely to give you a sensible answer.

We don't believe it is. Newton's theory of gravity predicts that it is, but even Newton himself wasn't comfortable with it. The problem is that relativity is completely incompatible with instantaneous action at a distance because both "instantaneous" and "distance" are not globally defined quantities. And efforts to patch Newtonian gravity to fit relativity failed in a "if the world worked this way the solar system wouldn't be here" kind of way.

The ultimate solution was a ground-up re-think of how gravity works, treating it as spacetime geometry instead of a force. Incidentally, this explained a fact that astronomers had been puzzling over for some time, that Mercury was very slightly in the wrong place compared to our Newtonian prediction.