Differences in Newtonian and Relativistic predictions of gravity

[espen180]

In what situations do Newton's law of universal gravity fail to predict correctly how matter moves? The examples I know of are frame dragging (around rotating BH), shifting of light paths, shifting orbits of planets.

But does Newton's law really predict the magnitude of gravity accuratly? How does the predicted Newtonian value compare to the predicted relativistic value in different situations?

Also, given the symmetry between the electromagnetic and gravitomagnetic forces, how come there isn't a spacetime-equivalent theory for electromagnetism? Am I wrong in my assuption of said symmetry?

I hope to get some answeres on my questions. Any help/feedback is appreciated.

 

[Matterwave]

Anywhere the gravitational fields are very very strong, GR will give different results than Newtonian gravity. Newtonian gravity is retrieved in GR in the weak field limit.

Since GR no longer has a notion of "force" of gravity, and instead replaces it with a curvature of spacetime, there is no way to say the "magnitude" of gravity is different between the two or not, only that they give different orbits or results.