General Relativity

Einstein was not content with the special theory of relativity, because it was in contradiction with Newton's theory of gravitation. Newtonian gravitation was a very successful theory. In particular, it accounted for the motion of the planets to high precision. However, it has a peculiar property that had even bothered Newton. It implies instantaneous transmission of the gravitation force between two objects across great distances.
Einstein knew this had to be wrong, because special relativity implies that no signal can be transmitted faster than the speed of light. Einstein provided the resolution around 1915 with a new theory of gravitation, which he called the general theory of relativity. It agrees with the Newtonian theory for low speeds and weak gravitational fields, but differs from it at high speeds and strong fields.
The theory had several immediate observational successes. First it implied a small correction to the orbit of the planet Mercury that accounted for a small discrepancy between the orbit implied by the Newtonian theory and the observed orbit. (The effect is too small to be observed for the other planets.) Second, it predicted that light from a distant star passing near to the limb of sun would be bent by a small but measurable angle. The measurement was made by Eddington during a solar eclipse.
The observations confirmed the theoretical prediction, and Einstein became an international celebrity. Other predictions of the theory -- such as the existence of black holes and of gravitational radiation -- have been confimed much more recently.
The thing that impressed other physicists most about the general theory of relativity is that it is based on very general physical principles -- the equivalence principle and general coordinate invariance -- and very beautiful mathematical concepts. The relevant mathematics is called differential geometry (specifically, Riemannian geometry). The idea is that gravity is a manifestation of the curvature of space-time. Also, the geometry of space-time is determined by the distribution of energy and momentum. The basic equation of motion is

In this equation Gmn describes the space-time geometry, G is Newton's constant characterizing the strength of gravitation, and Tmn describes the distribution of energy and momentum.

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