Having formulated his Special theory, Einstein wanted to generalize it to incorporate the gravitational interaction. It took him ten years to complete this task. The final version of the theory was published in 1916. It is a relativistic theory of gravitation (i.e. one consistent with Special Relativity), known as General Relativity .
The key principle on which General Relativity is built derives from Galileo's experiments in which he dropped bodies of different composition from the leaning tower of Pisa. These experiments showed that all bodies fall with the same acceleration irrespective of their mass and composition. This is known as the principle of equivalence .
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| This equivalence principle is best understood
in the context of Einstein's lift thought experiments where, neglecting
non-local effects, a body in a linearly accelerated rocket ship behaves
the same as one on the earth (experiencing the pull of gravitation).
On the other hand, a body in an unaccelerated rocket ship behaves the
same as one in free fall. |
In the absence of gravitation we get back to Special Relativity and a flat metric and so, in order to incorporate gravitation into the theory, Einstein proposed that the metric should become curved. This means that the geodesics become curved as well, which results in free bodies no longer moving in straight lines when affected by gravity. The reason that a satellite (like the Earth) orbits a central body (like the Sun) in Newtonian theory is a combination of two effects: uniform motion in a straight line (Newton's first law) and gravitational attraction between the two bodies (i.e. the satellite "falls" under the attraction of the central body). In General Relativity, the reason that a satellite orbits a central body is that the central body "curves up" space (and, in fact, time as well) in its vicinity, and the satellite travels on the "straightest path" which is available to it, namely on a curved geodesic. One major difference between the two theories is that whereas Newtonian theory describes how things move, and it does so remarkably accurately for ordinary bodies, it does not really explain what is the cause. Einstein's theory neatly provides answers for both these questions.
The General theory of Relativity can be stated mathematically as
These are the so-called Einstein field equations. They correspond to 10 coupled highly nonlinear partial differential equations. Their solution gives rise to a curved spacetime metric from which one can obtain the geodesics and hence investigate such things as the motion of free particles and light rays.
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The meaning of the Einstein equations can be summed up in the famous words of John Archibald Wheeler: "space tells bodies how to move and bodies tell space how to curve" |
General Relativity is concerned with studying the nature of these equations and their solutions. Since few of the known exact solutions to Einstein's equations describe physically relevant situations these studies are based on approximations, such as post-Newtonian expansions or perturbation techniques, or numerical simulations.
| One way to think of General Relativity is to use the idea of a "rubber sheet geometry" where in the absence of gravitation the sheet is flat, but a central massive body curves up the sheet in its vicinity so that a free body (which would otherwise have moved in a straight line) is forced to orbit the central body |  |
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