The happiest thought of Einstein's life

Finally, let us have a closer look at the place of the equivalence principle in the logical scheme of Einstein's general relativity theory.

First, Einstein new well, from Minkowski's geometric formulation of his own special relativity, that accelerated motions should be represented as curved lines in a flat space-time.

Second, the Galileo principle asserts that all bodies are accelerated in the same way in a given gravitational field, and consequently their motions are represented in the flat space-time by curved lines, all exactly in the same way.

Third, since all lines representing free motions are curved exactly in the same way in the flat space-time, one can say that the lines remain straight (as far as possible) but the space-time itself becomes curved.

Fourth, and last, since acceleration is (locally) equivalent to a gravitational field (here we have the equivalence principle), one is entitled to assert that it is the gravitational field (and not acceleration) that is represented as the curvature of space-time.

This looks almost like an Aristotelian syllogism. However, to put all the pieces of evidence into the logical chain took Einstein a few years of hard thinking. The result has been incorporated into the field equations which quantitatively show how the curvature of space-time and gravity are linked together.