Relativity

Abstract

This chapter studies one of the most important applications of Riemannian geometry: the theory of general relativity. This theory, which ultimately superseded the classical mechanics of Galileo and Newton, arose from the seemingly paradoxical experimental fact that the speed of light is the same for every observer, independently of their state of motion. In 1905, after a period of great confusion, Einstein came up with an explanation that was as simple as it was radical: time intervals and length measurements are not the same for all observers, but instead depend on their state of motion. In 1908, Minkowski gave a geometric formulation of Einstein’s theory by introducing a pseudo-inner product in the four-dimensional spacetime $\mathbb {R}^4$. While initially resisting this “excessive mathematization” of his theory, Einstein soon realized that curving spacetime was actually the key to understanding gravity. In 1915, after a long struggle with the mathematics of Riemannian geometry, he was able to arrive at a complete formulation of the general theory of relativity. The predictions of his theory were first confirmed in 1919 by a British solar eclipse expedition, led by Eddington, and have since been verified in every experimental test ever attempted.