General Relativity as a Geometric Theory of Gravity

Abstract

Abstract The mathematics of a curved manifold is Riemannian geometry. This chapter first presents some basic elements of Gaussian coordinates and the metric tensor. From the requirement that a curve have the shortest length, the calculus of variations is used to deduce the geodesic equation. A geometric description of the equivalence-principle-based physics of gravitational time dilation is presented. Einstein’s motivation for considering curved spacetime as a gravitational field is then discussed. In this geometric theory, the metric plays the role of the relativistic gravitational potential. The interpretation of curved spacetime as a gravitational field naturally suggests that the geodesic equation in spacetime is the general relativity equation of motion, which reduces to the Newtonian equation of motion in the limit of a nonrelativistic moving particle in a static and weak gravitational field. The gravitational redshift effect is shown to follow directly from a curved spacetime (with curvature in the time direction).