Riemann Spaces and General Relativity

Abstract

In this chapter we shall introduce the mathematical tools that are needed for the development of the theory of general relativity. They generalize to non-Euclidean spaces the tools that we developed in the last chapter for the tensor calculus in general coordinates. We begin with the characterization of a Riemannian space and the introduction of the Riemann-Christoffel curvature tensor, the Ricci tensor, the curvature scalar and the Bianchi identities. After stating the physical principles of the general relativity we analyze mechanics, electrodynamics and fluid dynamics in the presence of gravitational fields. We derive Einstein’s field equations and find the solution corresponding to a weak gravitational field and the Schwarzschild solution. We present also the solutions of Einstein’s field equations that correspond to the evolution of the cosmic scale factor in a universe described by the Robertson-Walker metric in the radiation and in the matter dominated periods.