The Geometry of Newton’s and Einstein’s Theories

Abstract

The aim of this paper is to present a simple, brief, mathematical discussion of the interplay between geometry and physics in the theories of Newton and Einstein. The reader will be assumed to have some familiarity with classical Newtonian theory, the ideas of special and general relativity theory (and differential geometry), and the axiomatic formulation of Euclidean geometry. An attempt will be made to describe the relationship between Galileo’s law of inertia (Newton’s first law) and Euclid’s geometry, which is based on the idea of Newtonian absolute time. Newton’s second law and classical gravitation theory will then be introduced through the elegant idea of Cartan and his space-time connection and space metric. This space metric will then be used to introduce Minkowski’s metric in special relativity and its subsequent generalization, by Einstein, to incorporate relativistic gravitational theory. The role of the principles of equivalence and covariance will also be discussed. Finally, a brief discussion of cosmology will be given. Stress will be laid on the (geometrical) concepts involved rather than the details of the mathematics, in so far as this is possible.