Geometry and Physics

This talk concerns the development and interrelations between geometry and physics since the development of Euclidean geometry. It briefly describes the ideas behind Euclid's work and moves quickly through to the groundbreaking work of Copernicus, Kepler, Galileo and Newton during which the foundations of classical (Newtonian) physics were laid down. These advances used the geometry of Euclid, as applied to 3-dimensional space, to act as the arena where the activities of physics took place and to measure and describe, for example, the motion of particles. In such a philosophy, the geometry was absolute and fixed (and Euclidean) and uninfluenced by the physics. Such ideas were developed in a different way by Lagrange, Hamilton, Routh and others who employed the use of "generalised" coordinates to shake off the idea of preferred inertial observers. However, Euclidean geometry still played the role of absolute background geometry. With the advent of Einstein's general relativity theory in the 20th century the sources of the physics (the matter and fields, etc) were now allowed to influence the geometry and, conveniently, the resulting (differential) geometry needed was then being developed by the Italian school. For Einstein the geometry was no longer background and Euclidean but rather a Riemannian-type geometry determined by the physics and, in this sense, dynamical and so there arose reciprocal actions of the geometry and the physics upon each other. This was the approach of Einstein and it led to his field equations which, given the physical situation, gave (somewhat complicated) field equations for determining the geometry and, from this, the physics. This is now the best available theory of gravitation and, when applied on a global scale, led naturally to the study of cosmology.