New systems of normal co-ordinates for relativistic optics

Abstract

The object of this paper is to introduce into the general theory of relativity some new systems of normal co-ordinates which are especially adapted to the discussion of problems of astronomical optics. In the classical system of normal co-ordinates introduced by Riemann the geodesics which start from a fixed base point in a metric manifold are represented by linear equations; in the new systems of normal co-ordinates null geodesics from points on a fixed base curve are represented by linear equations or form a system of parametric curves. If the system of null geodesics consists of the "forward" null geodesics starting from points on the fixed curve it represents the rays of light emitted by a source whose world line is the prescribed curve. If the system of null geodesics consists of the "backward" null geodesics ending at points on the fixed curve it represents the rays of light received by an observer whose world line is the given curve. It therefore seems appropriate to describe the new systems of co-ordinates as "optical co-ordinates". The use of optical co-ordinates considerably simplifies general optical theory in relativistic form. Fermat's Principle can be rigorously established and the treatment of the various astronomical determinations of distance can be put in a simple and concise form. Some previous discussions of these topics have been open to question in view of their unjustified use of Riemannian normal co-ordinates on the null cone of the base point. The general results obtained are applied to the case of greatest practical interest—the isotropic expanding universe.