Some Functional Equations in the Theory of Relativity. by Professor

IN transcribing the process of light-signalling which leads to his kinematic equations of transformation, Einstein* obtains relations, connecting the space-time coordinates in two systems of reference, in the form of functional equations. He solves these by passing to partial differential equations, and restricts the solutions to linearity by a rather obscure appeal to the homogeneity of time and space. In view of the fundamental character that his theory has proved to possess it is natural to ask more closely what is implied in the term homogeneity, which occurs in various senses in physical science, and whether it is necessary to assume differentiability. These and some related questions bearing on the fundamentals of that theory suggested the following analysis, which aims at a more complete transcription by means of functional equations and their solution without appeal to differentiation. The general setting of the problem is found in the supposition that for the set of space-time points there are various systems of reference, corresponding to observers and carriers of coordinate frames in uniform translatory motion with respect to each other; these systems being, under conditions to be described, all on a parity with each other in the sense to be defined as the meaning of relativity. It is understood that every space-time point is identifiable by each system in terms of a unique quadruple of coordinates, and conversely that each such quadruple that occurs at all identifies a unique space-time point. For a physical theory some limitation on the range of the coordinates might perhaps seem appropriate, and for the most part even if such a limitation of any natural kind were imposed the results would probably be unchanged if interpreted for corresponding regions. But in order to avoid the complication of boundary conditions that would accompany such a limitation it will be supposed that every