A class of transformations in special relativity

Abstract

Born & Biem have formulated a transformation from the co-ordinates used by an observer A at rest in an inertial frame, to the co-ordinates used by an observer B who has any given rectilinear motion relative to A , and who assigns co-ordinates as nearly as he can in the same way as A . The existence and properties of this transformation are discussed. Particular attention is devoted to an example proposed by Born & Biem in which B 's world-line in A 's co-ordinate system is part of a hyperbola meeting A ’s world-line in two events. The trans­formation in the example is here expressed in terms of elliptic integrals. Curves are drawn for a numerical case to show how A and B each plots the motion of the other and to show the Doppler effect observed by each when the other transmits radiation of given frequency. The example usually quoted in connexion with the clock paradox is discussed as a limiting case of the preceding one and it is shown that certain features are thereby clarified. The well-known case of uniformly accelerated motion in special relativity is also a particular case of Born & Biem’s example; its properties are discussed and illustrated by diagrams. Finally, a short discussion of the clock paradox, arising out of Born & Biem's treatment, is given.