Speed of Light in Vacuum in the Case of Arbitrarily Non-uniform Motion of Reference Frames; Examples

Abstract

In this paper we supply examples that illustrate or verify the ideas developed in the paper by the author titled “Speed of Light in Vacuum in the Case of Arbitrarily Non-uniform Motion of Reference Frames” and also submitted to the 43rd PIERS 2021 in Hangzhou. The first example chosen illustrates the fact that the time coordinate in one of a pair of arbitrary reference frames has to be a continuous function of the time coordinate in the other frame even when the relative velocity of the motion of the frames is discontinuous in time. This characteristic is the main ingredient of the proof taken up in above mentioned paper, of a set of necessary and sufficient conditions for the speed of light to attain an infinite velocity. Two examples consider a particle undergoing a motion for which the velocity is discontinuous at t=t’=0, but is uniform and rectilinear in t, t’>0. It is found that the Schrödinger and Dirac equations for this particle are covariant under the corresponding space-time coordinate transformation, with the additional side-result that for the reference frames of the motion, speed of light is infinite. This infinite speed relegates the related coordinate transformation to a Galilean transformation. These two examples constitute a verification of the above mentioned set of necessary and sufficient conditions for attaining an infinite speed of light. As another example of the application of these necessary and sufficient conditions, a coordinate transformation is presented for which the speed of light remains finite. Still as two other applications of the fact that the time coordinates of reference frames have to be continuous functions of each other regardless of whether the relative motion is uniform or not, nonlinear coordinate transformations for accelerated motion of reference frames are considered.