Relativistic transformations at variable velocities

Relativistic transformations at constant velocities V and v considered in [A. Einstein, Zur Elektrodynamik der bewegter Körper, Ann. Phys., 17 (1905) 891 921] are modified for the general case of variable speed V.:/ of synchronization signal and variable relative velocity v.t/ between two frames .K/ and .k/ linked through that signal. A differential approach to time synchronization is considered which makes clear that with variable velocities the observed time in the moving frame is represented by a nonlinear functional of those velocities V.:/ and v.t/ over the segments of trajectory covered by the synchronization (observation) signal, and the difference with respect to Einsteinʹs PDE at constant velocities is in the range of several decimal orders. Moreover, the application of linear transformations at constant velocities to the case of variable velocities may produce illusory effects of a relativistic mirage of expansion. For practical applications, and to preserve the simplicity of relativistic transformations, the averaging approach is developed, and a new form of relativistic transformations is obtained for discretized trajectories with on-line identification of average values for variable relative velocity v.t/ and in consideration of a nominal velocity V with possible variations V.:/: An estimation of errors is made through the -representation based on direct distance measurements. Natural time delays due to the finite speed of information transmittal are included into relativistic transformations at variable velocities, and equations for software development are presented to support computation of real time trajectories and error estimation. The results open new avenues for theoretical and experimental studies of special relativity in media moving with variable speed and for control of processes by means of actually available signals.