Special relativity for motions with variable speed

Relativistic transformations considered in [Albert Einstein, Zur Elektrodynamik der bewegter Körper. Ann. Phys. 17 (1905) 891–921] are extended for motions with variable speed through the introduction of the γγ-representation and continuous piecewise linear approximations which are used for velocity identification on the basis of discrete measurements of the distance function. This demonstrates that Einstein’s relativistic transformations originally established to explain some physical phenomena experimentally observed in systems one of which is moving with a constant relative velocity with respect to the other are, in fact, applicable (in some modified form) to systems with variable relative velocity. The method is illustrated in application to the control of a flight to Mars in a spacecraft with comfortable weight conditions on board, same as on Earth, moving with acceleration g=9.8 m/s2 at all times except for the lift-off from Earth and soft touchdown on Mars after 42 h flight. Natural time delays due to finite speed of information transmittal are included into relativistic transformations, leading to a 4D space–time frame in the form of a variable affinely connected time–space structure whose dynamics is defined by the relative velocity v(t)≠constv(t)≠const between the two systems. The results open new avenues in the theoretical and experimental studies of relativistic effects in media moving with variable speed.