Equal Relative Velocity Between Two Objects for Observers in Different Rest Frames

This paper discusses applications of Special Relativity Theory (SRT) in which the relative velocity of two objects needs to be determined from the vantage point of two different rest frames. It has generally been assumed without proof that such relative velocities are the same for all observers. 
In the present work, a general proof based on Einstein’s light-speed postulate is given. It is shown to be consistent with applications of the relativistic velocity transformation (RVT) introduced by Einstein in his 1905 paper. One consequence is that the average range of decay of metastable particles is directly proportional to the measured lifetime in each rest frame. The fact that an observer traveling with the particles will measure a shorter range of decay than his counterpart on Earth’s surface is traced to the increase in length of measuring devices in the former rest frame that results from the acceleration. It is pointed out that this conclusion runs counter to the SRT prediction of Fitzgerald-Lorentz length contraction accompanying time dilation.



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