velocity addition and the relativity of space and time intervals

A thought experiment first proposed by Sartori is analysed using the parallel velocity addition formula of special relativity. The spatial and proper-time intervals between some similarly defined spatial coincidence events are found to be widely different in different inertial frames. This relativity of space and time intervals is quite distinct from the well-known time-dilatation and length contraction effects of special relativity. Sartori's claimed derivation of the parallel velocity addition formula, assuming relativistic time dilatation, based on the thought experiment, is shown to be fortuitous. PACS 03.30.+p This paper analyses a thought experiment, that was orginally proposed by Sartori [1], involving two trains T1 and T2, moving at different speeds v and u respectively relative to a fixed platform P. It was claimed that the parallel velocity addition relation (PVAR) could be derived from the experiment by assuming the existence of the relativistic time dilatation effect [2]. The derivation will be critically examined below; however, the main purpose of the present paper is a different one. Assuming the PVAR, space and time intervals between spatial coincidences, in the direction of motion, of P, T1 and T2 are calculated in their respective rest frames S, S' and S " and compared.. In Fig.1 is shown the experiment as preceived by an observer in the rest frame of P. It is assumed that there is an array of synchronised clocks in each frame, each of which indicates a common 'frame time' (i.e. the proper time of any of the clocks) τ , τ ′ and τ ′′ in the frames S, S' and S ". At frame time τ 1 = 0 the train T1 is opposite the platform (Event1) and T2 is at a distance L1 from it (Fig1a). At frame time τ 2, T2 is opposite the platform (Event2) and T1 is at a distance L2 from it (Fig1b). Finally, since it is assumed that u > v, at frame time τ 3, T1 and T2 are opposite each other, at a distance L3 from P (Event3). Figs.2 and 3 show the same sequence of events as observed in the rest frames of T1 and T2 respectively. The corresponding frame times and distances are τ 1