special relativity and apparent, actual and synchronized positionsby bernhard.rothenstein Tags: actual, apparent, positions, relativity, special, synchronized 

#1
Jul2708, 04:23 AM

P: 997

Consider a particle that moves with speed V in the positive direction of the OX axis of the inertial reference frame I. At a time t=0 it is located at a point M_{a} characterized by a apace coordinate x_{a}. Using a terminology proposed by Deissler^{1} we call that position aparent. At t=0 a light signal is emitted from the origin O in the positive direction of the OX axis. Its arrival at the point M_{a} is associated with event E_{1} characterized by a space coordinate x_{a} and by a time coordinate x_{a} and by a time coordinate x_{a}/c. At that very time the moving particle arrives at its actual position associated with an event E_{2} characterized by a space coordinate x_{a}(1+V/c) taking into account that during the time interval x_{a}/c the moving particle has advances with Vx_{a}/c, and by a time coordinate x_{a}/c. The two events are simultaneous in I.
Performing the Lorentz transformation of the spacetime coordinates of event E_{2} to the rest frame I' of the moving partcle the results are x'=gx_{a}=gct_{a} (1) and t'=gt_{a}[1V/c(1+V/c)]=gx_{a}[1V/c(1+V/c)] (2) g standing for the Lorentz factor. If we neglect second order effects (2) becomes t'=t_{a}[(1V/c)/(1+V/c)]^{1/2} (3) Let E be the event associated with the fact that the world lines of the moving particle and of the propagating light signal intersect each other. Let x be its space coordinate. From x=x_{a}+Vx/c (4) we obtain that x=x_{a}/(1V/c) (5) event E being characterized by a time coordinate 



#2
Jul2708, 04:29 AM

P: 997

Do you consider that the derivations above are correct? Are they a simple exercise in handling the Lorentz transformations or there is some physics behind them. Thanks in advance for your help. 


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