Special relativity and apparent, actual and synchronized positions

  1. 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 Ma characterized by a apace coordinate xa. Using a terminology proposed by Deissler1 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 Ma is associated with event E1 characterized by a space coordinate xa and by a time coordinate xa and by a time coordinate xa/c. At that very time the moving particle arrives at its actual position associated with an event E2 characterized by a space coordinate xa(1+V/c) taking into account that during the time interval xa/c the moving particle has advances with Vxa/c, and by a time coordinate xa/c. The two events are simultaneous in I.
    Performing the Lorentz transformation of the space-time coordinates of event E2
    to the rest frame I' of the moving partcle the results are
    x'=gxa=gcta (1)
    and
    t'=gta[1-V/c(1+V/c)]=gxa[1-V/c(1+V/c)] (2)
    g standing for the Lorentz factor. If we neglect second order effects (2) becomes
    t'=ta[(1-V/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=xa+Vx/c (4)
    we obtain that
    x=xa/(1-V/c) (5)
    event E being characterized by a time coordinate
     
  2. jcsd
  3. t'=gxa/c (6)
    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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