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(adsbygoogle = window.adsbygoogle || []).push({}); _{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 space-time 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}[1-V/c(1+V/c)]=gx_{a}[1-V/c(1+V/c)] (2)

g standing for the Lorentz factor. If we neglect second order effects (2) becomes

t'=t_{a}[(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=x_{a}+Vx/c (4)

we obtain that

x=x_{a}/(1-V/c) (5)

event E being characterized by a time coordinate

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# Special relativity and apparent, actual and synchronized positions

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