Special relativity and apparent, actual and synchronized positions

[bernhard.rothenstein]

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 Deissler¹ we call that position apparent. 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 E₁ 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 E₂ 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 E₂
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

 

[bernhard.rothenstein]

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 Deissler¹ we call that position apparent. 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 E₁ 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 E₂ 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 E₂
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 x/c. Performing the Lorentz transformations to the rest frame of the moving particle we obtain
x'=gx_a

t'=gx_a/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.