Pentcho,
take your second paradigm event in detail:
| However take another event - the movement of the front of a beam
| along the y-axis. For this event, x = 0, x' = -vt' and by
| substituting these in Lorentz first equation we obtain
|
| t' = gamma*t /2/
The coordinates of this event are specified above to be:
x=0, t'=gamma*t, y=ct (To the unprimed observer a light pulse has
traveled from x=0 (at t=0) to the point (0,y) with a travel time t; so
that y=ct).
The Lorentz transformation then says that for this event:
y'=y=ct=(ct')/gamma , x'= -gamma*vt= -vt'
To the primed observer a light pulse has traveled from x'=0 (at t'=0) to
the point (x', y'), with a travel time t'
Do the algebra and notice that x'^2 + y'^2 = (ct')^2