# A question with the readings of synchronized clocks

1. Nov 12, 2007

### bernhard.rothenstein

Consider that from the origin O of the inertial reference frame O start at the origin of time a particle that moves with speed u<c and a photon with speed c both in the positive direction of the OX axis. At each point of that axis we find a clock all the clocks being synchronized following the procedure proposed by Einstein. After a given time t of motion the photon arrives at the location of a clock C(x)[x=ct,y=0). What is the reading of a clock located where the particle arrives when the photon arrives at the location mentioned above?
Thanks.

2. Nov 12, 2007

### Staff: Mentor

According to observers in frame O, when the photon arrives at point x = ct all frame O clocks, including the one at point x = ut, will read the same time t. Of course, since these points are spatially separated in O, other frames will disagree.

3. Nov 12, 2007

### bernhard.rothenstein

Thanks for helping me to change my question which becomes:"What is the distance travelled by the moving particle?" If it is ut=ux/c then consider the relative positions of the reference frames I and I' detected from I when the synchronized clocks of that frame read t. Let E(x=ct,y=0, t=x/c) and E'(x'=ct',y'=0, t'=x'/c) same events taking place on the overlapped axes OX(O'X'). Taking into account that relative motion distorts length and time intervals then adding only lengths measured in I we obtain
x-Vt=Ax' (1)
where A is a factor that accounts for the distorsion of lenghts. It could depend on the relative speed V but not on the space-time coordinates involved in the transformation process.
Considering the relative positions of I and I' detected from I' when the clocks of that frame read t' we obtain adding only lengths measured in I'
Ax=x'+Vt' (2)
Presenting (1) and (2) as
ct-Vt=Act' (3)
ct'+Vt'=Act (4)
the result is
A=sqrt(1-VV/cc) (5)
and so the Lorentz transformations for the space time coordinates are derived. Solving them for t and t' we obtain the Lorentz transformations for the time coordinates,
The problem is what was used above bedides Einstein's two postulates:
-Considering that the factor A is the same in I and I' as well we have involved linearity and simmetry.
-Considering that if I' moves with speed V relative to I and that I moves relative to I' with speed -V we took into account isotropy and reciprocity.
Did I use more?
A simillar derivation was presented recently by Levy (Am.J.Phys) consdiering that A is known from thought experiments.
Is the derivation free of flows?
Far from me to consider that was I have presented above is my contribution. I simply consider that it could be a standard, time saving and tranmsparent derivation of the LT. Please constribute to it in order to make it better!