Okay guys, let's get back to the subject.
My method of analyzing the OP scenario came in 4 stages;Stage 1;
The station frame should report that both clocks would stop at 18 μs (assuming the stationmaster had to perceive that clocks to be at 10 μs before he pressed the button).
From Einstein’s POV, the clocks are traveling at .5c.
The first clock (the closest to Einstein) is moving away from the button flash and thus it takes longer for the light to get to that clock.
t1 = time concerning the first stop-clock photon as it leaves the button.
x1 = position of the first photon.
x2 = position of the first stop-clock
Length/time contraction factor for .5c = 0.866
x1 = t1 * c
x2 = t1 * .5c + .866 * 4*10^-6
x1 = x2: the moment the photon strikes the first clock, E's POV
t1 * c = t1 * .5c + .866 * 4*10^-6
t1(c - .5c) = .866 * 4*10^-6
t1 = .866 * 4*10^-6/ (.5c)
t1 = .866 * 8*10^-6
From Einstein’s POV, that is how long it should have taken for the photon to get to the first clock.Stage 2;
The second clock is moving toward the flash and thus it takes less time for the light to get to that clock;
t2 * (-c) = t2 * .5c - .866 * 4*10^-6
t2(-c - .5c) = .866 * -4*10^-6
t2 = .866 * -4*10^-6/ (-1.5c)
t2 = .866 * 2.667*10^-6
So from Einstein’s POV for the clocks to show the same 18, they would have to have been out of sync by;
18 – 6.928*10^-6 = 11.072
μs for clock1
18 – 2.3096*10^-6 = 15.690
μs for clock2
That leaves 4.6184
μs time difference between them at the button press moment.Stage 3;
At some point from Einstein's POV;
Clock1 = 11.072
Clock2 = 15.690
So the question becomes;For what time t1 and other clock time t1+4.6184 separated by 6.928 μls would there be an x location from which to observe the difference of 4.6184 μs?
x' = γ(x-tβ)
x' = 1.1547(x - t1/2) ;for clock1
x’ = 1.1547((x+6.928) - (t1+4.6184)/2) ;for clock2
(x-t1/2) = ((x+6.928) - ( t1+4.6184)/2)
(x) = x + 6.928 - ( t1+4.6184)/2 + t1/2at this point x drops out meaning that any distance will do.
6.928 = ( t1+4.6184)/2 + t1/2
2*6.928 = ( t1+4.6184) + t1
2*t1 = 2*6.928 - 4.6184
t1 = 6.928 - 4.6184/2
So at any
location x, as long as t1 = 4.6188, the proper difference in time can be observed by Einstein at the button press.
If Einstein must see clock1 at 4.6184 at button press time. And the light coming from it telling the stationmaster to press the button had to be 4.6184 - 3.464 = 1.1544
by Einstein’s POV.
So in the station’s POV clock1 had to be reading 10
when it sent the message and in Einstein’s POV
that same clock1 had to be reading 1.1544
. Lorentz would suggest that Einstein’s POV would require that a clock in the station’s POV that reads 10 in that frame, must be reading 8.66 by Einstein’s POV, not 1.544.