If you do that, wouldn't you agree that observers on the train would always measure the proper length of the train to be unchanged and the clocks would still be synched? Meaning: fire off that light in the middle and both clocks would still read the same time when the photons hit. Am I wrong?
Yah... I started considering the situation earlier when Ram considered synching the clocks then accelerating the train. I could quickly see that it was obvious that the answer depends on how you do it, so I wondered just how you would have to do it to keep the clocks synched in their own frame.
Doing all of the analysis in the stationary frame, if the train is accelerated towards the right, then
no matter how the train gets acceelrated, if I consider two points that correspond to the same time in the train's frame, the right point has a greater temporal displacement and a lesser spatial displacement than the left point. Thus, if compute the proper time difference for each of the ends, the right end must have experienced more proper time than the left end.
For my example, if we switch to units where c = 1 (for simplicity), we consider the case where the left edge of the train lies along the worldline:
r(tau) = < cosh tau, sinh tau >
(exercise: check that tau is, indeed, the proper time along the worldline)
The velocity is given by
v(tau) = < sinh tau, cosh tau >
Thus, for the observer on the left edge his line of simultaneity corresponding to his proper time a is parallel to
< cosh a, sinh a >
because this is the unit vector orthogonal to v(a). The line is given, parametrically, by:
l(d) = r(a) + d < cosh a, sinh a>
= <cosh a, sinh a> + d <cosh a, sinh a>
= <(d+1) cosh a, (d+1) sinh a>
where d is the proper distance along the line of simultaneity. (aka the distance measured by the observer on the left edge of the train)
Setting d = 1 (corresonding to the train being constant length) and letting a vary gives us the worldline of the right edge:
r(s) = <2 cosh s, 2 sinh s>
A little calculus shows us that s is twice the proper time for the right edge.
In summary, according to the observer on the left edge of the train, the length of the train remains constant, but the clock on the right edge is running at double speed.
On a side note, the class of worldlines:
r(tau) = <k cosh (tau / k), k sinh (tau / k)>
has the property that each of them remain unchanged by any Lorentz boost that fixes the origin; in particular, these (and their translates) are the worldlines of observers that feel a constant acceleration.