There is nothing controversial in this post - I just want to finish off what I started in the the earlier thread which descended into argument.
I made the calculation based on the definition of rest length ##L=\int_0^L dx## which means we are adding up the notches on a ruler laid alongside the object we are measuring. Because there is no time in that definition, any time gaps between measuring the ends do not affect the result ( the object is not moving wrt the ruler).
When we define a similar quantity but from the frame of a moving ruler, we get ##L'=\int_0^{L/\gamma} dx'##. The change in the upper limit is because the projected length of ##L## onto the ##x'##-axis is ##L/\gamma## ( this can be shown).
From the LT joining the rod frame ( unprimed) and the moving ruler frame (primed) we have ##dt'=\gamma(dx+\beta dt)##. So the integral ##L'=\int_0^{L/\gamma} dx'## now has a time part, and to complete it we need two times ##t_0,t_1## for the limits of the integration. It seems obvious that these correspond to the times of the measurements of the ends of the rod.
Suppose we want the result of the measurement to be ##L##. This means that the integrand should integrate to ##\gamma L##, to cancel the ##\gamma## in the upper limit. The only solution of ##{dx'}=\gamma\beta dt + \gamma dx = \gamma dx## is ##dt=0##. To find the operational delay ( the delay in the ruler frame) we put ##dt=0## into ##dt'=\gamma dt + \gamma\beta dx## to get ##dt'=\gamma\beta L##. In the rod frame the measurements appear simultaneous (##dt=0##).
I think this agrees with other similar calculations.
Now I repeat the with requirement ##L'=L/\gamma##. This means the integrand in ##L'## must evalute to 1 (unity).
The rest is algebra and gives
##dt'=(1-\frac{L}{\gamma})/{\beta}##
This is fine if ##\beta^2>0## and is positive for ##L<c\gamma## (putting ##c## back into ##L##) which includes most laboratories. If it goes negative it means we waited too long and the rod is gone.
So according to this ##L/\gamma## can be the result of a suitable timed measurement.
There's nothing here that isn't known but I haven't seen it shown convincingly in a coordinate independent way.
What the experimenter chooses here is a time delay. I'm not sure how that relates to clock-synchronisation. Defining everything as integrals along curves removes the need for synchronisation, I think.
(I'd like to thank
@harrylin,
@A.T.,
@PAllen, WannaBeNewton and others who responded usefully in the previous thread)
