Dear Eckard,
My sincere apologies. I see I've been missing out on a very interesting discussion. Before I read through it and try to join back in, I want to refer back to your post on Jul. 7, 2013 @ 06:43 GMT. Since I've fallen out of sequence there, I'm posting this in a new thread.
The only thing that bothers me in what you wrote is the statement that you "consider it fallacious to have different perspectives at a time".
I think you're thinking too much about these people making observations and not enough about what their natural definition of space and time should be. I think we can get to the former easily enough after we deal with the latter--so I just want to ask you: do you think it's unreasonable for Albert to string out a measuring tape along the track and plot a graph of what he perceives as occurring along that axis of "space" through the course of "time"? Or to add a vertical tape measure and make a two-dimensional graph of an object's path through the two spatial dimensions? Or make a full three-dimensional graph?
Do you think it's any less reasonable for Henri to string out measuring tape along the floor and the wall of the train car and do something similar?
The principle of relativity says that they should both be able to do this, and the laws of physics should apply just as well in both of their frames.
Now, the clocks that they have with them in the first phase of the experiment have the same vertical distance between the mirrors, and a single photon bounces up and down. When this happens, the clock displays the number of "ticks", sending photons in all directions (and in these guys' frames of reference they can correct for the amount of time it takes those photons to reach their eyes by travelling through "space", etc.), so the experiment can actually be carried out in theory. But that really doesn't need to happen if you already just accept that their two coordinate systems are reasonable to use, because then you already know what's going to happen, from either perspective.
From Albert's perspective, H's clock moves to the right. The path of the photon in the clock on the train is therefore not vertical, but moves at an angle to the right. By the Pythagorean theorem, it travels a longer distance than the photon in Albert's clock, which just went up or down. Assuming c is a universal constant, this means that the "tick" of Henri's clock, as described by Albert, takes longer (t=d/c) than the "tick" of Albert's clock, as described by Albert.
The paradox is that the whole experiment can be *described* from Henri's perspective, where he just sits at rest and Albert moves gradually along his ruler. From Henri's perspective, the photon in his own clock goes vertically up and down, and it's Albert's that travels to the right. For all the same reasons as before, we conclude that as described in Henri's frame of reference, Albert's clock ticks more slowly than Henri's.
You said before that you "question the necessity to ascribe a different proper time to the observer" (meaning Albert in this example). This is the reason why. Relativistic time dilation is a real effect in the real world, and it's not a trivial problem to reconcile the fact with the ticking of an absolute clock. You also said that to you, "only the motion of the observer re train, or vice versa, is relevant", and not whether the train is actually moving. This was Einstein's way of thinking, and it's why I said before that you were thinking as a pure relativist. And the issue--the reason you and I are not seeing eye to eye, as far as I can tell--is that without defining a true cosmic frame of reference, actual or absolute motion, etc., and considering only relative motion as what matters, it's logically inconsistent to say that there is one absolute time, three-dimensional space is all that exists, time passes and all of reality is the three-dimensional present that exists "now", etc.
Time-dilation has to be admitted from a relativistic perspective, and it has to be reconciled with a global reference frame in order for it all to work. The way of actually allowing that their proper times tick at different rates while admitting an absolute time and the *existence* of a three-dimensional universe, is to define simultaneity as absolute, and therefore as something different from synchronicity. You and Paul are right that Einstein got simultaneity wrong, by defining it from an operationalist point of view. This is the point I keep trying to make. And I keep trying to show exactly how relativity works--and it works beautifully!--when it's understood in this way.
Cheers,
Daryl