Eckard
I do not understand this post, or even if you have got the example correct. You say the "distance between them steadily grows with velocity v", which means there is relative motion, or they are travelling in different directions?
Forget the clocks, these are irrelevant. And let us assume that the rate of change in reality is the same as that for light (this just makes the explanation simpler but does not interfere with the logic). [Note: you have presumed that physically A and B are identical over time, so the distance AB can always be measured using the same points on A & B, which will not be the case. Also, if there is relative motion, then there is the possibility that the differential in force incurred which is causing this could be causing dimension alteration].
As each reality occurs, which in this example means a degree of change in spatial position, (it cannot be the same reality if something is in a different position), a light based representation of that physically existent state is created. The existential sequence progresses, and the resulting existent lights travel (we are assuming perfect conditions, ie all the lights travel at c). Now, if the distance between A and B remains the same, then the sequence of lights will be received, after the initial delay, at the same rate as they were generated, and as reality altered, ie the perceived (received) rate of change (timing) will be the same as that which is physically occurring. If the distance is ever changing, which could be a function of A and B travelling in different directions, ie their relative momentum is the same but the effect in terms of distance is the same as if they were travelling in the same direction but at different speeds, then the duration between the receipt of each light will change on each occasion as the distance alters, so the perceived rate of change will be different to that which is occurring.
This might be what you are trying to explain, ie the caveat of 'uniform translatory motion' is wrong because it depends on direction of travel. But I am not sure, as the real point of his first postulate is that reality is unaffected by the reference used to calibrate it (albeit he invoked an unnecessary condition). [Incidentally, I have asked you for a translation of the sentence defining the first postulate in the Introduction, having realised that you are quoting the definition as in section 1 part 2 (On the relativity of lengths and times)].
Poincaré starts with an incorrect view as to how timing works, ie not that the real reference is a conceptual constant rate of change, and the ability to synchronise timing devices to this is a practical matter, and not an issue about time. He then goes on to develop his A B example, with light transmissions to enable syncronisation. Einstein realises this is the equivalent of observation, so his A B example was supposed to be about synchronisation of light received. But as he developed his theory he had no observational light, just an example of light which was used as a reference, nobody saw with it, so the relativity in effect was attributed to existence.
Poincaré, 1898, para 4:
"All this is unimportant, one will say; doubtless our instruments of measurement are imperfect, but it suffices that we can conceive a perfect instrument. This ideal can not be reached, but it is enough to have conceived it and so to have put rigor into the definition of the unit of time. The trouble is that there is no rigor in the definition. When we use the pendulum to measure time, what postulate do we implicitly admit? It is that the duration of two identical phenomena is the same; or, if you prefer, that the same causes take the same time to produce the same effects."
Poincaré, 1898, para 13:
"To conclude: We have not a direct intuition of simultaneity, nor of the equality of two durations. If we think we have this intuition, this is an illusion. We replace it by the aid of certain rules which we apply almost always without taking count of them.
But what is the nature of these rules? No general rule, no rigorous rule; a multitude of little rules applicable to each particular case. These rules are not imposed upon us and we might amuse ourselves in inventing others; but they could not be cast aside without greatly complicating the enunciation of the laws of physics, mechanics and astronomy. We therefore choose these rules, not because they are true, but because they are the most convenient, and we may recapitulate them as follows: "The simultaneity of two events, or the order of their succession, the equality of two durations, are to be so defined that the enunciation of the natural laws may be as simple as possible. In other words, all these rules, all these definitions are only the fruit of an unconscious opportunism."
Poincaré, 1900, page 20:
"It is the case that, in reality, that which we call the principle of relativity of motion has been verified only imperfectly, as shown by the theory of Lorentz. This is due to the compensation of multiple effects, but:...2. For the compensation to work, we must relate the phenomena not to the true time t, but to a certain local time t' defined in the following fashion.
Let us suppose that there are some observers placed at various points, and they
synchronize their clocks using light signals. They attempt to adjust the measured
transmission time of the signals, but they are not aware of their common motion, and
consequently believe that the signals travel equally fast in both directions. They perform observations of crossing signals, one travelling from A to B, followed by another travelling from B to A. The local time t' is the time indicated by the clocks which are so adjusted. If V = 1/√Ko is the speed of light, and v is the speed of the Earth which we suppose is parallel to the x axis, and in the positive direction, then we have: t' = t − v x/V2."
Poincaré, 1900, page 22:
"Suppose T is the duration of the emission: what will the real length be in space of the perturbation?...The real length of the perturbation is L = (V - v')T. Now, what is the apparent length?...the local time corresponding to that is T(1-vv'/V2). At local time t', it is at point x, where x is given by the equations: t ' = t − vx/V2,
x = v'T + V(t - T), from which, neglecting V2: x = [v'T + V(t - T)](1 + v/V)...The apparent length of the perturbation will be, therefore,
L' = Vt' - (x - vt') = (V - v')T(1 +v/V) = L(1 + v/V)."
Poincaré, 1902, para 90:
"1. There is no absolute space, and we only conceive of relative motion; and yet in most cases mechanical facts are enunciated as if there is an absolute space to which they can be referred.
2. There is no absolute time. When we say that two periods are equal, the statement has no meaning, and can only acquire a meaning by a convention.
3. Not only have we no direct intuition of the equality of two periods, but we have not even direct intuition of the simultaneity of two events occurring in two different places."
Poincaré, 1904, page 6:
"The most ingenious idea is that of local time. Let us imagine two observers, who
wish to regulate their watches by means of optical signals; they exchange signals,
but as they know that the transmission of light is not instantaneous, they are careful
to cross them. When station B sees the signal from station A, its timepiece should
not mark the same hour as that of station A at the moment the signal was sent,
but this hour increased by a constant representing the time of transmission. Let
us suppose, for example, that station A sends it signal at the moment when its
time-piece marks the hour zero, and that station B receives it when its time-piece
marks the hour t. The watches will be set, if the time t is the time of transmission,
and in order to verify it, station B in turn sends a signal at the instant when its
time-piece is at zero; station A must then see it when its time-piece is at t. Then
the watches are regulated."
"And, indeed, they mark the same hour at the same physical instant, but under
one condition, namely, that the two stations are stationary. Otherwise, the time
of transmission will not be the same in the two directions, since the station A, for
example, goes to meet the disturbance emanating from B, whereas station B sees
before the disturbance emanating from A. Watches regulated in this way, therefore,
will not mark the true time; they will mark what might be called the local time,
so that one will gain on the other. It matters little, since we have no means of
perceiving it. All the phenomena which take place at A, for example, will be
behind time, but all just the same amount, and the observer will not notice it since
his watch is also behind time; thus, in accordance with the principle of relativity
he will have no means of ascertaining whether he is at rest or in absolute motion."