Grimble said:
Alice's measure of Bob's time, measurement of time in another frame is coordinate time. Measure of Alice's time in Alice's frame is Proper time - the two differ by the Lorentz factor.
No, this is not correct. Look at the coordinates I gave for events. Alice, Bob, and Bob's light ray all start at (t, x, y) = (0, 0, 0) in Alice's frame. After 1 unit of time in Alice's frame--"time" meaning coordinate time in that frame--Alice is at event (1, 0, 0); Bob is at event (1, 0.6, 0); and Bob's light ray is at event (1, 0.6, 0.8). All three of these events have t = 1, i.e., they are at coordinate time 1. Alice's proper time between the two events is 1; Bob's is 0.8; and Bob's light ray has zero spacetime interval, which strictly speaking should not even be called its "proper time" since that term only applies to a timelike interval, not a null interval.
None of these involve "measurement of time in another frame". They all involve coordinate times in Alice's frame.
Grimble said:
Yet what is never specified in these diagrams is that those two times t and t' measure the interval between the same two spacetime events. That these are two measurements of the same interval.
Yes, they are; they are representations of the same spacetime interval (or more correctly, two successive ones on the same light ray's worldline) in two different frames. But what is this interval? It is the interval along the worldline of
Bob's light ray. It is
not the interval along Bob's worldline, or Alice's worldline.
Here is what the diagrams are telling you, expressed in the standard language of SR:
In Bob's frame, the light ray's worldline passes through the following events: (t, x, y) = (0, 0, 0), (L/c, 0, L), (2L/c, 0, 0). These are two segments, each of which obviously has a spacetime interval of zero.
In Alice's frame, the light ray's worldline passes through the following events: (t', x', y') = (0, 0, 0), (D/c, vD/c, L), (2D/c, 2vD/c, 0). Note that I have written the x' distance for each segment as vD/c, i.e., as v times the coordinate time. One can also use the Pythagorean theorem to show that ##D = \sqrt{v^2 D^2 + L^2}##, or, what is more useful, that ##D = \gamma L##.
These are, as you say, the
same set of three events, represented in two different frames. We can verify this by Lorentz transforming; the primed frame here is moving at velocity ##-v## in the ##x## direction relative to the unprimed frame (because the light clock is moving in the positive ##x## direction in the primed frame, so that frame itself must be moving in the negative ##x## direction relative to the unprimed frame).
But, once again, what intervals do these events represent? They represent the intervals traversed by the
light ray, not by Bob himself. And all of these intervals are
null intervals--their "length" in spacetime is zero. The events that lie along
Bob's worldline are different. In Bob's frame (the unprimed frame in the above), Bob's events are (0, 0, 0), (L/c, 0, 0), (2L/c, 0, 0). And in Alice's frame (the primed frame), Bob's events are (0, 0, 0), (D/c, vD/c, 0), (2D/c, 2vD/c, 0). And the spacetime "lengths" of the two intervals between these three events are each L/c, in both frames (because the spacetime interval between two events is invariant). This is easily verified by using the interval formula in both frames.
However, there is one glaring thing missing in all of this so far: where is Alice? No events are specified for Alice, so all of the above, as it stands, tells us
nothing whatsoever about the relationship between Bob's "time" and Alice's "time". To get that relationship, you need to add Alice's events and show how they are related to Bob's events. All of what I said above about spacetime intervals (and which is basically the same as what you say about them) does not say
anything about Alice's events. It only talks about Bob's events, and the events of Bob's light ray.
So now I have a question for you, to see if you actually do understand the physics: how would you add Alice and Alice's events to the discussion above (and to the diagrams emi_guy showed) to demonstrate "time dilation" of Bob relative to Alice?
Grimble said:
It seems important to me to recognise that it is one interval that is measured differently.
It is
Bob's interval, represented in two different frames, yes. But, as above, so far nothing at all has been said about Alice. And we were supposed to be showing how Bob is time dilated relative to Alice, by using the behavior of Bob's light clock. So, again, how would you add Alice and Alice's events to the picture given above to show that?