Micheth said:
AB are separated (let's say by 10 meters e.g.), and synchronized in a rest frame.
Yes.
Micheth said:
There are two points in AB's original rest frame (far, far away) that are C & D, also separated by 10 meters in that same frame.
No. C and D are two clocks that are moving, in A and B's original rest frame, with some speed ##v##. C and D never accelerate; they remain in inertial motion the whole time.
Micheth said:
AB then accelerate to some near-light speed toward CD.
No. A and B accelerate in such a way that they just match speeds with C and D, respectively, at the instants when C and D are spatially co-located with A and B, respectively. In other words, C and D are behind A and B, respectively, when A and B start accelerating; but C and D are moving faster, so they catch up to A and B, respectively (all of this is as seen by A and B), just as A and B match speeds with C and D, respectively.
Micheth said:
I was originally thinking you meant that AB's clocks would simply get out of sync because they accelerated.
That was mistaken, right?
No. A and B's clocks do get out of sync because they accelerated. Their clocks stay in sync in their original rest frame (meaning the inertial frame in which they were originally at rest and synchronized), but A and B do not remain at rest in their original rest frame. They accelerate, hence they start moving in that frame. When they finish accelerating, they are both moving relative to that original frame, so the fact that they are synchronized in that frame is no longer relevant. See further comments below.
Micheth said:
what's really happening is that the distance between AB has increased (from CDs viewpoint) as they pass CD
Please stop using the word "really". You keep on applying it to things that are frame-dependent. That's a very bad habit in relativity.
The correct way to state how things look in the frame in which C and D is at rest is basically what you say: A and B are moving towards C and D in this frame; then B starts decelerating; then A, a bit later, starts decelerating; then B comes to a stop right at D; then A comes to a stop right at C. Because B started decelerating first, the distance between A and B, in this frame, increases. But this is all relative to C and D's rest frame, so the word "really" is inappropriate.
(There are ways of describing the motion of A and B, and C and D, in frame-invariant form. For example, we could say that the worldlines of A and B, during the period when both are accelerating, have positive expansion; whereas the worldlines of C and D always have zero expansion. "Expansion" here is a technical term, referring to the frame-invariant mathematical description of these motions. In this sense, you could say that the distance between A and B "really" does increase as they accelerate. But then you would have to remove the phrase "from CDs viewpoint", because the expansion is frame-invariant.)
Micheth said:
Hence the non-simultaneity of the meeting of AC vs. BD.
Non-simultaneity in frame CD. The two meetings
are simultaneous in frame AB (the frame in which A and B were originally at rest). But once A and B are no longer at rest in frame AB, that frame's definition of simultaneity is no longer the right one to use to judge A and B's clock synchronization.
In other words, the reason A and B's clocks get out of sync is that they have changed inertial frames: they start out at rest in one inertial frame, and end up at rest in a different inertial frame. (They have to accelerate to do this, which is why I said above that they get out of sync because they accelerated.) That changes which definition of simultaneity is the one that determines whether their clocks are synchronized.
If all that is too abstract, consider how, physically, A and B would check that their clocks are synchronized. (This will also show how they can check the distance between them and verify that, as far as they are concerned, it has increased.) In the original rest frame, AB, before any acceleration has taken place, A and B can exchange light signals, and determine two things: (1) that the distance between them is in fact 10 meters (or whatever it turns out to be), based on the round-trip travel time of light signals between them; and (2) that their clocks are in fact synchronized, based on the fact that, when A receives a light signal from B, it shows B's clock reading exactly what A's clock read one light-travel time ago (so if they are 10 meters apart, when A receives a light signal from B, B's clock will read what A's clock read 33 nanoseconds ago, since it takes light about 33 nanoseconds to travel 10 meters). That is, the reading shown on B's clock in the light signal will be 33 nanoseconds before the reading shown on A's clock when the signal is received.
Now, once A and B have completed their acceleration, and are now co-located with C and D and moving inertially, at rest in frame CD instead of frame AB, they can repeat the above process. And when they do, they will find: (1) that the distance between them has increased, based on the round-trip travel time of light signals between them; and (2) that their clocks are no longer synchronized: when A receives a light signal from B, it shows B's clock reading something
later than what A's clock read one light travel-time ago, and when B receives a light signal from A, it shows A's clock reading something
earlier than what B's clock read one light travel-time ago.
For example, if A and B are now 20 meters apart (corresponding to a relative speed of about 0.87c between frames AB and CD), then when A receives a light signal from B, it will show B's clock reading, not a time 66 nanoseconds before A's (which would be one light-travel time before), but a time about 15 nanoseconds before A's, indicating that B's clock is about 51 nanoseconds ahead of A's. And when B receives a light signal from A, it will show A's clock reading, not a time 66 nanoseconds before B's, but a time about 117 nanoseconds before B's, indicating that A's clock is about 51 nanoseconds behind B's. So the two measurements are consistent (they both show the same offset between the clocks), and they clearly show the clocks not synchronized.
Micheth said:
I still can't understand why distance AB would have to increase (in anybody's frame)
Because, as I noted above, in frame CD, B starts decelerating before A does. This has to be true because of relativity of simultaneity; A and B both start accelerating simultaneously in frame AB, therefore they cannot start accelerating (or decelerating, depending on how you view it) simultaneously in any other frame, including frame CD. Therefore, the distance between them
must change in any other frame besides frame AB. (And, as the discussion above shows, they can verify that the distance has increased by exchanging light signals.)
Once again, I think part of the issue is that you are trying to reason from the wrong assumptions. Instead of starting with the known properties of relativistic spacetime (the main one here being relativity of simultaneity), and deducing what must happen in the scenario based on those properties, you are trying to start with your assumptions about how things "ought" to work, then wondering why relativity says they don't work that way.
