I Do clocks on Bell’s spaceships stay synchronized?


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So in their co-moving frame the clocks undergo the same acceleration and thus the same deviation to the original time in the rest frame, but nevertheless they also develop an offset of time in between them (to an observer traveling along in the commoving frame). How is that possible?
Whether two clocks are synchronized or not is frame-dependent. If two rockets are traveling at constant velocity, and the clocks on the rockets are synchronized in the frame of the Earth, then they will not be synchronized in the frame of the rockets. Why is that?

Well, Einstein gave an operational definition of when two clocks are synchronized in a frame in which they are at rest: Send a light signal from the rear clock to the front clock, and back again. Then there are three relevant events: (1) the signal is sent from the rear clock, (2) the signal is received by the front clock and a reply is sent, (3) the reply reaches the rear clock. The clocks are synchronized if the time of the second event, as shown on the front clock, is halfway between the times of the first and third events.

If we apply Einstein's definition of synchronization to moving rockets, then we find that in the trip from the rear clock to the front clock, the light signal takes longer (according to the Earth's frame) than in the return trip. That's because the front rocket is moving away from the light signal, so it takes longer for the light signal to catch up, and the rear rocket is moving toward the light signal, so it takes less time for the light signal to catch up. So if the clocks in the two rockets are synchronized, according to Earth's frame, then the front clock will be ahead of the rear clock, according to the rocket's frame (that is, the time shown on the front clock will be more than half-way between the first and third events, as described above). If the clocks are synchronized in the Earth's frame, then there will be a positive offset of the front clock in the rocket frame. If the rockets are traveling at constant velocity, then this offset will be constant.

If instead of the rockets traveling at constant velocity, the rockets are accelerating, then the offset will keep growing with time. So the riders in the rocket will interpret this as the front clock getting more and more ahead of the rear clock.


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I thought I'd generate some Minkowski diagrams, the rigorous version of the analogy from my earlier post. Here's one in the initial frame of the ships:

There is a red ship half a light year behind a blue ship. They are initially at rest, and they have synchronised clocks that tick every tenth of a year (marked on their worldlines by black crosses). At ##t=0## they both start accelerating with a proper acceleration of 1 ly/y2, which is about 1g, until they reach 0.6c compared to their initial state, then they switch off their engines and travel inertially. I've linked the clock ticks with fine grey lines, and you can see that they are simultaneous and the spacing increases as the ships accelerate.

This is the same situation from the final rest frame of the ships:

You can see that the ticks were never simultaneous in this frame, and the ships didn't accelerate simultaneously, which is why the distance changes. The grey lines do get closer together during the acceleration, but you'll probably have to take my word for that unless you've got a ruler.

What I haven't drawn is the point of view of the ships. That isn't inertial, of course, and needs to be drawn separately for the two ships since "during" the acceleration isn't the same block of spacetime for both ships. One set of reasonable simultaneity surfaces is shown in figure 6 of Dolby and Gull's paper on Radar Time.


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How is that possible?
Apparently there is a lot more possible in RT then I can imagine. However, I fear, at the cost of a specific possibility I always took for granted. But for overview convenience I’ll open a new thread on that. I sincerely thank you all for the efforts you make to enlighten me and hope to see you at my next post.

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