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I always assumed that you could explain the twin paradox in special relativity using Minkowski diagrams, such as the one shown in wikipedia. In that diagram it is shown how a change in reference frames from moving to the right to moving to the left creates a discontinuity, such that the accelerating observer perceives a sudden jump in time in the clock of the inertial observer. I always assumed that if the change in reference frames was smooth, instead of sudden, the accelerating observer would actually perceive that the clock in the inertial frame runs faster.

But the bottom of the diagram at the origin seems to show a different picture. The simultaneity lines agree for $t<0$, they are horizontal, and for the accelerating observer they change to downwards (from right to left) for $t>0$. It could be argued that if the acceleration is not sudden then this transition from horizontal to downward would be gradual. But if at the start the two coordinate systems are shifted in space (the accelerating one to the right), then it would look like the accelerating observer should see the inertial clock running backwards in time. What am I imagining wrong?

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  • $\begingroup$ When the observed object is outside of the accelerating observer's light cone then it is not visible. Any two separated points on the horizontal axis at t=0 are outside each other's light cones. $\endgroup$ – JMLCarter Mar 5 '17 at 4:12
  • $\begingroup$ What he said. You cannot see simultaneity, you see light, so you can only see down light cones. I made some videos of a twin paradox scenario which is full of various clocks, so you can SEE what they all say at all times. youtube.com/playlist?list=PLvGnzGhIWTGR-O332xj0sToA0Yk1X7IuI $\endgroup$ – m4r35n357 Mar 5 '17 at 10:49
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What the accelerating twin would actually observe when they accelerate up to some velocity in the direction away from the other twin is that everything about the twin's frame seems to slow down. The clocks would not appear to run backwards.

Then when they accelerate back to the direction towards the other twin everything would seem to speed up, and then even after they reach constant velocity time would continue to appear to run faster.

The reason why is the relativistic Doppler effect. If the twin on earth sends light pulses every second, the twin in the spaceship receives them at an interval greater than one second of their own proper time when they are moving away and less than one second when they are approaching.

The lines of simultaneity are not physical things, they are just coordinates in spacetime. They are nice coordinates because the metric takes a nice form, and you can calculate things easily if your proper time matches the coordinate time. For instance, say you want to send a signal to a spacetime point 2 light seconds away on the line of simultaneity at t=0, you see right away you need to send it at t=-2s.

But the accelerating twin hopefully wouldn't mistake the lines of simultaneity for something they aren't. Frankly, over 100 years after special relativity (or probably more in this scenario), the twin would just skip all this and calculate the other twin's proper time in the first place.

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