Answer to Question #112171 in Mechanics | Relativity for John Dawson

Question

This is a follow up question.
Assume in my first scenario that astronauts on B and C are asleep and unaware of A during the initial acceleration but awake during the second acceleration when the force is applied to B to stop its motion in relation to A. Astronaut C and B should at that point believe that the clock in B is ticking slower than the clock in C since both witness the acceleration of B with respect to C. But this contradicts our previous answer that the clock in C is ticking slower than the clock in B and A.
The only remedy to this contradiction that I can see is that the clocks in all different bodies would have to be compared to the point in which all bodies were initially together.

Answer 1

This contradictions may arise because the change in speed takes some time in reality. In your previous question you wrote that this change in direction occurs momentarily (takes no time at all), although it is practically impossible. Therefore, if we assume that this change in the velocity direction occurs absolutely momentarily, B and C will be synchronized right before B stops.

If that change occurs like in our real universe, say it takes a day to slow B down and accelerate it to sublight speed again backwards, indeed the clock in C will lag after B.

Also, let me comment one of your statements:

"Astronaut C and B should at that point believe that the clock in B is ticking slower than the clock in C since both witness the acceleration of B with respect to C".

Actually they should conclude that the clock in C is ticking slower than the clock in B because the clock in C spent more time moving at a relativistic speed (say A measured that C moves for 30 days and B for 29 days at a relativistic speed and 1 day decelerating and accelerating so no time dilation for one day).

If B and C move at 0.95c, the stationary observer in A will calculate that for the clock in C the 30 days became

$$t_{AC}=t_C\sqrt{1-v^2/c^2}=\\=30\sqrt{1-0.95^2}=9.37\text{ days},$$

while for B it was

$$t_{AB}=1+t_B\sqrt{1-v^2/c^2}=\\=1+29\sqrt{1-0.95^2}=10.06\text{ days},$$

which means that April 30 for A turned into April 9 for C (when B and A met) and April 10 for B (when A and B compared their time).

You might probably want to read about the Hafele–Keating experiment. Their experiment was similar to yours.