Does the Twin Paradox Challenge Classical Relativity?

[jujufactory]

I hate to bring this up again, but the twin paradox explanation based on acceleration does not hold water.

Let's go back to Object A mentioned on the now locked thread posted earlier.

Object A sees Object B fly by at close to the speed of light. A figures the clocks are moving more slowly on B. So A decides to verify this information. A turns on its thrusters and begins to catch up with B. As A accelerates, A notices that the clocks on B are now moving more and more normally. By the time A catches up with B, the clocks are moving at the same speed and at no point did B have a chance to catch up with A on his clocks. Therefore, A must be ahead of B. There is no way B can be anything but behind A since there is no opportunity for that to happen.

However, from B's point of view, all he saw was A fly by with slower clocks. He then seens A slow down, and stop. Here again, from B's point of view, A did not have the opportunity to catch up with the B clocks.

Here is the 2 million Euro question: At which point did the clocks agree on anything? When the two objects meet, the clocks must say something. What do they say?

 

[Garth]

When a body accelerates its surface of simultaneity slews round.

It is the slewing of the surface of simultaneity in the accelerating/decelerating spacecraft that explains the 'paradox'.

Garth

 

[JesseM]

I hate to bring this up again, but the twin paradox explanation based on acceleration does not hold water.

Let's go back to Object A mentioned on the now locked thread posted earlier.

Object A sees Object B fly by at close to the speed of light. A figures the clocks are moving more slowly on B. So A decides to verify this information. A turns on its thrusters and begins to catch up with B. As A accelerates, A notices that the clocks on B are now moving more and more normally. By the time A catches up with B, the clocks are moving at the same speed and at no point did B have a chance to catch up with A on his clocks. Therefore, A must be ahead of B. There is no way B can be anything but behind A since there is no opportunity for that to happen.

You're falsely generalizing from inertial frames to A's accelerating frame. It's true that at each moment that A is accelerating, an inertial observer who happens to have the same instantaneous velocity as A at that moment will say that B's clocks are running slower than A's at that moment; but this does not mean that in A's accelerating frame, B's clocks are running slower than his own throughout the acceleration! The reason has to do with the fact that A's definition of simultaneity is constantly swinging forward as A changes velocity, so that if the acceleration was very brief, the reading on B's clocks immediately after the acceleration in A's new inertial rest frame after the acceleration would be far ahead of the reading on B's clcoks immediately before the acceleration in A's inertial rest frame before accelerating. So, when you take this into account to figure out what is going on in A's accelerating frame during the acceleration (where the accelerating frame is constructed in such a way that at every moment, A's definition of simultaneity and distance matches that of A's instantaneous inertial frame at that moment), you actually find that B's clocks are ticking much faster than A's during the accelerating phase, in A's own accelerating frame.

Here is the 2 million Euro question: At which point did the clocks agree on anything? When the two objects meet, the clocks must say something. What do they say?

If we imagine A and B's clocks showed the same time at the moment they first passed, then after A accelerates and catches up with B, B's clocks will be far ahead of A's. As mentioned above, in A's non-inertial coordinate system this is because B's clocks ticked faster than A's while A was accelerating; in B's inertial frame this is just because A's clocks were ticking slower at all times until A caught up with B (except for a single instant when A's clock had the same instananeous rate of ticking as B's during the turnaround, since in B's frame A was initially moving away from B but then moving towards B after the acceleration, so there must have been a moment during the acceleration when A was instantaneously at rest in B's frame).

 

[HallsofIvy]

You didn't like the answers you got when you posted exactly this same thing 25 minutes earlier?