Twin paradox for (accelerated) dummies?

In summary, the conversation discusses the application of the Lorentz transformation to velocity and acceleration in the context of special relativity. It is explained that the equations for constant accelerated motion can be derived from the Lorentz transformation and that this type of motion is represented by a hyperbolic worldline on a spacetime diagram. The conversation also explores a scenario involving a spaceship with constant proper acceleration and the effects on time for a twin on board compared to a stationary observer on Earth. The equations for this scenario are derived and used to calculate the time and distance traveled by the spaceship, leading to a disagreement between the traveling twin and the stationary twin about their ages and the number of birthdays missed.
  • #36
Lluis Olle said:
for Bob perspective Alice is in free falling the whole time
The fact that Alice is free-falling the whole time is an invariant; it's true regardless of anyone's "perspective".

Lluis Olle said:
at the start position in Earth at T=t'=0, when Bob jumps into the spaceship, in Bob's non-inertial reference Alice is in free falling and is moving AWAY from Bob, but after some time after the acceleration switch event, Alice begins "free falling" TOWARDS Bob.
Yes, because you've changed the frame you're using. Nothing changed about Alice.

Lluis Olle said:
changed from "going away" to "coming towards", what is not something you can explain changing the sign of the x'-coordinate.
Yes, you can; in fact that's exactly the "explanation" for the "change"--because, as above, nothing about Alice changed. You can't magically change what Alice is doing by changing the way Bob is describing things. All you can do is change the way Bob is describing things, and that's what the sign change in the x coordinate does. It's purely a change in description.

The only actual physical change is that Bob changes the direction of his proper acceleration, from "away from Alice" to "towards Alice". And he chooses to change his description (what frame he is using) at the same time that he makes this physical change. But you still need to be careful to keep distinct the physical change in the direction of Bob's proper acceleration, which is an invariant, from the change in Bob's description, which is not--there is no corresponding change in Alice's description using the inertial frame in which she is at rest the whole time.
 
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  • #37
After some mentor deliberation, this thread will remain closed
 
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