There are a ton of videos that cover the twin paradox. They're all quite involved and don't really get to the heart of the matter, which is the fundamental asymmetry between the two observers (as explained by: https://youtu.be/nRuVGOm7560 ).
I have a very simple resolution of the paradox, but it's so simple that I think it's incorrect. I'm wondering where the flaw in my logic is. Does the explanation below explain something that countless Youtubers (including Fermilab and Minutephysics) have tried to explain? I have my doubts since my explanation is too simple, so please let me know what you think!
Here's the setup: Alice and Bob are twins and are standing next to each other in space. There is a planet far away. Bob, Alice and the planet are not moving relative to each other. Bob accelerates by using a rocket, travels to the far away planet, then accelerates in the opposite direction by using a rocket, retraces his steps and finally rejoins Alice. For the sake of simplicity, let's assume both Alice and Bob live in a special universe where there are no objects apart from the Alice and Bob, Bob's rocket and the far away planet.
Result: Alice is now older.
Question: From Alice's perspective, Bob accelerated, travelled far away and then returned. From Bob's perspective, it is Alice that accelerated, travelled far away then returned. It seems like a perfectly symmetric situation, but Bob has nonetheless aged more slowly. Why is that?
Source of asymmetry (absurdly simple, so possibly incorrect): Bob and Alice disagree on the distance that Bob covered. Due to length contraction, Bob thinks he travelled a shorter distance.
Let's put some numbers:
- The planet is originally 18.8 light years away
- Bob accelerates very quickly to 0.9c
- As soon as Bob reaches 0.9c, the far away planet is around 8.2 light years away from his perspective since 8.2 ~= 18.8 * sqrt(1-(0.9c)^2/c^2)
- From Bob's point of view, the round trip was 16.4 light years
- From Alice's point of view, the round trip was 37.6 light years
- In other words, Bob's odometer reads 16.4 light years but Alice clearly saw Bob travel to a planet 18.8 light years away and back
- Since bob travelled at 0.9c and his odometer reads 16.4 light years, only 18.2 years have passed for him.
- For Alice, since Bob travelled 37.6 light years, a total of 41.8 years have elapsed
Let's assume that Bob accelerated twice. Originally, Bob was stationary. Then, Bob started moving away from Alice. He went from 0 to 0.9c. let's assume he did so in 1 second. Then, when he reached the planet, he had to change the direction of his velocity so that he could return to Alice. Let's assume he also did that in 1 second.
Let's do the calculations from Bob's perspective:
- Bob was stationary the whole time, Alice and the planet are the ones that moved
- During the initial acceleration, from his perspective, the far away planet moved from 18.8 light years away to 8.2 light years away in the span of 1 second
- For the "first leg of the voyage", Bob observed the planet come towards him at 0.9c. The planet travelled 8.2 light years from Bob's perspective
- At the same time, Alice was moving away from Bob. Alice travelled X light years (I'm not sure how to calculate X) in the same time that it took the planet to reach Bob.
- When "the planet reaches Bob", Alice goes from being X light years away to 8.2 light years away, so Alice travelled (X - 8.2) light years in 1 second.
- Then Alice travels the 8.2 light years to be reunited with Bob
Could someone tell me if this is accurate from Bob's perspective? Also, how do I calculate X? Is it simply (lorentz factor * 18.8) = 43.1 ?