What is the figure of 6 years representing in resolving the Twin Paradox?

In summary, the conversation discusses the resolution of the "Twin Paradox" using the Lorentz relativistic velocity transformation equation. After several attempts at understanding the explanation, the speaker finally figures out that it takes 4 years for Bob to catch up with Ann at a speed of 15/17 C. The figure of 6 years mentioned later is a mistake and the correct figure is 4 years. The explanation involves calculating the time it takes for Bob to catch up in the inertial Bob frame and taking into account the time dilation factor.
  • #1
mrausum
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Resolving the "Twin Paradox"

http://mentock.home.mindspring.com/twins.htm

I've been trying to follow this unusual explanation to resolve twin paradox, which uses the lorentz relativistic velocity transformation equation to get the speed bob zooms off after Ann at,15/17 C. I can understand up to that point of the explanation and get the same value, however why does it take him 4 years to catch up with Ann at this speed? It's not explained. Maybe it's obvious? Help!

(please no explanations involving the relativistic dopler shift equation).
 
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  • #2


Nevermind, i resolved the problem :D. For anyone interested, I worked out the time it took for bob to catch up in the inertial bob frame, then divided this by the new gamma between bobs returning frame and bobs inertial frame (from v = 15/17 C), to get 4 years.
 
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  • #3


mrausum said:
Nevermind, i resolved the problem :D. For anyone interested, I worked out the time it took for bob to catch up in the inertial bob frame, then divided this by the new gamma between bobs returning frame and bobs inertial frame (from v = 15/17 C), to get 6 years.
What is the figure of 6 years supposed to represent? In your original post you were asking how they derived the figure of 4 years for Bob to catch up with Ann again, in the frame where Ann is moving at 3/5c and Bob is moving at 15/17c after Bob's acceleration.

Anyway, here's one way of deriving the 4 year figure. Before Bob accelerates, in this frame Bob is at rest while Ann is moving away at 3/5c, and this lasts for 4 years, so when Bob accelerates the distance between Ann and himself will be (3/5c)*(4 years) = 12/5 light years. Then after he accelerates, he'll be moving towards her at 15/17c while she continues to move away at 3/5c, so the "closing velocity" between them in this frame (the rate at which the distance between them is shrinking in this frame, which is different from the velocity of Ann in Bob's rest frame) will be 15/17c - 3/5c = 75/85c - 51/85c = 24/85c. So, if the distance between them is initially 12/5 light years at the moment Bob accelerates, the time in this frame for Bob to catch up will be (12/5 ly)/(24/85c) = 17/2 years. This is a lot longer than 4 years! But note that Bob's clock is running slow during the return journey, by a time dilation factor of sqrt(1 - (15/17)^2) = sqrt(289/289 - 225/289) = sqrt(64/289) = 8/17. So, during the 17/2 years it takes in this frame for Bob to catch up with Ann after accelerating, Bob's clock only ticks forward by (8/17)*(17/2) = 8/2 = 4 years.
 
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  • #4


JesseM said:
What is the figure of 6 years supposed to represent? In your original post you were asking how they derived the figure of 4 years for Bob to catch up with Ann again, in the frame where Ann is moving at 3/5c and Bob is moving at 15/17c after Bob's acceleration.

Anyway, here's one way of deriving the 4 year figure. Before Bob accelerates, in this frame Bob is at rest while Ann is moving away at 3/5c, and this lasts for 4 years, so when Bob accelerates the distance between Ann and himself will be (3/5c)*(4 years) = 12/5 light years. Then after he accelerates, he'll be moving towards her at 15/17c while she continues to move away at 3/5c, so the "closing velocity" between them in this frame (the rate at which the distance between them is shrinking in this frame, which is different from the velocity of Ann in Bob's rest frame) will be 15/17c - 3/5c = 75/85c - 51/85c = 24/85c. So, if the distance between them is initially 12/5 light years at the moment Bob accelerates, the time in this frame for Bob to catch up will be (12/5 ly)/(24/85c) = 17/2 years. This is a lot longer than 4 years! But note that Bob's clock is running slow during the return journey, by a time dilation factor of sqrt(1 - (15/17)^2) = sqrt(289/289 - 225/289) = sqrt(64/289) = 8/17. So, during the 17/2 years it takes in this frame for Bob to catch up with Ann after accelerating, Bob's clock only ticks forward by (8/17)*(17/2) = 8/2 = 4 years.

Sorry, i meant 4 years for this example. I'm writing this solution to the paradox as a short first year essay/report and so I'm using different values.

Thanks for the explanation, that's exactly what i did, except you explained it a lot clearer. Good to know I'm using the right method.
 
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1. What is the Twin Paradox?

The Twin Paradox is a thought experiment in special relativity that explores the concept of time dilation and how it affects two individuals who are moving at different speeds. It involves one twin traveling at high speed through space and returning to Earth, while the other twin remains on Earth. When the traveling twin returns, they will have aged less than the twin who stayed on Earth, leading to a paradox of one twin being younger than the other.

2. How is the Twin Paradox resolved?

The Twin Paradox is resolved by understanding that the traveling twin experiences acceleration and deceleration, which creates a difference in their frame of reference compared to the twin on Earth. This difference in frames of reference results in different perceptions of time and explains why the traveling twin experiences less time than the twin on Earth.

3. Is the Twin Paradox a real phenomenon?

While the Twin Paradox is a thought experiment, it is based on the principles of special relativity, which have been proven to be accurate through numerous experiments and observations. Therefore, while the scenario may seem hypothetical, the principles behind it are real and have been confirmed by scientific evidence.

4. Can the Twin Paradox be tested in real life?

Yes, the Twin Paradox can be tested in real life by using highly accurate atomic clocks. For example, in 1971, physicist Joseph Hafele and astronomer Richard Keating conducted an experiment where they flew atomic clocks around the world in opposite directions on commercial flights. When the clocks were compared to stationary clocks, they showed a slight difference in time, confirming the effects of time dilation predicted by the Twin Paradox.

5. What implications does the Twin Paradox have for space travel?

The Twin Paradox highlights the effects of time dilation on individuals traveling at high speeds. This has important implications for space travel, as astronauts who travel at high speeds for extended periods of time will experience a slower passage of time compared to those on Earth. This means that when they return to Earth, they will have aged less than their counterparts, which could have significant consequences for future space missions and intergalactic travel.

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