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Relativity: Twin Paradox
By Luqman Bajwa on Wednesday, September 19, 2001 - 10:46 pm:

Hi, can someone please shed some light (no pun intended) on how the twin paradox works. I’m going to assume everyone has heard of the twin paradox so I won’t explain it again. I've seen two different explanations on the internet but neither one gives me a satisfactory answer, for the following argument:

Consider three reference bodies, A (Anne), B (Bob) and M. Anne is travelling relative to M at a velocity of say 3/5 of the speed of light. Similarly Bob is travelling relative to M at a velocity of 3/5 c in the opposite direction Anne is moving in. After 4 years (relative to A and B) both Anne and Bob change their direction and speed to towards M at 3/5 c relative to M. When both Anne and Bob reach M after 8 years (relative to A and B), whether they experience any time dilation relative to M or not, due to the symmetry there should be no time dilation between Anne and Bob – ie Anne and Bob are of the same age.

Now lets consider the same situation from Anne´s reference frame (who is on earth). Relative to Anne Bob is travelling in his spaceship at some velocity between 3/5 c and c, and not at 6/5 c as relativity does not add up velocities this way (I think it adds up to 15/17 c but I´s not sure). After four years (relative to Bob) Bob reverses his direction and speed towards Anne. When Bob reaches Anne after another four years (relative to Bob) the results given for the twin paradox states that there is a time dilation between Anne and Bob, and Anne has actually aged more than her twin brother in this time period. However this would be in direct contradiction to the example given above.

As far as I can see, Bob is in no way unique from Anne. Both have symmetrical motion relative to M, and yet Anne supposedly ages more than Bob.

Can someone please help me to understand this problem.

Thanks

Luqman

By William Astle on Thursday, September 20, 2001 - 09:29 am:

Hi,

The problem here, I think, is that you have not adequately described when accelerations occur and so you have assumed some symmetry that does not exist.

In the first example it is implicit that M is fixed in a state of constant velocity relative to some inertial frame. It follows there is symmetry in the accelerations of A and B relative to an inertial frame and so, as you say, Anne and Bob age in the same way.

In the second example you have placed Anne on Earth (which I guess has roughly constant velocity on the scale being considered - at any rate that is your assumption). Anne undergoes no change in velocity relative to an inertial frame but in the first example she does, so the two situations are different.


I hope this is correct.

Will.

By Sean Hartnoll on Saturday, September 22, 2001 - 08:50 pm:

The first explanation is clearly correct because the symmetry is manifest. Now, in the second argument, the problem is that you have not treated the accelerations (the change in directions) carefully enough. You have to be very careful doing these calculations. The basic idea is that although from Anne´s point of view it is true that she is at any given instant aging faster than Bob, this is made up in that from Anne´s point of view, Bob ages for less time in total than she does. Strange, right?! The reason this happens is that in the moment of accelation for Anne, the is a jump in what happens to Bob. This is easiest to see if you know about spacetime diagrams. But the moral of the story is always to work in a frame that doesn't accelerate if you can possibly help it, even if it looks like the acceleration is innocuous.

So what I suggest is doing the second calculation again, you need to find the time elapsed before and after the poitn acceleration. But you also need to check where Bob and Anne are before and after the acceleration, and I think you find that there is a jump here.

Post again if still confused.

Sean