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Q & A: Twin Paradox and slow clocks in fast reference frames

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Most recent answer: 10/22/2007
Q:
In the question regarding special relativity, the twin paradox is mentioned. The twin paradox goes as such: When one twin gets in a rocket and blasts off from earth. he speeds off and comes back and since he has been traveling at a high rate of speed, he should be young. however, from the rocket twin’s reference frame, it looks like the earth twin flew off into space with the earth and then came back. both view points are 100% correct. so which one is older? The part which i don’t understand is "since he has been traveling at a high rate of speed, he should be young." Can u explain why travelling ar a high rate of speed will make that twin younger?
- Qihan and Wufan
A:
Hi Qihan and Wufan,

Special relativity predicts that clocks which are traveling quickly will run slower than those which are stationary. This is a difficult prediction to reconcile with our intuition, because one observer will think a second person's clocks are slow if the second person is moving, but according to the second person, the first person is doing the moving, and hence *his* clock should run slow. Puzzling over who's right in this case leads to the twin paradox.

The special theory of relativity starts with the hypotheses that

1) Space is the same everywhere and in all directions

2) The laws of physics are the same everywhere and to all observers, even if they are moving at constant velocities relative to each other.

(If the observers are accelerating, then the rules are broken, and that's what solves the twin paradox).

One set of laws of physics are the laws of electricity and magnetism. Out of what we know from electricity and magnetism, we can predict the speed of light. If the speed of light is something resulting from the laws of physics in one frame, it must be the same in all frames. This is very hard to match with intuition, because we are used to the velocities of objects in one frame add to the relative velocity of the frames to get the velocities of the same objects as seen by another observer with another point of view. But the physics has to be the same regardless of the point of view of the observer.

Imagine if you will a contraption consisting of two mirrors facing each other. They're perfectly shiny, so a pulse of light will bounce as many times as you like from one mirror to the other and back again. The time it takes for the pulse of light to go from one mirror to the other is fixed by the distance between them and the speed of light, and so this contraption functions as a clock.

Let's view this clock from the point of view of an observer traveling perpendicular to the path of the light pulse (if the clock is bouncing light vertically, say the observer is traveling horizontally past it). According to this observer, as the clock flies past, the light pulse bounces off of the bottom mirror at an angle so that it gets to the top mirror just when the top mirror makes it to the same location. The light pulse makes the same angle with the top mirror and heads for a spot where the bottom mirror hasn't gotten to yet, but will get to by the time the light pulse gets there.

According to this observer, the length of the path the light pulse takes is longer than the distance between the mirrors because the light beam travels diagonally instead of straight up and down. But the light pulse has the same speed in both frames, and so it must take longer in this observer's frame than it does to an observer resting on the clock.

 



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One of Einstein's great advances here was that this isn't just a property of this kind of clock, it is a general property of the time coordinate in moving frames.

"To study the properties of time, one must study the properties of clocks."

Of course the twin paradox is resolved because one twin accelerates when he turns around to go home. During this turnaround process, his stay-at-home twin's clock changes rapidly in the moving twin's accelerating frame, and then runs slow again on the return trip, but still ends up with much more time on it than the traveling twin's clock.

Tom

(published on 10/22/2007)

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