Relativity Conundrum

[aggle-rithm]

I was going over Special Relativity in my head the other night, and just as it was starting to make sense to me, I came across a mental conundrum that has me stumped. I was thinking about the twins paradox, where one twin goes out into space, traveling at relativistic speeds for a while, then comes back, while the other stays on Earth. When the first twin returns, he is considerably younger than the second twin. That part is easy.

However, I considered the case where the first twin, on his return to Earth, decided to "buzz" the Earth before landing, so his brother could see him as he went by. To the earth-bound twin, the one in the spaceship is both space- and time-dilated. The ship appears shortened, and the twin in the ship appears to be moving very slowly, as time is passing more slowly for the itinerate twin since he is moving close to the speed of light.

However, from the inertial frame of the twin in the spaceship, he is standing still, and the Earth is zipping by at relativistic speeds. Therefore, the twin on the Earth, from his vantage point, should also appear to be moving slowly, so that when the first twin lands, it is the second twin that is younger, not the first.

What am I missing here?

 

[Ziggurat]

What am I missing here?

First off, always keep in mind that there's a difference between what you "see" and what you "observe" in relativity. That distinction is important, because if you don't recognize it, things can get VERY confusing.

When you look at a moving clock, the rate at which you SEE the clock ticking is dependent not only on speed but also on direction, because there's a doppler shift (light emitted later will take less time to reach you if the clock is moving towards you or away from you). The doppler shift is a classical phenomenon, but it matters critically when asking what you "see". What you "observe" is what you would calculate about that clock after accounting for things like how far away it was when you saw it and how long the light took to reach you - and the rate at whcih you OBSERVE the clock ticking does not depend on direction.

This matters for your question, because you asked what the twins see, not what they observe. Well, when the twins are heading towards each other, they will see each others clocks run FAST, not slow, because of doppler shift. For the earthbound twin, the change from his twin's clock being seen to run slow as he travels away to being seen to run fast as he travels back happens MORE than halfway through the journey (since it takes time for light from the travelling twin's turnaround point to get back to earth). So for more than half his time, the earthbound twin sees the traveling twin's clock run slow, and for less than half the time it runs fast, and the total ends up being less than his clock.

For the traveling twin, however, he sees the earthbound twin's clock starts running fast as soon as he turns around, which IS halfway through his journey. So he sees the earthbound twin's clock run slow half the time, and fast half the time, and the total is more than his clock.

If you want to know what they OBSERVE, however, the question is a little different. Both twins "observe" each other's clocks running slow. However, observations are based on their equal-time axes (since he has to make calculations to figure out what he observes based on what he sees). As soon as the traveling twin switches directions, his equal-time axis tilts, so where it intersects the earth-bound twin's timeline suddenly changes. So he "observes" a sudden passage of a large amount of time for the earth-bound twin due to his change of reference frames, and that's where the "extra" time for the earthbound twin comes from. Note that there is no discontinuity in what he sees, ONLY in what he observes, when he changes direction.

 

[aggle-rithm]

If you want to know what they OBSERVE, however, the question is a little different. Both twins "observe" each other's clocks running slow. However, observations are based on their equal-time axes (since he has to make calculations to figure out what he observes based on what he sees). As soon as the traveling twin switches directions, his equal-time axis tilts, so where it intersects the earth-bound twin's timeline suddenly changes. So he "observes" a sudden passage of a large amount of time for the earth-bound twin due to his change of reference frames, and that's where the "extra" time for the earthbound twin comes from. Note that there is no discontinuity in what he sees, ONLY in what he observes, when he changes direction.

Thanks! I had a feeling it had something to do with the change in direction. And I did mean "observe" when I said "see", since I know that it would be pretty difficult to "see" your twin in a ship hundreds of miles away hurtling by at relativistic speeds.

 

[Tez]

First off, always keep in mind that there's a difference between what you "see" and what you "observe" in relativity. That distinction is important, because if you don't recognize it, things can get VERY confusing.
[snip]

Now this was a very nice and clear answer

 

[epepke]

I was going over Special Relativity in my head the other night, and just as it was starting to make sense to me, I came across a mental conundrum that has me stumped. I was thinking about the twins paradox, where one twin goes out into space, traveling at relativistic speeds for a while, then comes back, while the other stays on Earth. When the first twin returns, he is considerably younger than the second twin. That part is easy.

However, I considered the case where the first twin, on his return to Earth, decided to "buzz" the Earth before landing, so his brother could see him as he went by. To the earth-bound twin, the one in the spaceship is both space- and time-dilated. The ship appears shortened, and the twin in the ship appears to be moving very slowly, as time is passing more slowly for the itinerate twin since he is moving close to the speed of light.

However, from the inertial frame of the twin in the spaceship, he is standing still, and the Earth is zipping by at relativistic speeds. Therefore, the twin on the Earth, from his vantage point, should also appear to be moving slowly, so that when the first twin lands, it is the second twin that is younger, not the first.

What am I missing here?

Yes, each twin sees the other as aging more slowly during the fly-by, but the Earth twin still sees the flying twin as younger, and the flying twin sees the Earth twin as older, due to what happened through the entire trip.