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I think one catch in Twin Paradox was about the big acceleration that can turn back the traveling twin from light speed outward bound, to become light speed inward bound.

What if there is strictly no acceleration?

  1. Peter is on a space ship, traveling 99% of light speed. He is exactly 20 years old.
  2. Michael is on Earth (or a planet similar to Earth, but with a radius so small that any centripetal acceleration is negligible... or consider him standing just on a piece of concrete in space with an oxygen supply)
  3. Michael is also exactly 20 years old.
  4. According to time dilation, Peter's clock in the spaceship is slower than Michael's clock.
  5. According to time dilation, Michael's clock on Earth is slower than Peter's clock. (since motion is relative, if we consider Peter to be stationary, and Michael is traveling)
  6. Peter's spaceship is traveling towards Michael.
  7. After 30 years on Earth, Peter's spaceship went past Michael's face, so Peter and Michael are 1 cm apart, face to face and eye to eye.
  8. Now, would Peter see Michael quite older than him, and also, Michael sees Peter quite older than him?
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You say that both twins are "exactly 20 years old." I assume you mean that they are both 20 years old at the same time. But part of the point of special relativity is that a phrase like "at the same time" means different things in different reference frames.

To be specific, suppose that these two moments (Peter's birthday party and Michael's birthday party) are simultaneous in a reference frame in which the Earth is at rest. Then, performing the entire analysis in that same frame, we would say the following:

  • Peter's clock ticks slower.
  • Therefore, at the moment the two pass each other, Peter is younger than Michael.

Now let's look at things as measured in Peter's reference frame. In his frame, Michael's clock ticks slower. Therefore, from the time of his 20th birthday until the time the two of them meet, the amount of time as measured by Michael's clock is less than the amount of time as measured by Peter's clock. If we then concluded that Michael would be younger than Peter, we would indeed have a paradox. But that conclusion doesn't follow, because, in this reference frame, the two of them didn't start out the same age. To be specific, the event "Michael's 20th birthday" and the event "Peter's 20th birthday" were not simultaneous. Michael's birthday happened earlier. So in this reference frame, Michael started out older, and even though his clock ticked slower, he was still older when the two met.

All of that is the way things play out if the two birthdays were simultaneous in Michael's reference frame. If on the other hand the two events were simultaneous in Peter's reference frame, then you can just switch the names "Michael" and "Peter" throughout, and everything will work the same way.

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What if Michael and Peter synchronize their clocks at the moment Peter passes by. Then Peter goes on traveling. There would be no problem if they were never to meet up again, that's why I'll add this assumption (which really makes the question academic): assume that the universe has no boundary and that by traveling in one direction, you can go back to your starting point. (Kind of like in Asteroids) Whill they agree on the anniversary date of their meeting? – Raskolnikov May 1 '11 at 19:58
That's a good twist on the twin paradox! The answer is that such a universe does have a preferred "rest frame", and the one who's sitting still in that frame ages fastest. Why doesn't this violate special relativity? I guess the best way to think about it is that, in any spacetime other than true Minkowski spacetime, the large-scale properties are the province of general relativity. In GR, the laws of physics are Lorentz-invariant, but particular solutions to those laws need not be. – Ted Bunn May 1 '11 at 21:46
Thanks Ted, I suspected the answer would be one in GR. Maybe I should make this a question of its own so that you can elaborate on that? – Raskolnikov May 2 '11 at 8:55
@Ted Bunn : Einstein relativity forbids a preferred "rest frame" as you said ? or ... do not need it, under the terms choosen to do the analisys ? – Helder Velez May 3 '11 at 17:36
Oh, now I understand. Special relativity says that the laws of physics are the same in all inertial reference frames. You are perfectly free to pick one frame and call it "absolute," and sometimes that's a salutary thing to do, but it's you choosing it, not the physics. In the multiply-connected universe Raskolnikov asks about, there is a frame that's actually physically different from the others. That's what special relativity says shouldn't happen. – Ted Bunn May 4 '11 at 13:02

This is not an answer, just a re-phrasing of your question that eliminates potential misunderstandings.

One objection to this problem: if Peter and Michael are twins, they would've been born in roughly the same inertial frame (even if the mother accelerated rapidly between the two births, the not-yet-born twin would still age more slowly).

In order for them to have a relative speed of .99c, one or both must've accelerated at some point in their lives.

To work around this, suppose that, shortly after birth, both were accelerated in opposite directions (at exactly the same rate of acceleration) for a long time, and the decelerated (at the exact same rate) until they were at rest with respect to each other.

Since they both accelerated the same amount, they are now the same age (say 20) is the same inertial reference frame.

Then, they accelerate towards each other until their relative speed is .99c, at which point they stop accelerating and are both in an inertial (non-accelerating) reference frame. I realize this isn't exactly the question you posed, but hope my answer will answer your question as well.

At .99c, they will pass each other in about 30.3 years. The time dilation factor for .99c is about 1/7, so, when they pass, Peter will see Michael at age 20+30.3/7, about 24.3, and Peter will see himself (Peter) at age 20+30.3 or 50.3.

By symmetry, the same goes for Michael: when they pass, each twin will see the other as 50.3-24.3 or about 26 years younger.

Now, suppose they both accelerate equally into the same reference frame. Since they accelerate equally, they should age equally (say by y years) to each other, meaning both twins are now both 24.3+y and 50.3+y years at the same time.

How can this be?

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