# Twin Paradox - Acceleration vs Velocity

You know how it goes. You have 2 twins, A and B, who are magically the same age exactly. They start at the same point in space, each in their own spaceship. They then accelerate at the same rate until reaching the same constant velocity, somewhere close to the speed of light. After some time, A decelerates and stops. B continues at the same near-light velocity for some time longer, then decelerates in exactly the same manner as A and stops.

Both twins have undergone the same acceleration and deceleration. The only difference was that B stayed at a high velocity for longer. Is one twin older than the other?

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It depends on which twin's journey through space-time has the shorter proper length. And that's all it depends on according to SR.

Ich
Both twins have undergone the same acceleration and deceleration. The only difference was that B stayed at a high velocity for longer. Is one twin older than the other?
Yes, the one who decelerated earlier.
The twin paradox is calculated in the following way:
1. Choose an arbitrary inertial frame - and stick to it.
2. For each flight segment of duration dt (in that frame), calculate the proper time $d\tau=dt\sqrt{1-v^2/c^2}$.
This works in every inertial system, giving always the same result. Acceleration is not in the equation, and there is no paradox.
Sometimes people mistakenly assume that there is some "principle of relativity" that allows them to pick a different inertial system for each flight segment in their calculation, so as to make two segments with different velocities equivalent to a larger one with constant velocity. There isn't.
To drive home the point that the segments are not equivalent, people sometimes argue that there is acceleration in one case, and no acceleration in the other. While this is true(at least if there is no gravitation), it doesn't mean that acceleration does somehow "cause" time dilatation. It just means that there is a difference. You could see the difference as easily in a spacetime diagram, without referring to acceleration.

Fredrik
Staff Emeritus
Gold Member
Is one twin older than the other?
Yes. My standard answer is "Check out #3 and #142 (page 9) in this thread". Post #142 is just what Mentz and Ich are talking about. Post #3 contains a spacetime diagram that you should check out.

See also kev's argument and DrGreg's spacetime diagram here about why it doesn't make sense to say that it's the acceleration that causes the age difference.

It depends on which twin's journey through space-time has the shorter proper [strike]length[/strike] time. And that's all it depends on according to SR.
FYP

Thanks guys. "Acceleration is not in the equation," is pretty much what I wanted to hear.

mentz114 said:
It depends on which twin's journey through space-time has the shorter proper (length) time. And that's all it depends on according to SR.
Nitpicking proper length = c* proper time, so strictly it doesn't make any difference which we compare.

Fredrik
Staff Emeritus
Gold Member
Actually "proper length" (integral of $$\sqrt{-dt^2+dx^2}$$ along the curve) is only defined for spacelike curves (curves such that the thing under the square root is always positive), and "proper time" (integral of $$\sqrt{dt^2-dx^2}$$ along the curve) is only defined for timelike curves (curves such that the thing under this square root is always positive). So it does matter. These concepts are properties of two different classes of curves, and the world line of a massive "particle" (in this case a rocket containing an astronaut) is always timelike.

Demystifier