Another impure twin paradox

[kahoomann]

Suppose one twin travel to a distance L and turn around, another twin travel to a distance L/2 and turn around. When they reunite at home, the twin travel longer will age less?
Since they both experience acceleration, so acceleration is not the cause of age difference.
Can acceleration break the symmetry in this case? If not, what breaks the symmetry?

 

[lbrits]

The amount they age is determined by the proper length of their individual world-lines. The longer time you spend at higher velocity, the shorter your world-line is. The first twin spends more time at high speed relative to the two common points in their world-lines than the second twin does, and so ages less.

 

[Dale]

what breaks the symmetry?

one twin travel to a distance L and turn around, another twin travel to a distance L/2 and turn around.

what symmetry? Is there any inertial reference frame where their worldlines are symmetric?

 

[kahoomann]

what symmetry? Is there any inertial reference frame where their worldlines are symmetric?

The symmetry is this. If twin A measure twin B clock to run slowly, simply because B has a speed v relative to A, then the principle of relativity implies that B will also measure A's clock to run slowly, since the speed of A relative to B is also v. The question is what breaks this symmetry?

 

[Hurkyl]

The symmetry is if twin A measure twin B clock to run slowly, simply because B has a speed v relative to A, then the principle of relativity implies that B will also measure A's clock to run slowly, since the speed of A relative to B is also v

And it has the same resolution as in the original twin paradox -- the only difference is that in this case, you've had both twins make an invalid argument.

(In the usual twin paradox, this argument is actually correct for one of them)

 

[kahoomann]

And it has the same resolution as in the original twin paradox -- the only difference is that in this case, you've had both twins make an invalid argument.

(In the usual twin paradox, this argument is actually correct for one of them)

Why both twins make an invalid argument?
Do you mean that principle of relativity is invalid in this case?

 

[Hurkyl]

Why both twins make an invalid argument?
Do you mean that principle of relativity is invalid in this case?

You aren't invoking the 'principle of relativity'. You are trying to invoke the time dilation formula in some coordinate chart (that you haven't explicitly defined), and ignored the fact that your coordinate chart isn't inertial, so time dilation isn't the only relevant effect. i.e. you've made exactly the same mistake as the one in the usual twin paradox.

 

[kahoomann]

You aren't invoking the 'principle of relativity'. You are trying to invoke the time dilation formula in some coordinate chart (that you haven't explicitly defined), and ignored the fact that your coordinate chart isn't inertial, so time dilation isn't the only relevant effect. i.e. you've made exactly the same mistake as the one in the usual twin paradox.

>>So time dilation isn't the only relevant effect.
What else is the relevant effect?

 

[Hurkyl]

>>So time dilation isn't the only relevant effect.
What else is the relevant effect?

Familiar examples of this phenomenon should be the Coriolis force and centrifugal force. In special relativity, there are temporal analogues. For example, in the most 'natural' way to attach a comoving coordinate chart to an accelerating observer, you have the effect that when he accelerates towards a clock, it is observed to run faster. (The increase is proportional to coordinate distance, and probably also to the magnitude of the acceleration) And similarly, when accelerating away from a clock, it is observed to run slower, or even backwards.

Since you haven't actually specified which coordinate chart you're using, I couldn't say what actual effects would be seen.

 

[kahoomann]

Familiar examples of this phenomenon should be the Coriolis force and centrifugal force. In special relativity, there are temporal analogues. For example, in the most 'natural' way to attach a comoving coordinate chart to an accelerating observer, you have the effect that when he accelerates towards a clock, it is observed to run faster. (The increase is proportional to coordinate distance, and probably also to the magnitude of the acceleration) And similarly, when accelerating away from a clock, it is observed to run slower, or even backwards.

Since you haven't actually specified which coordinate chart you're using, I couldn't say what actual effects would be seen.

I don't think "accelerating observer" is the problem. since they both experience acceleration during flight, they can stop their clocks before engine ignition and restart their clocks after cutoff. So the clocks would show the total proper times they were in inertial motion.
So it must be something else that cause the age difference.

 

[Janus]

I don't think "accelerating observer" is the problem. since they both experience acceleration during flight, they can stop their clocks before engine ignition and restart their clocks after cutoff. So the clocks would show the total proper times they were in inertial motion.
So it must be something else that cause the age difference.

You didn't read Hurkyl's post closely enough. The acceleration doesn't effect how "their" clock runs, but instead what they determine is happening to the other twin's clock. Stopping their own clock during their own acceleration has no effect on their observation of what is happening to the other's clock. And remember, as Hurkyl pointed out in his post, this observation depends on the distance between the twins during the acceleration. The asymmetry comes in when one twin travels a shorter distance before turning around, thus he will measure a different distance between himself and his twin at this point than his twin will when he turns around.

 

[kahoomann]

Familiar examples of this phenomenon should be the Coriolis force and centrifugal force. In special relativity, there are temporal analogues. For example, in the most 'natural' way to attach a comoving coordinate chart to an accelerating observer, you have the effect that when he accelerates towards a clock, it is observed to run faster. (The increase is proportional to coordinate distance, and probably also to the magnitude of the acceleration) And similarly, when accelerating away from a clock, it is observed to run slower, or even backwards.

Since you haven't actually specified which coordinate chart you're using, I couldn't say what actual effects would be seen.

As DaleSpam indicated in another thread that acceleration does not cause time dilation by itself.

My recommendation is to forget the acceleration. It only serves to break the symmetry and does not cause time dilation by itself. Instead, use the spacetime interval which allows you to extend the analysis to arbitrarily accelerating twins.
.

 

[kahoomann]

You didn't read Hurkyl's post closely enough. The acceleration doesn't effect how "their" clock runs, but instead what they determine is happening to the other twin's clock. Stopping their own clock during their own acceleration has no effect on their observation of what is happening to the other's clock. And remember, as Hurkyl pointed out in his post, this observation depends on the distance between the twins during the acceleration. The asymmetry comes in when one twin travels a shorter distance before turning around, thus he will measure a different distance between himself and his twin at this point than his twin will when he turns around.

This is just gravitational time dilation. Actually, it is Not the g-force/acceleration that cause time dilation, it is the gravitational potential that causes time dilation