Time dilation and twins paradox

[Volcano]

"The confusion that arises in problems like Example 26.1 lies in the fact that movement
is relative: from the point of view of someone in the pendulum’s rest frame,
the pendulum is standing still (except, of course, for the swinging motion),
whereas to someone in a frame that is moving with respect to the pendulum, it’s
the pendulum that’s doing the moving. To keep this straight, always focus on the
observer who is doing the measurement, and ask yourself whether the clock being
measured is moving with respect to that observer. If the answer is no, then the observer
is in the rest frame of the clock and measures the clock’s proper time. If the
answer is yes, then the time measured by the observer will be dilated—larger than
the clock’s proper time.
This confusion of perspectives led to the famous “twin paradox.” "

Above is quoted from Serway physics. Like the last sentence above in quote I confused. Except last sentence, rest of explanation and problem solving help is not consider "twins paradox" results. Is not it?

 

[bcrowell]

Serway is a terrible book. Try this: http://en.wikipedia.org/wiki/Twin_paradox

 

[Volcano]

So you agree that the quoted problem solving tip is not true. Reaching to true knowledge is really hard.

 

[bcrowell]

So you agree that the quoted problem solving tip is not true. Reaching to true knowledge is really hard.

I didn't say that. I don't think it's false. It's just not very helpful.

 

[harrylin]

"The confusion that arises in problems like Example 26.1 lies in the fact that movement
is relative: from the point of view of someone in the pendulum’s rest frame,
the pendulum is standing still (except, of course, for the swinging motion),
whereas to someone in a frame that is moving with respect to the pendulum, it’s
the pendulum that’s doing the moving. To keep this straight, always focus on the
observer who is doing the measurement, and ask yourself whether the clock being
measured is moving with respect to that observer. If the answer is no, then the observer
is in the rest frame of the clock and measures the clock’s proper time. If the
answer is yes, then the time measured by the observer will be dilated—larger than
the clock’s proper time.
This confusion of perspectives led to the famous “twin paradox.” "

Above is quoted from Serway physics. Like the last sentence above in quote I confused. Except last sentence, rest of explanation and problem solving help is not consider "twins paradox" results. Is not it?

You are right, time dilation by itself is not the "twin paradox".

Moreover the explanation that "the time measured [...] will be dilated—larger" is ambiguous to say the least: with "dilated time" the writer seems to mean that the time period - the duration of a tick - is larger. But as a result the measured time - the number of ticks - is less.

On top of that I know of no historical evidence that supports the claim that a confusion of perspectives led to the twin paradox.

Note that I don't know that book, I can only comment on what you present.

Regards,
Harald

 

[Volcano]

Assume a pendulum on Earth swinging and an observer in moving spaceship measuring the period. In this case the pendulum is moving with respect to observer. Thus the observer(in spaceship) will measure the period shorter. This is what I understand from quoted part.

Perhaps I misunderstood. Let me ask two question.

1. Pendulum on Earth(period T1), for observer in spaceship(period T2). Which period is shorter(T1, T2)?
2. Pendulum in spaceship (period T3), for observer on Earth(period T4). Which period is shorter(T3, T4)?

 

[harrylin]

Assume a pendulum on Earth swinging and an observer in moving spaceship measuring the period. In this case the pendulum is moving with respect to observer. Thus the observer(in spaceship) will measure the period shorter. This is what I understand from quoted part.

Perhaps I misunderstood. Let me ask two question.

1. Pendulum on Earth(period T1), for observer in spaceship(period T2). Which period is shorter(T1, T2)?
2. Pendulum in spaceship (period T3), for observer on Earth(period T4). Which period is shorter(T3, T4)?

The quotation is ambiguous and evidently leads to misunderstanding, exactly as I feared.
A clearer term for "time dilation" is "clock retardation": "moving" clocks appear to run slow, so that the time between ticks appears to be longer.
1. T1>T2
2. T3>T4

 

[Volcano]

Thank you harrylin. Especialy your last message was helpful. Now I want to understand the relation between time dilation and twin paradox (difference or what it is). Above in question #2, T3>T4. Can we say someone in spaceship grow old T3 and the other on Earth grow old T4? If this is true, doesn't contradict with explanation of twin paradox?

 

[harrylin]

Thank you harrylin. Especialy your last message was helpful. Now I want to understand the relation between time dilation and twin paradox (difference or what it is). Above in question #2, T3>T4. Can we say someone in spaceship grow old T3 and the other on Earth grow old T4? If this is true, doesn't contradict with explanation of twin paradox?

That relates to your: "Pendulum in spaceship (period T3), for observer on Earth(period T4)."

In fact a pendulum doesn't work in a spaceship, so let's make it a mechanical watch with for example one tick per second.

Thus the time between each tick of the watch of the spaceship has increased according to the observer on Earth. As a result that moving watch will retard compared with the ones on Earth. That's correct.
It's easy to confuse between clock period T' (the duration between two ticks) and accumulated clock time t' (the total number of ticks).

 

[BruceW]

Here's a brief explanation of the 'twin paradox':
One twin (Jack) stays on earth, and the other (Jill) flies off in a spaceship at close to the speed of light, then returns back to earth. Imagine they both have watches that they synchronised before Jill set off.
The apparent paradox is that the twin on Earth (Jack) expects that Jill's watch will have ticked more than his watch when they meet up. This is because he thinks his watch represents the proper time between the twin's meetings, since he doesn't move.
But then Jill thinks that her watch also represents the proper time between their two meetings because the two meetings happen in the same place from her perspective (since she returns to the same place).
The answer to this paradox is that Jill must go through an acceleration in her spaceship, and therefore the laws of special relativity don't apply to this problem. You must use general relativity to explain it.
It is in fact Jill who grows older than Jack because Jill went through the acceleration.

 

[Volcano]

harrylin thank you very much. I think to my disappointment, the thing you point out about tick count and duration between two tick difference. Now something more clear and by the way need to think this a bit too. Regards

 

[harrylin]

harrylin thank you very much. I think to my disappointment, the thing you point out about tick count and duration between two tick difference. Now something more clear and by the way need to think this a bit too. Regards

You're welcome
And to be honest: the reason that I thought it may be useful to point that out, is that in the past I confused those two inversely related things myself - several times even.

 

[Volcano]

... in the past I confused those two inversely related things myself - several times even.

:) I believe with all my heart. Honestly I couldn't notice this. Big thanks again.

 

[Mentz114]

... and therefore the laws of special relativity don't apply to this problem. You must use general relativity to explain it.

Bruce, special relativity is sufficient to explain why different elapsed time are observed. It is just the proper length of the worldline between events on the worldline. It does not matter if the WLs curve ( ie accelerate) the proper length can still be found.

 

[BruceW]

Mentz, wow i never knew that.
We were taught that the proper time between two events is simply the time between two events when measured in a reference frame where the events happen in the same place.
But now I know that the proper time between two events is the arc length in 4-d spacetime of a particular worldline between two events.
Or in other words, for an accelerating particle, you sum up the proper times over each infinitesimally small part of the worldline to get the total proper time between two points on the worldline of an accelerating particle.
So does this mean that the proper time of an event at position x and time t, would equal:
$$\tau = \sqrt{ t^2 + \frac{x^2}{c^2} }$$
(which is a lorentz invariant quantity, which is why it is called 'proper')
And this would mean that the proper time between two events connected by a straight worldline would be equal to the time between those two events when measured in a reference frame in which the two events happen in the same place.
And then to calculate the proper time between two events connected by a curved worldline, you simply integrate the equation for proper time over whatever worldline you choose.
So basically, the proper time can be calculated in special relativity even when accelerations are involved!
This is huge. A whole new side to special relativity that I never even knew about...

 

[Mentz114]

Mentz, wow i never knew that.
We were taught that the proper time between two events is simply the time between two events when measured in a reference frame where the events happen in the same place.
But now I know that the proper time between two events is the arc length in 4-d spacetime of a particular worldline between two events.
Or in other words, for an accelerating particle, you sum up the proper times over each infinitesimally small part of the worldline to get the total proper time between two points on the worldline of an accelerating particle.
So does this mean that the proper time of an event at position x and time t, would equal:
$$\tau = \sqrt{ t^2 + \frac{x^2}{c^2} }$$
(which is a lorentz invariant quantity, which is why it is called 'proper')
And this would mean that the proper time between two events connected by a straight worldline would be equal to the time between those two events when measured in a reference frame in which the two events happen in the same place.
And then to calculate the proper time between two events connected by a curved worldline, you simply integrate the equation for proper time over whatever worldline you choose.
So basically, the proper time can be calculated in special relativity even when accelerations are involved!
This is huge. A whole new side to special relativity that I never even knew about...

Yes. Except the proper interval is $c\ d\tau\ =\ \sqrt{c^2dt^2-dx^2}$. It's the geometric invariant of Minkowski spacetime, so all observers agree on it.

 

[BruceW]

Oh, I get it now, for a particular event,
$$\sqrt{c^2 t^2 - x^2}$$
Is the lorentz invariant, therefore for the proper time between two events,
$$c\ d\tau\ =\ \sqrt{c^2dt^2-dx^2}$$
is integrated over a particular path to give a sum of invariants. (which would also be invariant).
So what all this tells us is that the shortest proper time between two events is a straight line in 4-d spacetime.
So does light follow the path of shortest proper time? And in general relativity, when a different metric is introduced, the shortest proper time is no longer a straight line in 4-d spacetime, but is warped around mass?

 

[Mentz114]

Oh, I get it now, for a particular event,
$$\sqrt{c^2 t^2 - x^2}$$
Is the lorentz invariant, therefore for the proper time between two events,
$$c\ d\tau\ =\ \sqrt{c^2dt^2-dx^2}$$
is integrated over a particular path to give a sum of invariants. (which would also be invariant).
So what all this tells us is that the shortest proper time between two events is a straight line in 4-d spacetime.
So does light follow the path of shortest proper time? And in general relativity, when a different metric is introduced, the shortest proper time is no longer a straight line in 4-d spacetime, but is warped around mass?

Have you seen this article http://en.wikipedia.org/wiki/Proper_time. It's worth a look.

Actually, geodesics have the longest proper interval ( which is why the traveling twin ages less compared with the stay-at-home twin).

Light travels along null geodesics - they have zero length.

Yes, gravity changes the metric so geodesics become curved. They are still the 'straightest' lines in curved spacetime, although differently straight .

 

[BruceW]

yeah, I looked at that wikipedia page after your first post, to see what the hell you were talking about, since I had only heard of the proper time in the context of straight worldlines.
My questions were my reaction to what I read on that page. P.S. you should have charged me for teaching me about curved worldlines in relativity ;)

 

[BruceW]

One last question: wouldn't the traveling twin age more?
Because the worldline of the traveling twin is longer, so the traveling twin would experience more time between the twin's meetings?

 

[PAllen]

One last question: wouldn't the traveling twin age more?
Because the worldline of the traveling twin is longer, so the traveling twin would experience more time between the twin's meetings?

No, that would be true if the metric were had all + signs. With:

ds^2 = c dt^2 - dx^2 - dy^2 - dz^2

the proper time is shorter for the object deviating from inertial motion. In special relativity, 'straight' world lines are paths of 'longest' time rather than the Euclidean situation where straight lines are shortest distance.

 

[BruceW]

Oh yeah, I forgot the minus sign. Sorry, guys!