Frame-dependency of aging rate in the twin paradox

[Queequeg]

Hi, from what I understand about the twin paradox, Is the resolution essentially that since the traveling twin undergoes acceleration when the ship reverses direction, so since the velocity of the twin is no longer constant, the inertial reference frames are no longer valid? Then does that also means that the twins are aging at the same rate only until the brother undergoes acceleration?

And I was also thinking about spacetime. Since time is a dimension, one brother is lying in a relatively constant distance through time, but the other brother is traveling much farther distance, so time must have slowed for him since he is traveling through two dimensions (space and time) ?

 

[PeterDonis]

Check out the Usenet Physics FAQ article on the twin paradox:

http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_paradox.html

If you still have questions after reading it, by all means come back and ask them, but I think it will address the issues you raise in your post.

 

[chogg]

Hi, from what I understand about the twin paradox, Is the resolution essentially that since the traveling twin undergoes acceleration when the ship reverses direction, so since the velocity of the twin is no longer constant, the inertial reference frames are no longer valid? Then does that also means that the twins are aging at the same rate only until the brother undergoes acceleration?

Yes, the acceleration resolves the paradox.

But be careful! One cannot say that the twins are "aging at the same rate", since "same rate" implies they can share a common definition of time. Since they're moving relative to one another, they can't.

You can pick a frame, and compare the rates, but the comparison will be frame-dependent (and hence not very meaningful). In either twin's rest frame (indeed, in most other frames too), they are aging at different rates.

And I was also thinking about spacetime. Since time is a dimension, one brother is lying in a relatively constant distance through time, but the other brother is traveling much farther distance, so time must have slowed for him since he is traveling through two dimensions (space and time) ?

TBH, I don't quite understand what you're saying here.
Sorry! :-)

 

[ModusPwnd]

TBH, I don't quite understand what you're saying here.
Sorry! :-)

It looks like he is getting at the idea that the space time interval must be constant for all observers. Since one has an increase in changing distance a decrease in time is needed for the space time interval to remain constant for all.

OP, if you haven't heard of this jargon, check it out.

 

[russ_watters]

Then does that also means that the twins are aging at the same rate only until the brother undergoes acceleration?

The brother undergoes an acceleration at the very start of the scenario: that's how they come to be traveling at different speeds!

 

[Queequeg]

It looks like he is getting at the idea that the space time interval must be constant for all observers. Since one has an increase in changing distance a decrease in time is needed for the space time interval to remain constant for all.

OP, if you haven't heard of this jargon, check it out.

Thanks, that's what I mean and I need to learn more about spacetime.

So if that's true, why bother explaining that the traveling brother undergoes acceleration and that there are actually 3 reference frames because his velocity isn't constant? The spacetime continuum explanation seems much more simpler.

 

[PeterDonis]

The spacetime continuum explanation seems much more simpler.

Many people (including me) agree with you. But there are still plenty of people who want an explanation in terms of reference frames and acceleration, which is why the Usenet Physics FAQ article I linked to gives multiple ways of understanding what's going on. (It does say that the spacetime explanation is the most general, since by drawing in appropriate curves on the spacetime diagram you can illustrate how all the other explanations work.)

 

[ghwellsjr]

The spacetime continuum explanation seems much more simpler.

Many people (including me) agree with you. But there are still plenty of people who want an explanation in terms of reference frames and acceleration, which is why the Usenet Physics FAQ article I linked to gives multiple ways of understanding what's going on. (It does say that the spacetime explanation is the most general, since by drawing in appropriate curves on the spacetime diagram you can illustrate how all the other explanations work.)

I'm confused. You say that you prefer the spacetime explanation over "an explanation in terms of reference frames and acceleration" but the link you provided in post #2 shows just that. What's the difference?

 

[Dale]

The spacetime continuum explanation seems much more simpler.

I agree. Also, the spacetime explanation works in General Relativity, where most of the other explanations fail.

 

[PeterDonis]

I'm confused. You say that you prefer the spacetime explanation over "an explanation in terms of reference frames and acceleration" but the link you provided in post #2 shows just that. What's the difference?

The link I provided in post #2 shows both kinds of explanation, the spacetime explanation (they call it the "Spacetime Diagram Analysis") and the explanation in terms of reference frames and acceleration (that's in the Introduction). They also say, as I noted, that the spacetime diagram analysis is the most general; as the FAQ says (on the "Too Many Analyses" page):

Most physicists feel that the Spacetime Diagram Analysis is the most fundamental. It does amount to a sort of "Universal Interlingua", enabling one to see how superficially different analyses are really at heart the same.

This is the point of view I was agreeing with.

 

[Ryoko]

Let me ask for some clarification about what it means to be in the same inertial frame of reference -- Does this mean that two observers are experiencing the same magnitude of acceleration or does it mean accel + velocity + direction?

 

[ghwellsjr]

Let me ask for some clarification about what it means to be in the same inertial frame of reference -- Does this mean that two observers are experiencing the same magnitude of acceleration or does it mean accel + velocity + direction?

Saying two observers are "in the same inertial frame of reference" is very sloppy terminology because everything and everybody is in every inertial frame of reference and there is no preferred frame. But the terminology is quite common and usually means that two observers are at rest according to a single inertial frame of reference, which, of course means, no acceleration, no velocity, and whatever distance in whatever direction they are from each other, it never changes. Next time someone uses that terminology, why don't you ask them what they mean?

 

[ghwellsjr]

I'm confused. You say that you prefer the spacetime explanation over "an explanation in terms of reference frames and acceleration" but the link you provided in post #2 shows just that. What's the difference?

The link I provided in post #2 shows both kinds of explanation, the spacetime explanation (they call it the "Spacetime Diagram Analysis") and the explanation in terms of reference frames and acceleration (that's in the Introduction).

I think when you say "reference frames and acceleration", you mean one reference frame before acceleration and another frame after acceleration, also known as "frame jumping", correct?

But as long as one frame is use throughout, then that is the same as the "Spacetime Diagram Analysis", correct?

They also say, as I noted, that the spacetime diagram analysis is the most general; as the FAQ says (on the "Too Many Analyses" page):

Most physicists feel that the Spacetime Diagram Analysis is the most fundamental. It does amount to a sort of "Universal Interlingua", enabling one to see how superficially different analyses are really at heart the same.

This is the point of view I was agreeing with.

Unfortunately though, the Spacetime Diagram Analysis is limited to in-line scenarios which means that it is no where near universal, wouldn't you say?

 

[PeterDonis]

I think when you say "reference frames and acceleration", you mean one reference frame before acceleration and another frame after acceleration, also known as "frame jumping", correct?

I'm trying to capture the options as described by the OP, for example in post #6.

But as long as one frame is use throughout, then that is the same as the "Spacetime Diagram Analysis", correct?

It depends on how you're using the frame. The key point of the Spacetime Diagram Analysis is that different curves between the same two points in spacetime can have different lengths (proper times), simply as a matter of geometry. Picking a single convenient frame in which to do the analysis makes it easy to compute the proper time along each worldline, and to draw the worldlines in a single spacetime diagram so that the relationships between them are visually evident. But that's not the only thing people use frames for.

Unfortunately though, the Spacetime Diagram Analysis is limited to in-line scenarios which means that it is no where near universal, wouldn't you say?

Why do you think that? You can draw a spacetime diagram of any scenario you like; there's no requirement that the time axis of the diagram has to correspond to any of the worldlines of interest.

 

[Dale]

But as long as one frame is use throughout, then that is the same as the "Spacetime Diagram Analysis", correct?

For me, the key part of the spacetime diagram analysis posted in the FAQ is the use of the spacetime interval. That is the metric in an inertial frame, and that is the key part that makes it generalizable to general relativity. The OP seems to be (correctly) focusing on the spacetime interval part of the explanation, which is commendable and is the point that I was reacting to.

 

[morrobay]

This excellent graph by Dr Greg is clarification on acceleration / velocity and time dilation for two travelers : https://www.physicsforums.com/showthread.php?t=354622&page=2
See Post # 36

 

[ghwellsjr]

This excellent graph by Dr Greg is clarification on acceleration / velocity and time dilation for two travelers : https://www.physicsforums.com/showthread.php?t=354622&page=2
See Post # 36

Here's Dr Greg's graph:

I can see the accelerations (in red) and the velocities (in blue) but where do you see any clarification on time dilation?

 

[morrobay]

With explanation. As Yuiop says in Dr Greg's graph: With identical acceleration events the difference in time dilation is accounted for only by time in relative higher velocities. And in also answering the OP question :
Then does that also mean the twins are aging at the same rate only until the brother undergoes
acceleration ?

 

[ghwellsjr]

Here's that graph again:

With explanation. As Yuiop says in Dr Greg's graph: With identical acceleration events the difference in time dilation is accounted for only by time in relative higher velocities.

Yuiop misspoke. When he said "time dilation", he meant and should have said "aging". The point of the graph and yuiop's explanation is that since A and B both experienced identical acceleration events and therefore attained identical velocities (and identical time dilations or aging rates), the difference in their aging is accounted for only because A spends more time at the higher velocity than B does.

And in also answering the OP question :
Then does that also mean the twins are aging at the same rate only until the brother undergoes acceleration ?

Neither the diagram nor its explanation addresses that question by the OP. Chogg answered that question in post #3 by saying that the aging rate (or the time dilation) is frame-dependent. So even before the turn-around acceleration, they are not aging at the same rate in the diagram above.

 

[morrobay]

Here's that graph again:

Yuiop misspoke. When he said "time dilation", he meant and should have said "aging". The point of the graph and yuiop's explanation is that since A and B both experienced identical acceleration events and therefore attained identical velocities (and identical time dilations or aging rates), the difference in their aging is accounted for only because A spends more time at the higher velocity than B does.


Neither the diagram nor its explanation addresses that question by the OP. Chogg answered that question in post #3 by saying that the aging rate (or the time dilation) is frame-dependent. So even before the turn-around acceleration, they are not aging at the same rate in the diagram above.

Time dilation and aging are related by Δt' = γΔt. For .6c, γ=1.25 So "time between ticks" of
travelers clock is longer. Hence less aging.

Regarding your second paragraph : I stated exactly the same thing : Even before turnaround point
they are not aging at same rate. The overall picture is that the traveling twins aging is dependent
on velocity before turnaround point and acceleration / frame change at turnaround point.
See this numerical example I did in post #3 www.physicsforums.com/showthread.php?t=734695

 

[Nugatory]

Time dilation and aging are related by Δt' = γΔt. For .6c, γ=1.25 So "time between ticks" of
travelers clock is longer. Hence less aging.

You cannot relate time dilation and aging that way. The $\Delta{t}$ in that expression is the difference between two coordinate times, not proper times.

Indeed, I could use exactly the same argument you're making here to show that stay-at-home will age less: during the entire journey, stay-at-home is moving relative to the traveler, so the "time between ticks" of stay-at-home's clock is longer ($\gamma$ may change as traveller accelerates and decelerates, but it's always greater the one), hence less aging for stay-at-home.

That's the paradox in the Twin Paradox: I can apply the time dilation formula in two seemingly equally valid ways, yet get different results.

 

[ghwellsjr]

Yuiop misspoke. When he said "time dilation", he meant and should have said "aging". The point of the graph and yuiop's explanation is that since A and B both experienced identical acceleration events and therefore attained identical velocities (and identical time dilations or aging rates), the difference in their aging is accounted for only because A spends more time at the higher velocity than B does.

Time dilation and aging are related by Δt' = γΔt. For .6c, γ=1.25 So "time between ticks" of travelers clock is longer. Hence less aging.

Yes, if you mean longer than the Coordinate Time of the particular frame that you are depicting the scenario in. But it isn't the aging rate (which is the inverse of the Time Dilation) that yuiop was talking about. Here's his quote

The issue of whether the differences of time dilation in the twins paradox is caused by acceleration or relative velocity has come up several times in this thread. The diagram below (created by Dr Greg a long time ago) shows a way to eliminate acceleration from the considerations. Observers A and B undergo identical acceleration events (The curved sections highlighted in red) and yet less proper time elapses for observer A than for observer B. The difference in time dilation in this example can only be accounted for by the difference in time spent at higher relative velocities.

There is no difference of time dilation for observers A and B during their times spent traveling. Yuiop didn't mean time dilation. He meant differences in accumulated age from the start of the scenario when A and B were colocated to the end of the scenario when they were again colocated. That's the only way you can compare differences in aging and it doesn't matter what frame you use to depict it whereas time dilation does depend on the frame.

Neither the diagram nor its explanation addresses that question by the OP. Chogg answered that question in post #3 by saying that the aging rate (or the time dilation) is frame-dependent. So even before the turn-around acceleration, they are not aging at the same rate in the diagram above.

Regarding your second paragraph : I stated exactly the same thing : Even before turnaround point they are not aging at same rate. The overall picture is that the traveling twins aging is dependent on velocity before turnaround point and acceleration / frame change at turnaround point.
See this numerical example I did in post #3 www.physicsforums.com/showthread.php?t=734695

In the particular frame depicted in the above diagram, it is true that the twins age at different rates before the next acceleration but we can transform to a different frame in which they are traveling away from each other at the same speed and they will be aging at the same rate. The OP needs to know that the aging rate (or time dilation) is dependent on the frame and not just on periods of acceleration. Here's his question:

Then does that also means that the twins are aging at the same rate only until the brother undergoes acceleration?

The OP needs to know that we can always pick a frame in which two observers age at the same rate at any particular time, even if they never accelerate, and we can pick another frame where either twin ages more than the other at any particular time.