1. Feb 1, 2008

### Fubini

Forgive me if this has been posted before, but looking through a page of search entries I found many topics that were related, but none that asked quite the same questions. I actually have two questions, but I feel they must be related. I hope you bear with me as I come to the subject from a math background.

The first question is about the resolution to the twin paradox. Mathematically this makes sense to me. When you consider the Minkowski metric as -dt^2+dx^2+dy^2+dz^2 the solution seems to be that the spacetime interval for the brother who stays at home is longer because the x, y, and z distances are all zero. Thus, in a way you can consider the brother at rest as experiencing "maximum time". Since the sign on dt^2 is negative, however, the traveling brother experiences a smaller spacetime interval since any movement in the x, y, or z directions will accumulate against movement in the t direction.

However, I have often read that the paradox is resolved by considering the acceleration of the traveling twin. I'm finding it very difficult to internalize why the acceleration has anything to do with the time difference. I've also read that this is related to how the traveling brother changes inertial frames mid-flight. However, when I try to think through to the same resolution I end up hopelessly confused. Is there anyone who can set the acceleration explanation in more mathematical terms? Maybe answer what is it about acceleration that causes the age difference?

My second (and more important) question is about the essential content of the Twin Paradox. I was told that the important idea behind the twin paradox is asking the question, "Why does the traveling twin age slower instead of the one who stays at home?" The idea being that when you take the point of view of either one of the twins, relatively the situations should be symmetric. Then I was posed a thought experiment where you consider the twins in their own spaceships in a totally empty universe, so the only two inertial frames to consider would be their own. If you repeat the experiment, which one is older and why? Again now we are in a space where the situations are seemingly perfectly symmetrical.

I've beat my head against the wall trying to think of explanations, but they always wash out someplace or another. Usually I end up trying to create some absolute frame you can use for comparison. Unfortunately my other ideas involve computations I don't think I'm mature enough to figure out, like what would happen if the brothers carried monochromatic signaling devices and measured the change in redshift repeatedly... but I don't know why that would help this situation, past there's not a lot else you can do in an empty universe.

Does anyone have a good explanation to the second scenario (empty universe)?

Thanks

2. Feb 1, 2008

### Hurkyl

Staff Emeritus
This resolution won't make sense unless you look at the (flawed) reasoning that leads to the paradox. That reasoning is:

"In A's frame, the time dilation formula says A ages more"
"In B's frame, the time dilation formula says B ages more"

The argument is invalid, because it presupposes both frames are inertial. (Otherwise, it couldn't apply the time dilation formula) Pointing out that one twin accelerates refutes the foundation upon which the paradoxical argument relies.

3. Feb 1, 2008

### JesseM

But that's only true in the frame where the stay-at-home twin is at rest, so it isn't a very general answer. After all, the whole point of the "paradox" is to consider the same situation as viewed in different frames--as usually stated, the paradox is something like "but from the travelling twin's perspective, he is at rest while the stay-at-home twin is moving, so why doesn't he predict that the stay-at-home twin ages less than himself"?
Because the answer to the question as I posed it above is that the traveling twin does not stick to a single inertial frame, and that the time dilation equation doesn't work in non-inertial frames. However, according to SR you can use the time dilation equation in any inertial frame you like and get correct predictions--so, for example, we could use the frame where the traveling twin is at rest during the outbound leg of the trip while the stay-at-home twin is moving, and then in this frame when the traveling twin turns around his speed will be even greater than that of the stay-at-home twin during the inbound leg of the trip. When you add up the amount the traveling twin aged during both legs, it still works out to less than the stay-at-home twin (by the same amount as if you used the stay-at-home's twin rest frame), even though the traveling twin was aging faster during the outbound leg. And it can be proved that different inertial frames will always give consistent answers about the total amount each twin ages between departing and reuniting if they use the same time dilation equation, even though they disagree about the speed of each twin during each section of the trip. It can also be proved that for any two events in the flat spacetime of SR, the path with the greatest amount of aging that goes through both events will be the inertial one, while a path that involves acceleration will involve less aging (this is pretty closely analogous to the fact that in ordinary plane geometry, a straight line is always the shortest path between two points, while a bent or curved path will have a greater length).
I'm actually not quite sure how to do a general proof that an inertial path between two events always has the greatest proper time, but then I'm also not sure how you'd prove in cartesian geometry that a path between two points with constant slope always has a shorter length when you integrate it than a path with non-constant slope...I think the proofs would be pretty similar though. In the relativistic case, since the instantaneous time dilation as a function of velocity is $$\sqrt{1 - v^2/c^2}$$, the total aging for a person whose velocity as a function of time is v(t) between coordinate times t0 and t1 would be $$\int_{t_0}^{t_1} \sqrt{1 - v(t)^2/c^2} \, dt$$. Similarly, in cartesian geometry if a path can be described by the function y(x), then the instantaneous value for the slope at a given value of x would be given by S(x) = dy/dx, and for any arbitrarily small interval dx on the x-axis the length of the path over that interval will be equal to $$\sqrt{dx^2 + dy^2}$$ by the pythagorean theorem, so we can substitute dy = S(x) * dx in there to find the length of the path over the interval must be $$\sqrt{dx^2 + S(x)^2 dx^2}$$ = $$\sqrt{1 + S(x)^2} \, dx$$, and thus the total length of the path between two x-coordinates x0 and x1 would be given by the integral $$\int_{x_0}^{x_1} \sqrt{1 + S(x)^2} \, dx$$. The two equations are pretty analogous, differing only slightly because distance in 2D space is given by $$\sqrt{dx^2 + dy^2}$$ while "spacetime distance" in 2D spacetime is given by $$\sqrt{c*dt^2 - dx^2}$$.
But there is no question about which one accelerates--that twin will feel G-forces during the acceleration. Of course, even if you ignore G-forces you could also define an inertial path through flat SR spacetime as the one that causes a clock to elapse the maximal time when it traverses it. Or you could define it as one in which the laws of Maxwell's laws of electromagnetism are observed to work consistently (an observer on an accelerating ship doing electromagnetic experiments on board would find that they didn't match the predictions of Maxwell's laws). The important thing to note is that acceleration is absolute in SR, even though velocity is not.

Last edited: Feb 1, 2008
4. Feb 2, 2008

### Fubini

Hmmm, many good things to think about. Thank you very much.

5. Feb 2, 2008

### Xeinstein

Minkowski spacetime interval is Lorentz invariant or frame independent but the proper time is not

Forget about the acceleration for a while.
The key question is how to compare time with a moving clock?
We all know moving clock slows down and the clock with the traveling twin is the moving clock. How do we know? Well, since traveling twin experience acceleration, so he is the one that is moving.

Last edited: Feb 2, 2008
6. Feb 2, 2008

### JesseM

Yes it is, proper time is the time elapsed along a given worldline between two events that lie on it. Maybe you mean "coordinate time" rather than "proper time"?
There is no objective truth about which clock is "moving" at any given moment. You are free to analyze the twin paradox from the point of view of a frame where the stay-at-home twin is moving at 0.99c for the whole time, for example.

7. Feb 2, 2008

### yogi

Fubini: "I've beat my head against the wall trying to think of explanations, but they always wash out someplace or another. Usually I end up trying to create some absolute frame you can use for comparison."

You are not alone. There are several explanations consistent with Special Relativity - they all give the correct (consistent with experiment) results for the time dilation - most of the resolutions turn on the fact that the traveling twin changes reference frames and this is observed by the two twins from their own perspective - when you dissect the problem by considering how a time difference can arise for a one way journey - it should get simpler - but it is actually more puzzling. Say the traveler takes off from earth and stops at the turn around point Altair and never returns. Just before decelerating to a stop on Altair, he compares his clock with the local time on Altare - if we assume Altare and the earth are not in relative motion - the only change in frame occurs for the traveler at the outset - during the initial acceleration. Altare and the earth are always in the same frame and the clocks on Altare should read the same as the earth clocks - but if you believe the traveler will age less during the journey (his watch will accumulate less time in going from earth to Altare than the clocks in the earth-Altare frame), then what is there about the situation that distinguishes the two frames. Some would say that in getting up to speed, the traveler is now in a different spacetime frame where he sees the journey as taking less time because the travel distance to Altare is contracted.

Myself, I find the problem forever amusing (hi Jesse - I know you cannot resist responding to my posts)

8. Feb 2, 2008

### JesseM

Er, if the traveler decelerates when arriving at Altair, isn't that a change in frame?
The clocks on Earth and Altair only "read the same time" in their common rest frame, but in SR of course other inertial frames disagree and say their clocks are out-of-sync due to the relativity of simultaneity, and SR says that all inertial frames are equally valid as far as physics is concerned.
If you "believe" there can be any physical basis for thinking there's a single objective truth about whether the traveler ages more or less than someone on Earth during the journey from Earth to Altair, you are rejecting SR. Different inertial frames disagree about whether the traveler's clock or the clock on Earth accumulates more time between the moment the ship leaves Earth and the moment it arrives at Altair, and as noted above, SR says that there is no physical basis for preferring one inertial frame's perspective over any other's.
Of course the prediction about what the traveler's clock reads when he reaches Altair isn't affected in the slightest by the initial acceleration, you'd get the same prediction if he had been moving inertially and just passed the Earth on his way to Altair.
It's not that I particularly enjoy responding to these kinds of posts by you, but the point of this forum is to answer people's questions about mainstream physics, not to confuse them with people's "alternative" ideas.

9. Feb 2, 2008

### yogi

As always Jesse, you misinterpret age difference as measured by two clocks that have been synchronized and put in relative motion, with the idea that one is right and representaive of an absolute time - that is not what is meant by age difference as used by Einstien (1905).

"Of course the prediction about what the traveler's clock reads when he reaches Altair isn't affected in the slightest by the initial acceleration, you'd get the same prediction if he had been moving inertially and just passed the Earth on his way to Altair"

I have come to that same conclusion. I wonder why Einstein made the fuss about starting both clocks in the same frame a la his first example of Part IV 1905

"If you "believe" there can be any physical basis for thinking there's a single objective truth about whether the traveler ages more or less than someone on Earth during the journey from Earth to Altair, you are rejecting SR. Different inertial frames disagree about whether the traveler's clock or the clock on Earth accumulates more time between the moment the ship leaves Earth and the moment it arrives at Altair, and as noted above, SR says that there is no physical basis for preferring one inertial frame's perspective over any other's."

This is at the heart of our disagreement - I will say it again - the Traveler's clock will accumulate less time during the journey - whether it is a one way journey ending in Altare or whether it goes in a circle and returns to the origin.

"Er, if the traveler decelerates when arriving at Altair, isn't that a change in frame?"

I think you misread my post - he can read the time on the clock tower on Altair before decelerating - he compares this time to his own watch -

"It's not that I particularly enjoy responding to these kinds of posts by you, but the point of this forum is to answer people's questions about mainstream physics, not to confuse them with people's "alternative" ideas."

Einstein is the one that says the clock that moved will lag the clock that remained at rest - when they are compared at the end of the trip, Einstein is talking about a definite age difference - which is not an alternative idea of mine

I would say that if you accept the plain meaning of his statement, SR is not internally consistent.

10. Feb 2, 2008

### JesseM

So you merely meant that the traveler aged less in the frame of Earth/Altair, nothing more? Of course this is a trivial prediction of SR. If not, what do you mean? And what specific quote are you referring to of Einstein's?
You mean this section of the Electrodynamics of Moving Bodies paper?
I don't really see that he "made a big fuss" about having the clocks at A and B be synchronized in the "stationary frame" K, he just picked that as an example. Remember that he was introducing relativity to an audience that was not already familiar with time dilation, so to fully illustrate how strange this phenomenon is, I think it was helpful to start out with two clocks that were synchronized in the frame we're working in, then show that merely by putting one into motion in this frame, this will cause it to be behind when the two clocks meet. It's a good example for pedagogical purposes, but I don't see any suggestion that the clock which was moved in this frame "objectively" ticked less time in a frame-independent sense.
But didn't you just say above "you misinterpret age difference as measured by two clocks that have been synchronized and put in relative motion, with the idea that one is right and representaive of an absolute time"? So when you say "the Traveler's clock will accumulate less time during the journey", you are not claiming that this is "right" in any frame-independent sense, or "representative of an absolute time"? Do you agree that in some inertial frame, less time elapsed on the Earth's clock than the traveler's clock between the moment the traveler left Earth and the moment the traveler arrived at Altair? Do you agree or disagree that this frame's perspective is no more or less "right" than the perspective of the Earth/Altair rest frame? (you have a tendency to ignore the questions I ask you, so please give specific answers to these questions)
Sure, and he can also compare the time on his watch to the time on Earth's clocks if he just passes Earth inertially as opposed to taking off from it, so both the initial acceleration and the final acceleration in your scenario are equally irrelevant to the outcome.
You are confusing a "definite age difference" when two clocks meet with the idea that there is some definite truth about which "ages more" during the trip (which Einstein definitely does not claim). All frames agree on the times on the two clocks when they meet, but most frames disagree that they started at the same age before one clock accelerated.

For example, suppose me and you start out 12 light years apart in our rest frame, and the same age in this frame, and then when I'm age 30, I accelerate and move towards you at 0.6c. In this frame it takes me 12/0.6 = 20 years to reach you, so you will be aged 50 when we meet, but due to time dilation I'll only have aged 20*0.8 = 16 years in this time, so I'll be age 46 when we meet. On the other hand, consider a frame where we were both already moving at 0.6c before I accelerated, and then when I accelerated I came to rest and you continued to move towards me at 0.6c. In this frame the distance between us was 12*0.8 = 9.6 light years when I accelerated, and while I was age 30 when I accelerated, you were already 37.2 at that moment because of the relativity of simultaneity. So, it takes 9.6/0.6 = 16 years for you to reach me after I come to rest, so I'll be aged 46, but you'll only have aged 16*0.8 = 12.8 years in this time, so you'll be 37.2 + 12.8 = 50 when we meet.

So, you can see that both frames agree on the "age difference" when we meet (I'm 46 while you're 50), but because they disagree on how old you were at the moment I accelerated at age 30 (in the first frame you were also 30 at the moment I accelerated, in the second you were already 37.2), they disagree on who "aged more" during the trip (in both frames I age 16 years, in the first frame you age 20 years but in the second frame you age only 12.8 years).
What statement of his are you referring to, and what is the "plain meaning" you read into it? Do you think Einstein's statements ever say anything about there being a definite truth about which of two clocks starting at different locations "ages more" when as they are brought together, as opposed to a truth about what time each one shows at the moment they come together? If so, please give a specific quote, because such a relativity-defying idea certainly can't be found in the section I quoted above.

11. Feb 4, 2008

### yogi

Jesse: "Do you agree that in some inertial frame, less time elapsed on the Earth's clock than the traveler's clock between the moment the traveler left Earth and the moment the traveler arrived at Altair? Do you agree or disagree that this frame's perspective is no more or less "right" than the perspective of the Earth/Altair rest frame?"

That is a good question - whether I believe it or not.

I am assuming you have in mind a situation where there is a third observer, call him "J" wherein the twin's velocity might be 0.5c relative to the earth-Altair frame and observer "J" has a speed of 0.55c wrt the earth-Altare frame, in the same direction as the twin - so the relative velocity between the observer "J" and the earth-Altare system is greater (0.55c) than the relative velocity between "J" and the twin (0.05c) - so from the perspective of J, the clock(s) in the earth-Altair frame should run slower than the traveling twin's clock. Have I stated a correct example of your question?

Last edited: Feb 4, 2008
12. Feb 4, 2008

### JesseM

Sure, any observer who measure the speed of the traveler as smaller than the speed of the Earth/Altair will do--in this observer's frame, the clocks on Earth and Altair are out-of-sync, but both elapse the same amount of time between the moment the ship leaves Earth and the moment it arrives at Altair, and this time is less than the time elapsed on the ship's clock between these moments (the actual reading on the Altair clock will be greater than the reading on the ship's clock, but from the perspective of this frame, this is only because the Altair clock had a 'head start' as its reading was already ahead of the reading on the ship's clock when the ship first left Earth). Do you agree that this frame's perspective is no more or less "right" than the perspective of the Earth/Altair rest frame? Also, would you agree that nothing in Einstein's 1905 paper contradicts the idea that every inertial frame's perspective is equally valid?

13. Feb 4, 2008

### yogi

Jesse: "For example, suppose me and you start out 12 light years apart in our rest frame, and the same age in this frame, and then when I'm age 30, I accelerate and move towards you at 0.6c. In this frame it takes me 12/0.6 = 20 years to reach you, so you will be aged 50 when we meet, but due to time dilation I'll only have aged 20*0.8 = 16 years in this time, so I'll be age 46 when we meet."

This is totally consistent with Einstein's example - and with all experiments - no problem

"On the other hand, consider a frame where we were both already moving at 0.6c before I accelerated, and then when I accelerated I came to rest and you continued to move towards me at 0.6c. In this frame the distance between us was 12*0.8 = 9.6 light years when I accelerated, and while I was age 30 when I accelerated, you were already 37.2 at that moment because of the relativity of simultaneity. So, it takes 9.6/0.6 = 16 years for you to reach me after I come to rest, so I'll be aged 46, but you'll only have aged 16*0.8 = 12.8 years in this time, so you'll be 37.2 + 12.8 = 50 when we meet."

In your second example, I assume you mean we are both initially in an inertial frame moving at 0.6 relative to some rest frame - then your acceleration brings you to rest in that frame - ok ...and if you impose the relativity of simultaneity to arrive at my initial age , you get the same result - very good - I had not seen that approach before

14. Feb 4, 2008

### yogi

Before going on, Can we agree then upon the following?

1) An age difference between twins for a one way trip, i.e., as per your statement ("So you merely meant that the traveler aged less in the frame of Earth/Altair, nothing more? Of course")

2) That acceleration per se has nothing to do with the age difference except to the extent it is involved in bringing about a velocity difference in those examples where the two clocks started out in the same inertial frame.

15. Feb 4, 2008

### JesseM

Yes, I'm just analyzing the same problem as in the section you quoted at the start of your post, but this time from the perspective of an observer moving at 0.6c relative to the two of us before I accelerated, so that when I accelerate, this observer sees me come to rest in his own frame.
Right, the relativity of simultaneity is key to understanding how both frames agree on our final age difference when we meet, even though they disagree on who aged more between the moment I accelerated and the moment we met.

16. Feb 4, 2008

### JesseM

Well, do you recognize the difference I pointed out between talking about the age difference when they meet, and talking about how much each twin ages over the course of the trip? All frames must make the same prediction about their ages when they meet, but because of the relativity of simultaneity they can disagree about which twin aged more over the course of the trip (in my example, I showed that there is a frame that agrees I am 46 and you are 50 when we meet, but which says I actually aged more, because at the moment I accelerated I was 30 while you were already 37.2). My statement "So you merely meant the traveler aged less in the frame of Earth/Altair" was talking about how much each twin aged over the course of the trip, which is frame-dependent (the traveler aged less in the Earth/Altair frame, but aged more in other frames), rather than the final age difference when they meet, which is not frame-dependent (all frames agree on who is older and who is younger when they meet, like in my example where I am 46 and you are 50 when we meet).
In the case of twins who start out at different locations and come together, I agree acceleration isn't really important--in my example above, instead of saying I accelerated at age 30, we could just say that I moved inertially past some marker which was at rest relative to you, and that in the marker's rest frame you were also 30 at the moment I passed it (while in my rest frame you were 37.2 when I passed it), and the problem would work out exactly the same, no matter what frame we used we'd still predict that I was 46 and you were 50 when we met. On the other hand, if two twins start at the same location, move apart, and then later reunite, acceleration is important, because regardless of the details of the velocities and the time at which the acceleration happened, every frame will agree that the twin that accelerated aged less than the inertial twin between the time they separated and the time they reunited.

17. Feb 12, 2008

### Xeinstein

If you draw worldlines for both twins in a space-time diagram, then it's clear which twin is in a inertial frame, and the twin in inertial frame will age the most, It's called the principle of maximal aging. The path of maximal aging is a straight worldline in space-time diagram

Last edited: Feb 12, 2008
18. Feb 12, 2008

### yuiop

Just being picky The path of maximal aging is a straight worldline (parallel to the time axis) in space-time diagram.

19. Feb 12, 2008

### JesseM

It doesn't have to be parallel to the time axis--maximal aging is about maximizing the proper time of a path between two points in spacetime, not maximizing the coordinate time in whichever frame you're working in.

20. Feb 12, 2008

### yuiop

Fair comment.