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I read a lot about the classical twin paradox recently. What confuses me is that some authors claim that it can be resolved within SRT, others say that you need GRT. Now, what is true (and why)?

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Great question and the answer depends on the point of view. Framework of SR will suffice to explain the paradox but you need to account for the acceleration of one of the twins and because of equivalence of acceleration and gravitation (in the elevator sense) this is best understood in the framework of GR. Will try to post an answer later if someone doesn't beat me to it. –  Marek Jan 6 '11 at 13:07
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Just one general point, not related specifically to the twin paradox: anyone who tells you that you need general relativity to treat acceleration is confused. General relativity is a theory of gravity. In special relativity, just as in Newtonian mechanics (with Galilean relativity), certain non-accelerating reference frames are preferred, but you can still perfectly well compute things involving accelerating objects! –  Matt Reece Jan 6 '11 at 14:51
    
@Matt: again, yes and no. General relativity is not only the theory of gravity, it is also a theory of general covariance and this topic is often lost in the standard expositions to SR because there SR is treated as a completely linear theory. That's why it's at least morally correct to say that GR (or its tools) is needed to understand acceleration in SR. –  Marek Jan 6 '11 at 16:03
    
Just today I was looking for a Twin paradox explanation and here it is. Wikipedia article has some nice pictures that help to understand how clocks are ticking on Earth and on the space ship. (en.wikipedia.org/wiki/Twin_paradox) –  gigacyan Jan 6 '11 at 19:05
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I concede defeat :). It is clear from the context. –  Joshua May 28 at 16:36
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5 Answers 5

up vote 21 down vote accepted

To understand this paradox it's best to forget about everything you know (even from SR) because all of that just causes confusion and start with just a few simple concepts.

First of them is that the space-time carries a metric that tells you how to measure distance and time. In the case of SR this metric is extremely simple and it's possible to introduce simple $x$, $t$ coordinates (I'll work in 1+1 and $c = 1$) in which space-time interval looks like this

$$ ds^2 = -dt^2 + dx^2$$

Let's see how this works on this simple doodle I put together

alt text

The vertical direction is time-like and the horizontal is space-like. E.g. the blue line has "length" $ds_1^2 = -20^2 = -400$ in the square units of the picture (note the minus sign that corresponds to time-like direction) and each of the red lines has length zero (they represent the trajectories of light).

The length of the green line is $ds_2^2 = -20^2 + 10^2 = -300$. To compute proper times along those trajectories you can use $d\tau^2 = -ds^2$. We can see that the trip will take the green twin shorter proper time than the blue twin. In other words, green twin will be younger.

More generally, any kind of curved path you might imagine between top and bottom will take shorter time than the blue path. This is because time-like geodesics (which are just upward pointing straight lines in Minkowski space) between two points maximize the proper time. Essentially this can be seen to arise because any deviation from the straight line will induce unnecessary space-like contributions to the space-time interval.

You can see that there was no paradox because we treated the problem as what is really was: computation of proper-time of the general trajectories. Note that this is the only way to approach this kind of problems in GR. In SR that are other approaches because of its homogeneity and flatness and if done carefully, lead to the same results. It's just that people often aren't careful enough and that is what leads to paradoxes. So in my opinion, it's useful to take the lesson from GR here and forget about all those ad-hoc SR calculations.

Just to give you a taste what a SR calculation might look like: because of globally nice coordinates, people are tempted to describe also distant phenomena (which doesn't really make sense, physics is always only local). So the blue twin might decide to compute the age of the green twin. This will work nicely because it is in the inertial frame of reference, so it'll arrive at the same result we did.

But the green twin will come to strange conclusions. Both straight lines of its trajectory will work just fine and if it weren't for the turn, the blue twin would need to be younger from the green twin's viewpoint too. So the green twin has to conclude that the fact that blue twin was in a strong gravitational field (which is equivalent to the acceleration that makes green twin turn) makes it older. This gives a mathematically correct result (if computed carefully), but of course, physically it's a complete nonsense. You just can't expect that your local acceleration has any effect on a distant observer. The point that has to be taken here (and that GR makes clear only too well) is that you should never try to talk about distant objects.

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Is that what you meant by using GR? See, you got me scared because I thought there were about to be a bunch of mathematical manipulations I couldn't follow. Nice answer. btw there is a typo in computing the interval - should be -300 instead of -300^2. –  Mark Eichenlaub Jan 6 '11 at 21:50
    
@Mark: haha, this is a simple problem, so GR only provides the more conceptual view of this stuff. But elsewhere one could imagine the use of the heavy-duty differential geometry machinery. Thanks for catching the typo. –  Marek Jan 6 '11 at 21:55
    
@Marek: Thank you - I understand these calculations and the doodle. What I still don't get is how you determine which world line belongs to which twin! Both twins could argue that they are the green ones and the other twin is moving (esp. after reading Chad's post which says that it is not the acceleration that resolves the paradox but the change of inertial frame). How could you determine which twin changed his/her inertial frame? –  vonjd Jan 7 '11 at 16:05
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@vonjd: the basic point is that the picture is geometrical in nature. It doesn't matter what coordinate system (or reference frame) you choose. Different systems will just made the horizontal and vertical coordinate lines curved. But the world-lines of the two twins are absolute and so are their space-time intervals and proper times. As for the acceleration: yes and no. What resolves the paradox is the fact that one path is straight (and this statement is again geometrical because straight lines are geodesics) and the other is curved. And we know curved lines correspond to acceleration. –  Marek Jan 7 '11 at 17:02
    
I don't see anything about this approach that is inconsistent with SR--computing path lengths in the minkowski background is totally allowed in special relativity. That's usually how you scaffold students up to GR in a GR class, actually. –  Jerry Schirmer Mar 5 '11 at 0:33
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The "paradox" in the twin paradox is that a naive view of the problem would suggest that the situation ought to be perfectly symmetric: each twin should believe that he or she is really at rest, while the other twin is the one who moves off at high speed then returns. This is incorrect, though, because one of the two necessarily accelerates, which provides a means of distinguishing between the two frames of reference.

You can understand the difference in times entirely from within special relativity, with no need to invoke general relativity. You can easily Google up a dozen different explanations of how it works, but the best I've seen is probably the one in Tatsu Takeuchi's An Illustrated Guide to Relativity. Sadly, being illustrated and not online, it's a bit beyond my ability to reproduce here, but it's worth a look.

The problem that needs to be resolved is that both twins should see the other's clock running slow during the trip, but somehow, the moving twin has to end up seeing less time pass than the Earthbound twin. The resolution is, very roughly, that in the moving frame of the twin on the return trip, the departure point is much farther in the past, so the trip has been going on for a longer time. An observer who has always been in that frame (say the captain of the interstellar transport the twin is hitching a ride on) would say that the Earth clock has always been slow, but it's been going for far longer than the twin's internal clock would suggest.

The details are somewhat subtle, though, which is why there are dozens of semi-contradictory explanations available via Google. It's important to note, though, that it is not the case that the acceleration causes the difference between the clocks (in which case you might actually need GR to work it out). It's the transitions between frames of reference that leads to the difference, not the acceleration itself, and you can get a twin-paradox-like difference between clocks even when both twins undergo exactly the same acceleration, as shown in this American Journal of Physics paper.

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+1 but not because of the answer which actually doesn't provide any solution at all, but because of the paper at the very end. You can also correctly read there that the effect of acceleration is the same as gravitational field (and in the case of identical acceleration leads to the equivalence with gravitational red-shift). Which it of course must be, by equivalence principle. So it's just a mystification to say that GR is not important to understand these effects. –  Marek Jan 6 '11 at 16:33
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I would say that GR is not necessary to explain the effects, in that you can get an entirely correct prediction of the effects just from thinking about the relativity of simultaneity. Those predictions are also in agreement with the predictions of general relativity because general relativity is the more complete theory. But it's not necessary to invoke GR for either of these cases, any more than it's necessary to use the relativistic 4-momentum to understand collisions of billiard balls. –  Chad Orzel Jan 6 '11 at 16:41
    
That's not a very good analogy you make. You need just a well-defined limit of relativity to understand those collisions. But to understand acceleration you need GR on the flat background. And I am saying that this is not the same thing as SR. Perhaps just a difference in taste, but for me SR is a linear theory of inertial frames. For non-linear effects you have to introduce almost all of the machinery of GR, so it's not honest to call the resulting theory still SR. E.g. to understand Thomas precession fully, you'd need Fermi-Walker transport, etc. –  Marek Jan 6 '11 at 17:24
    
Found a free copy of the paper here: cosmo.fis.fc.ul.pt/~crawford/papers/Accelerated_twins.pdf –  vonjd Jan 7 '11 at 17:34
    
Thank you for this interesting post. It seems to have two contradicting statements though: "one of the two necessarily accelerates, which provides a means of distinguishing between the two frames of reference." and "It's important to note, though, that it is not the case that the acceleration causes the difference between the clocks" Now what?!? –  vonjd Jan 7 '11 at 18:01
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The problem is the symmetry break caused by the traveling twin when he changes his direction. This break makes the two twins distinguishable from each other. Before the traveling one changes his direction, both think of their partner to be the younger one (because the time in your own system is always the fastest possible time). So IMHO this paradox has not much to do with general relativity - special relativity can of course not describe accelerated movements, but what really matters is that the twin reverses, not how he accelerates and so on.

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-1 this only explains that twins are distinguishable but doesn't explain anything about why one trajectory is faster than the other. –  Marek Jan 6 '11 at 13:15
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Special relativity can indeed describe accelerated movement. google.com/… –  Mark Eichenlaub Jan 6 '11 at 13:54
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@Marek Well, almost everything I have every read says you can do acceleration in SR, and also I have done calculation with acceleration in SR. See a blog post I wrote on it, for example, or look at the many explanations from google that I linked. arcsecond.wordpress.com/2010/10/25/… –  Mark Eichenlaub Jan 6 '11 at 19:16
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@Marek: Path lengths are perfectly well defined in SR, all you need is vector calculus and the minkowski signature. You don't need any of the machinery of General Relativity to do that. It's true that special relativists don't often do any of this, but you can certainly treat the problem. The whole second half of Jackson is basically a treatment of the acceleration of objects in special relativity. –  Jerry Schirmer Mar 5 '11 at 0:29
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For the sort of acceleration needed to reverse the direction of a twin (i.e. small value constant), it's easy to do the calculation in pure SR. I did it as a homework assignment years ago, I recall. In fact, I'd be willing to bet that the twin paradox predates the invention of GR. I'd +1 the answer except for the statement that this requires GR. No, the effect is entirely present for twins moving in flat Minkowski space and that is entirely governed by SR. –  Carl Brannen Mar 5 '11 at 0:50
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Its easy to resolve from the person on earth - he took the doppler effect out of the equation during his observations of the space ship and observed that the clock on the rocket ship was running slow, and confirmed this when the ship arrived back in port, by comparing the two clocks.

The observation according to the traveller is somewhat of a mystery. The traveller looks back at earth, removes the doppler effects and apparently views the earth time running slowly. Without taking his eye of the two clocks, at some point the doppler effect removed time on earth must speed up and overtake that of the travellers clock. When does this happen? - isn't this in contradiction to what we are taught is observed?

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The question is: what do you mean by "classical"? To me the Classical Twin Paradox is what Einstein called the Clock Paradox, i.e. the version without any accelerations. And this paradox has never been resolved.

If there are no accelerations involved in the situation, than the whole paradox results only from the simple equation for time dilatation:

$$ \Delta t' = \frac {\Delta t} {\sqrt{1 - \frac{v^2}{c^2}}} $$

As we can see, there is just constant $v$ there, and yet one of the clocks must be time-dilated.

If someone says that space-travel requires acceleration of a spaceship, they are absolutely right. The thing is that any time-dilatation due to acceleration is only on top of dilatation due to constant $v$.

We can imagine simply that both twins go on a space trip in different spaceships and travel in opposite directions. In such case we can easily assume that they both undergo exactly the same accelerations which effectively "cancel" each other out. And yet we still have the paradox resulting from time dilatation between inertial frames. The above equation still holds. Therefore, we need to answer the question: Which of the twins is younger and why?

As you can see, introducing accelerations only masks the real paradox, and not solves it. And perhaps that's why nobody solves the paradox with real numbers. Because if you make actual calculations, you will immediately see that you need to add time-dilatation due to acceleration to time-dilatation due to constant $v$. Which leaves the other one still unresolved.

In order to illustrate the classical paradox best, we can formulate it in two ways:

1) If one of the twins actually becomes older due to difference in constant $v$, than the axiom of Special Relativity that there are no preferred (inertial) reference frames is falsified. Because this means the reference frame with slower clock is moving with respect to the other in absolute terms. Therefore movement is absolute.

Someone can reply to this that I don't understand SR which allows us to revert the whole situation, i.e. switch the moving and the stationary frames. Sure, but then we have paradox 2):

2) If we claim that clock A is slower than clock B, and also that clock B is slower than clock A, than Special Relativity cannot be proved to be true. For in such case both clocks are late with respect to each other, which boils down to no real time-dilatation at all. If there is a difference measured, however, then we are back to paradox 1), i.e. one of the frames is clearly preferred, and therefore we proved absolute movement exists.

To sum up. There is no solution to the classical (strictly inertial) Twin Paradox. And also, any accelerations do not solve the paradox either - they only mask it.

EDIT: Professor Chad Orzel in his answer gave this paper as a source claiming that dilatation in SR (without accelerations) can be proved. That's simply astonishing, as anybody can see that this paper is based on a trick whereby the parents of the traveling twins stay on Earth and see there is some distance between them equal $γx_0$. However: 1) length calculations in SR are made from the two reference frames, and not from a 3rd one, 2) if twins travel in opposite directions (as in my thought experiment above) you can always claim the other twin is younger, and - most importantly - 3) this paper shows dilatation based on velocity $v$, and yet there is no velocity difference between the twins - they travel in two rockets in the same direction, having been accelerated in exactly the same way. The velocity $v$ is measured from their parents frame, i.e. Earth, and the author even admits that saying that effective velocity between the twins is null. Now how come a paper claiming SR time dilatation between reference frames stationary wrt. each other (and not accelerated) has been accepted for publication?!?! And how come it has been presented here as source by a professor?!

(And if vonjd didn't find this free paper, we might think that some smart guy proved the unprovable. And if I paid for it, I would be simply coned.)

NOTiCE: Any comments concerning this line of reasoning are more than welcome.

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