Twin Paradox in 3-sphere (S^3)

[MeJennifer]

In general these types of "paradoxes" are generally caused by the misunderstanding of the relativity and equivalence principle.

All objects have a definite path in space-time but due to the relativity principle and the equivalence principle an observer cannot always determine (locally) which definite path is taken by objects.

 

[KingOrdo]

In general these types of "paradoxes" are generally caused by the misunderstanding of the relativity and equivalence principle.

Indeed. Well, go tell all the professional physicists who are stumped by this one that the confusion is due to their "misunderstanding of the relativity and equivalence principle".

All objects have a definite path in space-time but due to the relativity principle and the equivalence principle an observer cannot always determine (locally) which definite path is taken by objects.

Well done.

Again: anyone have any ideas?

 

[MeJennifer]

...all the professional physicists who are stumped by this one that the confusion is due to their "misunderstanding of the relativity and equivalence principle".

Would you care to provide some references in the literature by people like Einstein, Hawking, Penrose, Wald, Schutz, Thorne, Misner, Wheeler or Weinberg writing they are "stumped" by this "problem"?

 

[KingOrdo]

Would you care to provide some references in the literature by people like Einstein, Hawking, Penrose, Wald, Shutz, Wheeler, Weinberg writing they are "stumped" by this "problem"?

No. Check the arXiv yourself.

Do any serious scholars--or any serious amateurs (i.e. people with non-crackpot theories)--have any ideas?

 

[Hurkyl]

That's exactly the point: it shouldn't. But in the cases at hand, you're postulating a physical change because of an arbitrary choice of coordinates, which is precisely what is disallowed by GR.

No, I'm not.

Suppose you naïvely try to put inertial (t, x) coordinates on flat RxS^1. Any such coordinate chart will be periodic: the coordinates

(t_0, x_0)​

and

(t_0 + d, x_0 + L)​

refer to the exact same point of RxS^1, for some d and L.

In observer X's coordinates, let's choose L positive, and assume for simplicity that d is positive and large.

Suppose X meets Y at (0, 0), in X's coordinates. Let's call that event E.

Event E also has coordinates (-d, -L). So, if X looks to his left, he finds that X and Y met a long time ago. (so that preimage of Y is much older than the one he just met)

Event E also has coordinates (d, L). So, if X looks to his right, he finds that X and Y will not meet for a long time. (so that preimage of Y is much younger than the one he just met)


I strongly urge you to work it out yourself. Draw a space-time diagram in X's coordinates. Start with the polar coordinates on the cylinder RxS^1, (which will be inertial for an observer whose worldline is parallel to the axis of the cylinder), and do Lorentz transformation.

(Yes -- happily the formulae of SR will work in these coordinates)


If Y is traveling inertially rightward around the universe (in X's coordinates), then they will meet again, say, at (s, 0) -- X traveled the straight line (0, 0) --> (s, 0), so he ages s between meetings.

Y traveled the straight line (0, 0) --> (s + d, L). Equivalently, we can consider the straight line (-d, L) --> (s, 0). So, he ages:

$\sqrt{(s + d)^2 - L^2}$​

which could be larger or smaller than s, depending on the actual values of everything.

If you're having trouble grasping exactly why the paradox reemerges in more complex topologies

I repeat, the (pseudo)paradox cannot emerge in a topology -- there is no such thing as the "age of an observer" or "inertial travel" or whatnot in a topology. You need a geometry before you can start talking about those things.

 

[yogi]

By specifying that Y follows a great circle, I presume this means circumnavigation of the Hubble sphere - which means a closed universe , and therefore defines a preferred frame (or at least a convenient frame) from which measurements of both x and y can be made to determine their respective cosmological world lines (their spacetime paths) with respect to a selected proper temporal interval as measured by a clock at rest wrt the CBR. Then use the principle of interval invariance to calculate which of the two clocks (X or Y) has accumulated more time during the two events that define the interval. Since the only events which are contained in both spacetime paths occur at the meeting points (the hi fives) the experiment will take a little while to collect the data - therefore:

Because of the practical importance of this subject, I suggest the contributors to this thread form a group to solicit money from the present administration to calculate from the information obtained over the course of the experimental period, which clock is older. We could call us the Cal-burton associates.

 

[KingOrdo]

Hurkyl: I'm just not sure how much clearer I can make things for you. If my precis is confusing (and this is very possible), I really recommend getting into the literature (good summary of the problem here: http://arxiv.org/abs/physics/0006039) .

By specifying that Y follows a great circle, I presume this means circumnavigation of the Hubble sphere - which means a closed universe , and therefore defines a preferred frame (or at least a convenient frame) from which measurements of both x and y can be made to determine their respective cosmological world lines (their spacetime paths) with respect to a selected proper temporal interval as measured by a clock at rest wrt the CBR.

Doesn't have to. Again, the problem is clearest in a matter-free universe, though that's not a necessary condition as long as X and Y are in inertial frames of reference. And it certainly doesn't have to be a spherical geometry (the 1+3 torus works just as well, e.g.) And I used 'great circle' rather than 'geodesic' because the original example was in S^3. Again, this is a thought experiment because the actual Universe isn't a compact space (well, probably not, anyway; and even if it is the experiment is practically impossible).

The only proposed resolution I can find in the literature (and none have been offered here) is the 'winding number' one, but that appears to have been debunked (cf. earlier link to the arXiv).

Any ideas? (Reminder, and pace some posters here, this isn't not a problem because Einstein/Feynman/Witten/et al. did/does not consider it so. We do not appeal to authority to resolve problems in physics; indeed, that's bad science. Please do not PM me saying 'Einstein wasn't worried about this, so there must be some obvious resolution.' I do not--and nor should you--take things on faith. Rigorous argumentation is what is welcome. Thanks!)

 

[K.J.Healey]

From Wikipedia : (its a long article and this is merely part of it concerning this topic. Maybe this will clarify something. Maybe not.)

http://en.wikipedia.org/wiki/Time_dilation

Time dilation is symmetric between two inertial observers

One assumes, naturally enough, that if time-passage has slowed for a moving object, the moving object would find the external world to be correspondingly "sped up." But counterintuitively, Einsteinian relativity predicts the opposite, a situation difficult to visualize. This is based on an essential principle of the overall theory: if one object is moving with respect to another (at an unchanging velocity), the other is equally moving with respect to it.

...

But if motion is thus understood as purely relative, it can be divided-up between "mover" and "benchmark" in any way one pleases, even allowing them to completely switch roles. All that matters is the rate at which they are approaching, or departing from, one another, a grand total which re-distributing the speed-contribution of each one doesn't change. And if that is true, the consequences of relative motion predicted by the theory must also "add up" to an unchanging total effect. If A finds that B has undergone a slowdown-in-time during the period of relative motion, it must work out that B will also find that A has a relatively slower "clock." It seems an inconceivable situation: yet the math works out, and actual tests confirm it.

With respect to constant relative motion between two "clocks", a measurement of relative time must choose one clock as being "stationary" in spacetime, and that clock is the basis of a temporal coordinate system where time throughout is treated as synchronized with the stationary clock. The other "moving" clock is in motion with respect to this treated-as-stationary coordinated system, and its relative motion is the velocity value used in the applicable equations.

In the Special Theory of Relativity, the moving clock is found to be ticking slow with respect to the temporal coordinate system of the stationary clock. And as indicated, this effect is symmetrical: In a coordinate system synchronized, by contrast, with the "moving" clock, it is the "stationary" clocks that is found (by all methods of measurement) to be running slow. (Neglecting this principle of symmetry leads to the so-called twin paradox being regarded as paradoxical.)

Note that in all such attempts to establish "synchronization" within the reference system, the question of whether something happening at one location is in fact happening simultaneously with something happening elsewhere, is of key importance. Calculations are ultimately based on determining what is simultaneous with what.

It is a natural and legitimate question to ask how, in detail, Special Relativity can be self-consistent if clock A is time-dilated with respect to clock B and clock B is also time-dilated with respect to clock A. It is by challenging the assumptions we build into the common notion of simultaneity that logical consistency can be restored. Within the framework of the theory and its terminology, the short answer is that there is a relativity of simultaneity that affects how the specified "benchmark" moments of "simultaneous" events are aligned with respect to each other by observers who are in motion with respect to one other. Because the pairs of putatively simultaneous moments are differently identified by the different observers (as illustrated in the twin paradox article), each can treat the other clock as being the slow one without Relativity being self-contradictory. For those seeking a more explicit account, this can be explained in many ways, some of which follow.

-----------------------------------------

So like I said, at the time when each one relatively see's the high 5, the other will appear younger(as if they had been moving). You can't talk about age relative to a specific even that can be measured from either coordinate system without taking into account the differences in relative time OF that even in each coordinate system.
I don't see any problems with this scenario. Theres no difference than ANY other Twin Paradox (which by the way aren't always resolved with the "acceleration" argument. You can do this without acceleration.)

Remember the twins will NOT agree upon the time at which they high-fived.

 

[Hurkyl]

Hurkyl: I'm just not sure how much clearer I can make things for you. If my precis is confusing (and this is very possible), I really recommend getting into the literature (good summary of the problem here: http://arxiv.org/abs/physics/0006039) .

This certainly doesn't help -- in fact, it looks as if its conclusion is diametrically opposed to what you're trying to argue.

I'm going to assume that you agree with everything I said in my last post (since you haven't said otherwise). Since your thesis appeared to be that there is no asymmetry, but I've clearly demonstrated how asymmetry can occur, I'm confused as to why this discussion is still going on.

 

[KingOrdo]

Healey01: You're misunderstanding the Twin paradox. The Twin paradox is *not* that "the twins will NOT agree upon the time at which they high-fived". That is trivally true. If you're finding the nature of relativity counterintuitive, this page might help: http://www.sc.doe.gov/Sub/Newsroom/News_Releases/DOE-SC/2005/THE_TWIN_PARADOX.htm. But the TP is resolved in SR by appeal to accelerations, and in GR by appeal to the fact that clocks run fast at large gravitational potentials, and vice versa.

This certainly doesn't help -- in fact, it looks as if its conclusion is diametrically opposed to what you're trying to argue.

Again, its conclusion has already been debunked in the literature (I previously cited the link for you). However, that paper does provide a good summary of the problem at hand, despite the falsity of its ultimate conclusion.

And again: the fundamental reason why this variant on the Twin paradox is stumping so many people--including professionals in global GR, etc.--is that in order to resolve the Twin paradox, one twin has to be 'preferred' in some sense. And that is definitionally *impossible* in the compact space cases unless you want to discard a central tenant of relativity theory. And again:

any ideas?

 

[masudr]

But the TP is resolved in SR by appeal to accelerations

The most intuitive way to resolve the TP in SR is to calculate the length of the two worldlines in Minkowski space - and it is trivial that the one who went to Pluto and back has aged more as he has traveled two sides of an isosceles triangle whereas the one on Earth has traveled along the longer side of the triangle, which is shorter than the sum of the other two lengths (trivially).

Note: this resolution has nothing to do with accelerations!

 

[KingOrdo]

The most intuitive way to resolve the TP in SR is to calculate the length of the two worldlines in Minkowski space - and it is trivial that the one who went to Pluto and back has aged more as he has traveled two sides of an isosceles triangle whereas the one on Earth has traveled along the longer side of the triangle, which is shorter than the sum of the other two lengths (trivially).

Note: this resolution has nothing to do with accelerations!

Again, that is *not* the Twin paradox. I really can't offer a primer on relativity theory--both because I have neither the time nor the talent--but I can certainly recommend some references; as mentioned previously, the papers in the arXiv are a good place to start. But, quoting from Wikipedia: "The perception of paradox, referred to as the twin paradox (sometimes called the 'clock paradox') is caused by the error of assuming that relativity implies that only relative motion between objects should be considered in determining clock rates. The result of this error is the prediction that upon return to Earth, each twin sees the other as younger -- which is clearly impossible." *That* is the Twin paradox, and it is resolved by citing a salient asymmetry between X and Y: the fact that X was in non-inertial frames of reference (e.g. when he turned his spaceship around).

Any ideas?

 

[StatusX]

If you put coordinates and a metric on the space, then you'll see that, in contrast to the case of minkowski spacetime, there is a priveleged frame which is at rest. The twin in this frame will be older. Exactly why are you so averse to using a metric?

 

[KingOrdo]

If you put coordinates and a metric on the space, then you'll see that, in contrast to the case of minkowski spacetime, there is a priveleged frame which is at rest. The twin in this frame will be older. Exactly why are you so averse to using a metric?

Again, I just can't explain it any clearer than I already have, nor than has been explained in the literature. I really do recommend checking out the papers on the arXiv, as they are especially perspicuous. I don't know who is "averse to using a metric"--I don't even know if you're talking to me--but the point to remember is that time dilation is a *real* phenomenon. When Y gets back to Earth, Y really is younger than X. It's a real, coordinate invariant phenomenon.

Any ideas?

 

[StatusX]

"Coordinate invariant" does not mean you can just ignore coordinates, you need them to define the inertial reference frames. The topology is not enough.

 

[KingOrdo]

"Coordinate invariant" does not mean you can just ignore coordinates, you need them to define the inertial reference frames. The topology is not enough.

Yes. That is precisely the point.

Any ideas?

 

[StatusX]

Right, so you can't solve this problem until you specify a coordinate system and metric on your space. Yes, without any other information about the system, this will be arbitrary, but that can't be avoided. And once you do this, the twins will no longer be equivalent. Moreover, you'll be able to see explicitly that the familiar rule from minkowski space time that moving observers appear to age slower does not hold exactly in more complicated spacetimes.

 

[KingOrdo]

Right, so you can't solve this problem until you specify a coordinate system and metric on your space. Yes, without any other information about the system, this will be arbitrary, but that can't be avoided. And once you do this, the twins will no longer be equivalent. Moreover, you'll be able to see explicitly that the familiar rule from minkowski space time that moving observers appear to age slower does not hold exactly in more complicated spacetimes.

StatusX, again: I can't make it any clearer. Consult the literature if you're not understanding why a problem arises. Links have been provided. All best,

Tom.

P.S. Anyone: any ideas?

 

[Hurkyl]

Exercise for KingOrdo: resolve the following paradox.

We have two numbers, x and y. Which is bigger? This problem is perfectly symmetric, so we cannot say that x is bigger than y. So, x and y have to be equal, which is paradoxical!​

P.S. Anyone: any ideas?

I already explained how to work out the problem in RxS^1. Either trying to understand it, or pick out an actual error is a good idea.

 

[KingOrdo]

Hurkyl: unlike you, I am not going to engage in polemic or insult (mods, please?).

If you do not understand the phenomenon at hand--which admittedly may be due to my unperspicuous treatment--I recommend you read the several excellent posts made by others, and especially the professional literature (i.e. the arXiv). Regards.

Anyone: any ideas?

 

[KingOrdo]

A-ha! I found a a proposed resolution to the paradox (and unlike the 'winding number' one it has not been, as far as I can find, refuted. Here's the link: http://arxiv.org/PS_cache/gr-qc/pdf/0101/0101014.pdf . But it seems to raise two problems:

(1) It only works in non-matter-free universes. Is that a price we're willing to pay? Should we just stipulate that there's something paradoxical about matter-free universes themselves?

(2) The mechanism to break the symmetry and resolve the paradox is fundamentally different than the one used in the simple cases (and indeed, in our own Universe). Does that seem intuitively right?

Any thoughts?

 

[Hurkyl]

(1) It only works in non-matter-free universes. Is that a price we're willing to pay? Should we just stipulate that there's something paradoxical about matter-free universes themselves?

Nowhere in that paper did they suggest their universe had matter. To wit, there were using a perfectly flat metric, and their space-time is locally isometric to Minkowski space.

But why does it matter? The cosmological twin "paradox" is merely a pseudoparadox because the conclusion does not follow from the premises: it is a logically flawed argument. The conclusion does not follow from the hypotheses. The merit of this paper is that it vividly demonstrates the logical flaw, so as to help those still stuck on the paradox.

(2) The mechanism to break the symmetry and resolve the paradox is fundamentally different than the one used in the simple cases (and indeed, in our own Universe). Does that seem intuitively right?

I think you're asking if it's intuitive that global topology should affect things. Well, it depends on how you've developed your intuition -- if you've studied topology, for example, it would be obvious that it should have some relevance. OTOH, if you've studied other things and never had reason to leave the world of affine space, it would be more surprising.

Hurkyl: unlike you, I am not going to engage in polemic or insult

What insult? I thought it sufficiently likely that you were making exactly that mistake. (But buried underneat a bunch of other stuff so you don't see it)

A-ha! I found a a proposed resolution to the paradox
...
http://arxiv.org/PS_cache/gr-qc/pdf/0101/0101014.pdf .
...
Any thoughts?

But in any case, I'm glad you've finally understand this demonstration of the flaw in the cosmological twin paradox.

 

[KingOrdo]

Nowhere in that paper did they suggest their universe had matter. To wit, there were using a perfectly flat metric, and their space-time is locally isometric to Minkowski space.

"A compact topology selects a preferred place and a preferred time so that some galaxy, if not our own, is at the center of the universe." "galaxy"=matter. This is the case for curved spaces, which is again I think the ones that bear especially important examination.

But why does it matter? The cosmological twin "paradox" is merely a pseudoparadox because the conclusion does not follow from the premises: it is a logically flawed argument. The conclusion does not follow from the hypotheses. The merit of this paper is that it vividly demonstrates the logical flaw, so as to help those still stuck on the paradox.

Well, if the paper is right, then yes: it is a pseudoparadox, just like the Twin "paradox" in the simple case is a pseudoparadox. Of course, if it fails--like the 'winding number' paper did--then the paradox would persist.

I think you're asking if it's intuitive that global topology should affect things. Well, it depends on how you've developed your intuition -- if you've studied topology, for example, it would be obvious that it should have some relevance. OTOH, if you've studied other things and never had reason to leave the world of affine space, it would be more surprising.

First, I don't know what "OTOH" is. Second, it is my no means obvious that global topology should change the physical laws of the universe. As a matter of fact, it's counter to standard experience: the laws of physics are the same on the surface of a sphere as they are in a flat space. If you want to argue for it, that's fine; you might be right. But that burden of proof is on you.

But in any case, I'm glad you've finally understand this demonstration of the flaw in the cosmological twin paradox.

Well, that's the question: is this the right way out? Are Barrow and Levin right where the other authors on the arXiv were wrong? The other papers postulated *different* mechanisms for resolving the case in complex spaces. At most one can be right. Do you think it's Barrow and Levin? Or the 'winding number' one (though that looks pretty conclusively refuted). But, again, please: *no faith-based arguments*. It's been nearly 100 posts and still no one is willing to say: 'here's the *right* way out: X, Y, and Z.' Or, 'Barrow and Levin are right; here's why I think that . . .'

Any ideas? Remember; operative word: *ideas*. . . .

 

[Hurkyl]

Well, if the paper is right, then yes: it is a pseudoparadox, just like the Twin "paradox" in the simple case is a pseudoparadox. Of course, if it fails--like the 'winding number' paper did--then the paradox would persist.

If you mean "persists" in the sense that it's a logical worry, then you're wrong. The cosmological twin "paradox" is a pseudoparadox because it is a logically flawed argument -- the conclusion does not follow from the premise. Whether or not anyone has presented a counter-example you like is irrelevant.

"A compact topology selects a preferred place and a preferred time so that some galaxy, if not our own, is at the center of the universe." "galaxy"=matter. This is the case for curved spaces, which is again I think the ones that bear especially important examination.

If you're going to get technical, the twins are matter too, so you can't have
the twin paradox without matter.

First, I don't know what "OTOH" is. Second, it is my no means obvious that global topology should change the physical laws of the universe. As a matter of fact, it's counter to standard experience: the laws of physics are the same on the surface of a sphere as they are in a flat space. If you want to argue for it, that's fine; you might be right. But that burden of proof is on you.

OTOH means "on the other hand".

And nobody said global topology changes physical laws. Global topology is relevant to global phenomena -- for example, what problems we might encounter when we take a local process like building coordinate charts isometric to Minkowski space, and try to extend globally across the entire universe.

I assume you've taken complex analysis? A lot of what you see in complex analysis is a simplified version of the issues we are seeing here. In the language of the paper you most recently linked, the fact that it is "impossible for H to synchronize her clocks" is the same sort of phenomenon as needing a branch cut for certain functions. And by golly, we see the same sort of pseudoparadoxes if we ignore that: as we trace counterclockwise around the unit circle, the imaginary component of log z is strictly increasing, so the imaginary component of log z must be larger at our ending point than at our starting point. This is seemingly paradoxical because our ending point can be our starting point if we go all the way around the circle.

But, again, please: *no faith-based arguments*.

Nobody's making a faith-based argument. People know there is no paradox because tensor analysis was defined precisely so that nothing depends on your choice of coordinates. (And that's why it was adopted for GR) So, when you analyze a situation in two different coordinates and get two different answers, we know one of the following is true:
(1) You made a mistake.
(2) The very foundation of mathematics is inconsistent.

And I don't mean (2) in the "oh, we got GR wrong" sense... I mean (2) in the "we now have a correct proof of 0 = 1" sense.

The "resolutions" people make of twin paradoxes are simply pedagogical devices: the author has a guess as to why people are confused by the twin pseudoparadox, and they try to make a vivid demonstration to help them out of their confusion.

 

[KingOrdo]

If you mean "persists" in the sense that it's a logical worry, then you're wrong. The cosmological twin "paradox" is a pseudoparadox because it is a logically flawed argument -- the conclusion does not follow from the premise. Whether or not anyone has presented a counter-example you like is irrelevant.

No. It's a pseudoparadox in the simple case because there is a way to resolve it: one twin accelerates, and the symmetry is broken. If Barrow and Levin are right, then it's a pseudoparadox in the complex case. But if they're wrong, like the 'winding number' paper was, then the paradox--or appearance of paradox--persists.

If you're going to get technical, the twins are matter too, so you can't have the twin paradox without matter.

Yes, that's true. Upon further reflection, I don't think it's an intelligible problem when using test particles.

And nobody said global topology changes physical laws. Global topology is relevant to global phenomena -- for example, what problems we might encounter when we take a local process like building coordinate charts isometric to Minkowski space, and try to extend globally across the entire universe.

Yes, you *are* claiming that physical law is variant on topology. If the mechanism for symmetry breaking in a complex space is essentially different than the mechanism in, say, the actual Universe (viz. acceleration), then--unlike in the actual Universe--law is not invariant with regard to topology. Now, I'm not saying you're *wrong*; indeed, a lot of (very smart) people believe exactly that. But we'll need some argument to overcome the prima facie implausibility.

I assume you've taken complex analysis? A lot of what you see in complex analysis is a simplified version of the issues we are seeing here. In the language of the paper you most recently linked, the fact that it is "impossible for H to synchronize her clocks" is the same sort of phenomenon as needing a branch cut for certain functions. And by golly, we see the same sort of pseudoparadoxes if we ignore that: as we trace counterclockwise around the unit circle, the imaginary component of log z is strictly increasing, so the imaginary component of log z must be larger at our ending point than at our starting point. This is seemingly paradoxical because our ending point can be our starting point if we go all the way around the circle.

The difference is, of course, that it's totally unintelligible to talk about the ages of points on the unit circle. Not so in the physical case. If you were right, then there would be no time dilation at all, even in the simple cases.

The "resolutions" people make of twin paradoxes are simply pedagogical devices: the author has a guess as to why people are confused by the twin pseudoparadox, and they try to make a vivid demonstration to help them out of their confusion.

No; you're confusing the counterintutiveness of time dilation with the Twin paradox itself. You are perfectly right in the former case; however, that's *not* what the Twin paradox is. The Twin paradox is *not*: 'Hey, this twin left Earth and came back to shake his brother's hand--when he did, he was younger than his brother!' It's clear that that's what you think it is, but you're wrong; again, I've been over this ad nauseum. I recommend checking out those links and especially the professional literature.

 

[pervect]

IMO Hurkyl isn't saying anything that conflicts with the literature. He is trying (rather patiently) to correct some of King Ordo's misunderstandings of what the literature is saying as far as what the cosmological twin "paradox" is about and what it is not about.

 

[KingOrdo]

IMO Hurkyl isn't saying anything that conflicts with the literature. He is trying (rather patiently) to correct some of King Ordo's misunderstandings of what the literature is saying as far as what the cosmological twin "paradox" is about and what it is not about.

But that's the problem: he's misconstruing the Twin paradox. He thinks (as do many people here, apparently) that the Twin paradox is just that it seems weird that when one twin leaves Earth and returns he's younger than his twin that stayed behind. But that's not paradoxical at all; rather, it's a straightforward implication of relativity theory (although it is a little weird, to be sure).

The Twin paradox is this: it's just as correct to say that the twin on Earth was the one that did the traveling, and the twin on the rocket stayed at rest. Therefore, when they get back together *both will see the other as older*, which is a logical contradicition. Now, it's a pseudoparadox in simple cases, because there is an asymmetry between the twins (viz. the one on the rocket accelerates). But that's not true in the complex cases. So an asymmetry needs to be found there, too. One candidate was the 'winding number'; however, that was debunked in the literature. Barrow and Levin have proposed another asymmetry. What do people think about this? Again, I know the topic is confusing, but the arXiv really does have several papers that make things pretty clear (links have been provided).

Any thoughts?

 

[Hurkyl]

No. It's a pseudoparadox in the simple case because there is a way to resolve it

I've been using pseudoparadox in the technical sense, rather than as a synonym for "aha, I'm no longer confused". (Of course, in this informal sense, this is a pseudoparadox for me, whether or not it's a paradox for you)

Yes, you *are* claiming that physical law is variant on topology. If the mechanism for symmetry breaking in a complex space is essentially different than the mechanism in, say, the actual Universe (viz. acceleration), then--unlike in the actual Universe--law is not invariant with regard to topology. Now, I'm not saying you're *wrong*; indeed, a lot of (very smart) people believe exactly that. But we'll need some argument to overcome the prima facie implausibility.

The symmetry demanded by Einstein was local, and it's still present even in these "complex spaces". It's not "broken".

But that symmetry was only demanded of the laws of physics -- it would be rather silly to demand everything be symmetric. The matter distribution, the metric, other fields we put on space-time... they aren't required, nor even expected to be symmetric.

But the diffeomorphism invariance of GR is not what's relevant here. We have the rather exceptional case that flat RxS^1 is locally isometric to 1+1 Minkowski space, and that flat Rx(S^1)^3 is locally isometric to 3+1 Minkowski space. We are interested in the question of whether a problem on RxS^1 can be treated as if it was a problem in 1+1 Minkowski space. This is a problem of piecing local information together, hoping to obtain global information, and whether we can do this is one of the big questions studied in topology.

That's how the winding number fits in, in the RxS^1 case -- if we follow observer X's path forward between meeting points, and then Y's path backwards between meeting points, we have a closed curve and can ask about its winding number. If that number is zero, we can treat the entire problem as if it were happening in Minkowski space. If that number is nonzero, then the global periodic nature of spacetime is relevant in an essential way -- in particular, the winding number is zero if and only if X and Y really do meet again, according to the Minkowski analysis.

To wit, if we take the cosmological twin paradox on RxS^1, then if the twins are both traveling inertially, they cannot possibly meet twice. In fact, that really should be the big clue that there are flaws in treating the situation with special relativistic methods.

The difference is, of course, that it's totally unintelligible to talk about the ages of points on the unit circle. Not so in the physical case. If you were right, then there would be no time dilation at all, even in the simple cases.

Age is a number. We're talking about numbers.

In Barrow & Levin, they observe that "it becomes impossible for H to synchronize her clocks" -- that is because she needs to have a branch cut in her coordinates. (Or use multi-valued coordinate functions, or use the universal cover...)

No; you're confusing the counterintutiveness of time dilation

It's not counterintuitive. (At least, it's not counterintuitive for me...)

 

[KingOrdo]

I've been using pseudoparadox in the technical sense, rather than as a synonym for "aha, I'm no longer confused". (Of course, in this informal sense, this is a pseudoparadox for me, whether or not it's a paradox for you)

No, you're quite wrong here; I've been using paradox in the precise, logical sense . . . you've been using it in the 'Hmm . . . that's weird.' sense. N.B. There's nothing wrong with your usage.

The symmetry demanded by Einstein was local, and it's still present even in these "complex spaces". It's not "broken".

No, it is broken. And it's broken in different ways; cf. the arXiv.

But that symmetry was only demanded of the laws of physics -- it would be rather silly to demand everything be symmetric. The matter distribution, the metric, other fields we put on space-time... they aren't required, nor even expected to be symmetric.

I'm not sure what you mean here; are you talking about symmetry as a component of a equivalence relation? Obviously some physical laws are symmetric, and some aren't.

But the diffeomorphism invariance of GR is not what's relevant here. We have the rather exceptional case that flat RxS^1 is locally isometric to 1+1 Minkowski space, and that flat Rx(S^1)^3 is locally isometric to 3+1 Minkowski space. We are interested in the question of whether a problem on RxS^1 can be treated as if it was a problem in 1+1 Minkowski space. This is a problem of piecing local information together, hoping to obtain global information, and whether we can do this is one of the big questions studied in topology.

That's how the winding number fits in, in the RxS^1 case -- if we follow observer X's path forward between meeting points, and then Y's path backwards between meeting points, we have a closed curve and can ask about its winding number. If that number is zero, we can treat the entire problem as if it were happening in Minkowski space. If that number is nonzero, then the global periodic nature of spacetime is relevant in an essential way -- in particular, the winding number is zero if and only if X and Y really do meet again, according to the Minkowski analysis.

This is why I have consistently directed you to the professional literature; this approach fails for the precise reasons it was implemented in the first place. I have been over this ad infinitum; again, if you are confused--and I don't blame you if you are (much of this is counterintuitive!)--consult the arXiv. The professionals say it much better than I do.

To wit, if we take the cosmological twin paradox on RxS^1, then if the twins are both traveling inertially, they cannot possibly meet twice. In fact, that really should be the big clue that there are flaws in treating the situation with special relativistic methods.

Why would we treat it using SR methods?

Age is a number. We're talking about numbers.

No, it's not. Age is a property that has to do with the physical composition of the entity in question. In the simple case, forget about age: when the twin gets back to Earth he *will be physically different* than the one who stayed behind. And so too in the complex case. Again, it is best to think of the two twins at the first point of intersection as one body with a symmetry.

In Barrow & Levin, they observe that "it becomes impossible for H to synchronize her clocks" -- that is because she needs to have a branch cut in her coordinates. (Or use multi-valued coordinate functions, or use the universal cover...)

Just so.

So again, I must ask: any ideas? To quote pervect:

"There is a general agreement about the broad details, which is that the two twins won't be the same age.

The disagreement as I read it is how to explain the age difference, i.e. to point to a particular mechanism. The first papers say that it can be addressed in terms of winding number (which is a topological property that doesn't even need a metric). The second papers say that this is not correct, that one needs more than the winding number to properly explain the age difference, that one needs the metric information.

Everyone agrees that there should be an age difference AFAIK."

So: what is the proper resolution? Is the 'winding number' approach right, despite its apparent refutation in the literature? Or do Barrow and Levin have the right idea? Is there perhaps another idea we're missing? And please: I can't explain it any better than I already have; consult the literature if you're unsure about why the paradox may persist in the complex cases.

Any ideas?

 

[Hurkyl]

The disagreement as I read it is how to explain the age difference, i.e. to point to a particular mechanism.

There is exactly one mechanism. If $\gamma$ is the worldline of a twin between the two events where the twins meet, parametrized by $u \in [0, 1]$, then the twin ages

$$\Delta \tau = \int_{\gamma} || \frac{d \gamma}{du} || \, du.$$

The amount each twin ages is a different integral, and thus can have different numerical values. We can compare those numbers to tell which twin ages more.


We can some prove some general facts about it $\Delta \tau$ (such as that geodesics yield a locally maximal value of $\Delta \tau$) And if we make some assumptions, we can derive shortcuts and specialized theorems for $\Delta \tau$ -- for example, the time dilation formula in an Minkowski inertial coordinate chart, or invoke the fact that in Minkowski space, there is only one geodesic between a pair of points to prove that an inertially traveling observer ages more than any other observer that he meets twice. But we should not mistake of assuming that the general case must also have such simple theorems.

 

[MeJennifer]

There is exactly one mechanism. If $\gamma$ is the worldline of a twin between the two events where the twins meet, parametrized by $u \in [0, 1]$, then the twin ages

$$\Delta \tau = \int_{\gamma} || \frac{d \gamma}{du} || \, du.$$

The amount each twin ages is a different integral, and thus can have different numerical values. We can compare those numbers to tell which twin ages more.

Exactly right, and really, that is all there is to say about this "paradox".

 

[KingOrdo]

I can make it no more perspicuous for you. Consult the arXiv if you need clarification.

So again, I must ask: any ideas? To quote pervect:

"There is a general agreement about the broad details, which is that the two twins won't be the same age.

The disagreement as I read it is how to explain the age difference, i.e. to point to a particular mechanism. The first papers say that it can be addressed in terms of winding number (which is a topological property that doesn't even need a metric). The second papers say that this is not correct, that one needs more than the winding number to properly explain the age difference, that one needs the metric information.

Everyone agrees that there should be an age difference AFAIK."

So: what is the proper resolution? Is the 'winding number' approach right, despite its apparent refutation in the literature? Or do Barrow and Levin have the right idea? Is there perhaps another idea we're missing? And please: I can't explain it any better than I already have; consult the literature if you're unsure about why the paradox may persist in the complex cases.

Any ideas?

 

[Hurkyl]

I can make it no more perspicuous for you. Consult the arXiv if you need clarification.

I'm not aware of anything I need clarified, except maybe precisely what you think, why you think that way, and what criteria an answer must satisfy before you would find it acceptable.

 

[KingOrdo]

I'm not aware of anything I need clarified, except maybe precisely what you think, why you think that way, and what criteria an answer must satisfy before you would find it acceptable.

I can make it no more perspicuous for you. Consult the arXiv (and my earlier posts) if you need clarification.

So again, I must ask: any ideas? To quote pervect:

"There is a general agreement about the broad details, which is that the two twins won't be the same age.

The disagreement as I read it is how to explain the age difference, i.e. to point to a particular mechanism. The first papers say that it can be addressed in terms of winding number (which is a topological property that doesn't even need a metric). The second papers say that this is not correct, that one needs more than the winding number to properly explain the age difference, that one needs the metric information.

Everyone agrees that there should be an age difference AFAIK."

So: what is the proper resolution? Is the 'winding number' approach right, despite its apparent refutation in the literature? Or do Barrow and Levin have the right idea? Is there perhaps another idea we're missing? And please: I can't explain it any better than I already have; consult the literature if you're unsure about why the paradox may persist in the complex cases.

Any ideas?

 

[JustinLevy]

The symmetry demanded by Einstein was local, and it's still present even in these "complex spaces". It's not "broken".

No, it is broken. And it's broken in different ways; cf. the arXiv.

You are misunderstanding the papers. And you are misunderstanding the posters here trying to teach you.

As pervect said:
"IMO Hurkyl isn't saying anything that conflicts with the literature. He is trying (rather patiently) to correct some of King Ordo's misunderstandings of what the literature is saying as far as what the cosmological twin "paradox" is about and what it is not about."

In short, people are trying to be very patient with you and help answer your questions. But you continue to ignore or misunderstand all the help presented to you.

I can understand that you do not believe you are misunderstanding anything, but please entertain the possibility to allow this discussion to move forward.

So again, I must ask: any ideas?

Yes, I have an idea to help this discussion. To clear up some misunderstanding and help everyone see where the root problem is coming from ... and to prevent the discussion from continuing in circles indefinitely ... KingOrdo, please answer these questions:

1) As pervect mentioned, even in a non-closed universe, two distinct inertial paths can cross in two places.
a] So before moving onto closed spaces, do you understand that there is no paradox about how much proper time elapsed on these two world lines?

b] If so, please explain your understanding of the resolution of this "paradox" to give others a starting point to build explanations from.

2) Do you agree that the question of how much proper time elapsed requires a geometry, ie. that until a geometry is defined we cannot ask for the distance between spacetime points? If not, please explain why.

3) Do you agree that specifying a geometry does not specify a coordinate system (ie. the description is still coordinate invarient)? If not, please explain why.

4) Do you agree that once the geometry is specified, there is a unique answer to how much proper time elapsed along a path in spacetime? And therefore there is no "paradox"? If not, please explain why.