Twin Paradox in 3-sphere (S^3)

[MeJennifer]

The problem is with your first sentence. Read it and ask, 'Is that a consequence I'm really willing to accept?'

Huh?
This is basic relativity theory. Are you perhaps questioning the validity of SR or GR?

 

[KingOrdo]

Huh?
This is basic relativity theory. Are you perhaps questioning the validity of SR or GR?

Yes, I know you're confused. But I don't know how else I can put it; like you said, "This is basic relativity theory". If my explanations aren't making sense, consult the arXiv; the relevant papers have been listed.

And yes we (physicsts) *are* questioning the "validity" of SR . . . well, at least we're not just going to take on *faith* that there's some explanation for what appears to be an anomalous result in an unusual topology (the example is easier to understand in the matter-free universe). *Evidence* and a *logical explanation* to the very interesting problem is both necessary and perhaps productive for the future of relativity theory. *One* explanation (the winding number theory) has been given, and that has shown to be false. So I will ask again:

any ideas?

 

[MeJennifer]

Yes, I know you're confused.

Why do you think I am confused? About what?

Again, I know how to calculate, in principle, the proper time interval differential between two space-time events for two observers. I don't know what more there is to say.

Perhaps it is that you simply are not willing to accept the reality of the properties of space-time.

the example is easier to understand in the matter-free universe

A matter-free universe is flat, expanding and obeys a hyperbolic geometry.

 

[KingOrdo]

Why do you think I am confused? About what?

Because you're not understanding the several really very basic examples I and other PF posters have given. And again, the papers I cited on the arXiv are especially good and clear in this regard. I recommend you consult them; and again, to keep it simple, imagine a matter-free universe of compact topology.

Again, I know how to calculate, in principle, the proper time interval differential between two space-time events for two observers. I don't know what more there is to say.

Well, I'm very pleased that you are able to calculate a proper time interval. That's a good first step, and really almost all you need to understand why the paradox persists in complex spaces.

A matter-free universe is flat, expanding and obeys a hyperbolic geometry.

Ah; now it's clear why you're not understanding the paradox. This statement of yours is false. There are lots of matter-free universes that are *not* flat; viz. the ones in question!

Perhaps it is that you simply are not willing to accept the reality of the properties of space-time.

Oh, yes: let's not worry about *evidence*; let's just take Jennifer's word for it that it's not a problem . . . we'll just *ignore* this lacuna because it's convenient. *One* good explanation has been provided; the 'winding number', and that was debunked in the literature. Again:

any ideas?

 

[MeJennifer]

imagine a matter-free universe of compact topology.

A matter-free universe cannot be compact.

You are presenting a case that is completely impossible.

 

[KingOrdo]

A matter-free universe cannot be compact.

You are presenting a case that is completely impossible.

Again, this is where you are making your mistake (although the paradox does persist in geometrically compact spaces (i.e. universes with matter)).

But one can have a compact matter-free universe (e.g. a compactified Kaluza-Klein manifold (equivalent to M^4xS^7, say)).

Again, any ideas?

 

[MeJennifer]

But one can have a compact matter-free universe (e.g. a compactified Kaluza-Klein manifold (equivalent to M^4xS^7, say)).

Like a fish caught in a net and trying to wiggle out of it.
I see there is no point in arguing with you.

 

[KingOrdo]

Like a fish caught in a net and trying to wiggle out of it.
I see there is no point in arguing with you.

Jennifer, what the heck? I'm asking a serious question about physics. I'm not looking for polemic, or faith-based arguments, or ad hominem attacks, or appeals to authority, or any other obfuscatory or magical mumbo-jumbo.

I have asked a question. The question has been asked extensively in the literature as well. I am soliciting opinions. If you do not understand it, or simply do not care, that is fine. But I would like to hear from people that have professional, considered opinions on the matter.

And again, there is such a thing as a matter-free compact manifold. Take the subject-mentioned S^3. You are confusing geometry with topology.

Again: any ideas?

 

[MeJennifer]

And again, there is such a thing as a matter-free compact manifold.

Yes there is, but such a manifold is not a possible in both SR and GR.

If there is no matter the Riemann curvature tensor is zero and this implies that the manifold cannot possibly be compact. Furthermore, as I wrote before, a matter-free flat space-time must be expanding. Think for instance about the Milne model of matter free space-time.

I have asked a question.

You imply there is a problem without arguing why you think there is a problem.

My best guess as why you think there is a problem is that you perhaps fail to distinguish between an observer's indeterminism of the space-time path taken by an object moving relative to it and the factual space-time path taken.

 

[KingOrdo]

Yes there is, but such a manifold is not a possible in both SR and GR.

If there is no matter the Riemann curvature tensor is zero and this implies that the manifold cannot possibly be compact. Furthermore, as I wrote before, a matter-free flat space-time must be expanding. Think for instance about the Milne model of matter free space-time.

I cannot keep going over things with you, though I certainly can recommend some excellent references. Though the paradox exists in universes with matter, it is easiest to envisage with test particles in a matter-free compact manifold. That is the original example given; and, in my mind, still the best.

You imply there is a problem without arguing why you think there is a problem.

You must be joking. The problem has been cited by me and others here at least a dozen times. And, again, on the arXiv. If you do not understand it I recommend the professional literature.

 

[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?