Twin Paradox in 3-sphere (S^3)

[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.

 

[KingOrdo]

I simply can't make it any clearer for you. If the fundamentals of relativity theory are still hazy to you, I can recommend some excellent references. Also, 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]

If you are not willing to entertain the possibility that you are misunderstanding, then you can never learn. In that case, it is pointless to even try to have a discussion about this with you. Is this what you are telling us?

I truly hope not. So please go back and answer the questions asked here:
https://www.physicsforums.com/showpost.php?p=1269327&postcount=105

 

[KingOrdo]

JustinLevy, I simply can't make it any clearer for you. If the fundamentals of relativity theory are still hazy to you, I can recommend some excellent references. Also, 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?

 

[MeJennifer]

Any ideas?

Yes, to close this topic, it's going nowhere.

 

[KingOrdo]

Yes, to close this topic, it's going nowhere.

Indeed; but not for want of trying on my part. I've been extremely patient, to the tune of five pages of posts. . . .

 

[JustinLevy]

JustinLevy, I simply can't make it any clearer for you.

Why do you insist on evading the questions? That would make your position abundantly clear. Which currently it is not as you keep stating things that contradict GR as well as the papers you are referring to.

The previous questions and related points have been brought up by other posters and you continue to ignore them or resist acknowledging them. In doing so you ignore the very discussion you repeatedly seek. So if you do not answer those questions, it will be nearly impossible for any poster to help you.

This is a simple request, and the questions are not difficult or time consuming. So please answer the specific questions previously given to you.

 

[KingOrdo]

JustinLevy, I simply can't make it any clearer for you. If the fundamentals of relativity theory are still hazy to you, I can recommend some excellent references. Also, 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]

Yes, to close this topic, it's going nowhere.

Indeed; but not for want of trying on my part.

How can you possibly say that? It is going no where directly because of your lack of trying.

I have asked repeatedly for you to answer some simple questions. You refuse to acknowledge questions were even asked of you, let alone actually try to answer them.

So stop avoiding the questions, as these relate to the heart of the matter.
Please answer the following:

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.

 

[KingOrdo]

JustinLevy, I simply can't make it any clearer for you. If the fundamentals of relativity theory are still hazy to you, I can recommend some excellent references. Also, 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?

 

[MeJennifer]

You have now repeated that last posting 6 times.
To me that is trolling. Hopefully a moderator can take some action here.

 

[KingOrdo]

I've asked for moderators to intervene pages ago. I ask a simple question and get only evasion. I can explain myself no better than I already have. Any insult/evasion/irrelevance/etc. will be met by boilerplate. You should expect that. I am asking a serious question about a serious topic.

 

[JustinLevy]

I ask a simple question and get only evasion.

You did not get evasion. You got answers from several posters which you preceded to ignore or misunderstand. When people took the time to help resolve this misunderstanding you refused to help in anyway.

I can explain myself no better than I already have.

Yes you can. You can answer the simple questions directly addressed to you to help others understand where the common ground lays and where the disagreement occurs.

The questions are not an insult, nor an evasion, nor irrelevant.

The are simple, reasonable, and relevant. Your evasion of them makes me question your motives here. If you are not here to just troll, please stop evading the questions.

 

[ZapperZ]

Obviously, this thread is going nowhere. Questions have either been answered, or not been addressed, or issues not clear, etc.. etc.

After 8 pages of responses, I believe it is time to stick a fork into this one. Please do not repost this question in another thread.

Zz.