Twin Paradox in 3-sphere (S^3)

[yogi]

No. Imagine you're on the Shuttle, orbiting the Earth. It may *appear* inertial to you, and indeed if you don't need hyper-precise measurements you can *assume* it's inertial, but it's not: it's an accelerated frame, which you can tell from the tidal forces due to the presence of the Earth. It's especially obvious if you substitute "neutron star" for "Earth".

.

The literature does not agree with your definition. See Road to Reality by Penrose at p 394 ..."our insects falling trajectory and our astronaunts motion about the Earth must both count as inertial motions"

Also see Spacetime Physics for a similar definition of a free float frame.

The fact that there are minor tidal effects should not obviate the thought experiment - The problem with the cosmological twins and the orbiting clocks in GPS is the same - the determination as to which twin ages the most will depend upon each orbiting twin setting up stations that allow each to determine how much distance the other twin has traveled between his two stations in a given amount of time as read by his own clocks. Previously this issue arose in a thread involving one clock in orbit and the other oscillating back and forth through the center of the Earth so the two clocks meet periodically and compare times. Both are in inertial frames and one will have accumulated more time than the other on each successive hi 5.

 

[KingOrdo]

The literature does not agree with your definition. See Road to Reality by Penrose at p 394 ..."our insects falling trajectory and our astronaunts motion about the Earth must both count as inertial motions"

Well, I've always used Tipler. But anyway, here is how Wikipedia defines things: "bodies are subject to so-called fictitious forces in non-inertial reference frames". And as we know, gravity is a fictitious force: as MTW says, 'space tells matter how to move'.

The fact that there are minor tidal effects should not obviate the thought experiment - The problem with the cosmological twins and the orbiting clocks in GPS is the same - the determination as to which twin ages the most will depend upon each orbiting twin setting up stations that allow each to determine how much distance the other twin has traveled between his two stations in a given amount of time as read by his own clocks. Previously this issue arose in a thread involving one clock in orbit and the other oscillating back and forth through the center of the Earth so the two clocks meet periodically and compare times. Both are in inertial frames and one will have accumulated more time than the other on each successive hi 5.

Yes, we have all agreed that "Both are in inertial frames." That's the whole problem. The choice of which one gets older then is purely arbitrary, because there is no asymmetry: and relativity tells us that that cannot be right.

Does anyone have any ideas?

 

[masudr]

The difference is that the GPS satellite is not in an inertial frame; it's orbiting, after all.

This is a geodesic: it is an inertial frame. Only if there are rockets firing on the satellite does it go to a non-inertial frame

 

[KingOrdo]

This is a geodesic: it is an inertial frame. Only if there are rockets firing on the satellite does it go to a non-inertial frame

No. A satellite is not an inertial frame. It is a very close approximation to one, but is technically only a freely falling frame. For a frame to qualify as inertial, a person in the frame cannot feel any forces; however, there is a small tidal force due to Earth's gravity which could be detected with a gradiometer. Cf., e.g. http://www.eftaylor.com/pub/chapter2.pdf. (There may also be a small pressure due to solar radiation.)

Again: any ideas for resolving the paradox?

 

[yogi]

To take the issue of satellites and free fall to its logical extreme, there is probably no such thing as a perfect inertial frame - this doesn't preclude making real experiments on Earth or in space where there is no appreciable acceleration - a slight bit of acceleration does not destroy the experiment - it only introduces a small correction to the clocks - the use of free fall frames and orbiting satellites are well accepted by generally recogonized authorities - Tipler has some very strange ideas that are more metaphysical than scientific -

In the case of orbits or boomeranging through the Earth (as Wheeler calls it), the answer to the question as to relative aging can probably best be approached by viewing the different trajectories from a thrid frame - such as the non rotating Earth centered system - the time difference between two different orbits for example can be calculated relative to the NRECRF and the results compared.

In the cosmological case - there is no obvious 3rd frame where one could place synchronized clocks that are on the intersect of the paths of both x and y. The worldline of x needs to be better defined

 

[Hurkyl]

From the moment they high-fived for the first time, neither has undergone any accelerations; yet my understanding is that the reason the Twin Paradox can be resolved in the canonical case is that one of the twins underwent acceleration (when his spaceship turned around), and that is why there is an asymmetry between the two twins. But in this case there seems to be no difference, and it really is as accurate to say that X's time dilates with respect to Y as it is to say that Y's time dilates with respect to X.

Which twin ages more always depends on the metric.

In the "canonical" case, it's specified that we're studying a Minkowski space-time, and we know a lot of shortcuts for analyzing things in a Minkowski space-time. (And the classical twin paradox appears when someone tries to use those shortcuts improperly)


You haven't specified any metric at all, so we cannot do the problem. I could assume that you intended a uniform metric on S^3 space, or even S^3 x R, but that still doesn't tell us everything we need to know.


Incidentally, you don't have to puzzle this out in your head. The 1+1-dimensional spacetime S^1 x R is readily accessible in the form of a cylinder. You can try drawing your example on, for example, a paper towel tube.

 

[George Jones]

Hurkyl has made two points that I intended to post.

Which twin ages more always depends on the metric.

It is impossible to talk about stuff like elapsed proper time, timelike, spacelike, lightlike, etc. without a metric.

You haven't specified any metric at all, so we cannot do the problem. I could assume that you intended a uniform metric on S^3 space, or even S^3 x R, but that still doesn't tell us everything we need to know.

To to do this problem needed are:

a spacetime manifold (not just a space manifold like S^3), like, for example the S^3 x R of a closed FRW universe;

a metric on the spacetime manifold (not just on space) so that we can talk about time;

a pair of events where the observers are coincident;

a pair of distinct geodesic worldlines between these events.

Again, the geodesics are geodesic in spacetime, *not* geodesics in S^3.

Only then can someone point out where the asymmetry, if any, lies. In a highly symmetrical spacetime like you (KingOrdo) want to use, it may be possible to set up a situation in where which the two elpased proper times are the same.

Where there is difference in elapsed times, there is always an asymmetry in worldlines. In standard special relativity, it just happens that this asymmetry can be characterized by 4-acceleration. In the Barrow-Levin compact spacetime, the asymmetry is characterized by homotopy (winding number). Etc. Give a complete setup, and an asymmtry likely will be apparent.

If you can't give us a complete setup, stop saying that there is a problem, and that no one is answering your questions.

Also, in your spacetime, the curvature tensor is non-zero, so there are tidal forces on observers that move on geodesics, just as is the case for an observer freely falling in orbit about a planet or star. You can't use tidal forces to say that your example is different than an orbital example

Incidentally, you don't have to puzzle this out in your head. The 1+1-dimensional spacetime S^1 x R is readily accessible in the form of a cylinder. You can try drawing your example on, for example, a paper towel tube.

I not sure that this spacetime is the best example, as it is not simply connected. I think King Ordo wants to use a simply connected spacetime.

 

[KingOrdo]

I'm afraid you guys are missing the point; someone earlier posted some very nice links to the literature that are interesting and perspicuous, and I recommend checking them out.

I'm not interested in *which* twin is older. Indeed, the reason that there's a problem in the first place is because of the stipulation that time dilation holds in situations like the one I described. Obviously we have no way to test that this is the case; it very well may be that time dilation is a contingent phenomenon. However, that's not the way we normally think in physics, and we have strong a priori grounds for believing that physical law is invariant with regard to the topology it's instantiated in.

The argument is this:
(1) Time dilation occurs in simply connected spaces (i.e. the Universe). We know this from experiment. The *explanation* is that one twin (the one that leaves Earth) undergoes accelerations. Fine: the twin asymmetry is due to the acceleration.

(2) There is no reason to think that the laws of physics would be different in a different, multiply connected universe: say, a compactified Kaluza-Klein spacetime R^3xS^1.

(3) In a compact spacetime the experiment *could* be set up such that X and Y undergo no accelerations. And so *that's* the stipulation of the experiment.

(4) But then the explanation that was given for the M^4 case doesn't work in the compact spacetime case.

(5) So either (1) or (2) is wrong. We have strong a priori grounds for (2), so the problem seems to lie in (1): acceleration is *not* the important criterion. But then what is it? (Plus, why do physical explanations change modulo the topology? Can that be right?)

The "winding number" argument has already been debunked in the literature (cf. earlier links), as it should intuitively: from the point-of-view of either observer, the other appears younger.

So what's the approach? Instead of looking for *real*, *testable*, *falsifiable* theory improvements, most people just want to *stipulate* that they'll find some asymmetry in the spacetime under consideration. That is, they just define away the problem. Well, I'm for one not willing to take this all on *faith*. And the literature makes apparent that this is a real problem upon which there is no professional consensus. We should be excited about this, because when lacunae are discovered in our theories, they can often lead to real discoveries. We should not try to sweep it under the rug.

So, again: any ideas?

 

[masudr]

For a frame to qualify as inertial, a person in the frame cannot feel any forces; however, there is a small tidal force due to Earth's gravity

Depends how big the satellite is...

 

[George Jones]

(3) In a compact spacetime the experiment *could* be set up such that X and Y undergo no accelerations. And so *that's* the stipulation of the experiment.

You don't really mean this. For example, R^3 x S^1 is not compact. Any compact spacetime contains closed timelike curves. https://www.physicsforums.com/showpost.php?p=1254758&postcount=82".

This does, however, provide a nice segue to ...

This thread seems to be stuck in a closed timelike curve, out which, I'm going to try my best to break.

 

[KingOrdo]

You don't really mean this. For example, R^3 x S^1 is not compact. Any compact spacetime contains closed timelike curves. https://www.physicsforums.com/showpost.php?p=1254758&postcount=82".

I'm giving you a multiply connected spacetime, per your request. Like I said, it's a *compactified* Kaluza-Klein manifold (equivalent to M^4xS^d, where d is 6, or 7, or 22, or whatever).

But we're getting rather far afield . . . again, I must ask: does anyone have any ideas?

 

[masudr]

There seems to be no dispute that one twin *will* be older in this experiment.

Have we really agreed on this?

The proper time measured between two events in spacetime is dependent on two things:

(i) the metric; and
(ii) the path taken between these two events.

In the original description of your problem, you have only specified (ii), but not (i).

Have I understood the problem correctly?

 

[KingOrdo]

Have we really agreed on this?

The proper time measured between two events in spacetime is dependent on two things:

(i) the metric; and
(ii) the path taken between these two events.

In the original description of your problem, you have only specified (ii), but not (i).

Have I understood the problem correctly?

I have provided several topologies that should generate the paradox.

Now, if you want to claim that X and Y's proper time are in fact identical in these cases, that's a very interesting claim that I'd like to hear more about. It seems to imply that relativity is false--or, at best, incomplete. But that there's nothing wrong with that: it might just have to be expanded the way Newton's theory had to be expanded in the high velocity limit.

But the one 'way out' proposed in the literature--the 'winding number'--was demonstrated to fail for the very reason the accounting of the twin paradox does: X sees Y's winding number as one thing, and vice versa (cf. the earlier links); and that cannot be right. So something very subtle and interesting seems to be going on. What do we do? Is it time to discard the equivalence principle? I'm very interested to hear any ideas people have.

 

[Hurkyl]

Well, I'm for one not willing to take this all on *faith*.

Neither is anyone else. The whole reason to use differential geometry is that you are guaranteed to get exactly the same answer, no matter what coordinates you use to do a computation.

(4) But then the explanation that was given for the M^4 case doesn't work in the compact spacetime case.

The explanation wasn't given for the M^4 case. The explanation was given for the "someone forgot about acceleration" case.

In the "someone forgot about the metric" case, the appropriate explanation is to remind them about the metric. :tongue:


FYI:

(1) Time dilation occurs in simply connected spaces (i.e. the Universe).

Time dilation is a coordiante-dependent phenomenon: it has no physical reality.

(Of course, it can be used to compute physical quantities -- e.g. the fact that an inertial worldline is the longest time between two timelike-separated points in Minkowski space)

And just to see if you're aware of it... if the spacebound twin travels to Mars and hangs out there long enough before traveling back to Earth, he will find out that he is older than the Earthbound twin. (Despite the fact the Earthbound twin traveled inertially the entire time)

The "winding number" argument has already been debunked in the literature (cf. earlier links), as it should intuitively: from the point-of-view of either observer, the other appears younger.

But why should your perception of the other's age be the same if you are looking forward around the universe or backward around the universe?

But the one 'way out' proposed in the literature--the 'winding number'--was demonstrated to fail for the very reason the accounting of the twin paradox does: X sees Y's winding number as one thing

(Assuming you project down onto space -- it doesn't make sense to speak of the winding number of a path that is not a closed curve)

Of course -- but the effects of a nonzero winding number are different in X's "coordinates" than they are in Y's "coordinates".

Consider flat RxS^1 again. Once around space in X-"coordinates" will be equivalent to going forward in time by some amount d_X. Similarly for Y. There's no reason to think d_X = d_Y: so when X and Y make their adjustments for the other's winding number, they are different.

(Incidentally, X and Y will also disagree on how long it is around the universe according to their respective coordinates)

I have provided several topologies that should generate the paradox.

But what about the geometry? You cannot even talk about the proper time an observer experiences until you have a geometry.

 

[KingOrdo]

The explanation wasn't given for the M^4 case. The explanation was given for the "someone forgot about acceleration" case.

Yes, it was given for the M^4 case. The problem is generated in the first place because the resolution in M^4 does not work for other topologies. And that seems, a priori, wrong to most people.

But why should your perception of the other's age be the same if you are looking forward around the universe or backward around the universe?

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.

But what about the geometry? You cannot even talk about the proper time an observer experiences until you have a geometry.

I have mentioned on several occasions to assume for simplicity that this is a matter-free universe. If you're having trouble grasping exactly why the paradox reemerges in more complex topologies, I recommend checking out the several papers on the arXiv that were cited earlier.

Again, for what must be the tenth time: does anyone have any ideas? A good response would say, e.g.: 'X is what I think is going on in the complex topologies, and Y is why there is an asymmetry between the simple and complex cases.'

 

[K.J.Healey]

How about commenting on the parts of Hurkyl's post that seem to answer the question such as:

Consider flat RxS^1 again. Once around space in X-"coordinates" will be equivalent to going forward in time by some amount d_X. Similarly for Y. There's no reason to think d_X = d_Y: so when X and Y make their adjustments for the other's winding number, they are different.

(Incidentally, X and Y will also disagree on how long it is around the universe according to their respective coordinates)

 

[K.J.Healey]

When they both get back to a position where they see the other as next to themselves, they will both see the other as older, correct?

 

[MeJennifer]

Time dilation is a coordiante-dependent phenomenon: it has no physical reality.

Perhaps you mean something else but the proper time interval differential between two objects is not coordinate dependent in the theory of relativity.

 

[KingOrdo]

When they both get back to a position where they see the other as next to themselves, they will both see the other as older, correct?

No. That's the point. You're citing the paradox; one says, 'Hey, X moved relative to Y, yeah; but Y moved relative to X! So shouldn't they both see each other as older when Y gets back to Earth?' And the answer is, 'Of course not: that's a logical contradiction.' But then how do we resolve it? In the simple case, we resolve it by citing the fact that Y left an inertial frame of reference (when his rocket thrusted, for example, and when he turned around, etc.), while X never did (we ignore the Earth's gravity). Hence, there is a real asymmetry and the paradox disappears.

But in these complex cases, the example can be given (I gave it) in which Y undergoes NO accelerations; and then it really is just as accurate to say that Y's time dilates with respect to X as X's time dilates with respect to Y. And when they get back together and high-five, both should see each as older. But that's an obvious absurdity, and precisely what Einstein was trying to avoid in the first place. We can dispose of the paradox in simple cases, but not, it seems, in the complex ones. And the one proposed resolution in the literature, the 'winding number', was debunked (both papers available in the arXiv, cited earlier). And interesting, it would debunked on isomorphic grounds! X sees Y's winding number as something (say, z), and Y sees X's as z . . . like Einstein told us, it's all relative!

So there appears to be a real problem. One solution is to assert, a priori, that the physics of our Universe would be different if the geometry of our Universe were different. But that is a very high price to pay, and in my view we're better off taking a look at how our contingent theories might be false. . . .

Any ideas?

 

[MeJennifer]

No. That's the point. You're citing the paradox; one says, 'Hey, X moved relative to Y, yeah; but Y moved relative to X! So shouldn't they both see each other as older when Y gets back to Earth?' And the answer is, 'Of course not: that's a logical contradiction.' But then how do we resolve it? In the simple case, we resolve it by citing the fact that Y left an inertial frame of reference (when his rocket thrusted, for example, and when he turned around, etc.), while X never did (we ignore the Earth's gravity). Hence, there is a real asymmetry and the paradox disappears.

But in these complex cases, the example can be given (I gave it) in which Y undergoes NO accelerations; and then it really is just as accurate to say that Y's time dilates with respect to X as X's time dilates with respect to Y. And when they get back together and high-five, both should see each as older. But that's an obvious absurdity, and precisely what Einstein was trying to avoid in the first place. We can dispose of the paradox in simple cases, but not, it seems, in the complex ones. And the one proposed resolution in the literature, the 'winding number', was debunked (both papers available in the arXiv, cited earlier). And interesting, it would debunked on isomorphic grounds! X sees Y's winding number as something (say, z), and Y sees X's as z . . . like Einstein told us, it's all relative!

So there appears to be a real problem. One solution is to assert, a priori, that the physics of our Universe would be different if the geometry of our Universe were different. But that is a very high price to pay, and in my view we're better off taking a look at how our contingent theories might be false. . . .

Any ideas?

Sorry but am I the only one who thinks there is no problem here at all?

The difference between the proper time interval of two observers between two space-time events is caused by the difference in path length. So what's the problem?

 

[KingOrdo]

Sorry but am I the only one who thinks there is no problem here at all?

The difference between the proper time interval of two observers between two space-time events is caused by the difference in path length. So what's the problem?

Jennifer, the easiest way to envisage it is by just imaging that the spacetime is closed. Either it's because of the geometry, or--this way is easier--because of the topology of the manifold.

Here is a good precis on the arXiv (http://arxiv.org/PS_cache/physics/pdf/0006/0006039.pdf) , although their proposed solution has been shown to be false.

Any ideas?

 

[MeJennifer]

Jennifer, the easiest way to envisage it is by just imaging that the spacetime is closed. Either it's because of the geometry, or--this way is easier--because of the topology of the manifold.

Ok, so I imagine a closed universe. Which, by the way, implies that both the temporal and spatial components are topologically closed.
Now I still do not see what you see as a problem.
Care to explain what you think is a problem here?

Again, you can calculate the difference between the proper time interval of two observers between two space-time events by comparing the traversed path lengths.

 

[KingOrdo]

Ok, so I imagine a closed universe. Which, by the way, implies that both the temporal and spatial components are closed.
Now I still do not see what you see as a problem.
Care to explain what you think is a problem here?

Jennifer, I'm just not sure I can make it any clearer. I'd refer you to my earlier posts; perhaps the first one is the most perspicuous.

Roukema and Bajtlik put it this way: "The paradox is the apparent symmetry of the twins' situations despite the time dilation effect expected due to their non-zero relative speed. It is difficult to understand how one twin can be younger than the other -- why should moving to the left or to the right be somehow favoured? Does the time dilation fail to occur?"

Perhaps time dilation *does* fail to occur. But that would mean physical law is not invariant of the universe in which it is instantiated. That price is, a priori, too high to pay, it seems. So however you slice it, an *explanation* is missing. Even if you believe--and this is really implausible--that "moving to the left" *does* matter, you still have to explain *why*. Most of us would, I think, claim that there's something analogous to the acceleration asymmetry in simple cases, but I've no idea what it is.

Any ideas?

 

[MeJennifer]

For starters, if the path length between the two space-time events is identical for both observers the proper time interval differential is obviously zero.

Sorry, but I fail to see what the problem really is here.

 

[KingOrdo]

For starters, if the path length between the two space-time events is identical for both observers the proper time interval differential is obviously zero.

Sorry, but I fail to see what the problem really is here.

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

Any ideas?

 

[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.