The Twin Paradox Revisited: A Local Case Study

[Garth]

There have been plenty of posts about the Twin Paradox. I don't think this version has been aired before except in my post #47 on the "Is Age Relative" thread.

The Twin Paradox in a closed universe.

If cosmic expansion slows down and reverses it would become hypothetically possible to circumnavigate the universe.

Take two twins, one twin stays put and the other is accelerated to
9.999…%c she passes her sister and they synchronise clocks. She continues at constant velocity along a geodesic path and circumnavigates the universe. She eventually meets up with her sister again.

She has aged only 50 years in this near light-speed voyage. However her sister has aged 10 billion years!

Or is it the other way round? How do you tell?

Just a thought - Garth

 

[robphy]

http://arxiv.org/abs/physics/0006039
http://arxiv.org/abs/gr-qc/0101014 ( http://dx.doi.org/10.1103/PhysRevA.63.044104 )

http://www.maa.org/pubs/monthly_aug_sep01_toc.html

The older twin has "winding number" zero.

 

[Garth]

Wow - straight in at the deep end! Thank you robphy.
It is important to see the consequences of this version of the paradox, and I quote from the Barrow and Levin paper "The twin paradox in compact spaces" http://arxiv.org/abs/gr-qc/0101014

In a compact space, the paradox is more complicated. If the traveling twin is on a periodic orbit, she can remain in an inertial frame for all time as she travels around the compact space, never stopping or turning. Since both twins are inertial, both should see the other suffer a time dilation. The paradox again arises that both will believe the other to be younger when the twin in the rocket flies by. The twin paradox can be resolved in compact space and we will show that the twin in the rocket is in fact younger than her sibling after a complete transit around the compact space. The resolution hinges on the existence of a preferred frame introduced by the topology,

and from the Uzan et al. paper "Twin paradox and space topology"
http://arxiv.org/abs/physics/0006039

Thus in Friedmann–Lemaıtre universes, (i) the expansion of the universe and (ii) the existence of a non–trivial topology for the constant time hypersurfaces both break the Poincare invariance and single out the same “privileged” inertial observer who will age more quickly than any other twin: the one comoving with the cosmic fluid – although aging more quickly than all her traveling sisters may be not a real privilege!

So a closed universe has a preferred frame of reference! It has so by virtue of its topology, which is finite yet unbounded.

In our discussions in the thread "Is age relative?" I argued

John S. Bell argued that quantum theory and GR could be integrated into QG if a Lorentzian view of relativity was taken. i.e. 3+1D rather than 4D space-time. This required some form of preferred frame, 'absolute' might be too absolute a term for it! (Speakable and unspeakable in quantum mechanics)
Isham and Butterfield reasoned the same, or similar, argument, calling for a ‘preferred foliation of space-time. (2001, Physics meets Philosophy at the Planck Scale, ed. by C. Callender and N. Huggett. Cambridge University Press.)

If these authors are right, and actually it is necessary for Quantum Gravity to have a preferred frame of reference, then perhaps by reversing the above conclusion, we might further conclude that the universe has to be closed.

As I have posted several times

I would argue too that if Mach’s Principle is brought into play then such a preferred foliation of space-time may be identified as selected by the distribution of mass and momentum in the universe, it is the one in which the CMB is globally isotropic.

This would seem to agree with the conclusions of the above two papers.

Just a thought.

Garth

 

[Garth]

The more I think about it the more the topological argument seems unsatisfactory.

Two observers with a mutual velocity of 0.999...c with clock-calendars closely pass each other when they start their clock-calendars. Say a flash bulb goes off midway between them and they start their clocks on receiving the pulse of light.

After a long time they circumnavigate the universe and pass each other again and record the time as they pass.

Each observer will believe they are at rest and the other has circumnavigated the universe.

The older twin has "winding number" zero.

Each will believe that they have a winding number of zero and the other has a winding number of one.

[Imagine the universe is a cylinder with time parallel to the axis. In each observer's frame of reference their world-line is straight up the cylinder and the other's winds around it. The winding number is the number of times the world-line is wrapped around the cylinder when the two encounter points are brought together and the "world-lines pulled taut". This is the topological argument.

Do the experiment from the point of view of one observer, now do it for the other, the cylinder itself rotates, exchanging the winding number values. How do we 'tie down' the cylinder so it doesn't rotate?]

There must be something else that distinguishes between the two observers.
I think it can only be the matter in the universe and the Machian frame of reference that matter defines that gives you something to 'hold onto' and enables you to 'tie the cylinder down'!

What do you think?
Garth

 

[robphy]

Here's an argument based on the article by Weeks (http://www.maa.org/pubs/monthly_aug_sep01_toc.html).

The twin with winding number zero is distinguished because his "surfaces of simultaneity" are closed curves. Twins with nonzero winding numbers have helical "surfaces of simultaneity".

 

[Garth]

Here's an argument based on the article by Weeks (http://www.maa.org/pubs/monthly_aug_sep01_toc.html).

The twin with winding number zero is distinguished because his "surfaces of simultaneity" are closed curves. Twins with nonzero winding numbers have helical "surfaces of simultaneity".

Yes, that was the argument in Barrow and Levin's paper. However I still do not think it answers the paradox.

Take my idealised thought experiment. Not two twins, one of whom has to be accelerated, and thereby infers a state of rest, but two independent inertial observers with clocks, with a high mutual velocity, who happen to pass by each other at which event they set their clocks. They are in a homogeneous and isotropic compact space universe. After a long time their geodesic paths cross again.

According to the Principle of Relativity, i.e. the doctrine of "no preferred frames of reference", each observer will consider that they are the one at rest and the other is moving, each will think that they have closed surfaces of simultaneity and the other has helical surfaces of simultaneity. Each will think their clock has recorded the longest duration between the two encounters.

This is of course nonsense.

What breaks down, and this is what the argument using surfaces of simultaneity depends on, is the Principle of Relativity. In fact all frames are not equivalent. The curvature of space-time that is required to close the universe and create a compact space requires the presence of a homegeneous ' representative gas' throughout the universe. The preferred frame, the one that does have a surface of closed simultaneity rather than any other frame, is that at rest with that representative gas, co-moving with the Hubble flow.

Does this not question the theoretical basis of GR?
Garth

 

[robphy]

Do the experiment from the point of view of one observer, now do it for the other, the cylinder itself rotates, exchanging the winding number values. How do we 'tie down' the cylinder so it doesn't rotate?

I believe that the winding number is a topological invariant... so this shouldn't be possible. I don't have a proof right now.

According to the Principle of Relativity, i.e. the doctrine of "no preferred frames of reference", each observer will consider that they are the one at rest and the other is moving, each will think that they have closed surfaces of simultaneity and the other has helical surfaces of simultaneity. Each will think their clock has recorded the longest duration between the two encounters.

This is of course nonsense.

What breaks down, and this is what the argument using surfaces of simultaneity depends on, is the Principle of Relativity. In fact all frames are not equivalent.

The "Principle of Relativity" is really only a local statement... that physical laws can be written down in a tensorial form, independent of the choice of coordinate axes.

Recall that the spacetime of "Special Relativity" is not merely flat... but is also topologically $R^4$.

Consider a strip... $R\times I$. Now, make the identification so that the manifold is now $R\times S^1$. This manifold is flat, i.e., has zero Riemann curvature. On this manifold, we draw geodesics as straight lines for our inertial observers. If I'm not mistaken, no Lorentz Transformation can transform the geodesic with winding number 1 into a vertical line (parallel to the axis of the cylinder) with winding number 0. Remember, the lightlike direction is unchanged. [Analogously, no rotation on that cylinder can make that geodesic vertical.]

Here's a physical experiment to consider in this closed cylindrical universe.
The "twin at rest" can exchange light signals with the "twin in inertial motion".
For a short enough time (short compared to the time for a light signal to circumnavigate the closed universe), the two twins are indistinguishable. However, consider light signals emitted to the right and to the left at the first meeting event. When those signals are received by the "twin at rest", those receptions coincide. However, those analogous reception signals for the "twin in inertial motion" are not coincident. Thus, the frames are not equivalent... but one had to wait until the topological features at the "large-scale" are encountered.

If I've made a mistake anywhere, please correct me.

 

[Garth]

robphy - Thank you for your post - it "has me thinking"!

And thank you too for that very illuminating thought experiment of the circumnavigating light signals. However are they not an example of the same paradox?

By the Relativity Principle each observer would think they are stationary and believe it is the other one who is moving, and each would expect to receive the two opposite circumnavigating light signals simultaneously. That they cannot is part of the same paradox as before; i.e. they cannot both have the longest proper times between their two meeting events.

Take simply a cylindrical space-time, at first empty of all matter apart from the two observers, i.e. test particles with their inertial frames of reference.
The topological system is the cylinder with the two world-lines, one straight (A) with a winding number of zero, and one wound round the cylinder (B) with a winding number of one. Represent the system as a flexible cylinder with two elastic strings A and B, fixed at the two meeting events.

Is not the topological invariant the winding number of the total system?
i.e. by continuous deformations one cannot 'unwind' both strings. However we can continuously deform the system to transform from A's frame of reference to B's by twisting the cylinder so that B becomes straight and A becomes wound round. The total winding number of the system is still one and that number has remained invariant under the transformation.

I believe I have shown the topological argument, on its own, does not work and leads to a nonsense! This cannot be right. The two frames of reference cannot be equivalent. One is a preferred frame.

The mistake in the above illustration is to assume the topological space is empty, in fact it is not and indeed it cannot be, matter, i.e. the cosmological representative gas, is essential to obtain a closed compact space, that is finite yet unbounded.

I believe the presence of the matter reasserts the importance of Mach's Principle. We can identify the 'stationary' observer, the one in the preferred frame of reference, as that observer who is stationary w.r.t. the representative gas, co-moving with the Hubble flow, in whose frame of reference the CMB is globally isotropic.

Now, is not the Principle of Relativity in SR is a necessary and sufficient condition for the Principle of Equivalence in GR? If so, in the cosmological twin paradox does not GR contain the seeds of its own destruction, i.e. it is internally inconsistent?

Garth

 

[robphy]

Take simply a cylindrical space-time, at first empty of all matter apart from the two observers, i.e. test particles with their inertial frames of reference.
The topological system is the cylinder with the two world-lines, one straight (A) with a winding number of zero, and one wound round the cylinder (B) with a winding number of one. Represent the system as a flexible cylinder with two elastic strings A and B, fixed at the two meeting events.

Is not the topological invariant the winding number of the total system?
i.e. by continuous deformations one cannot 'unwind' both strings. However we can continuously deform the system to transform from A's frame of reference to B's by twisting the cylinder so that B becomes straight and A becomes wound round.

I think that this operation cannot be performed while keeping the light-cones (light-like directions) invariant. So, the winding number for a given curve cannot be changed from, say, 1 to 0.

 

[Garth]

Then the question remains;
"According to the Principle of Relativity (PR), how do you distinguish between the two observers?"
They both think the other is circumnavigating the universe and they themselves are stationary.

As Barrow and Levin said in the paper you linked to:

The resolution hinges on the existence of a preferred frame introduced by the topology,

So
PR -> Equivalence Principle -> GR -> Friedmann Cosmology -> A possible compact space -> Twin Paradox -> preferred frame which contradicts the PR!

Is this therefore logically inconsistent?

Garth

 

[robphy]

Then the question remains;
"According to the Principle of Relativity (PR), how do you distinguish between the two observers?"
They both think the other is circumnavigating the universe and they themselves are stationary.

As Barrow and Levin said in the paper you linked to:

The resolution hinges on the existence of a preferred frame introduced by the topology,

As I alluded to before, the Principle of Relativity is a local statement.

http://zebu.uoregon.edu/~js/21st_century_science/lectures/lec07.html

"the relativity principle: local physics is governed by the theory of special relativity."

That is to say, "experiments done in an inertial `lab' can't distinguish themselves".

Since the "universe" in this scenario is not topologically R⁴, we are not dealing with Special Relativity any more. We should not expect everything from SR to carry over to here.

So
PR -> Equivalence Principle -> GR -> Friedmann Cosmology -> A possible compact space -> Twin Paradox -> preferred frame which contradicts the PR!

Is this therefore logically inconsistent?

Garth

So, the existence of a preferred [inertial] frame (due to a global [i.e. non-local] topological condition) in this non-SR spacetime does not contradict the "PR".

 

[Garth]

Thank you I do understand, but I'm still labouring over the point. Given the idealised scenario above, a empty compact space with two idealised test particles and their inertial frames of reference that happen to closely pass each other at speed. As they pass we are dealing with a local situation and are saying that one will end up with a winding number of zero and the other with a winding number of one.

Which is which?

What is it that gives a particular observer her winding number?

Put it another way, as they pass each other the first time can we predict which one will have the longer proper time interval between the two meeting events?

Do you see my problem?

Garth

 

[robphy]

Take again the experiment of sending out a light signal in both directions when they first separate. I have drawn the universe below... essentially unrolling the cylinder. (I think I might have missed an observer's worldline in the upper left corner.)


$$\begin{picture}(140.00,150.00)(0,0) \multiput(0.00,100.00)(0.12,-0.12){833}{\line(1,0){0.12}} \multiput(0.00,100.00)(0.12,0.12){417}{\line(1,0){0.12}} \multiput(0.00,140.00)(0.12,-0.12){1167}{\line(1,0){0.12}} \multiput(0.00,20.00)(0.12,-0.12){167}{\line(1,0){0.12}} \multiput(0.00,20.00)(0.12,0.12){1083}{\line(1,0){0.12}} \multiput(0.00,30.00)(0.12,0.18){667}{\line(0,1){0.18}} \multiput(0.00,60.00)(0.12,-0.12){500}{\line(1,0){0.12}} \multiput(0.00,60.00)(0.12,0.12){750}{\line(1,0){0.12}} \multiput(100.00,0.00)(0.12,0.12){333}{\line(1,0){0.12}} \multiput(100.00,0.00)(0.12,0.18){333}{\line(0,1){0.18}} \multiput(110.00,150.00)(0.12,-0.12){250}{\line(1,0){0.12}} \multiput(20.00,0.00)(0.12,0.12){1000}{\line(1,0){0.12}} \multiput(20.00,0.00)(0.12,0.18){833}{\line(0,1){0.18}} \multiput(30.00,150.00)(0.12,-0.12){917}{\line(1,0){0.12}} \multiput(60.00,0.00)(0.12,0.12){667}{\line(1,0){0.12}} \multiput(60.00,0.00)(0.12,0.18){667}{\line(0,1){0.18}} \multiput(70.00,150.00)(0.12,-0.12){583}{\line(1,0){0.12}} \put(20.00,0.00){\line(0,1){150.00}} \put(60.00,0.00){\line(0,1){150.00}} \put(100.00,0.00){\line(0,1){150.00}} \put(140.00,0.00){\line(0,1){150.00}} \put(35.99,24.05){\circle{2.00}} \put(51.97,48.03){\circle{2.00}} \put(67.89,72.11){\circle{2.00}} \put(83.95,96.05){\circle{2.00}} \put(100.26,119.47){\circle{2.00}} \end{picture}$$

As the signals travel around the closed universe,
one observer receives the incoming signals at the same event(s).
The other observer does not.

When they meet at the first reunion, they can compare the number of times they received that signal. In this example, the first observer received "a single signal from both directions", while the other received "two from the forward direction, but none from the backward direction".

When they meet at the second reunion, they can compare again. In this example, the first observer received "three signals from both directions", while the other received "four from the forward direction, then a fifth from both directions".

Clearly they disagree.

One can probably work out the number of signal receptions for arbitrary observer velocities and radii of the closed universe and relate that to the winding number.

 

[Alkatran]

Of course, this argument assumes it is the speed in the space you exist on that matters, maybe your 'acceleration' to go around the sphere matters?

 

[Garth]

As the signals travel around the closed universe,
one observer receives the incoming signals at the same event(s).
The other observer does not.

When they meet at the first reunion, they can compare the number of times they received that signal. In this example, the first observer received "a single signal from both directions", while the other received "two from the forward direction, but none from the backward direction".

When they meet at the second reunion, they can compare again. In this example, the first observer received "three signals from both directions", while the other received "four from the forward direction, then a fifth from both directions".

Clearly they disagree.

Yes, I have no problem with that, and as I have said there is clearly a preferred frame of reference, but can you tell which observer is in that frame? (Without waiting for the second encounter)

Consider my case presented above.

Take a topological compact space, a cylinder, empty except for two test particles in inertial frames of reference, with an arbitrary mutual velocity and which encounter each other at least twice.

Take the local space-time volume around the first encounter. Within that volume each test particle is equivalent, and the question is, "Is it possible to predict which one will experience the longest possible proper time between the two encounters, i.e. the one in a ‘preferred frame of reference’?" I would argue that in this case the answer is, “No, you have to wait to perform an experiment to find out, such experiments as the light ray experiment or the comparison of clocks at the second encounter.” This is because, as a simple topological space, otherwise empty of physics, the global structure can be continuously deformed (twisted) to give each observer the zero winding number.

Now convert to the real universe, add in the physics and consider two observers. Suppose one observer is more or less at rest w.r.t. the surface of last scattering of the CMB, and the other has a very high relative velocity. Wait and then perform the above experiments. I would find it incomprehensible to imagine that the first observer is not in the preferred frame of reference.

If this must be the case then the cosmological distribution of matter and energy has not only determined the global geometry but also locally affected the frames of reference, selecting one as preferred against all others.

But does this not undermine the Principle of Relativity?

Garth

 

[abbott]

Observation is the core necessity of an existence that operates as a balance of opposites. Our powers of observation will always end in unpercieveability (time or space "before" the big bang, the meaning of time or space in the face of the infinity we observe them approach).

 

[abbott]

answers?

Observation is the core necessity of an existence that operates as a balance of opposites. Our powers of observation will always end in unpercieveability (time or space "before" the big bang, the meaning of time or space in the face of the infinity we observe them approach).

 

[abbott]

We (classical philosophy "who am I?" and "what is everything?") must be the knowable universes understanding of itself, after all what is anything we have observed other than a reflection of the energies we've used to detect it. And what is a living thing? A converter of energies, we sustain ourselves by intaking things and changing them into other things (sounds like a process of observation to me).

I've just recently suggested to www.superstringtheory.com that perhaps light/energy (infinity in every "direction," and a string without concievable beginning or end) is oscilating through its opposite, nothingness. I am very curious to discover if this idea makes sense to superstring theory and was very dissopointed to find that shortly after my post the forum stopped being accesible. So if anyone here knows anything about string theory and can shed some light on the way I'm trying to understand it I would be very pleased to receive that feedback.

 

[abbott]

The paradox is maybe believing that the twins are in fact not the same thing. If they are just different versions of the same perspective of knowable existence, then differences they experience in time or anything else only serve to illuminate that the only things that change are the opinions of what is percieved.

 

[Garth]

Are all observers then "the same thing"? That you and I are the same? I think not.

Garth

 

[abbott]

why don't you think so? is it because your perception cannot distiguish the thought or perception of a singular existence indefinable by its lack of an opposite?

 

[abbott]

if perception is everything, then yes it reaches the end of its factual plausability at that thought (of you and i being the same "thing"). This is also analogous of the breakdown of factual plausability brought to mind by the thought of infinity (how could anything without end be definable?). We always reach a vanishing point of perception, a meaning where everything seems to lead but where meaning, by definition, cannot go.

 

[Garth]

But the Twin Paradox (after all that is what this thread is about) is saying the two separate observers are different. In the cosmological case - one has a winding number of zero, and the other of one.

Garth

 

[robphy]

Yes, I have no problem with that, and as I have said there is clearly a preferred frame of reference, but can you tell which observer is in that frame? (Without waiting for the second encounter)

Consider my case presented above.

Take a topological compact space, a cylinder, empty except for two test particles in inertial frames of reference, with an arbitrary mutual velocity and which encounter each other at least twice.

Take the local space-time volume around the first encounter. Within that volume each test particle is equivalent, and the question is, "Is it possible to predict which one will experience the longest possible proper time between the two encounters, i.e. the one in a ‘preferred frame of reference’?" I would argue that in this case the answer is, “No, you have to wait to perform an experiment to find out, such experiments as the light ray experiment or the comparison of clocks at the second encounter.” This is because, as a simple topological space, otherwise empty of physics, the global structure can be continuously deformed (twisted) to give each observer the zero winding number.

Now convert to the real universe, add in the physics and consider two observers. Suppose one observer is more or less at rest w.r.t. the surface of last scattering of the CMB, and the other has a very high relative velocity. Wait and then perform the above experiments. I would find it incomprehensible to imagine that the first observer is not in the preferred frame of reference.

If this must be the case then the cosmological distribution of matter and energy has not only determined the global geometry but also locally affected the frames of reference, selecting one as preferred against all others.

But does this not undermine the Principle of Relativity?

Garth

The CMB provides a family of worldlines in the spacetime, which can define a "frame of reference" from which physics can be described. However, this does not in any way invalidate the use of another family of worldlines to describe the same physics. The laws of physics are the same.

This may help:

Cosmic Microwave Background FAQ
http://www.astro.ubc.ca/people/scott/faq_basic.html

How come we can tell what motion we have with respect to the CMB?

Doesn't this mean there's an absolute frame of reference?

The theory of special relativity is based on the principle that there are no preferred reference frames. In other words, the whole of Einstein's theory rests on the assumption that physics works the same irrespective of what speed and direction you have. So the fact that there is a frame of reference in which there is no motion through the CMB would appear to violate special relativity!

However, the crucial assumption of Einstein's theory is not that there are no special frames, but that there are no special frames where the laws of physics are different. There clearly is a frame where the CMB is at rest, and so this is, in some sense, the rest frame of the Universe. But for doing any physics experiment, any other frame is as good as this one. So the only difference is that in the CMB rest frame you measure no velocity with respect to the CMB photons, but that does not imply any fundamental difference in the laws of physics.

 

[Garth]

Thank you, that is my point:
"Is not the allocation of a winding number a 'law of physics' that does depend on the frame of reference?"
Garth

 

[robphy]

Thank you, that is my point:
"Is not the allocation of a winding number a 'law of physics' that does depend on the frame of reference?"
Garth

I would not call that a '[local] law of physics'.

 

[Garth]

Right:- you observe two other observers with mutual velocity. One happens to be in the cosmological co-moving frame for whom the CMB is globally isotropic. Is there a preferred frame for this situation? I think our discussion has agreed that there is, because one frame has a winding number of 'zero' and the other of 'one'. It is agreed that this acquisition of a winding number is due to the cosmological embedding of the two observers. But that is just my point, Mach's Principle (in this case represented by the winding number)* endows the local situation with a preferred frame. It is a physically determinable frame, the only problem is that one has to wait a long time to find out which is which.

I believe this case indicates GR may need to be modified, by more fully including Mach's Principle. The theory of Self Creation Cosmology does just this, and it is being tested at the moment so the paradox may be experimentally resolved before long!

Garth
*[the winding number requires a closed compact space which in turn requires a 'representative cosmological density; so the topological situation is earthed in the distribution of matter in the universe i.e. Mach's Principle]
[Edit: A generalised definition of Mach's Principle may be that local laws of physics (inertial frames, inertial mass, Newton's gravitational parameter and here the "winding number") are determined by the distribution of matter in motion in the rest of the universe.]

 

[Garth]

I would not call that a '[local] law of physics'.

robphy - thank you for this discussion, I am finding it most illuminating. However, to tease out the question further, do you not think then that relative clock rate constitutes a 'local' law? For have we not established that the 'winding number' defines which observer has the 'slowest' proper time in an 'absolute' (cosmological) sense?

Garth

 

[Garth]

A paper published today (17th March) gr-qc/0503070] On the Twin Paradox in a Universe with a Compact Dimension [/URL] has this to say on the subject:

We consider the twin paradox of special relativity in a universe with a compact spatial dimension. Such topology allows two twin observers to remain inertial yet meet periodically. The paradox is resolved by considering the relationship of each twin to a preferred inertial reference frame, which exists in such a universe because global Lorentz invariance is broken. The twins can perform “global” experiments to determine their velocities with respect to the preferred reference frame (by sending light signals around the cylinder, for instance). Here we discuss the possibility of doing so with local experiments. Since one spatial dimension is compact, the electrostatic field of a point charge deviates from 1/r². We show that although the functional form of the force law is the same for all inertial observers, as required by local Lorentz invariance, the deviation from 1/r² is observer-dependent. In particular, the preferred observer measures the largest field strength for fixed distance from the charge.

So it does appear that the topology of the universe at large affects local physics so that local experiments can distinguish a preferred inertial reference frame.

My question again is, "Does the existence of an experimental determination of a preferred frame in such a universe constitute an inconsistency in the principles of GR?"

Garth

 

[JesseM]

So it does appear that the topology of the universe at large affects local physics so that local experiments can distinguish a preferred inertial reference frame.

My question again is, "Does the existence of an experimental determination of a preferred frame in such a universe constitute an inconsistency in the principles of GR?"

My guess is that the deviation from the inverse-square law would be explained by the fact that a charge's field will wrap around the universe and exert a force on the charge that's creating it. So although you can measure this effect locally, it is really a consequence of of an external field exerting on an influence on what's going on in your local region. The GR assumption that all observers moving along geodesics will observe the same local physics must include the tacit assumption that each observer's local region is shielded from all external nongravitational forces, like a measurement made within a small faraday cage--if one observer is moving through an applied EM field and the other is not, they will observe charges in their neighborhood behaving differently, but surely this does not in itself violate the equivalence principle! In the case of the compact universe, if the two observers are shielded from external fields I assume neither will see a violation of the inverse-square law.

 

[yogi]

A similar but not identical situation to the cosmological twin problem occurs if there are two satellite clocks A and B in identical orbits but moving in opposite directions - If they cross at point X and sync their clocks, then when they have each completed one orbit and pass again at X, each will believe the other clock is running slow - but with only this information, you cannot tell which if either is running slow. Suppose clock A and B are originally on the same satellite and synchronized togther while traveling at orbital velocity v East to West and when point X is reached, clock B is quickly accelerated to -v so that it now travels at orbital velocity v West to East. Which if either of the two clocks will have logged a greater time when they meet again at X?
As with the cosmological twin paradox, the conditions under which the initial synchronization takes place must be unambiguously defined.

 

[Ich]

yogi - both will read the same time, regardless of synchronization conditions.

 

[JesseM]

If they cross at point X and sync their clocks, then when they have each completed one orbit and pass again at X, each will believe the other clock is running slow

Do you mean each will believe the other clock is running slow in their instantaneous inertial rest frames at the moment they pass? Or do you mean each will think the other is running slow throughout the entire orbit in some sort of non-inertial coordinate system where it sees itself as being at rest throughout the journey? If the second, you're wrong--as long as you're using the laws of physics correctly, they can't get incorrect predictions about what the other clock will read when they pass each other, and we already know (by considering things from the inertial frame of the Earth's center) that they will read the same time when they pass each other. So depending on how you define your non-inertial coordinate system, each would either have to see the other running slow at some moments and fast at others, or each would see the other running at the same rate as itself throughout the trip.

 

[yogi]

What procedure is available for synchronising "on the fly" - none that I am aware of - and that is the problem with the cosmological twin and the orbiting twin - when they simply pass each other initially, they can read each others time at one instant - but that does not sync the clocks ...to sync as per Einstein, both clocks must be at rest in the same frame initially. When so synced, there is no paradox because only one will have accelerated, whether it be to circumnavigate the universe via a geodesic or to orbit a massive body...and once put in motion by an initial acceleration, the path integral for the worldline of the accelerated twin will always include a spatial component which must be subtracted from the temporal component, whereas the non accelerated twin's worldline only includes a temporal element.

 

[JesseM]

What procedure is available for synchronising "on the fly" - none that I am aware of - and that is the problem with the cosmological twin and the orbiting twin - when they simply pass each other initially, they can read each others time at one instant - but that does not sync the clocks ...to sync as per Einstein, both clocks must be at rest in the same frame initially.

There's an ambiguity in the word "synchronization", you're right that it is often only taken to apply to clocks at rest in a single inertial reference frame, but in the cosmological twin paradox it just means that the twins set their clocks to read the same time at the moment they pass each other, it doesn't mean their clocks stay synchronized in any sense.

When so synced, there is no paradox because only one will have accelerated, whether it be to circumnavigate the universe via a geodesic or to orbit a massive body

In the cosmological twin paradox, neither twin accelerates, unless you imagine they both start off on Earth and then one accelerates away...but the paradox works equally well (and is conceptually simpler) if you imagine both have been moving inertially since the beginning of time. Plus, even if one twin accelerates away from the earth, there's no reason that the Earth would necessarily be at rest in the preferred coordinate system determined by the topology of the universe, so it's possible that after accelerating the traveling twin will be moving slower in this coordinate system, meaning that he will have aged more when he meets the earth-twin again, not less. If you think the question of whether the traveling twin accelerates away from the Earth or just passes it while moving inertially is at all relevant here, then I think you're still suffering from the same confusion about the importance of acceleration that you were on the twins paradox thread...if you do think that, then please address my question on that thread about what you meant by the word "moved":

yogi:
The issue is not which frame accelerated most recently - it is simply this - two clocks in one frame are brought in sync - one is moved. It will be found to be out of sync with the one which did not move - and the amount of the discrepency is given in Part 4 of the 1905 paper.

JesseM:
What does "moved" mean? Does it mean "accelerated", or does it just mean that A is moving relative to the rest frame of B and E? If acceleration isn't relevant, why did you say "at least as I understand your question, I would have to know whether the relative velocity between A and the EB frame came about by acceleration of the EB frame or A"? If acceleration is relevant, yet it doesn't matter who was accelerated most recently, then in every possible experiment both A and EB will have been accelerated at some point in their past, it's not like there are any clocks in the universe which have been traveling at constant velocity since the Big Bang. So what does this lead you to conclude about which one was "moved", and which clock is "really" ticking slower?