A variation on the twin paradox

[dicerandom]

OK, so I'm pretty familliar with the regular twin paradox and the explinations of how you decide which twin is younger by noticing which one accelerated, and the geometrical view of this using k-calculus and minkowsky diagrams, etc. I just thought of something though, and I guess my relativity isn't strong enough to think of a solution, at least not in the period of time it took me to go make some tea

Suppose that we live in a closed universe, for simplicity's sake let's also assume that universe is balanced with a cosmological constant so that it is neither expanding nor contracting (ala Einstein's static universe model). If I'm in a spaceship and I go flying by my friend at a very high velocity, and then wait for some time, I should wrap around and go flying past him a second time. If we set both our clocks to zero at the first flyby, what will our clocks read when we pass each other the second time?

I'm at a loss to try and explain any differences in the clocks without attaching a preferred reference frame to "the surface of the universe" which my friend is allegedly still with respect to, which of course sets off all of my in-built relativity alarms. I'm tempted to say that we'll both have the same number on our watches when we pass the second time and ascribe it to a general relatavistic "you can't compare time or space intervals at a distance" effect.

 

[dicerandom]

I found this article: http://www.findarticles.com/p/articles/mi_qa3742/is_200108/ai_n8954244

And I'm bothered by this paragraph:

The statement of the closed universe twin paradox is completely symmetrical-- Albert and Betty travel in opposite directions from one another until they reunite-- yet the resolution is asymmetrical-Albert's erstwhile twin is now a little sister 10 years younger than he is. How can this be? The answer is that we broke the problem's symmetry when we constructed the spacetime diagram. Albert occupies a very special inertial frame. His is the only frame in which lines of constant time are closed circles. In all other frames the lines of constant time are helices. The faster an observer is moving relative to Albert, the steeper the pitch of the helix.

Couldn't we make the same argument if we chose to draw Betty's spacetime diagram instead?

 

[JesseM]

If you try to draw each observer's line of simultaneity in the usual way, the line of simultaneity in the +x direction won't necessarily match up with the line of simultaneity in the -x direction. So if one twin tried to use a badly-behaved coordinate system like this, she might conclude that her twin was "currently" at position x=8 light years and aged 35 years, but also "currently" at position x=-5 light years and aged 32 years, something like that. So maybe she's heading off in the +x direction and thinking that both "copies" of her twin are aging slower than she is, but the "copy" that already lies in her +x direction (as opposed to the one she just left behind in the -x direction) already has a "head start" in age, so even though that copy is aging slower too he can still be older when they reunite. On the other hand, there will be a "preferred frame" where the lines of simultaneity in both directions will meet up smoothly, so you don't have to worry about such problems in this coordinate system.

 

[Hurkyl]

a general relatavistic "you can't compare time or space intervals at a distance" effect.

You are getting close.

It is correct to note that the laws of special relativity are only guaranteed to hold on short scales, and that reference frames are only really meaningful at infinitessimal distances.

Special relativistic time dilation is, of course, a measurement that depends on a choice of reference frame.

I'm at a loss to try and explain any differences in the clocks without attaching a preferred reference frame to "the surface of the universe" which my friend is allegedly still with respect to, which of course sets off all of my in-built relativity alarms. I'm tempted to say that we'll both have the same number on our watches when we pass the second time and ascribe it to a general

But you can't just do that. The problem with this scenario is that there isn't a shortcut you can use. You (more or less) actually have to do the calculations to compare.

Couldn't we make the same argument if we chose to draw Betty's spacetime diagram instead?

To restate what JesseM said, what is happening is this:

The argument assumes that space-time is shaped like a cylinder -- i.e. if we took a flat (1+1)-dimensional space-time (i.e. the space-time of special relativity), and then we rolled it up into a cylinder.

In some sense, cylinders do have a distinguished direction: along the axis. Of course, this direction can only be distinguished with a global experiment: one that actually involves one observer having gone one more loop around the universe than another observer.

In other words, this distinguished direction has nothing to do with physics: it is simply related the shape of the universe.

For them to draw the space-time diagrams they did, they assumed that Albert was traveling in the distinguished direction. And if Albert is traveling in the distinguished direction, then Betty clearly is not, so the problem is not symmetrical. Of course, it could have been Betty (or neither) who is traveling in the distinguished direction.

 

[dicerandom]

Thanks for your comments everyone, I think I have a better grasp on this now. I do have one more related question though.

In the example discussed in the article I linked above, and more generally in the Robertson-Walker metric, space is treated as being curved whereas the temporal direction has no curvature. For reference the line element in the Robertson-Walker metric is:

ds^2 = -dt^2 + a(t)^2 \left[ dr^2 + S_{\kappa}(r)^2 d\Omega^2 \right]

And a(t)=1 for a static configuration, of course. Extending the arguments posed for the 1D spatial case I believe that this entire class of metrics will exhibit identical behavior, at least in the case when \kappa=1 (positive uniform curvature, closed universe).

Is it possible to have a metric for a closed universe where the temporal dimension is included on the surface of the hypersphere, and if so would this eliminate the preferred coordinate system?

 

[Garth]

This is the cosmological twin paradox, which has already been discussed several times.

It is not so easily dismissed. For example Barrow and Levin discuss it in The twin paradox in compact spaces in which they say:

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 Uzan et al. in Twin paradox and space topology in which they say:

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. That frame of reference is defined by the 'cosmic fluid'.

The paradox has arisen because we have treated the compact space as an empty topological space, in fact to get the universe to be closed it will have to have matter within it. That matter itself can be used to define the preferred frame of reference, the one co-moving with the cosmic fluid; however, you may feel that to do so relies on a use of Mach's Principle that GR cannot bear.

The link in the second post #2 above, contains the statement:

We have seen that the traditional lesson of special relativity--that all inertial frames are equivalent--applies only locally.

Yet GR takes this local symmetry of the equivalence of all inertial frames and through the conservation of energy-momentum and the stress-energy tensor treats it globally. Perhaps this is where the point is stetched too far?

Garth

 

[Zanket]

Suppose that we live in a closed universe, for simplicity's sake let's also assume that universe is balanced with a cosmological constant so that it is neither expanding nor contracting (ala Einstein's static universe model). If I'm in a spaceship and I go flying by my friend at a very high velocity, and then wait for some time, I should wrap around and go flying past him a second time. If we set both our clocks to zero at the first flyby, what will our clocks read when we pass each other the second time?

I assume that you and your friend were once at rest with respect to each other, and then you accelerated to make the flybys. Had you synchronized your watches before you initially accelerated, then just before re-synchronizing your watches on the first flyby, your watch will show less time elapsed. It will show less time elapsed on the second flyby too, because the cause of the difference in elapsed time is the same for both flybys—it is your acceleration.

The asymmetry in the twin paradox, including this cosmological twin paradox, is acceleration. Whoever accelerates experiences length contraction. Distances contract along your axis of motion when you accelerate in your spaceship, and expand from their contracted state when you decelerate to come to rest with respect to your friend. During your trip, both you and your friend are moving, relative to each other. But your friend does not experience length contraction of distances. Both of you travel at the same velocity relative to each other, but you travel less distance. Traveling less distance between flybys than your friend does, at the same velocity your friend has, takes less time for you than it does for your friend. It’s no more complicated than that.

 

[dicerandom]

The asymmetry in the twin paradox, including this cosmological twin paradox, is acceleration. Whoever accelerates experiences length contraction. Distances contract along your axis of motion when you accelerate in your spaceship, and expand from their contracted state when you decelerate to come to rest with respect to your friend. During your trip, both you and your friend are moving, relative to each other. But your friend does not experience length contraction of distances. Both of you travel at the same velocity relative to each other, but you travel less distance. Traveling less distance between flybys than your friend does, at the same velocity your friend has, takes less time for you than it does for your friend. It’s no more complicated than that.

I don't think what you've written here is in agreement with the solutions which have been discussed both here and in the papers linked, at least not in the cosmological version.

For instance, supposing my twin and I were initially in a frame such that our time axis was not aligned with the coordinate time, i.e. the vertical axis of the cylinder in the first paper I linked. If I accelerate until my time axis is aligned with the coordinate time, and then synchronize watches with my twin when we pass, he will then read a smaller proper time when we pass again.

 

[Garth]

I assume that you and your friend were once at rest with respect to each other,

No, they do not have to have been at rest.

Two inertial observers, moving mutually at high speed, happen to pass each other when they exchange signals and synchronise clocks.

A very long time later they pass close by each other again and compare clocks, one of them having circumnavigated the universe, but which one?

Garth

 

[Zanket]

For instance, supposing my twin and I were initially in a frame such that our time axis was not aligned with the coordinate time, i.e. the vertical axis of the cylinder in the first paper I linked. If I accelerate until my time axis is aligned with the coordinate time, and then synchronize watches with my twin when we pass, he will then read a smaller proper time when we pass again.

Sorry, talk of spacetime diagrams makes my eyes glaze over. I assume you’re talking about a situation where both you and your friend accelerated to attain high velocity relative to the more or less fixed galaxies, but stay at rest with respect to each other, and then you decelerated relative to the galaxies to make the flybys relative to your friend. Yes, in that case your friend’s clock elapses less time than yours does between flybys, because between flybys the friend travels less distance relative to you than you travel relative to your friend, and your velocities relative to each other are identical. When you decelerated relative to the galaxies then distances (like those between galaxies) along your axis of motion uncontracted from a contracted state, whereas distances remain contracted for your friend. This situation is just a fancy way of making the “stay at home” twin look like the traveling twin and vice versa.

If you’re talking about some other situation, can you put it in plain English?

 

[pervect]

The cosmological twin paradox has been discussed here before, and in certain hypothetical space-times (which are not expected to be our own) it is possible to circumnavigate the universe without ever accelerating. These sorts of hypothetical universe are where the paradox arises, and acceleration is not the solution to the paradox in these instances.

Thus I have to agree with Garth and Dicerandom.

 

[Zanket]

Two inertial observers, moving mutually at high speed, happen to pass each other when they exchange signals and synchronise clocks.

A very long time later they pass close by each other again and compare clocks, one of them having circumnavigated the universe, but which one?

That depends on what or who you choose the circumnavigation to be relative to. As Einstein said, there’s no universal hitching post. Regardless who is declared to be circumnavigating, the driver of the spaceship or the driver’s friend, the one who observes the greatest length contraction of distances between galaxies elapses less time between flybys.

 

[Hurkyl]

the one who experiences the greatest length contraction

There's no such thing as "experiencing" length contraction.

 

[Zanket]

Like it better now?

 

[JesseM]

That depends on what or who you choose the circumnavigation to be relative to. As Einstein said, there’s no universal hitching post. Regardless who is declared to be circumnavigating, the driver of the spaceship or the driver’s friend, the one who observes the greatest length contraction of distances between galaxies elapses less time between flybys.

There's no reason in principle that all the galaxies in this universe couldn't have some substantial velocity relative to the coordinate system whose time axis is the vertical axis of the cylinder. In this case, one twin could be at rest relative to the galaxies and the other could be in motion relative to them, yet it would be the one who was at rest relative to the galaxies who would elapse less time between flybys.

 

[Hurkyl]

Nobody in that example would "observe" length contraction -- for each observer, the distance between galaxies would remain constant.


Also, it is certainly possible for the twin who perceives the greater distance between galaxies to be the one who experiences less time between flybys.

 

[Zanket]

There's no reason in principle that all the galaxies in this universe couldn't have some substantial velocity relative to the coordinate system whose time axis is the vertical axis of the cylinder. In this case, one twin could be at rest relative to the galaxies and the other could be in motion relative to them, yet it would be the one who was at rest relative to the galaxies who would elapse less time between flybys.

I agree with that. Good luck accelerating all those galaxies though!

 

[Zanket]

Nobody in that example would "observe" length contraction -- for each observer, the distance between galaxies would remain constant.

How did you gather that from Garth's example?

Also, it is certainly possible for the twin who perceives the greater distance between galaxies to be the one who experiences less time between flybys.

Yes, if the galaxies are accelerated along with the traveling twin, as JesseM points out.

 

[JesseM]

I agree with that. Good luck accelerating all those galaxies though!

Why would they need to be accelerated? Why do you assume that their "initial" velocity would be at rest relative to the vertical cylinder axis coordinate system?

 

[Zanket]

Because if that assumption were false then, in tests of the twin paradox in special relativity, objects accelerated in some directions would show more time elapsed on their clocks relative to the unaccelerated clock, rather than the less time they do elapse when accelerated in any direction.

 

[JesseM]

Because if that assumption were false then, in tests of the twin paradox in special relativity, objects accelerated in some directions would show more time elapsed on their clocks relative to the unaccelerated clock, rather than the less time they do elapse when accelerated in any direction.

Only if they circumnavigated a closed universe. Any non-circumnavigating trip would still work the same way as in SR, regardless of the earth-twin's velocity relative to the vertical cylinder axis coordinate system.

 

[Zanket]

I disregard the "vertical cylinder axis coordinate system" stuff--makes my eyes glaze over. Whether circumnavigating the universe or a short trip, it is always possible in principle that all benchmark reference points, like the observable universe, can be accelerated beforehand so that a test of the twin paradox has opposite results (i.e. traveling twin elapses more time) when the traveling twin accelerates in the the direction opposite to that of the direction in which the benchmarks were accelerated.

 

[JesseM]

I disregard the "vertical cylinder axis coordinate system" stuff--makes my eyes glaze over. Whether circumnavigating the universe or a short trip, it is always possible in principle that all benchmark reference points, like the observable universe, can be accelerated beforehand so that a test of the twin paradox has opposite results (i.e. traveling twin elapses more time)

No. In flat spacetime, whether closed like the cylindrical universe or open and infinite like in SR, any twin experiment which doesn't involve circumnavigating the universe will have the same result (the inertial twin will always have aged more), regardless of each twin's velocity relative to "benchmark reference points".

 

[Zanket]

If that were true then real paradoxes would arise, so it cannot be true.

 

[JesseM]

If that were true then real paradoxes would arise

Like what? Consider the possibility that you are just misunderstanding something here.

so it cannot be true.

It is.

 

[Zanket]

Consider the possibility that you are just misunderstanding something here.

I think you're right. I now recall going over that same subject some time ago. So how does circumnavigation of the universe change things?

 

[JesseM]

I think you're right. I now recall going over that same subject some time ago. So how does circumnavigation of the universe change things?

Well, look over the explanations that were given earlier on this thread, like mine in post #3. Basically, if one twin heads out and turns around without circumnavigating (or even if one does travel far enough to circumnavigate the universe, but then turns around and returns to the earth-twin from the same direction he came) then you can look at the whole problem in terms of a single SR-like coordinate system where the traveling twin's poxition along the x-axis is increasing as he goes out, then decreasing as he returns. But when the traveling twin circumnavigates and returns from the opposite direction, if you wanted to treat the whole thing in an SR-like coordinate system you'd have to imagine a hall-of-mirrors effect where you have a series of repeating copies of each twin along your x-axis, with the copy of the traveling twin that returns to Earth being different from the one that departed, and with the clocks of all the twins only being synchronized in the frame whose time axis matches the cylinder's vertical axis. So I guess that's the key issue, whether the traveling twin returns to the same copy of the Earth that he departed from, in terms of this hall-of-mirrors effect you'd see in a finite closed universe.

 

[dicerandom]

But when the traveling twin circumnavigates and returns from the opposite direction, if you wanted to treat the whole thing in an SR-like coordinate system you'd have to imagine a hall-of-mirrors effect where you have a series of repeating copies of each twin along your x-axis, with the copy of the traveling twin that returns to Earth being different from the one that departed, and with the clocks of all the twins only being synchronized in the frame whose time axis matches the cylinder's vertical axis.

From my understanding of the discussion I believe this "hall of mirrors" effect will only happen in the coordinate system of the twin whose time axis is not vertical. The twin with the vertical time axis will have the standard closed universe effect where she can see many copies of a particular object, due to the fact that very old light has circumnavigated the universe multiple times, but she will not conclude that there is more than one simultaneous copy of either her or her twin.

 

[Hurkyl]

but she will not conclude that there is more than one simultaneous copy of either her or her twin.

Why not? The map from "coordinates" to the universe is still periodic, and the same event corresponds to many different points in her "coordinate chart".

It's exactly the same for her as it is for the other twin -- the only difference is that her coordinates are periodic along one of her line of simultaneity, but the other twin's coordinates are periodic along some different space-like vector.

IOW, if it's right to describe one twin's chart as seeing "multiple copies", then it is for the other too.

 

[dicerandom]

I think that the twin who has the vertical time axis has a coordinate distance that is simply a circle on the surface of the cylinder, it would be periodic in the sense that she could say that she was at x=2\pi R_0 n, where R_0 is the radius of curvature and n is an integer. All of these "copies" of herself will have the same coordinate t value though. I see this as being distinctly different from the other twin, her x-axis forms a helix on the surface of the cylinder and intercepts her own world line periodically, she sees multiple simultaneous copies of herself at different coordinate (and proper) t values.

 

[Garth]

If that were true then real paradoxes would arise, so it cannot be true.

There is a real paradox here - read my posts #6, #9 above.

Garth

 

[RandallB]

There is a real paradox here - read my posts #6, #9 above.

No, that’s not a paradox.
Just because they put the same problem in small dimensions and call it a “compact space”. It’s still no different than the spaceship twins or a GPS problem.
Expand the compact orbit to going around your house just down the street a few blocks and back behind the house on the return. Then the twin continues down along the back ally the other way, to come back in front of your house going the same way as always in front of the house.
When you notice them also zipping by going the other way though the back yard you realize it’s just like the spaceship twin going out, turning around to come back – continuing the other way out and back again, over and over.

Same deal with GPS, or if you shrink it down to orbit your living room table.
Unless you make a similar trip at the same speed, doesn’t matter which direction, you will age faster than the traveler, doesn’t matter how “compact” the space they travel in is.

And as far as figuring who is really circumnavigating some part of the universe, be it around your living room table or the galaxy. If the speed difference is near the speed of light, the one that measures the CBR with a big blue shift in one direction and an exaggerated red shift the other way is moving big time & aging slower, that much cannot be hidden.
It’s just basic SR.

 

[Garth]

SR it is not, the closed universe requires GR.

It seems that you have not understood the problem at all. Try reading the published paper The twin paradox in compact spaces by Barrow and Levin before making pronouncements like the one above.

The key to the problem is that in a finite universe (a topological compact space) not only will two observers moving at high speed relative to each other, (after first passing close by each other when they synchronise clocks,) each think the other is aging more slowly than they are, but also they can compare clocks again on a second close encounter, (because one of them has circumnavigated the universe), and check which is actually 'stationary' in the 'preferred' frame of reference.

But which one? Which observer has had the greatest time elapse?

Both observers have been in inertial frames of reference all along, they have been able to meet again only because of the large scale topology of the universe, not because they have 'gone round the block a few times'.

In an empty universe there is no way to distinguish between the two observers. Each will think they are the stationary one and the other is the one who has circumnavigated the universe.

The paradox hangs on, and is resolved by, the fact that, such a closed universe cannot be empty, it has to have matter or energy in it, and that by observing that other matter/energy in the universe one observer can establish that they are the one who is more or less stationary.

Local frames of reference are affected by the large scale distribution and motion of matter in the rest of the universe as Mach's Principle suggests, but which GR does not fully include.

Garth

 

[JesseM]

The paradox has arisen because we have treated the compact space as an empty topological space, in fact to get the universe to be closed it will have to have matter within it.

I don't think that's correct, the issue of topology is separate from the issue of curvature. You need matter to get a "closed universe" in the sense of the standard Friedmann-Robertson-Walker cosmological models, where the universe is only closed if it has positive curvature, but these models assume the simplest possible topology, it is also possible to have a universe with zero curvature that is closed due to some unusual topology. Some pages on this:

http://plus.maths.org/issue10/features/topology/
http://astro.uchicago.edu/home/web/olinto/courses/A18200/nbower.htm

In fact, some physicists have looked at the cosmological background radiation for possible evidence that our universe is closed due to such an unusual topology, even though they do not disagree with the evidence that the universe is spatially flat. See here:

http://www.hep.upenn.edu/~angelica/topology.html
http://www.etsu.edu/physics/etsuobs/starprty/120598bg/section7.htm
http://news.bbc.co.uk/2/hi/science/nature/3175352.stm

 

[robphy]

But which one? Which observer has had the greatest time elapse?

Both observers have been in inertial frames of reference all along, they have been able to meet again only because of the large scale topology of the universe, not because they have 'gone round the block a few times'.

In an empty universe there is no way to distinguish between the two observers. Each will think they are the stationary one and the other is the one who has circumnavigated the universe.

Let me remind you of my experiment https://www.physicsforums.com/showpost.php?p=367371&postcount=13 from the The Cosmological Twin Paradox thread you posted earlier.

 

[JesseM]

No, that’s not a paradox.
Just because they put the same problem in small dimensions and call it a “compact space”. It’s still no different than the spaceship twins or a GPS problem.

It is different, because in a compact space both twins can travel inertially away from each other and then reunite later without either needing to accelerate or turn around. An analogy that's sometimes used for a compact space is a video game like "asteroids", where if you go off the edge of the screen on the right you reappear on the right edge of the screen moving at the same speed and in the same direction. If the same is true about disappearing from the top edge of the screen and reappearing on the bottom, then your universe has the topology of a torus, as is discussed on http://astro.uchicago.edu/home/web/olinto/courses/A18200/nbower.htm that I linked to, and illustrated with this diagram:

http://astro.uchicago.edu/home/web/olinto/courses/A18200/fig4.gif

Expand the compact orbit to going around your house just down the street a few blocks and back behind the house on the return. Then the twin continues down along the back ally the other way, to come back in front of your house going the same way as always in front of the house.
When you notice them also zipping by going the other way though the back yard you realize it’s just like the spaceship twin going out, turning around to come back – continuing the other way out and back again, over and over.

Except that in a closed universe, there's no need for either twin to turn around, they can both travel away from each other at constant velocity and still meet up again later.

Unless you make a similar trip at the same speed, doesn’t matter which direction, you will age faster than the traveler, doesn’t matter how “compact” the space they travel in is.

Not true, if the Earth has a larger velocity in the frame where the size of the compact space is largest (it will be different in different frames due to Lorentz contraction), then it will be the traveling twin who is older when they reunite.

And as far as figuring who is really circumnavigating some part of the universe, be it around your living room table or the galaxy. If the speed difference is near the speed of light, the one that measures the CBR with a big blue shift in one direction and an exaggerated red shift the other way is moving big time & aging slower, that much cannot be hidden.

Only if you assume that the CBR's rest frame is also the frame where the size of the compact space is maximized, but there's no need to assume such a thing, or even to assume there is a CBR in this hypothetical universe (after all, if spacetime is flat as is usually assumed in the cosmological twin paradox, then this must be an eternal flat spacetime rather than an expanding universe).

 

[Hurkyl]

Not true, if the Earth has a larger velocity in the frame where the size of the compact space is largest (it will be different in different frames due to Lorentz contraction)

How do you measure distance around? I don't remember it being anywhere near as straightforward as that.

 

[JesseM]

How do you measure distance around? I don't remember it being anywhere near as straightforward as that.

I might be wrong that you can do it this way, I just got tired of saying the "cylinder vertical axis frame" and I figured this would be interchangeable...but if I'm wrong you can always substitute that phrase back in place of the thing about the distance being maximized. I had assumed that you could just look at the distance between different "copies" of a single marker like the earth, but of course if you used different markers that didn't have the same velocity you'd get different answers to which frame maximized the difference between copies, I didn't really think about this. The distance for a given marker would always be maximized in the frame where that marker is at rest, I guess...but if you looked at the distance between copies of a marker A in its own rest frame, then compared with the distance between copies of a marker B in B's rest frame, wouldn't these distances be different? Would there be a single frame that had the property that the distance between copies of a marker which are at rest in that frame (as measured in that frame) would be larger than the distance between copies of markers which were at rest in any other frame (as measured in their own rest frame)? And would this frame be the same as the "vertical cylinder axis frame"?

 

[RandallB]

It is different, because in a compact space both twins can travel inertially away from each other and then reunite later without either needing to accelerate or turn around. An analogy that's sometimes used for a compact space is a video game like "asteroids", where if you go off the edge

No a proper analogy would be two GPS orbiting in opposite directions.
Both age the same but slower than stationary.

Except that in a closed universe, there's no need for either twin to turn around, they can both travel away from each other at constant velocity and still meet up again later.

Again same as two GPS plus they will never meet in a real closed universe with the BigBang horizon limit. They can only go around a segment of it.

there's no need to even assume there is a CBR in this hypothetical universe

I'm assuming that if Garth wants to deal with a real paradox, it is in a real universe.
I'm sure we can write in a paradox in Star Trek as well.

 

[JesseM]

No a proper analogy would be two GPS orbiting in opposite directions.

Why is this a good analogy? They are moving non-inertially, so there's no reason each should see the other as ticking more slowly, which is the source of the apparent paradox in the cosmological twin paradox.

Again same as two GPS plus they will never meet in a real closed universe with the BigBang horizon limit. They can only go around a segment of it.

Depends on whether the horizon is larger or smaller than the distance you must travel to circumnavigate the universe.

I'm assuming that if Garth wants to deal with a real paradox, it is in a real universe.
I'm sure we can write in a paradox in Star Trek as well.

Paradoxes in the fundamental laws of physics would still expose problems with these laws, and thus be "real paradoxes", even if the paradox could only arise in a universe with a different set of initial conditions than ours had. Of course the paradox is only an apparent one in the cosmological twin paradox, but it's worth trying to understand why.

Anyway, even if you want to stick to discussing our universe, if our universe had an unusual topology that made it flat yet finite, could we be certain that the equivalent of the "cylinder vertical axis frame" would also be the frame of the CMBR? I'm not too sure.

 

[Garth]

Let me remind you of my experiment https://www.physicsforums.com/showpost.php?p=367371&postcount=13 from the The Cosmological Twin Paradox thread you posted earlier.

And my answer
https://www.physicsforums.com/showpost.php?p=367371&postcount=15

We agree, there is a real paradox and

I'm assuming that if Garth wants to deal with a real paradox, it is in a real universe.

I do.

Garth

 

[JesseM]

I'm assuming that if Garth wants to deal with a real paradox, it is in a real universe.

I do.

But do you agree that general relativity allows for the possibility of a closed universe where spacetime is flat like in SR? Do you agree that your earlier statement "in fact to get the universe to be closed it will have to have matter within it" is not correct, in other words?

 

[pervect]

Just my $.02. I don't see the cosmological twin paradoxes as being any worse than (or much different than) that of other similar issues arising with large scale topological issues. Wormholes come to mind.

 

[Garth]

But do you agree that general relativity allows for the possibility of a closed universe where spacetime is flat like in SR? Do you agree that your earlier statement "in fact to get the universe to be closed it will have to have matter within it" is not correct, in other words?

Nobody, AFAIK, has demonstrated how GR might allow for that possibility. If the universe has some unusual topology, a torus for example, and the cosmological principle still holds, then that would require a modification of GR.

Garth

 

[JesseM]

Nobody, AFAIK, has demonstrated how GR might allow for that possibility.

GR allows for flat spacetime, and I'm pretty sure it allows for any topology you like, as long as the Einstein field equations are satisfied everywhere (which wouldn't be a problem in an empty flat universe). For example, on p. 725 of Misner-Thorne-Wheeler's Gravitation, after discussing the "hypersurfaces of homogoneity" (slices of spacetime in which the distribution of matter/energy throughout space is homogoneous) for universes with positive, flat, or negative curvature in the Friedmann cosmological models, they write:

D. Nonuniqueness of Topology

Warning: Although the demand for homogeneity and isotropy determines completely the local geometric properties of a hypersurface of homogeneity up to the singl disposable factor K, it leaves the global topology of the hypersurface undetermined. The above choices of topology are the most straightforward. But other choices are possible.

This arbitrariness shows most simply when the hypersurface is flat (k=0). Write the full spacetime metric in Cartesian coordinates as

ds^2 = -dt^2 + a^2 (t) [dx^2 + dy^2 + dz^2 ] (16)

Then take a cube of coordinate edges L

0 < x < L, 0 < y < L, 0 < z < L,

and identify opposite faces (process similar to rolling up a sheet of paper into a tube and gluing its edges together; see last three paragraphs of 11.5 for detailed discussion). The resulting geometry is still described by the line element (16), but now all three spatial coordinates are "cyclic," like the \phi coordinate of a spherical coordinate system:

(t, x, y, z) is the same event as (t, x+L, y+L, z+L).

The homogenous hypersurfaces are now "3-toruses" of finite volume

V = a^3 L^3,

analogous to the 3-toruses which one meets under the name "periodic boundary conditions" when analyzing electron waves and acoustic waves in solids and electromagnetic waves in space.

Another example: The 3-sphere described in part A above (case of "positive curvature") has the same geometry, but not the same topology, as the manifold of the rotation group, SO(3) [see exercises 9.12, 9.13, 10.16, and 11.12]. For detailed discussion, see for example Weyl (1946), Coxeter (1963), and Auslander and Markus (1959).

They don't seem to be expressing any doubt here that the theory of "general relativity" as it's usually defined allows for such unusual topologies, although I suppose that's separate from the question of whether such topologies are allowed by the actual "laws of nature", whatever that means.

If the universe has some unusual topology, a torus for example, and the cosmological principle still holds, then that would require a modification of GR.

If by "cosmological principle" you mean the requirement that the universe be homogoneous and isotropic, this is not part of GR, it's an additional constraint on possible universes invented by cosmologists which doesn't follow directly from any fundamental principles of physics. But anyway, the quote above suggests you can still have "hypersurfaces of homogeneity" in a universe with a weird topology (and certainly an empty flat universe with the topology of a torus would be homogeneous and isotropic, no?)

 

[RandallB]

No a proper analogy would be two GPS orbiting in opposite directions.

Why is this a good analogy? They are moving non-inertially, so there's no reason each should see the other as ticking more slowly, which is the source of the apparent paradox in the cosmological twin paradox.

Because that’s not true. Only a misunderstanding of SR would expect one satellite to see the other as aging more slowly. They see themselves both aging the same and see the stationary Earth bound (especially the one at the same altitude as they are) observer as aging faster. Just like twins traveling in opposite directions from Earth return to see each other as the same age but their classmates older. That’s just basic SR.

As to “unusual topology” for the universe, I believe we have more than enough evidence of near homogeneity in all directions that only a near spherical universe can be assumed over some hypothetical torrid or cylinder shape. Some real observations implying otherwise would be required to warrant considering them.

 

[JesseM]

Because that’s not true. Only a misunderstanding of SR would expect one satellite to see the other as aging more slowly.

But that's because the satellites are not moving inertially. For two twins moving inertially, each twin will observe the other aging more slowly than themselves as long as neither one turns around. This is even true in the cosmological twin paradox, it's just that, as discussed before, each twin observe multiple copies of the other twin in a hall-of-mirrors effect, and although each copy is aging slower, they do not all start out at the same age.

Just like twins traveling in opposite directions from Earth return to see each other as the same age but their classmates older.

But only when they return, which requires them to turn around, accelerating in the process. As long as they are both moving away from the Earth inertially, each should observe the other to be aging slower.

As to “unusual topology” for the universe, I believe we have more than enough evidence of near homogeneity in all directions that only a near spherical universe can be assumed over some hypothetical torrid or cylinder shape. Some real observations implying otherwise would be required to warrant considering them.

"Spherical"? Current observations suggest the universe is flat, not spherical. Anyway, the thing you have to understand about a "toroidal shape" is that the embedding of a 2D torus in 3D space is misleading, it looks as though the surface is curved in this embedding, when in fact a torus can have a surface that is completely flat everywhere. Again, just think of the Asteroids video game where if your ship disappears off the right edge of the screen it reappears at a corresponding point on the right edge, and likewise if it disappears off the top it reappears on the bottom. This space has the topology of a torus, but it is obviously quite flat. And something similar could be true of a flat 3D space with the topology of a torus--see http://astro.uchicago.edu/home/web/olinto/courses/A18200/fig5.jpg may also be helpful in seeing why), so if we found identical circles in opposite parts of the CMBG this would be evidence that we lived in such a universe. In fact physicists really are looking for evidence of repeating circles in the CMBG data they got from the WMAP satellite, the possibility has definitely not been ruled out yet.

 

[Hurkyl]

Because that’s not true. Only a misunderstanding of SR would expect one satellite to see the other as aging more slowly.

Maybe I'm just being too nitpicky, but that's incorrect. During each flyby, each satellite will observe the other to be running slowly. Of course, they will both agree on the time between flybys.

 

[Garth]

GR allows for flat spacetime, and I'm pretty sure it allows for any topology you like, as long as the Einstein field equations are satisfied everywhere (which wouldn't be a problem in an empty flat universe). For example, on p. 725 of Misner-Thorne-Wheeler's Gravitation, after discussing the "hypersurfaces of homogoneity" (slices of spacetime in which the distribution of matter/energy throughout space is homogoneous) for universes with positive, flat, or negative curvature in the Friedmann cosmological models, they write: They don't seem to be expressing any doubt here that the theory of "general relativity" as it's usually defined allows for such unusual topologies, although I suppose that's separate from the question of whether such topologies are allowed by the actual "laws of nature", whatever that means.

Thank you - I had not remembered that caveat in MTW.

If by "cosmological principle" you mean the requirement that the universe be homogoneous and isotropic, this is not part of GR, it's an additional constraint on possible universes invented by cosmologists which doesn't follow directly from any fundamental principles of physics. But anyway, the quote above suggests you can still have "hypersurfaces of homogeneity" in a universe with a weird topology (and certainly an empty flat universe with the topology of a torus would be homogeneous and isotropic, no?)

Indeed. However, does not that make the paradox more intractable than ever?

Take a flat universe with such a global topology, how is the preferred frame selected by the global topology? What do you 'hang' such a frame of reference on?

My understanding of the paradox is that such topologies cannot be "physical" unless there is matter in the universe from which the 'preferred stationary' frame of reference may be defined.

Or, otherwise, there is a local absolute frame of refrence, defined purely by the global topology, in contradiction of the Principle of Relativity.

Garth

 

[JesseM]

Thank you - I had not remembered that caveat in MTW.
Indeed. However, does not that make the paradox more intractable than ever?

Take a flat universe with such a global topology, how is the preferred frame selected by the global topology? What do you 'hang' such a frame of reference on?

As I said in my last post to Hurkyl, I think you could probably define the preferred frame in terms of which frame had the extremal distance between multiple copies of the same object at rest in that frame. After thinking about it some more, I'm pretty sure that the "preferred" frame would be the one in which the distance between copies of an object at rest in that frame is minimized--if you look at the distance between copies at rest in any other frame (with the distance measured in that frame's coordinates), the distance would be larger.

My argument for this is that in the preferred frame, everything should basically look identical to how things would look in a spatially infinite universe (in which you could assume the normal rules of SR) where there were actual physical replicas of each object at regular intervals from each other, with every replica's clock synchronized in this frame. Let's say the interval is D. So for two copies at rest in this frame, their distance apart will be D in this frame. But it will also be true that for two objects in motion in this frame, their distance will be D in this frame--which means in the pair's own rest frame, their distance apart must be larger than D.

My understanding of the paradox is that such topologies cannot be "physical" unless there is matter in the universe from which the 'preferred stationary' frame of reference may be defined.

MTW included no such caveat about there needing to be matter in the universe, though. Certainly a flat and empty infinite universe is a valid GR solution, with an arbitrary flat hypersurface qualifying as a "hypersurface of homogeneity", so why can't you do the same trick they mentioned of identifying faces on a cube in such a universe?

Or, otherwise, there is a local absolute frame of refrence, defined purely by the global topology, in contradiction of the Principle of Relativity.

You are misunderstanding the principle of relativity here, I think. After all, the laws of physics would still work the same in every local inertial frame, and GR doesn't say anything about the equations of physics (written without using tensors) working the same in every non-local coordinate system, so that's all you need to satisfy the principle of relativity. But anyway, if you are sure to include the correct initial clock settings for each copy as seen in a given observer's coordinate system, then the laws of physics do work the same in every global coordinate system in the flat spacetime version of the cosmological twin paradox--when one twin leaves Earth and returns to Earth to find that his twin has aged more, he can say that he actually traveled from one copy of Earth to another, and although each copy was aging more slowly than himself (as required if the law of time dilation is to work the same way in his frame), the copy he was traveling towards started out older than the copy he left.