A variation on the twin paradox

[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.

 

[RandallB]

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.

You’re not being to nitpicky. You just haven’t looked at the Twins issue closely when you send them BOTH on the trip in opposite directions.

Don’t they see each other as aging more slowing at the start? Even on the return just making direct observations at a distance don’t they see the same thing? With the exception of when they do fly by Earth on the return comparing each other total age shows they are both THE SAME, only those on Earth have aged much more.
Since they are already flying-by let them retrace the other path and won’t you get the same result again on the next return? – and the next?
Just like counter orbiting GPS satellites.
Just like circumventing a “compact space”,
be it a living room, the Fermi Lab Ring, or galaxy etc. etc.

I see no reason for any of these to behave differently then the twins do.
Other than the acceleration to bring them back being applied differently,
what do they get the twins don’t?
None of them remain in a single reference frame.

 

[JesseM]

Since they are already flying-by let them retrace the other path and won’t you get the same result again on the next return? – and the next?
Just like counter orbiting GPS satellites.
Just like circumventing a “compact space”,
be it a living room, the Fermi Lab Ring, or galaxy etc. etc.

I see no reason for any of these to behave differently then the twins do.
Other than the acceleration to bring them back being applied differently,
what do they get the twins don’t?
None of them remain in a single reference frame.

There is no "acceleration to bring them back" in a compact universe, that's the whole point. They both travel inertially in opposite directions, each remaining in the same inertial rest frame, but they can still meet up again due to the weird topology of the universe. Again, just think of the game Asteroids, where you can fly away from the center of the screen to the right, then when you hit the right edge you reappear on the left edge of the screen still traveling to the right, so if you keep going you'll end up back at the center of the screen without ever having turned around or changed velocity.

 

[RandallB]

There is no "acceleration to bring them back" in a compact universe, that's the whole point.

Now you’re back to hypothetical universes that don’t have to deal with the light horizon of the Big Bang – As I said before I’m dealing with real universes here. Traveling inertially won’t get past that line and that’s not even ¼ the way around!

 

[JesseM]

Now you’re back to hypothetical universes that don’t have to deal with the light horizon of the Big Bang – As I said before I’m dealing with real universes here. Traveling inertially won’t get past that line and that’s not even ¼ the way around!

As I said, most physicists would still consider a paradox interesting even if it could only happen in a universe with different initial conditions. And you never addressed my point that the size of a compact universe can in fact be smaller than the horizon created by the big bang, and that in fact physicists are looking for evidence of this possibility in the cosmic microwave background radiation. For now we can't rule out the possibility that the universe could be small enough to circumnavigate, even with that horizon.

In any case, your argument is inconsistent. First you say, "the cosmological twin paradox is just like any other version of the twin paradox" and then I say "no it isn't, the special feature of the cosmological twin paradox is that the twins can depart and then later reunite without either accelerating" and your response is "yeah, but you could never circumnavigate the universe anyway!" Circumnavigating the universe without accelerating is the essential feature of the cosmological twin paradox, so you're free to dismiss the cosmological twin paradox as irrelevant if you think it'll turn out to be impossible to do this in our universe, but your comparison with the GPS satellites or other situations involving acceleration is still off-base.

 

[RandallB]

Garth
I see my problem
The term “Compact Space” must mean:
A large enough periodic orbit moving fast enough in an inertial straight line to circumvent the complete universe.

Then it’s the term “Compact Space” I’ve been having a problem with – my mistake.
Just substitute “any periodic orbit circumventing any part of the real universe large or small” where I may have used the term.

Until someone shows they can even be such a thing, no need for me to deal with “Compact Space”. Sorry if I intruded on just a hypothetical.
Thought you were dealing with a real thought experiment.

 

[dicerandom]

Until someone shows they can even be such a thing, no need for me to deal with “Compact Space”. Sorry if I intruded on just a hypothetical.
Though you were dealing with a real thought experiment.

There certainly can be, the question is just whether or not the universe we live in is closed. Although just because it's closed doesn't mean that you can circumnavigate it, the space could have an accelerating expansion such that you'd never be able to circumnavigate it. In the Robertson-Walker metric I posted earlier the parameter $\kappa$ dictates whether the space is positively curved and closed, flat and open, or negatively curved and open.

Which of these three possibilities our universe truly is depends on the relative densities of matter, radiation, and a possible value for a cosmological constant. As it turns out we are somewhere extremely near the critical value which is the "tripple point", meaning that our universe is basically flat on large scales, but it could be very slightly positively or negatively curved and we simply don't have the accuracy available to measure which it is.

 

[JesseM]

Which of these three possibilities our universe truly is depends on the relative densities of matter, radiation, and a possible value for a cosmological constant. As it turns out we are somewhere extremely near the critical value which is the "tripple point", meaning that our universe is basically flat on large scales, but it could be very slightly positively or negatively curved and we simply don't have the accuracy available to measure which it is.

But as I said before, the question of the topology of the universe is actually independent of the curvature issue. It's true that if you assume the simplest possible topology, a positively-curved universe would be finite while a flat or negatively-curved universe would be infinite; but by choosing other topologies you can have a finite universe that is flat or negatively curved (not sure whether a positively-curved and infinite universe is possible, though).

 

[Garth]

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?

You can; however, I would question the physical reality of such a hypothetical extrapolation of testable physics. On the other hand, if 'circles in the sky' are observed I would have to revise this opinion. I am ready to acknowledge that not only is the universe more weird than I imagined, but more weird than I can imagine!

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.

You have a local laboratory belonging to one observer A. A second observer B momentarily passes through at high speed and both synchronise clocks. A very long time later B passes through A's laboratory again after inertial circumnavigation of the universe, and clocks are compared.
The fact that the global topology imparts a preferred frame which says that it is A's clock that will register the greatest time elapse means that A and not B can, at the initial local encounter, think of their time as being 'absolute' in some sense. This I understand to be in contradiction to 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.

Are these 'copies' of the Earth the actual one Earth experienced after circumnavigations of the universe, or are we saying that the world we know is itself replicated many/infinite number of times? I cannot swallow the second interpretation. That interpretation, IMHO, seems too high a price to pay, stretching physical reality too far, in order to resolve the paradox.

I would argue, contrary to MTW, that mass in the universe is essential to resolve this paradox and that A's 'absolute' frame of reference is that defined by the Centre of Mass/momentum of the matter in the universe at large.

Garth

 

[RandallB]

I am ready to acknowledge that not only is the universe more weird than I imagined, but more weird than I can imagine!

Don’t give up on your imagination based on a hypothetical paradox.
If a paradox only exists in a hypothetical universe then the paradox isn’t real.
Just because a parameter can define a “hyperspace” curved universe, doesn’t make the idea of a curved universe real. No more than being shown a simple SR-GPS problem masquerading as a house of mirrors.
Until there is REAL evidence of ‘compact space’ there is no reason to let this “paradox” control your imagination no matter how good a hypothetical argument may sound. The idea that reality might act like a computer screen has only the idea to support it, nothing real.

I would argue, contrary to MTW, that mass in the universe is essential to resolve this paradox and that A's 'absolute' frame of reference is that defined by the Center of Mass/momentum of the matter in the universe at large.

Very good point, and the way to bring your idea into reality is to use the CBR. And that 'absolute' frame will look like a “preferred” frame. But, contrary to the Lorentz R fans out there, it cannot be preferred because if you move a significant distance away to define another 'absolute' frame using the same CBR, it will not be the same as the first CBR defined 'absolute' frame. Thus still no “preferred” frame, just as relativity demands.
Using your imagination on a real paradox like this is much more valuable than giving up on it over some hypothetical paradox.

I disagree with Eddington on what we can do with imagination.

 

[Garth]

Rather than take the hypothetical case of non-trivial topologies I prefer to keep it simple and consider this paradox in the case of the topologically simple compact space of a closed universe, i.e. the spherical or cylindrical universes of Friedmann or Einstein.

I am willing to ignore the practicality of circumnavigating such a universe for the sake of the 'gedanken'.

Garth

 

[dicerandom]

There were some comments earlier about the non-physicalness of the cylindrical spacetime, I'd just like to point out that it is a case of the Friedman-Robertson-Walker metric, it just hapens to be a 1+1 temporal-spatial case. The problem breaks down to a two dimensional one anyway if you only consider inertial observers.

 

[JesseM]

You can; however, I would question the physical reality of such a hypothetical extrapolation of testable physics.

Sure, I don't have a problem with that, as long as you acknowledge that the mathematical theory of GR allows such things. What you're saying is that not all spacetimes that are valid according to GR may actually be possible, and I agree we have no way of knowing for sure.

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.

You have a local laboratory belonging to one observer A. A second observer B momentarily passes through at high speed and both synchronise clocks. A very long time later B passes through A's laboratory again after inertial circumnavigation of the universe, and clocks are compared.

No, I don't think that counts as a valid test of the "principle of relativity" in GR. After all, even without picking out weird topologies you can do something similar in the neighborhood of the earth--have two observers, one orbiting the Earth and the other shot up from the surface at slightly less than escape velocity, in such a way that the shot-up observer passes right next to the orbiting observer on his way up, moves away from the Earth for a while and finally begins to fall back down, with everything timed so that on his way down he will again pass right next to the orbiting observer who has just completed one orbit. In this case both observers are moving on geodesics, so in any local neighborhood of a point on their path it should look like they are moving inertially with all the normal rules of SR applying in this local neighborhood; and yet, if they synchronized their clocks at the first moment they passed, I don't think their clocks would still be synchronized at the second moment they pass. Surely this does not mean that this simple situation violates the principle of relativity, or implies a "preferred frame" in terms of the fundamental laws of physics? I'm pretty sure you can't compare two separate local regions like this as if they are two crossings in SR flat spacetime, the principle of relativity as applied to GR just means that if you look at a single local region of spacetime, an observer following a geodesic through that region will locally observe the laws of physics to work just like an inertial observer would see them work in SR.

The fact that the global topology imparts a preferred frame which says that it is A's clock that will register the greatest time elapse means that A and not B can, at the initial local encounter, think of their time as being 'absolute' in some sense. This I understand to be in contradiction to the Principle of Relativity.

I'm pretty sure you're wrong. Again, in GR the principle of relativity as I understand it only says that if you look at a single local region of spacetime, within that region the laws of physics must work just like in SR, including the symmetry between different locally inertial observers' view of events within that local region. But if you look at things non-locally, then even without invoking weird topologies you can still have situations where two different geodesic paths cross at two different points, and the geometry of spacetime tells you which of two observers traveling along these paths will have elapsed more time on their clock. If the cosmological twin paradox was a violation of the principle of relativity, then any such situation would have to be one too, even the simple one I outlined above with one observer orbiting the Earth and the other shot upwards from the surface and then falling back down.

Are these 'copies' of the Earth the actual one Earth experienced after circumnavigations of the universe, or are we saying that the world we know is itself replicated many/infinite number of times?

The actual one earth. I was just describing how things would look in each observer's coordinate system, assuming they construct their coordinate systems in the same way as in SR, but allow the spatial axes to keep wrapping around the closed space over and over, so that each event would have multiple coordinates. For example, the departure of the rocket from the Earth in the Earth's coordinate system might have coordinates x=0 l.y., t=0 y, but also x=5 l.y., t=0 y, x=10 l.y., t=0 y, x=15 l.y., t=0 y, and so on.

Also, with a sufficiently powerful telescope you could see multiple images of the same object at different distances from you, with each image being caused by light that has circumnavigated the universe a different number of times before reaching your telescope, so this is another sense in which there'd be "copies" of the earth. Visually, if a rocket circumnavigated the universe and returned to earth, it would look like the rocket that departed "my" Earth landed on the distant image of the Earth on my right, while the rocket that landed on my Earth would appear to be the one that had departed the distant image of the Earth on my left.

I would argue, contrary to MTW, that mass in the universe is essential to resolve this paradox and that A's 'absolute' frame of reference is that defined by the Centre of Mass/momentum of the matter in the universe at large.

But if you agree that an empty compact universe is a valid solution in GR--whatever the definition of "valid solution" is used, perhaps a spacetime manifold where the Einstein field equations are obeyed at every point--then how can you argue that there's a genuine paradox without saying that the paradox is inherent to GR itself, or denying that GR really does respect the principle of relativity?

 

[Garth]

No, I don't think that counts as a valid test of the "principle of relativity" in GR. After all, even without picking out weird topologies you can do something similar in the neighborhood of the earth--have two observers, one orbiting the Earth and the other shot up from the surface at slightly less than escape velocity, in such a way that the shot-up observer passes right next to the orbiting observer on his way up, moves away from the Earth for a while and finally begins to fall back down, with everything timed so that on his way down he will again pass right next to the orbiting observer who has just completed one orbit. In this case both observers are moving on geodesics, so in any local neighborhood of a point on their path it should look like they are moving inertially with all the normal rules of SR applying in this local neighborhood; and yet, if they synchronized their clocks at the first moment they passed, I don't think their clocks would still be synchronized at the second moment they pass. Surely this does not mean that this simple situation violates the principle of relativity, or implies a "preferred frame" in terms of the fundamental laws of physics?

I'm pretty sure you can't compare two separate local regions like this as if they are two crossings in SR flat spacetime, the principle of relativity as applied to GR just means that if you look at a single local region of spacetime, an observer following a geodesic through that region will locally observe the laws of physics to work just like an inertial observer would see them work in SR. I'm pretty sure you're wrong. Again, in GR the principle of relativity as I understand it only says that if you look at a single local region of spacetime, within that region the laws of physics must work just like in SR, including the symmetry between different locally inertial observers' view of events within that local region.

This experiment relies on the Earth's gravitational field. Extend this gedanken by drilling holes through the centre of the Earth. Now include in these inertial observers the one at the centre of the Earth, in free fall and yet stationary wrt the Earth. Let the orbiting observers now fly through this COM laboratory. The observer whose clock records the longest interval between all such encounters will be the stationary one, and this may be defined therefore as an 'absolute' frame, different from all the rest as having the greatest proper time interval. It is a frame of reference that is defined by the presence of the Earth's mass. At the COM laboratory the field is locally flat, in a 'small enough' region around the COM, and a particular frame of reference is different from all the rest. The Prinicple of Special Relativity does not hold!

But if you agree that an empty compact universe is a valid solution in GR--whatever the definition of "valid solution" is used, perhaps a spacetime manifold where the Einstein field equations are obeyed at every point--then how can you argue that there's a genuine paradox without saying that the paradox is inherent to GR itself, or denying that GR really does respect the principle of relativity?

I believe the paradox is inherent to GR itself, that is why in my work http://en.wikipedia.org/wiki/Self_creation_cosmology I include Mach's Principle and violate the Principle of Relativity - the COM Machian frame is the preferred frame.

Garth

 

[JesseM]

This experiment relies on the Earth's gravitational field. Extend this gedanken by drilling holes through the centre of the Earth. Now include in these inertial observers the one at the centre of the Earth, in free fall and yet stationary wrt the Earth. Let the orbiting observers now fly through this COM laboratory. The observer whose clock records the longest interval between all such encounters will be the stationary one, and this may be defined therefore as an 'absolute' frame, different from all the rest as having the greatest proper time interval. It is a frame of reference that is defined by the presence of the Earth's mass. At the COM laboratory the field is locally flat, in a 'small enough' region around the COM, and a particular frame of reference is different from all the rest. The Prinicple of Special Relativity does not hold!

OK, then you are defining the "principle of special relativity" in such a way that it is violated in GR. However, most physicists would not define it this way, I think. I don't think it even makes sense to talk about the "principle of special relativity" in GR except in a local sense, since this principle only says the laws of physics should look the same in coordinate systems constructed in a certain way (a system of measuring-rods and clocks at rest with respect to each other, with the clocks synchronized using the Einstein synchronization convention) which is not really possible in curved spacetime where the measuring-rods cannot remain rigid. It is still true that if you look at only a single local region of spacetime, and don't compare multiple regions as you are doing, that within that region the laws of physics work just like they do in SR (in the limit as the size of the region goes to zero).

 

[pervect]

I'll have to take back my previous remark (which I've deleted to minimize confusion, since I think now I was wrong).

Lets' try to do the calculation of which time is longer.

(I tried to be careful, but I could use a double check. Having an annoying cold isn't helping my accuracy).

First we need a maetric for an object inside of the Earth. We can use the Newtonian approximation, where

goo = 1-2U
g11 = 1

-U is the potential enregy, which in the inside of the planet is
$$-U = -\frac{3}{2}\frac{M}{r_0}+\frac{1}{2}\frac{M}{r_0} \left( \frac{r}{r_0} \right)^2$$

Geometric units are used so G=c=1

M is the mass of the planet
r_0 is the radius of the planet
0<r<r_0 is the position coordinate

This gives a force, dU/dr, of
$$\frac{M}{{r_0}^2}\frac{r}{r_0}$$

which is a hooke's law force with the correct value at the surface $r=r_0$. In addition U has the proper value at the surface as well.

So we have

$$g_{00} = 1 -\frac{3M}{r_0} + \frac{M}{r_0}\left(\frac{r}{r_0}\right) ^2$$

Now we need to integrate $d\tau = \sqrt{g_{00}(t) - v^2(t)}$

We can approximate \sqrt{1+a+b+c} = 1 + 1/2(a+b+c) to make our job of integration easier.

We know that an object experience a hooke's law force will move in sinusoidal motion, thus r(t) = r_0 sin(wt).

Thus

$$\tau = \int (1 - \frac{3}{2}\frac{M}{r_0}) dt + \frac{1}{2}\int \frac{M}{r_0}\left( \frac{r(t)}{r_0}\right)^2 dt - \frac{1}{2} \int v^2(t) dt$$

This adds a positive term that's equal to

$$\frac{1}{2}\frac{M}{r_0} T \int_0^\pi sin^2$$

and a negative term equal to

$$-\frac{1}{2}v_{max}^2 T \int_0^\pi cos^2$$

where T = $\int dt$

Now $$\frac{v_{max}^2}{2} = \frac{M}{2r_0}$$ therefore v_{max}^2 = M/r_0

This means that the postive and negative terms are equal, and the two times are the same (?!).

I think this makes sense from the virial theorem. T=V for a hooke's law force, T=-V/2 for an inverse square law force (I looked this up in Goldstein). Here T = time avg of kinetic energy, V = time avg of potential energy.

Under the square root, we have 2U twice the potential energy and v^2 which is twice the kinetic energy. Since T=V for a hooke's law force, the terms cancel.

Outside the planet, with an inverse square law the terms shouldn't cancel, and I believe the object thrown upwards will have the longer time. But I haven't double checked this.

 

[Garth]

pervect That is an interesting calculation, however I don't think it has correctly dealt with the situation:

This means that the postive and negative terms are equal, and the two times are the same (?!)

Is not this result simply the consequence of:

First we need a metric for an object inside of the Earth. We can use the Newtonian approximation, where

goo = 1-2U
g11 = 1

i.e. by starting with the Newtonian approximation you end up with a Newtonian result?

I am quite sure that with a (-+++) metric any extra motion of a particle on a geodesic between two events will result in its proper time duration being shortened.

OK, then you are defining the "principle of special relativity" in such a way that it is violated in GR. However, most physicists would not define it this way, I think. I don't think it even makes sense to talk about the "principle of special relativity" in GR except in a local sense, since this principle only says the laws of physics should look the same in coordinate systems constructed in a certain way (a system of measuring-rods and clocks at rest with respect to each other, with the clocks synchronized using the Einstein synchronization convention) which is not really possible in curved spacetime where the measuring-rods cannot remain rigid. It is still true that if you look at only a single local region of spacetime, and don't compare multiple regions as you are doing, that within that region the laws of physics work just like they do in SR (in the limit as the size of the region goes to zero).

I do think my understanding of the Principle of Relativity as being "no preferred frames of reference" ( in a local laboratory) is enshrined in the foundations of GR, particularly in the conservation laws and the use of the principle of Least Action in 4D space-time.

The point about using Lagrangian methods is that they work for generalised coordinates, in particular in a (-+++) metric, they work for all Lorentzian frames of reference, and they thereby guarantee the conservation of energy-momentum and the conservation, wrt covariant differentiation, of the stress-energy tensor.

The Bianchi identities guarantee the conservation, wrt covariant diffrentiation, of the Einstein tensor in the GR field equation and the constant G guarantees the consistency of that field equation's conservation properties.

NB: the Brans-Dicke theory allows G to vary and has to introduce extra scalar field terms to maintain the conservation qualities of its field equation.

If, however, there are preferred frames of reference, defined by the presence of local or cosmological mass, or the global topology of a compact space, then this restriction of the conservation of energy-momentum need not necessarily be enforced. This might then allow mass creation for example as it appears in http://en.wikipedia.org/wiki/Self_creation_cosmology .

Garth

 

[pervect]

pervect That is an interesting calculation, however I don't think it has correctly dealt with the situation: Is not this result simply the consequence of:i.e. by starting with the Newtonian approximation you end up with a Newtonian result?

I don't agree - the effect of g11 should be second order, as it only modifies the velocity term slightly. The approximations should be perfectly resonable, and adequate to demonstrate relativistic effects (the apprxoimations include the two important terms of gravitational time dilation and relativistic time dilation).

I am quite sure that with a (-+++) metric any extra motion of a particle on a geodesic between two events will result in its proper time duration being shortened.

BOTH paths are following geodesics. They are local maximums of the elapsed time, but you can't tell which is the longest unles you calculate them and compare them.

Consider the following example on the 2D surface of a curved 3D space

           x
          xxx
         xxxxx
       xxxxxxxx
     xxxxxxxxxxx
    xxxxxxxxxxxxx
A  xxxxxxxxxxxxxxx  B

Points A and B are separated by a tall mountain. There is a "straight line" path from A to B over the mountain. It is satisfies the geodesic equation. However, the existence of this geodesic path does not mean that there is not a shorter path around the mountain!

This shorter path must also be a geodesic, of course, if it is truly the shortest.

The shortest distance between two points is always a straight line, but a given straight line connecting two points is not always the shortest distance - there may be another line connecting them that is shorter in curved spaces.

Another example. On a sphere, a great circle is a geodesic path. You can get from point A on the great circle to point B on the great circle by heading in either direction. One direction will generally be shorter than the other, however. Both paths are geodesics - but one path is shorter than the other.

 

[Garth]

pervect I see what you are saying, both observers, the one remaining at the Centre of the Earth and the other one on a rectilinear orbit are in free fall and think it is the other that is moving. However, the metric of the Schwarzschild solution used is anchored to the COM of the system, and I am sure that the COM inertial observer will have the longest proper time duration.

Therefore, I would like to redo the calculation in the Post Newtonian approximation. Remember in the calculation of the deflection of light the spatial curvature term, g₁₁, makes an equal contribution as the time dilation term, g₀₀, to the total result.

Garth

 

[JesseM]

I do think my understanding of the Principle of Relativity as being "no preferred frames of reference" ( in a local laboratory) is enshrined in the foundations of GR, particularly in the conservation laws and the use of the principle of Least Action in 4D space-time.

If I'm understanding your argument correctly, I think your interpretation of what "in a local laboratory" means is incorrect. You can't have two small laboratories pass each other, then travel on different paths for significant time periods, then reunite later, and compare what has happened to each laboratory between the two meetings, saying that any asymmetry in how much time has elapsed on each lab's clock between these meetings indicates a failure of the principle of relativity. The principle of relativity only applies to measurements in a single small region of spacetime, not a comparison of two distinct small regions of spacetime separated by a large spacetime interval.

 

[Garth]

If I'm understanding your argument correctly, I think your interpretation of what "in a local laboratory" means is incorrect. You can't have two small laboratories pass each other, then travel on different paths for significant time periods, then reunite later, and compare what has happened to each laboratory between the two meetings, saying that any asymmetry in how much time has elapsed on each lab's clock between these meetings indicates a failure of the principle of relativity. The principle of relativity only applies to measurements in a single small region of spacetime, not a comparison of two distinct small regions of spacetime separated by a large spacetime interval.

I'm not using two laboratories.
Barrow and Levin say in The twin paradox in compact spaces

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

When the two observers pass close by the first time in the single local laboratory already one of them is maked off as being in the 'preferred frame'. It is true that you have to wait until the second encounter to do the experiment and discover which observer it is, or you could simply look out and see what the matter in the rest of the scenario is doing and discover which observer is at 'rest'.

Garth