There is a real paradox here - read my posts #6, #9 above.Zanket said:If that were true then real paradoxes would arise, so it cannot be true.
No, that’s not a paradox.Garth said:There is a real paradox here - read my posts #6, #9 above.
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:Garth said: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.
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.Garth said: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.
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:RandallB said: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.
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.RandallB said: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.
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.RandallB said: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.
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).RandallB said: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.
How do you measure distance around? I don't remember it being anywhere near as straightforward as that.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)
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"?Hurkyl said:How do you measure distance around? I don't remember it being anywhere near as straightforward as that.
No a proper analogy would be two GPS orbiting in opposite directions.JesseM said: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
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.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.
I'm assuming that if Garth wants to deal with a real paradox, it is in a real universe.there's no need to even assume there is a CBR in this hypothetical universe
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.RandallB said:No a proper analogy would be two GPS orbiting in opposite directions.
Depends on whether the horizon is larger or smaller than the distance you must travel to circumnavigate the universe.RandallB said: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.
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.RandallB said: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.
And my answerrobphy said: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.
I do.RandallB said:I'm assuming that if Garth wants to deal with a real paradox, it is in a real universe.
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?Garth said:I do.RandallB said:I'm assuming that if Garth wants to deal with a real paradox, it is in a real universe.
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.JesseM said: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?
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:Garth said:Nobody, AFAIK, has demonstrated how GR might allow for that possibility.
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.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).
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?)Garth said: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.
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.JesseM said: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.No a proper analogy would be two GPS orbiting in opposite directions.
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.RandallB said:Because that’s not true. Only a misunderstanding of SR would expect one satellite to see the other as aging more slowly.
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.RandallB said:Just like twins traveling in opposite directions from Earth return to see each other as the same age but their classmates older.
"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.RandallB said: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.
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.Because that’s not true. Only a misunderstanding of SR would expect one satellite to see the other as aging more slowly.
Thank you - I had not remembered that caveat in MTW.JesseM said: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.
Indeed. However, does not that make the paradox more intractable than ever?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?)
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.Garth said: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?
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?Garth said: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.
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.Garth said:Or, otherwise, there is a local absolute frame of refrence, defined purely by the global topology, in contradiction of the Principle of Relativity.
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.Hurkyl said: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.
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 said: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.
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 said:There is no "acceleration to bring them back" in a compact universe, that's the whole point.
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.RandallB said: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!
RandallB said: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.
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).dicerandom said: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.
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!JesseM said: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 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.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.
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.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.
Don’t give up on your imagination based on a hypothetical paradox.Garth said:I am ready to acknowledge that not only is the universe more weird than I imagined, but more weird than I can imagine!
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.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.