1. Sep 13, 2006

### michael879

famous paradox, everyone should knows it. Whenever Ive heard it though its been dismissed as not really being a paradox since the two twins paths arent identical. The twin in the rocket accelerates off earth and then accelerates back. With a hyper-spherical universe however you could have this done without any acceleration (to make it perfect you can replace the twins with clocks calibrated as the rocket passes earth at some high velocity) since anything going away from earth in a straight line will eventually arrive back at earth. I did some quick calculations earlier with a 10^10 m circumference of the universe and a rocket going at .8c. I got that the rocket twin arrives back at earth in 25 seconds (distance is shortened to 6x10^9 which divided by .8c is 25) while the twin on earth ages 70 seconds (time goes faster for him as earth moves away and dt = 5/3).

I may of gotten the second part wrong because if you think about it from the earth twins frame he just watches the other twin travel 10^10m at .8c which is about 40 seconds. Can someone do the math right for me before I comment on the paradox? (if there is one).

2. Sep 13, 2006

### MeJennifer

Well good point, but, a hyperspherical universe implies not just closed spatial dimensions it also includes a closed time dimension. So you go forward in time and reach the past!
So it looks that special relativity works here as well, since who is really older here?

3. Sep 13, 2006

### michael879

what do you mean you go forward in time and reach the past? The 2d equivalent of the model Im thinking of is a balloon where the universe is the 2d surface and the radius is time.

4. Sep 13, 2006

### MeJennifer

You can think of such a model but you might want to ask yourself if such a model is uberhaupt possible in GR.

As I wrote before in a hyperspherical universe not just the spatial dimensions are closed but also the time dimension. In other words you can make a full roundtrip in time as well.

5. Sep 14, 2006

### robphy

6. Sep 14, 2006

### George Jones

Staff Emeritus
Eventually, I plan to say a lot more about the stuff in these links.

But that will be another day, or week, or month, or ...

7. Sep 14, 2006

### michael879

so your talking about a 5 dimensional universe, what I was thinking of was a 4d one. Is a 4 dimensional universe impossible?

8. Sep 15, 2006

### michael879

wow those links suggest that the two frames arent equal... They say that the rocket twin ages less than the earth twin. wtf? doesnt that go against relativity? you shouldn't be able to distinguish between two inertial frames.

9. Sep 15, 2006

### michael879

does this show that a closed universe cant exist? or maybe that moving reference frames in a closed universe arent actually intertial??

10. Sep 15, 2006

### JesseM

Well, you can't talk about "inertial frames" in curved space (outside of an infinitesimally small region of spacetime), but instead of considering a hyperspherical universe, you could consider a universe which has flat spacetime but an unusual topology which makes it finite, a bit like the old video game "asteroids" where if you disappear off one edge of the screen you reappear at the same position on the opposite edge. In this sort of flat but finite universe, the key is to realize that in such a universe you'll have a hall-of-mirrors effect where each observer sees an infinite number of copies of himself at regular intervals, and that because different observer's lines of simultaneity tilt at different directions, there will be only one frame where an observer at rest in that frame will judge all his mirror-image copies to have their clocks synchronized with his own. For observers in other frames, each successive copy will be out-of-sync with the previous one by a constant amount. So if the earth is at rest in the "special" frame (special only relative to the topology of the universe, not a preferred frame in the sense of the laws of physics working differently), and a ship travels at constant velocity around the universe, what this will look like is a ship leaving one copy of the earth and returning to a different copy, while an infinite number of copies of the ship do the same thing (each returning to a copy one increment to the right of the one they left from, say). In the earth's frame, all the ships leave simultaneously and return simultaneously, and all their clocks are ticking at the same slowed-down rate; in the ship's frame, the earth-copies' clocks are the ones that are slowed down, but earth-copies' clocks are all out of sync, so that the clock of the earth-copy it left started out well behind the clock of the earth-copy it returns to, so the ship's clock will still be behind the clock of the earth-copy it returns to even though the clock was ticking slow throughout the journey.

11. Sep 15, 2006

### JesseM

I wouldn't interpret it that way--I would understand "hyperspherical" just to mean the shape of curved space, not the shape of spacetime. But I don't know if the phrase "hyperspherical universe" has any standard definition in physics, so I guess you're free to interpret it however you like. Still, note that when people talk about "closed" or "open" or "flat" universes, terms which do have standard definitions in cosmology, they are referring only to the curvature of space.

12. Sep 15, 2006

### michael879

theres no inertial frames in a hyperspherical universe? Just because things travel in "circles" (or helixes) doesnt mean they acclerate.. Are you saying anything that appears to have a constant velocity is actually accelerating?

13. Sep 15, 2006

### JesseM

I'm not saying they "accelerate" any more than any other freefalling object following a geodesic in curved spacetime (would you say a person in a falling elevator is accelerating in any absolute sense, even though they feel no G-forces?), it's just that you can't assume the traditional laws of SR would still apply to a coordinate system covering a large region of curved spacetime--if they did, why would you even need to worry about spacetime curvature?

Last edited: Sep 16, 2006
14. Sep 16, 2006

### michael879

so your saying there is a global reference frame in a closed universe?

15. Sep 19, 2006

### michael879

ok forget my original question. Lets say you have two people each traveling at .5c away from earth in opposite directions in a closed universe. They are both in inertial frames the whole time (not that it would matter if they werent) and have completely equal reference frames. What happens when they each go halfway around the universe and reach each other again? each will have watched the other's clocks slow down.

*edit*
as they move away they keep looking at each other (through earth) with very powerful telescopes. When they are within 10 second of each other they turn their telescopes around 180 degrees.

16. Sep 19, 2006

### JesseM

Are you talking about traveling around a closed universe with positive curvature? In this case you can't have "inertial frames" that cover the whole spacetime, in curved spacetime you can only talk about locally inertial frames in an infinitesimal region of spacetime. But if you're talking about a flat spacetime that has a weird topology that makes it finite, like I talked about in my earlier post, then just read that earlier post for the answer--there would be a "hall of mirrors" effect with the appearance of an infinite number of earths and an infinite number of rockets, and each rocket would actually appear to return to a different earth than the one it departed from. Only an observer who was at rest in the special frame defined by the topology of the universe would observe all his copies to have their clocks synchronized with him, see that previous post for how this could explain why each would see all the copies of the other one running slow yet they could both agree on who is older when they reunite.

17. Sep 19, 2006

### michael879

no I mean a curved space-time "with positive curvature". Ok yes, the frames won't be intertial I guess but does it matter? They both still see time distortion in the other frame. No frame is preferred so it seems like theres still a paradox.