1. Feb 27, 2005

### touqra

Let say, the universe is a torus, a doughnut shape. Regarding the twin paradox, it is possible, that the moving twin is not in an accelerating frame, and can still come back to Earth to compare her time with the twin stationary on Earth.
Then, both the twins are equivalent in a torus universe.
Right?

2. Feb 27, 2005

### chronon

This is an interesting question. If the universe is finite due to spacetime curvature, I would guess that it works out OK as the situation is non-symmetrical. The curvature is due to mass in the universe, and one of the twins is moving relative to that mass, the other one isn't.

However, you can assume finiteness due to the topology of the universe, rather than the geometry (this is like the universe of the Asteroids game). This point of view is put forward by Janna Levin, in the book How the universe got its spots. Then there's more of a problem. It's difficult to see how to get consistent Newtonian physics in such a universe, and maybe the inconsistency isn't solved by General Relativity.

3. Feb 27, 2005

### Garth

This is the cosmological twin paradox and has been discussed on these Forums here in depth.

Again I quote from a paper by Barrow and Levin "The twin paradox in compact spaces" http://arxiv.org/abs/gr-qc/0101014
You do not have to have an exotic topology like a torus for this paradox, an ordinary spherical closed universe will do just as well. Two inertial observers, travelling at speed relative to each other, pass closely and set their clocks. After a very long time they pass each other closely again and compare clocks a second time.

One observer has circumnavigated the universe at speed and so her clock has recorded less proper time lapse than the other one who has remained ‘stationary’. But which one of the two is this, and how do you tell?

The resolution of this paradox is that the presence of matter in the universe has closed the universe making it a 'compact topological space'. Furthermore this topology has “introduced a preferred frame” so that one definite observer ends up younger than the other. This is because the preferred frame is determined by the mass in the universe so the stationary observer is stationary relative to the centre of momentum of the rest of the matter-energy in the universe.

However is the existence of this ‘preferred frame’ not inconsistent with the principles of GR?

We find that the paradox is resolved only at the expense of the consistency of GR!

Garth

Last edited: Feb 27, 2005
4. Mar 2, 2005

### RandallB

I've seen a lot of circumnavigated the universe posts & most wonder about how to tell which one traveled - I don't see the paradox.

This shouldn’t be such a mystery - send me on the trip in my sleep and I’ll figure out if I'm the traveler or not.

Assuming a uniform universe - when I start noticing Galaxies coming at me and zipping by me, I’ll at least guess I’m not in Kansas any more! And Ask to use the Wizard’s equipment - making a quick check of the background radiation should prove that what seems to be coming at me and going by so fast must be closer to “stationary” (but not preferred now) than I.

Seems to me it would be obvious I’m moving, in any shape universe, as long as it had a CBR like ours. Observers should be able use the same references and SR to figure out the same. In a similar manner that a GPS satellite and Earth station can tell each other apart by looking at the background star positions.
RB

5. Mar 2, 2005

### JesseM

I think in GR, the notion of "no preferred reference frame" only applies at a local level, there is no guarantee that if you create different coordinate systems which cover large regions of spacetime, the laws of physics will work the same in each coordinate system.

6. Mar 4, 2005

### Garth

RandallB I quite agree that it will be possible to work out which is travelling and which is not, the problem is the equivalence principle would have us believe you should not be able to.

The issue I am getting at is GR ought to include Mach's Principle whereas in fact MP is inconsistent with the equivalence principle, that is with the principle of 'no preferred frame'.

The equivalence principle was set up in SR in flat, empty space-time, and works in, and is appropriate for, that scenario; however once you extend the theory to curved space-time and introduce matter, you introduce a means of determining a 'preferred' frame, that which co-moves with the Centre of Momentum.

In so doing you break the conditions required for the equivalence principle. This frame is preferred in the sense that it is the one in which one of our twins in stationary and records the greatest elapsed proper time between such encounters.

JesseM In a similar way the notion of "no preferred reference frame" applies to the equivalence principle which applies wherever GR is applied, i.e. cosmologically and not just at a local level.

What I am actually arguing is that this cosmological twin paradox exposes an inconsistency in GR cosmology. The topology of a compact cosmological model is in conflict with the principles of 'no preferred frames' and therefore equivalence. It is this conflict that I have tried to address in A New Self Creation Cosmology.

Garth

Last edited: Mar 4, 2005
7. Mar 4, 2005

### RandallB

I don't know Garth I think you may want to go back and reread some of Einstin's stuff.

He did not use "the equivalence principle" for SR at all. He only agreed with and followed Newtons lead - that the Laws shouold apply the same everywhere. Einstin was first to figure what that meant to apply them to near light speeds. SR even gave him E=mc^2

It was not till the equivalence principle that he started on GR.

And without opening the window on the ship the non move twin, the earth station or the GPS satilite I think the equivalence principle would stand up OK.

RB

8. Mar 4, 2005

### Garth

Let me reiterate; Einstein founded the theory of SR on the 'no preferred frames' concept.

In the presence of gravitational fields the Einstein Equivalence Principle (EEP) is a necessary and sufficient condition for the Principle of Relativity, (PR). Here I summarise PR as the doctrine of no preferred frames of reference. In the absence of such fields the EEP becomes meaningless, although then the PR does come into its own and is appropriate in Special Relativity (SR), which was formulated for such an idealised case. However the presence of matter and the gravitational fields that it generates allows a particular frame to be identified, the co-moving centroid or centre of momentum.

The question is, "Does this constitute a 'preferred frame'?" The cosmological twin paradox reveals that it does. Of all inertial observers passing and re-passing each other after circumnavigations of the compact space of a closed universe one will have an absolutely longest proper time passage between encounters. This will be that observer in the co-moving frame, stationary wrt 'the rest of the universe,' in apparent contradiction to the Principle of Relativity.

Garth

Last edited: Mar 4, 2005
9. Mar 5, 2005

### RandallB

Compact Space?

Maybe what I'm missing is the definition of "Compact Space" What is that.

In the case of a GPS satellite and it's base station set on the North pole. Does compact space mean that neither would be able to look at the rest of the earth nor the stars behind each other?
Under those conditions the Earth station would seem to be preferred by both as long as they never become aware of the other parts of earth or stars.
Is that the meaning of "compact space".

10. Mar 5, 2005

### Garth

Compact space = closed universe, finite yet unbounded.

Garth

11. Mar 5, 2005

### JesseM

Garth, I'm not sure the notion of a "reference frame" is even meaningful in GR, because I don't think there's a unique way of defining the "relative velocity" of two objects which aren't in the same local neighborhood. In SR, your reference frame is defined by the readings on a network of rulers and clocks which are at rest relative to you; in GR you can define various abstract global coordinate systems, but I don't think they'd necessarily have this sort of physical meaning. Note that even in SR, it's certainly not true that the laws of physics work in any coordinate system where the origin is moving inertially, they only work the same in coordinate systems defined by a network of rulers and synchronized clocks. For example, say you have defined coordinates x,y,z,t based on readings on rulers and synchronized clocks at rest relative to yourself, and then I define a new coordinate system x',y',z',t' using the following abstract mathematical transformation:

$$x^{'} = 12x$$
$$y^{'} = \pi y$$
$$z^{'} = z^9$$
$$t^{'} = e^t$$

Since there's a one-to-one mapping between the two coordinate systems, then any object with a fixed coordinate x',y',z' must also have a fixed x,y,z coordinate, so it must be moving inertially; but this does not qualify as a valid "inertial reference frame" because it isn't based on physical measurements, and the laws of physics certainly would not look the same if expressed in this coordinate system. In GR I don't think there's this sort of natural distinction between "physical" coordinate systems and "abstract mathematical" ones (although locally there is of course, since GR reduces to SR in local neighborhoods).

12. Mar 5, 2005

Staff Emeritus
It could be bounded, within the definition of compact, provided it included the boundary. What it can't be is open. Compact requires every infinite sequence of points to have a cluster point (the advanntage of "compact" is that you don't have to say "bounded sequence"; the topology bounds it for you). So in a flat or hyperbolic universe you could define a sequence going "out to infinity" wth each point a fixed step away from all the preceding ones, so it would never cluster.

Since we tend to assume the universe has no boundary, we also tend to skip over the fine points of the definition.

13. Mar 5, 2005

Staff Emeritus
"Inertial" means non-accelerated. Your example has a highly nonlinear curved worldline, hence an observer experiencing it is accelerated, therefore not inertial. Inertial observers in flat Minkowski space have straight worldlines, in GR their worldlines are geodesics. In SR Lorentz transforms obtain between all inertial frames.

14. Mar 5, 2005

### JesseM

What do you mean "experiencing it"? An observer whose x',y',z', coordinates don't change as t' varies (one who is at rest in this coordinate system) will also have x,y,z coordinates that don't change as t varies, no? That means the observer will have a straight wordline as seen in any inertial frame, so he is moving inertially. But while you can call the x',y',z',t' system an "inertial coordinate system" in the sense that an observer at rest in these coordinates must be moving inertially, it's not a valid inertial reference frame, as I understand the definitions.
I know that, but I wasn't talking about worldlines, I was talking about coordinate systems. My point is that in GR I don't think it would make sense to ask whether a global coordinate system (say, Hubble coordinates) is "inertial" or "non-inertial".

Last edited: Mar 5, 2005
15. Mar 6, 2005

### Alkatran

Why don't we ask a similar question:

If, say on my basketball at home, we had a two dimensional finite yet unbounded universe: how does the twin paradox play out there? Well, obviously, we can equate a twin travelling around a 2d universe (sphere) to me walking in a circle. The twin that goes around ends up younger because he is 'accelerating' around the ball

16. Mar 6, 2005

### Garth

Thank you

compact = not open; finite.

The qualifier "unbounded" is not necessary.

Garth

Alkatran Alternatively Use a cylinder instead.
The long axis represents the time axis and the circumference represents space.

Have two pins on the outer surface at either end and connect with two elastic strings, one of which is straight between the pins and the other twists round the cylinder between the two.

Allow one end of the cylinder to be rotated relative to the other end.

The straight string represents the world-line of a 'stationary' observer and the twisted string a moving observer. Obviously the model is set up in the frame of reference of the first observer. If we rotate the cylinder the twisted string can be made straight and the other now twists around the cylinder. We are now in the frame of the second observer.

You cannot straighten out both strings, there is always a difference of one complete twist between them, this is the 'winding number’, which is a topological invariant.

However how do you tell the difference between the two? As we have set it up you cannot, each scenario is equivalent to the other and there is no preferred frame or observer. However in a real closed or 'compact space' universe one observer will definitely have run up a longer elapsed time between encounters than the other, her frame can therefore be said to be 'preferred'.

My point is that this preferred frame is introduced by the presence of mass in the universe. It is the distribution of matter in motion that determines this special frame of reference and that is in accordance with Mach's Principle rather than those of Einstein's relativity.

JesseM A 'physical' coordinate system, as opposed to a merely 'mathematical' one, is defined by a system based on physical measurements, i.e. scales, clocks and rulers. Nothing you have said has convinced me that this paradox does not reveal an inconsistency in GR. The observer is in a physical preferred frame in the sense that it is determined by the measurement of her clock.

Garth

Last edited: Mar 6, 2005
17. Mar 6, 2005

### JesseM

Well, how do you propose an observer should set up a network of clocks and rulers throughout space to define a "physical" coordinate system? There are all kinds of problems that will crop up in GR--for example, in curved spacetime you can't assume that clocks in different locations will all tick at the same rate, because gravitation causes time dilation (not to mention the stretching of rulers). What's more, in SR we assume that all the clocks and rulers are at rest relative to each other, but as I said before there is no unique way to define the "relative velocity" of two distant objects in GR; the only way to compare distant velocities is by doing a parallel transport of the velocity vector at one point along a geodesic to another point, but there can be multiple geodesics between two points (think of gravitational lensing), so the result of the parallel transport is path-dependent. Imagine if you tried to define a coordinate system in SR using clocks which were not at rest relative to one another--the laws of physics certainly wouldn't look the same in such a coordinate system as they do in normal inertial coordinate systems, even if each clock in the system is moving inertially. So despite the fact that this coordinate system is still based on "physical measurements" in some sense, it doesn't qualify as a valid inertial frame. In GR there doesn't seem to be any way to define the notion of a network of clocks which are all at rest wrt one another, so how do you distinguish between coordinate systems that qualify as "reference frames" and those that don't?

One more point: gravitational lensing shows that in GR, unlike SR, light from an event can take multiple paths to reach you. In SR, different clocks are synchronized using light signals--If I look through my telescope and see a clock one light-year away that reads "12:00, Jan. 1, 2105" then at the moment I receive that light my clock should read "12:00, Jan. 1, 2106" if the two are "synchronized" according to the SR definition. But if light from a single event can reach me at two different times, how are clocks to be synchronized in GR? I suppose if you have some unique way of defining the "distance" that light travelled along a geodesic, you could divide the distance along a particular geodesic by c to define "how long ago" the event happened, but I don't know if such a unique definition of "distance" is possible, and I'm pretty sure that the answer you'd get for "how long ago" the event happened using this procedure would be not necessarily be the same along two different paths (As an extreme case, imagine a wormhole that allows an event to send light into its own past light cone, something that is permitted by GR although quantum gravity may rule it out; in this case, an observer might first receive light that travelled 'backwards in time' through the wormhole, then later receive light from the same event that travelled along a more normal path.)

Last edited: Mar 6, 2005
18. Mar 6, 2005

### jcsd

JesseM -The coordinate system doesn't have any real physical significance (even thoguh it's time axis corresponds to the worldline of an inertial observer) as the basis vector fields don't correspond to anything of real physical signifcance. In SR arbiartry coordiante transformations are not particularly interesting as the laws of SR refer to specifc sets of coordinate sytems only and it only really makes sense to call these coordinate systems inertial.

In GR it certainly does make sense to tlak of reference frames, though two obsrevres are usually considerd to be in different refernce frmaes not only if they are travelling at different velocities, but if they are spatially separted. The laws of GR apply to all coordiante systems, so in this sense there is not distinction between coordinate systems with physical and those without signifacnce, though it doesn't mean that coordinate systems with physicla signifcance don't exist.

Garth - the term compact manifold is probably what you're looking for as it implies usually that the manifold is boundaryless (I think the term boundaryless is better than unbounded as in actually fact these universe are bounded metric spaces!).

Cleraly this cosmological twin paradox doesn't actual breach GR in any tangible way, the worse it could be claimed is that such a result brecahes the philosphical aims of GR, howver even then I would disagree as the phislophy of GR does not extend to measuring everything the same in all coordinate sytems. Look at it this way: just like in the original twin paradox the two apparently symmertical obsrevers are not symmertical, in the cosmological version this is due to the fact that they both see different universes i.e. their spatial slices are different, so it should be no supridse when they measure different times for the round trip.

19. Mar 6, 2005

### JesseM

But what I was saying is that there is no global notion of an observer's "reference frame" in GR, in the sense of a global coordinate system which tells you stuff like how fast a distant object is moving "in your reference frame". You seem to be saying that it only makes sense to say two objects are in "the same reference frame" if they are in the same local neighborhood, which agrees with that.
But when you say the same laws of GR apply to all coordinate systems, does that actually mean the equations you'd use to make physical predictions would look the same in different coordinate systems? In the case of the twins circumnavigating a flat finite universe, if you define each twin's coordinate system the same way you would in SR, it doesn't seem like both can use the same equations to predict the other twin's time dilation as a function of speed in their own coordinate system, since each one sees the other moving at the same constant speed v in their coordinate system. Maybe the answer is that in GR you can't assume the equations describing things like time dilation are just functions of the coordinate values for position, time and velocity, maybe the equations would also have to be functions of the metric at each point in space...but in the case of a flat space whose topology allows it to be finite, doesn't the metric look the same in both coordinate systems?

Last edited: Mar 6, 2005
20. Mar 6, 2005

### jcsd

Well an obsrever can observe distant objects and assign them a velocity based on these obsrevations (Though this velcoity is most defintelty diffefernt from the concept of the relative velocity of two objects which are not spatially seperated) and you cna construct a coordinate system to reflect the obsrevations of an obsrever (though it's entirely possible that the coordianate system assigns multiple sets of coordinate sto the same points or that it contains singularities). generally speaking you can't regard spatially seprated observers as having the same reference frame.
The equations of GR look the same in all coordinate systems in GR as they are tensorial.

For a universe that is geometrically flat, but finite the universe does not look the same to both twins, though it would appear flat to both of them (e.g. lets say this universe is a cylinder, they both still see the universe as a cylinder, but one of the twins would measure it's circumference to be less than the other one would)