Twin Paradox Time Dilation in Positively Curved Universe

[Hiero]

I’m sure the resolution is something to the effect of “we can only apply special relativity in flat spacetime” but I’m hoping someone can explain in more detail.

Disclaimer: I don’t know general relativity.

So in a positively curved universe, if you keep traveling (let us neglect expansion) you will eventually come back to where you started?

This seems to present a paradox. Consider two inertial(?) observers in relative motion. They both see each other moving in slow motion (time dilation). This is normally not paradoxical because they can never compare clocks to see who really slowed down without one accelerating (thus breaking the symmetry) like in the twin ‘paradox.’ But in a positively curved universe they would come back to each other without ever changing their state of motion. So they would both expect the other to have had less time elapse on their clocks.

The only resolution I can think of is to say that a curved universe implies a universal rest frame? So that the one who is truly at rest (or I suppose, moving slower) is the one who ages more?

Appreciate any insight into this revised version of the twin paradox.

 

[PeroK]

I’m sure the resolution is something to the effect of “we can only apply special relativity in flat spacetime” but I’m hoping someone can explain in more detail.

Disclaimer: I don’t know general relativity.

So in a positively curved universe, if you keep traveling (let us neglect expansion) you will eventually come back to where you started?

This seems to present a paradox. Consider two inertial(?) observers in relative motion. They both see each other moving in slow motion (time dilation). This is normally not paradoxical because they can never compare clocks to see who really slowed down without one accelerating (thus breaking the symmetry) like in the twin ‘paradox.’ But in a positively curved universe they would come back to each other without ever changing their state of motion. So they would both expect the other to have had less time elapse on their clocks.

Appreciate any insight into this revised version of the twin paradox.

What happens if you apply your thinking to motion on the surface of a sphere?

 

[Hiero]

What happens if you apply your thinking to motion on the surface of a sphere?

Then (at least) one of the observers is accelerating, hence why I said it seems to imply a universal rest frame. Is that the case?

 

[PeroK]

Then (at least) one of the observers is accelerating, hence why I said it seems to imply a universal rest frame. Is that the case?

I think you know that's not the case!

Consider instead two satellites in the same natural orbit around a star or planet, but moving in opposite directions.

The time recorded by a clock is the spacetime distance along its worldline. Two clocks in the above orbits have worldlines of the same length between passing each other. But, in curved spacetime there is no such thing as a universal inertial frame. You can't apply the simple notion of time dilation based on relative velocity alone. You also have to take the curvature of spacetime into account.

 

[PeterDonis]

in a positively curved universe, if you keep traveling (let us neglect expansion) you will eventually come back to where you started?

If you keep traveling along a geodesic, yes.

Consider two inertial(?) observers in relative motion.

Why the question mark? Either they're inertial, or they're not. Which is it? You're setting up the scenario, so you have to pick one. For the rest of this post, I'm going to assume that you want to pick "inertial".

They both see each other moving in slow motion (time dilation).

They will to start out, but that won't be the case indefinitely. See below.

in a positively curved universe they would come back to each other without ever changing their state of motion.

Yes, that's correct.

So they would both expect the other to have had less time elapse on their clocks.

No, they wouldn't, because spacetime is not flat and the rules of flat spacetime do not apply. They can see that spacetime is not flat from the very fact that, even though both of them are inertial the whole time, they can meet twice.

What you should be asking is what these observers will actually see--as in, what light signals will each receive from the other? The answer is that each one will initially see redshifted light signals from the other, indicating that they are moving apart. But after a certain amount of time, each one will start seeing blueshifted light signals from the other, coming from the other direction--ahead of them, not behind them. These light signals will have gone "around the universe", so to speak. These light signals indicate that the two observers are moving closer together--until, of course, they meet again.

The overall conclusion from all of the data is that each observer has the same elapsed time in between meetings.

The only resolution I can think of is to say that a curved universe implies a universal rest frame? So that the one who is truly at rest (or I suppose, moving slower) is the one who ages more?

It is actually true that a positively curved universe has something like a "universal rest frame"; but your scenario as it stands doesn't show that. To see that, you have to add one more observer to your thought experiment.

Consider a third observer, call him O, who remains at the common starting point of the two observers in your scenario, call them A and B. A and B go in opposite directions from O, meet on the other side of the universe, continue on past each other there, and come back around to meet again at O. When all three meet up again, A, B, and O, A and B will each have the same elapsed time on their clocks, but O will have more elapsed time on his clock. In fact, O will have the most possible elapsed time on his clock of any observer whose worldline passes through the two events where all three of A, B, and O meet. This is a manifestation of the fact that O is the one who is at rest "relative to the universe".

Note that none of this violates the principle of relativity, because what picks out O as being at rest "relative to the universe" is not any preferred rest frame built into the laws of physics, but a particular symmetry of this particular solution to the laws of physics. This solution has a particular time translation symmetry which is much more limited than the time translation symmetries of the flat Minkowski spacetime of special relativity; unlike the latter, where every inertial observer's worldline is an integral curve of a time translation symmetry of the spacetime, here only O's worldline is; A's and B's worldlines are not. That is why O has more elapsed time than A and B, or indeed, as I said before, than any other observer whose worldline passes through both events where all three of A, B, and O meet.

 

[PeterDonis]

Two clocks in the above orbits have worldlines of the same length between passing each other.

Yes, but both of those worldlines have less length than a worldline which is an integral curve of the time translation symmetry of the spacetime--the worldline of an observer who simply "hovers" at the point in space where the two clocks pass each other, so they pass that observer again after completing one orbit.

Note that in this particular case, since the worldline which is an integral curve of the time translation symmetry is not a geodesic (the observer has to have nonzero proper acceleration to "hover"), there will be another inertial worldline which has even greater length (the maximal length of any worldline between the two meeting events): the worldline of an observer who free-falls radially outward from one event where the two clocks pass each other (and pass the "hoverer"), and then free-falls back radially inward to just pass both clocks again when they pass each other (and the "hoverer") after completing one orbit.

In the case of the positively curved universe, the integral curve of the time translation symmetry--the worldline of O in my previous post--is already a geodesic, so there is no curve between the same two events that has greater length.

 

[Hiero]

I think you know that's not the case!

Without knowing GR I know nothing

Consider instead two satellites in the same natural orbit around a star or planet, but moving in opposite directions.

The time recorded by a clock is the spacetime distance along its worldline. Two clocks in the above orbits have worldlines of the same length between passing each other. But, in curved spacetime there is no such thing as a universal inertial frame. You can't apply the simple notion of time dilation based on relative velocity alone. You also have to take the curvature of spacetime into account.

I do know that worldlines are a measure of proper time, and by the symmetry of your example I would agree that they see the same time pass, but I still don’t feel like I understand.

What if one satellite were not moving, would they still see the same time? I would suppose yes because we could take a frame of reference rotating at half the rate of the satellite to restore symmetry. Is this a valid thing to do right?

I guess I just generally don’t understand the effect of curvature.

 

[PeterDonis]

What if one satellite were not moving, would they still see the same time

See my post #6.

 

[Ibix]

You can consider the twin paradox in a flat spacetime that is rolled up into a cylinder. That does pick out a particular frame, the one that finds the circumference of the universe to be maximal. Locally it's no different from any other, but the non-trivial topology has an effect when you look globally. The twin who was at rest in the frame closer to that picked out frame will be older.

In an actual closed FLRW universe, I seem to recall someone (@PeterDinis?) saying that light can circumnavigate the universe once in the entire lifetime of the universe. So you can't do a circumnavigation on a timelike trajectory and can't do a similar experiment.

 

[PeterDonis]

In an actual closed FLRW universe, I seem to recall someone (@PeterDinis?) saying that light can circumnavigate the universe once in the entire lifetime of the universe.

For the case of a closed FRW universe with zero cosmological constant, yes, this is true. But this universe is not static, it is expanding (and then contracting), and the OP wanted to consider a case where there is no expansion (or contraction). The solution that fits that specification is the Einstein static universe (which has a nonzero cosmological constant of just the right size to balance the matter density and keep the universe from expanding or contracting), so that's the solution I was using in my previous posts.

 

[Ibix]

that's the solution I was using in my previous posts.

Thanks Peter. Post #5 was the last one visible when I started typing my last...

 

[Hiero]

Why the question mark? Either they're inertial, or they're not. Which is it? You're setting up the scenario, so you have to pick one. For the rest of this post, I'm going to assume that you want to pick "inertial".

Yes I meant inertial; the question mark was because I wasn’t sure if “inertial” is even meaningful in a curved spacetime (as I was thinking in analogy with a sphere’s surface which requires acceleration to stay on, and hence is non-inertial).

As for the rest of your post(s), well that is a lot to digest and it’s nearly 1am so I will have to sleep on it and read it again tomorrow.

Thank you for the food for thought. Good night.

 

[PeroK]

What if one satellite were not moving, would they still see the same time? I would suppose yes because we could take a frame of reference rotating at half the rate of the satellite to restore symmetry. Is this a valid thing to do right?

I guess I just generally don’t understand the effect of curvature.

I think considering the whole universe complicates matters. A simpler example is planetary orbits, where you can have symmetry of motion and repeated meetings. That's why I suggested you think about that first. Time dilation in SR doesn't generalise to curved spacetime. That's the main point. You can't just measure relative velocity and use a time dilation formula.

Flat spacetime is just a special case, where only relative velocity matters. If you start with curved spacetime where you have potentially multiple paths that meet twice or more without proper acceleration, then there is no mystery. SR is just a special case where this can't happen.

From that point of view, there is no reason that the simplicity of flat spacetime geometry extends to any curved spacetime.

 

[A.T.]

I guess I just generally don’t understand the effect of curvature.

As @Ibix points out, the key is not really the curvature, but rather the global topology, which can break the symmetry between inertial frames:

You can consider the twin paradox in a flat spacetime that is rolled up into a cylinder. That does pick out a particular frame, the one that finds the circumference of the universe to be maximal. Locally it's no different from any other, but the non-trivial topology has an effect when you look globally. The twin who was at rest in the frame closer to that picked out frame will be older.

On the cosmological scale, SR is a local concept.

 

[Dale]

This is normally not paradoxical because they can never compare clocks to see who really slowed down without one accelerating (thus breaking the symmetry) like in the twin ‘paradox.’ But in a positively curved universe they would come back to each other without ever changing their state of motion.

It still isn’t paradoxical. To achieve the specified curvature requires a particular distribution of matter. That matter breaks the symmetry. Both observers will agree on which of them is at rest wrt the matter and which is moving wrt the matter. As such they can both calculate who will be older.

 

[PeterDonis]

I wasn’t sure if “inertial” is even meaningful in a curved spacetime

It is. It just means "freely falling", or "feeling no force", or "feeling no weight". The same things it means in flat spacetime.

I was thinking in analogy with a sphere’s surface which requires acceleration to stay on

In what context?

and hence is non-inertial

If you mean, it takes acceleration to "hover" at a constant altitude above, say, a spherical planet, that doesn't make the planet non-inertial, it makes the person hovering non-inertial.

 

[kent davidge]

It just means "freely falling", or "feeling no force", or "feeling no weight"

Sorry for this, but are you sure? What about a observer inside an elevator freely falling towards our planet. Newton's first law is not satisfied in such frame.

 

[Ibix]

Sorry for this, but are you sure? What about a observer inside an elevator freely falling towards our planet. Newton's first law is not satisfied in such frame.

Yes it is, assuming the elevator is small enough that tidal effects may be neglected. What experiment did you have in mind where you think Newton's law is not respected?

 

[kent davidge]

Yes it is, assuming the elevator is small enough that tidal effects may be neglected. What experiment did you have in mind where you think Newton's law is not respected?

I had in mind a ordinary elevator falling in. I should have qualified my comment with what you said.
But @PeterDonis seemed to have been considering it without restrictions (in our case, that would be, the person inside the elevator makes measurements over arbritarily large periods of time, or the elevator not having small dimensions compared to our planet).

 

[PeterDonis]

@PeterDonis seemed to have been considering it without restrictions

I was talking about an observer, not an extended region. Whether or not an observer is inertial is determined exactly as I said. The OP was talking about inertial observers.

 

[Hiero]

Thanks for all the replies. I like the explanations by @Ibix and @Dale about how the topology of the cylinder universe and the matter causing curvature break the symmetry and give a kind of universal reference.

Also thanks @PeroK for emphasizing the flaw in applying SR to curved spacetime with the non-universal example.

I really appreciate how @PeterDonis was detailed and responded to every aspect of my questions, though some of it (like the integral time-translational symmetry stuff) went over my head.

Sorry if I didn’t respond to everything but I’m content enough and should get back to studying statistical mechanics.

Take care everyone, and feel free to continue discussing.

 

[PeterDonis]

the integral time-translational symmetry stuff

Just to clarify that bit, it is saying the same thing, in more technical language, as @Ibix and @Dale were saying about the presence of matter breaking the symmetry--the matter is all at rest in the "universal rest frame", so that is another way of picking out that frame physically for that particular solution.

 

[pervect]

I’m sure the resolution is something to the effect of “we can only apply special relativity in flat spacetime” but I’m hoping someone can explain in more detail.

Disclaimer: I don’t know general relativity.

So in a positively curved universe, if you keep traveling (let us neglect expansion) you will eventually come back to where you started?

This seems to present a paradox. Consider two inertial(?) observers in relative motion. They both see each other moving in slow motion (time dilation). This is normally not paradoxical because they can never compare clocks to see who really slowed down without one accelerating (thus breaking the symmetry) like in the twin ‘paradox.’ But in a positively curved universe they would come back to each other without ever changing their state of motion. So they would both expect the other to have had less time elapse on their clocks.

The only resolution I can think of is to say that a curved universe implies a universal rest frame? So that the one who is truly at rest (or I suppose, moving slower) is the one who ages more?

Appreciate any insight into this revised version of the twin paradox.

As I recall, but haven't been able to confirm with a reference (I think it was in MTW, but I couldn't find it in a brief search), an object or a light signal in a closed universe won't have time to come back to it's starting point as the paradox demands before the universe recollapses.

There are some other ways of creating the "paradox", but the one that comes to mind seem artifical. This involves changing the topology of space-time, and curvature is not needed. In two dimensions, it's a lot like gluing the left and right sides of a sheet of paper together, wrapping it in a tube, then gluing the top and bottom together. The 4-d case is harder to visualize, but uses the same basic idea. One is "cutting and gluing" together space-time, however, rather than sheets of paper.

One can do this in flat space-time (no curvature needed). But I don't see how it qualifies as a paradox. Is the analogy of having geodesics (straight lines) on a torus (think- donught) be shorter when the wrap around in a circle one way "paradoxical"? To me, it's just a feature of the geometry of the torus/ donught, and the same can be said for the space-time analogy. Though I will admit that I'm not particular fond of geometries with closed time-like curves, which some of these examples do have.

 

[PAllen]

I hope this doesn't confuse the OP too much, but you can create such a scenario in flat spacetime with non-trivial topology. For example, there is such a thing as a flat 3-torus. Then the topological product of this with the R, produces a spacetime that everywhere can have the standard Minkowski metric, but from every point there are 3 directions in which geodesics will return to their starting point. Any of these free fall paths will be younger on re-crossing with the geodesic that remains 'stationary' in the defining foliation. What breaks the symmetry in this case is the topology; there is no curvature anywhere.

 

[PeterDonis]

an object or a light signal in a closed universe won't have time to come back to it's starting point as the paradox demands before the universe recollapses

See post #10.