I Would circumnavigating the Universe allow one way measurement of light's speed?

[Flatland]

Suppose we're in a closed and non-expanding universe. You shine a beam of light in one direction such that it circumnavigates the universe and returns to its starting point from the other side. Now you put a detector at this starting point. Would this be a one way measurement of the speed of light?

 

[sysprog]

How would we detect that the beam had arrived after a 'supra-universal' traversal? Wouldn't it seem to have come from 'somewhere over there'? If something from the cosmic background radiation that started at the big bang is ever seen coming in from the opposite direction, how will we know its origin?

 

[Flatland]

How would we detect that the beam had arrived after a 'supra-universal' traversal? Wouldn't it seem to have come from 'somewhere over there'? If something from the cosmic background radiation that started at the big bang is ever seen coming in from the opposite direction, how will we know its origin?

I suppose you can do it by encoding a message on that beam of light.

 

[Ibix]

Depends what you mean by "a closed and non-expanding universe".

Assuming you mean a Minkowski spacetime that's wrapped into a cylinder, then there's only one global inertial chart that can cover this spacetime, and you are committed to using that. If you use a non-inertial chart then the speed of light isn't constant over its path and there isn't "a" speed to measure; if you use any inertial chart except the one picked out by the axial symmetry of the spacetime then you have a break in your coordinates and your answer depends on how you address that. So you've forced a coordinate choice on yourself due to the physics of your model universe. I think that means that you have a one-way speed of light measurement. However, note that local one-way measurements still remain ambiguous - it's only this "loop" speed that is so defined.

If you don't mean a cylindrical Minkowski spacetime then it depends what spacetime you do mean.

 

[PeroK]

Suppose we're in a closed and non-expanding universe. You shine a beam of light in one direction such that it circumnavigates the universe and returns to its starting point from the other side. Now you put a detector at this starting point. Would this be a one way measurement of the speed of light?

There's also an (albeit unstable) circular lightlike orbit at a radius of $3M$ around a Schwarzschild black hole.

 

[Ibix]

How would we detect that the beam had arrived after a 'supra-universal' traversal? Wouldn't it seem to have come from 'somewhere over there'? If something from the cosmic background radiation that started at the big bang is ever seen coming in from the opposite direction, how will we know its origin?

If you have a CMB then it's not a non-expanding universe.

 

[sysprog]

If you have a CMB then it's not a non-expanding universe.

Well, the OP postulated hypothetically in post #1: "Suppose we're in a closed and non-expanding universe."

 

[PeterDonis]

Suppose we're in a closed and non-expanding universe.

Aside from the suggestions already made, a possible spacetime geometry that meets this description is the Einstein static universe. This universe has a constant density of matter everywhere and a positive cosmological constant; the effects of the two just balance. Note that this solution is unstable to small perturbations; any small perturbation will either cause it to collapse or cause it to expand.

You shine a beam of light in one direction such that it circumnavigates the universe and returns to its starting point from the other side. Now you put a detector at this starting point. Would this be a one way measurement of the speed of light?

I would say no, because it's not confined to a localized region of spacetime.

 

[PeterDonis]

you've forced a coordinate choice on yourself due to the physics of your model universe. I think that means that you have a one-way speed of light measurement.

I don't think you can call it that since, as you point out, it's not localized.

 

[russ_watters]

Suppose we're in a closed and non-expanding universe. You shine a beam of light in one direction such that it circumnavigates the universe and returns to its starting point from the other side. Now you put a detector at this starting point. Would this be a one way measurement of the speed of light?

How exactly are you measuring speed? d/t? How do you know d?

 

[Flatland]

There's also an (albeit unstable) circular lightlike orbit at a radius of $3M$ around a Schwarzschild black hole.

From what I understand that would not be a one way measurement of light.

How exactly are you measuring speed? d/t? How do you know d?

You take a spaceship and a measuring tape and loop that measuring tape around the universe.

 

[PeterDonis]

From what I understand that would not be a one way measurement of light.

If that is the case, the same argument would apply to your scenario. If you are sitting at rest at $r = 3M$ above a black hole, and you send a light beam out in the tangential direction, it will come back to you (from the opposite tangential direction) after going around the hole once.

 

[Flatland]

If that is the case, the same argument would apply to your scenario. If you are sitting at rest at $r = 3M$ above a black hole, and you send a light beam out in the tangential direction, it will come back to you (from the opposite tangential direction) after going around the hole once.

I asked a very similar question in this thread

https://www.physicsforums.com/threa...way-speed-of-light.995539/page-5#post-6423799

and one user pointed out that it's no different than using mirrors to bounce that light back to you.

 

[PeterDonis]

one user pointed out that it's no different than using mirrors to bounce that light back to you

Which, as @Nugatory pointed out in that post, corresponds to a two-way measurement, not a one-way measurement.

I'll have to think over whether I agree that such a measurement can count as a two-way measurement; I'm still not sure that's valid given that the measurement is not localized.

 

[Ibix]

I don't think you can call it that since, as you point out, it's not localized.

I'm not sure I agree. It's definitely not a local one-way speed measure, but it's a one-way speed measure. There's no turnaround - in a conventional two-way speed measurement (and also the black hole photon sphere situation) you can always find an observer who will say that the pulse is traveling left-to-right at some time and right-to-left at another.

You're probably right that there's a distinction there that needs to be emphasised more than I did: local one-way speed measures aren't possible in the cylindrical Minkowski spacetime either (or, indeed, any other). So the corollary is that the existence of a "global" one-way speed measure, even if you are happy to call it that, doesn't invalidate the claim that you can't measure one-way speeds of light.

I'll have to think over whether I agree that such a measurement can count as a two-way measurement; I'm still not sure that's valid given that the measurement is not localized.

It will clearly return the same value as a two-way speed measurement. Isn't the point here that, like the cylindrical Minkowski example, the structure of the spacetime picks out a special group of inertial observers: Schwarzschild observers at the photon sphere are the only ones who see equal light orbit times in opposite directions. Because of that they can Einstein synchronise to their own clocks (which other types of observer can't do) and Einstein synchronise to any other clock along the light path (ditto). So they can make a one-way measure and it's guaranteed to be the same as a two-way measure because that's what the physics picks out.

 

[PeterDonis]

It's definitely not a local one-way speed measure, but it's a one-way speed measure. There's no turnaround - in a conventional two-way speed measurement (and also the black hole photon sphere situation) you can always find an observer who will say that the pulse is traveling left-to-right at some time and right-to-left at another.

No, you can't find any such observer in the black hole photon sphere situation either. Or in the closed Einstein static universe situation.

In all three cases, what we basically have is a light ray traveling on a helical worldline in a submanifold with topology $R^1 \times S^1$. The "direction of travel" along the $S^1$ dimension of the submanifold is constant along the worldline; this is the key difference from an ordinary two-way measurement with a mirror, where the direction of travel along the spacelike dimension of the 2-D submanifold in question changes when the mirror reflection occurs. (I have not given a precise technical definition of "direction of travel"; hopefully it's obvious enough what I mean. A precise definition would involve the inner product of the worldline tangent vector with some appropriate family of spacelike vectors.)

This type of trajectory doesn't really correspond to the usual intuitive picture of either a one-way or a two-way measurement. Perhaps the best way to describe it in ordinary language would be to say that both descriptions are applicable in some sense--they both capture part of what is going on. "One-way" captures the fact that the "direction of travel" doesn't change. "Two-way" captures the fact that the spatial starting and ending points are the same. But both terms also have ordinary implications that do not apply: "one-way" ordinarily implies different starting and ending points, but they're the same; "two-way" ordinarily implies a changing direction of travel, but it doesn't change.

 

[Dale]

Would this be a one way measurement of the speed of light?

No, but it would be a measurement of the size of the universe in meters.

 

[Flatland]

No, but it would be a measurement of the size of the universe in meters.

If you can measure the size of the universe then you can also measure the time it took the light to loop around and thus measuring its speed.

 

[Dale]

If you can measure the size of the universe then you can also measure the time it took the light to loop around and thus measuring its speed.

Not independently.

 

[A.T.]

This type of trajectory doesn't really correspond to the usual intuitive picture of either a one-way or a two-way measurement.

Instead of arguing about the intuitive meaning of "one-way" or "two-way", one could reframe the question in terms of physical functionally:

Does the proposed method allow to detect anisotropic propagation speed?

 

[PeterDonis]

Does the proposed method allow to detect anisotropic propagation speed?

I would say no, because an observer at rest relative to the $S^1$ dimension of the submanifold must see an isotropic propagation speed (if he sends two light signals in opposite directions, they will both return to him at the same instant), and there is no way to set up any valid global coordinate chart centered on an observer who is not at rest relative to the $S^1$ dimension of the submanifold. (Again, I haven't given a precise definition of this notion of "at rest", hopefully it is understood what I mean; a precise definition would involve picking out integral curves of a Killing vector field that is orthogonal to a particular family of $S^1$ curves that foliates the spacetime in a particular way.)

 

[PeterDonis]

Not independently.

For a particular family of observers, it is possible to independently measure the distance around the $S^1$ dimension of the submanifold, by placing a circle of rulers (or a circular tape measure, or something similar) at rest relative to the observers that goes around the $S^1$ dimension and closes back on itself.

 

[Sagittarius A-Star]

Because of that they can Einstein synchronise to their own clocks (which other types of observer can't do) and Einstein synchronise to any other clock along the light path (ditto). So they can make a one-way measure and it's guaranteed to be the same as a two-way measure because that's what the physics picks out.

A one-way measure cannot be guaranteed to be the same as a two-way measure. For a one-way measure you must always have the freedom to define a coordinate sheet for anisotropic light speed:

The observer in the light sphere could have two clocks $A$ and $B$ at the same location and measure the start time of a light pulse in clockwise direction with clock $A$ and the arrival time with clock $B$. Then clock $B$ can be used to measure the start time of a light pulse in counter-clockwise direction and clock $A$ to measure the arrival time.

Before that, an Einstein synchonization or a different synchonization must be carried out with clocks $A$ and $B$.

 

[PeterDonis]

For a one-way measure you must always have the freedom to define a coordinate sheet for anisotropic light speed

No, you don't always have this freedom. As I have already noted, in all of the cases under discussion, since they all amount to having a submanifold with topology $R^1 \times S^1$, it is impossible to define a valid coordinate chart covering all of the worldline of one "circumnavigation" (one full circle around the $S^1$ dimension) for any case except the "at rest" case I described in an earlier post.

Note, btw, that this is the case even without trying the (invalid) "two clocks" thing you describe (see further comments below about that).

The observer in the light sphere could have two clocks $A$ and $B$ at the same location

Which, unless both keep exactly the same time (and if they do, there's no point), does not define a valid coordinate chart, since a valid coordinate chart can only assign one coordinate time to any event, but your scheme here with clocks $A$ and $B$ keeping different times would assign two different times to every event.

 

[PeterDonis]

As I have already noted, in all of the cases under discussion, since they all amount to having a submanifold with topology $R^1 \times S^1$, it is impossible to define a valid coordinate chart covering all of the worldline of one "circumnavigation" (one full circle around the $S^1$ dimension) for any case except the "at rest" case I described in an earlier post.

Actually, this is too strong as I state it. I should have said it is impossible to define a valid orthogonal chart (i.e., one with no "cross terms" in the metric) except for the "at rest" case. If one accepts non-orthogonal charts, one can define a non-orthogonal chart for any observer (other than the "at rest" one) by simply using the same time coordinate as the "at rest" chart, but "tilting" the worldlines that are at constant space coordinates (the way the Born chart does).

In one of these non-orthogonal charts, the coordinate speed of light as measured by "circumnavigation" will be anisotropic, because of the Sagnac effect. But this has nothing to do with having multiple clocks on a given observer's worldline running at different rates.

 

[Sagittarius A-Star]

Which, unless both keep exactly the same time (and if they do, there's no point), does not define a valid coordinate chart, since a valid coordinate chart can only assign one coordinate time to any event, but your scheme here with clocks $A$ and $B$ keeping different times would assign two different times to every event.

A counter-example would be Born coordinates in a rotating frame.

 

[PeterDonis]

A counter-example would be Born coordinates in a rotating frame.

No. Born coordinates do not have multiple clocks on the same worldline running at different rates. See my post #25.

 

[Sagittarius A-Star]

No. Born coordinates do not have multiple clocks on the same worldline running at different rates. See my post #25.

They do not have multiple clocks on the same worldline running at different rates. But they do have two clocks at the same rate with an offset, at the "same" location(s) $\phi = 0$ and $\phi = 2\pi$, assigned to different events.

 

[A.T.]

For a particular family of observers, it is possible to independently measure the distance around the $S^1$ dimension of the submanifold, by placing a circle of rulers (or a circular tape measure, or something similar) at rest relative to the observers that goes around the $S^1$ dimension and closes back on itself.

Let's say one these observers sends 2 light signals simultaneously in opposite directions, and checks if they reach him again simultaneously. Is this not a test for isotropic propagation speed?

 

[Dale]

For a particular family of observers, it is possible to independently measure the distance around the $S^1$ dimension of the submanifold, by placing a circle of rulers (or a circular tape measure, or something similar) at rest relative to the observers that goes around the $S^1$ dimension and closes back on itself.

Except that in theories where c is anisotropic you can have length contraction in certain directions at rest. I have not worked out the math here, but I think that would be the case.

 

[Dale]

Is this not a test for isotropic propagation speed?

I would need to work out the math, but I think it would be a test for rotation, not isotropy.

 

[A.T.]

I would need to work out the math, but I think it would be a test for rotation, not isotropy.

Wouldn't rotation also imply a proper acceleration? You could choose the family of observers with no proper acceleration.

You could also pick the family of observers who needed the least tape to circumvent the universe.

 

[Dale]

Wouldn't rotation also imply a proper acceleration? You could choose the family of observers with no proper acceleration.

You could also pick the family of observers who needed the least tape to circumvent the universe.

That should ensure a null result, meaning that you have no rotation.

 

[A.T.]

... I think it would be a test for rotation, not isotropy.

That should ensure a null result, meaning that you have no rotation.

Are you saying that any anisotropy in the propagation is per definition due to rotation? Then I don't understand your first statement.

The test itself is directly checking for isotropy of propagation. We can interpret the result as a measure of rotation. But I don't understand why you say that it is not a test of isotropy of propagation.

And if it is rotation, which direction does the centripetal proper acceleration have?

 

[Dale]

Are you saying that any anisotropy in the propagation is per definition due to rotation? Then I don't understand your first statement.

The test itself is directly checking for isotropy of propagation. We can interpret the result as a measure of rotation. But I don't understand why you say that it is not a test of isotropy of propagation.

And if it is rotation, which direction does the centripetal proper acceleration have?

I am thinking of the Sagnac effect which measures only rotation and does not measure anisotropy of the one way speed of light. I am not sure that the restriction generalizes to curved spacetime. But since in flat spacetime it definitively measures rotation and not anisotropy I am immediately skeptical of any claim that in curved spacetime it measures anisotropy.

 

[A.T.]

I am thinking of the Sagnac effect which measures only rotation and does not measure anisotropy of the one way speed of light.

I would say the Sagnac effect measures the anisotropy of the one way speed of light in a rotating reference frame.

 

[Dale]

I would say the Sagnac effect measures the anisotropy of the one way speed of light in a rotating reference frame.

Consider a rotating reference frame and a family of Sagnac interferometers throughout the rotating reference frame and all at rest with respect to the reference frame. For convenience we will use cylindrical $(r, \theta, z)$ coordinates with $\hat r$, $\hat \theta$, and $\hat z$ being the radial, tangential, and longitudinal unit vectors respectively, and of course the rotation is about the $r=0$ axis with an angular velocity $\omega \hat z$ and the interferometers are limited to the region $r<c/\omega$.

Now, at each point the one way speed of light in this frame is anisotropic about the $\hat \theta$ direction, and the degree of anisotropy increases with increasing $r$. In contrast, at each point the rotation is about the $\hat z$ direction and the rate of rotation is $\omega$ independent of $r$.

Now, consider the measurements of the Sagnac interferometers. The Sagnac interferometers detect no signal when they are oriented in the $\hat r$ or $\hat \theta$ directions, they only detect signal when oriented in the $\hat z$ direction. So the Sagnac signal is anisotropic about the $\hat z$ direction. Furthermore, the magnitude of the Sagnac signal is independent of $r$ and is proportional to $\omega$. Therefore, the Sagnac signal is strictly a measure of rotation and definitively not a measurement of the anisotropy of the one way speed of light in the rotating frame.

This is an occasional source of confusion because the Sagnac effect can be derived in the rotating frame using the one-way speed of light in that frame, but despite the derivation, it does not possesses the right signal behavior to be considered a measurement of the one way speed of light. This should not be terribly surprising since the Sagnac effect can also be derived in an inertial frame with no anisotropy in the one way speed of light. So it must be measuring something else, and that something is the rotation which is present in both derivations and which fits the observed signal behavior.

 

[Sagittarius A-Star]

This should not be terribly surprising since the Sagnac effect can also be derived in an inertial frame with no anisotropy in the one way speed of light. So it must be measuring something else, and that something is the rotation which is present in both derivations and which fits the observed signal behavior.

Also, the relativistic calculation shows, that the $\Delta t$ of the Sagnac effect is independent of the signal speed. It is a relativistic effect, related to the Born coordinates.

Rather remarkably, this is independent of the signal velocity! A beam of light sent around the disk each way, or two people strolling slowly around the disk each way, will encounter the same difference in the time it takes to go all the way around.

Source:
http://www.physicsinsights.org/sagnac_1.html

 

[Sagittarius A-Star]

Let's say one these observers sends 2 light signals simultaneously in opposite directions, and checks if they reach him again simultaneously. Is this not a test for isotropic propagation speed?

I would say no.

Suppose you have an observer $A$ with a clock $A$ in the light sphere and a second observer $B$ with a clock $B$ on the opposite side of the black hole in the light sphere.

Clock synchonization:
Observer $A$ sends a light pulse at 01:00 A.M. according to clock $A$ via the light sphere to observer $B$, who sets his clock $B$ to 01:00 A.M. just as the light pulse arrives and reflects with a mirror the light pulse back along the same semi circle.

Then the light moved instantaniously from $A$ to $B$ and back with c/2.

The same experiment is repeated via the other semi circle. Also then the light moved from $A$ to $B$ instantaniously and back with c/2.

If you define a related sphere-coordinate system, then $A$ can send a light pulse around the whole circle, which moves instantaniously in the 1st semi circle and with c/2 in the 2nd semi circle. But $A$ measures then only the 2-way speed of light c.

 

[Vanadium 50]

Let's go back to basics for a moment.

SR says that c is c. A different speed in different directions would mean SR is wrong. GR builds on SR, and cosmology builds on GR. So it's not clear that this question even makes sense.

Actually, I think it doesn't make sense as asked, although it may be possible to pose a well-defined similar question. Or maybe not? If you don't have spacetime, what does it mean for spacetime to be curved and the universe closed? If you have spacetime without SR, how exactly does this work?

Next, let's examine why you can't measure the one-way speed of light. You need a distance, a time, and a synchronization convention.

Sagredus: We don't need a synchronization convention because the light comes back to you.
Salvatus: Provided you are at the same point. How do you know that?
Sagredus: Um, absolute space?
Salvatus: There is no such thing. How do you know you receive the light at the same place you emitted it? After all, there are two parts to a synchronization convention - space and time.
Sagredus: If you move from A to B, you feel the acceleration as you start and stop.
Salvatus: Not if you are moving at constant velocity.
Sagredus: But you can tell if you are moving!
Salvatus: How?
Sagredus: Relative to the fixed stars! Or, if you like, in a frame where the totality of the rest of the universe has no net momentum with respect to you.
Salvatus: That is picking a synchronization convention. One can do this even in expanding FRW universe, if one could imagine such a crazy thing. The time is measured in comoving frames relative to the big bang. A person at one place shines a light at another person, and tells that person how many seconds it has been since the big bang in his comoving frame. The second person knows the time the pulse was sent, how to synchronize their clocks, the distance between himself and the source, and off he goes to calculate. He will of course get c, because that's a consequence of his synchronization convention.

So, I believe that if it were possible to replace the question asked by a similar question that is well-defined, the answer would be "you are picking a synchronization convention. The most obvious one, so obvious that you don't even know you are picking it, gives c as the result of the proposed experiment."

 

[PeterDonis]

They do not have multiple clocks on the same worldline running at different rates. But they do have two clocks at the same rate with an offset, at the "same" location(s) $\phi = 0$ and $\phi = 2\pi$, assigned to different events.

If $\phi = 0$ and $\phi = 2 \pi$ are the same location (same worldline), then the two clocks with different offsets, even if they run at the same rate, must still be assigning different times to the same event on that worldline. Which is not a valid coordinate chart.

 

[PeterDonis]

Let's say one these observers sends 2 light signals simultaneously in opposite directions, and checks if they reach him again simultaneously. Is this not a test for isotropic propagation speed?

Not unless you make an appropriate coordinate choice. What you are describing is a test for the Sagnac effect, and the only "objective" (i.e., invariant) property that tests for is, heuristically, whether or not your worldline has nonzero angular velocity relative to a particular global symmetry of the spacetime. With an appropriate choice of coordinates, you can interpret this nonzero angular velocity relative to the global symmetry as implying an anisotropic speed of light; but nothing requires you to adopt such coordinates.

 

[PeterDonis]

Wouldn't rotation also imply a proper acceleration?

No, in the sense of the simple presence of proper acceleration being an independent test for rotation. The "rotation" being referred to here is not the same as following a circular path in ordinary Minkowski spacetime.

In the "Minkowski rolled up into a cylinder" case and the Einstein static universe case, none of the observers in question have any proper acceleration, whether they are "rotating" with respect to the global symmetry of the spacetime or not.

In the "photon sphere around the black hole" case, all of the observers have proper acceleration, whether they are "rotating" or not. Their proper acceleration does vary with angular velocity--it is greatest for zero angular velocity, and approaches zero in the limit as the angular velocity approaches the maximum possible, the angular velocity corresponding to a light ray in the photon sphere circular orbit--but the proper acceleration is never exactly zero except for the light ray itself (its circular orbit is a geodesic).

 

[PeterDonis]

You could also pick the family of observers who needed the least tape to circumvent the universe.

There is only one family of observers who can even have a tape at rest relative to them that circumnavigates the universe. That is the family of observers whose worldlines are integral curves of the timelike Killing vector field that I have been calling a "global symmetry" of the spacetime--the one that is orthogonal to a set of closed $S^1$ curves that foliate the $R^1 \times S^1$ submanifold.

 

[PeterDonis]

SR says that c is c.

Actually, that's not what SR says in the modern view. In the modern view, you are perfectly free to do SR in a coordinate chart in which the coordinate speed of light is not always $c$. You don't even have to get into all the weird possibilities; the Rindler coordinate chart is a simple, well-known example.

What SR says in the modern view is that spacetime has an invariant geometric light cone structure and zero curvature (the latter condition is what distinguishes SR from GR, in which the curvature can be nonzero). How that structure "appears" in quantities like the speed of light can depend on your choice of coordinates; only the underlying geometric structure itself is invariant. As you note, the most common choice of coordinates when spacetime is flat is the one that makes the coordinate speed of light invariant and always $c$.

 

[Dale]

There is only one family of observers who can even have a tape at rest relative to them that circumnavigates the universe.

So, here is the issue in geometric terms as I see it. If you have such a tape then that constitutes a cylinder in spacetime. The tape can be given marks, each of which becomes a line along the length of the cylinder, and a pulse of light of the type we are interested in is a helix wrapping around the cylinder.

Now, if you take one of the mark lines and the two events where the light intersects it then there are only two relevant frame invariant measures you can make: the invariant length of the two paths between the two events. One is the proper time, but the other is null, not the distance.

The cylinder has timelike lines on it from the marks, but you can draw rather arbitrary spacelike lines around it. And many different sets of spacelike lines are valid, each representing a different synchronization convention. Those spacelike lines will give different values for the total circumference as well as different values for the local one way speed of light at each point.

So, in the end, whether this is a “one way” measurement or not it still suffers from the same ambiguity as a typical one way measurement. Namely, you still need to define a synchronization convention which changes the result both locally along the trip and overall.

 

[PeterDonis]

If you have such a tape then that constitutes a cylinder in spacetime.

Yes; the "worldsheet" of the tape is precisely the $R^1 \times S^1$ submanifold I have been referring to.

The tape can be given marks, each of which becomes a line along the length of the cylinder

Yes; these are the "at rest" worldlines I have been describing.

a pulse of light of the type we are interested in is a helix wrapping around the cylinder

Yes.

if you take one of the mark lines and the two events where the light intersects it then there are only two relevant frame invariant measures you can make: the invariant length of the two paths between the two events.

Yes, these are frame invariant measures, but not the only ones. As you note, the invariant length along the mark line defines an invariant proper time between the two "light intersection" events. But you can also define an invariant proper distance around the cylinder: just take the invariant length of the closed spacelike circle that intersects the mark line at one of the two"light intersection" events (either one will do--and in fact any event along the mark worldline will do--since the metric of the submanifold is stationary). And dividing that invariant proper distance by the invariant proper time gives a measurement of the speed of light.

many different sets of spacelike lines are valid, each representing a different synchronization convention

No. Only one set of spacelike lines is valid, because it is the only one that is a set of closed circles that each intersect every mark line at precisely one event. Every other set of spacelike lines intersects each mark line at multiple events, because the spacelike lines are helixes, not closed circles. And that means you can't use those spacelike lines to define a valid synchronization convention, because any such convention must lead to a foliation by spacelike curves that each only intersect a timelike curve once.

In other words, the $R^1 \times S^1$ submanifold in question has a unique foliation in terms of worldlines of marks on the tape and closed spacelike circles, picked out by its geometry. And that unique foliation can be used to measure the speed of light that circumnavigates the manifold.

 

[Dale]

But you can also define an invariant proper distance around the cylinder: just take the invariant length of the closed spacelike circle that intersects the mark line at one of the two"light intersection" events (either one will do--and in fact any event along the mark worldline will do--since the metric of the submanifold is stationary)

The problem is that “the closed spacelike circle” is not uniquely required. You can take any number of closed spacelike paths.

No. Only one set of spacelike lines is valid, because it is the only one that is a set of closed circles that each intersect every mark line at precisely one event. Every other set of spacelike lines intersects each mark line at multiple events, because the spacelike lines are helixes, not closed circles

This is not correct. There are also spacelike ellipses which are closed and intersect each mark line at one event. Think of slicing a baguette straight crosswise (one of your circles) or at an angle (one of my ellipses).

There are also more complicated slicings which are permissible, but more difficult to describe (like Pringles potato chips or waffle fries)

Every other set of spacelike lines intersects each mark line at multiple events, because the spacelike lines are helixes, not closed circles.

Are you thinking of rotating frames here?

 

[A.T.]

Sagredus: But you can tell if you are moving!
Salvatus: How?
Sagredus: Relative to the fixed stars! Or, if you like, in a frame where the totality of the rest of the universe has no net momentum with respect to you.

My understanding is that in a closed (cylindrical universe) there is a preferred rest frame:

https://www.tandfonline.com/doi/abs/10.1080/00029890.2001.11919789

We can find it by sending clocks around the universe, and comparing the proper times between two meetings. No need to refer to fixed stars or other references.

 

[PeterDonis]

There are also spacelike ellipses which are closed and intersect each mark line at one event.

Yes, you're right, this is true; my statement as I gave it was too strong.

I think these ellipses will not work as a synchronization convention, but I'll have to think some more to see if I can come up with a rigorous way to express why not.

Are you thinking of rotating frames here?

No. I'm thinking of the natural way a Lorentz transformation acts. Suppose we have a frame with the mark lines and closed circles (not ellipses--the fact that they are circles and not ellipses does geometrically pick out this particular foliation) as its timelike and spacelike axes. If we now transform to a "moving frame" whose time axis is a helical timelike worldline that wraps around the cylinder, with some velocity $v$ relative to the mark lines of the original frame, then the corresponding spacelike axis of this "frame" is a helical spacelike curve like the ones I described. (To see why this must be the case, unroll the cylinder.) And of course that means this "frame" is not actually a valid coordinate chart globally; it can only be used on a local patch that does not wrap all the way around the cylinder.