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

[Dale]

However, the ellipses are not what you get when you do a standard Lorentz transformation on the unrolled cylinder.

Agreed, but the whole point is to not limit ourselves to the simultaneity convention of the Lorentz transform.

It just won't be a coordinate chart that looks like an "inertial frame".

Agreed. Inertial frames include the isotropic one way speed of light assumption by definition, so we should be explicitly clear that these anisotropic c frames are not inertial.

I also don't think the surfaces of constant coordinate time in such a chart will correspond to any realizable clock synchronization among observers at rest in the chart, i.e., whose worldlines have unchanging spatial coordinates.

I disagree here. Any of these conventions are perfectly realizable. Given anyone of these conventions you know the distance to your neighboring clock and the one way speed of light from them to you. So when you receive a time stamped signal from your neighbor you can correct it for the known distance and speed to synchronize your clock.

Would it be the "vertical" axis (i.e, the same as the original "vertical-horizontal" chart)?

I would leave the time axis unchanged. I don’t know if it is possible to use a tilted axis in this sort of spacetime. I would need to see a proof before being willing to try it with a tilted axis.

it is impossible to pick a single "tilt" for the timelike curves that would be orthogonal to the ellipses everywhere

Yes, clearly. But with anisotropic one way speed of light discussions that is expected.

Or would the timelike curves have to have varying "tilt" as well (so they would also be sinusoids in the unrolled cylinder) to try to make them always orthogonal to the spacelike curves (if that is even possible)?

That is interesting. I had not considered that.

 

[PeterDonis]

I would leave the time axis unchanged.

Ok. That's the simplest way to do it, although I think it would also be possible to use a "tilted" time axis--it would just complicate the construction by clock offset that I describe below. (The "tilt" would also have to be limited, since we can't have more than one intersection between any timelike "grid line" and any spacelike "grid line"; I think that means the "tilt" of the timelike lines, relative to the "vertical" ones, would have to be no greater than the "tilt" of the spacelike ellipses relative to the spacelike circles.)

Given anyone of these conventions you know the distance to your neighboring clock and the one way speed of light from them to you.

If you need to already know the one-way speed of light in order to do the synchronization, then you can't use the synchronization convention in order to measure the one-way speed of light. I thought the point was to define different synchronizations that could be independently set up, so that they could be used as a basis to measure the one-way speed of light.

That said, I think I was wrong about there being no way to setup such a synchronization (without prior knowledge of the one-way speed of light). The "ellipse" synchronization convention basically amounts to giving a space-dependent offset to each clock; the offset will be $A \cos \left ( x / C \right)$, where $x$ is the spatial coordinate of the clock in the original "rest" frame (the one whose spacelike surfaces of constant time are circles), $C$ is the "circumference of the universe" (which is also the circumference of the spacelike circles), and $A$ is the maximum amplitude of the offset (which corresponds to the "tilt" of the ellipses relative to the circles). So given a set of clocks synchronized in the original "rest" frame, it is easy to adjust them to any desired "ellipse" synchronization. However, note that this construction makes no use of light signals (more precisely, none once the synchronization in the original "rest" frame is complete).

 

[Dale]

If you need to already know the one-way speed of light in order to do the synchronization, then you can't use the synchronization convention in order to measure the one-way speed of light.

Yes, of course. It is just an arbitrary choice of convention. The OP I think intended it to be a direct measurement, but I am convinced that it requires a simultaneity convention.

 

[PeterDonis]

The OP I think intended it to be a direct measurement, but I am convinced that it requires a simultaneity convention.

I agree, but I think it's worth explicitly laying out some of the complexities.

We have an invariant timelike interval: the interval between the emission and reception, by the same observer, of a light ray that "circumnavigates the universe" one time.

In order to obtain a value for "the speed of light" (leaving aside the question, which has been discussed elsewhere in this thread, of whether it is appropriate to call this speed a "one-way" speed, a "two-way" speed, both, or neither) from that, we need to have a distance. That means we need to have some way of specifying a closed spacelike curve (or family of curves all of the same circumference that foliate the $R^1 \times S^1$ submanifold--since the manifold is stationary, that is possible) whose circumference will be the distance we need. This is equivalent to specifying a simultaneity convention.

There are at least two ways of obtaining such a family of curves:

(1) We could impose some kind of constraint on the speed of light; for example, we could impose the constraint that it must be isotropic; or we could impose some kind of precisely specified anisotropy. The "isotropic" specification would give us the circles as our spacelike curves; different specifications of anisotropy would give us different sets of ellipses.

(2) We could impose some other kind of constraint that has nothing to do with the speed of light. For example, we could say that the spacelike curves must all be orthogonal to the timelike worldlines of our family of observers (or the segments of our tape measure that runs around the universe), or that they must have some specified "tilt" with respect to those worldlines. The "orthogonal" specification would give us the circles; different specifications of "tilt" would give us different sets of ellipses.

Method #1 would correspond to what you were describing in post #61: we know what we want the speed of light to look like and we deduce everything else accordingly.

Method #2 would correspond to what I was describing in post #62: we have an independent way of obtaining the simultaneity convention, and we deduce the speed of light from that, however it comes out.

Either way, though, as I said, I agree you need the family of curves, which is equivalent to a simultaneity convention, because that's the only way to obtain the distance you need to convert the invariant timelike interval you have into a speed.

 

[anuttarasammyak]

The issue is that we don’t know the radius. Or rather, defining the radius requires an assumption and a different assumption can be made.

Thank you . So I know we do not have sufficient information to answer the question of OP.

I imagine if we emit light to all the directions as the start, we would observe lights from all the directions at the same time or not as the goal to know anisotropy of universe or speed of light.

 

[A.T.]

So being able to set up a single global Einstein clock synchronization is a property that only the $O_A$ family of observers has.

So the $O_B$ observers can also have a continuous tape $T_B$ at relative rest to them around the universe (build as described below), they just cannot synch their clocks in the standard way?

$O_A$ could build two new tapes $T_{B1}$ and $T_{B2}$. Accelerate $T_{B1}$ segments until they are half their proper length, then accelerate $T_{B2}$ to fill the gaps in $T_{B1}$, to form a continuous moving $T_B$.

If $O_B$ rely only on their resting tape $T_B$ to measure the size of the universe (no clocks needed), they would get twice the value that $O_A$ gets?

But $O_B$ would also see the moving contracted tape $T_A$ with just half the number of meter marks spanning the same distance?

 

[PeterDonis]

So the $O_B$ observers can also have a continuous tape $T_B$ at relative rest to them around the universe (build as described below), they just cannot synch their clocks in the standard way?

If $O_B$ rely only on their resting tape $T_B$ to measure the size of the universe (no clocks needed), they would get twice the value that $O_A$ gets?

But $O_B$ would also see the moving contracted tape $T_A$ with just half the number of meter marks spanning the same distance?

The short answer is: it's more complicated than that.

The worldlines of the $O_A$ observers, and of their segments of tape, are timelike curves that go vertically up the cylinder, and don't wrap around it at all. If they Einstein synchronize their clocks, then a snapshot of their tape at an instant of time is just a closed circle around the cylinder. The length of the tape is then just the circumference of the circle.

The worldlines of the $O_B$ observers, and of their segments of tape, are then timelike helixes that wrap around the cylinder. They can't Einstein synchronize their clocks, as I have already said; but that means they also cannot define any set of surfaces of constant time, which would be needed to define the "length of the tape" according to these observers, that matches up with the local notion of "rest" of any of these observers. That is because, as I have already noted, the local notion of "rest" of each of these observers (which is the one that you are implicitly using when you talk about the $O_B$ tape being "contracted") is the spacelike curve that is orthogonal to their worldlines, and that curve is not closed--it is a spacelike helix that winds around the cylinder.

So there is no closed spacelike surface that defines a global notion of "space" for the $O_B$ observers and has the properties you are assuming. So the only way to define such a global notion of "space" is as a quotient space (similar to what is done with a rotating disk in discussions of how to resolve the Ehrenfest paradox). I think that the "circumference of the universe" in this quotient space will indeed be longer (by a factor of 2 in your specific example) than the circumference of the circle for the $O_A$ observers, but I have not done the math to check. However, this "space" also has weird properties such as the speed of light being anisotropic (it is slower in the "co-rotating" direction than the "counter-rotating" direction), and I believe at least one of those speeds of light will be significantly greater than $c$.

Because the global "space" for the $O_B$ observers is a quotient space, not a spacelike surface in the actual spacetime, it's not clear to me how the $O_A$ tape will look in this space, or if that notion is even well-defined.

 

[A.T.]

So the only way to define such a global notion of "space" is as a quotient space (similar to what is done with a rotating disk in discussions of how to resolve the Ehrenfest paradox). I think that the "circumference of the universe" in this quotient space will indeed be longer (by a factor of 2 in your specific example) than the circumference of the circle for the $O_A$ observers, but I have not done the math to check. However, this "space" also has weird properties such as the speed of light being anisotropic (it is slower in the "co-rotating" direction than the "counter-rotating" direction), and I believe at least one of those speeds of light will be significantly greater than $c$.

I agree that this very similar to the rotating frame, but here you have no proper acceleration. So locally you have perfect symmetry between members of $O_A$ and $O_B$.

It's only when $O_B$ starts counting, when they realize that a moving tape-loop $T_A$, has 1/2 the number of segments, each only 1/2 the length, compared to the resting tape-loop $T_B$, laying side by side to $T_A$.

Weird indeed.

 

[StandardsGuy]

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

How would you know that you are at rest when you make this measurement, or does it matter?

 

[PeterDonis]

How would you know that you are at rest when you make this measurement

By the absence of a Sagnac effect: light pulses that you send in opposite directions around the universe come back to you at the same instant.