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.