Rotating Disk Physics: SR Reconciles Fast-moving Points?

[PeterDonis]

The rulers are at constant positions and have constant lengths.

But the spacelike curves realized by the rulers are *not* curves in the quotient manifold; they are curves in spacetime. More precisely, if you look at the spacelike curve that represents all the points on a given ruler at a given instant of that ruler's proper time, that curve is a curve in spacetime, *not* the quotient manifold. So I think Fredrik's uneasiness with calling the quotient manifold "space" is well founded.

 

[yuiop]

No one has yet explicitly mentioned that it is impossible to synchronise clocks all the way around the perimeter of a spinning disc using the Einstein convention, so I thought I would . (If it has already been mentioned, then I missed it and my apologies in advance). If we assume the speed of light is isotropic and use a suitable set of mirrors all the way around the perimeter together with a single clock on the perimeter to measure the distance around the disc then the distance depends on which direction we send the timing signal. If we take the average distance of the clockwise and anticlockwise measurements (effectively a radar distance measurement) we obtain the ruler distance obtained by laying rulers end to end all the way around the perimeter. It soon becomes clear, using a single clock on the perimeter, that the one way speed of light in a rotating reference frame is not c and is direction dependent.

 

[DrGreg]

But the spacelike curves realized by the rulers are *not* curves in the quotient manifold; they are curves in spacetime. More precisely, if you look at the spacelike curve that represents all the points on a given ruler at a given instant of that ruler's proper time, that curve is a curve in spacetime, *not* the quotient manifold. So I think Fredrik's uneasiness with calling the quotient manifold "space" is well founded.

But to measure the length of a stationary object, you chop it up into "infinitesimal" bits, measure each bit and add all the bits together (to put it in crude terms -- mathematically you perform an integral). As the object is stationary, each bit has a constant length over time, so there's no need to measure all the bits simultaneously (whatever your definition of simultaneity). So there's no need to stick the bits together in 4D spacetime. You just take each bit's proper length and add the proper lengths together.

 

[DrGreg]

I haven't seen any published papers take this particular approach, but it makes sense to me. Furthermore, when you take this approach that rulers measure distances between worldlines and not points, it becomes clear that the circumference is measured by adding together the distances between successive wordlines until one returns to the beginning worldline.

You still won't have a closed curve for the circumference, but you'll have a well-defined notion of its length.

See what I said last year...

This topic is covered in more detail in Rindler W (2006, 2nd ed), Relativity: Special, General, and Cosmological, Oxford University Press, ISBN 978-0-19-856732-5 Section 9.7 p.198. Earlier in that chapter, Rindler proves a more general result to show an arbitrary stationary spacetime's metric can be expressed in the form<br /> ds^2 = e^{2\Phi/c^2}\left( c\,dt - \frac{w_i}{c^2}dx^i\right)^2 - k_{ij}\,dx^i\,dx^j<br />where \Phi,\,w_i,\,k_{ij} are all independent of t. Here, k_{ij}\,dx^i\,dx^jis the time-independent metric of 3D space, but c\,dt - w_i\,dx^i/c^2 isn't the differential of any coordinate, except in the static case when \textbf{w} = \textbf{0}.

 

[PeterDonis]

As the object is stationary, each bit has a constant length over time, so there's no need to measure all the bits simultaneously (whatever your definition of simultaneity). So there's no need to stick the bits together in 4D spacetime.

So basically, the quotient manifold isn't a "spacelike slice" taken directly out of the spacetime, but it does represent a "spatial geometry" indirectly derived from ruler measurements via this procedure.

 

[TrickyDicky]

So as long as we keep the time-independence (stationary) of the rotating disk scenario we get a non-euclidean spatial geometry as HallsofIvy was saying almost a year ago. What a strange detour to get to the same conclusion.

 

[A.T.]

The rulers are at constant positions and have constant lengths. They are measuring a time independent metric. How much more "spatial" can it get?

But the spacelike curves realized by the rulers are *not* curves in the quotient manifold; they are curves in spacetime.

Who cares what they are in space-time, or in space-temperature or in space-pressure or whatever 4D-space one might postulate? They are rulers at rest. They measure spatial distances.

So I think Fredrik's uneasiness with calling the quotient manifold "space" is well founded

This uneasiness comes from defining space via time based on some abstract geometrical interpretation of physics, and completely ignoring the physical reality that would be observed on the rotating disc. To me, physical quantities are defined by stating how you measure them:

- time is what clocks measure
- space is what rulers measure

Every observer uses clocks and rulers at rest relative to him. Any measurement that involves only rulers at rest is trivially measuring only the spatial geometry.

These definitions are the basis for Relativity and far more fundamental than hypersurfaces in 4D spaces which were introduced later on. If some 4D-interpretation conflicts with these fundamental definitions in some cases, one should change the 4D-interpretation or limit it's applicability.

 

[DrGreg]

To put it yet another way, a "point in space" isn't really an event on a spacelike surface of simultaneity, it is actually a worldline in spacetime. So the manifold representing "space" does not consist of events in a 3D sub-manifold of 4D spacetime, but consists of worldlines (each "point" in the manifold is a worldline).

The quotient-space metric tells you the distance between two neighbouring worldlines (which is constant over time). You use a local definition of simultaneity (4D orthogonality, compatible with comoving inertial observer using Einstein sync) to determine this distance, but there's no requirement to extend that local simultaneity to global simultaneity, which you can't for non-static coordinates like these.

 

[PeterDonis]

They are rulers at rest. They measure spatial distances...


- time is what clocks measure
- space is what rulers measure

a "point in space" isn't really an event on a spacelike surface of simultaneity, it is actually a worldline in spacetime...

The quotient-space metric tells you the distance between two neighbouring worldlines (which is constant over time).

I see what you both are saying here, but it seems to me that this viewpoint only works for "stationary" situations. If points in space are actually worldlines, then the "ruler" is now a bundle of worldlines, not a 3-D object. So the notion of a "ruler at rest" only applies if the distance between worldlines remains constant. In general, with a real ruler, this condition won't hold, although there are certainly a wide range of scenarios where it is a good approximation.

The notion of "spatial distance" as "the integral of the metric along a given spacelike curve", however, applies in any situation. And in fact, the "distance between worldlines" approach requires it under the hood, so to speak, because in order to define the distance between two worldlines, you have to specify pairs of points on the worldlines that are "matched up". In other words, you have to adopt an implicit criterion of "simultaneity" between the worldlines. This can be local, by only directly pairing up points on adjacent worldlines, as DrGreg says, but when you add up all the little local pieces, in order to calculate the distance between pairs of worldlines that are not adjacent, what you are actually doing is integrating the metric along a spacelike curve, the curve that results when you link together all the little segments whose length you calculate. The curve is not a straight line because the local definitions of simultaneity don't match up, so the little segments are not "lined up" with each other, but it's still a perfectly good spacelike curve.

So perhaps the real problem is that all the different spacelike curves over which I am measuring distance between different pairs of worldlines don't form a single spacelike hypersurface in the scenario we've been discussing. That means we can no longer conflate two different ways of defining "space"--the quotient manifold defined by identifying "points in space" with worldlines, and the single spacelike hypersurface in which various segments define the "distance" between points--because they no longer lead to the same answer. So we have to pick one or the other. DrGreg and A.T. are picking the first option, while Fredrik is picking the second. The first option has the advantage that it gives you a single manifold that captures all "ruler distances", but it only works for "stationary" situations. The second option is completely general, but you may end up with a spacelike hypersurface that doesn't match up with local "ruler distances".

 

[DrGreg]

I agree that the "quotient space" method works only for a collection of mutually "stationary" observers (and never claimed otherwise) but Fredrik's method, as I understand it, seems rather arbitrary in that (as far as I can tell) you just choose some arbitrary spacelike surface which bears no relationship to the observers in question. So you end up with a choice of lots of different "spaces" with no criterion for choosing just one to represent the "space" of the observers. (In the "stationary" but "non-static" situation we are discussing, it's not possible to find a "space" simultaneously orthogonal to all of the observers' worldlines.) Or am I missing something?


(By the way, I'll be here for another hour or two today but then I'll be offline for 3 days.)

 

[PeterDonis]

Fredrik's method, as I understand it, seems rather arbitrary in that (as far as I can tell) you just choose some arbitrary spacelike surface which bears no relationship to the observers in question.

In this particular scenario, Fredrik's spacelike surface is simply the surface of simultaneity for an observer who is not rotating with the disk but who is at rest relative to the center of the disk. So the choice is certainly not arbitrary in this case; it's picked out by an obvious property of the object. The same sort of choice should be possible for any "stationary" but not static object.

(In the "stationary" but "non-static" situation we are discussing, it's not possible to find a "space" simultaneously orthogonal to all of the observers' worldlines.)

Yes, but that's true regardless of the method used to define "space", so it doesn't help us to choose which method to use in stationary but non-static cases. Basically, that definition of "space" *only* applies to static objects.

 

[graviton1]

I have a similar thought, suppose one have a rot expanded to inf. into to universe and for some reason the rod remains rigid and nothing obstructs it, on Earth we move the rod 1 degree, what would happen at the tip? answer exist...but isn't it a lovely day dream experiment.

 

[A.T.]

Fredrik's spacelike surface is simply the surface of simultaneity for an observer who is not rotating with the disk...

If that is the case, then it simply misses the point. The question was about the spatial geometry in the rotating reference frame.

Basically, that definition of "space" *only* applies to static objects.

That is no problem, because we don't want to measure moving objects. We want to measure the spatial geometry in a certain frame, which is given by the distances between coordinates as measured in that frame. And those coordinates are static per definition.

 

[PeterDonis]

If that is the case, then it simply misses the point. The question was about the spatial geometry in the rotating reference frame.

But there is no single spacelike hypersurface *within the spacetime* that is a "surface of simultaneity" for the "rotating reference frame" as a whole. Each point on the rotating disk has a different surface of simultaneity. If you insist on picking out a single such surface *within the spacetime*, as Fredrik does, you have to pick a single point of the disk to serve as the "reference", and the center of the disk is the obvious choice.

That is no problem, because we don't want to measure moving objects. We want to measure the spatial geometry in a certain frame, which is given by the distances between coordinates as measured in that frame. And those coordinates are static per definition.

You're misinterpreting what I mean by "static". A static object is an object for which all points share a single set of surfaces of simultaneity; static coordinates would be coordinates that match up with that single set of surfaces (so those surfaces are "surfaces of constant time"). The rotating disk is not static in this sense, nor are the coordinates attached to it, even though an observer at a particular point on the rotating disk can view himself as "at rest".

 

[A.T.]

But there is no single spacelike hypersurface *within the spacetime* that is a "surface of simultaneity" for the "rotating reference frame" as a whole.

Well, then the concept of "surface of simultaneity" is useless to define "space" in a rotating reference frame.

What about the Schwarzschild space time? It there a single "surface of simultaneity" for the entire region around a massive sphere? Or is the "spatial geometry", given by the Schwarzschild-metic as Flamm's-paraboloid, simply what rulers placed at rest around that region would measure? Why not use the same definition of "spatial geometry" for the rotating frame, if "surfaces of simultaneity" fail there?

The rotating disk is not static in this sense, nor are the coordinates attached to it, even though an observer at a particular point on the rotating disk can view himself as "at rest".

Not only himself. He can view the entire disc as at rest. But I still don't know what you mean by "Basically, that definition of "space" *only* applies to static objects.". The definition defines spatial geometry using objects (rulers) at rest. But it applies in any frame.

 

[PeterDonis]

Well, then the concept of "surface of simultaneity" is useless to define "space" in a rotating reference frame.

I don't necessarily disagree (although it seems like Fredrik would). I'm only pointing out that by defining "space" in a rotating frame as you are doing it, you are in fact discarding the idea of "space as a surface of simultaneity".

I still don't know what you mean by "Basically, that definition of "space" *only* applies to static objects.".

I meant that only "static" objects (as I defined that term) have a single "surface of simultaneity" that is shared by all points within the object (i.e., that is orthogonal to the worldlines of all points within the object), so only such objects can have a definition of "space" as such a surface of simultaneity. I wasn't talking about the quotient manifold definition of "space" (your definition) there.

The definition defines spatial geometry using objects (rulers) at rest. But it applies in any frame.

I'm not sure what you mean by "applies in any frame". The "space" defined by the quotient manifold has nothing to do with reference frames; it defines a "spatial geometry" by considering the worldline of each point of the rotating disk as a single point and defining the "distance" between points as DrGreg described, which corresponds to using "rulers at rest" to define the distance between *adjacent* points, and then defining the distance between non-adjacent points by summing up small distance elements between pairs of adjacent points between them. But there is nothing corresponding to "time" anywhere in this, so it doesn't define a "reference frame" in the usual sense of that term.

 

[A.T.]

I'm only pointing out that by defining "space" in a rotating frame as you are doing it, you are in fact discarding the idea of "space as a surface of simultaneity".

I would like to hear your comments on Schwarzschild space time regarding this. Do we use a surface of simultaneity to define "spatial geometry" there?

I'm not sure what you mean by "applies in any frame".

That for any frame you can use rulers at rest in that frame to do the following :

using "rulers at rest" to define the distance between *adjacent* points, and then defining the distance between non-adjacent points by summing up small distance elements between pairs of adjacent points between them.

it doesn't define a "reference frame"

Of course not. It defines the "spatial geometry" in the frame where those rulers are rest.

 

[PeterDonis]

I would like to hear your comments on Schwarzschild space time regarding this. Do we use a surface of simultaneity to define "spatial geometry" there?

Schwarzschild spacetime isn't really the best example because it's static--the surfaces of constant Schwarzschild time are orthogonal to the worldlines of "hovering" observers. So we can, in fact, use those surfaces of simultaneity to define a "spatial geometry", which is, as you note, the geometry described by the Flamm paraboloid.

There is one interesting wrinkle here, though: we could also use at least one other set of surfaces of simultaneity, those of Painleve observers, who free-fall into the hole from "rest at infinity". These surfaces of simultaneity define a spatial geometry that is flat--Euclidean 3-space.

Kerr spacetime would be a better example since the surfaces of constant coordinate time are *not* orthogonal to the worldlines of static "hovering" observers--observers who don't change their spatial coordinates with time. In Kerr spacetime the surfaces of constant coordinate time are orthogonal to the worldlines of "zero angular momentum observers", or ZAMOs, who are rotating around the hole with an angular velocity that depends on their height above the horizon. So there is the same sort of disconnect as there is in the rotating disk example: the set of spacelike surfaces that matches the time translation symmetry does *not* allow us to define a "spatial geometry" because of the non-orthogonality. I have not really seen a discussion of how "spatial geometry" is defined in Kerr spacetime, but I think it would indeed be relevant to the situation we're discussing here.

Of course not. It defines the "spatial geometry" in the frame where those rulers are rest.

Can you be more specific about what you mean by "the frame where the rulers are at rest"? Do you mean Born coordinates?

http://en.wikipedia.org/wiki/Born_coordinates

I ask because your use of the term "frame", as here...

That for any frame you can use rulers at rest in that frame to do the following:

...seems to indicate that by "frame" you mean "inertial frame", or at least "momentarily comoving inertial frame", or something like that. But there is no single "frame" in this sense in which more than one ruler on the disk is at rest at any instant. So you can't use a single "frame in which the rulers are at rest" in this sense to calculate distances between non-adjacent points.

If, OTOH, by "frame" you mean Born coordinates, then, as noted on the Wiki page, the "time" coordinate in Born coordinates is Fredrik's--the "surfaces of constant time" are the surfaces of simultaneity of an observer moving with the center of the disk but *not* rotating with the disk. The Wiki page discusses the problems (some of which we've already alluded to here) in using these surfaces to define a "spatial geometry" for the rotating disk.

So neither of the above meanings of "frame" seems satisfactory for defining a single "frame in which the rulers are at rest" that we can use to define a spatial geometry. Do you have another meaning in mind?

 

[yuiop]

I don't necessarily disagree (although it seems like Fredrik would). I'm only pointing out that by defining "space" in a rotating frame as you are doing it, you are in fact discarding the idea of "space as a surface of simultaneity".

You seem to be forgetting that simultaneity is just a convention (ie we normally use the Einstein clock synchronisation convention) and seem to treating simultaneity as an absolute quantity. Einstein clock synchronisation assumes the one way speed of light is isotropic but there is in fact no way even in principle to measure the one way speed of light so that is just an assumption. In the non inertial rotating reference frame of the observers on the disc, we can use a light signal at the centre of the disc to synchronise all the clocks on the perimeter of the disc, and then we obtain a surface of simultaneity that is not a spiral shape, but a regular plane in which the ends of the rulers around the perimeter are all simultaneous. This way the space measured with rulers agrees with the idea of "space as a surface of simultaneity" and so there is no conflict or discarding of that notion.

When A.T, says "It defines the "spatial geometry" in the frame where those rulers are rest." we can define "at rest" as the end points of the rulers have a coordinate position that does not change over time in a given reference frame, so there is no problem with that notion for rulers in a rotating reference frame.

The above methods require that the notion of isotropic speed of light be discarded, but since there is no way to measure the one way speed of light, there is no experimental support for that notion.

 

[PeterDonis]

In the non inertial rotating reference frame of the observers on the disc, we can use a light signal at the centre of the disc to synchronise all the clocks on the perimeter of the disc, and then we obtain a surface of simultaneity that is not a spiral shape, but a regular plane in which the ends of the rulers around the perimeter are all simultaneous. This way the space measured with rulers agrees with the idea of "space as a surface of simultaneity" and so there is no conflict or discarding of that notion.

And if you do this, the "spatial geometry" you come up with will either be flat (if you adopt Fredrik's approach and define "spatial geometry" as simply the restriction of the metric of the overall spacetime to a particular surface of simultaneity defined as above) or won't be well-defined (if you try to do it along the lines described in the Wiki page). That's why A.T. and DrGreg objected to using this method of defining "simultaneity".

When A.T, says "It defines the "spatial geometry" in the frame where those rulers are rest." we can define "at rest" as the end points of the rulers have a coordinate position that does not change over time in a given reference frame, so there is no problem with that notion for rulers in a rotating reference frame.

If you use Born coordinates as your "reference frame", yes, it meets the requirement that the rulers are "at rest" in that frame. But again, you run into problems defining "spatial geometry" using this frame, which is why A.T. and DrGreg have objected to doing it this way. That's why I asked A.T. what his definition of "frame in which the rulers at rest" was; if he is actually using the definition you just gave, then the "spatial geometry" in that frame is not what he has been calling the "spatial geometry of the rotating disk"; the latter is a different object altogether.

 

[A.T.]

Can you be more specific about what you mean by "the frame where the rulers are at rest"?

The rotating frame is rotating around some axis at some angular velocity wrt to an inertial frame. The rulers are rotating around the same axis at the same angular velocity wrt to that inertial frame.

 

[PeterDonis]

The rotating frame is rotating around some axis at some angular velocity wrt to an inertial frame. The rulers are rotating around the same axis at the same angular velocity wrt to that inertial frame.

Did you read the Wiki page on Born coordinates that I linked to, and my comments in post #48? I'm not sure the "rotating frame" you have just defined has quite the properties you think it has.

 

[PeterDonis]

This way the space measured with rulers agrees with the idea of "space as a surface of simultaneity" and so there is no conflict or discarding of that notion.

Actually, I should clarify my comment on this: the notion of "surface of simultaneity" that you have defined does *not* agree with "the space measured with rulers", which is the notion of "space" that A.T. and DrGreg have described. That's part of the problem that we have been discussing.

 

[Fredrik]

This thread has been moving along fast. I haven't had time to reply. I'll post few replies now. I'll start with pervect and the hypersurface of simultaneity spirals.

If you draw the set of points that are Einstein-synchronized on a rotating cylinder, you get a non-closed spiral.

See for instance http://en.wikipedia.org/wiki/File:Langevin_Frame_Cyl_Desynchronization.png

Talking about the spatial geometry of such a non-closed surface (which I gather can be thought of also as a quotient manifold) is definitely odd and tends to cause confusion. Specifically, it's generally assumed the circumference of a disk is a closed curve, and we can clearly see from the diagram that this is not the case if one uses Einstein clock synchronization.

I'm not so sure that this spiral surface has the same metric as the quotient manifold. I haven't tried to work out the details, but I would expect them to be different. If I'm right, then the geometry of this spiral surface is not what people are referring to as "the spatial geometry".

Note by the way that while the spiral curves (like the blue line in the picture you linked to) can be thought of as simultaneity lines of component parts of the disk, other curves in the surface aren't simultaneity lines of the same observer. So the surface isn't really a surface of simultaneity.

 

[Fredrik]

Change? I for my part have always understood "spatial geometry" as "what rulers at rest measure".

And I have always understood "geometry" as a mathematical term. In this case, I'd say that it's a synonym of "metric". So the term "spatial geometry" should refer to the metric of a manifold that we can call "space". In pre-relativistic physics, "space" was a slice of spacetime. It's obvious that SR and GR needs to generalize the term, but it's very far from obvious that we're going to need a definition that's so general that "space" doesn't even need to be a subset of spacetime.

If there's a good argument for why the word "space" should refer to a congruence of timelike curves rather than a spacelike 3-dimensional hypersurface of spacetime, then I would have to say that "spacetime" was inappropriately named from the beginning.

the concept of "surface of simultaneity" is useless to define "space" in a rotating reference frame.

If the term "rotating reference frame" refers to the orthonormal frame field* associated with the congruence of curves that represent the motion of the disk, then this is true. But the term "reference frame" often refers to an orthonormal frame field associated with a single observer, and is sometimes used informally as a synonym for "coordinate system". If it refers to an orthonormal frame field associated with (only) the point at the center, or a coordinate system that's rotating with the disk but has its spatial origin at the center, then there certainly is a hypersurface of simultaneity associated with it (and it's flat).

*) A frame field can be defined as a function that takes each member p of some subset of spacetime to a basis for the tangent space at p.

 

[Fredrik]

To me, physical quantities are defined by stating how you measure them:

- time is what clocks measure
- space is what rulers measure

This is how most physicists think about these things. I assume that you would also use the term "time" for the mathematical thing that corresponds to time, because that's what physicists usually do. I dislike this approach, because there is certainly a better way to specify how to interpret the mathematics as predictions about results of experiments than to simply use the same term for two very different things. I think this approach makes it unnecessarily hard for students to understand the difference between physics and mathematics.

I prefer to define terms (only) mathematically, and then explicitly state the rules that tell us how to interpret the mathematics as predictions about results of experiments. These correspondence rules are what turn a piece of mathematics into a theory of physics. No theory of physics is fully defined without a set of correspondence rules.

Note that the same term can have different definitions in different theories of physics. For example, in classical electrodynamics, "light" is an electromagnetic wave. In QED, it's a state that involves photons.

To deal with "what clocks measure", I would first define "proper time" as a coordinate-independent property of a curve given by an integral that I'm not going to write down here, and then I would state the correspondence rule that says that the difference between the numbers displayed by a clock at two events A and B, is the proper time of the curve that represents its motion from A to B. This correspondence rule is an essential part of the definitions of both SR and GR.

I like this approach better because it makes it easy to understand that a) mathematics doesn't say anything about reality, b) a theory of physics consists of a purely mathematical part and a set of correspondence rules (that do say something about reality), and c) how the specific theory we're talking about is defined.

 

[yuiop]

- time is what clocks measure
- space is what rulers measure

Initially I liked this definition, mainly because of its simplicity, practicality and the apparent dismissal of the requirement to consider definitions of simultaneity. However it is obvious from the posts in this thread that even with this approach, that simultaneity issues refuse to go away quietly. Another physical problem that is present here that is not present when purely inertial motion is considered, is the stress that the rulers or rods are under in a non inertial rotating reference frame. When we rotate a measuring rod from a tangential to a radial position in the rotating frame, we have to be sure that the rod is not distorted by purely Newtonian forces, resulting in a distortion of what we are trying to measure. One way to achieve this, is to check the radar length of the measuring rods in the two orientations. When we do this, the first thing we might notice is that radar length of a radial rod depends upon whether the clock used to measure the radar length of the rod is located at the end nearer the centre or the end nearer the disc perimeter. This can be minimised to a certain extent by using infinitesimal measuring rods.

I would now like to propose a method to measure the geometry of the disc that is indisputably independent of simultaneity issues. We use a single clock that is fixed to a point on the rim of the rotating disc to make all measurements. First we measure the circumference of the disc by sending a signal all the way around the disc back to the single clock and then all the way back again in the opposite direction, using suitably placed mirrors, to obtain the radar circumference of the disc. Doing this we find the circumference is is gamma times longer than the circumference measured in the inertial reference frame at rest with the centre of the disc. Next we measure the radar radius of the disc, by sending a signal from the rim to a mirror at the centre and back out to the clock on the rim again. This time we obtain that the result that the radius of the disc is gamma times shorter than the radius measured in the non rotating inertial reference frame. The end result is that in the rotating reference frame, the ratio of the circumference to the radius is 2*pi*gamma^2.

This method uses the same clock that is at rest in the rotating reference frame to measure both radar circumference and radius, so that the two measurements can be compared in a consistent way without any simultaneity issues or concerns about physical distortions of measuring rods due to "centrifugal forces". Interestingly, this alternative analysis obtains a different result from the usual interpretations, that the radius measured in the rotating frame is the same as the radius measured in the non rotating reference frame at rest with the spin axis of the disc.

 

[PeterDonis]

First we measure the circumference of the disc by sending a signal all the way around the disc back to the single clock and then all the way back again in the opposite direction, using suitably placed mirrors, to obtain the radar circumference of the disc. Doing this we find the circumference is is gamma times longer than the circumference measured in the inertial reference frame at rest with the centre of the disc.

You have to assume a speed of light to transform this "round-trip" travel time into a distance, of course. What speed of light are you assuming?

Next we measure the radar radius of the disc, by sending a signal from the rim to a mirror at the centre and back out to the clock on the rim again.

Again, you need to use an assumed speed of light to transform this into a distance measurement. Do you use the same speed as you used for the circumference measurement?

Also, because the clock on the rim is moving, relative to an inertial observer's clock, the travel time it measures will be *smaller* than the travel time an inertial observer would measure. This will result in a *smaller* result for the radius of the disk for the moving observer, *not* the same result as the inertial observer.

 

[Austin0]

You have to assume a speed of light to transform this "round-trip" travel time into a distance, of course. What speed of light are you assuming?



Again, you need to use an assumed speed of light to transform this into a distance measurement. Do you use the same speed as you used for the circumference measurement?

Also, because the clock on the rim is moving, relative to an inertial observer's clock, the travel time it measures will be *smaller* than the travel time an inertial observer would measure. This will result in a *smaller* result for the radius of the disk for the moving observer, *not* the same result as the inertial observer.

Is there some reason not to use c??

Is there some reason to assume a different speed for the radial measurement??

As I read it, yuiop said exactly what you are stating here. The radial evaluation taken from the rim would be smaller than the measurement by an inertial observer.

 

[PeterDonis]

Is there some reason not to use c??

I don't think so, but be aware that if you measured the two travel times separately for the beams going around the circumference (with the rotation, then back opposite to the rotation), the individual times would be *different*, because of the Sagnac effect:

http://en.wikipedia.org/wiki/Sagnac_effect

Some people interpret this as the speed of light being different for light going with the rotation and light going opposite to the rotation. (I don't agree with this interpretation, btw.)

There's also the more general point that the coordinate speed of light in non-inertial reference frames may not be c when evaluated over significant distances. See below.

Is there some reason to assume a different speed for the radial measurement??

Again, in this case I don't think so, but in general, in non-inertial reference frames, the coordinate speed of light may not be c, and may not be isotropic, when evaluated over significant distances. The *local* speed of light (evaluated using proper distances and proper times in a local inertial frame) is always c, but that doesn't automatically allow you to assume that it will always be c for a non-local measurement in a non-inertial frame, such as those yuiop has proposed.

As I read it, yuiop said exactly what you are stating here. The radial evaluation taken from the rim would be smaller than the measurement by an inertial observer.

Hm, you're right. I must have misread his post somehow.