Lorentz Contraction Circular Motion

[Fredrik]

I define a metric in terms of distances which are measured using rulers. See my quote above again.

Still seems to me that you're trying to define a mathematical object in terms of physical objects. (The book you're quoting isn't).

I mean what is described here in more detail (last paragraph of page 91):
http://books.google.com/books?id=Iy...=Let us now see how the non-Euclidean&f=false

I find it pretty strange that they're using the term "spatial geometry" when they're not even talking about "space". They've made it obvious that "space" is flat, and that the hypersurface with negative curvature that they're considering doesn't consist of simultaneous events, and therefore doesn't deserve to be called "space". Hmm...now I see that they're doing something funny on page 90. They're not considering space at all. They're considering an entirely different concept which they call "rest space". If I understand them correctly, it's the metric of "rest space" that says the circumference is 2πγr (because they're really calculating the length of a spiral), while the metric of "space" says the circumference is 2πr (because now we're dealing with the length of a circle).

 

[pervect]

How is that preventing you from measuring spatial geometry with rulers?

If you want to talk about the spatial geometry of a set of points, you do need to specify the set of points that you're talking about, if you expect to get an answer that everyone agrees with.

For instance, you can slice up the space-time around a massive body with Schwarzschild coordinates in which the spatial slices are not flat - or in Painleve coordinates, in which the spatial slices are flat. So in general, it doesn't make sense to talk about a spatial geometry unless you also specify the manner in which you create the time slices.

Now, you specify "with rulers", so perhaps you aren't taking the same basic approach that I am of considering a geometry as something that applies to a set of points. *If* you regard a ruler as measuring the distance between worldlines, I believe you can get a well -defined answer for the circumference of a rotating disk. (You have to make some basic assumptions that the distance between worldlines is the shortest worldline connecting them, and that this distance is static because the geometry is static, and that you take the limit for closely space worldlines). Note that what happens is that you start at one point on one worldline to measure the circumference, and trace out some path through space-time. This path ends on the same worldline it started from, by definition, but it does NOT end at the same point on that worldline - it ends on the same worldline at a different time than when it started out.

This defines a circumference, but it's not clear that this approach actually defines a "geometry". The "circumference" defined by this means is not a closed curve!

I'm not aware of anyone using this particular approach in the literature - though there may be someone, I'm not familiar with all of the literature on the topic by any means, it's quite large.

You can't synchronize clocks around a massive object either, but you can still say that the spatial geometry is non-Euclidean there. What is different in a rotating frame?

You can syncyhronzize clocks around an equipotential surface of a massive non rotating object, I'm not sure why you think you can't?

 

[heldervelez]

...
The length of moving objects measured with ruler at rest is less than measured with a co-moving ruler.
...
.

I see your point. My error, because I did not saw the 'cross-polinization'

But each observer, with each respective ruler, will have to measure the same value.
As we can not find a single object in the whole Universe that has an extension of 2*Pi, I will measure in a physical way:
Suppose I've a ruler made of 1 atoms of Hidrogen at rest and the circular perimeter by suposition, measures 100 Hidrogen atoms, then when in motion I'will have to measure 100 atoms (contracted) with my ruler (also 1 H contracted).
But the the object, does not have two distinct realities, but only one and it will be perceived with distinct shapes.

 

[A.T.]

I think the issue here is that although you can use your definition in terms of laying down rulers, it's not quite as absolute and natural as it might seem at first.

It is 'absolute' in the sense that it is frame invariant: Everyone will agree what the rulers will measure.

When the cars meet up on the far side of the disk, their clocks will be out of sync due to the d\theta' dt term in the metric, even though they've traveled an equal distance at an equal speed. You could just accept this, but it's uncomfortable, because it leaves you wondering where the funny asymmetry comes from.

Yes, "funny things" happen with clocks in non-inertial frames, but the good news is that you don't have to use clocks to measure spatial geometry.

it's the metric of "rest space" that says the circumference is 2πγr (because they're really calculating the length of a spiral),

They are calculating what http://img688.imageshack.us/img688/4590/circleruler.png" would measure, when placed at rest in the rotating frame. And this ruler is a circle, not a spiral.

You can't synchronize clocks around a massive object either, but you can still say that the spatial geometry is non-Euclidean there. What is different in a rotating frame?

You can syncyhronzize clocks around an equipotential surface of a massive non rotating object, I'm not sure why you think you can't?

Now you added a restriction, which also applies to clocks in a rotating frame: You can synchronize clocks which are equidistant to the rotation axis in a rotating frame.

So my above question still stands: Why is it correct to call the spatial geometry around a massive object non-Euclidean, but "controversial" to say the same about the spatial geometry in a rotating frame? Clock synchronization issues arise in both cases, so what is the difference?

 

[Fredrik]

They are calculating what http://img688.imageshack.us/img688/4590/circleruler.png" would measure, when placed at rest in the rotating frame. And this ruler is a circle, not a spiral.

Yes, but they're doing it by calculating the length of a spiral in spacetime.

So my above question still stands: Why is it correct to call the spatial geometry around a massive object non-Euclidean, but "controversial" to say the same about the spatial geometry in a rotating frame? Clock synchronization issues arise in both cases, so what is the difference?

The controversial part is to use the term "spatial geometry" about the geometry of a surface that isn't "space".

 

[A.T.]

They are calculating what http://img688.imageshack.us/img688/4590/circleruler.png" would measure, when placed at rest in the rotating frame. And this ruler is a circle, not a spiral.

Yes, but they're doing it by calculating the length of a spiral in spacetime.

That is what they do in chapter 5.1. While in chapter 5.2 they arrive at the non-Euclidean spatial geometry just trough Lorentz contraction. You don't need to consider spirals in space-time to predict what the ruler will measure. In space the ruler is just a circle and it measures spatial distances, which determine the spatial geometry in the ruler's rest frame.

The controversial part is to use the term "spatial geometry" about the geometry of a surface that isn't "space".

So in your opinion, rulers at rest in the rotating frame don't measure "spatial geometry" in that frame ? Fine, we can use the term "proper spatial geometry" for what these co-rotating rulers measure, in analogy to "proper length" which is measured by a co-moving ruler.

For me this "proper spatial geometry" is the physically relevant spatial geometry:

If I want to build a huge structure near a massive object, I have to the take the non-Euclidean spatial geometry around the mass into account, when calculating the lengths of the structure's segements.

Analogously:

If I want to build a fast rotating structure, I have to the take the non-Euclidean "proper spatial geometry" in the rest frame of the structure into account, when calculating the lengths of the structure's segements.

 

[bcrowell]

This paper http://www.phys.uu.nl/igg/dieks/rotation.pdf gives a very nice treatment that I think is both more transparent than Gron's and more directly related to the question that Fredrik and A.T. are debating. There are problems with applying ruler measurements directly to this case, because the rulers are subject to Coriolis and centrifugal forces. You might say that this is no big deal, because we can just use nice, rigid rulers. However, there is a relativistic limit to how rigid the rulers can be. (If they were perfectly rigid, then vibrations would propagate along them at v>c.) If you bring your rulers out to r=c/\omega, then their velocity relative to the axis equals the speed of light, which is impossible. Physically, they must be torn apart by centrifugal forces before they get there, even if they are as rigid as relativity allows any material object to be. Even supposing maximum-rigidity rulers, you are going to get dynamical effects at smaller values of r, and therefore you can't use rulers to measure the spatial geometry quite as directly as A.T. is claiming, or as Einstein believed in 1912. The way to get around all these issues is to use radar measurements to establish the spatial geometry, and that requires clock synchronization.

Another way of getting at this is suggested by Wald, near p. 119. When we want to split the metric into separate spatial and temporal parts, with the form (\ldots)dt^2-(\ldots)dx^\mu dx^\nu, that means we're claiming the spacetime can be put in what's technically known as static (as opposed to stationary) form. Static is more strict than stationary. All static metrics have to have time-reversal symmetry. An example that Wald gives is that a rotating fluid can't have a static metric applied to it. If you want to put material objects like rulers on the disk, they're analogous to the rotating fluid. They have their own stress-energy tensors, etc. You clearly don't have time-reversal symmetry, and therefore you can't measure a static metric using material objects. Again, this can be sidestepped by not using material objects to measure the geometry, but then you have to do clock synchronization.

 

[A.T.]

This paper http://www.phys.uu.nl/igg/dieks/rotation.pdf gives a very nice treatment that I think is both more transparent than Gron's and more directly related to the question that Fredrik and A.T. are debating.

That's the reference from post #2. It also describes the spatial geometry in the rotating frame as non-Euclidean (Chapter 6) :

The spatial geometry defined by the line element (5) is non-Euclidean, with a
negative r-dependent curvature

And if you do a search on "non-Euclidean spatial geometry rotating frame" you find a lot of references using this interpretation, way back to Einstein:
http://books.google.de/books?id=DH7...idean spatial geometry rotating frame&f=false

There are problems with applying ruler measurements directly to this case, because the rulers are subject to Coriolis and centrifugal forces.

The rulers are supposed to be Born rigid as described here: http://books.google.com/books?id=Iy...q=Let us now see how the non-Euclidean&f=true
Which I guess means they are not subject to inertial forces?

BTW: Coriolis force for rulers at rest in the rotating frame?

The way to get around all these issues is to use radar measurements to establish the spatial geometry, and that requires clock synchronization.

Can you synchronize rotating clocks along the same r-coordiante and then radar-measure the circumference at r?

 

[bcrowell]

Re #38 by A.T. -- Ah, thanks for pointing out that you'd already posted a reference to the Dieks paper. I wish I'd paid more attention to you and read it earler :-) I agree with you that the spatial geometry is non-Euclidean -- were you under the impression that I disagreed on that point? Unfortunately the books.google.com links aren't helping me much; their software is blocking me from seeing the relevant parts of the Rizzi anthology and the Gron book, presumably because they want to make sure I don't see too much through the keyhole without paying money. (Well, the Rizzi anthology is only $359 on amazon; maybe we should all buy copies.) Because of that I'm not able to make much of this: "The rulers are supposed to be Born rigid as described here: [...] Which I guess means they are not subject to inertial forces?"

Re Born rigidity, the information I have available is in the Gron paper and the WP article, http://en.wikipedia.org/wiki/Born_rigidity . The thing to realize is that Born rigidity isn't a physically possible attribute of real objects. E.g., the Gron Am. J. Phys. paper (p. 872) says:

"By definition a Born rigid motion of a body leaves lengths unchanged, when measured in the body's proper frame. As made clear by Cavallieti and Spinelli, and by Newburgh, a Born rigid motion is not a material property of abody, but the result of a specific program of forces designed to set the body in motion without introducing stresses. The result of the analysis given above shows that a transition of the disk from rest to rotational motion, while it satisfies Born's definition of rigidity, is a kinematic impossibility. This is the kinematic resolution of Ehrenfest's paradox."

I think you may be under the impression that I'm taking sides with Fredrik in the debate you two have been having. Actually there are some points where I agree with you, and some points where I agree with Fredrik. I agree with you about the non-Euclidean spatial geometry, and that mathematical descriptions need operational definitions to tie them to physical reality.

BTW: Coriolis force for rulers at rest in the rotating frame?

I think Coriolis forces are at least potentially relevant here. If you want to form an operational definition of non-Euclidean geometry in this situation, using rulers, then you have to have some way of comparing radial and azimuthal distances. This requires rotating rulers, and then Coriolis forces will compress or expand the rulers, depending on whether you rotate them in the same direction as the disk's rotation or the opposite direction. If you rotate them slowly enough, you're probably okay, but this is an example of how you really can't get away with ignoring the dynamics of the rulers. The Coriolis force is also what prevents you from transporting a ruler past r=c/\omega along a radial line.

 

[Fredrik]

...and that mathematical descriptions need operational definitions to tie them to physical reality.

You're saying this as if it's something you expect that I'd disagree with. If there's anything in what I've said that suggests that I would, it was 100% unintentional. I was just objecting to the idea that mathematical objects can be defined by statements about physical objects. That's what A.T. seemed to be doing.

An operational definition is something else entirely. It's actually a poorly stated axiom of a theory of physics. For example, the statement "time is what you measure with a clock" is often described as an operational definition of "time", but it isn't really an attempt to define a term. It's an attempt to explain how something in the real world corresponds to something in the mathematical model. To really do that, we at least have to be precise about what mathematical quantity we have in mind. This is how I would say it: "A clock measures the proper time of the curve in spacetime that represents its motion". It's misleading to characterize this statement as a "definition". It's an axiom of a theory of physics.

The statement I used as an example is one of the axioms of special relativity. We clearly need a similar axiom about length measurements, but it's surprisingly hard to state such an axiom in a satisfying way. It's hard enough to write down an axiom that's valid for measuring devices doing inertial motion, and I have no idea what an axiom that's valid for measuring devices in an arbitrary state of motion would look like. I definitely haven't seen one.

That last part is the main reason why I don't like A.T.'s approach. He talks about this stuff as if it's trivial, and it certainly isn't. When we use a method that neither of us understands, the result is likely to be wrong. The standard axioms are however perfectly clear. The geometry of a set of simultaneous events (i.e. "space" at some time t) is Euclidean. I really don't see the point of defining a hypersurface that consists of a bunch of spirals in spacetime and call it "rest space", just so we can describe its geometry as "spatial geometry".

 

[A.T.]

I agree with you that the spatial geometry is non-Euclidean -- were you under the impression that I disagreed on that point?

No I didn't want to imply this. I just wanted to point out that it is something I read in several sources.

The thing to realize is that Born rigidity isn't a physically possible attribute of real objects.

The way I understand it: It is an idealized ruler. When measuring with a real ruler you would have to account for the elastic deformations, to calculate the result of the idealized ruler.

Gron Am. J. Phys. paper (p. 872) says:
"The result of the analysis given above shows that a transition of the disk from rest to rotational motion, while it satisfies Born's definition of rigidity, is a kinematic impossibility.
"

Two points on this:

1) Do we have to use a solid disk? I proposed a http://img688.imageshack.us/img688/4590/circleruler.png" , which can change it's proper circumference without introducing any tangential stresses. I think it could satisfy Born's rigidity (in the tangential direction) without a kinematic impossibility.

2) Do we have to care about the transition from rest to rotational motion? We could build the rotating ruler from small Born's rigid parts in the rotating state already.

This requires rotating rulers, and then Coriolis forces will compress or expand the rulers, depending on whether you rotate them in the same direction as the disk's rotation or the opposite direction.

Okay, that is an issue during transport of the rulers. But once they are at rest in the rotating frame the only problem I see is the centrifugal force. But who says that the centrifugal force has to be countered by the rigidness of the ruler? You could support the structure with small rocket engines facing outwards. The rigid parts from point (2) above don't even have to be fixed to each other, they just use their rockets to form a rotating circle. Then you measure the circumference by counting how many of them you needed.

Another idea: making r very large and omega small (while still keeping a relativistic tangential velocity), should make the centrifugal force sufficiently small.

But all this thinking about how to deal with inertial forces on the rulers seems a bit weird: Don't we have to assume massless rulers anyway? Because otherwise the ruler would also produce gravitation, that could counter the centrifugal force. But we don't want our measuring device to change the scenario and introduce non-Euclidean geometry by it's own mass.

 

[A.T.]

I really don't see the point of defining a hypersurface that consists of a bunch of spirals in spacetime and call it "rest space", just so we can describe its geometry as "spatial geometry".

If you don't like using the name "spatial geometry" for it, then it is just about semantics. I didn't make this name up.

But I understand the pragmatic reason why this is considered the "spatial geometry in the rotating frame" by many authors, way back to Einstein: The physical consequences of this "rest space geometry" are the same as those in other cases that involve non-Euclidean spatial geometry (e.g. due to a massive object).

 

[bcrowell]

A.T. -- The thrust of your #41 is that you're proposing a variety of ways of handling the dynamics of the rulers. I would make the following general comments:

This is likely to be extremely difficult and complicated. E.g., the WP article on the Ehrenfest paradox has the following: "1981: Grøn notices that Hooke's law is not consistent with Lorentz transformations and introduces a relativistic generalization." You're going to run into lots and lots of issues like this. It seems like your original motivation for the treatment using rulers was that it seemed conceptually simple, but it looks to me like it is actually much more conceptually complicated than the treatment using light rays.

I have no doubt that it is possible, by picking crafty approximations, to make the dynamical treatment of the rulers work, to some approximation, and that within this approximation you will get the non-Euclidean spatial geometry that we've all been convinced was right ever since Einstein first thought about the example in 1912.

I suspect that the ruler method can achieve either one or the other, but not both, of the following: (1) an exact result, or (2) a method that avoids clock synchronization. The reason is that material rulers can't be used at r>c/\omega, so it takes quite a leap of faith to imagine that they could be made to work perfectly at r=(.999)c/\omega. I suspect that if you wanted to use rulers at (.999)c/\omega, they'd be under so much strain that in order to correct for all the dynamical effects you'd need an explicit model of their behavior in terms of relativistic quantum mechanics. But it seems unlikely to me that a QED model of a ruler can be carried out without dealing with time as a variable, which would obviate the goal of avoiding clock synchronization.

Re Born rigidity, do you have access to the Gron Am. J. Phys. paper? The main point of the paper is that Born rigidity is a kinematical impossibility. E.g., when you say, "You could support the structure with small rocket engines facing outwards," this is exactly the kind of thing that Gron is proving is kinematically impossible, and it's kinematically impossible because of issues relating to clock synchronization.

 

[atyy]

Doesn't the ruler method does involve clock synchronization to define the radial direction, ie. the radial line is the line along which clocks can be synchronized?

 

[bcrowell]

Doesn't the ruler method does involve clock synchronization to define the radial direction, ie. the radial line is the line along which clocks can be synchronized?

You can synchronize clocks along a non-straight curve that connects the axis to an off-axis point. You just can't do a global synchronization without discontinuities.

I think the ruler method can be used to define a radial line as the shortest curve connecting the axis with an off-axis point. This does assume that you can locate the axis using nothing but static ruler measurements, but I think that is possible. The Ricci scalar curvature of the spatial metric (which I guess is probably some constant multiple of the Gaussian curvature?) is R=6/(r^2-2r^2+1), where \omega=1. So since you can determine R with static ruler measurements, I think you can locate the axis by looking for where R has a local minimum value of R=6.

 

[A.T.]

The thrust of your #41 is that you're proposing a variety of ways of handling the dynamics of the rulers.

Yes but I end with:

Don't we have to assume massless rulers anyway? Because otherwise the ruler would also produce gravitation. But we don't want our measuring device to change the scenario and introduce non-Euclidean geometry by it's own mass.

Doesn't this make all discussions about problems with inertial forces acting on the rulers kind of pointless?

The reason is that material rulers can't be used at r>c/\omega, so it takes quite a leap of faith to imagine that they could be made to work perfectly at r=(.999)c/\omega.

To determine that the spatial geometry is non-Eclidean it would suffice if they worked at r=(0.1)c/\omega.

Re Born rigidity, do you have access to the Gron Am. J. Phys. paper? The main point of the paper is that Born rigidity is a kinematical impossibility. E.g., when you say, "You could support the structure with small rocket engines facing outwards," this is exactly the kind of thing that Gron is proving is kinematically impossible, and it's kinematically impossible because of issues relating to clock synchronization.

I will have a look at that. What is the exact title? Or can you summarize his argument?

BTW, This chapter is by Gron as well:
http://books.google.de/books?id=DH7...idean spatial geometry rotating frame&f=false
It deals also with methods to synchronize clocks along a circumference in the rot. frame.

 

[atyy]

Won't massless rulers travel at the speed of light?

 

[atyy]

You can synchronize clocks along a non-straight curve that connects the axis to an off-axis point.

Interesting!

I think the ruler method can be used to define a radial line as the shortest curve connecting the axis with an off-axis point. This does assume that you can locate the axis using nothing but static ruler measurements, but I think that is possible. The Ricci scalar curvature of the spatial metric (which I guess is probably some constant multiple of the Gaussian curvature?) is R=6/(r^2-2r^2+1), where \omega=1. So since you can determine R with static ruler measurements, I think you can locate the axis by looking for where R has a local minimum value of R=6.

Operationally, how is a particular "off-axis point" identified?

 

[A.T.]

Won't massless rulers travel at the speed of light?

I'm just pointing out, that if placing rulers at rest in the rotating frame is seen as problematic due to the rulers' inertia, you could just as well make a problem of the rulers own gravitation curving spacetime. And measuring distances with light doesn't help, because the energy of the light beam curves spacetime as well.

 

[atyy]

I'm just pointing out, that if placing rulers at rest in the rotating frame is seen as problematic due to the rulers' inertia, you could just as well make a problem of the rulers own gravitation curving spacetime. And measuring distances with light doesn't help, because the energy of the light beam curves spacetime as well.

If we assume special relativity then gravity disappears. Not sure about inertial forces though.

 

[Fredrik]

A.T., you should clear the search before you link to a page at Google Books.

Regarding the "massless" rulers, it's sufficient to say that we're talking about what predictions special relativity would make, because spacetime is a fixed mathematical structure (Minkowski spacetime) in that theory. (So neither mass nor anything else has any influence on it).

Edit: I see now that I didn't really need to tell you that.

 

[atyy]

BTW, I'm a bit mixed up about where the discussion is. Is it that we all agree "something" is non-Euclidean, and the only problem is what that "something" is, and how to demonstrate that an an ideal experiment?

 

[A.T.]

Is it that we all agree "something" is non-Euclidean, and the only problem is what that "something" is,

That is the semantical part.

and how to demonstrate that an ideal experiment?

That is the physically relevant part.

 

[bcrowell]

BTW, I'm a bit mixed up about where the discussion is. Is it that we all agree "something" is non-Euclidean, and the only problem is what that "something" is, and how to demonstrate that an an ideal experiment?

I think the remaining disagreement is about whether you can determine the spatial geometry by purely static measurements using material rulers, and without clock synchronization.

A few more thoughts:

Suppose you have a ruler that's as rigid as relativity allows (meaning it's less than Born-rigid). You bring it very close to r=c/\omega. Rotations and translations of rulers are necessary in order to carry out these measurements. (If a bunch of rulers are just left in place since the beginning of time, then you have no way to verify that they're all the right lengths in relation to one another.) When this maximally-rigid ruler is oriented radially and positioned near r=c/\omega, the speed at which vibrations propagate along it equals c. It's going to vibrate, because you're moving it into place. The vibrations are going to be large, because the ruler is very close to the radius where it would be destroyed by centrifugal forces, meaning that the vibrations almost exceed its elastic limit. You can't just wait for the vibrations to die out, because in the limit of r \rightarrow c/\omega, the time dilation becomes infinite, and therefore your thesis adviser (who is back at some smaller value of r) would have to wait an infinite amount of time to hear about the data. The best you can do is to use the propagation of the vibrations to probe the geometry of spacetime. But now the ruler is really just functioning as a sort of waveguide for the electromagnetic fields that bind it together. In other words, all you've done is replace the material rulers with radiometric measurements.

Re clock synchronization, suppose I make measurements with rulers, and I find out that at r=1 m, the Gaussian curvature (as determined from the angular deficit of triangles per unit area) is -1x10^-23 m^-2. I call up a friend, and he says that when he did the same experiment at r=1 m, the Gaussian curvature was -4x10^-23 m^-2. How can we explain the discrepancy? Well, the most likely reason is that he's circling the axis at a frequency that's twice as big as the frequency at which I'm circling the axis. Our measuring apparatus is going around in circles like the hand of a clock, and the problem arose because the hand on his clock was going around at twice the speed at which mine was. So in this sense, you really do need clock synchronization. If there is a physical, rotating "discworld" (with apologies to Terry Pratchet), then all we're doing by bringing the apparatus to rest with respect to the surface of the disk is to synchronize the lab-clock with the discworld clock.

So in summary, I'm convinced that:

(1) Clocks and clock synchronization are necessary, but can be reduced to something relatively trivial. The impossibility of global synchronization is only a big deal because it shows that Born-rigidity is a kinematical impossibility, so you can't have Born-rigid rulers.

(2) Material rulers are probably not nearly as easy to use for this as you'd naively think, even at small r, and at large r it's not even theoretically possible to use them.

 

[pervect]

Now you added a restriction, which also applies to clocks in a rotating frame: You can synchronize clocks which are equidistant to the rotation axis in a rotating frame.

Unfortunately, the whole point I'm trying to make is that you *can't*. To be more specific, you cannot synchronize all the clocks equidistant from the rotation axis according to the Einstein convention. Working your way around the circle, pairwise, when you finally get to the starting point, the last clock you synchronize won't be synchronized with the clock that you started with.

see for instance http://arxiv.org/abs/gr-qc/9805089

the circumference of the disk is treated as a geometrically well defined entity,that
possesses a well defined length without worrying about the fact that no transitive synchronism exists along the said circumference.

Transitive means that if A is sync'd to B, and B is to C, A is syncd'd to C.

 

[bcrowell]

This defines a circumference, but it's not clear that this approach actually defines a "geometry". The "circumference" defined by this means is not a closed curve!

I'm not aware of anyone using this particular approach in the literature - though there may be someone, I'm not familiar with all of the literature on the topic by any means, it's quite large.

For a historical discussion of Einstein's original approach, see p. 11 of this paper: http://philsci-archive.pitt.edu/archive/00002123/01/annalen.pdf To me it seems very close to what A.T. is talking about.

For Einstein's popular-level description of the idea: http://en.wikisource.org/wiki/Relat...easuring-Rods_on_a_Rotating_Body_of_Reference "If, then, the observer first measures the circumference of the disc with his measuring-rod and then the diameter of the disc, on dividing the one by the other, he will not obtain as quotient the familiar number p = 3.14 . . ., but a larger number,[4]** whereas of course, for a disc which is at rest with respect to K, this operation would yield p exactly. This proves that the propositions of Euclidean geometry cannot hold exactly on the rotating disc, nor in general in a gravitational field, at least if we attribute the length I to the rod in all positions and in every orientation."

For a mathematical derivation of the spatial metric: p. 6 of http://www.phys.uu.nl/igg/dieks/rotation.pdf

There is a somewhat different treatment in Rindler's "Relativity: Special, General, and Cosmological" (the long one, not the "Essential" version), p. 198. "The metric of the lattice is the negative of the last three terms in (9.26) and represents a curved three-space..." Amazon will let you peek at the two relevant pages if you use the "look inside" feature and search for "uniformly rotating lattice." Rindler introduces a trick of putting metrics into a certain canonical form, and then uses it in this example. In this canonical form, there's once piece of the metric that's always interpreted as the spatial geometry.

 

[atyy]

There is a somewhat different treatment in Rindler's "Relativity: Special, General, and Cosmological" (the long one, not the "Essential" version), p. 198. "The metric of the lattice is the negative of the last three terms in (9.26) and represents a curved three-space..." Amazon will let you peek at the two relevant pages if you use the "look inside" feature and search for "uniformly rotating lattice." Rindler introduces a trick of putting metrics into a certain canonical form, and then uses it in this example. In this canonical form, there's once piece of the metric that's always interpreted as the spatial geometry.

I see - Rindler's p198 does what you were trying to do in post #8.

In a footnote on p72, Rindler agrees with A.T.'s approach (the main points, not sure about not needing clocks): http://books.google.com/books?id=MuuaG5HXOGEC&dq=rindler+relativity&source=gbs_navlinks_s

 

[pervect]

You can synchronize clocks along a non-straight curve that connects the axis to an off-axis point. You just can't do a global synchronization without discontinuities.

Let me expand on this a bit...

Global synchronizations *which follow the Einstein convention* don't exist. Someone, possibly AT, is probably going to object "but I can use non-Einstein synchronizations" at some point in the discussion. However, one wouldn't want to use such synchronizations to measure velocities. To illustrate the point, I'll exaggerate it. If one has a jet that takes of at noon in Chicago (CST) and lands at noon in San Diego (PST), it makes no sense to say that it has an infinite velocity because it landed at the same time it took off. Using non-Einstein synchronizations in general needs to be handled carefully to avoid mistakes. Sometimes one can't avoid it, but it is a breeding ground for confusion, and the rotating disk is a prime example of the sorts of confusion that arrise.

It's also possible to mess up distance measurements by using non-Einstein synchronizations, this is more a matter of taking proper care. Personally, I think the best approach for defining distance is to use radar measurements, which is what the SI standard more or less does anyway by defining 'c' as a constant. If we can get a general agreement that any good distance measurement scheme is equivalent to a radar measurement for "close enough" points, I'll feel that we are all on the same definitional page.

 

[A.T.]

clocks, clocks, clocks

Okay, here another naive idea to determine that space is non-Euclidean, without dealing with clock synchronization issues:

Place three observers at rest in the rotating frame and connect them with laser beams. They will find that the sum of the triangle angles is less than PI and therefore conclude negative spatial curvature.

 

[atyy]

Okay, here another naive idea to determine that space is non-Euclidean, without dealing with clock synchronization issues:

Place three observers at rest in the rotating frame and connect them with laser beams. They will find that the sum of the triangle angles is less than PI and therefore conclude negative spatial curvature.

Wouldn't one need clocks to define "at rest in the rotating frame", since clocks are needed to define a frame - say first define an inertial frame, there you have clocks, then define a rotating frame relative to that?