Lorentz Contraction Circular Motion

[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?

 

[yuiop]

...
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.

I thought you had better intuition than that, Pervect. You should have known it would be me! First of all, the Einstein synchronisation method has the explicit assumption that the speed of light is constant and isotropic in all directions. This is clearly not the case as far as a rotating disk is concerned. If a disk is rotating clockwise in the non rotating frame, then a clockwise going light signal sent from a source fixed to the disk, takes longer to go around the perimeter and return to the source than a signal sent from the same source going in the opposite direction. Therefore the speed of light is not isotropic in the rotating frame even when using a single clcok and the Einstein synchronisation method is invalid and simply does not work.
Now we can find a synchronisation method that has the property you described in post 55, of being "transitive". It simply requires all clocks fixed to the perimeter of the rotating disk to be started by a single start signal sent from an omni-directional source located at the axis of the disk. Of course, using such a synchronisation scheme, will mean that the speed of light is not isotropic according to observers in the rotating frame even on a small local scale.

...
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.

The radar method will not work, or at least it will not work any better than the proper distance measurement of space by using rulers at rest in the rotating frame as championed by A.T. For example, if a radar light source located on the rim sends a signal to a mirror located further along the rim of the disk in a clockwise direction, we could adjust the location of the mirror until it takes 2 femto-seconds for the radar pulse to return to the radar source and define the location of the finely positioned mirror as being 1 femto-lightsecond from the radar source. Using this radar method produces identical length measurements to those produced by simply using a ruler at rest with the rotating disk. Additionally we see a failure of the SI standard of defining length as the distance traveled by a light in a given time interval. By timing the period it takes a light signal to go around the perimeter of the disk and return to the source located on the rotating disk, the SI method defines the clockwise circumference to be greater than the anti-clockwise circumference. At least the A.T. proper ruler measurement produces the same distance measurement in either direction. It would seem that your desire for a definition of length that "is equivalent to a radar measurement for 'close enough' points" would be more sympathetic to the proper definition of length as championed by A.T. than the formal definition of length championed by Fredrick.

It would seem that Fredrick has the moral/formal high ground in his definition of spatial distance as being the difference between two events that are measured simultaneously in the a given reference frame and the proper ruler method of measuring the disk circumference does not meet that requirement.

In SR, the purely spatial interval dx is defined as the interval between two events when dt is zero. This definition coincides with proper distance being defined as the distance measured by a physical ruler at rest in the reference frame. This equivalence between spatial interval dx and proper distance breaks down in accelerating reference frames and the arguments in this thread seem to be basically about which method is the better definition of spatial distance in an accelerating reference frame.

Perhaps the arguments in this thread could be made clearer by considering an ideal numerical experiment and asking what the various parties predict the numerical spatial circumference of the rotating disk to be.

Experiment:
Disk radius = 1 light second in the non rotating frame.
Instantaneous velocity of a point of the rim of the rotating disk is 0.8c clockwise, relative to an observer just outside the disk in the non rotating frame.
Gamma = 1/0.6
Circumference = 2*pi*r = 6.28318531 lightseconds in the non rotating frame.

A.T. proper distance circumference:

2*pi*r*gamma = 10.4719755 lightseconds. (Same in both directions.)

Transitive sychronisation method:

Same circumference as A.T. proper distance.

Speed of light is anisotropic (0.555555556 c clockwise and 5.0 c anti-clockwise).

Radar circumference distance (SI standard):

2*pi*r*c/(c-v)/gamma = 18.8495559 lightseconds (Clockwise).

2*pi*r*c/(c+v)/gamma = 2.0943951 lightseconds (Anti-clockwise).

Speed of light is isotropic (c).

Fredrick circumference (dt=zero):

I will let Fredrick work out what this predicts ;)

 

[bcrowell]

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.

I think this still requires local clock synchronization of the relatively trivial type I defined in #54. That is, verifying that the measuring apparatus is at rest with respect to the rotating frame is itself a kind of clock synchronization.

If you want to be able to compare the results with theory, you're also going to need to be able to measure r. The Ricci scalar is R=-6\omega^2/(1-2\omega^2r^2+\omega^4r^4). The angular excess of a triangle is \epsilon=\Sigma\theta-\pi. The Gaussian curvature is K=\lim \epsilon/A=R/2. So the theoretical prediction you'd need to verify is
<br /> \lim\epsilon/A = -\frac{3\omega^2}{1-2\omega^2r^2+\omega^4r^4}<br />
To measure r you're going to need something more than a purely local measurement, and the method that would probably realistically work would involve radar and clock synchronization.

 

[bcrowell]

Kev, re your #61, thanks for going into more detail about clock synchronization. This cleared up some things for me. I was conceiving of the difficulties with clock synchronization as being ones that would only apply to a region that wrapped all the way around in \theta. I can see now that that was incorrect. Measuring the Sagnac effect locally shows that Einstein synchronization fails (or fails to have all the desired properties like transitivity) even locally.

The radar method will not work[...]

The radar method for determining the spatial geometry is described by Ø. Grøn, Relativistic description of a rotating disk, Am. J. Phys. 43 869 (1975), at pp. 873-874. The spatial distance d\sigma between two nearby points is defined as half the round-trip time for a beam of light.

[...] or at least it will not work any better than the proper distance measurement of space by using rulers at rest in the rotating frame as championed by A.T.

I disagree with you here. There are serious difficulties with using rulers, as described in my #54. Basically you want a Born-rigid ruler, but all you can really have is a ruler that's as rigid as allowed by the fundamental limits that relativity places on the properties of materials. Both Grøn and Dieks ( http://www.phys.uu.nl/igg/dieks/rotation.pdf ) discuss this. For conceptual simplicity, we'd like to be able to describe the spatial geometry as the one that would be measured by rulers. This is what Einstein did in his popularization of GR ( http://en.wikisource.org/wiki/Relat...easuring-Rods_on_a_Rotating_Body_of_Reference ). But in fact that's an oversimplification. Re uniqueness, see Grøn, p. 873, 1st paragraph of section B. Re the issues with the dynamics of actual rulers, see p. 7 of Dieks.

Additionally we see a failure of the SI standard of defining length as the distance traveled by a light in a given time interval. By timing the period it takes a light signal to go around the perimeter of the disk and return to the source located on the rotating disk, the SI method defines the clockwise circumference to be greater than the anti-clockwise circumference.

This objection doesn't apply to Grøn's definition. Since d\sigma is defined in terms of round-trip time, you get the same answer regardless of whether you perform the integral \int d\sigma in the clockwise or counter-clockwise direction. One way to see that the difficulty is eliminated is that the Sagnac effect is proportional to the area of the loop, but Grøn's definition uses a loop of zero area.

Perhaps the arguments in this thread could be made clearer by considering an ideal numerical experiment and asking what the various parties predict the numerical spatial circumference of the rotating disk to be.

Is there any actual disagreement on this? Grøn's equation 42 on p. 874 for the circumference is equivalent 2\pi R\gamma, which is what I think everyone agrees is correct. If you go back and look at pervect's #58, he was not proposing anything like the "Radar circumference distance (SI standard)" that you seem to be ascribing to him. He says, '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.' The part about "close enough" is clearly equivalent to Grøn's definition, which uses a differential, and inequivalent to what you've labeled "Radar circumference distance (SI standard)."

 

[A.T.]

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?

Here the setup:

- The mother ship is moving inertially and not rotating, as verified by accelerometers and Sagnac interferometers.

- The mother ship sends out 3 space ships, at 120°-step angles.

- All 3 ships run the same acceleration program, that brings and keeps them in an orbit around the mother ship. Now all 3 are at rest in the same rotating frame.

- Each of the 3 ships is observing the other 2 ships through telescopes and measuring the apparent angle between them.

 

[bcrowell]

Re # 64 by A.T. -- I think this works fine, except for the issue raised by my #62. If the ships are at r &lt;&lt; c/\omega, then the angular deficit should equal -3\omega^2A, where A is the area of the triangle. But you're going to need some kind of distance measurement in order to determine A, and if you want to test theory at values of r that are not <<c, you'll also need to measure r. It's not a totally static measurement, in the sense that they need to determine \omega, which requires clocks. So it seems to me that you can prove the non-Euclidean nature of the geometry by purely static measurements without using radar for distances, but I don't yet see how you can do any kind of quantitative test of theory under those very strict constraints. Maybe you could make a rotating network of triangles, and there could be relationships between the angles in the different triangles?

I think I've now succeeded in clearing up the glitches in the derivation of the spatial metric I gave in my #8. The result is here http://www.lightandmatter.com/html_books/genrel/ch03/ch03.html#Section3.4 , in subsection 3.4.4. I would be grateful for any comments.

 

[Fredrik]

Fredrick circumference (dt=zero):

I will let Fredrick work out what this predicts ;)

Proper length is a coordinate independent property of the curve, so there's nothing to work out. It's 2\pir.

I haven't seen any comments about an issue that I raised on page 1. We all agree that it's not possible to get a disc spinning without stretching the material, right? When it's streched, the sum of the internal forces on any atom should be towards the center, while the centrifugal force is in the opposite direction. Has anyone worked out which one of these forces "wins"? Does the radius of the disc get larger or smaller when we give it a spin?

 

[atyy]

I haven't seen any comments about an issue that I raised on page 1. We all agree that it's not possible to get a disc spinning without stretching the material, right? When it's streched, the sum of the internal forces on any atom should be towards the center, while the centrifugal force is in the opposite direction. Has anyone worked out which one of these forces "wins"? Does the radius of the disc get larger or smaller when we give it a spin?

The assumption is that the disc doesn't get bigger or smaller - or we set it up so that it is flat and circular and spinning in the inertial frame - where one needs clock synchronization to verify the "flat, circular and spinning".

For a real disc, I imagine it could warp, so the radius would stay the same, but the disc would not be flat. Or it could break, in which case you could not define a radius. Or ...

 

[pervect]

Re: discs that contract when you spin them faster.

There was some discussion of rotating hoops in which I took part - see Greg Egan's webpage http://gregegan.customer.netspace.net.au/SCIENCE/Rings/Rings.html and the link on that webpage back to physics forums. The specific model we analyzed was a "hyperelastic" model.

Under certain conditions the hyperelastic model could predict the radius decreasing as you increased the spin - but this led to some bad behavior, including equations of motion that had no solution (the Lagrangian became singular). I feel now, and I think Greg Egan agrees, that this particular prediction is in a realm where the hyperelastic model fails - one symptom of the failure is the speed of sound exceeding 'c' - though it took some time to notice this.

My own conjecture at this point is that the moment of inertia of the disk must always increase, or you'll get non-physical runaway effects and general bad behavior. If the moment of inerta decreases with increasing angular velocity the disk must spin faster, making it collapse further, making it spin faster, leading to a runaway effect.

Having the moment of inertia not decrease probably implies that the radius also doesn't decrease, it's hard to see how the radius could decrease and the moment of inertia increase. But I haven't analyzed that point very carefully.

But I don't think there's any proof - at this point, I'd just say it's conjecture, though an informed conjecture.

 

[pervect]

I thought you had better intuition than that, Pervect. You should have known it would be me! First of all, the Einstein synchronisation method has the explicit assumption that the speed of light is constant and isotropic in all directions. This is clearly not the case as far as a rotating disk is concerned. If a disk is rotating clockwise in the non rotating frame, then a clockwise going light signal sent from a source fixed to the disk, takes longer to go around the perimeter and return to the source than a signal sent from the same source going in the opposite direction. Therefore the speed of light is not isotropic in the rotating frame even when using a single clcok and the Einstein synchronisation method is invalid and simply does not work.

I would agree that the Einstein synchronization method is only possible locally on a rotating frame, and not possible globally.

However, I would disagare that this makes it invalid.

Note that Tartaglia has much the same view, he has influenced my thinking on the topic.

From http://arxiv.org/abs/gr-qc/9805089
It is often taken for granted that on board a rotating disk it is possible to operate a {global}3+1 splitting of space-time, such that both lengths and time intervals are{uniquely} defined in terms of measurements performed by real rods and real clocks at rest on the platform. This paper shows that this assumption, although widespread and apparently trivial, leads to an anisotropy of the velocity of two light beams traveling in opposite directions along the rim of the disk; which in turn implies some recently pointed out paradoxical consequences undermining the self-consistency of the Special Theory of Relativity (SRT). A correct application of the SRT solves the problem and recovers complete internal consistency for the theory. As an immediate consequence, it is shown that the Sagnac effect only depends on the non homogeneity of time on the platform and has nothing to do with any anisotropy of the speed of light along the rim of the disk, contrary to an incorrect but widely supported idea.

Now we can find a synchronisation method that has the property you described in post 55, of being "transitive". It simply requires all clocks fixed to the perimeter of the rotating disk to be started by a single start signal sent from an omni-directional source located at the axis of the disk. Of course, using such a synchronisation scheme, will mean that the speed of light is not isotropic according to observers in the rotating frame even on a small local scale.

Yes, other authors have pointed this out - I don't have the exact reference handy.

The radar method will not work, or at least it will not work any better than the proper distance measurement of space by using rulers at rest in the rotating frame as championed by A.T. For example, if a radar light source located on the rim sends a signal to a mirror located further along the rim of the disk in a clockwise direction, we could adjust the location of the mirror until it takes 2 femto-seconds for the radar pulse to return to the radar source and define the location of the finely positioned mirror as being 1 femto-lightsecond from the radar source. Using this radar method produces identical length

I fail to see why this is bad. The fact that the radar method and the perhaps less-easily defined idea of using rulers arive at the same answer is a plus in my view, suggesting that they both measure what is meant by "distance", as long as the points are close enough.

measurements to those produced by simply using a ruler at rest with the rotating disk. Additionally we see a failure of the SI standard of defining length as the distance traveled by a light in a given time interval. By timing the period it takes a light signal to go around the perimeter of the disk and return to the source located on the rotating disk, the SI method defines the clockwise circumference to be greater than the anti-clockwise circumference. At least the A.T. proper ruler measurement produces the same distance measurement in either direction. It would seem that your desire for a definition of length that "is equivalent to a radar measurement for 'close enough' points" would be more sympathetic to the proper definition of length as championed by A.T. than the formal definition of length championed by Fredrick.

I support keeping the SI notion of distance, and reject the notion of an anisotropic velocity of light. And I'm not quite sure what you are proposing to replace the SI notion of distance? Anyway, if we can't come to an operational agreement on how to measure distance, we have some problems. I was really hoping that everyone would think it was clear that the SI notion was the correct one, at least for nearby points.

 

[bcrowell]

[re kinematic impossibility of Born rigidity]
I will have a look at that. What is the exact title? Or can you summarize his argument?

Ø. Grøn, Relativistic description of a rotating disk, Am. J. Phys. 43 869 (1975). My shorter presentation of some of the same ideas is here http://www.lightandmatter.com/html_books/genrel/ch03/ch03.html#Section3.4 in subsection 3.4.4, at "Impossibility of rigid rotation, even with external forces."

[re testing R(r)]Operationally, how is a particular "off-axis point" identified?

I would do it by using radar measurements to determine infinitesimal proper distance dr, and then integrating to find r. The "radar ruler" notion of proper distance is defined in the Grøn reference above, and summarized in my own treatment linked to above.

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.

I think a general statement is that you can synchronize clocks on any open curve, or along any topologically connected set of points that doesn't surround any region with finite area. One way to see this is that the Sagnac effect is proportional to area, and an open curve doesn't enclose any area. Another way to see it is that you can do a chain of Einstein synchronizations, and there will be no self-contradiction if the curve doesn't close back on itself. BTW, the Greg Egan article is a real tour de force! If you were a significant enough contributor to that for him to single you out for credit, then clearly you understand a heck of a lot about this topic.

Proper length is a coordinate independent property of the curve, so there's nothing to work out. It's 2\pir.

Hmm...so are you saying that Rindler, Grøn, and Dieks are all wrong? If so, then it would be interesting to see what you think is the error in their treatments.

 

[bcrowell]

*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).

Can you explain in a little more detail what you mean by this? It seems to me that given any two world-lines A and B, you could have other world-lines connecting them that would have any length you want. Say you're using a +--- metric. Then to get a world-line C with big positive length that connects A and B, you could start at a point on A, maintain a coordinate velocity of 0, wait until B is about to hit you, then run back toward A at the speed of light. Lather, rinse, repeat. World-line D could be one that goes from A to B at the speed of light, giving a length of 0. World-line E, which can't be physically realized, runs back and forth between A and B at speeds that are always greater than c; it gives as big a negative length as you like.

 

[pervect]

Can you explain in a little more detail what you mean by this? It seems to me that given any two world-lines A and B, you could have other world-lines connecting them that would have any length you want. Say you're using a +--- metric. Then to get a world-line C with big positive length that connects A and B, you could start at a point on A, maintain a coordinate velocity of 0, wait until B is about to hit you, then run back toward A at the speed of light. Lather, rinse, repeat. World-line D could be one that goes from A to B at the speed of light, giving a length of 0. World-line E, which can't be physically realized, runs back and forth between A and B at speeds that are always greater than c; it gives as big a negative length as you like.

I think I got the sign wrong. But it's probably safer to say "extremizes" the distance, that way the sign doesn't matter :-). And I did mean to say that path along which we measure the distance was to be a straight line in SR (or in general a geodesic if space-time is curved) even if I forgot to specify it in my last post. We don't need to define how to synch clocks to define the notion of a straight line either - it's coordinate independent.

suppose we have two curves C1 and C2

C1 is (x=0, t=lambda)
C2 is (x=1, t=lambda)

The distance between C1 and C2 will in general depend where on C1 (or C2) we are. Let's say we want the distance between C1 and C2 at the point (0,0) on C1

Lets pick a point on C2, (0,tau). Then the square of the straight line distance will be the loretnz interval between them

d^2 = (1-0)^2 - (0-tau)^2 = 1 - tau^2

this is extremized when tau=0, making the distance one.

 

[Fredrik]

Hmm...so are you saying that Rindler, Grøn, and Dieks are all wrong? If so, then it would be interesting to see what you think is the error in their treatments.

I'm just saying that the closed continuous curve in Minkowski spacetime that satisfies t=0, x^2+y^2=r^2, z=0 in an inertial frame has proper length 2\pi r, and that proper length is a coordinate independent property of a spacelike curve. I don't think any of those guys would disagree with that.

Regarding what they're doing, I just don't see the point. I don't see why they feel their calculations are worth doing, or why they insist on using terms like "spatial geometry". I also don't see a way to write down a rule that describes how to make length measurements using a radar device that's in an arbitrary state of motion. If the purpose of these calculations is to find such a rule, then I think it's worth doing, but I didn't get that impression.

 

[pervect]

The standard treatment is more abstract, but it should be equivalent to taking the radar distances between nearby points on the circumference and summing them.

 

[heldervelez]

CMB photons as standards clocks and rulers

Here the setup:

- The mother ship is moving inertially and not rotating, as verified by accelerometers and Sagnac interferometers.

- The mother ship sends out 3 space ships, at 120°-step angles.

- All 3 ships run the same acceleration program, that brings and keeps them in an orbit around the mother ship. Now all 3 are at rest in the same rotating frame.

- Each of the 3 ships is observing the other 2 ships through telescopes and measuring the apparent angle between them.

We can use CMB photons as standards clocks and rulers, when at rest in the mother ship.
CMB provides a commom reference.

Suppose we dispose radially in the circular disk a set of equal surface (when at rest) photodetectors.
For calibration purposes, when at rest, we measure the count of incoming photons in each device. The count must be equal in each other.
When moving the circle the count in each device will differ as the device is more inner or outer.
The number counts at each can be used to infer the equivalent area of each device or the geometry?

 

[bcrowell]

I also don't see a way to write down a rule that describes how to make length measurements using a radar device that's in an arbitrary state of motion.

See pp. 6-8 of Dieks, http://www.phys.uu.nl/igg/dieks/rotation.pdf

The standard treatment is more abstract, but it should be equivalent to taking the radar distances between nearby points on the circumference and summing them.

I don't know if one treatment is more standard than another. Dieks uses radar distances between nearby points. Rindler defines what he calls the "first standard form" of a stationary metric (section 9.3), and then interprets it as being equivalent to radar distance and ruler distance (first page of section 9.4).

 

[bcrowell]

Thanks, pervect, for the explanation in #72. I think I follow you now.

A.T., I've been thinking some more about the three-spaceships idea, and I've come to the conclusion that the ground rules we've set for that are not likely to be fruitful. The general idea here has been that there are issues with global clock synchronization and issues with dynamics of rigid rulers, so we want to find a purely static method of measurement that doesn't require clocks or rigid rulers. This has the flavor of the classic geometrical constructions, like compass and straightedge constructions. There are other families of constructions like these in plane geometry. There are constructions that can be done with a compass and no straightedge. There are others that can't be done with compass and straightedge, but can be done with origami. In general, such a system of constructions corresponds to a certain geometry with constructive axioms. Compass and straightedge are the instruments that are explicitly referred to in Euclid's axioms. If you restrict yourself to a system where there is only a straightedge and a method for drawing parallels, then you have affine geometry.

What we've been doing is essentially trying to perform a certain construction in a system where there is angular measure but no distance measure. That doesn't really work. There is no interesting geometrical system that is like this. Euclidean geometry has both angular and distance measure. Affine geometry has distance measure but no angular measure (and distances along non-parallel lines aren't comparable). But there is no system that has angular measure without distance measure, and that's because if you have angular measure, then your system is at least as strong as affine geometry (which only defines parallelism), and affine geometry allows a distance scale to be constructed using a ladder construction.

I came up with a somewhat more explicit way of showing this in the context of the spaceship experiment you've been talking about. The objection I raised earlier is that you can't do much of a quantitative test of GR unless you can measure r, since GR's only testable prediction about the spatial geometry is
<br /> \lim\epsilon/A = -\frac{3\omega^2}{1-2\omega^2r^2+\omega^4r^4}<br />
for the angular deficit of a triangle per unit area. You can sort of get around this by the following trick. Start with a single spaceship in some randomly chosen state of inertial motion, which means that with unit probability it's rotating. The ship has two identical rocket engines. Fire one engine with constant thrust, construct a triangle in a lab inside the ship, and measure \epsilon using protractors. During this measurement, there exists some inertial frame such that the ship is moving in a circle, but you can't directly measure what the circle's radius is. Now fire both engines simultaneously, and measure \epsilon again. The circle now has some other radius. Although you don't know r, GR does predict that \lim\epsilon/A=f(n), where n is the number of engines firing, and f is some function, which has adjustable parameters in it. By doing measurements with enough values of n, you can determine the adjustable parameters. It now seems like you have a clearcut test of GR, because you can measure f, which is essentially a test of the radial dependence of the Gaussian curvature. The problem with all this is that you don't know A, and A isn't constant. (To see that A can't be constant, note that \epsilon/A diverges as n approaches infinity, whereas -\pi &lt; \epsilon \le 0). To measure A, you need either clocks+radar, or some other kind of ruler. I think this is a symptom of the fact that, for the reasons I outlined above, the particular set of measuring instruments we've been discussing is not a fruitful one to talk about. Having a protractor is actually equivalent to having a ruler (by the affine-geometry ladder construction), so you might as well just use radar rulers.

Another issue that occurred to me is that geodesics of the spatial metric are not the same as geodesics of the spacetime metric, so you can't use laser beams to define the sides of your triangle. For example, a laser beam going outward from the axis is actually a curve in the rotating frame, and it's not a geodesic of the spatial metric.

 

[yuiop]

Proper length is a coordinate independent property of the curve, so there's nothing to work out. It's 2\pir.

I haven't seen any comments about an issue that I raised on page 1. We all agree that it's not possible to get a disc spinning without stretching the material, right? When it's streched, the sum of the internal forces on any atom should be towards the center, while the centrifugal force is in the opposite direction. Has anyone worked out which one of these forces "wins"? Does the radius of the disc get larger or smaller when we give it a spin?

I now see that your assertion that the proper length of the circumference is 2*pi*r is conditional upon the outcome of the question that follows it, which perhaps can be rephrased as "What will be the proper length of the circumference, if the disc is slowly and carefully slowed down so that it is no longer rotating?"?

My answer to that question, is that the proper circumference of the disc when brought to rest is numerically the same as the result obtained by 2*pi*r*gamma when the disc was rotating, or put another way the radius gets smaller when the disc is given a spin.

Of course, in normal circumstances, a disc is torn apart by "centrifugal force" long before its its peripheral velocity reaches a tiny fraction of the speed of light, as can be seen by the difficulty in designing high speed, energy storage flywheels. To be able to spin a disc up in a way that demonstrates the radius gets smaller as it spins faster, an idealised form of angular acceleration would have to be set up that is similar to the ideal of linear Born rigid acceleration. For example a number of stations could be placed on spokes radiating from a common hub. The stations are equally spaced and equidistant from the hub and there are enough stations to aproximate the circumference of a disc. The stations are free to slide along the spokes if required and each station has its own rocket thrusters to maintain position. This is similar to the ideal of a Born rigid rocket having a micro thruster attached to every single atom of a linearly accelerating rocket. Now as the disc is spun up, the controllers of the stations are instructed to operate their outward thrusters so that they maintain constant radar distance with their immediate neighbours. Under these conditions, the final radius of the disc will be less than the initial radius and proper circumference of the disc will remain constant as measured by local radar measurements (or by ideal co-moving rulers as specified by A.T.) for any angular velocity.

I hope by the above argument, you will concede that the coordinate independent proper circumference of the disc is 2*pi*r*gamma where r is the final radius of the rotating disc and gamma is a function of the final peripheral velocity of the disc and this distance is distance measured by local rulers that are at rest with the rotating circumference as described by A.T. (i.e. the location of the end points of the rulers do not change over time according to the reference frame of the accelerating rotating disc riders). This circumference is also the distance that would be obtained by the sum of chained local radar distance measurements. Put yet another way, the proper circumference is 2*pi*r/sqrt(1-v^2/c^2) for any angular velocity, where r is the instantaneous radius (measured in the non rotating frame) and v is the instaneous peripheral velocity at any given time and this length value remains constant at all times, under the form of Born rigid angular acceleration I have described.

 

[Fredrik]

I now see that your assertion that the proper length of the circumference is 2*pi*r is conditional upon the outcome of the question that follows it, which perhaps can be rephrased as "What will be the proper length of the circumference, if the disc is slowly and carefully slowed down so that it is no longer rotating?"?

It isn't. It's only based on the fact that we have specified that the distance from the center to the edge in the center-of-mass inertial frame is r. It doesn't matter what the radius was before we gave the disc a spin.

The gamma in the claim that the circumference is 2*pi*gamma*r "in the rotating frame" comes from the fact that if the circumference is measured by a large number of co-moving inertial observers, the result is 2*pi*gamma*r. (Each inertial observer measures the length of a short segment that he's approximately co-moving with, and then you add up the results. It's approximately 2*pi*gamma*r, because their measuring devices are Lorentz contracted by a factor of gamma, and the approximation becomes exact in the limit of infinitely many co-moving observers). Why anyone would call this a measurement of the circumference in the rotating frame is beyond me.

The stations are free to slide along the spokes if required and each station has its own rocket thrusters to maintain position. This is similar to the ideal of a Born rigid rocket having a micro thruster attached to every single atom of a linearly accelerating rocket. Now as the disc is spun up, the controllers of the stations are instructed to operate their outward thrusters so that they maintain constant radar distance with their immediate neighbours. Under these conditions, the final radius of the disc will be less than the initial radius

You're describing a scenario where we can exactly cancel the centrifugal force on each component part, but you have also removed the internal forces between the component parts. The way I see it, they are the reason why the disc would get smaller when you give it a spin (and magically compensate for the centrifugal force).

This is what I'm thinking: Suppose that the radius at rest is s. If we could add an inward force that exactly cancels the centrifugal force on each atom, we would have r<s, because the forceful stretching of the circumference that occurs when we increase the angular velocity will create internal forces in the inward direction. The shrinking of the disc will produce an outward force that grows as the disc get smaller (and the pressure increases), and at some point an equilibrium is reached. This must happen before the radius has shrunk to s/gamma, because that's what we would expect the radius to become if we neglect that compressing the disc will increase the pressure in the material. So the r is going to be somewhere between s and s/gamma (probably much closer to s than s/gamma).

and proper circumference of the disc will remain constant as measured by local radar measurements (or by ideal co-moving rulers as specified by A.T.) for any angular velocity.

Here you seem to be neglecting that the co-moving rulers get Lorentz contracted.

I hope by the above argument, you will concede that the coordinate independent proper circumference of the disc is 2*pi*r*gamma where r is the final radius of the rotating disc and gamma is a function of the final peripheral velocity of the disc and this distance is distance measured by local rulers that are at rest with the rotating circumference as described by A.T.

No, the rulers would measure 2*pi*r*gamma, because they're Lorentz contracted by a factor of gamma, so you need more of them to fill up the entire circumference. The coordinate independent proper length of the circumference is 2*pi*r, because that's what it is in the center-of-mass inertial frame. The curve that has coordinate-independent proper length 2*pi*gamma*r is a spiral in spacetime, not a circle in space. (It's not even a circle in the spiral-shaped hypersurface that Grøn calls "rest space", since the endpoint isn't the same as the starting point).

 

[Ich]

Why anyone would call this a measurement of the circumference in the rotating frame is beyond me.

Because you can actually and comfortably stuff 8 rulers in the circumference of a 1 ruler radius. For an indefinitely long time, with anyone on board having time to accurately measure length. No synchronization issues there.

If I somehow lost track of what this discussion is about, my apologies.

 

[A.T.]

No, the rulers would measure 2*pi*r*gamma, because they're Lorentz contracted by a factor of gamma, so you need more of them to fill up the entire circumference.

But in the rotating frame the rulers are not contracted, yet you still need more of them to fill up the entire circumference. So the circumference is more than 2*pi*r in the rotating frame.

The curve that has coordinate-independent proper length 2*pi*gamma*r is a spiral in spacetime, not a circle in space.

So a spiral in spacetime cannot be a circle when projected onto space? And why should I even care how the circular ruler is represented in spacetime, if the ruler is at rest in my rotating frame, and I read off what it measures.

 

[Fredrik]

But in the rotating frame the rulers are not contracted, yet you still need more of them to fill up the entire circumference.

How do you make sense of that statement? I assume the rulers are supposed to be "at rest" in the rotating frame. What can that possibly mean other than that they're at rest in a sequence of co-moving inertial frames? If that's what they are, then they're Lorentz contracted. If the rulers are instead being held in place by a circular rim, then they will get bent and squeezed in addition to being Lorenz contracted.

So a spiral in spacetime cannot be a circle when projected onto space?

Of course it can, but who's talking about projections? The projection onto space has circumference 2*pi*r, not 2*pi*gamma*r. Your guys defined "rest space" to be that spiral surface just to get a larger circumference, but the price we pay is a) that we're not even talking about a closed curve anymore (so why call it "circumference"?), and b) that we're not talking about a set of simultaneous events (so why call it "space"?)

And why should I even care how the circular ruler is represented in spacetime,

One reason why you should care about these things is that they're needed to justify the terminology. This isn't just "semantics". I still feel like these calculations that "prove the non-euclidean geometry of the rotating frame" are the equivalent of claiming that pigs can fly and then redefining "fly" until the statement is true.

A circular ruler is by the way, at any time in the center-of-mass inertial frame, represented by a circle, which has a coordinate independent proper length 2*pi*r.

if the ruler is at rest in my rotating frame, and I read off what it measures.

Is there any reason why you wouldn't describe this result as obtaining the wrong result because the rulers have been deformed from their rest shapes?

 

[yuiop]

It might help to return to the numerical example to clarify a few points:

Experiment:
Disk radius = 1 light second in the non rotating frame.
Instantaneous velocity of a point of the rim of the rotating disk is 0.8c clockwise, relative to an observer just outside the disk in the non rotating frame.
Gamma = 1/0.6 = 1.666666

Disk rotating with a rim velocity of 0.8c:

Circumference = 2*pi*r = 2*pi*1.00 = 6.28 lightseconds (Non rotating observer.)
Circumference = 2*pi*r*gamma = 2*pi*1.00*1.67= 10.47 lightseconds. (Observer on the disc.)

Disk not rotating.

Circumference = 2*pi*r = 2*pi*1.67 = 10.47 lightseconds (Non rotating observer.)
Circumference = 2*pi*r*gamma = 2*pi*1.67*1.00 = 10.47 lightseconds. (Observer on the disc.)

Note that the observer on the disc always measures the circumference as 10.47 lightseconds. This is the proper length of the circumference whether measured by rulers or local radar measurements. When the disc is spinning the non rotating observer sees the disc circumference length contracted to 6.28 lightseconds. I am assuming the Born rigid rotation method I described in post #78.

Here you seem to be neglecting that the co-moving rulers get Lorentz contracted...

No, the assumption of length contraction was built in.

I understand your concern that the location of two ends of the tape measure wrapped around the disc do not seem to be measured simultaneously because of the spiral path that a point on the rim of the disc takes through spacetime. However, this an artifact of the Einstein synchronisation method, It is easy to establish that spatial distance around the rim of the disc does in fact have a dt of zero by placing a single clock at the location where the two ends of the tape measure overlap each other. Since the two ends are at the same location in space and time there should be no question that the two ends are in fact measured simultaneously. This makes more sense when the clocks on the rim are synchronised by a single common signal from the centre of the disc.

 

[atyy]

I still feel like these calculations that "prove the non-euclidean geometry of the rotating frame" are the equivalent of claiming that pigs can fly and then redefining "fly" until the statement is true.

No, it's like redefining "pig" until the statement is true

 

[Fredrik]

No, it's like redefining "pig" until the statement is true

I approve of this post.

Disk rotating with a rim velocity of 0.8c:

Circumference = 2*pi*r = 2*pi*1.00 = 6.28 lightseconds (Non rotating observer.)
Circumference = 2*pi*r*gamma = 2*pi*1.00*1.67= 10.47 lightseconds. (Observer on the disc.)

I'd rather say it like this:

Circumference = 2*pi*r*gamma = 2*pi*1.00*1.67= 10.47 lightseconds. (Lots of observers on the disc who add their results)[/color]

Disk not rotating.

Circumference = 2*pi*r = 2*pi*1.67 = 10.47 lightseconds (Non rotating observer.)
Circumference = 2*pi*r*gamma = 2*pi*1.67*1.00 = 10.47 lightseconds. (Observer on the disc.)

Did you really mean to have the radius shrink by a factor of gamma when the disc is given a spin (from being initially at rest)?

I am assuming the Born rigid rotation method I described in post #78.

Ah, but this has very little to do with an actual solid disc, and I don't think the term "Born rigid" is appropriate here (but I'd have to think that through to be sure).

I understand your concern that the location of two ends of the tape measure wrapped around the disc do not seem to be measured simultaneously because of the spiral path that a point on the rim of the disc takes through spacetime.

That isn't my concern at all.

It is easy to establish that spatial distance around the rim of the disc does in fact have a dt of zero

Sounds like you're considering a circle of radius r in the hypersurface of constant time coordinate. That's what I'm doing. That's the curve that has coordinate independend proper length 2*pi*r, which is not equal to the sum of the results of lots of distance measurements by co-moving rulers on the edge.

 

[yuiop]

Did you really mean to have the radius shrink by a factor of gamma when the disc is given a spin (from being initially at rest)?

Yes, that is what I was hinting at when I earlier said "put another way the radius gets smaller when the disc is given a spin".

I agree the Born rigid rotation method is artificial, and the shrinking of the radius will not happen spontaneously or naturally. It is a bit like accelerating a rocket from rest to a new relative velocity. The clocks onboard the rocket will not stay synchronised naturally and we have to articificially resync the clocks again. If we desire that the speed of light should remain constant and isotropic to observers onboard the rocket we have to fiddle with the clocks. If we desire the proper circumference to remain constant on the disc we have to fiddle with the radius. Nature does not care about our desires.

Ah, but this has very little to do with an actual solid disc, and I don't think the term "Born rigid" is appropriate here (but I'd have to think that through to be sure).

Even better, consider a solid rotating cylinder. Spinning a solid cylider up while trying to maintain a constant radius will produce enormous stress in the surface of the cylinder even if centrifugal/centripetal forces did not exist and the cylinder will inevitably tear itself apart as its circumference tries to length contract. The nearest example in nature is a spinning neutron star which is held together at very high rotational velocities by gravity, but the gravity makes that example difficult to analyse and distill out the purely kinematic aspects.

If you insist on the radius remaining constant when the solid cylinder is spun up, then if the circumference was 10.47 lightseconds when the cylinder was not rotating, then the circumference would be measured as 2*pi*r*gamma = 2*pi*1.67*1.67 = 17.52 light seconds by observers on the surface of the cylinder, when the cylinder is rotating with a rim velocity of 0.8c.

That isn't my concern at all.

Well I am now not sure what your concerns are. Perhaps the best way forward would be for you to describe what practical method, the observers onboard the disc, would use to measure the circumference as being simply 2*pi*r.

I think most people on this forum would agree that science is more about what you would measure than about what is "really" happening. I have described several measurement methods, but none of them come up with a circumference of 2*pi*r for the proper circumference of a rotating disc.

I'd rather say it like this:
"Circumference = 2*pi*r*gamma = 2*pi*1.00*1.67= 10.47 lightseconds. (Lots of observers on the disc who add their results)"

Or (One observer on the disc with one long tape measure.)


Pigs fed and watered and limbering up ready for takeoff.

 

[A.T.]

I assume the rulers are supposed to be "at rest" in the rotating frame.

That is the point.

What can that possibly mean other than that they're at rest in a sequence of co-moving inertial frames?

Still true, but irrelevant, because I want to consider just one frame: the rotating one

If that's what they are, then they're Lorentz contracted.

In which frame are they Lorentz contracted? Obviously not in the rotating frame, in which they are at rest. Objects at rest are not Lorentz contracted.

Is there any reason why you wouldn't describe this result as obtaining the wrong result because the rulers have been deformed from their rest shapes?

The rulers are at rest and so they preserve their rest lengths.

One reason why you should care about these things is that they're needed to justify the terminology.

The justification for calculating this non-Euclidean geometry in a rotating frame and calling it "spatial" is the same justification as for calculating the Centrifugal/Coriolis-forces and calling them "forces", in classical mechanics:

I want to do calculations in the rotating frame, but use the physics laws that were designed for inertial frames.

 

[Fredrik]

OK. After thinking some more, I agree that it makes sense to say that the circumference is measured to be 2*pi*gamma*r in the rotating frame. I'll try to explain why. A classical theory is a set of statements that makes predictions about results of experiments. A mathematical structure (like Minkowski spacetime) can never define a theory by itself. The theory is defined by a set of axioms that tells us how to interpret the mathematics as predictions about results of experiments. This means that even if we have an operational procedure (like using a tape measure) that associates a number with dimensions of length with the circumference of a disc, it isn't possible to relate this to a mathematical quantity in the theory unless there's an axiom that describes how to do that. The theory may not consider what we have done to be a measurement. Because of that, it doesn't make much sense to discuss these things without properly defining the theory first. The theory I have in mind when I use the term "special relativity" is defined by the following three axioms:

1. Physical events are represented by points in Minkowski spacetime. (A consequence of this is that motion is represented by curves, and this suggests the definition of a "particle" as a system the motion of which can be represented by exactly one curve).
2. A clock measures the proper time of the curve in Minkowski spacetime that represents its motion.
3. A radar device measures infinitesimal lengths in the following way: If the roundtrip time is T, then cT/2 is the approximate proper length of the spacelike geodesic from the midpoint of the timelike geodesic through the emission event and the detection event to the reflection event. The approximation becomes exact in the limit T→0. (I haven't found a way to say this that isn't really awkward).

Actually these just define a framework in which we can define classical special relativistic theories of matter and interaction (in several different ways). I won't go into details about those things here. Note that I could have chosen to define 3 in a different way:

3'. A radar device moving as represented by a timelike geodesic measures lengths in the following way: If the roundtrip time is T, then cT/2 is the proper length of the spacelike geodesic from the midpoint of the worldline between the emission event and the detection event to the reflection event.

With 3', we have a theory that's at least as worthy of the name "special relativity" as anything Einstein could have written down in 1905, but it doesn't make any prediction at all about the circumference of the disc in the rotating frame. This theory simply doesn't tell us how to make measurements with non-inertial measuring devices. This is of course exactly why we should prefer 3 over 3'. If we put lots of tiny radar devices along the edge of the rotating disc, have them measure the distance to the next device, and then add up the results, the total will clearly be 2*pi*gamma*r (in the limit of infinitely many radar devices).

My first thought was that it doesn't make sense to call the result obtained this way "the measured circumference in the rotating frame". I thought that it made no sense to describe the sum of many measurements made by measuring devices in different states of motion as the result of a single measurement in a frame where all devices have constant spatial coordinates. But then I realized that this is exactly what we do when we claim to have used axiom 3 to measure something (non-infinitesimal) in an inertial frame. All the measuring devices have the same velocity, but not the same world lines, so we're definitely adding up results from measuring devices in different states of motion. If we allow ourselves to say that we have measured a non-infinitesimal length in an inertial frame (using axiom 3 rather than 3'), then we have no reason not to allow ourselves to say that we have measured non-infinitesimal lengths in the rotating frame.

What I did before is the equivalent of using 3' for measurements in inertial frames, and 3 only for measurements in non-inertial frames. But if we include 3 in the definition of the theory, we don't need 3'.

I still disagree that it makes any sense to interpret this as a non-euclidean spatial geometry, because the term "spatial geometry" can only refer to the geometry of a hypersurface of points that are all assigned the same time coordinate. Such a hypersurface is flat, and the circumference of the disc in that hypersurface is 2*pi*r. This is a coordinate independent proper length of a closed curve.

 

[Fredrik]

In which frame are they Lorentz contracted?

In the center of mass inertial frame. When you describe these things from that frame, the reason why the result is 2*pi*gamma*r is that the rulers are Lorentz contracted, and therefore give you a result that isn't the proper length of the closed curve that defines the circumference of the disc in "space".

The rulers are at rest and so they preserve their rest lengths.

You keep talking about the rulers being at rest in the rotating frame, but you also keep ignoring that they would be crushed by the centrifugal force. (The whole disc would of course be messed up, but the deformations of the measuring devices are more important since they're what we use to measure things). You also haven't said anything that indicates that any of your ideas are based on an actual definition of the theory. To avoid the centrifugal forces, we're going to have to make the rulers inertial, and it's very far from clear that measurements made using inertial rulers can be called a measurement in the rotating frame. This is something that would have to be derived from the axioms of the theory (and first you would have to specify what the axioms are).

The justification for calculating this non-Euclidean geometry in a rotating frame and calling it "spatial" is the same justification as for calculating the Centrifugal/Coriolis-forces and calling them "forces", in classical mechanics:

I don't follow you at all here, but I haven't ruled out that it's because I'm tired and need to get some sleep.

 

[A.T.]

To avoid the centrifugal forces, we're going to have to make the rulers inertial,

Fine, for practical purposes let's make them inertial and let them meet at the circumference for a moment. In the rotating frame they will be at rest in that moment, so they will measure the correct circumference in the in the rotating frame.

The justification for calculating this non-Euclidean geometry in a rotating frame and calling it "spatial" is the same justification as for calculating the Centrifugal/Coriolis-forces and calling them "forces", in classical mechanics:
I want to do calculations in the rotating frame, but use the physics laws that were designed for inertial frames.

I don't follow you at all here, but I haven't ruled out that it's because I'm tired and need to get some sleep.

It is very simple:

Let's take Newtons 1st & 2nd law. They are very useful and simple, but hold only in inertial frames and fail in rotating frames. So you have two choices:

1) Never use rotating frames for calculations

2) Assume Centrifugal/Coriolis-forces in the rotating frame and still use Newtons 1st & 2nd law.

That trick works fine as long as omega * r << c, otherwise you need to assume more things to keep your simple "inertial frame physics" working. One of these things is a non-Euclidean spatial geometry.

The question if the spatial geometry is really non-Euclidean here is analogous to the question if those inertial forces are really forces. Some say "No, it is just a math-trick", others say "If it walks like a duck and quacks like a duck, let's call it 'duck'". I personally don't really care.

 

[bcrowell]

I still disagree that it makes any sense to interpret this as a non-euclidean spatial geometry, because the term "spatial geometry" can only refer to the geometry of a hypersurface of points that are all assigned the same time coordinate.

This is what I originally thought about this example, but actually it's incorrect. Rindler has a nice description of this in ch. 9 of Relativity: Special, General, and Cosmological. You can define two properties: stationary and static. Stationary means there's a timelike Killing vector. Static means that in addition, there's no rotation (i.e., no Sagnac effect). In a stationary case, you have a preferred time coordinate, which is defined by placing a master clock at some point in space, sending out a carrier wave from that clock, and adjusting all other clocks at other points so as to eliminate Doppler shifts, i.e., calibrating them so that they agree with the master clock on the frequency of the sine wave. In the static case, you can also globally match the phase of the master clock, and you have Einstein synchronization, and this Einstein synchronization is independent of your choice of where to place the master clock. Our example is stationary but not static. Because of the symmetry of the problem, you probably want to put the master clock at the center. Then the moving clocks will all be running at different rates. Clocks at the same theta are Einstein-synchronized, but clocks at different thetas aren't.

If you do all this, a surface of simultaneity is simply a light-cone centered on an event at the axis. A cone has zero intrinsic curvature.

However, the spatial geometry determined by obervers using co-moving radar rulers is not the same as the (flat) spatial geometry obtained by restricting to that surface of simultaneity. The reason is that if you orient a radar-ruler in the azimuthal direction, you are measuring the spatial separation between events that are Einstein-synchronized, whereas the surface of simultaneity isn't Einstein-synchronized azimuthally.

Such a hypersurface is flat, and the circumference of the disc in that hypersurface is 2*pi*r. This is a coordinate independent proper length of a closed curve.

This is correct, but restriction to the hypersurface doesn't correspond to the geometry measured by co-moving observes with radar rulers.

 

[yuiop]

It occurs to me that we can carry out an adaptation of A.T's angle measurement on a more local scale on the disc. This is the set up:

Two points A and B are locations on the rim of the disc and the distance from A to B is a small fraction of the total circumference of the disc. Two theodolites are placed at A and B and lined up to view each other and measure the angles. The actual angles are not important here. Once the theodolites are lined up accurately, they are welded so that they can no longer be adjusted. The theodolites are then swapped with each other and it will be noticed that they no longer line up with each other. By such a method the inhabitants of the inside of a cylinder will be able to establish that light paths are not isotropic in their world and if they are clever enough, maybe even figure out they are rotating, even if they do not have any view outside their cylindrical world and even if they do not have access to the entire inner circumference. The main cause of the difference in the angular measurements is abberation. They could also shine laser beems at each other. It would be noticed that they can block one beem at a time conclusively proving the two light paths do not follow the same path.

 

[bcrowell]

By such a method the inhabitants of the inside of a cylinder will be able to establish that light paths are not isotropic in their world and if they are clever enough, maybe even figure out they are rotating, even if they do not have any view outside their cylindrical world and even if they do not have access to the entire inner circumference.

This is basically a measurement of the Sagnac effect, which is certainly a method that works for detecting by local measurements whether you're in a rotating frame.

The restriction to angle measurements, rather than distance measurements, is not a useful one, as I pointed out in #77: Any method of angular measurement can also be used to determine distances. Macroscopic, solid rulers will not work (because Born rigidity is kinematically impossible), but radar rulers will.

 

[Fredrik]

I read a couple of pages of Rindler's book (the stuff surrounding (9.26) (which is the metric I'm including below)). He talks about a rotating lattice, rather than a rotating disc. The substitution \phi\rightarrow\phi-\omega t puts the metric in the form

ds^2=(1-r^2\omega^2)\bigg(dt-\frac{r^2\omega}{1-r^2\omega^2}d\phi\bigg)^2-dr^2-\frac{r^2}{1-r^2\omega^2}d\phi^2-dz^2

This is still just the metric of Minkowski spacetime expressed in a funny way, but then he says that "the metric of the lattice is the negative of the last three terms". This is an interesting comment. The only way I can make sense of it is this: Instead of considering Minkowski spacetime, he decides to consider another manifold. Its underlying topological space is \mathbb R\times[0,\infty)\times[0,2\pi)\times\mathbb R with the topology induced by the topology on \mathbb R^4. Note that the simplest definition of Minkowski spacetime uses \mathbb R^4 as the underlying topological space, so we can say that we're dealing with a proper subset of Minkowski spacetime. The coordinate systems are defined to be the coordinate systems of Minkowski spacetime restricted to that proper subset. The metric is defined to have components

g_{00}=1-r^2\omega^2,\ g_{11}=1,\ g_{22}=\frac{r^2}{1-r^2\omega^2},\ g_{33}=1

g_{\mu\nu}=0 \mbox{ when }\mu\neq\nu

in the coordinate system defined by the inclusion map for that subspace. (I:M&#039;\rightarrow\mathbb R^4,\ I(x)=x,\ \forall x\in M&#039;)

I don't have a problem with saying that "space" in this spacetime has a non-euclidean geometry. It just seems so incredibly pointless to introduce a different spacetime when the one we had was just fine. Think of the definition of SR in my post #88. SR is a theory about one specific spacetime, Minkowski spacetime. What Rindler is really saying here (without actually saying it) is that there's another theory that can reproduce the predictions of SR when we're dealing with uniform rotation.

I don't think this is very interesting. Some of you guys (and the sources you've referenced) have been talking about this stuff as if this is the correct way of treating rotation in SR, but it isn't even SR.

 

[DrGreg]

I read a couple of pages of Rindler's book (the stuff surrounding (9.26) (which is the metric I'm including below)). He talks about a rotating lattice, rather than a rotating disc. The substitution \phi\rightarrow\phi-\omega t puts the metric in the form

ds^2=(1-r^2\omega^2)\bigg(dt-\frac{r^2\omega}{1-r^2\omega^2}d\phi\bigg)^2-dr^2-\frac{r^2}{1-r^2\omega^2}d\phi^2-dz^2

This is still just the metric of Minkowski space expressed in a funny way, but then he says that "the metric of the lattice is the negative of the last three terms".

I'm struggling to make sense of this myself, but you need to read what Rindler says before this, at the start of section 9.6. From what I can gather, the first term in the above metric takes account of the moving lattice's simultaneity (which isn't dt=0). Or, to put it another way, the radar distance measurement of infinitesimally close lattice points implies a pair of events for which the first term vanishes, I think (?).

 

[A.T.]

SR is a theory about one specific spacetime, Minkowski spacetime.

I rather see it the other way around: Minkowski spacetime is one possible geometrical interpretation of SR.

What Rindler is really saying here (without actually saying it) is that there's another theory that can reproduce the predictions of SR, when we're dealing with uniform rotation

Another theory would imply different predictions. But so, it is at most a different geometrical interpretation.

Some of you guys (and the sources you've referenced) have been talking about this stuff as if this is the correct way of treating rotation in SR, but it isn't even SR.

Most sources rather indicate that historically this problem motivated the development of GR. So it might be correct that this is not SR anymore.

 

[atyy]

I don't think this is very interesting. Some of you guys (and the sources you've referenced) have been talking about this stuff as if this is the correct way of treating rotation in SR, but it isn't even SR.

To mix metaphors, aren't there many ways to slice a pig, or parts of a pig? It's just some noninertial coordinates covering a piece of Minkowski spacetime.

Edit: Actually, I'm a bit puzzled on reading Wiki's http://en.wikipedia.org/wiki/Born_coordinates which says that there Langevin observers are not hypersurface orthogonal, so what is the hypersurface that Rindler has defined?

Edit: I took a look at Poisson's http://www.physics.uoguelph.ca/poisson/research/agr.pdf . It's not exactly relevant, since it deals with timelike geodesics (section 2.3), and my guess is the Langevin observers are not geodesic. However, he says that although there are congruences that are not hypersurface orthogonal (section 2.3.3), these congruences still have a "transverse metric" which is purely "spatial" in the limited sense that it is "locally spatial" (section 2.3.1, that last term is mine, see his notes for the real maths).

 

[bcrowell]

I don't think this is very interesting.

Nobody can force you to be interested. Einstein, historically, thought it was interesting -- in fact, he considered it a crucial example that helped him to develop GR. Ehrenfest thought it was interesting. Gron and Rindler thought it was interesting enough to put it in their textbooks. Dieks thought it was interesting enough to write a paper about. Rizzi and Ruggiero thought it was interesting enough to edit an entire anthology on the topic. I think it's interesting, and so do A.T., kev, and others who have kept this thread going for nearly 100 posts. But if you don't think it's interesting, that's up to you. Diff'rent strokes.

Some of you guys (and the sources you've referenced) have been talking about this stuff as if this is the correct way of treating rotation in SR,

I don't think anybody said it was "the" correct way to treat rotation in SR. I'm sure there are many different ways of looking at it. As far as correctness, please feel free to point out anything you think is correct in the treatments by Rindler, Dieks, and Gron -- but the last time I asked you this, you said it was correct but not interesting.

but it isn't even SR.

Historically, Einstein considered it interesting as a bridge from SR to GR. Whether a treatment like Rindler's or Dieks' is SR or GR depends on your point of view, and on how you define SR. From the perspective of the year 2010, my opinion is that the most reasonable way to define SR is that it deals with 3+1 spacetime that's flat. By that definition, this example is SR, because the underlying 3+1 spacetime is flat. Historically, there was some doubt about whether accelerating observers could be incorporated into SR, but I think the clear answer from a modern point of view is that they can.

The coordinate systems are defined to be the coordinate systems of Minkowski space restricted to that proper subset.

No, this is incorrect. See #91.

I'm struggling to make sense of this myself, but you need to read what Rindler says before this, at the start of section 9.6. From what I can gather, the first term in the above metric takes account of the moving lattice's simultaneity (which isn't dt=0). Or, to put it another way, the radar distance measurement of infinitesimally close lattice points implies a pair of events for which the first term vanishes, I think (?).

Yeah, this is pretty much right. The point about the moving lattice's simultaneity is subtle, since there's no way of doing an Einstein synchronization over any finite area. However, all you need for a radar-ruler measurement of distance is Einstein synchronization on a line segment, and since that doesn't enclose any area, you're OK. I personally think Dieks' treatment is a lot more transparent than Rindler's, and I would suggest starting there. Rindler's is more general and abstract. The thing that confused me at first was that I thought there was a surface of simultaneity, which would appear as an equation of constraint that would eliminate one variable from the metric. That's incorrect, for the reasons outlined in #91. If you read Dieks' treatment, I think it's more clear what's going on. There's no variable whose differential is dt in the rotating frame. You just substitute in for dt, which mathematically represents doing a local Einstein synchronization between the two ends of a radar-ruler of length dl. So this is the only thing that I think might not be quite right in your quote above. In the last sentence, it's not that you're making the time term vanish, it's that you're eliminating the variable dt.

 

[Fredrik]

I rather see it the other way around: Minkowski spacetime is one possible geometrical interpretation of SR.

If we use my definitions of the terms "theory" and "special relativity", those statements don't make much sense. So your definitions must be different than mine.

Another theory would imply different predictions.

Since I define a theory as a set of statements that makes predictions about results of experiments*, I have to consider two different sets of statements that lead to the same predictions different but equivalent theories.

*) That's just a part of it actually. I'll post the full definition if someone requests it.

From the perspective of the year 2010, my opinion is that the most reasonable way to define SR is that it deals with 3+1 spacetime that's flat.

Yes, I agree. (See #88 for a more complete definition).

By that definition, this example is SR, because the underlying 3+1 spacetime is flat.

As I said in my previous post, I can't interpret Rindler's statement that way. Gron appears to be considering a hypersurface in Minkowski space that consists of a bunch of spirals, but Rindler appears to be considering a different spacetime altogether. I think the "space" part of the spacetime Rindler is describing must be isomorphic to the submanifold of Minkowski space that Gron is considering. So it appears that Gron is doing SR, but ends up using a very weird terminology, like using the term "circumference" about the proper length of a spiral and the term "space" about a hypersurface of events that aren't assigned the same time coordinate by the coordinate system we're using. Rindler is not doing SR, but avoids the terminology issues.

Historically, there was some doubt about whether accelerating observers could be incorporated into SR, but I think the clear answer from a modern point of view is that they can.

The real issue is whether we can describe a way to perform measurements of proper length using non-inertial measuring devices. Axiom 3 in #88 takes care of that, so we didn't really need to study the rotating disc problem to understand accelerating "observers".

No, this is incorrect. See #91.

I don't think you understood what I meant. In order to define a manifold, you have to specify a set, a topology on that set, and all the coordinate systems that we're allowed to use. That's what I was doing. #91 is about something else entirely.

OK. After thinking some more, I agree that it makes sense to say that the circumference is measured to be 2*pi*gamma*r in the rotating frame.

I think I may have to retract this comment, or at least elaborate a bit. In order to get this to be true, we either have to use the term "circumference" about the proper length of a spiral (or an even uglier curve), or define the "rotating frame" as the spatial part of an entirely different spacetime. (See my previous two posts). It "makes sense" to redefine "fly" or "pig" to make the statement "pigs can fly" true, but that doesn't mean that we should. SR allows us to do these things, but doesn't give us a reason why we should.

 

[bcrowell]

The coordinate systems are defined to be the coordinate systems of Minkowski space restricted to that proper subset.

I don't think you understood what I meant. In order to define a manifold, you have to specify a set, a topology on that set, and all the coordinate systems that we're allowed to use. That's what I was doing. #91 is about something else entirely.

I think I did understand what you meant, and what you meant was incorrect. In the first quote, you're defining a space consisting of a subset of Minkowski space. That isn't what you want to do in this example, because there is no global time synchronization, and therefore no natural way to pick such a subset, and it doesn't define the metric you want by the usual process of restricting a space to a lower-dimensional subspace. To define the spatial manifold successfully, you just want to define the set as the set of coordinates (r,\theta), the same way you would do if you were just going to talk about ordinary plane polar coordinates. The reason I can tell that you're not understanding this correctly is that you keep on talking about spirals. The only reason to talk about spirals is if you're imagining this as a process where you induce a new metric by restricting the parent space to a surface of simultaneity. All of your previous objections (that the subspace is flat) make perfect sense with this approach. That's why it's not a fruitful approach.