Lorentz Contraction Circular Motion

[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'\rightarrow\mathbb R^4,\ I(x)=x,\ \forall x\in M')

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.

 

[yuiop]

I have found another worm to add to the can. To a non rotating observer the radius of the disc is r or 1 lightsecond in the numerical example. To an observer on the rotating disc, the radar measured radius (R) is r/gamma = 0.6 lightseconds. This makes the situation even less Euclidean to the disc observer, because by his measurements of circumference and radius using radar measurements, the circumference is 2*pi*R*gamma^2.

Fredrik, you have still failed to provide a practical method by which an observer on the disc will measure the proper circumference as being simply 2*pi*r and when you do find a method, I predict it will not have the desirable properties of transitive time synchronisation, isotropic speed of light, same distance measured in either direction, a local speed of light of c, etc, etc.

 

[atyy]

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).

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.

I just read Gron's treatment, which is identical to Rindler's as far as I can tell, except he gives a picture of the surface, and also states in the preceding chapter "... there does not exist a single space of simultaneity encompassing the "rest spaces" of all observers in an arbitrary reference frame. In this sense the 3-dimensional space described by the spatial metrical tensor is local." So I suppose this is all very opaque from the SR point of view, but apparently is a useful precursor to thinking about things in GR, just like Fredrik's favourite index notation for tensors in SR

 

[Fredrik]

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.

I was really confused by the first comments, but the last one shows that you have definitely misunderstood what I'm doing. I'll try to explain it in a different way. Minkowski spacetime can be defined in the following way:

* Choose the set \mathbb R^4.
* Choose the standard topology.
* Choose the standard coordinate systems (i.e. all C^\infty injective functions from \mathbb R^4 into \mathbb R^4)
* Choose the metric to have components \eta_{\mu\nu} in the coordinate system I, where I is the identity map on \mathbb R^4.

What I'm doing is to define a different spacetime manifold M':

* Choose the set \mathbb R\times [0,\infty)\times [0,2\pi)\times\mathbb R, i.e. the range of the specific coordinate system on Minkowski spacetime that we've been considering.
* Choose the topology inherited from \mathbb R^4.
* Choose the coordinate systems to be the same as the ones used in the construction of Minkowski spacetime, but restricted to this smaller subset of \mathbb R^4.
* Do not choose the metric that would be inherited from Minowski spacetime if we thought of this as a submanifold (it would have the same components as the Minkowski metric), but instead choose the metric that has 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 I, where I is now the restriction of the identity map of \mathbb R^4 to M'.

Edit: This may not be enough. Maybe we should use [0,2\pi] instead of [0,2\pi), and then identify the line \phi=0 with the line \phi=2\pi.

The reason I can tell that you're not understanding this correctly is that you keep on talking about spirals.

Not sure if that means that I've misunderstood something, or if it means that you have.

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.

Then we agree about that. I thought this was the approach taken by Grøn & Hervik in their book. Are you saying it isn't?

I thought I made it clear that this is how I interpreted what they were doing, and no one has objected against that until now. A.T. even defended calling the spiraling hypersurface "space".

As I've been saying, Rindler appears to be doing something different. To be more precise, he appears to be doing what I described for the second time earlier in this post.

 

[atyy]

Concerning the curved "spatial" hypersurface, are these statements true?
(i) it is everywhere locally orthogonal to the rotating observers
(ii) it is not a spacelike hypersurface, which is what disqualifies the rotating observers from being "hypersurface orthogonal" in the technical sense

 

[Fredrik]

If we're talking about the "spiral" hypersurface of Minkowski spacetime, then (i) is true and (ii) is not. (That's actually implied by (i)). I think it's also true for the alternative approach that I've been describing (as my interpretation of what Rindler is doing), but I'm not 100% sure about that one.

 

[atyy]

If we're talking about the "spiral" hypersurface of Minkowski spacetime, then (i) is true and (ii) is not. (That's actually implied by (i)). I think it's also true for the alternative approach that I've been describing (as my interpretation of what Rindler is doing), but I'm not 100% sure about that one.

Sorry, just to clarify, did you mean that (a) the spiral surface is not spacelike or (b) the spiral surface is spacelike?

BTW, I think Rindler and Gron are doing the same thing. Gron's 4.70, 4.71 and 4.76 are the same as Rindler's equations.

Edit: I see you meant that (b) the spiral surface is spacelike. So is Wiki wrong that the Langevin observers are not hypersurface orthogonal?

 

[Fredrik]

Yes, I meant that it's spacelike.

I don't know what a Langevin observer is. I haven't been reading all the articles and books that have been referenced. Just a couple of pages here and there.

Rindler and Grøn/Hervik are using the same definition of the rotating coordinate system. That's why they get the same result when they express the Minkowski metric in that coordinate system. But this doesn't mean that they're doing the same thing. They might be doing the same thing. Right now I'm not sure what either of them are doing. I think the choices are a) to take the last three terms to be the metric of a spacelike hypersurface of Minkowski spacetime, which consists of a bunch of spirals, and b) to define a new spacetime with a metric that has the same components in a cartesian coordinate system that the Minkowski metric has in rotating polar coordinates, and then consider the 3-dimensional submanifold (of the new spacetime) that's defined by a constant 0th coordinate.

 

[Fredrik]

If no one says anything in like two days, does that mean I won?

 

[yuiop]

If no one says anything in like two days, does that mean I won?

Nope, it just means we are waiting for you to answer this question:

Fredrik, you have still failed to provide a practical method by which an observer on the disc will measure the proper circumference as being simply 2*pi*r ...

As I understand it, the proper length of something, is a measurement that all observers will agree on and yet the learned members of this forum do not seem to be able to agree on what that measurement would be.

 

[DrGreg]

The "spiralling surface of simultaneity" (in spacetime) that some people have referred to doesn't exist.

Yes, if you consider simultaneity around a rotating circular ring, you get a spiral. And you can join lots of "concentric spirals" together to get a surface. But the surface you get coincides with the local definition of simultaneity (viz. of comoving inertial observer) only in the tangential direction and not in the radial direction. In the radial direction, a line of simultaneity would have to be horizontal (in the usual way of drawing things, with the centre of the circle's worldline vertical). The spirals don't join together "horizontally". It's not just a "global" problem of a discontinuity after a complete revolution, there are discontinuities around smaller loops, which vanish only when the loops shrink to zero.

So it's impossible to come up with a coordinate system in which

1. all points in the spinning disk have constant spatial coordinates
2. each surface of constant time coordinate coincides with every comoving inertial observer's local definition of simultaneity

In case you can't picture this geometrically, look at the maths of this. If (t,r,\theta) are inertial polar coordinates with the disk centre at r=0, and (t',r',\theta') are rotating polar coordinates which are supposed to meet the two criteria above (we can ignore the third spatial dimension), by considering a Lorentz transformation we require:

<br /> \begin{array}{rcl}<br /> dt&#039; &amp; = &amp; \gamma(r) (dt - r^2\omega\,d\theta / c^2)\\<br /> r\,d\theta&#039; &amp; = &amp; \gamma(r) (r\,d\theta - r\omega\,dt)\\<br /> dr&#039; &amp; = &amp; dr<br /> \end{array}<br />​

where

\gamma(r) = \frac{1}{\sqrt{1 - r^2 \omega^2 / c^2}}​

There is no solution to these equations.

For example, the second equation above is another way of saying

<br /> \begin{array}{rcl}<br /> \partial \theta&#039; / \partial \theta &amp; = &amp; \gamma(r)\\<br /> \partial \theta&#039; / \partial t &amp; = &amp; \gamma(r) \omega\\<br /> \partial \theta&#039; / \partial r &amp; = &amp; 0<br /> \end{array}<br />​

So you can't find the 3D space-metric of the lattice points as a restriction of the 4D Minkowski spacetime-metric, as no suitable coordinates exist.

Distance along any space-curve has to be determined by "local radar" between infinitesimally close points, which you then integrate to get the macroscopic length. Local radar implies two events that Einstein-simultaneous relative to the lattice points that are deemed to stationary in our frame. That means that in the equation quoted by Fredrik in post #94, the first term is zero, between any neighbouring events, and that's why the other terms represent the space-metric for the lattice points. Just don't bother trying to extend this 3D space-metric to a 4D spacetime-metric, because (as far as I can tell) that wouldn't mean anything useful.

 

[A.T.]

A.T. even defended calling the spiraling hypersurface "space".

I rather meant: Even if the 3D-spatial geometry as measured by rulers at rest in the rotating frame looks weird in 4D space-time, this doesn't stop me from calling it "spatial".

 

[Fredrik]

Nope, it just means we are waiting for you to answer this question:

Fredrik, you have still failed to provide a practical method by which an observer on the disc will measure the proper circumference as being simply 2*pi*r

He could set up a bunch of radar devices along the edge and have them all measure the distance (i.e. cT/2) to the next radar device at the precise moments when they receive the same spherical light signal from the point at the center, and then divide the results by \gamma before he adds them up (to compensate for the fact that each radar measurement gives him the proper length of a curve in spacetime that doesn't end where the next curve begins).

If you think this is somehow less valid than simply adding up the length measurements performed by a sequence of co-moving rulers (or radar devices) to 2\pi\gammar, then I request that you show me how this follows from a definition of "special relativity", or a reasonable definition of "length" or "circumference". See #88 for my definition of SR, and post your own if you don't like mine. The reason I consider my "measurement" a more appropriate representation of "the circumference in the rotating frame" is that it gives us the proper length of a continuous closed curve (a circle) in a hypersurface of constant time coordinate in the rotating frame. (Note that the rotating frame is defined with the same time coordinate as the inertial frame that's co-moving with the center).

As I understand it, the proper length of something, is a measurement that all observers will agree on and yet the learned members of this forum do not seem to be able to agree on what that measurement would be.

To me "proper length" is a mathematical concept that doesn't have anything to do with measurements until we have included an axiom in the theory that tells us how the two are related. If you prefer to use the term differently, that's not a real problem (except for you, who would have to think of another term for the mathematical concept), but you should at least agree that the issue here isn't what numbers the measuring devices are "displaying" to us, but how those numbers are related to things in the mathematical model.

 

[Fredrik]

The "spiralling surface of simultaneity" (in spacetime) that some people have referred to doesn't exist.

Yes, if you consider simultaneity around a rotating circular ring, you get a spiral. And you can join lots of "concentric spirals" together to get a surface. But the surface you get coincides with the local definition of simultaneity (viz. of comoving inertial observer) only in the tangential direction and not in the radial direction. In the radial direction, a line of simultaneity would have to be horizontal (in the usual way of drawing things, with the centre of the circle's worldline vertical). The spirals don't join together "horizontally". It's not just a "global" problem of a discontinuity after a complete revolution, there are discontinuities around smaller loops, which vanish only when the loops shrink to zero.

So it's impossible to come up with a coordinate system in which

1. all points in the spinning disk have constant spatial coordinates
2. each surface of constant time coordinate coincides with every comoving inertial observer's local definition of simultaneity

Yes, we have to drop number 2. I think what we should do is to define a new "time" coordinate

t&#039;=t-\frac{r^2\omega}{1-r^2\omega^2}\phi

Note that when \phi=0, we have t=t'. So the hypersurface of constant t' can be described as follows. Start with a straight line from the center to the radius, in a hypersurface of constant t. The hypersurface of constant t' is the union of all the spirals through that line, that have the property that at every event E on the spiral, the tangent of the spiral is Minkowski orthogonal to the world line of the point in the disc whose world line goes through E.

These spirals are "going up faster" near the edge than near the center. I can't really tell if this implies that the hypersurface of constant t' has positive or negative curvature.

 

[heldervelez]

I saw no reference to the 2002 paper of Rizzi & Ruggiero "Space geometry of rotating platforms: operational approach" at arxiv:gr-qc/0207104v2 13 Sep 2002
with their position and a review of methodologies, and at page 6:

"(s2) both the radius and circumference contract, so that their ratio remains 2*Pi (f.i. Lorentz, Eddington)"

...

 

[yuiop]

"(s2) both the radius and circumference contract, so that their ratio remains 2*Pi (f.i. Lorentz, Eddington)"

...

That should be qualified by "as measured in the non-rotating frame".

What the ratio would be measured as, by an observer in the rotating frame is still being debated here.

 

[yuiop]

... Suppose that you have two cars with odometers. I think the odometers match up with your notion of d\ell defined by laying down rulers. The cars also have clocks on their dashboards. You send one car out around the disk in the clockwise direction, and the other in the counterclockwise direction. When the cars meet up on the far side of the disk, their clocks will be out of sync due to the d\theta&#039; 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. Someone who doesn't like your laying-down-rulers definition can say, "See? I told you that definition would lead to no good!"

I just did some quick calculations and concluded that a car going clockwise around the rim of the rotating disk will not return to the start point at the same time as a car that starts out at the same time and goes anti-clockwise, when both cars have the same velocity relative to the rim. Since they are both returning to the same point on the rim, there should be no issues about how clocks are synchoronised and they either return at the same time or not in any reference frame. The conclusion seems a bit shocking and counter intuitive and maybe I made a mistake. Can anyone confirm or refute this observation?

... Suppose that you have two cars with odometers. I think the odometers match up with your notion of d\ell defined by laying down rulers...

Actually, the odometers will read d\ell\ \gamma where \gamma = \sqrt{(1-u^2/c^2)} and u is the velocity of the car relative to the disc rim. In other words in linear example, two cars traveling from point A to point B at different velocities (where A and B are at rest wrt each other), will read different distances on their mechanical odometers and this will therefore not agree with laying down of rulers method. We analysed how a relativistic wheel rolls in an old thread and for each complete revolution of the wheel moving with linear velocity u relative to a road, the wheel moves forward a distance x in the frame comoving with the wheel axis and a distance x/ \gamma in the frame that the road is at rest in.

 

[DrGreg]

I just did some quick calculations and concluded that a car going clockwise around the rim of the rotating disk will not return to the start point at the same time as a car that starts out at the same time and goes anti-clockwise, when both cars have the same velocity relative to the rim. Since they are both returning to the same point on the rim, there should be no issues about how clocks are synchronised and they either return at the same time or not in any reference frame. The conclusion seems a bit shocking and counter intuitive and maybe I made a mistake. Can anyone confirm or refute this observation?

I agree with your conclusion and can explain it without any calculation.

To ease the explanation, I'd like to consider a slight modification. Instead of a circular disk, consider a tightly-wound helical rod. By "tightly-wound" I mean the distance between two adjacent turns is tiny, we can consider it negligible compared with the radius (and ultimately we can consider the limit as the distance drops to zero and the helix "collapses" to a circle). The helix is rotating about its axis. The reason I want to introduce this is so that I can unambiguously refer to "simultaneity along the rod", determined by local Einstein-simultaneity between pairs of nearby points and then "daisy chaining" along the rod. By this definition, the simultaneity is the same as if it were unwound into a straight line and moving along its length. Clearly the rod-simultaneity is not the same as the simultaneity of an inertial observer fixed on the axis, because of the relative motion.

Consider two points on the helix which are exactly one circumference apart on the rod: the points are right next to each other in 3D space. It should be clear that the local 3D definition of simultaneity disagrees significantly with daisy-chained-round-the-rod-simultaneity for these two points. (When the two points are a negligible distance apart we can talk of absolute local simultaneity instead of relative simultaneity.)

So now we consider kev's experiment above adapted to my helix. Two cars start at the same point and each travel one revolution in opposite directions at the same speed relative to the rod. They end up two turns apart on the helix, but right next to each other. It should be clear that they reach their destinations rod-simultaneously. (Consider the equivalent journey on an unwound rod; they travel the same distance, one circumference, at the same speed.) But as we've seen, rod-simultaneous is not locally-simultaneous for points that are as good as coincident in 3D space, so the cars don't actually get to the destination at the same event, as kev asserted.

 

[Fredrik]

I have another explanation. Suppose the two cars both start their laps at the same event A. We either imagine that they pass right through each other on the opposite side, or that the disc is so large that we can neglect the little detour that one of them has to make to avoid a collision. Let B be the event where they meet up for the second time after they separated. (The first is the near collision). Now look at the events from the inertial frame that's co-moving with the center. Let t be the time between event A and event B. Since both cars have been traveling for t seconds, they will have traveled the same distance if and only if they've been traveling at the same speed. But the relativistic velocity addition law implies that they haven't. (Because of that, they can't be at the starting point on the disc at B, so this isn't exactly the scenario that kev described).

 

[DrGreg]

(Because of that, they can't be at the starting point on the disc at B, so this isn't exactly the scenario that kev described).

It's pretty much the same scenario, because if they get to the starting point at two different times then (assuming the one who gets there first doesn't stop but keeps on going) they will meet somewhere other than the starting point i.e. after unequal distances measured on the disk.

 

[yuiop]

I have another explanation. Suppose the two cars both start their laps at the same event A. We either imagine that they pass right through each other on the opposite side, or that the disc is so large that we can neglect the little detour that one of them has to make to avoid a collision. Let B be the event where they meet up for the second time after they separated. (The first is the near collision). Now look at the events from the inertial frame that's co-moving with the center. Let t be the time between event A and event B. Since both cars have been traveling for t seconds, they will have traveled the same distance if and only if they've been traveling at the same speed. But the relativistic velocity addition law implies that they haven't. (Because of that, they can't be at the starting point on the disc at B, so this isn't exactly the scenario that kev described).

One small problem with this cut down analysis, is that it might lead a casual reader to the wrong conclusion. For a clockwise rotating disc, a clockwise going car returns first in your scenario, while in my scenario, the anti-clockwise going car returns first (still a clockwise rotating disc).

This leads on to an interesting alternative view of the circumference of a rotating disc. Since light and cars going clockwise take longer to go around the disc than light and and cars going anticlockwise (for a clockwise rotating disc) there is a sense that the clockwise circumference is longer than the anticlockwise circumference of the same clockwise rotating disk. Inhabitants of the disc might place mileage signs on roads saying things like "New York to Old Town 50 miles" and "Old Town to New York 10 miles" pointing in opposite directions. This might be pleasing to those who desire the speed of light to be the same in all directions, but the fact that it requires different size rulers to prove that, might not be convincing to everyone. This directional dual definition of length might be in agreement with distances defined by measuring angles using theodlites, but that is a bit complicated to work out.

Of course we could just stick to the usual cT/2 radar measurement of length and just accept that the speed of light is anisotropic in a non inertial frame over non infitesimal distances. Inhabitants of a rotating frame would be aware of the strange behavior of light in there system by observing that a photon going from A to B does not pass a photon going from B to A, as they follow two distinct spatial paths. To clarify what I mean, I have shown the light path AB in red and the light path BA in blue on a clockwise rotating disc in the attached sketch. (This is the viewpoint of observers on the disc.)

He could set up a bunch of radar devices along the edge and have them all measure the distance (i.e. cT/2) to the next radar device at the precise moments when they receive the same spherical light signal from the point at the center, and then divide the results by \gamma before he adds them up (to compensate for the fact that each radar measurement gives him the proper length of a curve in spacetime that doesn't end where the next curve begins).

It seems that all the observer does by "compensating" is work out what the disc circumference would be from the point of view of a non rotating observer. This is a bit like an observer in an accelerating rocket in in flat space time claiming that since he knows he is accelerating he should adjust the proper length measurement, of his rocket because he must be length contracting. It also requires the disc observer to have knowledge of his rotational velocity to work out out proper lengths in in his reference frame. In SR, the observer in a reference frame does not need any knowledge of his velocity to work out distances in his own reference frame.

One disadvantage of your compensated length measurement is that is the rotational velocity of the disk is changed to a new constant rotational velocity, all the road mileage markings have to be changed. The same applies to my dual directional method of defining circumferential distances. The simple uncompensated cT/2 method means that any road marking can remain permanent and are accurate even when the rotational velocity of the disc is changed. It only requires that the rulers remain unstressed along their length before and after any change. In SR, the proper length of a rocket remains unchanged before and after an acceleration period using either the cT/2 radar method or the layed down rulers method and surely the same should apply in the case of the disc?

If you think this is somehow less valid than simply adding up the length measurements performed by a sequence of co-moving rulers (or radar devices) to 2\pi\gammar, then I request that you show me how this follows from a definition of "special relativity", or a reasonable definition of "length" or "circumference". See #88 for my definition of SR, and post your own if you don't like mine.

Well your definition in #88 seems fine and I quote it again here:

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).
...
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'.

Neither method 3 nor 3' as defined by you, mention that the observer needs to work out his velocity and compensate by gamma to work out proper lengths in his own reference frame.

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...

Consider the following practical demostration. Draw the worldlines of points A and B moving at constant and equal velocity in a straight line in flat space on a time space diagram. Indicate the proper length distance between A and B taking simultaneity into account on the same diagram. Now roll the paper the diagram is drawn on into a cylinder. Essentially nothing has changed and the methods for calculating proper length in Minkowski spacetime apply equally in the rotating disc case. I know you do not actually need to draw the diagram, so count it as a thought experiment.