Lorentz Contraction Circular Motion

[Fredrik]

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

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.

3 implies that we have to do that, if we're talking about a circle in a hypersurface of simultaneity of the rotating coordinate system. I still don't see a reason why anyone would use the term "the circumference of the disc in the rotating coordinate system" about something completely different, like the proper length of a spiral in Minkowski spacetime, or the arc length of a circle in a 3-dimensional Riemannian manifold that isn't a hypersurface of Minkowski spacetime. (I have been told that this manifold is actually the quotient space of Minkowski spacetime and the set of world lines of points on the disc. I'm expecting to receive more information about that soon).

 

[Demystifier]

I said a lot on this stuff more than 10 years ago in
http://xxx.lanl.gov/abs/gr-qc/9904078 [Phys.Rev. A61 (2000) 032109].
I hope someone may still find it usefull.

 

[yuiop]

I said a lot on this stuff more than 10 years ago in
http://xxx.lanl.gov/abs/gr-qc/9904078 [Phys.Rev. A61 (2000) 032109].

All quotes below are from your paper:

If the disc is constrained to have the same radius r as the same disc when it does not rotate, then L is not changed by the rotation, but the proper circumference L0 is larger than the proper circumference of the nonrotating disc. This implies that there are tensile stresses in the rotating disc.

Correct, there would be tensile stresses in the rotating disc, if the radius is constrained to be the same as when it is not rotating.

However, there is something wrong with this standard resolution of the Ehrenfest paradox.

I don't think there is.

Consider a slightly simpler situation; a rotating ring in a rigid nonrotating circular gutter with the radius r = r0. ...
This means that an observer on the ring sees that the circumference is L' = \gammaL. The circumference of the gutter seen by him cannot be different from the circumference of the ring seen by him, ...

I beg to differ. Let us consider the numerical example I mentioned in an earlier post. A ring of radius 1 lightsecond rotating clockwise with an instantaneous rim velocity of 0.8c. Here I will assume the rotating and non-rotating observers measure the radius to be the same. Using your statement "This means that an observer on the ring sees that the circumference is L' = \gammaL" the circumference of the ring measured by the observer on the ring is 2*pi*r*gamma = 10.4719. Now to measure the circumference of the gutter, the observer on the ring would note a mark on the gutter and time how long it takes the mark to complete a revolution and multiply the time by the relative velocity of the gutter to him. This method uses one clock so we can avoid argument about how spatially spearated clocks on a ring are synchronised. The result is 2*pi*r/gamma = 3.7699. I am dividing by gamma because the time measured by the observer on the ring is reduced by time dilation of his clock. The circumference of the gutter seen by him is not the same as the circumference of the ring seen by him, as you claim.

... so the observer on the ring sees that the circumference of the relatively moving gutter is larger than the proper circumference of the gutter, ...

The observer sees the circumference of the relatively moving gutter as 3.7699 and the proper circumference of the gutter is 2*pi*r = 6.2831 so your above statement is simply wrong.

... whereas we expect that he should see that it is smaller. This leads to another paradox.

It does not lead to another paradox, because your assumptions and calculations are wrong in the first place.

Note that the observer on the ring sees the gutter as length contracted and the observer on the gutter sees the ring as length contracted, exactly as relativity predicts. No paradox.

 

[Demystifier]

The observer sees the circumference of the relatively moving ring as 3.7699 and the proper circumference of the gutter is 2*pi*r = 6.2831 so your above statement is simply wrong.

The ring completely fills the gutter, so one observer should see that the two circumferences are equal. That's why I disagree with you.

 

[yuiop]

The observer sees the circumference of the relatively moving ring as 3.7699 and the proper circumference of the gutter is 2*pi*r = 6.2831 so your above statement is simply wrong.

The ring completely fills the gutter, so one observer should see that the two circumferences are equal. That's why I disagree with you.

I made a typo in my last post. I should have said:

The observer (on the ring) sees the circumference of the relatively moving gutter as 3.7699 and the proper circumference of the gutter is 2*pi*r = 6.2831

(The edits are in bold.)

To summarise what the various observers measure:

Observer on the rotating ring measures:
Circumference of the ring = 10.4719
Circumference of the gutter = 3.7699

Observer on the non rotating gutter measures:
Circumference of the ring = 6.2831
Circumference of the gutter = 6.2831

From the above measurements, it can be seen that only the observer on the gutter measures the circumference of the rotating ring and the non rotating gutter to be the same. The observer on the rotating ring does not see it that way. I admit that does seem slightly unintuitive, but that is relativity for you.

 

[Fredrik]

Looks like you're both confusing "an observer on the ring" with "a rotating observer". The coordinate system we would associate with the motion of a little guy riding the ring is a co-moving inertial frame. The ring isn't even circular in such a frame. A "rotating observer" on the other hand, isn't an observer at all. In operational terms, it's a bunch of measuring devices spread out all over the ring, and mathematically, it isn't represented by a coordinate system, but by a frame field.

 

[A.T.]

A "rotating observer" on the other hand, isn't an observer at all.

I guess the rotating observer would disagree with that.

In operational terms, it's a bunch of measuring devices spread out all over the ring,

Isn't every observer who observes more than his local surrounding a "bunch of spread out measuring devices"?

and mathematically, it isn't represented by a coordinate system, but by a frame field.

What prevents the rotating observer from using a rotating coordinate system.

 

[Fredrik]

I guess the rotating observer would disagree with that.

If you mean a guy riding on the disc/ring/whatever, then no, he wouldn't, as he only performs local measurements which agree with the coordinate assignments of a co-moving inertial frame. The only coordinate systems that can be naturally associated with his motion are co-moving inertial frames.

Isn't every observer who observes more than his local surrounding a "bunch of spread out measuring devices"?

You could say that, but there's definitely a coordinate system (a co-moving inertial frame) that we can associate with an inertial observer's motion and orientation (his notion of "right", "forward" and "up"). In addition to that, all of those other devices would agree with that coordinate system about all lengths and times.

What prevents the rotating observer from using a rotating coordinate system.

He could use any coordinate system he wants of course, but the measurements of the "bunch of spread out measuring devices" won't agree with the lengths assigned by the rotating coordinate system, as I have pointed out many times in this thread. The rotating coordinate system assigns the length 2*pi*r to the circumference of the disc at all times, while the measurements add up to 2*pi*gamma*r.

The motion and orientation of each of the measuring devices determine a co-moving inertial frame at each point on its world line. The inertial frame defines an orthonormal basis for the tangent space at that point. Orthonormal bases are bijective with frames at that point. (The technical definition of a "frame" at a point p in a manifold M is a linear function f:\mathbb R^n\rightarrow T_pM, where n=dim M). So the collection of world lines determine a frame at each point on any of the world lines. Hence, we have a "frame field" defined on the region of spacetime that's filled up with world lines of points in the disc.

 

[yuiop]

Looks like you're both confusing "an observer on the ring" with "a rotating observer". The coordinate system we would associate with the motion of a little guy riding the ring is a co-moving inertial frame. The ring isn't even circular in such a frame. A "rotating observer" on the other hand, isn't an observer at all. In operational terms, it's a bunch of measuring devices spread out all over the ring, and mathematically, it isn't represented by a coordinate system, but by a frame field.

I made it clear that I was specifying what an observer on the ring would measure and I do not think I was getting that mixed up with anything else.

You have probably noticed we have narrowed the discussion down to an infinitessimally thin ring rather than a disc, because I am sure we would all agree that there is no natural way for observers at different radii to synchronise clocks with each other and hopefully that will simplify things a bit.

Now we come to issue of specifying a common frame for observers spread out on a ring that are all equidistant from the axis of the ring. In linear SR, observers that are not moving parallel to each other, obviously do not share a common reference frame even if the magnitude of their velocities are equal. However in this situation it is also obvious that these observers do not measurethemselves to be at rest with each other as there spatial separation is continually changing over time. The case of the rotating ring is different because multiple observers on the ring can consider themselves to be at rest with respect to each other, as their mutual spatial separation does not change over time, so it should not be too difficult to construct a common coordinate system for those observers "on the ring".

Using the EP as a guiding principle it might be well to consider the issue from a gravitational point of view. Do you agree that observers equidistant from a central gravitational mass can consider themselves to be sharing a common reference frame? Do you agree that the Schwarzschild coordinate system is a "coordinate system" that suitably describes the measurements of a bunch of observers/measurement devices spread out all over the place at different radii with clocks running at different rates or would you say the Schwarzschild coordinate system or Schwarzschild metric would be better described as the "Schwarzschild frame field"? Please do not assume I am saying you are wrong, because I readily concede your knowledge of the terminology is much greater than mine and I am just trying to understand better what you are saying and maybe learn something.

 

[yuiop]

Here is a observation about rotational motion that may be of interest.

In linear SR, if a series of synchronised clocks spread out along a rod, are accelerated using Born rigid acceleration, the clocks naturally go "out of sync" because the clocks do not follow "parallel" paths through spacetime.

If a ring with a similar series of synchronised clocks spread out on it, is rotationally accelerated, the clocks remain in sync with each other naturally, because all the clocks follow a similar path through spacetime.

In other words, "transitive sychronisation" (peripheral clocks synchronised by a central clock) rather than "Einstein synchronistation" would seem to be the natural condition of rotating clocks. This seems to be an important difference between the linear and circular cases, but I am not quite sure what the significance of that observation is at the moment.

 

[Fredrik]

I made it clear that I was specifying what an observer on the ring would measure and I do not think I was getting that mixed up with anything else.

The observer on the ring only measures an infinitesimal segment at his own location, and you talked about the measured value of the circumference, so you're clearly not just talking about what he is measuring. Also, the ring isn't even shaped like a circle in the coordinate system that we would associate with his motion and orientation.

so it should not be too difficult to construct a common coordinate system for those observers "on the ring".

The common coordinate system is the rotating coordinate system, and it doesn't agree with their measurements.

Do you agree that observers equidistant from a central gravitational mass can consider themselves to be sharing a common reference frame?

Are they in orbit? In free fall? Held at fixed spatial coordinates in the Schwarzschild coordinate system? And what's a "reference frame"? In SR it's usually used synonymously with "inertial frame", which really means "inertial coordinate system". But then we usually only consider observers that are, always have been, and always will be, moving with the same constant velocity. When we're talking about an observer (still in SR) with a world line that isn't a geodesic, the obvious generalization is the concept of local inertial frame, which works in GR too. A local inertial frame in SR is actually just a co-moving global inertial frame.

"Reference frame" doesn't seem to be a well-defined concept to me. Maybe there is a definition, but if there is one, I'm still unaware of it. If we really want to consider a bunch of measuring devices spread out all over the place, then I think we need to be talking about frame fields instead of coordinate systems, for the reasons I've mentioned.

In other words, "transitive sychronisation" (peripheral clocks synchronised by a central clock) rather than "Einstein synchronistation" would seem to be the natural condition of rotating clocks.

Why? It doesn't work all the way round. I also wouldn't call them "rotating clocks". They're just clocks on circular paths (in "space", as defined by the rotating coordinate system, or equivalently, by the inertial frame in which the point at the center is at rest).

 

[Demystifier]

The coordinate system we would associate with the motion of a little guy riding the ring is a co-moving inertial frame. The ring isn't even circular in such a frame. A "rotating observer" on the other hand, isn't an observer at all. In operational terms, it's a bunch of measuring devices spread out all over the ring, and mathematically, it isn't represented by a coordinate system, but by a frame field.

I agree with you.

 

[Demystifier]

I
Observer on the rotating ring measures:
Circumference of the ring = 10.4719
Circumference of the gutter = 3.7699

If, by observer, one means a local observer staying at a single point of the ring, then I disagree. Such an observer observes that the circumference of the ring is equal to the circumference of the gutter.

 

[Demystifier]

The observer on the ring only measures an infinitesimal segment at his own location

That is exactly one of my points in the paper I mentioned above.

Yet, it is not in contradiction with my post #133, because this local observer may also watch the whole ring/gutter, which may also be counted as a sort of measurement. Of course, this is a measurement of a different kind.

 

[Demystifier]

What prevents the rotating observer from using a rotating coordinate system.

He can use any coordinates he wants. However, the particular coordinate system you mention does NOT correspond to his PROPER frame of coordinates, unless he is sitting in the center of rotation.

 

[Demystifier]

The common coordinate system is the rotating coordinate system, and it doesn't agree with their measurements.

More precisely, the rotating coordinate system agrees only with the measurements of the observer sitting in the center of rotation.

What many people here (Fredrik is excluded) do not seem to understand is that rotation is not the same as circular motion. The observer sitting in the center of the rotating disc rotates but does not move circularly. The observer sitting at the rim of the rotating disc both rotates and moves circularly. A gyroscope orbiting around a planet moves circularly but does not rotate.

 

[A.T.]

Held at fixed spatial coordinates in the Schwarzschild coordinate system?

Assume that for example. Can they consider themselves to be sharing a common "reference frame"?

And what's a "reference frame"? In SR it's usually used synonymously with "inertial frame",

It is used inaccurately then. The general term "reference frame" includes "non-inertial reference frames" as well. Rotating frames for example.

"Reference frame" doesn't seem to be a well-defined concept to me.

That is interesting, given the wide use of the term in physics.

 

[Demystifier]

Assume that for example. Can they consider themselves to be sharing a common "reference frame"?

There is a general way to associate a proper reference frame to a local observer moving in a specified way. (See Misner, Thorne, Wheeler, Sec. 13.6)
In particular, observers (static with respect to a Schwarzschild black hole) who sit at DIFFERENT points do not share a common proper reference frame.

 

[A.T.]

Assume that for example. Can they consider themselves to be sharing a common "reference frame"?

There is a general way to associate a proper reference frame to a local observer moving in a specified way. (See Misner, Thorne, Wheeler, Sec. 13.6)
In particular, observers (static with respect to a Schwarzschild black hole) who sit at DIFFERENT points do not share a common proper reference frame.

By "proper reference frame" do you mean "local inertial frame for freely falling and non-rotating observer"? I'm asking about a "reference frame" in general, not specifically an inertial one. The one defined by the Schwarzschild coordinates seems to be widely applied and useful. Is it a "reference frame"?

The concept of "rotating reference frames" seems to be accepted and used in physics as well. Inertial forces were introduced to make them handleable, and Relativity added some extra issues to them. But a rotating reference frame is still a "reference frames" isn't it?

 

[Demystifier]

By "proper reference frame" do you mean "local inertial frame for freely falling and non-rotating observer"?

No, I mean local noninertial frame for arbitrarily moving and rotating observer.

The one defined by the Schwarzschild coordinates seems to be widely applied and useful. Is it a "reference frame"?

It is a reference frame, but not a proper reference frame associated with a local observer.

But a rotating reference frame is still a "reference frames" isn't it?

Yes it is. Moreover, it is the proper reference frame associated with a local observer who, e.g., sits in the center of a rotating disc.

 

[Fredrik]

So a "reference frame" is a synonym for "coordinate system", i.e. a function from an open subset of the spacetime manifold into \mathbb R^4, and a "proper reference frame" is the non-inertial coordinate system that's associated with the motion and orientation of an object, in the most natural way. (The orientation of the object defines an orthonormal basis of the tangent space at each point on the world line. One of the basis vectors is equal to the tangent vector of the curve. The time axis of the coordinate system is the world line, labeled by proper time. A hypersurface of constant time consists of geodesics through a point on the curve, that are orthogonal to the curve at the point where they intersect it. The x,y,z axes are defined as the geodesics in the t=0 hypersurface that have the three remaining basis vectors as tangent vectors. They are labeled by the usual synchronization procedure).

I'm OK with that terminology. No real surprises there.

Note that this coordinate system will not extend very far from the world line. It's only well-defined in a region where the spacelike geodesics I just talked about don't intersect. Also note that "a bunch of measuring devices spread out over the ring/disc" don't define a proper reference frame according to this definition.

 

[Demystifier]

Fredrik, that's all correct.
(Provided that an inertial coordinate system is viewed as a special case of non-inertial coordinate systems.)

 

[Fredrik]

Yes, that's how I meant it. I wrote "non-inertial" mainly because I had previously said that a local inertial frame is the natural coordinate system to associate with a physical observer's motion and orientation, and I wanted to emphasize that I wasn't still talking about that. I guess I did it in a confusing way.

(Yes, I realize that this "proper reference frame" is at least as natural as the local inertial frame).

 

[Demystifier]

I'm glad that I allways eventually get to an agreement with you. :-)

 

[yuiop]

The ring completely fills the gutter, so one observer should see that the two circumferences are equal. That's why I disagree with you.

This is far too simplistic. This is like saying that because the observer at rest with the barn in the "pole and barn paradox" sees the pole completely filling the barn, that the observer on the pole should also conclude that the pole and barn are the same length. In fact the pole rider measures the barn to be shorter than the pole and he reasons that he passes safely through the barn because the front and back doors do not shut simultaneously from his point of view.

In the case of the ring and the gutter, the observer at rest with the gutter does indeed measure the ring and gutter to have the same circumference (analogous to the barn observer) and the observer on the ring measures the gutter to have a smaller circumference than the ring (analogous to the observer on the pole).

One other (not directly related) observation that may be of interest. If the observer in the gutter measures the velocity of the ring to be v relative to the gutter, the observer on the ring will not measure the velocity of the gutter relative to the ring to be exactly v, if he uses transitive (synchronised by a central clock) rather than Einstein clock synchronisation.

 

[heldervelez]

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I will take back the mention to the paper because when I checked again the publication, I realized that I've messed up with other paper with similar title in that Journal (the title also started by NONINVARIANT...and is from Bari Univ." .
My apologies to everyone.
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