Lorentz Contraction Circular Motion

[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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