The rotating disk of Albert Einstein

[PAllen]

And to clarify, there is nothing wrong with speaking of the geometry of spatial surface defined in some rotating coordinates (with a 4-metric given for the coordinates, and 3-metric derived for any chosen coordinates of the spatial surface). Where you get into trouble is assuming such a surface can always be constructed so as to say anything meaningful about spatial geometry of an object described by a timelike congruence. For not sufficiently well behaved congruences, there is no possible spatial geometry for the 'object' that has any expected meaning. And what is surprising, is that a simple rotating congruence is already a case that is ill behaved in this regard.

 

[Nana Dutchou]

Do you have a reference for this definition of "proper geometry"? I remind you that the rules of this forum require that everything discussed here must stem from the current professional literature. So you really need to provide a professional reference for this "proper geometry" concept or the discussion will be closed per the rules.

I can not give you a reference but I think it is a definition intuive even in the context of classical physics.


#In classical physics all observers observe the same spatial distances between pairs of simultaneous events.
Are you agreement DaleSpam?


#In classical physics each observer possesses a unique three-dimensional physical space which is a particular famiy of world lines. This unique three-dimensional physical space is the set of fixed points according to the observer.
Are you agreement DaleSpam?


#To define a geometry on a three-dimensional physical space we have to allocate a numerical value (a length) to each of its segments of curves (mathematically, a segment of curve defined on this space is not a world line segment but it can be represented by a particular family of world line segments).
Mathematically we can define several euclidean geometries on a unique three-dimensional physical space, and we can also define non-euclidean geometries on the same space.
Are you agreement DaleSpam?


#In classical kinematics we can easily build a coherent family of world lines that are not a set of fixed points according to a unique observer. Such a coherent family of world lines is not a three-dimensional physical space but it can mathematically be used to perform a spatial location of events.
Are you agreement DaleSpam?


Cordially.

 

[WannabeNewton]

#In classical physics each observer possesses a unique three-dimensional physical space which is a particular famiy of world lines.

The point is in special relativity this is no longer true-there is no unique family of simultaneity surfaces available to a given observer. It depends on the choice of simultaneity or synchronization convention. Of course for inertial observers there is a natural choice of such space-like hypersurfaces and these are just the 4-velocity orthogonal hyperplanes built from Einstein synchronization.

Coming back to the observers on the rotating disk, we can use local Einstein simultaneity between neighboring observers on the disk in order to construct a (spatial) metric relative to these observers. This clearly characterizes the local (spatial) geometry of the disk relative to these observers and tells us the disk curvature is hyperbolic under this metric. We can integrate this metric along different curves on the disk and obtain values for its circumference and radius and find $C > 2\pi R$ but what we're doing is adding up the value of the (spatial) metric on the local rest space of each observer on the disk along the chosen curve of integration. We are not defining the entire spatial shape of the disk at an "instant of time" relative to these observers using local Einstein simultaneity because this is impossible.

Of course this doesn't stop us from defining the entire spatial shape of the disk at an "instant of time" relative to these observers using the simultaneity surfaces of the inertial observers but as already noted this will just give us back the overall flat circular shape of the disk because all we've done is synchronize the standard clocks at rest on the disk with the inertial clock at the center by changing them to coordinate clocks. This isn't an interesting alternative by any means but it is practical.

 

[JVNY]

Can I ask a simplistic question about this paper:

I think it is worth linking the paper Pervect provided in a related thread. . .

http://arxiv.org/abs/gr-qc/9805089

The relevance here is the clear argument in section two, of the paradox that results if you try to treat the disc as having any fixed spatial geometry. That is, if you assume that measuring the circumference is providing you with information about a fixed spatial geometry of the disc. . .

How does the analysis in the paper differ from the typical explanation, which states that the receptor moves during the light's transit? Here is an example of a typical explanation:

Clearly the pulse traveling in the same direction as the rotation of the loop must travel a slightly greater distance than the pulse traveling in the opposite direction, due to the angular displacement of the loop during the transit. As a result, if the pulses are emitted simultaneously from the “start” position, the counter-rotating pulse will arrive at the "end" point slightly earlier than the co-rotating pulse. http://mathpages.com/rr/s2-07/2-07.htm​

The typical explanation seems to assume that there is a fixed circumference that is the same regardless of the transit direction. The light flash travels a longer path in one direction than the other because the receptor moves, not because of the geometry of the circumference.

The linked paper also concludes at one point that "there are two different traveled distances for the two light rays" (page 27), but seemingly for a different reason. At one point it states that the difference in transit times is due to "the nonuniformity of time on the rotating platform, and in particular to the 'time lag' arising in synchronizing clocks along the rim" (page 20). At another point its seems to reject the view that "the length of a round trip is related to a univocally defined geometric object" (page 14).

But then, in the example of the traveler the paper concludes on page 30 that the same number of rods fits the circumference when laid down traveling in either direction, and on page 32 that "the circumference of the rotating disk can be considered as a geometrically well defined entity, with a well defined length in the reference frame of the disk" (although it could also be considered not to be well defined, but rather direction dependent).

Thanks.

 

[Dale]

I can not give you a reference

Then we cannot discuss it here. If you do find a modern professional reference then please start a new thread on the topic and we can discuss that reference.

#In classical physics all observers observe the same spatial distances between pairs of simultaneous events.
Are you agreement DaleSpam?

Yes. They also agree on which pairs of events are simultaneous and on the time interval between non simultaneous events.

#In classical physics each observer possesses a unique three-dimensional physical space which is a particular famiy of world lines. This unique three-dimensional physical space is the set of fixed points according to the observer.
Are you agreement DaleSpam?

No, the 3D space of Newtonian mechanics is not a family of world lines. Particles don't have worldlines in any meaningful sense in Newtonian mechanics. They have positions which evolve over time. Also, it doesn't make sense to talk about fixed points in a 3D space since there is no time evolution for a single 3D space and therefore all points are fixed.

#To define a geometry on a three-dimensional physical space we have to allocate a numerical value (a length) to each of its segments of curves (mathematically, a segment of curve defined on this space is not a world line segment but it can be represented by a particular family of world line segments).
Mathematically we can define several euclidean geometries on a unique three-dimensional physical space, and we can also define non-euclidean geometries on the same space.
Are you agreement DaleSpam?

Again, it doesn't make sense to talk about worldlines inside 3D space. I also disagree about defining different geometries on the space. You can define different coordinate charts, but the geometry is the same regardless of the coordinates.

#In classical kinematics we can easily build a coherent family of world lines that are not a set of fixed points according to a unique observer. Such a coherent family of world lines is not a three-dimensional physical space but it can mathematically be used to perform a spatial location of events.
Are you agreement DaleSpam?.

I think I agree. I think that you are saying that in Newtonian physics you could have a coordinate chart which was smoothly varying as a function of time and that you could use constant coordinates to define the location of moving objects.