# The rotating disk of Albert Einstein

• Nana Dutchou
In summary, Albert Einstein discusses the idea of a non-Euclidean three-dimensional space in his book "Relativity: The Special and the General Theory." He explains how an observer on a rotating disc, who experiences a centrifugal force, can interpret it as a gravitational field according to the general theory of relativity. The observer conducts experiments with clocks and measuring rods, but faces difficulties in defining space and time coordinates due to the effects of rotation on these objects. This leads to the conclusion that Euclidean geometry does not hold in a gravitational field.
DaleSpam said:
Yes, but what if they are not spinning at the same rate?
Then you get different hyperbolic geometries and therefore different amounts of paint needed. The amount of paint needed depends only on the disc radius and its spin rate.

A.T. said:
Then you get different hyperbolic geometries and therefore different amounts of paint needed. The amount of paint needed depends only on the disc radius and its spin rate.
Paint can be strained also. The amount of paint depends on its strain.

DaleSpam said:
Paint can be strained also.
Assume it is not.

A.T. said:
Assume it is not.
Then that needs to be part of the definition. As I said above that is a problematic part of your proposed definition and needs to be addressed.

DaleSpam said:
Then that needs to be part of the definition. As I said above that is a problematic part of your proposed definition and needs to be addressed.
The paint amount calculation wasn't meant to be a definition, just an example for why the hyperbolic geometry is a useful definition of "spatial geometry" of a rotating disc.

A.T.,

What you are really saying, I think, is that the amount of paint is uniquely specified as follows:

Paint is applied somehow (doesn't matter) such that each disc observer (described by one of the world lines of the specified congruence) sees the paint as uniform right near them and of a specified color saturation.

I would agree, given the congruence and the other parameters, this is a unique definition for amount of paint. However, I disagree that it says anything about disk spatial geometry 'from a rotating disc' perspective. What it specifies is a summation over a collection of disjoint local frames, each with a separate locally orthogonal space-time slicing. Each such disc observer would claim the paint is non-uniformly applied at any distance from them, using the extension of local orthonormal frame.

If you pick a different definition of how to apply the paint, then it will vary in saturation when observed even locally.

A.T. said:
The paint amount calculation wasn't meant to be a definition, just an example for why the hyperbolic geometry is a useful definition of "spatial geometry" of a rotating disc.
Then we are back to post 20 and have just wasted the last 22 posts. I still don't know what "proper geometry" is. As far as I know it is not a standard term in the literature.

DaleSpam said:
I still don't know what "proper geometry" is.
For the rotating disc I would associate it with the hyperbolic geometry. But I didn't introduce it, so I will let the OP explain.

I'm afraid I'm not following a lot of the thread. If the OP can comment whether his idea is a personal theory, or has already been discussed in the literature (for instance one of the papers in Gron) it would be helpful. (Though if it's a personal theory it might not be fit for discussion at PF).

I'm not following AT's ideas on the "proper geometry", like Dale - though the idea sounds interesting.

I think it is worth linking the paper Pervect provided in a related thread. I could not find it linked in this 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.

(This argument was very enlightening for me - I had never seen it before; it is so much more compelling than (correct) statements about a spatial slice not being orthogonal to the congruence, therefore not representing a meaningful geometry for the object; and that no such geometry can be defined, because the required slices can't exist).

PAllen said:
I would agree, given the congruence and the other parameters, this is a unique definition for amount of paint. However, I disagree that it says anything about disk spatial geometry 'from a rotating disc' perspective.
To me, a useful definition of "spatial geometry of the object in its rest frame", is one that would (for example) allow to determine the amount of paint needed to cover the object.

PAllen said:
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.
I understand that you cannot apply standard Special Relativity to a rotating disc.
PAllen said:
That is, if you assume that measuring the circumference is providing you with information about a fixed spatial geometry of the disc.
Since the amount of paint is fixed over time, I would say that it is determined by a "fixed spatial geometry of the disc".

Nana Dutchou said:
How does one do to demonstrate that the proper geometry of the rotating disk is not euclidean?
WannabeNewton said:
There is no such thing as "proper" geometry.
DaleSpam said:
What is a "proper geometry"?
DaleSpam said:
I have never heard the term "proper geometry" before. In keeping with the usual meanings of proper time and proper length, I would guess that it is the geometry in the object's rest frame. So I think that may be related to what you are saying A.T.
However, one thing that I see as problematic in your "amount of material" bit is that the amount of material depends on the strain that the structure is under.
DaleSpam said:
Then we are back to post 20 and have just wasted the last 22 posts. I still don't know what "proper geometry" is. As far as I know it is not a standard term in the literature.
Defining a geometry on a three-dimensional physical space consists in attributing an intrinsic measure to each of its segments of parametrized curves and the proper geometry of this physical space is the one which reports the character superposable or not superposable of its segments of parametrized curves (which can be paths of rays of light) by a simple comparaison of their measures.

To define a geometry on a three-dimensional physical space (each point of this space is a world line of an observer) we have to allocate a numerical value (a length) to each of its segments of curves. Mathematically we can define several euclidean geometries on a unique three-dimensional physical space. We can also define non-euclidean geometries this space! It's easy! But this space has only one proper geometry (knowing that we can change the choice of the standard of lengths).In classical physics each observer possesses a unique three-dimensional physical space on which there is a unique proper geometry and this proper geometry is assumed to be euclidean (this is an intuitive assumption of classical physics).
Are you agreement dear all?In special relativity each inertial coordinate system possesses a unique three-dimensional physical space on which there is a unique proper geometry and this proper geometry is assumed to be euclidean (this is an mathematical assumption of special relativity).
Are you agreement dear all?Mathematically, a segment of parametrized curve (a segment of curve) defined on a three-dimensional physical space is not a segment of world line segment. Each point of this segment of curve is a world line and mathematically, it can be represented by a particular family of world line segments.
Are you agreement dear all?In his text of the post #1, Eintein Albert wants to establish that we can use the theory of special relativity to show that:
(i) We can highlight a non-inertial obserser which possesses a unique three-dimensional physical space.
(ii) The proper geometry of this three-dimensional physical space is not an euclidean geometry.
Are you agreement dear all?
pervect said:
I'm afraid I'm not following a lot of the thread. If the OP can comment whether his idea is a personal theory, or has already been discussed in the literature (for instance one of the papers in Gron) it would be helpful. (Though if it's a personal theory it might not be fit for discussion at PF).
I'm not following AT's ideas on the "proper geometry", like Dale - though the idea sounds interesting.
In his text, Albert Einstein supposes that he is able to specify the equation (in an inertial coordinate system) of each point of the unique three-dimensional physical space of the rotating observer (the equation of each world line which constitutes this three-dimensional space).
Albert Einstein supposes that it is enough to copy out the equations which are used in classical physics to describe a three-dimensional physical space in rotation with respect to a cartesian and inertial coordinates system.
In the post #1 and #2 at the beginning of this thread I explain that a physicist can argue with that.
Are you agreement dear all?The metric tensor of general relativity is mathematically defined to allocate a numerical value (a length) to each segment of world line which represents a trajectory of matérial body and such a segment is the trajectory of a body in free fall if its measure is optimal.
In his text, Albert Eintein did not define such a metric tensor to show that the three-dimensional physical space of the rotating observer possesses a non-euclidean proper geometry.
Are you agreement dear all?
Nana Dutchou said:
I wrote: why try to verify the accuracy or the inaccuracy of the relation circumference = diameter * pi ?
WannabeNewton said:
It is not inaccurate. The spatial metric geometry of the disk relative to the inertial observers obeys ##C = \pi D## whereas the spatial metric geometry of the disk relative to observers at rest on the disk obeys ##C > \pi D## when the disk is rotating. Neither is any more correct than the other because spatial metric geometry is relative.

Indeed, the proper geometry of a three-dimensional physical space is relative to this three-dimensional physical space ! Mathematically, we cannot compare a segment of parametrized curve defined on a three-dimensional physical space E and a segment of parametrized curve defined on a different three-dimensional physical space E' because mathematically, a point of E is not a point of E' (in other words, because these segments of curves are not represented by the same families of world lines segments). Each theory in physics can only make assumptions to compare the proper lengths of these segments of curves.

Albert Einstein is working with two separate three-dimensional physical space:
- The first is the family of world lines of observers constanly at rest in an inertial coordinate system.
- The second is the family of world lines of observers constantly at rest on the rotating disc.

Each of these three-dimensional physical space has its own proper geometry (its own spatial metric geometry you said !). The theory of relativity assumes that the proper geometry of the first is euclidean and Albert Einstein wants to show that we can infer from the special relativity that the proper geometry of the second is not euclidean.
Mathematically we can define several geometries on a each three-dimensional physical space.
Are you agreement dear all?

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Nana Dutchou said:
the proper geometry of this physical space is the one which reports the character superposable or not superposable of its segments of parametrized curves (which can be paths of rays of light) by a simple comparaison of their measures.
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.

Nana Dutchou said:
Mathematically we can define several euclidean geometries on a unique three-dimensional physical space. We can also define non-euclidean geometries this space! It's easy! But this space has only one proper geometry (knowing that we can change the choice of the standard of lengths).
I assume that the reference that you will provide also demonstrates that the "proper geometry" is unique. It certainly is not clear to me.

Nana Dutchou said:
Are you agreement dear all?
No. How can there be agreement on the subject of "proper geometry" without an accepted definition?

A.T. said:
To me, a useful definition of "spatial geometry of the object in its rest frame", is one that would (for example) allow to determine the amount of paint needed to cover the object.
But there is no such rest frame. And the amount of paint computed is the sum of amounts in a collection of different rest frames. You can pretend otherwise, but it it is still pretending.
A.T. said:
I understand that you cannot apply standard Special Relativity to a rotating disc.
It seems you missed the whole argument. It is that assuming measurements made in a collection of disjoint rest frames adds up to a measurement of spatial geometry leads to a severe logical contradiction. You end up being able to disprove SR. It is not a question of applying standard SR to a rotating disc. The disproof is ameliorated only by letting go of the idea that a collection of independent local measurements can be treated as a geometry of the whole. Changing from summing rulers to summing paint changes absolutely nothing about the logic of the problem. Yes, there is a circumferential measurement that can be made, and a paint measurement that can be made. But it leads to a contradiction to assume these are measurements of a static geometry.
A.T. said:
Since the amount of paint is fixed over time, I would say that it is determined by a "fixed spatial geometry of the disc".

Then you fall into the paradox described.

A.T. said:
I understand that you cannot apply standard Special Relativity to a rotating disc.

Sure you can. You just can't assume that "spatial geometry" is always Euclidean, or that that concept even has a unique meaning. But you don't need those assumptions to apply SR. The congruence of worldlines describing the rotating disk is perfectly well defined, and can be described using standard SR.

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.

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DaleSpam said:
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.

Nana Dutchou said:
#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.

PAllen said:
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.

Nana Dutchou said:
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.

Nana Dutchou said:
#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.

Nana Dutchou said:
#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.

Nana Dutchou said:
#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.

Nana Dutchou said:
#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.

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