# Spin Length Contraction

#### Dedocta

Registered Member
This theory proposes that the contraction of the length of a spinning object is due to the change in the tangential velocity of its surface, as opposed to the traditional explanation of the contraction occurring solely in the direction of rotation.

The Ehrenfest paradox, which highlights a discrepancy between classical and relativistic mechanics, arises from the fact that as an object spins faster, the circumference of its rotating frame contracts, while its radius remains static. This results in an apparent contradiction of the conservation of energy and momentum, as well as the geometry of space-time.

My theory suggests that the contraction occurs along the radial direction, as the radius of the object decreases with the decrease in tangential velocity, while the circumference contracts according to the Ehrenfest Paradox. This new perspective allows for a more intuitive explanation of the paradox, without violating any laws of physics.

The mathematical derivation of the spin length contraction formula is straightforward, involving only the Lorentz factor and the original dimensions of the object.

One possible solution to the Ehrenfest paradox was proposed by Richard Muller in 1991, in which he suggested that the rotating disk should be considered as a collection of rotating rods, rather than a single object. This perspective leads to the conclusion that the circumference of the disk contracts by the same factor as its radius, as the rods change shape due to the centrifugal forces.

However, my proposed solution, based on the contraction of the radius due to the change in tangential velocity, is a more intuitive and straightforward explanation of the paradox. It is consistent with the predictions of special relativity, which have been extensively verified by experiments and observations.

To fully understand the mathematical framework of this theory, we can consider a disk with radius $R_0$ at rest in the laboratory frame. In the frame of reference of the disk, its circumference is given by $2\pi R_0$, and its radius is $R_0$. Now, let us assume that the disk is set into rotation with angular velocity $\omega$. According to the theory of special relativity, the circumference of the disk in the laboratory frame is given by:

$$C = 2\pi R_0 \sqrt{1 - \frac{v^2}{c^2}}$$

where $v = \omega R_0$. From this, we can see that the circumference of the disk has decreased due to its rotation.

Now, let us assume that the radius of the disk has also decreased by the same factor. In other words, the radius in the laboratory frame is given by:

$$R = R_0 \sqrt{1 - \frac{v^2}{c^2}}$$

where $v$ is as defined above. With this assumption, we can calculate the circumference of the disk in its own frame of reference:

$$C' = 2\pi R \sqrt{1 - \frac{v^2}{c^2}} = 2\pi R_0 \sqrt{1 - \frac{v^2}{c^2}} = C$$

Thus, we can conclude that the proposed contraction of the radius of the disk is consistent with the contraction of the circumference.

The proposed solution is just one of several possible explanations for the Ehrenfest paradox. Another solution involves the use of the concept of a rigid body and a modification of the laws of mechanics to account for the rotation of the body. However, this solution has been shown to be inconsistent with special relativity and the observation of high-energy particles. The proposed solution using spin length contraction does not require any modification of the laws of physics and is consistent with special relativity.

The implications of this theory extend beyond the resolution of the Ehrenfest paradox. It may help us better understand the behavior of high-energy particles and their interactions with electromagnetic fields. Furthermore, it could have applications in the design of high-speed rotating machinery, such as turbines and centrifuges.

It is important to note that this theory is not the only proposed solution to the Ehrenfest paradox. Other solutions include the use of non-linear transformations of spacetime and modifications to the laws of physics, such as those proposed by Richard Muller in 1991.

However, the beauty of the spin length contraction theory is its simplicity and its adherence to the fundamental principles of special relativity. The theory provides a clear and intuitive explanation for the paradox, without requiring any modifications to the laws of physics or the geometry of spacetime.

In conclusion, the spin length contraction theory proposes a new perspective on the behavior of spinning objects and provides a possible solution to the Ehrenfest paradox. While more research is needed to fully understand the implications of this theory, it has the potential to deepen our understanding of the fundamental laws of nature and may have practical applications in engineering and physics.

Where I am now, is trying to incorporate this concept with General Relativity, how would this affect Minkowski space?

References:

Ehrenfest, P. (1909). “In what way does it become manifest in the fundamental laws of physics that space has three dimensions?” Proceedings of the Amsterdam Academy, 11, 584-592.

Einstein, A. (1905). “Zur Elektrodynamik bewegter Körper.” Annalen der Physik, 17, 891–921.

Muller, R. A. (1991). “The gyroscope and the equivalence principle.” American Journal of Physics, 59(7), 619-628.

Gron, O. (2009). “The Ehrenfest paradox.” American Journal of Physics, 77(4), 296-297.

Rindler, W. (1969). “Visual horizons in world models.” Monthly Notices of the Royal Astronomical Society, 138, 393-404.

Last edited:
The wording seems really good, like it is just telling everyone what they want to hear.

The math doesn’t appear to be self consistent.

It appears that R = R_o and C = C’ from the math you provided in the third set of equations, but this contradicts the first and second equation.

In Minkowski Spacetime, a prime notation should never equal the same variable without a prime notation. I always prefer to see only one equals sign per line.

It appears you just slapped the proper distance around a circle and called it good. That is not Lorentz Invariant. It would mean there is some warping of spacetime.

My theory suggests that the contraction occurs along the radial direction, as the radius of the object decreases with the decrease in tangential velocity, while the circumference contracts according to the Ehrenfest Paradox.

You don't need to have your own theory, because the so-called Ehrenfest paradox is already solved within Special Relativity theory. SR does not have any actual paradoxes because it is internally consistent. The word "paradox" is only meant to imply that there is an "apparent paradox" even though there is not actually a paradox.

For the Ehrenfest scenario, one must consider how the rotating wheel was constructed. Assuming the wheel is made of only spokes and no rim, then there is nothing to contract, so no paradox. If the wheel must have a rim, we could add the rim while the wheel is already rotating at constant speed. Otherwise the rim would have to stretch like an elastic while the wheel spins up to speed.

You don't need to have your own theory, because the so-called Ehrenfest paradox is already solved within Special Relativity theory. SR does not have any actual paradoxes because it is internally consistent. The word "paradox" is only meant to imply that there is an "apparent paradox" even though there is not actually a paradox.

For the Ehrenfest scenario, one must consider how the rotating wheel was constructed. Assuming the wheel is made of only spokes and no rim, then there is nothing to contract, so no paradox. If the wheel must have a rim, we could add the rim while the wheel is already rotating at constant speed. Otherwise the rim would have to stretch like an elastic while the wheel spins up to speed.

A problem is that people have begun to conflate "paradox" with "self-contradictory". A paradox is a situation that gives the appearance of being self-contradictory or unreasonable, but is actually based on sound logic.

You’re right, something’s wonky with the mathematical formulation. I will edit and fix. The other framework I’ve used is:

Equations for Modified Lorentz Transformation :
Δx' = γ(Δx - vt)
Δy' = γ_t(Δy - v_t*t)
Δz' = Δz
Δt' = γ(Δt - vx/c^2)

Here, v is the relative velocity (utilizing a combined angular and linear motion while maintaining the speed of light,) c is the speed of light, γ is the Lorentz factor, v_t is the tangential velocity of a spinning object and γ_t incorporates solely v_t and no linear velocity

edit: I don’t think I can edit the original post. I’ll work on the framework and respond with a reply!

Last edited:
You’re right, something’s wonky with the mathematical formulation. I will edit and fix. The other framework I’ve used is:

Equations for Modified Lorentz Transformation :
Δx' = γ(Δx - vt)
Δy' = γ_t(Δy - v_t*t)
Δz' = Δz
Δt' = γ(Δt - vx/c^2)

Here, v is the relative velocity (utilizing a combined angular and linear motion while maintaining the speed of light,) c is the speed of light, γ is the Lorentz factor, v_t is the tangential velocity of a spinning object and γ_t incorporates solely v_t and no linear velocity

edit: I don’t think I can edit the original post. I’ll work on the framework and respond with a reply!
By using delta as the change in position along with the Lorentz Factor, the best hope you would have is developing a close approximation.

From what I recently found out on Reddit about this is that it actually inspired Einstein to develop the General Theory of Relativity. That is basically what you would have to derive in order to accurately describe the situation.

This was done by creating tensors or combining different vectors together in a framework. The equations you originally posted was Einsteins first crack at the problem. It doesn’t include things like the vector for it’s forward momentum. That would have to be considered.

One could express the circumference C and diameter D in terms of the uncontracted radius R0:

$C = 2\pi R_0 \sqrt{1 - v^2/c^2}$

$D = 2R_0 \sqrt{1 - v^2/c^2}$

Dividing the first equation by the second, we get:

$C/D = 2\pi R_0 / 2R_0 = \pi$

This shows that the value of pi remains the same in the moving reference frame, despite the Lorentz contractions of the radius and circumference.

One could express the circumference C and diameter D in terms of the uncontracted radius R0:

$C = 2\pi R_0 \sqrt{1 - v^2/c^2}$

$D = 2R_0 \sqrt{1 - v^2/c^2}$

Dividing the first equation by the second, we get:

$C/D = 2\pi R_0 / 2R_0 = \pi$

This shows that the value of pi remains the same in the moving reference frame, despite the Lorentz contractions of the radius and circumference.
Part of Erhenfest’s main arguments was that special relativity didn’t claim to have length contraction in the direction perpendicular to the direction of motion. The idea was that he was not supposed to be able to achieve this result.

It just seems to be a happy coincidence that it turns out that way, and that is what should be expected. I never heard any reports of large gravitational bodies shrinking due to their speed of rotation. They actually bulged outwards at the equator.

General Relativity actually predicted this, even though that may have been unknown to them for sure at the time. That is the reason they theorized that black holes should form into flat disk. That statement alone was viewed as being controversial at the time.

Honestly, I don’t believe I even have the mental capacity to try to wrap my head around how this paradox actually works out. I believe that it may just mean that the centrifugal force used to describe General Relativity becomes the more dominant force in the tensor field, somehow.

I know, it’s a lot! This was my pandemic project, been working on it for 5 years now, trying to disprove it - getting busy with something else at work (biology research) and my boss wants me to drop the physics. So I dropped it here for feedback!

Mike

Part of Erhenfest’s main arguments was that special relativity didn’t claim to have length contraction in the direction perpendicular to the direction of motion. The idea was that he was not supposed to be able to achieve this result.

It just seems to be a happy coincidence that it turns out that way, and that is what should be expected. I never heard any reports of large gravitational bodies shrinking due to their speed of rotation. They actually bulged outwards at the equator.

General Relativity actually predicted this, even though that may have been unknown to them for sure at the time. That is the reason they theorized that black holes should form into flat disk. That statement alone was viewed as being controversial at the time.

Honestly, I don’t believe I even have the mental capacity to try to wrap my head around how this paradox actually works out. I believe that it may just mean that the centrifugal force used to describe General Relativity becomes the more dominant force in the tensor field, somehow.

Does the bulge come from the objects growing over time, or is it a prediction of GR without an increase in mass?

Does the bulge come from the objects growing over time, or is it a prediction of GR without an increase in mass?
It is supposed to be a principle that affects the Earth as the moon orbits around it. That is supposed to be the reason why the Earth shifts chaotically.

The rotation of a black hole is supposed to be one of the only things that allow it to occupy real space. It counteracts the force of gravity.

Length contraction only becomes relevant at speeds close to the speed of light. It seems that if you really wanted to dig into what actually takes place in that situation then you would have to describe a black hole. The amount of speed required to create length contraction would have to be at a speed so great that the centrifugal force as well already taken over.

The derivation of the theory is supposed to make an object travel in a straight line with spatial curvature. This line completely wraps around the object, so your first equation could be a close approximation if it was in orbit.

Dedocta:
This theory proposes that the contraction of the length of a spinning object is due to the change in the tangential velocity of its surface, as opposed to the traditional explanation of the contraction occurring solely in the direction of rotation.

The tangential direction is the direction around the circumference of a rotating disc.

The "traditional explanation" (from relativity) is that the length of the circumference contracts (in the tangential direction).

You are proposing, instead, that the radius contracts (i.e. contraction happens in the radial direction, perpendicular to the direction of rotation). Is that correct?
The Ehrenfest paradox, which highlights a discrepancy between classical and relativistic mechanics, arises from the fact that as an object spins faster, the circumference of its rotating frame contracts, while its radius remains static. This results in an apparent contradiction of the conservation of energy and momentum, as well as the geometry of space-time.
You haven't said why you think this breaks conservation of energy or momentum yet. As for the spacetime geometry, general relativity handles that just fine, doesn't it?
My theory suggests that the contraction occurs along the radial direction, as the radius of the object decreases with the decrease in tangential velocity, while the circumference contracts according to the Ehrenfest Paradox.
The radius would have to increase with decreasing tangential velocity, would it not?
This new perspective allows for a more intuitive explanation of the paradox, without violating any laws of physics.
You're explicitly violating the theory of relativity. Specifically, the result that lengths contract only in the direction of motion.
The mathematical derivation of the spin length contraction formula is straightforward, involving only the Lorentz factor and the original dimensions of the object.
You're postulating it, rather than deriving it, aren't you?
One possible solution to the Ehrenfest paradox was proposed by Richard Muller in 1991, in which he suggested that the rotating disk should be considered as a collection of rotating rods, rather than a single object. This perspective leads to the conclusion that the circumference of the disk contracts by the same factor as its radius, as the rods change shape due to the centrifugal forces.
Why does Muller say that the radius contracts?
However, my proposed solution, based on the contraction of the radius due to the change in tangential velocity, is a more intuitive and straightforward explanation of the paradox.
I don't understand. Are you saying that the change in rotational speed somehow causes the radius to contract? What's the mechanism for that?
To fully understand the mathematical framework of this theory, we can consider a disk with radius $R_0$ at rest in the laboratory frame. In the frame of reference of the disk, its circumference is given by $2\pi R_0$, and its radius is $R_0$. Now, let us assume that the disk is set into rotation with angular velocity $\omega$. According to the theory of special relativity, the circumference of the disk in the laboratory frame is given by:

$$C = 2\pi R_0 \sqrt{1 - \frac{v^2}{c^2}}$$

where $v = \omega R_0$. From this, we can see that the circumference of the disk has decreased due to its rotation.
Okay.
Now, let us assume that the radius of the disk has also decreased by the same factor. In other words, the radius in the laboratory frame is given by:

$$R = R_0 \sqrt{1 - \frac{v^2}{c^2}}$$

where $v$ is as defined above.
What's your justification for making this assumption? Does it apply to anything other than the particular example of a rotating disc? Is there some set of physical postulates from which this assumption can be derived?
With this assumption, we can calculate the circumference of the disk in its own frame of reference:

$$C' = 2\pi R \sqrt{1 - \frac{v^2}{c^2}} = 2\pi R_0 \sqrt{1 - \frac{v^2}{c^2}} = C$$
There must be an error there, because this implies that $R=R_0$, contrary to your hypothesis. Do you mean:
$C' = 2\pi R = 2\pi R_0 \sqrt{1 - \frac{v^2}{c^2}} = C\sqrt{1-\frac{v^2}{c^2}}$?
Thus, we can conclude that the proposed contraction of the radius of the disk is consistent with the contraction of the circumference.
It ensures that the relationship $C=2\pi R$ remains the same whether the disc is rotating or not. Is that all you mean by consistent?
The proposed solution is just one of several possible explanations for the Ehrenfest paradox. Another solution involves the use of the concept of a rigid body and a modification of the laws of mechanics to account for the rotation of the body. However, this solution has been shown to be inconsistent with special relativity and the observation of high-energy particles. The proposed solution using spin length contraction does not require any modification of the laws of physics and is consistent with special relativity.
Your solution is not consistent with special relativity. In special relativity, length contraction is observed in the direction of motion - i.e. only in the tangential direction here, not the radial direction.

In reality, things are more complex, because we'd have to consider the forces and stresses on the material of the disc. For instance, a centripetal force is required to make the circumference revolve around the centre, which is usually provided by stresses in the material.
The implications of this theory extend beyond the resolution of the Ehrenfest paradox. It may help us better understand the behavior of high-energy particles and their interactions with electromagnetic fields. Furthermore, it could have applications in the design of high-speed rotating machinery, such as turbines and centrifuges.
Unless you decide to discuss these things in more detail, I'm going to leave them for now.

I think the first thing you need to do is to provide some kind of justification for your assumptions. Why, for instance, do you require the relationship $C=2\pi R$ to remain true regardless of the rate of rotation? This seems to be the motivation for your postulate about the radius.
However, the beauty of the spin length contraction theory is its simplicity and its adherence to the fundamental principles of special relativity. The theory provides a clear and intuitive explanation for the paradox, without requiring any modifications to the laws of physics or the geometry of spacetime.
You're attempting to modify the theory of relativity, whether you realise it or not.
Where I am now, is trying to incorporate this concept with General Relativity, how would this affect Minkowski space?
The geometry of a rotating disc is not Minkowskian. That's kinda the point of this example. If you recognise that $C \ne 2 \pi R$, that implies that the disc is not sitting in a flat space. In terms of general relativity, the rotation of the disc curves spacetime around it.

For the Ehrenfest scenario, one must consider how the rotating wheel was constructed. Assuming the wheel is made of only spokes and no rim, then there is nothing to contract, so no paradox. If the wheel must have a rim, we could add the rim while the wheel is already rotating at constant speed. Otherwise the rim would have to stretch like an elastic while the wheel spins up to speed.

I should have stated that I am assuming the spokes are not compressible, so the radius remains constant. The circumference of the rotating wheel is measured to be 2piR in the inertial reference frame in which the axle is stationary and the wheel is rotating. In the non-inertial 'frame' of the rim itself, the circumference would be larger by a factor of gamma.

I wonder if the OP is considering the spokes to be compressible or telescoping, and perhaps that is why they are allowing the radius to shrink?

I prefer to use my suggested method of a wheel made of only rigid spokes. Then we can build the rim while the wheel is rotating at constant speed. In that case, the people in the inertial frame in which the axle is stationary say that material becomes length contracted as it is carried up the spokes and placed tangentially to form the rim. In the non-inertial frame of a piece of that rim material, the material is not length contracted locally. This explains how the non-inertial 'frame' measures the circumference to be larger by a factor of gamma.

I wonder if the OP is considering the spokes to be compressible or telescoping, and perhaps that is why they are allowing the radius to shrink?
That's a good question.

It's not clear whether this is supposed to be about an idealised wheel, or whether we're supposed to be considered the actual likely forces and stresses inside some kind of realistic wheel material.

Ovyind Gron covered many options in his paper https://www.researchgate.net/public...ating_Reference_Frames_A_Historical_Appraisal

The only place I have seen a figure 9 part C solution was nearly 10 years ago and the spoke length was length contracted. Gron does not provide a method or working for his solution in the paper. https://www.thephysicsforum.com/special-general-relativity/5577-relativistic-rolling-wheel-ii-3.html#post12704

Years ago I realized that if there’s a circumferential contraction due to contraction within the x y plane of the spin, then there’s omnidirectional length contraction in said plane. That’s where this resolution would arise.

Dedocta:

Unless and until you can produce a coherent argument that points to the resolution you claim to have, people won't much care what you believe you realised years ago.

Are you going to respond to my post, above - in particular to answer the questions I asked you?

Sorry for the delay:

I mean tangential velocity of a spinning object.

The "traditional explanation" (from relativity) is that the length of the circumference contracts (in the tangential direction).

You are proposing, instead, that the radius contracts (i.e. contraction happens in the radial direction, perpendicular to the direction of rotation). Is that correct?
Yes

You haven't said why you think this breaks conservation of energy or momentum yet. As for the spacetime geometry, general relativity handles that just fine, doesn't it?
In the frame of spinning objects, lengths are closer between two objects relative to two at rest in the framework of this theory. Assuming Sag A* was spinning faster early in the universe, the distance between two spin aligned objects would be different than we currently believe - potentially leading to increase in distance as objects grow / slow down in tangential velocity.

The radius would have to increase with decreasing tangential velocity, would it not?
No, c/d = pi so to conserve pi, with a contracted circumference, one would need a smaller diameter in order to compensate and maintain pi.

You're postulating it, rather than deriving it, aren't you?
Ahh, guilty as charged! The idea is to derive eventually.

Why does Muller say that the radius contracts?
Thomas precession: Where is the torque?
Richard A. Muller
Department of Physics and Lawrence Berkeley Laboratory, University of California, Berkeley, California 94720

I don't understand. Are you saying that the change in rotational speed somehow causes the radius to contract? What's the mechanism for that?
As I responded before, an object that has motion omnidirectionally in the xy plane has length contraction in the xy plane within this framework.

What's your justification for making this assumption? Does it apply to anything other than the particular example of a rotating disc? Is there some set of physical postulates from which this assumption can be derived?
Yes, as galaxies grew they spun slower - this could affect rotational dynamics and affect distance measures over time. It would affect a lot of things - including spin aligned objects forming large-scale structures. Many rabbit holes to go down (too many!)

Last edited:
No, c/d = pi so to conserve pi, with a contracted circumference, one would need a smaller diameter in order to compensate and maintain pi.

There is no reason to maintain pi as the ratio between Circumference (C) and Diameter (D). We can have C/D=pi in the inertial reference frame in which the wheel is only rotating, and we can also have C/D=pi*gamma in the non-inertial reference 'frame' of a small piece of the rim of the rotating wheel.

This is perfectly analogous to the length of a meter stick measuring less than 1 meter in an inertial reference frame where it is moving along the direction of its length, yet still measuring 1 meter long in its own reference frame, where it is not moving.

Dedocta,

I assume you are having trouble visualizing how a virtual observer fixed to the rim of the rotating wheel would be able to justify that the piece of rim he is on can possibly be its full rest length, while an observer fixed to the center of the wheel would say that same bit of the rim is length-contracted. How can both be correct?

I had the same question years ago. Online sources just say that Euclidean geometry does not hold on the perimeter of a rotating disk, but that was not explanatory enough for me. I had to figure out a way to reason it that would make more sense than that. I will share that idea here when I have more time...