I Car, wheels and Lorentz contraction of the road: Is this a Paradox?

[Dale]

New Perspectives on the Relativistically Rotating Disk and Non-time-orthogonal Reference Frames Robert D. Klauber

Unfortunately, Foundations of Physics Letters is not a recognized professional scientific journal. It is not listed in the Clarivate master journal list. This article is not a valid reference per the forum rules. In particular, it has several problems which explain why it was not able to be published in any reputable journal.

One is the claim “special relativity is restricted to inertial systems”. That is clearly false and even Einstein’s seminal 1905 paper included an analysis of a non-inertial system. Even non-inertial reference frames are part of special relativity. The math used to transform between arbitrary coordinates is well known and is not anathema to SR.

His 4 postulates of general relativity are non-recognizable.

He states “these results should be expected”, as though they were not. They were expected and predicted by Einstein who encouraged Michelson and Gale in their efforts on this topic.

His entire section 2.1 entirely misrepresents both the scientific community’s understanding of these topics as well as the compatibility of these concepts with special relativity.

His phrase “heretofore seemingly sacrosanct” is designed to convey the impression that he is challenging a religious community’s doctrine. In fact, the coordinate system he uses is completely standard and used routinely by the scientific community.

Most importantly for this discussion, the claim “No tension arises in the disk as it is spun up” is completely unjustified. The angular velocity in the entire paper is always a fixed $\omega$. The author never even considers angular acceleration and so can make no claims about what happens as a disk is “spun up”. And nowhere is a single calculation about the expansion tensor performed which is what would be required to claim that there is no material strain. So neither the “no tension arises” nor the “as it is spun up” parts of the claim are supported by any of the math anywhere in the paper.

In short, this is not a valid source. You should not use it as support for any posts here at PF.

 

[Sagittarius A-Star]

New Perspectives on the Relativistically Rotating Disk and Non-time-orthogonal Reference Frames Robert D. Klauber
Some excerpts:
...
The Lorentz contraction is a direct result of non-agreement in simultaneity between frames. If there is agreement in simultaneity, there is no Lorentz contraction.

Thank you for providing the link instead of only writing "I have been reading more about the Ehrenfest paradox and there is also another interpretation ...". However, as @Dale wrote, this is not a valid source.

R. Klauber's argument contains a logical flaw.

3.2.2 No Lorentz Contraction.
The Lorentz contraction is a direct result of non-agreement in simultaneity between frames. If there is agreement in simultaneity, there is no Lorentz contraction. To show this we need one additional, presumably inviolable, postulate. That is,

3. The proper spacetime length of any path is invariant under any transformation, i.e., it is the same for all observers.

Hence, for two frames in relative motion (notation should be obvious)
$$(∆s)^2 = − c^2(∆t)^2 + (∆l)^2 = − c^2(∆t^′)^2 + (∆l^′)^2\ \ \ \ \ (5)$$
For a rod at rest in the primed frame, an observer in the unprimed frame sees that rod such that its endpoints are events which for him occur simultaneously, i.e., $∆t = 0$. But in the primed system those events are not, according to standard relativity theory, simultaneous and $∆t^′ \neq 0$. This means $∆l \neq ∆l^′$, and results in Lorentz contraction [27]. If, however, the same two events could also appear simultaneous in the primed system, then $∆t^′ = 0$ and $∆l$ must equal $∆l^′$.

His equation (5) is based on both coordinate systems (primed and unprimed) being time-orthogonal. But for $(∆t = 0) \Rightarrow (∆t^′ = 0)$ he uses a non-time-orthogonal coordinate system (clock synchronization based on anisotropic one-way-speed of light) along the rim of the rotating disk. That is logically inconsistent.

 

[Ibix]

Most importantly for this discussion, the claim “No tension arises in the disk as it is spun up” is completely unjustified. The angular velocity in the entire paper is always a fixed $\omega$. The author never even considers angular acceleration and so can make no claims about what happens as a disk is “spun up”. And nowhere is a single calculation about the expansion tensor performed which is what would be required to claim that there is no material strain. So neither the “no tension arises” nor the “as it is spun up” parts of the claim are supported by any of the math anywhere in the paper.

As a matter of curiosity I set up a simple "constant angular acceleration" model and computed the expansion scalar. Consider a disk that is initially at rest in some inertial frame. At time $t=0$ it starts to spin with (in this frame) constant angular acceleration $\dot\omega$, so at time $t$ all points on the disc have the same angular velocity $\dot\omega t$. This is not a particularly plausible acceleration model in a relativistic case (as discussed above) but it's easy to work with.

Noting that the three velocity magnitude of each point on the disc is $\dot\omega tr$ then in polar coordinates ($ds^2=-dt^2+dr^2+r^2d\phi^2+dz^2$, $c=1$) the four-velocity field is $$X^a=\left(
\frac{1}{\sqrt{1-\dot\omega^2t^2r^2}},
0,
\frac{\dot\omega t}{\sqrt{1-\dot\omega^2t^2r^2}},
0\right)$$where I have used $X^a$ and a -+++ sign convention to be consistent with the notation in the Wikipedia page on congruences. Grinding through the maths there leads to the expansion scalar$$
\theta = \frac{\dot\omega^2r^2t}{\left(1-\dot\omega^2t^2r^2\right)^{3/2}}
$$Note that there's an assumption that $X^a$ is timelike so this is only defined for $\left(\dot\omega tr\right)^2<1$, which is a technical way of saying that the rim of the disc must be travelling slower than light. It also means that the division by zero at $\dot\omega tr=1$ is in the region where this expression isn't valid. This expansion scalar is manifestly positive for positive $t$, so two nearby points on the rim of the disc will indeed see the distance between them grow. (It being negative for negative $t$ is also expected since extending this congruence to negative times would represent the disc slowing its spin at that time.) Or, to put it another way, short rulers along the rim of the disc will indeed separate from one another, contrary to the claim in the paper.

Edit: in fact, the entire expansion tensor is interesting. In these coordinates the only non-zero components are $\theta_{tt}$, $\theta_{\phi t}$, $\theta_{t\phi}$ and $\theta_{\phi\phi}$, which shows that there's only expansion/contraction in the time and tangential directions - nothing in the radial or vertical directions. This is exactly what you'd expect from the naive momentarily co-moving inertial frame analysis.

Here is a Maxima batch file to compute all of the above.

/* Compute the expansion scalar of a simple "constant angular acceleration" case in flat */
/* spacetime. Procedure follows Wikipedia link below, including an assumption of -+++    */
/* metric signature.                                                                     */ 
/* https://en.wikipedia.org/wiki/Congruence_(general_relativity)#Kinematical_description */

load(ctensor);
load(eigen);

/* Minkwoski metric, -+++, cylindrical polars */
dim: 4;
ct_coords: [t, r, phi, z];
lg: matrix([-1, 0, 0,   0],
           [ 0, 1, 0,   0],
           [ 0, 0, r^2, 0],
           [ 0, 0, 0,   1]);

/* Compute the inverse metric, ug, and the Christoffel symbols, mcs. */
/* NB: mcs[i,j,k] is \Gamma^k_{ij}.                                  */
cmetric();
christof(true);

/* Some useful functions for later */
tensorProduct(u,v) := block(
  /* Takes two one-index tensors u^a and v^b (expressed as matrices) and */
  /* returns u^a v^b. Either or both tensors may have lower indices -    */
  /* the resulting tensor has the implied indices.                       */
  [i, j, ans],
  ans: zeromatrix(4,4),
  for i: 1 thru 4 do block(
    for j: 1 thru 4 do block(
      ans[i, j]: u[i][1] * v[j][1]
    )
  ),
  ans
);
covdif(x) := block(
  /* Computes the covariant derivative of the lower-index tensor x_b, */
  /* (expressed as a matrix). That is, this returns \nabla_a x_b.     */
  [i, j, k, ans],
  ans: zeromatrix(4,4),
  for i: 1 thru 4 do block (
  	for j: 1 thru 4 do block (
  		ans[i, j]: diff(x[i][1], ct_coords[j]),
  		for k: 1 thru 4 do block (
  			ans[i, j]: ans[i, j] - mcs[j, i, k] * x[k][1]
  		)
  	)
  ),
  ans
);
symmetrise(T) := block (
  /* Returns the symmetric part of a two-index tensor T^{ab} or T_{ab}. */
  (T + transpose(T)) / 2
);

/* Construct the "accelerating rotation" congruence, which represents a */
/* three-velocity field with magnitude (dotomega * t) * r.              */
v: dotomega * t * r;
gamma: 1/sqrt(1-v^2);
X: columnvector([gamma, 0, dphi, 0]);
assume(dotomega^2*r^2*t^2 < 1);
solve(-1 = transpose(X).lg.X, dphi);
X: substitute([%[2]], X);

/* Compute the projection operator h^a{}_b and the symmetrised covariant */
/* derivative of X_a, \nabla_{(a}X_{b)}.                                */
hul: ug.lg + tensorProduct(X, lg.X);
sdXll: symmetrise(covdif(lg.X));

/* Compute the expansion tensor */
theta_ll: zeromatrix(4,4);
for a: 1 thru 4 do block(
  for b: 1 thru 4 do block(
    for m: 1 thru 4 do block(
      for n: 1 thru 4 do block(
        theta_ll[a, b]: theta_ll[a, b] + hul[m, a]*hul[n, b] * sdXll[m, n]
      )
    )
  )
);

/* Compute the mixed-index version of the expansion tensor and its trace, */
/* the expansion scalar, \theta.                                          */
theta_ul: ug.theta_ll;
theta:0;
for a: 1 thru 4 do block(
  theta: theta + theta_ul[a, a]
);
theta: ratsimp(theta);

 

[Ibix]

R. Klauber's argument for this contains a logical flaw.

And his "postulate 3" is unnecessary since it is a direct consequence of the way the interval is defined.

 

[A.T.]

New Perspectives on the Relativistically Rotating Disk and Non-time-orthogonal Reference Frames Robert D. Klauber

Some excerpts:

Rods in inertial frames with velocities equal to the tangent velocities at a given disk radius can not be used to measure the circumference, since the ”surrogate frames postulate” for equivalence of inertial and non-inertial standard rods is not valid for the rotating frame, and is generally invalid for any non-time-orthogonal frame, and doing so leads to a discontinuity in time.

Given that the centripetal acceleration at the rim can be made arbitrarily small (by increasing the radius), I wonder what Klauber thinks happens to tangentially moving inertial rods, when they are given even the smallest amount of centripetal acceleration. Do they suddenly expand to their proper length?

That would be a problematic discontinuity, unlike the one he worries about stemming from mere conventions of simultaneity.

 

[Sagittarius A-Star]

New Perspectives on the Relativistically Rotating Disk and Non-time-orthogonal Reference Frames Robert D. Klauber

He writes:

4.1 Rotating Frame Metric and Transformations

This coordinate transformation, where upper case coordinates represent the inertial frame
K, lower case denote the rotating frame k, and the axis of rotation is coincident with both the
Zand z axes, is
$cT = ct \ \ \ \ \ (8a)$
$R = r \ \ \ \ \ (8b)$
$Φ = φ + ωt \ \ \ \ \ (8c)$
$Z = z \ \ \ \ \ (8d)$
...
The transformation (8) seems Galilean in nature, rather than relativistic, and if it is valid
(as most researchers today feel that it is), we should not be surprised to find the disk exhibiting
at least some Galilean characteristics.

Maybe he took his transformation (8) from the Wikipedia article about Born coordinates, "Transforming to the Born chart":
https://en.wikipedia.org/wiki/Born_coordinates#Transforming_to_the_Born_chart

I'm not familiar with Born coordinates. But in the Wikipedia article about it, they continue with a complete relativistic transformation to a Langevin observer frame on the rim of the rotating disk, while R. Kolb only adds later in his chapter 4.3.1 a $\gamma$-factor for time dilation.

R. Klauber also refers to F. Selleri, but doesn't use the Selleri-transformation (=modified Lorentz transformation without the term for "relativity of simultaneity") for a non-time-orthogonal frame on the rim of the rotating disk. Instead he uses a wrong "shortcut".

 

[Ibix]

Grinding through the maths there leads to the expansion scalar$$
\theta = \frac{\dot\omega^2r^2t}{\left(1-\dot\omega^2t^2r^2\right)^{3/2}}
$$

Replying to myself, sorry, but one other comment on this is that in this case $\theta=\frac{d}{dt}\gamma$ where $\gamma$ is the Lorentz gamma factor associated with the velocity $\dot\omega tr$ - i.e., the linear velocity of a point at radius $r$ at a time $t$ into the acceleration. The interpretation of $\theta$ is that it's the fractional increase in volume of a small region - and since we've established that the expansion is one dimensional, this is saying that the rate of growth of that 1d stretch is equal to the rate of growth of $\gamma$, exactly as you'd expect if rulers were length contracting exactly like in linear acceleration.

This case is a bit special to this particular congruence, I think, but it's a rather nice result.

 

[Sagittarius A-Star]

Interestingly, in the case that the wheel undergoes a Lorentz contraction of its radius when it rotates, it would always hold that gear and track mesh and N is conserved at any speed. Contraction of radius is something that seems to agree in some obscure way with SR. Wheel rim shrinks as apparently expected due to shrinkage of radius.

• If you allow in a thought experiment the rim (circumference) to freely contract while spinning-up, i.e. by having a disk made of very elastic rubber, and only the circumference is made of a thin layer of metal and ignoring centrifugal force, then the conclusion in this statement is correct, except the word "obscure".
• If you force in another thought experiment the rim (circumference) to not contract while spinning-up, i.e. by having only the rim of a wheel made of rubber and ignoring centrifugal force, then obviously gear and track don't mesh and N is not conserved at any speed. In this case the rim would get internal mechanical stress (for the same reason as the thread in the Bell spaceships paradox). This case also agrees with SR.
  • 

Source of the picture:
https://www.spacetimetravel.org/tompkins/7

If you are interested in understanding SR, I recommend not to waste your time with the paper of R. Klauber, because it describes a different physics than SR, by introducing a strange combination of Galilei transformation and time dilation. SR was never refuted by any experiment.

 

[A.T.]

Interestingly, in the case that the wheel undergoes a Lorentz contraction of its radius when it rotates, it would always hold that gear and track mesh and N is conserved at any speed.

The shrinking of the radius is not Lorentz contraction, which happens only in the tangential direction. But you can allow the radius to shrink physically (e.g. proper length change of telescoping radial beams), to preserve the proper length of the rim. That would indeed allow to the gears to mesh at any speed, while preserving the number of teeth and thus revolutions per given proper length of track.

 

[Sagittarius A-Star]

But then, Ehrenfest Paradox is false. In hub of wheel IFR there is no any perimeter length contraction. At any rotation speed perimeter is 2πR, and pitch of teeth keeps constant. In this IFR there is absolutely no difference in the wheel from its state of rest, except that it is spinning. However, at the point of contact wheel-track, at same location-time, track, and pitches between teeth, are contracted. This difference can only arise because track moves straight while wheel slice at that point is rotating. If there is a paradox it is this one.

I can't comment on this because I don't understand your text, and you did not answer my related question in posting #53.

 

[Sagittarius A-Star]

New Perspectives on the Relativistically Rotating Disk and Non-time-orthogonal Reference Frames Robert D. Klauber

The following paper contains a critical review of R. Klauber's personal theory:

In contrast to this, Klauber(61) claimed that the length of the rim must be measured along a closed curve in spacetime. We have shown that such a measurement would not be in agreement with relativity. Klauber(61) measures
lengths on the rotating disk as invariant space-time intervals not along lines of isotropic simultaneity (“non-timeorthogonal”), but along lines of constant global time, i.e. in our notation along dT = 0, where the synchronization is
defined as in Sec. 5.7 with A = v. This leads directly to the absence of Lorentz contraction in the case of the rotating
disk. It is clear that neither the concept nor the consequence is compatible with SR, and indeed the theory is meant
as a testable alternative to relativity, which moves the subject outside the scope of this paper.

Source (on page 23):
https://arxiv.org/abs/gr-qc/0403111

via:
https://de.wikipedia.org/wiki/Ehrenfestsches_Paradoxon

 

[Dale]

The following paper contains a critical review of R. Klauber's personal theory:

Source (on page 23):
https://arxiv.org/abs/gr-qc/0403111

via:
https://de.wikipedia.org/wiki/Ehrenfestsches_Paradoxon

Note, however, that this paper is also not published in a reputable journal. That in itself is not terribly surprising since Klauber's paper is so obscure that most professional journals would not consider it something that even needs to be rebutted.

 

[Dale]

Since the OP is continuing to push their same non-mainstream views using yet more unpublished sources from the same questionable author, their most recent response is deleted and this thread is closed.

Please review the forum rules prior to posting any further.