Translation:On v. Ignatowsky's Treatment of Born's Definition of Rigidity

by Paul Ehrenfest

On v. Ignatowsky's Treatment of Born's Definition of Rigidity.

By Paul Ehrenfest.

§ 1. A body which satisfies Born's requirement of relative-rigidity^[1], can in no way be transferred from the state of rest into the state of uniform rotation about a fixed axis. – This theorem was proven so clear and succinctly^[2], that it was accepted without further ado by all authors who (until recently) have commented on the relative-rigidity question, and this was independent of the position taken by the authors at other occasions regarding the rigidity problem. For clarities sake, the relevant comments shall be shortly summarized in chronological order:

M. Born^[3] admits the previous result, and consequently decides himself to the following assumptions: 1. The electrons actually satisfy the rigidity requirement, yet they don't rotate. The theory of electrons nowhere unconditionally requires the assumption of rotating electrons. 2. The macroscopic "rigid" bodies are molecular aggregates, thus they are elastically deformable, so they don't satisfy the rigidity requirement and are able to rotate.

M. Planck^[4]: 1. The macroscopic "rigid" bodies are to be seen as elastic, and the theory of their motion must be founded by stating the kinetic potentials of elastic deformations. "The attempt, to make the abstraction of the rigid body (which is so important for ordinary mechanics) also useful for the theory of relativity, doesn't appear to me as promising any real success." 2. The maintenance of the rigidity requirement for the single electron, would deprive itself from any real verification as to "whether it brings us physically further than the general principles of relativity theory".

M. Abraham^[5]: "The attempt undertaken .... by Born, to generalize ... Einstein's definition of length, must be considered as failed, after Ehrenfest, Herglotz, Noether und Levi-Civita have demonstrated, that the employed concept of rigidity is not applicable to rotational motions."

M. Born^[6], in a later paper, sticks to the idea (in opposition to Planck) that the extension of the rigidity concept to relativity theory is necessary, and he develops – by reseting his original rigidity definition as too narrow – a new rigidity definition. There he accepts, that any rigid body must be imagined as having – with respect to the rigidity definition – a preferred point once and for all.^[7] Yet also in this way, by far not all difficulties of the rotation problem can be totally resolved: As demonstrated by Born, a stationary (!) rotation about a fixed axis takes place here, so that the resting observer sees the multi-axial layers rotating with larger angular velocity as the peripheral layers, thus the "rigid" body (starting from the axis) increasingly stirs itself.^[8]

§ 2. Now – in a paper recently published^[9] – also v. Ignatowsky comments on the problem of rotation of relative-rigid bodies. His standpoint is mainly characterized by the following statement (§ 5, end):

"Let us again consider the cylinder of § 3, namely the passage from rest to uniform rotation. It will be set in motion under the influence of external forces, yet always under maintenance of condition (20) § 2. The effect of these forces will propagated with a certain velocity, so that the angular velocity for a certain moment $t$, is not the same for all angles. At the same time it will apparently be compressed, until its motion passes into a uniform motion after some time."

Does v. Ignatowsky – to arrive at this claim – probably uses a rigidity definition deviating from the (original) one of Born?

No! His initial equation

$\left(dw_{0}\right)^{2}=\left(dw'_{1}\right)^{2}$

(6)

is identical to the (original) requirement of Born, and his equation

${\frac {d}{dt}}\left(dw'_{1}\right)^{2}=0$

(7)

emerges out of it by total differentiation. Eq. (13) derived from it by conversion, is the temporal derivative of that form of Born's rigidity equation, with which I and Herglotz have operated, and finally the previously mentioned eq. (20) is the additional conversion of eq. (7) and (13) from the Lagrangian into the Eulerian from, according to the procedure of Noether.

Also the following statement regarding the stationary rotating cylinder, made by v. Ignatowsky at the end of § 3, is in best agreement with that:

"Let us consider now a spherical cylinder, rotating with constant angular velocity about a resting axis. It is easy to demonstrate, that eq. (20) is satisfied identically in this case."

Indeed: that Born's rigidity definition allows stationary rotation, is proven for a long time. It's known that the difficulty exclusively lies in the question after the passage from rest to uniform rotation^[10]. I have formulated this difficulty in the following way:

"Let a relative-rigid cylinder of radius $R$ and height $H$ be given. A rotation about its axis which is finally constant, will gradually be given to it. Let $R_{1}$ be its radius during this motion for a stationary observer. Then $R_{1}$ must satisfy two contradictory conditions:

a) The periphery of the cylinder has to show a contraction compared to its state of rest:

$2\pi R_{1}<2\pi R\,$

.....;

b) ..... the elements of a radius cannot show a contraction compared to the state of rest. It should be:"

$R_{1}=R\,$

Compare with this the specifications given by v. Ignatowsky: "The distance between two points of the cylinder, not lying on the same diameter, and measured when the cylinder was still at rest, won't be equal to the distance between the same points measured synchronously, when the cylinder is rotating (§ 3, end)."

"In my view, the whole thing seems of be based on a misunderstanding. If we measure a line element along the circumference of the" – stationary rotating – "disc in a synchronous way, then we obtain a value which is smaller than $2\pi R$, where $R$ means" – synchronously measured at the rotating disc – "the radius of the disc. However, in this lies absolutely no contradiction, but everything explains itself from the definition of synchronous measurement .... in general we can define the true shape and dimension of a rigid body by measurement, when (and only when) the body is at rest. The measurements on moving bodies give only apparent values ...." (concluding remark).

§ 3. In the interest of explaining the meaning of the words:

"However, in this lies absolutely no contradiction, but everything explains itself from the definition of synchronous measurement"

I allow myself to respectfully request v. Ignatowsky, that he comments on two questions immediately to be formulated: – For the sake of the precise formulation of these questions, I predefine a convention and an assertion:

Let the spherical disc be equipped upon its entire surface with infinitely many marks that are individually recognizable.

While the disc is at rest, the resting observer $B$ holds a tracing paper above it, and traces the marks upon the resting paper.

While the disc is rotating stationarily, the resting observer $B$ holds a tracing paper $P_{1}$ above it, and in the moment when his clock indicates $t$, he traces with one stroke all marks upon the resting paper.

Eventually, the resting observer $B$ measures the marking distribution upon the resting tracing-images $\Pi$ and $\Pi _{1}$.

I assert: The periphery- and radius length upon $\Pi _{1}$ measured in this way, coincides in the present example exactly with that, what was called by v. Ignatowsky as the disc circumference or disc radius at the stationary rotating disc, "synchronously measured" by the resting observer at moment $t$. (See definition of "synchronous measurement" in § 2 of the work of v. Ignatowsky.)

My questions to v. Ignatowsky are^[11]:

Question 1: Is the last formulated assertion correct? If not – wherein is then the difference between the result obtained by the resting observer by "synchronous measurement" of the rotating disc, and the result obtained by the measurement of the resting tracing-image $\Pi _{1}$?

Question 2: If my assertion is correct, then the statements put together by v. Ignatowsky about periphery and radius being "synchronously measured", and that are strangely denoted by him as absolutely without contradiction, are transformed into the following statements concerning the tracing-images: The tracing-image $\Pi _{1}$ has the same radius as $\Pi$ while its periphery is shorter. How can one imagine tracing-images of such properties without contradiction?

Petersburg, October 4, 1910.

(Received October 7, 1910.)

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1. M. Born, Ann. d. Phys. 30, I, 1909.
2. P. Ehrenfest, this journal. 10, 918, 1909; G. Herglotz, Ann. d. Phys. 31, 393, 1910; F. Noether, Ann. d. Phys. 31, 919, 1910.
3. M. Born, this journal 11, 233, 1910.
4. M. Planck, this journal 11, 294, 1910.
5. M. Abraham, this journal 11, 527, 1910.
6. Gött. Nachr. 1910, 28. May.
7. l. c, § 3, M. Born calls this point "center".
8. l. c., § 5, M. Born thus remarks: "By that, the globe were hardly a "rigid" body in this sense."
9. Ann. d. Phys. 33, 607, 1910.
10. In other words: It was trivial to prove that eq. (20) is satisfied by stationary rotation. What had to be proven and what was claimed without proof at the of § 5, is the following: that a motion of the cylinder can be given which transfers it from rest to stationary rotation, and at that occasion – as opposed to the result of Noether – constantly satisfies eq. (20)!
11. It would be very desirable in the case of further discussion, to avoid the expressions "true" and "apparent" shape of the rotating disc throughout, or when this should be impossible, to define these expressions by a convention as simple and strict as possible.