The Einstein-Varicak Correspondence on Relativistic Rigid Rotation

TL;DR

This paper reviews the historical development of relativistic rigid rotation by analyzing Einstein and Varicak's correspondence from 1909 to 1913, highlighting its significance in the context of relativity theory.

Contribution

It uncovers the historical insights and the evolution of ideas on relativistic rigid rotation through primary correspondence analysis.

Findings

Historical correspondence sheds light on early ideas of relativistic rotation.

The significance of Einstein's and Varicak's discussions in the development of relativity.

Highlights the importance of historical context in understanding scientific concepts.

Abstract

The historical significance of the problem of relativistic rigid rotation is reviewed in light of recently published correspondence between Einstein and the mathematician Vladimir Varicak from the years 1909 to 1913.

Full text

1 Introduction

The rigidly rotating disk has long been recognized as a crucial ‘missing link’ in our historical reconstruction of Einstein’s recognition of the non-Euclidean nature of spacetime in his path toward general relativity.[1, 2] Relativistic rigid rotation combines several different but related problems: the issue of a Lorentz-covariant definition of rigid motion, the number of degrees of freedom of a rigid body, the reality of length contraction,[3] as well as Ehrenfest’s paradox[4] and the introduction of non-Euclidean geometric concepts into the theory of relativity.[5]

2 Relativistic rigid motion

A relativistic definition of rigid motion was first given by Max Born.[6] The definition was given in the context of a theory of the dynamics of a model of an extended, rigid electron, and defined a rigid body as one whose infinitesimal volume elements appear undeformed for any observer that is comoving instantaneously with the (center of the) respective volume element. The definition and its implications were discussed at the 81st meeting of the Gesellschaft Deutscher Naturforscher und Ärzte in Salzburg in late September 1909.

Gustav Herglotz and Fritz Noether, in papers received by the Annalen der Physik on 7 and 27 December, respectively, further elaborated on the mathematical consequences of Born’s definition.[7] Herglotz, in particular, reformulated the definition in more geometric terms: A continuum performs rigid motion if the world lines of all its points are equidistant curves. The analysis showed that Born’s infinitesimal condition of rigidity can only be extended to the motion of a finite continuum in special cases. It implied that a rigid body has only three degrees of freedom. The motion of one of its points fully determines its motion. Translation and uniform rotation are special cases. In particular, the definition does not allow for acceleration of a rigid disk from rest to a state of uniform rotation with finite angular velocity.

In view of these consequences, various other definitions of a rigid body were suggested, e.g. by Born and Noether,[7, 8] until it became clear that special relativity does not allow for the usual concept of a rigid body. In other words, a relativistic rigid body necessarily has an infinite number of degrees of freedom.[9]

On 22 November 1909, a short note appeared by Paul Ehrenfest pointing to a paradox that follows from Born’s relativistic definition of rigid motion of a continuum.[10] He considered a rigid cylinder rotating around its axis and contended that its radius would have to meet two contradictory requirements. The periphery must be Lorentz-contracted, while its diameter would show no Lorentz contraction. The difficulty became known as the “Ehrenfest paradox.” In a polemic exchange with von Ignatowsky,[11] Ehrenfest devised the following thought experiment to illustrate the difficulty. He imagined the rotating disk to be equipped with markers along the diameter and the periphery. If their positions were marked onto tracing paper in the rest frame at a fixed instant, with the disk both at rest and in uniform rotation, the two images should show the same radius but different circumferences.

3 The Einstein-Varićak correspondence

Immediately after the 1909 Salzburg meeting, Einstein wrote to Arnold Sommerfeld that “the treatment of the uniformly rotating rigid body seems to me of great importance because of an extension of the relativity principle to uniformly rotating systems.”[12] This was a necessary step for Einstein following the heuristics of his equivalence hypothesis, but only in spring 1912, a few weeks before he made the crucial transition from a scalar to a tensorial theory of gravitation based on a general spacetime metric,[5] do we find another hint at the problem in his writings.[1, 2]

The Collected Papers of Albert Einstein recently published[13] nine letters by Einstein to Vladimir Varićak (1865–1942), professor of mathematics at Agram (now Zagreb, Croatia). Varićak had published on non-Euclidean geometry[14] and is known for representing special relativistic relations in terms of real hyperbolic geometry.[15, 16] The correspondence seems to have been initiated by Varićak asking for offprints of Einstein’s papers. In his response, Einstein added a personal tone to it with his wife Mileva Marić, a native Hungarian Serb, writing the address in Cyrillic script in order to raise Varićak’s curiosity. After exchanging publications, Varićak soon commented on Einstein’s (now) famous 1905 special relativity paper, pointing to misprints but also raising doubts about his treatment of reflection of light rays off moving mirrors. These were rebutted by Einstein in a response of 28 February 1910 in which he also, with reference to Ehrenfest’s paradox, referred to the rigidly rotating disk as the “most interesting problem” that the theory of relativity would presently have to offer. In his next two letters, dated 5 and 11 April 1910 respectively, Einstein argued against the existence of rigid bodies invoking the impossibility of superluminal signalling, and also discussed the rigidly rotating disk. A resolution of Ehrenfest’s paradox, suggested by Varićak, in terms of a distortion of the radial lines so as to preserve the ratio of $\pi$ with the Lorentz contracted circumference, was called interesting but not viable. The radial and tangential lines would not be orthogonal in spite of the fact that an inertial observer comoving with a circumferential point would only see a pure rotation of the disk’s neighborhood.

About a year later, Einstein and Varićak corresponded once more. Varićak had contributed to the polemic between Ehrenfest and von Ignatowsky by suggesting a distinction between ‘real’ and ‘apparent’ length contraction. The reality of relativistic length contraction was discussed in terms of Ehrenfest’s tracing paper experiment, but for linear relative motion. According to Varićak, the experiment would show that the contraction is only a psychological effect whereas Einstein argued that the effect will be observable in the distance of the recorded marker positions. When Varićak published his note, Einstein responded with a brief rebuttal.[17]

Despite their differences in opinion, the relationship remained friendly. In 1913, Einstein and his wife thanked Varićak for sending them a gift, commented favorably on his son who stayed in Zurich at the time, and Einstein announced sending a copy of his recent work on a relativistic theory of gravitation. The Einstein-Varićak correspondence thus gives us additional insights into a significant debate. It shows Einstein’s awareness of the intricacies of relativistic rigid rotation and bears testimony to the broader context of the conceptual clarifications in the establishment of the special and the genesis of the general theory of relativity.