7 The Einstein-Varićak Correspondence on
Relativistic Rigid Rotation∗
Tilman Sauer
Einstein Papers Project
California Institute of Technology 20-7
Pasadena, CA 91125, USA
tilman@einstein.caltech.edu
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 Varićak from the years 1909 to 1913.
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 paradox4 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 instanta-
neously 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 mathe-
matical consequences of Born’s definition.7 Herglotz, in particular, reformulated
∗To appear in: Proceedings of the Eleventh Marcel Grossmann Meeting on General Rela-
tivity, ed. H. Kleinert, R.T. Jantzen and R. Ruffini, World Scientific, Singapore, 2007.
http://arxiv.org/abs/0704.0962v1
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 rel-
ativity 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 Som-
merfeld 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 heuris-
tics 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 published13 nine letters by
Einstein to Vladimir Varićak (1865–1942), professor of mathematics at Agram
(now Zagreb, Croatia). Varićak had published on non-Euclidean geometry14 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 ask-
ing 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 treat-
ment 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 π 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 sug-
gesting a distinction between ‘real’ and ‘apparent’ length contraction. The real-
ity of relativistic length contraction was discussed in terms of Ehrenfest’s tracing
paper experiment, but for linear relative motion. According to Varićak, the ex-
periment 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 signifi-
cant debate. It shows Einstein’s awareness of the intricacies of relativistic rigid
rotation and bears testimony to the broader context of the conceptual clarifica-
tions in the establishment of the special and the genesis of the general theory
of relativity.
References
[1] J. Stachel, Einstein and the Rigidly Rotating Disk, in General Relativity
and Gravitation: One Hundred Years after the Birth of Albert Einstein.
Vol. 1, ed. A. Held (Plenum, 1980), 1–15; see also “The First Two Acts,”
in J. Stachel. Einstein from ‘B’ to ‘Z’ (Birkhäuser, 2002), 261–292.
[2] G. Maltese and L. Orlando. Stud. Hist. Phil. Mod. Phys. 26, 263 (1995).
[3] M. Klein et al. (ed.) The Collected Papers of Albert Einstein. Vol. 3. The
Swiss Years: Writings, 1909–1911. (Princeton University Press, 1993),
478–480.
[4] M. Klein. Paul Ehrenfest: The Making of a Theoretical Physicist. (North-
Holland, 1970), 152–154.
[5] M. Janssen, J. Norton, J. Renn, T. Sauer, J. Stachel. The Genesis of Gen-
eral Relativity: Einstein’s Zürich Notebook. Vol. 1. Introduction and Source.
Vol. 2. Commentary and Essays. (Springer, 2007).
[6] M. Born. Ann. Phys. 30, 1 (1909); Phys. Zs. 10, 814 (1909).
[7] G. Herglotz, Ann. Phys. 31, 393 (1910); F. Noether, Ann Phys. 31, 919
(1910).
[8] M. Born, Nachr. Königl. Ges. d. Wiss. (Göttingen) 161 (1910).
[9] A. Einstein, Jahrb. Radioaktiv. Elektr. 4, 411 (1907); M. Laue, Phys. Zs.
12, 85 (1911).
[10] P. Ehrenfest, Phys. Zs. 10, 918 (1909).
[11] P. Ehrenfest, Phys. Zs. 11, 1127 (1910); 12, 412 (1911); W.v.Ignatowsky,
Ann. Phys. 33, 607 (1910); Phys. Zs. 12, 164, 606 (1911).
[12] M. Klein et al. (ed.) The Collected Papers of Albert Einstein. Vol. 5. The
Swiss Years: Correspondence, 1902–1914. (Princeton University Press,
1993).
[13] D. Buchwald et al. (ed.) The Collected Papers of Albert Einstein. Vol. 10.
The Berlin Years: Correspondence, May–December 1920 and Supplemen-
tary Correspondence, 1909–1920. (Princeton University Press, 2006).
[14] V. Varićak. Jahresber. dt. Math. Ver. 17, 70 (1908); Atti del Cong. inter-
nat. del Mat. 2, 213 (1909).
[15] V. Varićak. Phys. Zs. 11, 93, 287, 586 (1910); Jahresber. dt. Math. Ver.
21, 103 (1912).
[16] S. Walter. The Non-Euclidean Style of Minkowskian Relativity, in The
Symbolic Universe. ed. J. Gray (Oxford University Press, 1999), 91–127.
[17] V. Varićak, Phys. Zs. 12, 169 (1911); A. Einstein. Phys. Zs. 12, 509 (1911).
	Introduction
	Relativistic rigid motion
	The Einstein-Varicak correspondence

  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.

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 paradox4 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 instanta-
neously 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 mathe-
matical consequences of Born’s definition.7 Herglotz, in particular, reformulated
∗To appear in: Proceedings of the Eleventh Marcel Grossmann Meeting on General Rela-
tivity, ed. H. Kleinert, R.T. Jantzen and R. Ruffini, World Scientific, Singapore, 2007.
http://arxiv.org/abs/0704.0962v1
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 rel-
ativity 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 Som-
merfeld 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 heuris-
tics 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 published13 nine letters by
Einstein to Vladimir Varićak (1865–1942), professor of mathematics at Agram
(now Zagreb, Croatia). Varićak had published on non-Euclidean geometry14 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 ask-
ing 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 treat-
ment 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 π 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 sug-
gesting a distinction between ‘real’ and ‘apparent’ length contraction. The real-
ity of relativistic length contraction was discussed in terms of Ehrenfest’s tracing
paper experiment, but for linear relative motion. According to Varićak, the ex-
periment 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 signifi-
cant debate. It shows Einstein’s awareness of the intricacies of relativistic rigid
rotation and bears testimony to the broader context of the conceptual clarifica-
tions in the establishment of the special and the genesis of the general theory
of relativity.
References
[1] J. Stachel, Einstein and the Rigidly Rotating Disk, in General Relativity
and Gravitation: One Hundred Years after the Birth of Albert Einstein.
Vol. 1, ed. A. Held (Plenum, 1980), 1–15; see also “The First Two Acts,”
in J. Stachel. Einstein from ‘B’ to ‘Z’ (Birkhäuser, 2002), 261–292.
[2] G. Maltese and L. Orlando. Stud. Hist. Phil. Mod. Phys. 26, 263 (1995).
[3] M. Klein et al. (ed.) The Collected Papers of Albert Einstein. Vol. 3. The
Swiss Years: Writings, 1909–1911. (Princeton University Press, 1993),
478–480.
[4] M. Klein. Paul Ehrenfest: The Making of a Theoretical Physicist. (North-
Holland, 1970), 152–154.
[5] M. Janssen, J. Norton, J. Renn, T. Sauer, J. Stachel. The Genesis of Gen-
eral Relativity: Einstein’s Zürich Notebook. Vol. 1. Introduction and Source.
Vol. 2. Commentary and Essays. (Springer, 2007).
[6] M. Born. Ann. Phys. 30, 1 (1909); Phys. Zs. 10, 814 (1909).
[7] G. Herglotz, Ann. Phys. 31, 393 (1910); F. Noether, Ann Phys. 31, 919
(1910).
[8] M. Born, Nachr. Königl. Ges. d. Wiss. (Göttingen) 161 (1910).
[9] A. Einstein, Jahrb. Radioaktiv. Elektr. 4, 411 (1907); M. Laue, Phys. Zs.
12, 85 (1911).
[10] P. Ehrenfest, Phys. Zs. 10, 918 (1909).
[11] P. Ehrenfest, Phys. Zs. 11, 1127 (1910); 12, 412 (1911); W.v.Ignatowsky,
Ann. Phys. 33, 607 (1910); Phys. Zs. 12, 164, 606 (1911).
[12] M. Klein et al. (ed.) The Collected Papers of Albert Einstein. Vol. 5. The
Swiss Years: Correspondence, 1902–1914. (Princeton University Press,
1993).
[13] D. Buchwald et al. (ed.) The Collected Papers of Albert Einstein. Vol. 10.
The Berlin Years: Correspondence, May–December 1920 and Supplemen-
tary Correspondence, 1909–1920. (Princeton University Press, 2006).
[14] V. Varićak. Jahresber. dt. Math. Ver. 17, 70 (1908); Atti del Cong. inter-
nat. del Mat. 2, 213 (1909).
[15] V. Varićak. Phys. Zs. 11, 93, 287, 586 (1910); Jahresber. dt. Math. Ver.
21, 103 (1912).
[16] S. Walter. The Non-Euclidean Style of Minkowskian Relativity, in The
Symbolic Universe. ed. J. Gray (Oxford University Press, 1999), 91–127.
[17] V. Varićak, Phys. Zs. 12, 169 (1911); A. Einstein. Phys. Zs. 12, 509 (1911).
	Introduction
	Relativistic rigid motion
	The Einstein-Varicak correspondence
