ABSOLUTE ROTATION AND EHRENFEST PARADOX
BY A ROTATING LIGHT CLOCK
Giovanni Zanella
Dipartimento di Fisica e di Astronomia dell’Università di Padova and
Istituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8,
35131 Padova, Italy
Abstract
In this paper it is showed that using a transversal light clock, integral with a
rotating disk, we can measure locally, and by a kinematic method, the
velocity of the disk relatively to the velocity of the light. Furthermore, it is
possible recognize, by the invariance of the path of the light rays between
the mirrors of the clock, that the Lorentz distortions on time-space take
place only in case of translation, but not in case of rotation, solving the
known Ehrenfest paradox.
Introduction
The rotation of the bodies, contrary to the translation, has an absolute
character and can be locally investigated by kinematical measurements.
The first experiment, aimed at measuring the angular velocity of the Earth
by an interferometry method, was performed by G. Sagnac in 1913. Its
purpose was to detect the effect of the relative motion of the aether, in
practice to decide between the idea of a stationary aether and an aether
which is completely dragged by the Earth [1][2].
Sagnac method can be applied also to the measurement of the absolute
angular velocity of a rotating rigid disk. The setup, integral to the disk, split
a beam of coherent light in two beams which are made to follow an optical
loop in opposite directions. Both beams end up at the same detector where
the phases of the two signals are compared, so the angular velocity of the
apparatus is related to the time difference on the arrival of the light beams on
the detector.
Alternately, we display in this paper the possibility to measure
kinematically the angular velocity of a rigid disk using a transversal light
clock, having the advantage to verify the space-time invariance in a pure the
1
rotation motion. Consequently, the Ehrenfest paradox, which concerns the
rotation of a rigid disk in the special theory of relativity [3], finds a simple
solution.
2. Absolute rotation measurement
We can suppose to install, integrally to a rotating rigid disk, a mass-less
transversal light clock (Fig.1). A mirror is put on the rim of the disk and the
other in its rotation center, so the path of the light ray between two
consecutive reflections corresponds always to the radius R of the disk.
A local observer can measure the angular velocity of the disk by the angle
 described during the time t elapsed between two consecutive
reflections of the light ray. Indeed, while the direction of the reflected light
ray cannot rotate during its fly, the disk (and the mirrors) rotate of an angle
 (Fig.1).
Therefore, being the time t=R/c, the angular velocity will be

  c

t
R
3. Ehrenfest paradox and its solution
The paradox essentially concerns a rotating rigid disk which radius R’,
always perpendicular to its motion, should be equal to its stationary value R,
on the point of view of the special relativity. On the contrary, the external
circumference should appear Lorentz-contracted according to a radius R’<R
[3].
Vice versa, the insertion of a transversal light clock, as in Fig.1, permits us
to verify the invariance of the path of the light ray between two consecutive
reflections and consequently the invariance of the clock time at any angular
velocity of the disk. Hence, we can conclude that in a rotating rigid body do
not appear Lorentz distortions on time and space, confirming that the
Lorentz transforms work only for inertial translatory motions.
On the contrary, if we limit our measurement on a portion of disk, for
example on a peripheral portion as in Fig.2, this portion is affected both by a
translatory and rotatory motion, so the Lorentz transforms can be applied
only to the translation. Indeed, as we can see in Fig.2, the path of the light
ray depends on the velocity of the disk.
2
4. Conclusions
In this paper we demonstrate that, using a transversal light clock, fixed to a
rotating rigid disk in such a manner that the path of the light ray corresponds
exactly to the radius of the disk, it is possible to measure locally and by a
kinematical method the angular velocity of the disk itself. In practice we find
that the absolute velocity of the disk is related to the velocity of the light,
confirming the result obtained by G. Sagnac by an interferometry method
[1][2].
Furthermore, the transversal light clock permits us to verify operationally
the invariance of the time associated to the rotating disk and consequently
the non-applicability of the Lorentz transforms to the rotating bodies,
according to the arguments of A. Grünbaum and A.J. Janis [4], R.D. Klauber
[5] and A. Tartaglia [6] aimed to solve the Ehrenfest paradox.
References
[1] G. Sagnac, Comptes Rendus 157 (1913) 708.
[2] G. Sagnac, Comptes Rendus 157 (1913) 1410.
[3] P.Ehrenfest, Physikalische Zeitschrift 10 (1909) 918.
[4] A. Grünbaum and A.J. Janis, Synthèse 34 (1977) 281.
[5] R.D. Klauber, Am. J. Phys. 67 (2) (1999) 158.
[6] A. Tartaglia, Fund. Phys. Lett. 12 (1999) 17.
Figure captions
Fig.1 Pure rotation of a transversal light clock, integral in a rotating rigid
disk. The path of the light rays happens always between the center of
the disk and its rim.
Fig.2 Translation and rotation motion of a transversal light clock, integral in
a portion of a rotating rigid disk.
3

Reflecting
mirrors
R

Fig. 1
Reflecting
mirrors
Fig. 2
4