The Ehrenfest Paradox troubled Einstein and was a spur to his development of General Relativity. It seems Einstein thought his Special Relativity was incomplete in its approach to rotating and accelerating reference frames. In view of the inability of modern physics to explain the rotational speeds of stars in spiral galaxies (unless dark matter exists) any issues related to orbits and rotations are worth careful consideration. This essay suggests that while linear motions are relative, rotations are more complicated.
Introduction
Paul Ehrenfest (1880-1933) was friends with Albert Einstein and a successor to Heindrik Lorentz at Leiden University.
The Ehrenfest paradox was formulated by Ehrenfest in 1909 and is a thought experiment involving Special Relativity and an idealized rigid disc which is rotating about its axis very rapidly. The disc radius R as seen in the laboratory frame is always perpendicular to the motion and should therefore remain equal to its value R0 as measured when stationary, so 2πR = 2πR0. However, the circumference (2πR) is moving very rapidly in the surrounding reference frame and so should be contracted by the Lorentz factor γ. Even if it is measured properly (i.e. both ends of the measuring tape are noted at the same well coordinated time within the refernce frame). So 2πR < 2πR0. But it is a contradiction that R=R0 and R<R0.
The paradox was considered by Einstein. He wondered if the Euclidean formula circumference = 2πR stayed true for a rapidly rotating system and this may have influenced his interest in using non-Euclidean geometry within physics. The rotating disc was important for Einstein in developing his theory of General Relativity. He referred to it in publications in 1912, 1916, 1917 and 1922 and his conclusion was that the geometry of the disc becomes non-Euclidean for a co-rotating observer.
The Ehrenfest paradox has attracted the attention of many great scientific thinkers from 1909 to the present day, including Max Born, Fritz Noether, Max Planck, Max von Laue, Arthur Eddington, Paul Langevin, Nathan Rosen, Øyvind Gron, Hrvoje Nikolic and many others.
Theoretical physicists have struggled with the description of relativistic effects in rotating systems for most if not all of the last hundred years. The resolution of the Ehrenfest paradox is still being debated today, over a hundred years after Ehrenfest first posed it!
This is a pity because nearly everything in the Universe is spinning, rotating or orbiting. When the description of something as basic as a spinning disc is so complicated, one might hope that someone will one day invent a better way of looking at the physics of rotating systems, just as Copernicus, Kepler and Galileo improved our mathematical model of the Solar System by making the model heliocentric.
Of course if a real disc is spun up so fast that its circumference approaches relativistic speeds, any material that it is made of will surpass its elastic limits and the disc will disintegrate. So, with the possible exception of some sort of neutron star, the paradox is probably destined to remain a thought experiment.
The fact that this paradox is so problematic suggests that there is something astray in our thinking about it, just as there was something missing in our understanding of infinite processes in the two centuries that followed the paradoxes put forward by Zeno of Elea.
Suggestions for the Ehrenfest Paradox
Imagine that the disc is spinning in an inertial reference frame. Put its centre at (0,0,0,t) and align its axis of rotation with the z axis. Surround it with inertial observers and synchronised clocks. Cut a hole in the middle of the disc and stand a few stationary observers there as well.
Before the disc starts spinning, paint its circumference with blocks of colour at meter intervals. Suppose there are exactly 10 of these intervals. Measure and confirm that the radius of the disc is 10/2π meters long. Put a circular ruler around the disc marked in exactly the same way and attach this ruler to the floor. Now set the disc spinning so that its circumference reaches an appreciable proportion of the speed of light. Say c/2.
In any real disc the material will be stretched apart by the enormous centrifugal forces, but we are imagining a completely rigid disc. Let us try to avoid being drawn into a discussion about the rigidity or otherwise of hypothetical discs, even though that was the focus in the early days of this paradox.
The essence of the paradox is that in Special Relativity the inertial observers standing in the hole cut out in the middle of the disc are supposed to see the radius stay the same but the circumference shrink. Does something weird happens to the geometry of the disc?
Conjecture: What would happen is that the disc would stay completely flat and that observers riding on the disc, or standing in a hole in the middle of the disc, or standing next to the disc or positioned above or below it would observe (i.e. properly measure) a disc which is flat in the usual 3 dimensions of space, and they would all observe that the circumference is unchanged in length and that the ratio of the circumference to the diameter is still exactly equal to π. In other words there would not be any contraction of any lengths at all. The inertial observers would see nothing shrink on the disc and the observers on the disc would see nothing shrink either on the disc or in the external environment.
Note that the rotating disc itself is not an inertial reference frame and any observations made by observers riding on the disc are complicated. Not least because they will have great difficulty in agreeing on the time duration between any two given events. And if they cannot agree on time durations they will stuggle to agree on speeds as well.
However, the resolution is probably simple. The paradox only arises because there is something misleading in the way it is presented it. There is no such thing as perfectly rigid disc. As the disc starts spinning it tries to expand. This is resisted by increasing tension in the intramolecular forces holding it together. As the disc reaches very high tangential speeds it starts to gain relativistic mass. The increasing desire of the rim to stretch and expand counters the Lorentzian contraction of ‘measuring rods’ in the rim of the disc. This is why the disc stays flat. Lorentz and Ehrenfest’s contemporaies were on the right track all along.
The Rotating Disc and Time
If the disc does not display Lorentz contractions, what happens to time as shown on any clocks attached to the disc? Any clock attached to the rotating disc would slow down by a Lorentz factor involving the particular speed of that clock, which is its distance from the centre of the disc multiplied by the rate of rotation. This has been demonstrated by experiments involving very fast centrifuges.
For example, an experiment by Kundig (Kundig, 1963) makes use of the Mossbauer effect and a high speed centrifuge.
There is an emitter of very precisely defined gamma rays at the centre and a very precise absorber of these gamma rays mounted on the rotating rim. There are also detectors fixed outside of the centrifuge and these measure whether the gamma rays from the centre are passing through the absorber on the rim or not. When the centrifuge is in motion, the characteristic resonance absorption frequency of the moving absorber at the rim decreases due to time dilation, and instead of being absorbed the gamma rays pass through to stationary detectors mounted outside the rim.
But is this a Special Relativistic effect or a General Relativistic effect? Clocks mounted on the disc undergo significant forces. They want to travel in a straight tangential line but are constantly forced towards the centre.
Consider clocks at rest in a laboratory on Earth. Such clocks experience a constant acceleration due to gravity and thiscauses a gravitational time dilation. Since General Relativity considers gravity to be locally equivalent to uniform accelerations (and vice versa) and it is proven that gravity causes gravitational time delay, perhaps the centrifuge experiment is demonstrating a General Relativity effect? But it seems not. Only the tangential speed has an effect. The experiment is at a constant potential in the Earth’s graviational field and this affects the source and abd sorber equally. The radial acceleration due to the rapid rotation does not seem to matter and just the tangential speed of the absorber accounts for the reducing efficiency of absorptions as the rotor speed increases.
Life on the Rim
Explanations of what stationary observers can observe about the rapidly rotating disc are complicated. This issue is also present in rigorous interpretations of the Michelson-Morley experiment since the motion of the apparatus through the elusive aether is not uniform rectilinear motion but is instead a combination of surface speed of the Earth’s rotation at the latitude of the experiment, the Earth’s rotation about the Sun, the Sun’s rotation around the galactic core and the overall rotation of the galaxy in the cosmos.
Predicting what observers stationed on the rim of the disc might observe seems to be contentious. Refer to Wikipedia article on the History of the Ehrenfest paradox.
Suppose the external fixed observers see points on the rim moving at V million meters per second in an anti-clockwise direction. The Principle of Classical Relativity says that observers on the rim should see the countryside immediately outside the rim moving at the same speed but in the opposite direction. But will this in fact be true?
Conjecture: The passage of time for the rim based observers will slow down and so they will see or calculate that the outside world going past faster. Here is the argument.
The external observers see one rim circumference go by in T seconds. But a rim based observer sees one rim circumference go by in T’ seconds where T’ is the duration on their local clock. The conjecture is that the observers on the rim will have a slow clock. So when the observers on the rim get back to the measuring point their clock will give a low a low reading, say 9 microseconds instead of 10 microseconds. T’ < T.
The external inertial observers and the rim based observers are both measuring the circumference properly. The external observers can just run a tape measure around it. The rim based observers cannot dispute that this is in fact the length of the rim. All the observers can see both ends of the tape in the same place at the same instant. There is no reason to think the tape has shrunk or stretched just because a nearby disc has been set in motion.
Dividing the circumference by the smaller time value T’ gives a bigger result for the calculated speed at which the scenery is passing by. The rim based observers see the outside world going fast by a factor of γ, and this is simply because their clocks are running slow by a factor of γ.
The Principle of Classical Relativity may seem obvious but it seems that it has not been tested experimentally at relativistic speeds. Nor for that matter has Lorentzian length contraction. It has been inferred to exist from the scattering patterns resulting from collisions in high speed accelerators but that is about all.
Conjecture: Lorentzian time dilation arises in physical materials moving at very high speeds relative to a universal background field. If two systems move at a very high speed relative to each other, the one that is moving the fastest with respect to the universal background will have the greater slow down in their time keeping devices.
The Clock Postulate
Anyone further interested in this paradox might be interest in something called the Clock Postulate. This suggests that time dilation effects depend only upon the relative speed of the clock at any particular instance, and not upon any acceleration of the clock (except that over time the acceleration will affect the relative speed). A corollary to this postulate is that elapsed time durations will depend upon how the clock gets from one speed to the other. An acceleration that is big at first and small later on will give a different result than if the acceleration was small early and bigger later on.
Comment
The Ehrenfest paradox is an important thought experiment. The fact that a simple rotating disc system causes so much complexity and confusion is a concern. If we cannot understand the physics of a simple rotating disc, what hope do we have for understanding a Universe in which nearly everything is rotating? The complexities and confusions seem to reduce if it is admitted that rotations are relative to a Univesal background.
References
Van de Vusse, Sjoerd B.A., 2024, Some ideas and experiments for issues affecting modern physics, https://hereticalphysics.com.au
Kundig, W, 1963, Measurement of the transverse Doppler effect in an accelerated system, Phys Rev 129 : 2371-2375
Author contact: SBAvan@utas.edu.au
Author’s location: Hobart, Australia
