Ehrenfest paradox

The Ehrenfest paradox is a paradox of relativity theory and was first discussed in 1909 by Paul Ehrenfest. It states that according to the theory of relativity can exist no rigid body and assumes a non-Euclidean geometry for a co-rotating observer of the room.

Rigid body and Relativity

Max Born in 1909 tried to integrate the concept of a rigid body in the description of accelerated motion in special relativity. The Born rigidity condition states that the distances in the infinitesimal neighborhood of the observer remain constant in a mitbeschleunigten reference system. From the perspective of an inertial system, however, these distances the relativistic length contraction are subject. However, this leads to a contradiction in principle, which was demonstrated in 1909 by Paul Ehrenfest. In its original formulation, he expects a " rigid " cylinder, which is rotated. The description is made in the inertial frame, which will be referred to as " laboratory system " hereinafter. The radius of the cylinder does not change in acceleration. But according to the born between rigidity condition is subject to the scope of the Lorentz contraction. This results in the laboratory system, in which the Euclidean geometry must continue to be valid, the contradictory relation:

Regardless of the limited validity of the Ehrenfest born between rigidity condition has also been recognized by Gustav Herglotz and Fritz Noether (1909). They noted that a shear- born " rigid body " has only 3 degrees of freedom, so the analogy is to the " rigid body " of classical mechanics very limited. Max Planck ( 1910), furthermore, pointed out that this paradox is to be treated in the context of the theory of elasticity. For while the acceleration would occur stresses and deformations are considered. Finally, Max von Laue (1911 ) showed in a simple way, that of rigid bodies can no longer be spoken, because each change of direction immediately causes deformations in the body and therefore can not be a restriction of the degrees of freedom as in the Newtonian mechanics.

This apparent contradiction, therefore, shows that rigid bodies are in general relativity theory in contradiction. This is in connection with the consequence of relativity theory that effects are not faster than the speed of light can propagate, while in a perfectly rigid body, the sound velocity would be infinite. It result from the fact generally the following consequences:

• A disc can not be moved like a " rigid body " from the resting state into rotation, and consequently, there are no rigid body. And also carefully selected forces that act on each point of the body, can be only in selected cases, a deformation avoid. Born's definition of the rigid body can be used only in a very small number of cases. The accelerated rotation is not one of those cases.
• Common materials are therefore in the phase in which they are displaced from the stationary state to perform an accelerated rotation or rotation, different deformations to be subjected, which in turn depend on the nature of the materials. If the disk in the rotating state is greater or less than in the rest, not only depends on the length of contraction, but also by centrifugal force and mechanical stresses.

To this end, the following special case will be considered in the laboratory system: At the edge of a disk several rods to be arranged loosely. The disc is to be deformed, during the phase of accelerated rotation such that the disc periphery is constant in spite of the contraction length to reach the uniform rotation. However, since the present thereon rods are not linked together, are to them as opposed to the disc hardly deformations occur, and they can contract freely. Their mutual distance on the disc is the same size remaining, therefore, be greater. This is analogous to Bell's spaceship paradox: Had some spaceships circularly arranged and connected to each other with ropes and would be from the perspective of the laboratory system the spaceships are accelerating at the same time, then both ships as well as the ropes of length contraction and various deformations would be subject. The spacecraft would be due to their greater resilience to withstand these deformations and be subject only to the length contraction. In contrast, the ropes would be at least stretched, so that the scope of the spaceship -rope circle remain the same tear through the deformations or. So the original idea is Ehrenfests that contracts from the perspective of a laboratory system with a constant radius, the entire circumference, in the framework of the theory of relativity not possible.

Rotation and non-Euclidean geometry

So far, the question has been treated, how from the perspective of the non- co-rotating laboratory system distinguishes a static slice of a rotating disk. However, when a disk is already in uniform rotation, now raises the purely kinematic question, what are the differences in the measurement of the disk occur when the measurement is made either in the laboratory or in a rotating frame of reference. Identical to rods to be used both in the laboratory and in the disk system. Is now measured with these dormant in each system bars the wheel periphery and the disk radius, the result is:

• As demonstrated above, in the laboratory frame the perimeter of the disk is not contracted in proportion to the radius. He is thus in accordance with Euclidean geometry. That are on the disk, however, the moving rods are in tangential, but not subject to the radial direction in accordance with contraction of the length. So you observed in the laboratory system, that the co-rotating observer must invest their rods in tangential direction more often than the observer in the laboratory frame, whereas there is no difference in the radial direction. This means that the measured the contracted rods scope no longer has a ratio of the radius, but.
• As the co-rotating observer on the disk not notice anything of the contraction ( as they themselves are just as subject to length contraction as the rods ), they must assume that the rod lengths are the same in both the radial and tangential directions. That their measurement gives a ratio for them is an expression of the extent of the disk is larger than in the laboratory system, and is interpreted as a result of non-Euclidean geometry in the disk system.
• So Ehrenfest was originally assumed that the disk circumference in the rotating reference system remains the same and in the laboratory frame is smaller. In fact, however, the scope in the laboratory system remains the same and is greater in the rotating reference system.

Formal solutions

Since gravitational forces play no role here, this paradox or the non-Euclidean geometry in the rotating reference system can be quite treated with the agents of the special theory of relativity. Contrary to a popular misconception, this theory is also valid for all accelerations - the general theory of relativity is only needed if gravity is in play. It is essential that the Poincaré -Einstein synchronization of clocks on the whole system, but can be applied locally in rotating frames of reference not only for synchronous idle watches lose during rotation or when accelerating their synchronization.

In 1910, Theodor Kaluza suggested that the geometry is on a slice of non-Euclidean in the sense of hyperbolic geometry. The formal standard solution for the description of non-Euclidean geometry in the rotating reference systems, which in addition to the Ehrenfest 's paradox and the Sagnac effect is to be called, goes back to Paul Langevin (1935 ), and was among others continued by Christian Møller, Lev Davidovich Landau, Evgeny Mikhailovich Lifshitz and Øyvind Grøn ( " Langevin -Landau - Lifschitz metric "). What will also be discussed and where deviations still exist, are questions of detail in the application and interpretation of the Langevin -Landau - Lifschitz metric.

In addition to the special theory of relativity, this problem can be treated with the general theory of relativity, of course, since the former theory is contained in the latter as a limiting case. In fact, this paradox was of great importance for Albert Einstein in the development of the general theory of relativity. For in the special theory of relativity accelerated reference systems and inertial systems are not equal. In general relativity theory, however, Einstein tried to represent all reference systems as equal. For example, should accelerated reference systems ( at least locally ) equivalent to the free fall in a gravitational field be ( principle of equivalence). It was the realization that a non-Euclidean geometry must be used in rotating systems, a key indication that this is necessary also in gravitational fields.

In addition, the complexity of the problem or the ignorance of the above-mentioned formal solution meant that over the decades repeatedly erroneous statements were published. For example, Weinstein (1971 ), the hypothesis held that due to the Thomas precession radial lines would be distorted on the rotating disk, and this effect would be cumulative. Phipps led in 1973 an experiment with a rotating disc for months through to detect tartar effect, with negative results. Whitmire (1972 ), however, already showed previously that such an effect (if it occurs at all ) would immediately thereby occurring voltages are balanced and thus would not be measurable from the outset. Moreover pointed Grøn (1975 ) pointed out that in his development of the relativistic kinematics of rotating discs, no such effect occurs. The theory of relativity is thus in accordance with the negative result.