Trouton–Noble experiment

The Trouton–Noble experiment was an attempt to detect motion of the Earth through the luminiferous aether, and was conducted in 1901–1903 by Frederick Thomas Trouton (who also developed the Trouton's ratio) and H. R. Noble. It was based on a suggestion by George FitzGerald that a charged parallel-plate capacitor moving through the aether should orient itself perpendicular to the motion. Like the earlier Michelson–Morley experiment, Trouton and Noble obtained a null result: no motion relative to the aether could be detected. This null result was reproduced, with increasing sensitivity, by Rudolf Tomaschek (1925, 1926), Chase (1926, 1927) and Hayden in 1994. Such experimental results are now seen, consistent with special relativity, to reflect the validity of the principle of relativity and the absence of any absolute rest frame (or aether). See also Tests of special relativity.

In the experiment, a suspended parallel-plate capacitor is held by a fine torsion fiber and is charged. If the aether theory were correct, the change in Maxwell's equations due to the Earth's motion through the aether would lead to a torque causing the plates to align perpendicular to the motion. This is given by:

τ = − E ′ v 2 c 2 sin ⁡ 2 α ′

where τ is the torque, E the energy of the condenser, α the angle between the normal of the plate and the velocity.

On the other hand, the assertion of special relativity that Maxwell's equations are invariant for all frames of reference moving at constant velocities would predict no torque (a null result). Thus, unless the aether were somehow fixed relative to the Earth, the experiment is a test of which of these two descriptions is more accurate. Its null result thus confirms Lorentz invariance of special relativity.

However, while the negative experimental outcome can easily be explained in the rest frame of the device, the explanation from the viewpoint of a non-co-moving frame (concerning the question, whether the same torque should arise as in the "aether frame" described above, or whether no torque arises at all) is much more difficult and is called "Trouton-Noble paradox", which can be solved in several ways (see Solutions below).

Right-angle lever paradox

The Trouton–Noble paradox is essentially equivalent to a thought experiment called "right angle lever paradox", first discussed by Gilbert Newton Lewis and Richard Chase Tolman in 1909. Suppose a right-angle lever with endpoints abc. In its rest frame, the forces f y towards ba and f x towards bc must be equal to obtain equilibrium, thus no torque is given by the law of the lever:

τ ′ = L 0 ( f x ′ − f y ′ ) = 0

where τ is the torque, and L 0 the rest length of one lever arm. However, due to length contraction, ba is longer than bc in a non-co-moving system, thus the law of the lever gives:

τ = f x ⋅ L 0 − f y ⋅ L 0 1 − v 2 c 2 = L 0 ( f x − f y 1 − v 2 c 2 )

It can be seen that the torque is not zero, which apparently would cause the lever to rotate in the non-co-moving frame. Since no rotation is observed, Lewis and Tolman thus concluded that no torque exists, therefore:

f x f y = 1 − v 2 c 2

However, as shown by Max von Laue (1911), this is in contradiction with the relativistic expressions of force,

f x = f x ′ , f y = f y ′ ⋅ 1 − v 2 c 2

which gives

f x f y = 1 1 − v 2 c 2

When applied to the law of the lever, the following torque is produced:

τ = − L 0 ⋅ f x ′ ⋅ v 2 c 2

Which is principally the same problem as in the Trouton-Noble paradox.

Solutions

The detailed relativistic analysis of both the Trouton-Noble paradox and the Right-angle lever paradox requires care to correctly reconcile, for example, the effects seen by observers in different frames of reference, but ultimately all such theoretical descriptions are shown to give the same result. In both cases an apparent net torque on an object (when viewed from a certain frame of reference) does not result in any rotation of the object, and in both cases this is explained by correctly accounting, in the relativistic way, for the transformation of all the relevant forces, momenta and the accelerations produced by them. The early history of descriptions of this experiment is reviewed by Janssen (1995).

Laue current

The first solution of the Trouton-Noble paradox was given by Hendrik Lorentz (1904). His result is based on the assumption that the torque and momentum due to electrostatic forces are compensated by the torque and momentum due to molecular forces.

This was further elaborated by Max von Laue (1911), who gave the standard solution for these kind of paradoxes. It was based on the so-called "inertia of energy" in its general formulation by Max Planck. According to Laue, an energy current connected with a certain momentum ("Laue current") is produced in moving bodies by elastic stresses. The resulting mechanical torque in the case of the Trouton–Noble experiment amounts to:

τ = E ′ v 2 c 2 sin ⁡ 2 α ′

and in the right-angle lever:

τ = L 0 ⋅ f x ′ ⋅ v 2 c 2

which exactly compensates the electromagnetic torque mentioned above, thus no rotation occurs on both cases. Or in other words: The electromagnetic torque is actually necessary for the uniform motion of a body, i.e., to hinder the body to rotate due to the mechanical torque caused by elastic stresses.

Since then, many papers appeared which elaborated on Laue's current, providing some modifications or re-interpretations, and included different variants of "hidden" momentum.

Reformulations of force and momentum

Other authors were unsatisfied with the idea that torques and counter-torques arise only because different inertial frames are chosen. Their aim was to replace the standard expressions for momentum and force and thus equilibrium by manifestly Lorentz covariant ones from the outset. So when there is no torque in the rest frame of the considered object, then there are no torques in other frames as well. This is in analogy to the 4/3 problem of the electromagnetic mass of electrons, where similar methods were employed by Enrico Fermi (1921) and Fritz Rohrlich (1960): In the standard formulation of relativistic dynamics the hyperplanes of simultaneity of any observer can be used, while in the Fermi/Rohrlich definition the hyperplane of simultaneity of the object's rest frame should be used. According to Janssen, deciding between Laue's standard model and such alternatives is merely a matter of convention.

Following this line of reasoning, Rohrlich (1966) distinguished between "apparent" and "true" Lorentz transformations. For example, a "true" transformation of length would be the result of a direct application of the Lorentz transformation, which gives the non-simultaneous positions of the endpoints in another frame. On the other hand, length contraction would be an example of an apparent transformation, since the simultaneous positions of the endpoints in the moving frame must be calculated in addition to the initial Lorentz transformation. Furthermore, Cavalleri/Salgarelli (1969) distinguished between "synchronous" and "asynchronous" equilibrium conditions. In their view, synchronous consideration of forces should only be used for the object's rest frame, while in moving frames the same forces should be considered asynchronously.

Force and acceleration

A solution without compensating forces or redefinitions of force and equilibrium was published by Richard C. Tolman and Paul Sophus Epstein in 1911. A similar solution was re-discovered by Franklin (2006). They alluded to the fact that force and acceleration do not always have the same direction, that is, the relation of mass, force and acceleration has tensor character in relativity. So the role played by the concept of force in relativity is very different from that of Newtonian mechanics.

Epstein imagined a massless rod with endpoints OM, which is mounted at point O, and a particle with rest mass m is mounted at M. The rod encloses the angle tan ⁡ α with O. Now a force towards OM is applied at M, and equilibrium in its rest frame is achieved when f x ′ f y ′ = tan ⁡ α ′ . As already shown above, these forces have the form in a non-co-moving frame:

f x = f x ′ , f y = f y ′ ⋅ 1 − v 2 c 2 , tan ⁡ α = tan ⁡ α ′ 1 − v 2 c 2

Thus f x f y = tan ⁡ α 1 − v 2 c 2 .

So the resultant force does not directly point from O to M. Does this lead to a rotation of the rod? No, because Epstein now considered the accelerations caused by the two forces. The relativistic expressions in the case, where a mass m is accelerated by these two forces in the longitudinal and transverse direction, are:

a x = f x m γ 3 , a y = f y m γ , where γ = 1 1 − v 2 c 2 .

Thus a x a y = tan ⁡ α .

Thus no rotation occurs in this system as well. Similar considerations are also to be applied to the right-angle lever and Trouton-Noble paradox. So the paradoxes are resolved, because the two accelerations (as vectors) point to the center of gravity of the system (condenser), although the two forces do not.

Epstein added, that if one finds it more satisfying to re-establish the parallelism between force and acceleration with which we are accustomed in Newtonian mechanics, one has to include a compensating force, which formally corresponds to Laue's current. Epstein developed such a formalism in the subsequent sections of his 1911 paper.

References

Trouton–Noble experiment Wikipedia

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