Trouton-Noble experiment
Encyclopedia
The Trouton–Noble experiment (also connected to thought experiment
s like the "Trouton-Noble paradox", "Right-angle lever paradox", or "Lewis-Tolman paradox") attempted 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.
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:
where
is the torque, 
the energy of the condenser, 
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".
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
towards ba and 
towards bc must be equal to obtain equilibrium, thus no torque is given by the law of the lever:
where
is the torque, and 
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:
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:
However, as shown by Max von Laue
(1911),
this is in contradiction with the relativistic expressions of force,
which gives
When applied to the law of the lever, the following torque is produced:
Which is principally the same problem as in the Trouton-Noble paradox.
(1904). His result is based on the assumption, that the torque and momentum due to electrostatic forces, is 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. And the resulting mechanical torque in the case of the Trouton-Noble experiment amounts to:
and in the right-angle lever:
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 and provided some modifications or re-interpretations. Other authors criticized Laue's solution, and tried to find another solution by redefining the concepts of force, equilibrium etc. (see the list at "further reading" below.)
(1911).
He alluded to the fact, that force and acceleration are not necessarily having the same direction any more in relativity, that is, the relation of mass, force and acceleration has tensor
character. So the role played by the concept of force in relativity is very different from that of Newtonian mechanics.
For example: Imagine 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
with O. Now a force towards OM is applied at M, and equilibrium in its rest frame is achieved when 
. As already shown above, these forces have the form in a non-co-moving frame:
Thus
.
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:
, where 
.
Thus
.
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.
Thought experiment
A thought experiment or Gedankenexperiment considers some hypothesis, theory, or principle for the purpose of thinking through its consequences...
s like the "Trouton-Noble paradox", "Right-angle lever paradox", or "Lewis-Tolman paradox") attempted to detect motion of the Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...
through the luminiferous aether
Luminiferous aether
In the late 19th century, luminiferous aether or ether, meaning light-bearing aether, was the term used to describe a medium for the propagation of light....
, and was conducted in 1901–1903 by Frederick Thomas Trouton
Frederick Thomas Trouton
Frederick Thomas Trouton FRS was an Irish physicist known for Trouton's Rule and experiments to detect the Earth's motion through the luminiferous aether.- Life and work :...
(who also developed the Trouton's ratio) and H. R. Noble. It was based on a suggestion by George FitzGerald
George FitzGerald
George Francis FitzGerald was an Irish professor of "natural and experimental philosophy" at Trinity College in Dublin, Ireland, during the last quarter of the 19th century....
that a charged parallel-plate capacitor
Capacitor
A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...
moving through the aether should orient itself perpendicular to the motion. Like the earlier Michelson–Morley experiment, Trouton and Noble obtained a null result
Null result
In science, a null result is a result without the expected content: that is, the proposed result is absent. It is an experimental outcome which does not show an otherwise expected effect. This does not imply a result of zero or nothing, simply a result that does not support the hypothesis...
: no motion relative to the aether could be detected.
This null result was reproduced, with increasing sensitivity, by Rudolf Tomaschek
Rudolf Tomaschek
Rudolf Karl Anton Tomaschek was a German experimental physicist. His scientific efforts included work on phosphorescence, fluorescence, and gravitation. Tomaschek was a supporter of deutsche Physik, which resulted in his suspension from his university posts after World War II...
(1925, 1926), Chase (1926, 1927) and Hayden in 1994.
Such experimental results are now seen, consistent with special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
, to reflect the validity of the principle of relativity
Principle of relativity
In physics, the principle of relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference....
and the absence of any absolute rest frame (or aether). See also Tests of special relativity.
Trouton–Noble Experiment
In the experiment, a suspended parallel-plate capacitorCapacitor
A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...
is held by a fine torsion fiber and is charged. If the aether theory were correct, the change in Maxwell's equations
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
due to the Earth's motion through the aether would lead to a torque
Torque
Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....
causing the plates to align perpendicular to the motion. This is given by:
where
is the torque, the energy of the condenser, 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".
Right-angle lever paradox.
The Trouton–Noble paradox is essentially equivalent to a thought experimentThought experiment
A thought experiment or Gedankenexperiment considers some hypothesis, theory, or principle for the purpose of thinking through its consequences...
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
towards ba and towards bc must be equal to obtain equilibrium, thus no torque is given by the law of the lever:where
is the torque, and the rest length of one lever arm. However, due to length contractionLength contraction
In physics, length contraction – according to Hendrik Lorentz – is the physical phenomenon of a decrease in length detected by an observer of objects that travel at any non-zero velocity relative to that observer...
, ba is longer than bc in a non-co-moving system, thus the law of the lever gives:
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:
However, as shown by Max von Laue
Max von Laue
Max Theodor Felix von Laue was a German physicist who won the Nobel Prize in Physics in 1914 for his discovery of the diffraction of X-rays by crystals...
(1911),
this is in contradiction with the relativistic expressions of force,
which gives
When applied to the law of the lever, the following torque is produced:
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 LorentzHendrik Lorentz
Hendrik Antoon Lorentz was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect...
(1904). His result is based on the assumption, that the torque and momentum due to electrostatic forces, is compensated by the torque and momentum due to molecular forces.
This was further elaborated by Max von Laue
Max von Laue
Max Theodor Felix von Laue was a German physicist who won the Nobel Prize in Physics in 1914 for his discovery of the diffraction of X-rays by crystals...
(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
Max Planck
Max Karl Ernst Ludwig Planck, ForMemRS, was a German physicist who actualized the quantum physics, initiating a revolution in natural science and philosophy. He is regarded as the founder of the quantum theory, for which he received the Nobel Prize in Physics in 1918.-Life and career:Planck came...
. According to Laue, an energy current connected with a certain momentum ("Laue current") is produced in moving bodies by elastic stresses. And the resulting mechanical torque in the case of the Trouton-Noble experiment amounts to:
and in the right-angle lever:
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 and provided some modifications or re-interpretations. Other authors criticized Laue's solution, and tried to find another solution by redefining the concepts of force, equilibrium etc. (see the list at "further reading" below.)
Force and Acceleration
A simple solution without using compensating forces, and without the need of redefining the relativistic concepts of force, equilibrium etc., was published by Paul Sophus EpsteinPaul Sophus Epstein
Paul Sophus Epstein was a Russian-American mathematical physicist...
(1911).
He alluded to the fact, that force and acceleration are not necessarily having the same direction any more in relativity, that is, the relation of mass, force and acceleration has tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
character. So the role played by the concept of force in relativity is very different from that of Newtonian mechanics.
For example: Imagine 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
with O. Now a force towards OM is applied at M, and equilibrium in its rest frame is achieved when . As already shown above, these forces have the form in a non-co-moving frame:Thus
.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:
, where .Thus
.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.
Further reading
- Michel Janssen, "A comparison between Lorentz's ether theory and special relativity in the light of the experiments of Trouton and Noble, Ph.D. thesis (1995). Online: TOC, pref., intro-I, 1, 2, intro-II, 3, 4, refs.
External links
- Kevin Brown, "Trouton-Noble and The Right-Angle Lever at MathPages.
- Michel Janssen, "The Trouton Experiment and E = mc2," Einstein for Everyone course at UMNUniversity of MinnesotaThe University of Minnesota, Twin Cities is a public research university located in Minneapolis and St. Paul, Minnesota, United States. It is the oldest and largest part of the University of Minnesota system and has the fourth-largest main campus student body in the United States, with 52,557...
(2002).