Trouton–Noble experiment

The Trouton–Noble experiment (also connected to thought experiments 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.

Trouton–Noble Experiment

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:

$\tau =-E'{\frac {v^{2}}{c^{2}}}\sin 2\alpha '$

where $\tau$ is the torque, $E$ the energy of the condenser, $\alpha$ 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 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:

$\tau '=L_{0}\left(f'_{x}-f'_{y}\right)=0$

where $\tau$ 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:

$\tau =f_{x}\cdot L_{0}-f_{y}\cdot L_{0}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}=L_{0}\left(f_{x}-f_{y}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\right)$

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:

${\frac {f_{x}}{f_{y}}}={\sqrt {1-{\frac {v^{2}}{c^{2}}}}}$

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

$f_{x}=f'_{x},\ f_{y}=f'_{y}\cdot {\sqrt {1-{\frac {v^{2}}{c^{2}}}}}$

which gives

${\frac {f_{x}}{f_{y}}}={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$

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

$\tau =-L_{0}\cdot f'_{x}\cdot {\frac {v^{2}}{c^{2}}}$

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 and momenta. 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, 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:

$\tau =E'{\frac {v^{2}}{c^{2}}}\sin 2\alpha '$

and in the right-angle lever:

$\tau =L_{0}\cdot f'_{x}\cdot {\frac {v^{2}}{c^{2}}}$

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 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 Epstein (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 $\tan \alpha \!$ with O. Now a force towards OM is applied at M, and equilibrium in its rest frame is achieved when ${\tfrac {f'_{x}}{f'_{y}}}=\tan \alpha '$. As already shown above, these forces have the form in a non-co-moving frame:

$f_{x}=f'_{x},\ f_{y}=f'_{y}\cdot {\sqrt {1-{\frac {v^{2}}{c^{2}}}}},\ \tan \alpha =\tan \alpha '{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}$

Thus ${\frac {f_{x}}{f_{y}}}={\frac {\tan \alpha }{1-{\frac {v^{2}}{c^{2}}}}}$.

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:

$a_{x}={\frac {f_{x}}{m\gamma ^{3}}},\ a_{y}={\frac {f_{y}}{m\gamma }}$, where $\gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$.

Thus ${\frac {a_{x}}{a_{y}}}=\tan \alpha$.

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.

See also

• History of special relativity

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.

• Tolman, R.C. (1917), "The Right-Angled Lever", The theory of relativity of motion, Berkeley: University of California press, pp. 152–153

• Pauli, Wolfgang (1921), "Die Relativitätstheorie", Encyclopädie der mathematischen Wissenschaften, 5 (2): 685–689 {{citation}}: |chapter= ignored (help)

• Butler, J. W. (1968). "On the Trouton-Noble Experiment". American Journal of Physics. 36 (11): 936–941. Bibcode:1968AmJPh..36..936B. doi:10.1119/1.1974358.

• Aranoff, S. (1969). "Torques and Angular Momentum on a System at Equilibrium in Special Relativity". American Journal of Physics. 37 (4): 453–454. Bibcode:1969AmJPh..37..453A. doi:10.1119/1.1975612.

• Furry, W. H. (1969). "Examples of Momentum Distributions in the Electromagnetic Field and in Matter". American Journal of Physics. 37 (6): 621–636. Bibcode:1969AmJPh..37..621F. doi:10.1119/1.1975729.

• Newburgh, R. G. (1969). "The relativistic problem of the right-angled lever: The correctness of the Laue solution". Il Nuovo Cimento B. 61 (2): 201–209. Bibcode:1969NCimB..61..201N. doi:10.1007/BF02710928.

• Butler, J. W. (1970). "The Lewis-Tolman Lever Paradox". American Journal of Physics. 38 (3): 360–368. Bibcode:1970AmJPh..38..360B. doi:10.1119/1.1976326.

• Aranoff, S. (1972). "Equilibrium in special relativity" (PDF). Il Nuovo Cimento B. 10 (1): 155–171. Bibcode:1972NCimB..10..155A. doi:10.1007/BF02911417.

• Sears, Francis W. (1972). "Another Relativistic Paradox". American Journal of Physics. 40 (5): 771–773. Bibcode:1972AmJPh..40..771S. doi:10.1119/1.1986643.

• Aranoff, S. (1973). "More on the Right-Angled Lever at Equilibrium in Special Relativity". American Journal of Physics. 41 (9): 1108–1109. Bibcode:1973AmJPh..41.1108A. doi:10.1119/1.1987485.

• Grøn, Ø. (1973). "The asynchronous formulation of relativistic statics and thermodynamics". Il Nuovo Cimento B. 17 (1): 141–165. Bibcode:1973NCimB..17..141G. doi:10.1007/BF02906436.

• Nickerson, J. Charles; McAdory, Robert T. (1975). "The Trouton-Noble paradox". American Journal of Physics. 43 (7): 615–621. Bibcode:1975AmJPh..43..615N. doi:10.1119/1.9761.{{cite journal}}: CS1 maint: multiple names: authors list (link)

• Grøn, Ø. (1978). "Relativistics statics and F. W. Sears". American Journal of Physics. 46 (3): 249–250. Bibcode:1978AmJPh..46..249G. doi:10.1119/1.11164.

• Cavalleri, G.; Grøn, Ø.; Spavieri, G.; Spinelli, G. (1978). "Comment on the article "Right-angle lever paradox" by J. C. Nickerson and R. T. McAdory". American Journal of Physics. 46 (1): 108–109. Bibcode:1978AmJPh..46..108C. doi:10.1119/1.11106.{{cite journal}}: CS1 maint: multiple names: authors list (link)

• Holstein, Barry R.; Swift, Arthur R. (1982). "Flexible string in special relativity". American Journal of Physics. 50 (10): 887–889. Bibcode:1982AmJPh..50..887H. doi:10.1119/1.13002.{{cite journal}}: CS1 maint: multiple names: authors list (link)

• Aguirregabiria, J. M.; Hernandez, A.; Rivas, M. (1982). "A Lewis-Tolman-like paradox". European Journal of Physics. 3 (1): 30–33. Bibcode:1982EJPh....3...30A. doi:10.1088/0143-0807/3/1/008.{{cite journal}}: CS1 maint: multiple names: authors list (link)

• Prokhovnik, S. J.; Kovács, K. P. (1985). "The application of special relativity to the right-angled lever". Foundations of Physics. 15 (2): 167–173. Bibcode:1985FoPh...15..167P. doi:10.1007/BF00735288.{{cite journal}}: CS1 maint: multiple names: authors list (link)

• Singal, Ashok K. (1993). "On the "explanation" of the null results of Trouton-Noble experiment". American Journal of Physics. 61 (5): 428–433. Bibcode:1993AmJPh..61..428S. doi:10.1119/1.17236.

• Teukolsky, Saul A. (1996). "The explanation of the Trouton-Noble experiment revisited". American Journal of Physics. 64 (9): 1104–1109. Bibcode:1996AmJPh..64.1104T. doi:10.1119/1.18329.

• Jefimenko, Oleg D. (1999). "The Trouton-Noble paradox". Journal of Physics A. 32 (20): 3755–3762. Bibcode:1999JPhA...32.3755J. doi:10.1088/0305-4470/32/20/308.

• Jackson, J. D. (2004). "Torque or no torque? Simple charged particle motion observed in different inertial frames". American Journal of Physics. 72 (12): 1484–1487. Bibcode:2004AmJPh..72.1484J. doi:10.1119/1.1783902.

• Ivezić, Tomislav (2005). "Axiomatic Geometric Formulation of Electromagnetism with Only One Axiom: The Field Equation for the Bivector Field F with an Explanation of the Trouton-Noble Experiment". Foundations of Physics Letters. 18 (5): 401–429. arXiv:physics/0412167. Bibcode:2005FoPhL..18..401I. doi:10.1007/s10702-005-7533-7.

• Franklin, Jerrold (2006). "The lack of rotation in the Trouton Noble experiment". European Journal of Physics. 27 (5): 1251–1256. arXiv:physics/0603110. Bibcode:2006EJPh...27.1251F. doi:10.1088/0143-0807/27/5/024.

• Ivezić, Tomislav (2006). "Trouton Noble Paradox Revisited". Foundations of Physics. 37 (4–5): 747–760. arXiv:physics/0606176. Bibcode:2007FoPh...37..747I. doi:10.1007/s10701-007-9116-x.

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 UMN (2002).