Trouton–Noble experiment

The Trouton–Noble experiment 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. (Such thought experiments are also referred to as "Trouton-Noble paradox", "Right-angle lever paradox", or "Lewis-Tolman paradox"). 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.

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. 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.

Like any experiment measuring very small forces, the Trouton–Noble experiment is very difficult to control so as to eliminate outside influences. For example, based on standard electromagnetism and relativity, Nieves et al. predict that a very slight deflection could result from the interaction with the Earth's magnetic field and its axial rotation. However, this effect would be far smaller than that originally sought by Trouton and Noble.

Right-angle lever paradox.

The Trouton–Noble experiment is essentially equivalent to the so-called "right angle lever paradox" thought experiment, 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:

$T'=L_{0}\left(f'_{x}-f'_{y}\right)=0$

where $T$ 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, and $f_{x}$ and $f_{y}$ are different, thus the law of the lever gives:

$T=L_{0}\left(f'_{x}-f'_{y}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\right)$

It can be seen that the torque is not zero, which apparently would cause the lever to rotate in the non-co-moving frame.

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). He argued, that the torque arising in this experiment is compensated by the momentum of the electromagnetic field.

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" (based on the formulation of Max Planck) according to which any energy flow is connected with momentum. That is, in moving bodies an energy current connected with a certain momentum ("Laue current") is initiated, which exactly compensates the torque, thus no rotation occurs.

Force and Acceleration

A more formal solution is due to Paul Sophus Epstein (1911). He alluded to the fact, that force and acceleration are not proportional (or parallel) any more in relativity. For example: Imagine a rod with endpoints OM, which is mounted at point O. The rod encloses the angle $\tan \alpha \!$ with this point. Now a force towards OM is applied at M, and equilibrium in its rest frame is achieved when ${\tfrac {f'_{x}}{f'_{y}}}=\tan \alpha '$. 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 forces do 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 these forces. The relativistic expressions are:

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

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

So the paradox is resolved, and similar considerations are also to be applied to the right-angle lever and Trouton-Noble paradox. Epstein added, that the role played by the concept of force is therefore very different in relativity. However, if one finds it more satisfying to re-establish the parallelism between force and acceleration, one has to include a compensating force, which formally corresponds to Laue's current.

See also

• History of special relativity

Further reading

• 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. 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. 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. doi:10.1007/BF02710928.

• Butler, J. W. (1970). "The Lewis-Tolman Lever Paradox". American Journal of Physics. 38 (3): 360–368. 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. 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. doi:10.1119/1.1987485.

• Grøn, Ø. (1973). "The asynchronous formulation of relativistic particle mechanics". Il Nuovo Cimento B. 34 (1): 127–140. doi:10.1007/BF02723345.

• Nickerson, J. Charles; McAdory, Robert T. (1975). "The Trouton-Noble paradox". American Journal of Physics. 43 (7): 615–621. 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. 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. 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. 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. 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. 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.

• 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.

• 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. 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. 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).