Trouton–Noble experiment

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.[12]

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:

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

and in the right-angle lever:

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

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.[10] [13] [14] [15]

Since then, many papers appeared which elaborated on Laue's current, providing some modifications or re-interpretations, and included different variants of "hidden" momentum.[16]

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.[17] 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.[18] According to Janssen, deciding between Laue's standard model and such alternatives is merely a matter of convention.[18]

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.[19]

A solution without compensating forces or redefinitions of force and equilibrium was published by Richard C. Tolman[20] and Paul Sophus Epstein[21][22] in 1911. A similar solution was re-discovered by Franklin (2006).[23] 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 ${\displaystyle \tan \alpha \!}$ with O. Now a force towards OM is applied at M, and equilibrium in its rest frame is achieved when ${\displaystyle {\tfrac {f'_{x}}{f'_{y}}}=\tan \alpha '}$. As already shown above, these forces have the form in a non-co-moving frame:

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

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

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:

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

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

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.