Trouton–Noble experiment
With the Trouton -Noble experiment Frederick Thomas Trouton and Henry R. Noble tried in 1903 to measure in a different way than the Michelson -Morley experiment, the state of motion of the earth relative to the ether. The negative outcome of the Trouton - Noble experiment was next to the Michelson -Morley experiment one of the most important early confirmations of special relativity, and was repeated several times with the same result (see tests of special relativity ).
Relatedly, there are also a number of paradoxes of relativistic statics, which are known for example as " Trouton -Noble paradox" or " angle lever paradox". It involves giving, whether entering through the change of the inertial torque or even a measurable rotation in a static system. There were to propose a series of solutions that agree that no rotation occurs.
Trouton -Noble experiment
In this experiment, a charged plate capacitor was used. This is designed so that it can rotate freely about an axis parallel to the plates, he would be subjected to a torque. Now, when the earth and the capacitor have a speed each charged capacitor plate represents a current, the magnetic field B, should exert a Lorentz force F, and thus a torque in the other plate. An " ether wind speed " at an angle to the vertical line connecting the two plates would the ( ultimately from the angle independent ) to torque
(: Field energy in the capacitor; : speed of light). In the experiment, however, no torque could be detected. It was hence (together with the Michelson - Morley experiment ) a significant objection to the view of a stationary ether or a preferred reference system dar. Similar experiments were later also with even greater precision but the same negative result, by Rudolf Tomaschek (1925, 1926 ), Carl T. Chase (1926, 1927) and Howard C. Hayden (1994 ) repeated.
This result is consistent with the following from the special theory of relativity expectancy agree that the experimental arrangement in accordance with the principle of relativity as in an inertial system can be considered dormant, and consequently can occur no positive result.
This must also apply to all other inertial frames, since a Lorentz transformation (which the coordinates of the inertial ties together ) does not change the result. However, their application to static and dynamic problems proved to be quite difficult and there were different models proposed for the " Trouton -Noble paradox" ( namely, whether a torque in relatively moving inertial system occurs or not) to solve.
Angle lever paradox
The Trouton -Noble paradox is essentially equivalent to the so-called " angle lever paradox" ( Right-angle lever paradox ), which was treated first by Gilbert Newton Lewis and Richard C. Tolman (1909). There was a bell crank with Endpunktion given abc with legs of equal length to the length. In its rest frame, the forces in the direction ba with attack point c and towards bc are there with a point of the same size so that balance, so there is no torque according to the lever rule:
This, however, is viewed from a relatively moving the x-axis system, shrinks due to the length BC is longer than contraction and Ba BC. The law of the lever results in this case:
The torque is not in this frame of reference zero, which would enable the angle lever apparently in rotation. Since this is due to the conservation of angular momentum can not be the case, joined Lewis and Tolman, no torque is present. Consequently, they concluded:
However showed Max von Laue (1911 ) that this is in contradiction with the transformation law of force at coordinate transformation:
Resulting in instead
Results. Applied to the lever principle, results in the following torque:
Which is the same problem is the same as with Trouton -Noble paradox.
Solutions
The detailed relativistic analysis of these paradoxes requires a careful consideration of the relevant forces and impulses. For this, different approaches have been presented, all of them agree that no rotation occurs.
Laue - current
The first solution of the Trouton -Noble paradox was given by Hendrik Antoon Lorentz ( 1904). It is based on the assumption that the pulse and torque can be compensated by the electrostatic forces of the torque pulse, and molecular bonding forces.
This was Max von Laue (1911 ) continued, gave the standard solution to this paradox. It was based on the " inertia of energy ," according to which a flow of energy is generated by elastic stresses, which is also equipped with a pulse ( " Laue - current"). The resultant is ( mechanical ) torque in the case of the Trouton -Noble experiment
And in the case of the angle lever:
Which the above-mentioned electromagnetic torque exactly compensated, so that no rotation occurs. Or, in other words, the electromagnetic torque is even necessary to allow for the uniform movement of a strained body, that around the body to prevent them to rotate due to the mechanical torque.
Since then a number of works have been published, the Laue solution further developed or modified, and for different problems "hidden" pulses ( "hidden momentum" ) introduced.
Reformulations of force and momentum
Other authors were unhappy with the idea of respect system-dependent torques. If in the rest frame of the object no torque occurs, this should not be the case in all other inertial frames. Therefore, an attempt was made to replace the standard expressions for momentum and energy with those that were manifest from the outset Lorentzkovariant. This method is analogous to the solution of the electromagnetic 4/3-Problem mass of electrons according to Enrico Fermi (1922 ) and Fritz Rohrlich ( 1960). Contrary to the standard method where forces and impulses is based on the simultaneity hyperplanes of each observer, merely simultaneity hyperplanes of the rest system of the object to be used in the Fermi - Rohrlich definition. According Jannsen the difference between Laue 's standard solution and such alternative formulations is thus based only on different conventions for selecting the simultaneity hyperplane.
Similarly, different Rohrlich (1967 ) between "apparent " and " true" Lorentz transformations. Direct application of the Lorentz transformation, where the non- concurrent positions of the end points of a route is determined in a moving system, would be a "true" transformation. The Lorentz contraction, however, the result would be an apparent transformation, since additionally the simultaneous positions of the end points have to be calculated in addition to the Lorentz transformation. Additional languages Cavalleri / Salgarelli (1969 ) of " synchronous " versus " asynchronous " formulation of static equilibrium. In your opinion, however, forces and impulses should be considered only in the rest frame of the object synchronously, asynchronously in the moving system.
Force and acceleration
A simple solution that did without compensation and without forces redefinitions, was commissioned by Richard C. Tolman and Paul Sophus Epstein (1911 ). A similar solution was rediscovered by Franklin ( 2006). They pointed to the fact that force and acceleration in relativity theory does not necessarily have the same direction, ie the relationship between mass, force and acceleration tensor. The notes played by the force role in relativity theory is very different from the classical mechanics.
An example: Let there be given a massless rod with endpoints OM. This is secured to the point O, wherein a body of mass m is attached to M. The entire staff includes the angle with O. Now, in M, a force acts in the direction OM, with balance in the rest frame then prevails when. As mentioned above, the shape of these forces is in a moving system relative thereto:
So.
Thus, the resultant force is not directly from O to M. As Epstein showed, however, this does not lead to a rotation, for now he looked at the problems caused by the forces of acceleration. The relativistic expressions for the relationship between mass, acceleration and force are in longitudinal and transverse direction:
So.
Consequently, also in this system, there occurs no rotation. Similar considerations also apply to the Trouton -Noble and the angle lever paradox. The paradoxes are so so resolved, because the two accelerations show ( as vectors ) to the gravity of the system ( capacitor in the Trouton -Noble ), although the forces do not.
Epstein added that if one finds it more satisfying, even in relativity theory, the proportionality between force and acceleration restore ( as in the familiar Newtonian mechanics ), then compensation forces have to be introduced, which coincide formally with Laue current. Epstein developed such a formalism in the following sections of its 1911er work.