Posting #26:- "The Right Angle Lever Paradox"

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"Great spirits have always encountered violent oppositions from mediocre minds." - A. Einstein

The "Right Angle Lever Paradox" is a classic construct which is taught in most courses in Special Relativity. As with all paradoxes, it reveals that and error has been made in our thinking. It may be interesting then to examine this paradox and the means that is conventionally used for its resolution.

The Right Angle Lever paradox reveals itself when we consider a right angle lever with forces applied to the ends in two different velocity reference frames. The arrangement is diagrammed in the figure below. In this diagram the lever is shown as observed in its own reference frame in Figure B and as observed in a reference frame which is moving at velocity V with respect to the lever in figure A. The lever is aligned with one of its arms parallel to the velocity vector between the reference frames and in both reference frames the lever is observed not to rotate in response to the forces applied to its ends.

In order for the lever not to rotate in response to the forces applied to the ends of the lever, it is necessary that the torques generated on each of the arms be equal and opposite, as observed in both reference frames (A and B). Because of the relativistic contraction observed for the parallel arm, as observed in reference frame B, the moment applied to the parallel arm is observed to be reduced by the factor (1-V^2/C^2)^0.5 multiplied by the Lorentz Transformation for Parallel Force as compared to the moment observed in reference frame A. In the transverse axis there is no relativistic shortening of the lever arm and the moment applied to the transverse arm, as observed in reference frame B is equal to the transverse force multiplied by the Lorentz Transformation for Transverse Force and it would seem that, for the lever not to rotate in either reference frame, the Lorentz Transformation for Transverse Force would have to be (1-V^2/C^2)^0.5 times the Lorentz Transformation for Parallel Force. These transformations were derived (Minkowski) and, most embarrassingly, the required relationship was not obtained. The Lorentz Transformation for Transverse force was found to be the inverse of what was required to prevent the rotation of the lever, or (1-V^2/C^2)^0.5!

It was obvious early on that the paradox required a further explanation. Either the derivation of the Parallel and/or Transverse Transformations for force were faulty or the moments applied to the arms of the lever did not have to balance in order to prevent rotation. Instead of accepting that there was a flaw in the derivation(s) of the Parallel and/or Transverse Transformations, a different and highly creative approach was taken. It was asserted that, in reference frame B, the force applied to the end of the parallel lever added energy to it at the rate of Fp*V and added angular momentum to the lever at the rate of Fp*L. It was then argued that the rate at which energy was added to the lever and the rate at which angular momentum was added to the lever produced equal and opposite effects and the lever did not rotate in either reference frame! IT SHOULD BE NOTED THAT THE DISCUSSION UP TO THIS POINT IS IN COMPLETE AGREEMENT WITH STANDARD TEXTS ON THE SUBJECT. From this point on , however, the discussion diverges from the texts.

If one examines the expression for the angular momentum of an object one will note that its angular momentum about an axis is the product of the moment of inertia about that axis and the angular velocity about that axis. Since the lever is observed not to rotate about its pivot pin axis in either reference frame, one must conclude that, since its moment of inertia is not infinite, ITS RATE OF CHANGE OF ANGULAR MOMENTUM MUST BE ZERO as observed in both reference frames! Next, if one examines any text on basic mechanics one observes that, in order for a torque to exist, a couple must ales exist. (A couple is defined by the presence of equal and opposite forces separated by a distance. The torque is equal to the product of the separation between these forces and their magnitudes.) In the case of the lever, the couple results from the presence of the force at the end of the lever and the resulting reaction force component at the hinge pin which is equal in magnitude and opposite in direction to the force at the end of the lever. (This is a requirement of classical mechanics. Advanced physics and cannot be by-passed by the use of more advanced physics.) When these effects are considered, the supposedly elegant solution to the Right Angle Lever Paradox breaks down to the statement that zero=zero. This is most certainly true, BUT IT IS HARDLY MEANINGFUL.

The Lorentz Transformations for Parallel and Transverse Force are readily derived without the use of advanced mathematics or Electromagnetic Theory (apparently used by Minkowski and which has the potential for introducing error). All that is needed are the well known Lorentz Transformations of the Special Theory of Relativity, the recognition that E=M*C^2, and simple algebra. It is readily shown that the Lorentz Transformation for Parallel Force as currently provided is correct but the correct value for the Lorentz Transformation for Transverse Force is the reciprocal of the accepted value. The correct transformation is 1/ (1-V^2/C^2)^)0.5. With this transformation, the right angle lever paradox is no longer a paradox. What it signified is that the accepted derivation of the Transformation for Transverse Force was erroneous. Apparently this error was not recognized because it was inconceivable that a mathematical approach could produce a faulty conclusion. Lesson:- anyone or anything can screw up.

The material which derives the writer's conclusions is provided in "Corrections to Residual Errors in Special Relativity" for your reference. The writer has received an E-mail from an individual which asserted that he had derived the Lorentz Transformation for Transverse Force using Maxwell's Equations and found its accepted value to be correct. He probably used the method used by Minkowski. That method, since it involves using the velocity of light, would probably produce the observed error since the velocity of light is must be considered in both reference frames and velocity is measured using both length and time. The writer doesn't know the exact nature of his error and frankly, he doesn't care.

The source material for this posting may be found in "The Einstein Hoax" (1997) ; "Gravity" (1987) ; and "Corrections to Special Relativity" (1997). EVERYTHING WHICH WE ACCEPT AS TRUE MUST BE CONSISTENT WITH EVERYTHING ELSE WE HAVE ACCEPTED AS TRUE, IT MUST BE CONSISTENT WITH ALL OBSERVATIONS, AND IT MUST BE MATHEMATICALLY VIABLE. PRESENT TEACHINGS DO NOT ALWAYS MEET THIS REQUIREMENT. THE WORLD IS ENTITLED TO A HIGHER STANDARD OF WORKMANSHIP FROM THOSE IT HAS GRANTED WORLD CLASS STATUS.

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