2011-01-19 15:34:48 UTC
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
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
http://einsteinhoax.com/re511.htm. 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 also 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
http://einsteinhoax.com/relcor.htm 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
http://einsteinhoax.com/hoax.htm (1997); http://einsteinhoax.com/gravity.htm
(1987); and http://einsteinhoax.com/relcor.htm (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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