"The Limitations of Mathematics in Physics"

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Klim - 30 May 2006 02:08 GMT

"The Limitations of Mathematics in Physics"

The present philosophy in theoretical physics is to search for
mathematical relationships that predict experimental results that are then
used to verify the validity of the mathematics in defining our physical
Universe, from the microcosmic to the macrocosmic. The philosophy was
expressed succinctly by Dr. Hawking in Scientific American when he stated
that he didn't care about "reality" because he didn't know what "reality"
was, all that he cared about was whether the mathematics correctly predicted
the results of experiment. Unfortunately, requiring that the results of
observation and of mathematics be in agreement is NECESSARY BUT NOT
SUFFICIENT in our study of the Universe around us.

As a research tool, mathematics suffers from two basic limitations:

A:- It is a linear logic process rather than an area logic process. Any
error along the line of logic being used can easily propagate throughout the
remaining logic chain without giving warning of its presence. (A solution
using an area logic process, on the other hand, is similar to the solution
of a crossword or jigsaw puzzle. An error made anywhere in such a process is
revealed by an incongruity somewhere in the solution. As such, such a logic
process is self-correcting because it forces one to go back and rethink the
problem.)

B:- The logic associated with the mathematical treatment of a process
can lead to results in regions of the problem domain where another
constraint is imposed by Nature. To a mathematically trained physicist such
a constraint may well go unrecognized because it is not inherent in the
mathematics. The existence of such a constraint may render the mathematical
conclusions involved useless.

The limitation described in paragraph A is illustrated by the by the
derivation of the Lorentz Transformation for Transverse Force. Unlike the
derivation of the Lorentz Transformation for Parallel Force that provides
the correct result, the derivation of the Lorentz Transformation for
Transverse Force is incorrect. (The derivation of this transformation is
straightforward and is easily accomplished using the conventional Lorentz
Transformations for Mass, Length and Time. The infamous
http://einsteinhoax.com/rf511.gif, provided a "common sense" warning that an
error had been made in that mathematical derivation. In this though
experiment shown by the diagram, if forces are applied at the ends of the
right angle lever and the lever is observed not to rotate in both the
"moving" and "stationary" reference frames, then the moments applied to each
arm must be equal in magnitude and opposite in direction in each of the
reference frames.

With the conventionally accepted values, the product of the Lorentz
Transformation for Parallel Length and the alleged Transformation for
Parallel Force is not equal to the product of the Lorentz Transformation for
Transverse Length and the alleged Transformation for Parallel Length. This
inequality implies that, if the torques applied to the arms are equal in one
of the reference frames ("moving" or "stationary") they cannot be equal in
the other reference frame ("stationary " or "moving") and the lever would
then rotate in one of the reference frames and not the other. Since the
lever does not seem to know of this requirement, it remains stationary, and
it was necessary to reconcile the dilemma that resulted.

One would think that rational men would accept that an error had been
made in the derivation of the force transformations and search for that
error. Instead, it was assumed that the derivation of the Lorentz
Transformations for the Forces had to be correct, after all they were
derived mathematically. To account for the difficulty, a convoluted
explanation was devised which asserted that the rate of increase of energy
in the transverse lever produced by the parallel force was balanced by the
rate of increase of the angular momentum supplied by the torque difference
between the two directions. The explanation conveniently ignored the fact
that for moments to be applied to the lever by the forces applied to its
ends, equal and opposite forces would result at the hinge pin. As a result,
any energy that is added to the transverse lever arm by the applied parallel
force at its end will be removed at the hinge pin, and the rate of change of
energy in the lever is exactly zero.

What was also ignored was the fact that the angular momentum of the
lever is the product of its moment of inertia and its angular velocity.
Since the lever was observed not to rotate in either reference frames, the
rate of change of the lever's angular momentum is also zero. Properly done,
the mathematical explanation amounts to the statement that 0 = 0. This is
certainly true but is also meaningless. It is frightening that the advanced
academic community could overlook such an error and appear in a large number
of postgraduate level texts. It is even more frightening that individuals
who would embrace such an explanation are in a position to teach our best
and brightest. For a derivation of the correct Lorentz Transformations for
Forces see http://einsteinhoax.com/relcor.htm.

The type of error illustrated by paragraph B occurs in regard to the
idea of "action at a distance" being produced by the exchange of "virtual
particles". Mathematically, such particles are capable of providing both
attractive and repulsive forces. In reality, in the absence of an overlying
substrate for space (e.g.- the classical Aether or Dirac's "Sea of Negative
Energy") with which the "virtual photons" could interchange momentum, the
production by "virtual photons, of attractive forces acting at a distance
would seem to be impossible. But then, if such a substrate exists, why do we
need the concept of "virtual photons"?

Perhaps it would be constructive to illustrate this type of limitation
on the reliability of a purely mathematical treatment by considering the
analysis of a conventional FET transistor circuit, the cascode. This circuit
consists of two FET transistors with the drain of the first one driving the
source of the second one and with the gate of the second biased at a
constant positive voltage. For low frequency signals, the operation of such
an amplifier is completely defined mathematically by a property known as
"transconductance". The electron flow in the collector of the second FET is
the product of the signal voltage applied to the gate of the first FET times
the transconductance of the first FET. The mathematics tells us that a
positive signal will produce a positive flow of electrons at the output and
a negative signal will produce a negative flow of electrons at the output.

What the mathematics doesn't tell us is that the process will not work
unless there was a bias flow of electrons in the amplifier to which the
output signal was added. With the bias current, the amplifier can produce
both positive and negative electron flow signal outputs by adding to or
subtracting from the bias current. Without the bias current, the amplifier
can only produce positive electron flow outputs. The mathematics is
NECESSARY BUT NOT SUFFICIENT to define the process. Similarly, the writer
asserts that the mathematics associated with "virtual photons" is also
NECESSARY BUT NOT SUFFICIENT. A substrate (e.g.- the classical Aether or
Dirac's "Sea of Negative Energy" or one of the several "background energy of
space" concepts) would seem to be necessary for the idea of "virtual
photons" as a force carrier to work, but, if the substrate exists, why is
the concept of "virtual photons" as force carriers required?

Whether "virtual photons" exist can be evaluated by a rather simple
experiment as shown in the diagram hyyp://einsteinhoax.com/cf43.gif. There
is an even simpler means of evaluating the concept. A recent news item
revealed that a loose oxygen cylinder was captured by the magnetic field of
an MIR machine and flew across the room to crush the skull of a child being
examined. If such a strong magnetic field existed in the room and that field
involved "virtual photons", those photons would be sufficiently numerous to
produce an amount of electromagnetic noise that would be easily measured. I
doubt very seriously if such noise would
be found if s search were made.

Remember that Dr. Einstein warned "we have not proven that the Aether
doesn't exist, we have merely proven we do not need it (for calculations).
Also remember that it took 25 years of peer pressure to cause Dr. Einstein
to relinquish the idea of "absolute time" (equivalent to believing the
Aether) and accepting the unproven (and rather ludicrous) consensus
viewpoint of space and time as a single entity.

Theoretical physics has taken the easy position that only mathematics
and experiment are required to deal with reality. Unlike every other branch
of science, it is asserted that "mechanism" is irrelevant in physical
theory. Since we live in the "mechanism" represented by our Universe, the
prime goal of any responsible theoretician should be an understanding of the
workings of that "mechanism" with the mathematical and experimental results
used to iteratively improve our understanding of that "mechanization". What
we have today, with the idea of "mechanism" eliminated, is more akin to
group quasi-religious charlatanism (how many Angels can dance on the head of
pin?) practiced by a controlling cadre of self promoting "scientists" than
it is to an organized science.

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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http://einsteinhoax.com/postinglog.htm.

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please do not raise objections that are not related to material that you
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Vert - 31 May 2006 00:30 GMT

VERGON
Excellent post (as usual).

Another example of mathematics gone bad is a matter of interpretating
the math.
Einstein made such an error.--- as shown in the monograph below.

TIME DILATION

Vertner Vergon

We examine time dilation as set forth in Einstein's paper and find it
has an inconsistency and a reductio ad absurdum. We ascertain a correct
time variation and illustrate its viability.

There is always an open question as to the accuracy of Scientific
articles. So the question arises, how can we judge the accuracy of an
article that involves a theory?

Firstly, the article (theory) must be consistent both internally and
with empiricism. If it isn't it is flawed.

Secondly, it must not violate any known physics laws. Nor, obviously,
can it lead to a reductio ad absurdum.

Theories necessarily contain speculation - else they are not theories
but simply logical constructs. Such constructs must also meet the above
requirements.

Articles generally contain both theory and reference to facts (such as
experiments). These must be verified.

If an article ( or theory) meets these requirements it cannot be
rejected. If it is, it is done so out of sheer obstinancy and/or
stupidity.

Let us examine the concept of Time Dilation as contained in the Special
Theory of Relativity.

We start our examination of the time dilation concept by going to its
source -- Einstein's paper, On the Electrodynamics of Moving Bodies. We
refer to his gedanken experiment of moving clocks. One of two
synchronized identical clocks remains at rest while the other moves
away and returns. When Einstein perceived the difference of the clock
readings in his calculations, he stated the moving clock "was slow by
...". The immediate perception by the public was that he meant if a
clock was "slow by" - it had to have run slower. He also said the
moving clock was "behind" the inertial clock by ... . These two
statements do not mean the same thing. If one clock is running slower,
then it is running slower, and that has only one meaning.

On the other hand to say one clock is behind the other is open to
alternative explanations, e.g., the moving clock could have traveled a
shorter world line -- or may have traveled faster than observed. In
either case the clock would maintain its normal (proper) rate but for a
shorter duration than the inertial clock and thus be behind. At any
rate the accepted version is that the clock ran slower and thus was
born the concept of time dilation.

This concept is usually presented by an illustration that says if one
were to observe a clock on a fast moving spaceship, they would observe
it to run slowly. It must be emphasized that this running slowly occurs
regardless of the direction or vector of the ship. That is to say it
matters not whether the spaceship is receding or approaching, time runs
slower. Also to be emphasized is that this slowness of time is not just
a matter of observation but actually takes place -- for when the clock
returns it is actually 'behind" the stationary (or Earth) clock.

Since this is the declared case, we are inevitably drawn to only one
conclusion, the moving clock has to be running slowly in its own
coordinate system. Here we are faced with an unforgiving contradiction
for basic relativity states that all clocks keep proper time in their
respective coordinate systems. It is inconsistent that a clock can
actually run slowly in its own coordinate system - and also, the
while, keep proper time; in short the clock would have to keep two
different rates simultaneously, a reductio ad absurdum. This
contradiction requires that the time dilation concept, i.e., t' = t (1
- v2/c2) 1/2 be discarded.

The question arises, is there something to take its place? The answer
is yes.
To lay the foundation for this replacement we note that any known
constant frequency is a clock. The scientific community has chosen the
excited cesium atom as the standard atomic clock. This is an arbitrary
choice. Next, we note that the cesium clock (or any other constant
frequency atom) will never vary in its rate. So they will always keep
proper time. However, observations of these clocks will show a
variation due to the Doppler effect. We declare this variation in
frequency is a variation in observed time. Since the frequency is a
clock, an observed variation in the frequency is an observed variation
in time. Call it Doppler or what one will, that is merely a description
of the mechanics. In the final analysis it is a variation in observed
time.

THINGS ARE NOT ALWAYS WHAT THEY SEEM.

Next, we examine a counter argument which states that the relativistic
Doppler rate is the result of the following: If one takes the
non-relativistic Doppler rate and modifies it by the time dilation
they, come up with the relativistic Doppler rate. This, supposedly,
confirms time dilation. Upon further examination, we perceive that even
if we were to accept that explanation, what we have is the situation
whereby an approaching clock is observed to run fast -- not as fast as
non-relativistic mechanics would have -- but fast. This is contrary to
time dilation which requires an approaching clock to run slowly. That a
reversal in direction results in a reversal of time rate is to be
expected since time variation is the result of velocity vectors - and
vectors are directional.
If a theory conflicts with empiricism, it has to be discarded.
Using the Doppler rate as the time rate will yield the time dilation
effect. And that is the proper way to refer to it -- "time dilation
effect". The reason: The net time differential when the movement of the
clock is complete is the same as though the dilation time was operative
throughout.

What then? What is the explanation for the "effect", but not the
actuality? Particle accelerator operators and Ives & Stillwell claim to
have "directly observed" time dilation. What they have observed is
transverse Doppler rates. And it so happens, by some extraordinary
coincidence, that the transverse Doppler rate is the same as the time
dilation rate.

And now to display the Doppler time resolution of the Twins Paradox:
What is displayed here is the round trip Twins experience with no
paradox. The upward arrow signifies outward bound, the downward arrow
signifies inward bound. The double arrows signify the ship being
observed going outbound while it in actuality is traveling inbound.
(second chart). This, of course, is due to the time it takes light to
travel. The observed velocity is v(.75c). At this velocity the so
called time dilation rate is 1/2 . The distance to far-point is sqrt(3
light seconds). So the elapsed time for the ship's transit is one
second due to the fact its actual velocity is sqrt(3 light seconds/sec)
( See The Dual VelocitiesTheory of Relativity.)*
( For seconds or years, the figures hold.)

This means that four years on earth would be only two years on ship.
The time rates shown in the second chart are the time rates of the ship
as observed by earth. They are Doppler rates.

They have to be. Any invariable frequency -- clock or atom -- will be
observed at Doppler rates. The same is true for observations of earth
by the ship (first chart). So there is a parity of rate observations as
required by the principle of relativity.
Notice, there is no time dilation rate observed (1/2 in this case) --
but when the trip is concluded the difference in the clock readings is
as though the ship clock had run at half time to the earth clock, i.e.
, run at dilation time. When Einstein saw that in his calculations
(1905 paper) he said the A clock "ran slow" by ... . This created the
impression that time really ran slower for the ship's clock. The table
below shows this isn't so. It runs at the normal (proper) rate but is
observed to run at the Doppler rate.

It is worth mentioning that according to the Special Theory of
Relativity a clock in the approach mode runs slow - but astronomers
observe clocks in the approach mode, and they are running fast.
Consequently, there is no time dilation observed. There is, however, a
time dilation effect, i.e., the transit time differential on the two
clocks -- and these were created by Doppler time.

OBSERVE:
The program on this NG scatters the tables all over the place. If one
wants to see them go to http://www.wbabin.net --- find the pull down
LIST OF AUTHORS and click on Vertner Vergon. This monograph appears
there.

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Natural Science Forum / Physics / Relativity / May 2006

"The Limitations of Mathematics in Physics"