Examining Mathematical Approaches

From: Asuzaiq (asuzaiq_at_aol.com)
Date: 06/03/04 

Date: 03 Jun 2004 13:08:56 GMT

Examining Mathematical Approaches

     I have received communication from a mathematician who asserted that my
statements about the validity of non-Euclidean Geometry were erroneous. He
apparently objected to the assertion that the definition of a straight line was
faulty for all geometries and that any non-Euclidean Geometry of N dimensions
could be contained in a Euclidean geometry of N+1 dimensions. (The latter
objection might be considered important because it implies that GR has a
responsibility to describe the Euclidean geometry of 4 spatial dimensions
implied by its assertion that our space of three dimensions is curved. The
failure to include a description of such a space does General Relativists
little credit.)

     One of the arguments presented is that the geometry represented by a
circle, a mobius strip or a klein bottle cannot be embedded in a three
dimensional Euclidean geometry but can only described isometrically. If I were
to take his arguments literally I, as someone who IS embedded in a three
dimensional Euclidean space, could not draw a circle on a piece of paper, and
could not construct a klein bottle or a mobius strip! Obviously I, and anyone
else, can do so. It is apparent then that the correspondent's mathematical
approach is faulty. What he should be doing is making sure that his mathematics
is sufficient to do the job before asserting that someone else's understanding
is deficient.

     To illustrate the second conclusion, that the true definition of a
straight line for any geometry is the shortest distance between two points
WHICH REMAINS WITHIN THE GEOMETRY, consider the distance between New York and
Los Angeles. In the two dimensional geometry which represents the surface of
the Earth, the shortest distance is along a great circle. It will be noted that
this meets the revised definition of the straight line for that geometry. In
terms of the three dimensional geometry represented by the Earth as a whole,
the definition also is valid. The shortest distance which remains within the
three dimensional geometry is though a tunnel which passes about 200 miles
below the Mississippi River. The revised definition works for all geometries.

     It is asserted that the force which we sense as gravity results from the
geometry of space. If one would take the trouble to examine this conclusion he
would recognize that there is no way that this would occur. Imagine a perfectly
smooth spherical planet sufficiently isolated so that external gravitational
effects could be ignored. On this planet are scribed two great circles at right
angle to each other. At the intersection of the two lines rests a perfectly
smooth steel ball and along the first line, perhaps 30 degrees away, rests an
identical ball. To simplify the argument let us consider that the balls can
roll over the planet's surface with zero friction. If the balls are started
rolling in a direction at right angles to the first line, they will follow
great circle paths and will approach each other. A GR geometer would assert
that, due to the curved space represented by the two dimensional geometry, the
balls experienced a force which would cause them to approach each other since
they followed a curved path. There is a difficulty with this idea however. The
balls could be stopped anywhere along the path and they would remain where they
were showing that an attractive force between them did not exist! The force
that caused them to approach each other is a vector component of the force of
gravity which is at all times normal to the surface and therefore cannot be
sensed in the two dimensional geometry of the surface. To make the statement
more forcefully, THERE IS NO WAY IN @#$% THAT GEOMETRY CAN PRODUCE THE FORCE OF
GRAVITY!

     The source material for this posting may be found in "Gravity" (1987),
"The Einstein Hoax" (1997), and "Corrections to Residual Errors in Special
Relativity (1999) located at http://www.members.aol.com/einsteinhoax/site.htm.
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.

     Please make any response via E-mail as Newsgroups are not monitored on a
regular basis. Objective responses will be treated with the same courtesy as
they are presented. To prevent the wastage of time on both of our parts, please
do not raise objections that are not related to material that you have read at
the Website. This posting is merely a summary.

     E-mail:- einsteinhoax@aol.com

     The material at the Website has been posted continuously for over 5 years.
In that time THERE HAVE BEEN NO OBJECTIVE REBUTTALS OF ANY OF THE MATERIAL
PRESENTED. There have only been hand waving arguments by individuals who have
mindlessly accepted the prevailing wisdom without questioning it. If anyone
provides a significant rebuttal that cannot be objectively answered, the
material at the Website will be withdrawn.
     
     

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