"Examining Mathematical Approaches"

I have received communications 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.

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