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Joined: 26 Jun 2006
|Posted: Fri Jun 30, 2006 2:10 pm Post subject:
"Examining Mathematical Approaches"
"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
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
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