Sunday, November 4, 2001
The Galilean theorem of
addition of velocities is proved
hereunder as a mathematical theorem.
all Relativistic formulae are derived
from the postulate of the constancy
of the speed of light, which
contradicts this Galilean theorem, and
in mathematics no axiom, postulate,
proposition or theorem may contradict
any other, it is proved hereunder that
the postulate of the constancy of
the speed of light cannot logically
form a part of mathematics as we know
The "mathematics" -- including the "geometry" -- of Relativity are based, not only on the axioms of mathematics (such as those of Peano, or those enunciated by Zermelo and Fraenkel, later extended by John von Neumann) and on the postulates and propositions of geometry, Euclidean or otherwise, but also on the postulate that light propagates in a vacuum at a speed which is constant for all observers, regardless of the speed of the observer relative to the source of the light.
Thus for their formulation, the "mathematics" and "geometry" of Relativity require a postulate additional to the axioms, postulates and propositions from which the rest of mathematics and geometry (as we know them to be) are formulated.
However, logically speaking, no axiom, postulate or proposition in mathematics and geometry may contradict another; nor may it -- nor any theorem derived from it -- contradict any other theorem. If this occurs, that particular axiom, postulate or proposition cannot logically be a part of mathematics and/or geometry.
We logically and mathematically prove hereunder the Galilean theorem of addition of velocities. Since the so-called Relativistic "theorem" of addition of velocities contradicts the Galilean theorem, it is demonstrated logically that the postulate on which Relativistic "mathematics" and "geometry" are based -- namely the postulate of the constancy of the speed of light -- cannot be a part of mathematics and/or geometry as we know them.
Let there be an inertial frame of reference F in which there is an observer O possessing a clock C for measuring time, as well as other instruments -- such as rods -- for measuring distances. Let two point-like bodies B1 and B2 be moving rectilinearly and uniformly past the observer O in opposite directions, at their closest point each body passing at a negligible distance from O and from the other body. Let both B1 and B2 pass O at a single time instant t0 as indicated by the clock C, the body B1 moving at a velocity v1 relative to O, and the body B2 moving at a velocity v2 relative to O.
Let the following be defined:
V = D/T
= (d1 + d2)/T
= (v1T + v2T)/T
= (v1T)/T + (v2T)/T
= (v1 + v2).
This logically and
addition of velocities.
Note that in the above calculation, there is no restriction whatsoever placed on the magnitude of the velocities v1 and v2. Thus they can even be so-called "Relativistic" velocities -- i.e., velocities approaching that of light.
If it were correct that V = (v1 + v2)/(1 + v1v2/c2), which is less than V = (v1 + v2), then the distance between B1 and B2, namely D = VT = [(v1 + v2)/(1 + v1v2/c2)]T, would be less than D = VT = (v1 + v2)T = [(v1T) + (v2T)] = (d1 + d2) -- or in other words, the distance, as measured by the observer O, between B1 and B2 would be less than the sum of the distances between B1 and O on the one hand, and O and B2 on the other, also as measured by the observer O ... which is impossible.