"Appendix II of the Gravitics Situation"
(Return to Index Page)
Main Document |
Appendix I |
Appendices III - VI
A QUANTUM MECHANICAL APPROACH TO THE EXISTENCE
OF NEGATIVE MASS AND ITS UTILIZATION IN THE
CONSTRUCTION OF GRAVITATIONALLY NEUTRALIZED BODIES
Since the overwhelming majority of electrostatic quantum mechanical effects
rely for their existence on an interplay of attractive and repulsive forces
arising from two types of charge, few if any fruitful results could come
from a quantum mechanical investigation of gravity, unless there should be
two types of mass. The first type, positive mass; (hereafter denoted as
posimass) retains all the properties attributed to ordinary mass, while the
second type, negative mass (hereafter denoted as negamass) differs only in
that its mass is an inherently negative quantity.
By considering the quantum mechanical effects of the existence of these two
types of mass, a fruitful theory of gravity will be developed. Theory will
explain why negamass has never been observed, and will offer a theoretical
foundation to experimental methods of detecting the existence of negamass
and utilizing it in the production of gravitationally neutralized bodies.
To achieve these results, recourse will be made to Schroedinger's time
independent equation with the center of mass motion removed. This equation
is:
where all symbols represent the conventional quantum mechanical quantities.
Particular attention will be paid to the reduced mass µ=(m1m2)/(m1+m2)
where m1 and m2 are the masses of the two interacting bodies.
GS page - 30 -
One can approach the first obstacle that any theory of negamass faces,
namely the explanation of why negamass has never been observed, by a
consideration of how material bodies would be formed if a region of empty
space were suddenly filled with many posimass and negamass quanta. To
proceed along these lines, one must first understand the nature of the
various possible quantum mechanical interactions of posimass and
negamass.
Inserting the conventional gravitational interaction potential into
Schroedinger's equation and solving for the wave function ß, yields the
result that the probability of two posimass quanta being close together is
greater than the probability of their being separated. Hence, there is said
to be an attraction between pairs of posimass quanta. By a similar
calculation it can be shown that while the potential form is the same two
negamass quanta repel each other. This arises from the fact that the
reduced mass term in Schroedinger's equation is negative in this latter
case. The type of negamass posimass interaction is found to depend on the
relative sizes of the masses of the interacting posimass and negamass
quanta, being repulsive if the mass of the negamass quantum is greater in
absolute value than the mass of the posimass quantum, and attractive in the
opposite case. If the two masses are equal in absolute value the reduced
mass is infinite and Schroedinger's equation reduces to (V - E)ß - 0. Since
the solution ß - 0 is uninteresting physically, it must be concluded that V
- E, and, hence, there is no kinetic energy of relative motion. Thus, while
there is an interaction potential between the equal mass posimass and
negamass quanta, it results in no relative acceleration and thus, no mutual
attraction or repulsion while much could be said about the philosophical
implications of the contradiction between this result and Newton's Second
Law, such discussion is out of the scope of the present paper, and the
author shall, instead, return with the above series of derivations to a
consideration of the construction of material bodies in a region suddenly
filled with many posimass and negamass quanta.
GS page - 31 -
Because of the nature of the posimass-posimass and negamass-negamass
interactions, the individual posimass quanta soon combine into small
posimass spheres, while nothing has, as yet, united any negamass quanta.
Since it is reasonable to assume that a posimass sphere weighs more than a
negamass quantum in absolute value, it will attract negamass quanta and
begin to absorb them. This absorption continues until the attraction
between a sphere and the free negamass quanta becomes zero due to the
reduced mass becoming infinite. The reduced mass becomes infinite when the
sphere absorbs enough negamass quanta to make the algebraic sum of the
masses of its component posimass and negamass quanta equal to the negative
of the mass of the next incoming negamass quantum. Thus the theory predicts
that all material bodies after absorbing as many negamass quanta as they
can hold, weigh the same very small amount, regardless of size.
Since this prediction is in violent disagreement with experimental fact,
one must conclude that the equilibrium arising as a result of the reduced
mass becoming infinite has not yet been reached. That is, assuming that
negamass exists at all, there are not enough negamass quanta present in the
universe to allow posimass spheres to absorb all the negamass they can
hold. One is thus able to explain the experimental fact that negamass has
never been observed by deriving the above mechanism in which the smaller
amounts of negamass that may be present in the universe are strongly
absorbed by the greater amounts of posimass producing bodies composed of
both posimass and negamass, but which have a net positive, variable, total
mass
GS page - 32 -
Having thus explained why negamass has never been observed in the pure
state, it is next desirable to derive an experimental test of the existence
of negamass through considering the internal quantum mechanical problem of
small amounts of negamass in larger posimass spheres. One is able to gain
much physical insight into this problem by simplifying it to the
qualitatively similar problem of one negamass quantum in the field of two
posimass quanta that are fixed distance apart. Further simplification from
three dimensions to one dimension and replacement of the posimass quanta
potentials by square barriers, yields a solution in which the ground state
energy E0 of the negamass quantum in the field of one posimass quantum, is
split into two energy levels in the field of the two posimass quanta. These
two levels correspond to even and odd parity solutions of the wave equation
where Eeven lies higher and Eodd lower than E0. The magnitudes of the
differences Eeven-E0 and E0-Eodd depend on the separation distance between
the two posimass quanta, being zero for infinite separation and increasing
as this separation distance is decreased.
Since the energy of a system involving negamass tends to a maximum in the
most stable quantum mechanical configuration, the negamass quantum will
normally be in state Eeven. When the system is excited into state Eodd, the
negamass quantum will favor the situation in which the two posimass quanta
are as far apart as possible, since Eodd increases with increasing
separation distance between the two posimass quanta, and the system tends
toward the highest energy state. Thus independent of and in addition to the
attractive posimass posimass gravitational interaction, there is a
repulsive quantum mechanical exchange interaction between pairs of posimass
quanta when the system is in state Eodd. The result of these two oppositely
directed interactions is that the two posimass quanta are in stable
equilibrium at some separation distance.
GS page - 33 -
Since this equilibrium occurs between all posimass pairs in an elementary
particle, a necessary consequence of the existence of negamass is that when
in the first excited state elementary particles have a partial crystal
structure.
This theoretical conclusion is capable of experimental verification by
performing a Bragg analysis of the elementary particle crystal structure
through shining high energy gamma rays on hydrogen. Part of the gamma ray
energy will be utilized in lowering the system from energy Eeven to Eodd,
and if selective reflection is observed, it will constitute a striking
verification of the existence of negamass. An order of magnitude
calculation shows that, if the equilibrium distance between pairs of
posimass quanta is one one-millionth the radius of an electron, 100 bev
gamma rays will be required to perform this experiment.
Having discussed why negamass has never been observed, and having derived
an experimental test of its existence it is next desirable to develop an
experimental method of utilizing negamass in the production of
gravitationally neutralized bodies by further consideration of some ideas
previously advanced. It has been pointed out that if a source of negamass
is present, aposimass sphere continues to absorb negamass quanta until
equilibrium is reached as a result of the reduced mass becoming infinite.
Because the sphere thus produced is practically massless and because the
gravitational interaction between two bodies is proportional to the product
of their respective masses, it follows that the sphere is practically
unaffected by the presence of other bodies. And thus, the problem of making
gravitationally neutralized bodies is reduced to the problem of procuring a
source of negamass quanta. This will be the next problem discussed.
The binding energy of a negamass quantum in a posimass sphere
GS page - 34 -
may be obtained as one of the eigenvalue solutions to Schroedinger's
Equation. If the negamass quanta in a body are excited to energies in
excess of this binding energy by shining sufficiently energetic gamma rays
on the body these negamass quanta will be emitted and negamass source will
thus be obtained.
To estimate the gamma ray energy required to free a negamass quantum from a
posimass body, certain assumptions must be made concerning the size and
mass of posimass and negamass quanta. Since these quantities are extremely
indefinite, and since the whole theory is at best qualitative, attempting
to estimate the energy would be a senseless procedure. Suffice it to say
that because of the intimate, sub-elementary particle nature of the
posimass-negamass interaction, it seems reasonable to assume that quite
energetic gamma rays will be required to break this strong bond.
To briefly review what has been shown a quantum mechanical theory of
negamass has been developed based on the assumptions that gravitational
interactions obey the laws of quantum mechanics and that all possible
interactions of negamass and posimass with themselves and each other follow
the well known inverse square law. This theory explains the experimental
fact that negamass has never been observed, and outlines plausible
experimental methods of determining the existence of negamass and utilizing
it in the construction of gravitationally neutralized bodies. While these
experimental methods may perhaps be out of the realm of practicality at the
present, there is every reason to hope that they will be performable in the
future. At that time, the plausibility of the existence of negamass and the
theory behind the construction of gravitationally neutralized bodies from
it, will meet their final tests.
GS page - 35 -
SUMMARY PARAGRAPH
A quantum mechanical theory of negative mass is developed, based on the
assumptions that gravitational interactions obey the laws of quantum
mechanics, and that all possible interactions of negative and positive mass
with themselves and each other follow the well-known inverse square law.
This theory explains the experimental fact that negative mass has never
been observed, and outlines plausible experimental methods of determining
the existence of negative mass and utilizing it in the construction of
gravitationally neutralized bodies.
Prof. F. Mozer
GS page - 36 -
