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The paper, "Demonstrating the
Relationship Between Quantum Mechanics and Relativity"
by John Cipolla unifies
Einstein's theory of general relativity, which governs
gravitational fields around massive objects and quantum
mechanics, or the wave nature of matter and energy. This
paper demonstrates that Newton's Law of Gravitation
can be derived from string theory and that general
relativity and quantum mechanics can be unified under
one theory. In
this analysis we attempt to relate the gravitational
forces predicted by string theory, Einstein’s theory of
general relativity and classical Newtonian gravity. Establishing the rudimentary relationship between
general relativity and quantum
mechanics requires a simple "model equation" that
represents the basic physics
but at a simplistic level. In this analysis the only
major assumption is that particle velocity is much less
than the speed of light, c.
Figure-1, Matter-wave trapped in a
gravitational potential well
Using the concept of a
"model equation" for this
analysis is required because the full string theory
equations are far too
complex at the present time. This basic technique is
used in computational fluid
dynamics (CFD) where solution methods are tested using
one-dimensional "model equations" and later extended to
the full three dimensional nature of fluid flow. For
this analysis the one-dimensional string illustrated in
Figure-2 represents the wave equation in Figure-3 and
its solution. This is a notional string theory solution
and not the "real" solution. But, the basic physics of
any problem can sometimes be revealed using the “model
equation” technique. The solution for the wave equation
in Figure-3 reveals the traveling wave velocity of a
disturbance on a string equals the square root of the
tension on a string divided by linear string mass
density.
Figure-2, String segment
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Figure-3, String wave equation and
solution |
Figure-4, Schrödinger's wave equation for
a free particle in a potential well and probability
function
After acknowledging the wave equation in Figure-3 and
its solution for wave
velocity, represents the physics of a vibrating string
it becomes clear the
Schrödinger's wave equation for a free particle in
Figure-4 has a clear analogy
with the physics of a vibrating string. Therefore, its
seems reasonable to
approach the unification of general relativity and
quantum mechanics by using
physics of the string wave equation as a starting point
to develop a model
equation for a Theory of Everything (TOE).
GRAVITATIONAL FORCE OF ATTRACTION FOR A STRING
This analysis assumes a heterotic or closed superstring
circles the Sun at a
distance equal to the orbital radius of the Earth around
the Sun. The total mass
of the circular superstring is equal to the mass of the
Earth having uniform
mass-energy density along the string. Motion of the
Earth particle is notionally
like that of matter-wave trapped in a potential well as
depicted in Figure-1. In
this case the probability of finding the Earth in any
particular location around
the Sun is described by Schrödinger's wave equation of a
free particle in a
potential well where the probability function and its
solution is described in
Figure-4. In this case the potential well where the
Earth is “trapped” is the
gravitational potential well generated by the mass of
the Sun which curves
space-time as described by the Einstein field equation. Also, the
energy required for the Earth to escape the potential
well where it is “trapped”
is equal to the escape velocity from the Earth’s
location, orbiting the Sun.
The escape velocity is critical to describe the exact solution
required for the
string solution to match classical theories of
gravitation. In addition, the length of
the planetary matter-wave is derived using
the geometrized mass of
the Earth. This results prove that string theory force of
gravitational attraction and string theory gravitational
acceleration are equal to Newtonian theory and general
relativity when v << c. The equations for force of gravitational attraction
and acceleration
between each planet in the solar system and the Sun
indicate excellent agreement
between string theory and classical gravitational
theory. After some algebra string theory gravitational
force of attraction is.
The string theory force of gravitational attraction
derived using string theory is
identical to the Newtonian force of gravitational
attraction except for the vanishingly small term in the
denominator. These results indicate
Newtonian, general relativity and string theory forces
of gravitational
attraction are equal. The classical Newtonian equation
of gravity is presented below for comparison with the
result derived using string theory.
SUMMARY OF RESULTS
This simple thought experiment indicates the tangential
force acting in tension; t
on a heterotic or closed string having radius equal to Rsun
from the Sun and planetary mass, m has an inward
gravitational force, Fstring equal to
the gravitational force, FNewtonian
acting on the planet. These results were achieved using geometrized mass units for planetary string
length and then by setting planetary wave velocity equal
to escape velocity at each planetary location, Rsun.
These results indicate string theory has an embedded
field theory relationship to general relativity and
quantum mechanics when v << c. Finally, these results
demonstrate that a relationship exits between quantum
mechanics and general relativity and that string theory
may provide a basis for describing the relationship
between the gravity, electromagnetism, strong force
(nuclear force) and weak force for a theory of
everything (TOE).
VALIDATION OF THE RELATIONSHIP USING THE SOLAR SYSTEM
The following figure plots the ratio of gravitational force and
acceleration computed using
general relativity at the Newtonian limit (v << c) to the
gravitational force and
acceleration computed by string theory for each of nine planets
in the solar system.
Please notice the amazing result that the force of
gravity and gravitational
acceleration for Newtonian gravity and string theory are
equal making the ratio
of the gravitational results displayed identically equal. These
amazing results indicate the effects of gravity may be
modeled as a heterotic or
closed string as a first order model equation solution
for a unifying theory for
general relativity and quantum mechanics.
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