DEMONSTRATING
A THEORY OF EVERYTHING

New Research by
John Cipolla.

"Demonstrating the Relationship
Between Quantum Mechanics and Relativity"  

Please see the viXra e-print archive (4.5 MB)

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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


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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