time. Ψ(x,t) - Natural PhilosophyGeneral Relativity and Quantum Mechanics are the twentieth...

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THE SIXTH PROBLEM LECTURES Copyright © 2009

A testable solution to mathematics and physics greatest problem is presented for the first time.

Ψ(x,t)

Author: Mr. Jamahl Peavey, BSME M. Ed Physics and Mathematics Instructor

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PREFACE Mega experiments and abstract mathematics dominance in modern physics has provided scientist with a wealth of information. This project was developed to increase our understanding related to information. Over one hundred years ago David Hilbert, proposed an axiomatic approach to physics. With this approach, logic would insure understanding such that most phenomena could be generalized by fundamental principles or axioms. In THE SIXTH PROBLEM lectures, this approach is applied to Newtonian Mechanics, General Relativity, and Quantum Mechanical concepts simultaneously. The results provide the simplest testable solutions. “All things being equal, the simplest solution is most likely the answer”, this is Occam Razor as well as Hilbert’s expectations related to an axiomatic approach. The binary star systems used to test these solutions are phenomenal. Until now, these magnificent structures have never been adequately explained. “Physics most endearing ideas exist in a domain where logic, mechanics, mathematical form and experimental confirmation are inseparable.” Jamahl Peavey

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THE LECTURE INDEX

THE SIXTH PROBLEM– INTRODUCTION 4-5

THE SIXTH PROBLEM – AXIOM I 6-7

THE SIXTH PROBLEM – AXIOM 2 8-9

THE SIXTH PROBLEM – THE COUPLING CIRCUIT 10-12 THE SIXTH PROBLEM – ORBITS 13-14

THE SIXTH PROBLEM – FINE STRUCTURE CONSTANT 15-16

THE SIXTH PROBLEM – GRAVITATIONAL COUPLINGS 17-18

THE SIXTH PROBLEM – PRINCIPLE QUANTUM NUMBER 19-20

THE SIXTH PROBLEM – THE SUN AND MERCURY 21-25

THE SIXTH PROBLEM – INTERFACING FIELDS 26-27

BINARY STAR SYSTEMS SOLVED

THE SIXTH PROBLEM – PSR B1913+16 28-29

THE SIXTH PROBLEM – PSR B1534+12 30-31 THE SIXTH PROBLEM – PSR B2127 +11C 32-33

THE SIXTH PROBLEM – PSR J1756-2251 34-35

THE SIXTH PROBLEM–PSR J0737-3039A 36-37

THE SIXTH PROBLEM–PSR J1518+4904 38-39 THE SIXTH PROBLEM–B2303+46 40-41 THE SIXTH PROBLEM–V541 CYGNI 42-43 THE SIXTH PROBLEM–DI HERCULIS 44-45 THE SIXTH PROBLEM– AS CAMELOPARDALIS 46-47

CONSEQUENCES THE SIXTH PROBLEM – PREDICTIONS 48-50

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THE SIXTH PROBLEM-INTRODUCTION

How to unite a community?

David Hilbert was considered the most important mathematician of the last century. Invariant theory and axiom development for geometry was among his most significant contributions. With the goal of setting the agenda for the next century, he addressed the International Congress of Mathematicians at Paris in 1900. During this assembly, he proposed 23 problems that he considered most important for mathematicians to solve during the century. Among these was the sixth problem and it states, “Mathematical Treatment of the Axioms of Physics. The investigations on the foundations of geometry suggest the problem. To treat in the same manner, by means of axioms, those physical sciences in which already today mathematics plays an important part; in the first rank are the theory of probabilities and mechanics” [1].

The sixth problem is virtually untouched by mathematicians. Many mathematicians then

and now considered it a very strange proposal because of its connection to physical science. David Hilbert on the other hand considered the proposal very significant. He said during the assembly, “Further, the mathematician has the duty to test exactly in each instance whether the new axioms are compatible with the previous ones. The physicist, as his theories develop, often finds himself forced by the results of his experiments to make new hypotheses, while he depends, with respect to the compatibility of the new hypotheses with the old axioms, solely upon these experiments or upon a certain physical intuition, a practice which in the rigorously logical building up of a theory is not admissible. The desired proof of the compatibility of all assumptions seems to me also of importance, because the effort to obtain such proof always forces us most effectually to an exact formulation of the axioms.” [1]. David Hilbert’s words would be prophetic.

What divided a community? Max Planck’s position in the physics community was comparable to David Hilbert’s position in mathematics. In the same year David Hilbert spoke, Max Planck solved a problem as perplexing as any of Hilbert’s twenty-three. He developed an equation that described the energy released by heated bodies or black body radiation. The equation was experimentally confirmed, yet there was a problem. In order for the equation to work, Max Planck had to assume the unthinkable. He introduced an assumption that was incompatible with the experimentally confirmed classical electromagnetic wave theory. Max Planck assumed radiation could exist as particles and with that assumption Quantum Mechanics was born. Max Planck’s assumption would be experimentally confirmed with Einstein’s work on the Photoelectric Effect and Bohr’s mathematical explanation for the hydrogen atom spectra. The particle verses wave interpretation for radiation was an old debate dating back to Isaac Newton and Christian Huygens but this was very different. Newton believed light or radiation was a particle and Christian Huygens believed it was a wave. Thomas Young and James Clerk Maxwell experimentally proved light was a wave and now Einstein and Bohr experimentally proved light was a particle. Experimentation is to science what the gavel is to law. With two experimental verdicts, scientist embraced a dual nature of light. In a continued search for meaning, scientist turned to mathematics or the language of science. They hoped to find the reality that had escaped them in

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experimentation. Instead, they found complex abstract mathematics and the lost of reality. Worst of all, they would witness the redefining of physics itself. The cause and effect that had defined physics as deterministic was replaced by the wave function 𝜓 𝑥, 𝑡 and its statistical reality. This statistical reality is known as the Copenhagen Interpretation. Lead by Bohr, Heisenberg, Born, Pauli and John von Neumann physics became unreal and without any deep meaning. Einstein, Schroedinger, De Broglie, Planck, Bohm, Landé, Popper and Russell would oppose this view in an attempt to save the universe of cause and effect. Many of these figures would meet at the Solvay Conference of 1927 to determine the nature of a statistical reality. They remained as divided as the mathematics that came to define their views. The split was so deep it became philosophical, so much that Einstein would declare, “God does not play dice”. Statistical verses determinism and particle verses wave. These two views of reality represent the greatest problems in physics with one solution. Solve The Sixth Problem! The Sixth Problem and Unification General Relativity and Quantum Mechanics are the twentieth century’s greatest sciences. General Relativity or the modern interpretation of gravity is mechanical and deterministic while Quantum Mechanics or the description of atomic systems is probabilistic. Each employs the use of advance mathematics to describe reality. The mathematics is so advance, David Hilbert would say, “Physics is obviously too difficult to be left to the physicist”. With this, he actively encouraged mathematicians to enter the field of physics. Einstein viewed their entrance as an invasion saying, “Every since mathematicians invaded General Relativity, I don’t know what it means anymore”. Adding to Einstein’s displeasure, it was David Hilbert who published the correct Einstein Field Equations. The abstract mathematics that came to define the unobservable atom was an open invitation to mathematicians. It was not long before the Hilbert Space came to define, dominate and claim Quantum Mechanical systems. Over the years, General Relativity would remain unchanged as an advanced classical model for gravity. Quantum Mechanics on the other hand would continue to grow as a unifying concept for reality. Richard Feynman would develop a Quantum Mechanical version of electromagnetism called QED or Quantum Electrodynamics. Abdus Salam, Sheldon Glashow and Steven Weinberg would unify electromagnetism and the weak nuclear force to produce the electroweak. David Gross would develop a quantized strong force or Quantum Chromodynamics. The continued success of Quantum Mechanics, lead many to believe it was time to quantize Einstein’s gravitational field theory. For this to happen, Einstein’s deterministic universe would have to die statistically. String Theory, Super Gravity, Twistor Theory, Non-commutative geometry and others represent the final quest. This quest continues to claim the careers of many notable physicist and mathematicians. Yet in the shadow of this attractive quest hides the heart of the problem which is all but forgotten, The Sixth Problem. REFERENCE [1] Mathematical Problems: Lecture delivered before the International Congress of Mathematicians at Paris in 1900.

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THE SIXTH PROBLEM – AXIOM I

AXIOM I – Physic is applied mathematics Why are axioms important? Axioms are givens forming the foundations for proofs. The consequences related to these axioms lead to general conclusions. This logical approach is the most effective way to solve problems. The axiomatic approach to geometry transformed geometry from a science to a mathematical system. As a natural science, geometric conclusions were reached experimentally. For example, Pythagoreans Theorem could be tested experimentally. Just draw right triangles and test the equation. How many test would you need to complete before the statement was considered valid? The answer is no number of experiments can establish validity for certain. A logical proof is the only way to establish this certainty. Pythagoras was the first known to present such a proof. Because the axiomatic approach to geometry was successful in transforming geometry, Hilbert believed physical sciences depending on those geometries could be transformed into mathematical systems.

Experimental conclusions are sometimes inconsistent. Experimental conclusions are not always consistent. The experimental conclusions below were reached by classical physicist.

The universe can be visualized as a machine. All motion has a cause. An object’s motion is deterministic. Light is an electromagnetic wave

These conclusions are inconsistent with the experimental conclusions reached by quantum physicist. A few quantum mechanical conclusions are below.

The universe is probabilistic. Outcomes related to motion are statistical. Given the same conditions outcomes can be

different. Light is a particle. Objects are defined by a wave function.

Finding axiomatic based solutions to physical problems would remove these inconsistencies and reveal more general forms for describing physical systems.

In 1905, Albert Einstein found such an axiomatic solution. He postulated that the laws of physics were invariant or unchanged in any reference frame and that the speed of light was constant with respect to any reference frame. The consequences related to these axioms allowed Einstein to generalize James Clark Maxwell’s equations for electromagnetism. The generalized forms represent the discovery of expressions free from reference frame inconsistencies. These forms also represent a perfect example of how mathematics, physics and conceptual analysis are interwoven constructively. In fact, defining mathematics is critical to establishing our first axiom.

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What is pure mathematics? Pure mathematics is the study of mathematical systems. Understanding how mathematical systems function requires a strong knowledge related to logic and proofs. Excluding Hilbert’s 6th Problem, the other problems proposed are purely mathematical. For example, Hilbert’s 11th Problem is to solve quadratic forms with any algebraic numerical coefficients. He stated, “Our present knowledge of the theory of quadratic number fields puts us in a position to attack successfully the theory of quadratic forms with any number of variables and with any algebraic numerical coefficients. This leads in particular to the interesting problem: to solve a given quadratic equation with algebraic numerical coefficients in any number of variables by integral or fractional numbers belonging to the algebraic realm of rationality determined by the coefficients. The following important problem may form a transition to algebra and the theory of functions [1].” Unlike the 11th Problem, the 6th Problem is applied mathematics. What is applied mathematics? In general we can consider applied mathematics to be mathematic forms coupled to a physical system. Based on this definition, physics is applied mathematics such that mechanics and mathematics are discovered to exist in the same domain. For example, Equation 1 is the general form for a quadratic equation and Equation 2 is how that quadratic form is applied to a physical system. The mathematics and mechanics for exist in the same realm. Defining physics as applied mathematics means there are no abstract structures in physics.

(1)

(2)

The variables and coefficients for the general quadratic form become real under applied conditions. In this case, the variables and becomes position and time respectively. The coefficients A, B and C also correspond to physical structures. Axiom 1 is scale independent. This means mathematical descriptions in the atomic world have physical meaning because atoms are physical systems. Therefore, the wave function 𝜓 𝑥, 𝑡 , the principle quantum number , the fine structure constant and all orbits have physical meaning. Unlocking the meaning of these structures is essential to solving The Sixth Problem. REFERENCE [1] Mathematical Problems: Lecture delivered before the International Congress of Mathematicians at Paris in 1900.

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THE SIXTH PROBLEM – AXIOM 2

Axiom 2: Space coupled harmonic systems exhibit optimized and synchronized motion.

How is space defined?

In Newtonian Mechanics, space is a rigid background in which physical processes occur. In General Relativity, space is a flexible structure that is shaped by matter and energy. Einstein described this flexible structure as a space-time continuum. There is mathematical and experimental proof that Einstein defined space in a more accurate manner than Newton. If space-time characteristics are harmonic and acts between harmonic systems then synchronized motion exist between fields, objects and space-time. The synchronized conditions between these structures are best illustrated by the coupled harmonics model. How does the coupled harmonics model work? This diagram represents two balls suspended by two springs. The springs have different lengths to represent different harmonic frequencies and they are oscillating independently. If space-time connects objects and has flexible characteristics similar to a spring then this diagram describes a dependently oscillating system. Space-Time Once connected, each object must move according to the harmonics describing by the system. In this case, an equilibrium oscillation will form over time until the objects are moving In Phase with

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one another as described below. Binary star systems are excellent structures for testing Axiom 2. A gravitational field and a gravitating object is model as an object oscillating from a spring. An electromagnetic field and an electron are modeled as an object oscillating from a spring. A neutron star exists as the coupling of both systems in space and time. This means a synchronized motion exist between neutron stars and their electromagnetic fields. This synchronized motion can only exist if both systems are deterministic. Therefore, the statistical forms related to motion must exist as potentials in the space-time continuum. The relationship between the statistics of motion potentials and actual motion is a solution to Hilbert’s 6th Problem. The goal is to identify and test the structure describing this relationship. Gravitation Electromagnetism Space-Time

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THE SIXTH PROBLEM – THE COUPLING CIRCUIT

Why are visual models important? No matter how powerful the mathematical model, visualization is critical to understanding physical systems. Without visual models, equations become abstract and hard to conceptualize. For example, Newton’s spatial view is more visual than Einstein’s but less accurate.

Einstein’s space-time continuum cannot be visualized because it exists in four

dimensions. In quantum mechanics, the wave function is even less visual because it has no

real meaning according to quantum interpretation. The wave function is said to exist as a statistical reality. In fact, Werner Heisenberg said the atomic world described by could not be

visualized. He and others went so far as to say there is no reality beyond measurement. How to visualize basic coupling? Basic coupling is easy to visualize and explain because you are simply mapping a pure mathematical expression onto a physical system. Newton’s universal law of gravitation is basic coupling because the gravitational force is mapped onto a closed ellipse. The projectile discussed earlier is another example of basic coupling because the motion of the projectile is simply mapped onto a quadratic form. Basic coupling is a good mathematical way for modeling systems assuming the geometries describing the system are not effect by the mechanics of the system. How to visualize mechanical coupling? Mechanical coupling is also easy to visualize and explain. Einstein said, “Matter tells space-time geometry how to curve and those space-time curves tell matter how to move”. This is not basic coupling because the geometries are active within the mechanical process. The space we generally regard as three dimensional graph paper is really dynamic and interactive. Mechanical coupling is the best way to model the interaction between geometry and physical systems acting on those geometries. How do describe the radiation-matter interaction mechanically? The dynamic variables associated with radiation are active within the mechanical processes describing matter. This interaction is best modeled and visualized as a parallel resistive circuit.

Light speed signal Potential

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The node is a point in space and time carrying the structural and dynamic variables for

radiation. The node is a point in space and time carrying the structural and dynamic variables for

a particle. These nodes are never equal in space and time so the interaction between radiation and a particle always has and resistance. Radiation’s resistance to localization when absorbed or released by a particle is defined as or Planck’s constant. The geometric resistance to localization is superposition or . This superposition maps the radiation’s wavelength onto the particle’s position and radiation’s velocity onto the particle’s velocity. This mapping is symmetric. For example, and are not equal because one is the energy for a particle and the other is energy for radiation. Equating the two is an interaction described by Equation 3.

(3)

Planck’s constant is angular momentum and it can be compared to the classical description for angular momentum as described by Equation 4. (4)

The symmetry between both equations show radiation’s speed or is superimposed on the particle’s velocity and the radiation’s wavelength is superimposed on the particle’s position R. The radiation matter interaction is symmetric such that equal resistance is defined as for

as the product of two dynamic variables. Heisenberg’s Uncertainty Principle ≥

is a masterful manipulation of equivalence and superposition characteristics which exist between radiation-matter interactions. The statistical conditions he placed on the object’s motion as uncertainty is really a feature of space-time potentials. The wave functions indentified in binary star system will confirm this assumption. What is the circuit potential? The circuit potential is space-time capacity. For example, a container has a potential or capacity based on its size. The actual state is how much the container is holding. An interaction is completely described when the potential and actual state are identified with certainty. The wires and signal

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The wire is a model for the space-time connective feature. The invariant speed of light is the signal which ensures symmetry between space-time and the matter radiation interaction. The coupling circuit is a simple visual aid for understanding how matter and radiation interacts in a closed system.

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THE SIXTH PROBLEM – ORBITS

How are orbits described? In general, an orbit is a path one object scribes in space around another object. The Moon has an orbit around the Earth, the Earth has an orbit around the Sun and the Sun has an orbit within our galaxy. Kepler, Newton and Einstein successfully described celestial orbits mathematically. In Newton and Kepler’s mathematical description, orbits are closed elliptical paths. Newton’s Universal Law of Gravitation describes the force between a smaller mass m and a larger mass M at some distance r apart. The form described by this equation is elliptical. Einstein described orbits as open elliptical paths. The open elliptical path allows the orbit to rotate in space. This rotation or precession exists in all celestial orbits so Einstein’s mathematical expression is more accurate then Newton’s and Kepler’s. Newton described orbits as projectiles in continuous free fall and Einstein described orbits as the shortest distance through curved space-time, but neither explained why orbits exist mechanically. Why do orbits exist? The mechanical relationship between objects and satellites can be derived from their mathematical relationship. In Newton’s Law of Universal Gravitation, objects and satellites are virtual singularities within an ellipse. For example, the Sun and Mercury do not have volume in the ellipse describing their relationship. The Sun is at a foci and Mercury is at a point on the ellipse. If mass is treated as if it is at a point then we can consider it a virtual singularity. Newton’s Second Law also describes colliding objects as virtual singularities. Each object is just a point on a vector.

What happen to the volume that defined these objects and how is it related to orbits? These answers not only explain the mechanical nature of orbits but it also explains space-time. How is volume and orbits related? To understand the relationship between volume and orbits, classical constraints must be placed on space. These constraints are listed below.

1. Mass cannot exist at a point. 2. Masses cannot be superimposed. Such attempts produce volume displacement. 3. Boundaries can be place on any system such that conservation laws can be applied to the

system.

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The Universal Law of Gravitation and Newton’s Second Law must be valid under the three spatial constraints. Their equivalence as described below must also support the validity of the three spatial constraints.

The equivalence shows the acceleration due to gravity is independent of the smaller mass. This must be true under the three spatial constraints.

Point A

Point A can have any mass and will remain constant at the surface of the object. If Point A

has mass then it violates the first constraint. If Point A is allowed to expand to avoid violating the first constraint, it will violate the second constraint unless volume displacement destroys the boundaries of the system as defined by the third constraint. The spatial constraints are valid for Point A of any mass if virtual singularities are impact points on the harmonic nature of space. Orbits exist as extensions of these impact points or virtual singularities. These orbits harmonic structure is sinusoidal. The sinusoid is In Phase with Point A’s path around the object. All harmonic systems have frequencies. This frequency or inverse time cannot be separated from space. This connection forms what Einstein identified as space-time.

The sinusoidal orbit is universal. Sinusoidal orbits are essentially extensions of three dimensional volumes. This extension delocalizes mass into a four dimensional harmonic space. This delocalization ensures that singularities do not exist physically or mathematically. This mechanical description for orbits is scale independent. The sinusoidal nature of electron orbits was first identified by French physicist Prince Louis Debroglie. Electron orbits are mechanically extensions of the atomic nucleus just as Mercury’s orbit is an extension of the Sun. The electron does not collapse into the atomic nucleus because in harmonic space, the electron orbit is a spatially extended nucleus. More important, this means orbits are the best way to describe an object’s dimensions in harmonic space or space-time. The sinusoidal orbit extents objects and divides the harmonic space into an interior and exterior region. These regions are essential to constructing the sinusoid.

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THE SIXTH PROBLEM – FINE STRUCTURE CONSTANT

What is the fine structure constant? The fine structure constant is a coupling constant and it describes electromagnetic strength. This unit less natural constant is and it is critical to atomic spacing in spectral lines. It is

also essential to Quantum Electrodynamics or QED. This constant holds a mysterious place in physics. Richard P. Feynman, QED founder said, "It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it. Immediately, you would like to know where this number for a coupling comes from: is it related to π or perhaps to the base of natural logarithms? Nobody knows. It's one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the 'hand of God' wrote that number, and 'we don't know how He pushed his pencil.' We know what kind of a dance to do experimentally to measure this number very accurately, but we don't know what kind of dance to do on the computer to make this number come out, without putting it in secretly!”[1] According to Axiom 1, mysterious or abstract numbers do not exist in physics. This means has mechanical meaning beyond electromagnetic strength. A Photo-Ionized Hydrogen nucleus is constructed to reveal the mechanics for the fine structure constant. How is the Photoelectric Effect related to the fine structure constant? The Photoelectric Effect occurs when light is projected onto a photo sensitive metal. The incident photon ejects an electron from the metal. As a result, the metal is charged and the electron travels at a velocity away from the metal. The electron’s kinetic energy and the incident photon’s energy are related by Einstein’s equation.

The symbol is the work function for a specific metal or the amount of energy needed to work the electron out of the metal. The Photoelectric Effect is actually macroscopic ionization. The system boundary is the metal’s surface. The electron is removed from the system producing a net charge. The same logic is used to evaluate the Hydrogen nucleus. Photo-Ionized Hydrogen Nucleus The Photo-Ionized hydrogen nucleus is the Photoelectric Effect applied to the Hydrogen nucleus. Equation 5 describes radiation released or absorbed by the Hydrogen atom.

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(5)

Rydberg Constant is and the structure controls the hydrogen

radiation sequence. Ionization occurs when such that and ∞ for

an electron ejected from the nucleus. This assumes no energy is needed to work the electron out of the nucleus. Equation 6 describes the same logic used for the Photoelectric Effect.

(6)

Nucleus The ejected electron leaves the nucleus with a net positive charge similar to a beta decay process. Rydberg’s Constant is related to the fine structure constant and the relationship is described by Equation 7.

(7)

The Compton wavelength is an X-Ray wavelength. Equation 8 shows the wavelength’s relationship to the electron’s mass.

(8)

If is substituted into Equation 7 the result is Equation 9.

(9)

If is substituted into Equation 6 then the result is Equation 10.

(10)

The fine structure constant is described by Equation 11.

(11)

The fine structure constant’s squared form is the common ratio k for a converging

geometric series. The Photo-Ionized evaluation shows that the electron is ejected outside the nucleus and reaches a position closest to the nucleus or ground state. The electon has a velocity at the ground state position.

Reference

[1] The Strange Theory of Light and Matter, Princeton University Press, p. 129

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THE SIXTH PROBLEM – GRAVITATIONAL COUPLINGS

What is Gravitational Coupling? Gravitational coupling is how a gravitational field holds objects together. The gravitational field strength and electromagnetic field strength have the same form. Remember, the fine structure constant describes electromagnetic strength where is the electron’s ground state

velocity in the hydrogen atom. Gravitational coupling is weaker than electromagnetic coupling and varies with the distance between objects. Newton’s Universal Law of Gravity forms the bases for three gravitational coupling configurations. What is Escape Velocity Coupling? Imagine a rocket of mass escaping at from the Earth’s gravitational field. The diagram below illustrates the coupling. In this coupling, harmonic space is reacting to kinetic energy. The smaller mass transfers kinetic energy into the gravitational potential between both masses. The equation below illustrates the process.

Let represent the gravitational coupling configuration for this form. The coupling form

squared is the common ratio k for a converging geometric series. Equation 12 is the

squared form.

(12)

What is Orbital Velocity Coupling?

Imagine a satellite of mass orbiting at a velocity about the Earth. Orbiting objects have different coupling effect on space. In this coupling, harmonic space is reacting to centrifugal force.

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The centrifugal force produced by is balanced by the gravitational force between both masses. The equation below illustrates the process.

Let represent the gravitational coupling configuration for this form. Equation 13 is the

squared form and it is the common ratio for a converging geometric series.

(13)

What is Collapse Velocity Coupling? Imagine an object or system of objects falling into themselves with or collapse velocity. The harmonic space has a potential sink that is absorbing the kinetic energy of the system. This coupling configuration is not part of common experiences but it is the configuration for Contact and Semi-Detached binary star systems. A signature feature for this configuration is the accretions disk or vortex of hot gases which form around one of the masses. The equation below illustrates the process.

Let represent the gravitational coupling configuration for this form. Equation 14 is the

squared form and it is the common ratio for a converging geometric series. (14)

Gravitational couplings and electromagnetic coupling are operating on the geometric structure of harmonic space.

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THE SIXTH PROBLEM – PRINCIPLE QUANTUM NUMBER

What is the Principle Quantum Number? Principle quantum numbers are defined as whole numbers ranging from one to infinity. In the hydrogen sequence they have an initial and final state. The principle quantum

numbers exist within a system of interactions modeled by the coupled harmonics model. Principle quantum numbers adequately describe the nodes associated with oscillating or synchronized fields. The objects generating those fields have continuous motion that is not described by nodes. Mechanical logic requires a generalized form for nodes which supports both fields and objects. This generalized form is a geometric principle quantum number. What is the Geometric Principle Quantum Number? The geometric principle quantum number is a converging geometric series

where and the object’s velocity. The geometric principle quantum number and the

principle quantum number exist in the same pattern. Consider a series of bound oscillating signals. No matter how the signals are disturbed a base can be established to produce nodes and continuous distributions.

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The node and distribution patterns are symmetric. This means every node corresponds to a distribution. This correspondence between nodes and distributions is evident in the hydrogen atom because each is associated with a series. The Lyman, Balmer, Paschen, Brackett, Pfund and Humphrey’s series corresponds to the nodes respectively. This means the node pattern described by Equation 15 has a corresponding distribution pattern described by Equation 16.

(15)

(16)

Both patterns are associated with a nuclear or interior harmonic space. The interior and exterior harmonic spaces are described next. Ultimately, each harmonic space must establish harmonic stability between an object’s motion and its associated fields. If an object is at rest then represents the stable state. If an object is in motion then harmonic space spins to establish the

state. On a celestial scale this spatial spin is detected as orbital precession and light path deflections.

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THE SIXTH PROBLEM – THE SUN AND MERCURY

What established General Relativity as an advanced model for gravity? General Relativity is Einstein’s theory for gravitation. In this theory, gravity is not a force as described by Newton. Instead, gravity is space-time curvature. According to Einstein, what we experience as a gravitational force is actually the object taking the shortest distance through curved space-time. When asked to explain the process, Einstein said, “Matter tells space-time how to curve and space-time curves tell matter how to move”. The equation below describes the physical reality Einstein imagined.

This equation appears deceptively simple. In reality, it took Einstein three years to master the mathematics of curved space needed to produce the equation. Einstein’s hard work would be validated by two amazing astrological insights. Mercury’s orbit had long been a mystery for astronomers. The elliptical orbit Newton described is fixed but Mercury’s orbit actually rotates or experiences precession. After calculating the gravitational effects of surrounding planets, Mercury’s orbit shifts 43.1 arc seconds per one hundred years. Many scientists explained the additional rotation by speculating the existence of another planet. For Einstein, there was no other planet. He proved Mercury’s precession was a feature of space-time curvature. He spent days overwhelmed with joy after calculating Mercury’s precession. Later, Einstein would predict the bending of star light as it moved through curved space. With the 1919 solar eclipse and 1.75 arc seconds of bending star light, Einstein overthrew Newton’s theory of gravity and became an international celebrity. Not one to rest on his laurels, Einstein set the task of unifying his new theory of gravity with Maxwell’s electromagnetic field theory. This task consumed his remaining days. Until his death in 1955, Einstein believed a gravitational and electromagnetic unification based on classical principles would reveal the greatest secrets about the universe. What is “spooky action” at a distance?

Among the many things General Relativity attempted to explain was “spooky action at a distance”. How does the Sun reach out across empty space to hold planets in orbit? According to the mechanical description of orbits, this space is not empty. Harmonic space is regional and objects have sinusoidal orbital representation. These orbits connect an interior and exterior spatial region. It is these regions under spin that causes star light deflection and Mercury to experience precession. This means there is no real distance between objects in harmonic space. The distance we define in three dimensions is connected to harmonic space as orbits not three dimensional volumes. How is star light deflection related to the interior spin? In the illustration below, the Sun is a white sinusoidal boundary separating the black interior region from the blue exterior region. In this case, the black interior region has a diameter equal to the Sun’s diameter.

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The Sun’s interior region has the sequential pattern as do most stars. The pattern

corresponding to the geometric principle quantum numbers is expressed below.

The Sun’s interior region spins to establish stability for object motion and field harmonics. The angle is the regional rotation needed to produce this stability. Equation 17 describes this process and Equation 18 describes the inverse form for the process. (17)

(18)

The precession or is valid for Equation 17. The Sun’s mass is , its radius is , the universal gravitational constant is

and is the speed of light. Escape velocity coupling defines the attachment to the

Sun’s interior space. Therefore the following is true.

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The deflection is degrees or arc seconds for starlight passing just inside the interior. The observed deflection is arc seconds.

How is Mercury’s precession related to the exterior spin?

The illustration below shows Mercury’s position in the exterior region at Perihelion or its closest distance to the Sun. Orbital Velocity coupling defines Mercury’s attachment to the exterior space with an orbital velocity. Let represent an object’s precession in the exterior space just outside the Sun. Let represent the precession drop towards Mercury’s perihelion. Let

represent Mercury’s precession at perihelion. Unlike the interior region, the exterior region is unrestricted.

This exterior region spins to produce stability for moving objects and their harmonic fields. Equation 19 describes the exterior spin needed to achieve this stability. The geometric series corresponding to exterior coupling is and is the geometric series corresponding to interior coupling. The notation difference is because their respective couplings are different. Equation 19 describes the stabilizing process and Equation 20 describes the inverse process. The precessions

and are all measure with respect to the vertical.

(19)

(20)

The precession is calculated below.

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The precession degrees or 0.437 arc seconds per cycle. After 100 cycles or years, arc seconds.

The precession is calculated below. Let represent the distance at

perihelion.

The precession arc seconds per 100 years. Mercury’s precession or

is 43.034 arc seconds per 100 years. This is very close to the observed precession of

43.1 arc seconds per 100 years. More important, it means the interior region is expanded by orbits as indicated by the precession drop. The boundary between the regions is an interface which defines the object in 4-D harmonic space.

How to construct the sinusoid from interfacing regions?

The equations below describe the interface between the interior and exterior regions. Equation 21 and 22 represent the precession for the entire system.

Derivation

A tornado is an excellent model for visualizing these equations. The tornado has an interior and exterior pressure differential which drives the entire system. In this case, the vortex is driven by coupling differences.

(21)

(22)

These are precession equations for circular orbits. The general forms consider orbital size, shape and amplitude.

THE GENERAL FORMS

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The general forms are written as frequency equations where cycle or year.

The shape factor is where e is the eccentricity for the orbit. The shape factor is 1 for circular

orbits. The size factor is for expanding systems and for contracted

systems. Let be the distance at perihelion and is the distance at aphelion. Equation 23 and

24 represent positive and negative fractional change in orbit respectively.

(23)

(24)

The size factor is until expansion or contraction is determined. Let represent the

initial position before expansion or contraction. Equations 25 and 26 are the general forms.

(25)

(26)

The intensity or potential for space-time is described by amplitude. These space-time amplitudes are unrestricted. The gravitational-electromagnetic interface configurations will determine the conditions for amplitudes.

26

THE SIXTH PROBLEM – INTERFACING FIELDS

If Field A is coupled to the interior space and Field B is coupled to the exterior space then the sinusoidal orbit interfaces or unifies both fields. This is the method used to unite the gravitational and electromagnetic fields. This method is a classical unification or unification using a continuous structure. This interface is ultimately a solution to Hilbert’s Sixth Problem. How to test the solution to Hilbert’s 6th Problem?

Binary star systems are excellent systems for testing the interface. The neutron stars within

these systems produce violent gravitational and electromagnetic fields. The interface method described is used to calculate precession in binary star systems. The primary or larger mass

and the secondary or smaller mass are orbiting a common center. These binary star systems come in the three forms listed.

1. In Contact Binary Systems, and are so close they share the same gaseous envelope.

Collapse velocity base coupling or describes the system. Let be the initial orbital distance for the primary mass.

2. In Semi-Detached Binary Systems, or is filling the gaseous envelop. Mass transfer

occurs between and whereby producing an accretion disk around one mass.

Collapse velocity coupling or describes this system. Let be the

initial orbital distance for the primary mass.

3. In Detached Binary Systems, is dominated by the gravitational field. Each mass

exist within its own gaseous envelope and little or no mass transfer occurs between masses. Orbital velocity coupling or commonly describes this system. Let be the

initial orbital distance for the primary mass. If acts on a region then a specific acts on the other region. This condition produces the four interface configurations described by Equations 27, 28, 29 and 30. If and then for a Variation 1 system to exist.

(27)

If and then ≥ 10 for a Variation 2 system to exist.

(28)

If and then ≥ 10 for a Variation 3 system to exit. Equation 28

and 29 are inverses because the electromagnetic and gravitational coupling changed regions.

(29)

27

If and then for a Variation 4 system to exist. Equation 27

and 30 are inverses because the electromagnetic and gravitational coupling changed regions. (30)

Amplitudes are fundamental to all waves including harmonic space waves. The ≥ 10 condition not only allows certain interfaces to exists, these elevated amplitudes explain other natural phenomena. The harmonic space wave is the most general wave function. In 1926, Erwin Schrodinger introduced an equation which successfully described atomic systems. In that equation, the wave function appeared in physics for the first time. It was

not only new, it had no real interpretation. Scientist did not know what was waving or what the wave amplitude represented. The final interpretation defined as a statistical reality and the

amplitude was its probability density. This interpretation logically restricts the wave function’s amplitude probability modeling. Equations 31 and 32 describe the minimum and maximum amplitudes for the gravitational and electromagnetic interfaces. These amplitudes are visually like ripples rising up from the surface of a pond. These precession amplitudes exist without making the harmonic space field discontinuous and values are whole numbers ranging from zero to infinity.

(31)

(32)

If describes precession and these wave functions have superposition characteristics then

superposition must also be a characteristic of some precessions. In binary star systems, the superposition principle does exist in precessions. Let and represent the

minimum and maximum precession respectively. The equations below describe the average and normalized precessions due to superposition.

±

These mean values are produced when the wave function interacts with itself. This feature is consistent with what is observed in the double slit experiment. The object has one path within the precession distribution. This path is connected to harmonic space so path potentials are always present before a measurement. A measurement stops precession and the wave function describing the precession distribution collapses. After a series of measurements, the experimenter will reproduce the precession distribution because that one physical condition can produce a range of outcomes in harmonic space. __________________________________________________________________________________________________________________ Mechanical statements constructively integrated forms the equivalence of a geometric proof. The mechanical approach to derivation produced the equations and the next phase is the test.

28

THE SIXTH PROBLEM – PSR B1913+16

This Contact Binary system was discovered in 1974 by Hulse and Taylor. They later won the 1993 Nobel Prize. The Pulsar component of the binary system emits a signal that sweeps past the Earth periodically. The periodic signal emission has been used to study Einstein’s General Theory of Relativity. The study revealed orbital decay consistent with the release of gravitational radiation. Einstein predicted this radiation would be released as the mutual orbits decreased into a death spiral.

[Common Orbit Center – Same Size Orbit] General Data Notation:

kg / Sun Mass km / Sun Radius

represents projected semi-major axis represents distance at aphelion

represents orbit eccentricity represents distance at perihelion

/Light Speed primary or larger binary mass

/ Gravity Constant secondary or smaller binary mass

[1] PSR B1913+16

Calculate PSR B1913+16’s precession. Identify the electromagnetic-gravitational variation for the system. This binary system is a Variation 1 configuration. If and then is the minimum

and only amplitude for the system to exist whereby and for this system. The system is described by the equation below.

The system is expanding from the initial position and is defined by collapse velocity coupling.

This means and for the system. Based on this

information the follow is true.

29

The electromagnetic coupling squared is and is the corresponding geometric series. The gravitational coupling squared is and is the corresponding geometric series. The fractional expansion is for the system.

The precession is or cycle. The observed precession or apsidal motion

is approximately [1] which is very close to the calculated precession.

Comments:

A neutron star’s typical diameter is about 3 to 4 kilometers across. With masses larger than our Sun, these systems produce some of the most violent gravitational and electromagnetic fields in the universe. High precession speeds is a feature of these extreme conditions. If gravitational radiation and black hole entropy are also features of these extreme conditions then they must be detectable to validate their associated theories. Data Reference [1] http://www.johnstonsarchive.net/relativity/binpular.html. Astronomy and Space: Johnson Archive.

http://www.johnstonsarchive.net/relativity/binpular.html

30

THE SIXTH PROBLEM – PSR B1534+12 This Semi-Detached binary star system is similar to PSR B1913+16. Researchers believe the neutron stars may have existed as a detached system. The system was discovered by the Arecibo Observatory in 1990. PRS 1534+12 and PRS 1913+16 have similar orbit times and precession mechanics. The measurement accuracy for PSR 1534+12 is higher than PSR 1913+16 because the system is closer to the Earth.

[Common Orbit Center – Same Size Orbits] General Data Notation:






[1] PSR B1534+12

Calculate PSR B1534+12’s precession. Identify the electromagnetic-gravitational variation for the system. This binary system is a Variation 1 configuration. If and then is the minimum

and only amplitude for the system to exist whereby and for this system. The system is described by the equation below.




31


The system has the superposition configuration ± where

is normalized by a fraction of itself.

The precession is ± or cycle. The observed precession or apsidal

motion is approximately [1] which is close to the destructive superposition

or cycle.

Comments:

The precession calculation indicates that PSR B1534 +12 may have started as a Contact Binary system and separated. PSR 1913+16 does not have this distribution because it is still a contact system. If PSR 1913+16 separated into a semi-detached system, its precession would be

± or cycle.

Data Reference [1] The Astrophysical Journal, 446:L87-L90, 1996 August 1 PSR J1518 +4904: A Mildly Relativistic Binary Pulsar System.

32

THE SIXTH PROBLEM – PSR B2127 +11C This semi-detached binary system has precession mechanics similar to PSR B1534+12. Researchers discovered orbital decay in both systems. General Relativity predicts orbital decay but must also predict precession. It was precession not gravitational radiation that established General Relativity as a superior gravitational model.







[1] PSR B2127+11C

Calculate PSR B2127+11C precession. Identify the electromagnetic-gravitational variation for the system. This binary system is a Variation 1 configuration. If and then is the minimum

and only amplitude for the system to exist whereby and for the system. The system is described by the equation below.




33


The system has the superposition configuration ± where

is normalized by a fraction of itself.


motion is approximately [1] which is the destructive superposition for the precession

distribution. The destructive superposition is or cycle. This precession is

close to the observed precession. Comments

The precession calculation indicates that PSR B2127+11C also started out as a contact binary system and separated whereby producing destructive precession superposition. The destructive and constructive superposition’s ability to normalize creates a potential

precession distribution with features similar to a normalized wave function. If really

describes a precession distribution then it is obvious binary systems cannot have multiple precessions simultaneously until measured.

Data Reference [1] The Astrophysical Journal, 446:L87-L90, 1996 August 1 PSR J1518 +4904: A Mildly Relativistic Binary Pulsar System.

34

THE SIXTH PROBLEM – PSR J1756-2251 This Semi-Detached binary system is similar to B1534+12 and B2127+11C. The Parkes Multi-beam Pulsar Survey discovered the system in 2004. These systems also have accretion disk or a hot gaseous region around a component within the system. The precession mechanics shows the similarities between these systems.







[1] PSR J1756-2251

Calculate PSR J1756-2251precession. Identify the electromagnetic-gravitational variation for the system. This binary system is a Variation 1 configuration. If and then is the minimum

and only amplitude for this configuration where and for this system. The system is described by the equation below.




35


The system has the superposition configuration ± whereby

is optimized by a fraction of its precession.


motion is approximately [1] which is the constructive superposition for the

precession distribution. The constructive superposition is or cycle. The

calculated precession is in excellent agreement with the observed precession. Comments:

The precession calculation indicates that PSR J1756-2251 also started out as a contact binary system and separated whereby producing a constructive precession superposition. The standard deviation ± increases with precession speed and orbital elongation. This

structure is normalizing the precession. The precession speed and orbital elongation’s effect on standard deviation is what Heisenberg identified as uncertainty. This is clearly not uncertainty but a space-time or harmonic space capacity or potential. This is the acceptable range of precessions which can be supported by harmonic space for a given condition. This means one condition can produce a number of physical outcomes in space-time. The system does not have all possible precessions until measured so the system is still deterministic. This is the real relationship between mechanics and statistical distributions. Data Reference [1] PSR J1756-2251: A new relativistic double neutron star system. A. J. Faulkner arxXiv:astro-ph/0411796 v1 30 Nov. 2004

36

THE SIXTH PROBLEM – PSR JO737-3039A This Contact Binary system is the only double Pulsar system known. It is also the fastest. The system was discovered in 2003 by Dr Marta Burgay at the CSIRO’s Parkes radio telescope. This system is similar to PSR B1913+16 except the electromagnetic and gravitational coupling changed regions. The symmetry between the systems is broken by this coupling relocation.







[1] PSR JO737-3039A

Calculate PSR J0737-3039A precession. Identify the electromagnetic-gravitational variation for the system. This binary system is a Variation 2 configuration. If and then ≥ 10 for the V2 system

to exist. This system has minimum and maximum amplitudes described by where and respectively. The system is expanding from the initial

position and is defined by collapse velocity coupling. This means and

for the system. Based on this information the follow is true.

The precessions related to and are described by Equation 33 and 34 respectively. (33)

37

(34)

The electromagnetic coupling squared is and is the corresponding geometric series. The gravitational coupling squared is and is the corresponding geometric series The system has superposition configuration where

is an average precession.

The maximum precession is and the minimum precession is

. The average precession is ≈ or cycle. In this

electromagnetic-gravitational configuration, the system precession is slowed by higher amplitudes. The observed precession or apsidal motion is approximately or cycle [1]. The

observed precession is close to the average precession. Comments This system has a weaker electromagnetic field than PSR 1913+16. This means the electromagnetic field is weaker if is coupled to the exterior region. More important, it indicates that the symmetry between electromagnetism and the weak nuclear force is broken by the same regional displacement of electromagnetic coupling. Data Reference [1] Test of General Relativity from Timing the Double Pulsar Originally published in Science Express on 14 September 2006 Science 6 October 2006. Vol. 314. No. 5796, pp. 97-102

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THE SIXTH PROBLEM – PSR J1518 + 4904 The Green Bank Northern Sky Survey recently discovered this Semi-Detached system. This system presents a challenge to General Relativity because its precession is slower than the predicted value. In fact all, V3 electromagnetic-gravitational configurations have precessions that are inconsistent with General Relativity.







[1] PSR J1518+4904

Calculate PSR J1518+4904 precession. Identify the electromagnetic-gravitational variation for the system. This binary system is a Variation 3 configuration. If and then ≥ 10 for the V3 to exist

as a V2 inverse. This system has minimum and maximum amplitudes described by where and respectively. The system is expanding from the initial

position and is defined by collapse velocity coupling. This means and


The precessions related and are described by Equation 35 and 36 respectively.

39

(35)

(36)

The electromagnetic coupling squared is and is the corresponding geometric series. The gravitational coupling squared is and is the corresponding geometric series The system has the superposition configuration where

is an optimized or average precession.

The maximum precession is and the minimum precession is

. The average precession is . In this

electromagnetic-gravitational configuration, precessions increase with increasing amplitudes. The observed precession or apsidal motion for the system is ± or cycle [1]. The

average precession is close to the observed precession. Comments PSR J1518+4904 and J0737-3039A both have average precessions within the precession distribution. This distribution is the system capacity and the measurement is the actual state for the system. They are related but exist independently. This statistical-mechanical relationship is what Hilbert sought to understand mathematically. Data Reference [1] The Astrophysical Journal, 446:L87-L90, 1996 August 1 PSR J1518+4904: A Mildly Relativistic Binary Pulsar Syste

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THE SIXTH PROBLEM – PSR B2303+46 This Semi-Detached binary system is believed to have a component that is a White Dwarf. This system is similar PSR J1518+4904 as they are both V3 configurations. The observed precession is also slower than that predicted by General Relativity.







[1] PSR B2303+46

Calculate PSR B2303+46’s precession. Identify the electromagnetic-gravitational variation for the system. This binary system is a Variation 3 configuration. If and then ≥ 10 for the V3 to exist


position and defined by collapse velocity coupling. This means and


The precessions related to and are described by Equation 37 and 38 respectively.

(37)

41

(38)

The electromagnetic coupling squared is and is the corresponding geometric series. The gravitational coupling squared is and is the corresponding geometric series The maximum precession is and the minimum precession is

for the system. The system has no superposition configuration. This

system is operating at minimum precession. The observed precession or apsidal motion is approximately or cycle [1]. The calculated precession is around the observed

precession. Comments This system and PSR J1518+4904 are high amplitude binary systems. This system can reach a maximum precession of even through it operates at the

minimum precession. It is logical to assume on an atomic level a system can cycle between minimum and maximum. A measurement stops the precession whereby collapsing the wave function which describes the precession distribution. Wherever the object was within the cycle when the collapse occurs is the actual state. If describes precession then no system has

multiple precessions simultaneously. Data Reference [1] The Astrophysical Journal, 446:L87-L90, 1996 August 1 PSR J1518+4904: A Mildly Relativistic Binary Pulsar System.

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THE SIXTH PROBLEM – V541 CYGNI

This detached binary system is considered an excellent test for the General Theory of Relativity. However, the theoretical precession predicted is not consistent with the observed precession. This is another V3 configuration.

[Common Orbit Center – Different Sized Orbits] General Data Notation:






[1] V541 CYGNI Primary Secondary

Calculate V541 CYGNI precession. Identify the electromagnetic-gravitational variation for the system. This binary system is a Variation 3 configuration. If and then ≥ 10 for the V3 to exist


position and is defined by orbital velocity coupling. This means and



43

(39)

(40)

The electromagnetic coupling squared is and is the corresponding geometric series. The gravitational coupling squared is and is the corresponding geometric series The minimum precession is and the maximum precession is

for system. The originally observed precession was ±

and the new observed precession is ± [1]. The original and

new precessions are consistent with the maximum and minimum precessions respectively. Comments A more accurate tool was used to make the new measurement for this system. It is logical to assume the device did not make a more accurate measurement but actually identified the minimum precession for the system. Without a direct observation, it is impossible determine the actual state. Again, we know for certain this system cannot have two precessions. Data Reference [1] The Astronomical Journal. 115:801-808, 1998 February. Absolute Dimensions and Masses of V541 CYGNI and The General Theory of Relativity. Received 1997 July 28.

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THE SIXTH PROBLEM – DI HERCULIS This system and V541 CYGNI are both detached binary systems. Unlike V541 CYGNI, DI HERCULIS was discovered fifteen years after Einstein published his General Theory of Relativity. This binary system is the most studied because the General Relativistic precession prediction is not consistent with observation. For over ninety years this system’s precession was a mystery.

[Common Orbit Center – Different Size Orbits] General Data Notation:






[1] DI HERCULIS Primary Secondary

Calculate DI HERCULIS precession. Identify the electromagnetic-gravitational variation for the system. This binary system is a Variation 3 configuration. If and then ≥ 10 for the V3 to exist


position and is defined by orbital velocity coupling. This means and



45

(41)

(42)

The electromagnetic coupling squared is and is the corresponding geometric series. The gravitational coupling squared is and is the corresponding geometric series The minimum precession is and the maximum precession is

. The originally observed precession was approximately

and the new observed precession is approximately [1]. The original

and new observed precessions are consistent with the minimum and maximum precessions respectively. Comments DI HERCULIS precession condition is similar to V541 CYGNI. Either precession is valid for the given condition. Without direct visual observation, DI HERCULIS could be operating at a minimum or maximum precession but certainly not both. These measurements were taken using Light Curve Spectography. This measurement method is obviously able to detect the boundaries for the precession distribution. Data References [1] Astronomy and Astrophysics-“Some notes on the relativistic motion of DI HERCULIS”: Villanova University Department of Astronomy and Astrophysics: Annual Report: October 2003 to September 2004. [S0002-7537 (95)00201-0].

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THE SIXTH PROBLEM – AS CAMELOPARDALIS AS CAMELOPARDALIS is a detached binary system. As with DI HERCULIS, many attempts have been made to describe the system in a three body configuration. This third body has yet to be observed.

[Common Orbit Center – Different Size Orbits] General Data Notation:






[1] AS CAMELOPARDALIS Primary Secondary

Calculate AS CAMELOPARDALIS precession. Identify the electromagnetic-gravitational variation for the system. This binary system is a Variation 4 configuration. If and then for Variation 4

to exist as a V1 inverse. The minimum amplitude is the only amplitude for this configuration and is described by where for this contracting system. The equation below describes the system.

The system is contracting from the initial position and is defined by orbital velocity coupling. This means and for the system. Based on this information

the follow is true.

47

The electromagnetic coupling squared is and is the corresponding geometric series. The gravitational coupling squared is and is the corresponding geometric series. The fractional contraction is for the system.

The precession is . The observed precession or apsidal motion is

approximately ± [1] which is very close to the calculated precession.

Comments: AS CAMELOPARDALIS is proof contracting systems exist. There is no question the measured precession is the precession because there is only one for this system. Did AS CAMELOPARDALIS start contracting after an expansion or has it always been in a state of contraction? __________________________________________________________________________________________________________________ The binary stars tested represent the broadest precession range. These tests prove a mechanically based axiomatic approach is the best way to produce successful generalizations for physical systems. Hilbert predicted an axiomatic approach would yield a general theory that could explain a broad range of phenomena. Data Reference [1] Eclipsing binary stars tests of gravity theories: New Solutions for AS Camelopardalis. Journal: Astronomical Journal (ISSN 0004-6256), vol. 102, July 1991, p.256-261.

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THE SIXTH PROBLEM – PREDICTIONS

The sinusoidal orbital equations are solutions to Hilbert’s 6th Problem. In general, an object’s motion acts as a field coupler. As the object moves, the fields oscillate whereby producing potential configurations in what Einstein described as space-time and quantum physicist described as the wave function. The potential distributions determine what the object(s) can do in

space-time. Pure classical mechanics identifies the object(s) mechanical motion and pure quantum mechanics identifies the motion potential for the object(s). The sinusoidal orbital equations identify the object(s) mechanical motion and the motion capacity or potential distribution. This statistical-mechanical connection is fundamental to structures on a galactic and atomic scale. The sinusoidal orbital equations constructively merge structures that are fundamental to classical and quantum mechanics without producing inconsistencies. An axiomatic solution tends to remove inconsistencies and produce a general solution capable of explaining a broad range of phenomena. What is Dark Matter? In general, a galaxy is a system of stars and by this definition binary star systems are the smallest possible galaxies. The minimum amplitude and the maximum amplitude

will exist in the smallest possible galaxy as well as the largest. These

amplitudes are the intensity for what Einstein described as space-time. This intensity or potential can increase or decrease the precession. This intensity is logically detected in large galaxies as Dark Matter. Four important facts support this hypothesis.

1. Dark Matter is detected by bending light and Einstein’s space-time was detected by bending light. It is logical to assume the space-time field and its associated intensity or amplitudes are detected in the same way.

2. Dark Matter to ordinary matter ratio is consistent with a base ten structure. 3. Additional mass increases the frequency of a system in the way an amplitude increase

would increase frequency. 4. If Dark Matter is the imprint of elevated amplitudes on galaxies then means

there are galaxies that exhibit no dark matter effects. Researchers from the Polish Academy of Science in Karakow were unable to detect Dark Matter in the spiral galaxy NGC 4736. The system logically has no space-time intensity beyond unity. This discovery is not explained by current cosmological models. What is Dark Energy and Dark Flow? Unlike Dark Matter, Dark Energy and Dark Flow are related to the dynamics of the sinusoidal orbital equations. The expansive condition described by and the

contractive condition described by are detected as Dark Energy and Dark Flow

respectively. The sinusoidal orbital equations and are essentially the same structure. These

wave functions expand or contract without energy because the objects are not moving through harmonic space. Harmonic space is actually being produce between the galaxies in the expansive

49

case and harmonic space is being removed in the contractive case. Three important facts support this hypothesis.

1. Wave functions do not carry energy, so phenomena associated with wave function dynamics do not carry energy.

2. Dark Energy is calculated make up 75 percent of the expanding universe. The size

factors and have fractional orbit changes reduced by 75

percent. 3. Contracted binary star systems are rare so galactic motion related to contraction must

be rare as well. AS CAMELOPARDALIS dynamics prove these conditions must exist.

In 2008, researcher Alexander Kashlinsky used NASA’s (WMAP) to identify galaxy clusters like 1E0657-56 as they moved against the cosmic microwave background. This regional contraction in space was described as Dark Flow. Binary star systems exist in one universe so this discovery simply means there are regions of the universe that we identify as expanding and other parts that are contracting. This discovery is not explained by current cosmological models. The neutron stars in binary systems are entangled. Entanglement is a non-local phenomena which is experimental confirmed to exist between photons and other small structures. Cosmic scale entanglement cannot be tested but it must logically exist. Four important facts support this hypothesis.

1. The sinusoidal orbital equations are non-local and generalizations for the wave function. The harmonic space removes “spooky action at a distance”.

2. Both neutron stars are described by the same wave function. This is a fundamental

characteristic of entangled structures. 3. The neutron stars in the system are believed to have been born together in a supernova

explosion. Most entangled structures have the same origin. 4. The neutron stars orbit in opposite directions. This is also a fundamental characteristic

of entangled structures. Information between the stars is transmitted in a non-local harmonic space. This makes information appear to be instantaneously transmitted in localized space. Einstein’s Special Theory of Relativity describes objects in a local space, so instantaneous transmission does not violate STR. If you could reverse the orbit direction for one neutron star, the other neutron star would change direction instantaneously. Consequences related to electromagnetic and gravitational symmetry. In General Relativity, gravity is not a force but space-time curvature. The sinusoidal orbital equation’s structure shows the symmetry between the electromagnetic and gravitational couplings. This symmetry proves gravity and electromagnetism are forces operating on what can be described as a harmonic space, space-time or generalized. Electrons are acting as particles in this

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symmetry. The wave characteristics associated with electrons is actually characteristics of harmonic space. If gravity is an ordinary force then gravitational waves and gravitational radiation must be reevaluated. Gravitational Waves Reevaluated: Gravitational Waves are described as space-time ripples. The sinusoidal orbital equations describe a space-time that is produced by the gravitational-electromagnetic interaction. This synthesis means there are no waves exclusively associated with gravity. The sinusoidal orbital equation’s amplitudes can be considered space-time ripples but they are not moving.

Gravitational Radiation Reevaluated: Gravitational Radiation is postulated to be associated with accelerating masses. Just as accelerated charge releases photons, Einstein postulated radiation must be associated with accelerated masses. Radiation is not associated with one force but the coupling of forces. For example, light or radiation is produced by coupling an electric and magnetic force. There is no precession in this electromagnetic field so the radiation is continuous. If precession exists within the field as it does with gravitational and electromagnetic coupling then radiation is localized or quantized. Continuous or quantized, the radiation released from stars is essentially the same as that released or absorbed by atoms. The sinusoidal orbital equations characteristics must exist on a subatomic level. The characteristics described by the quantum numbers and all have parallel characteristics in the sinusoidal orbital equations. The characteristic defined by and

are unparalleled by atomic description. These amplitudes make

the sinusoidal orbital equations generalized wave functions. On an atomic level, elevated amplitudes are closely related to the color force. Color Force: Protons and Neutrons are composed of quarks which are held together by the color force. The color force reaction to quark separation is similar to the reaction which is experience by separating binary stars in the V3 gravitational-electromagnetic configuration. These ≥ 10 conditions produce higher amplitudes the farther the binary stars are separated. These increased amplitudes confine binary stars just as the color force confines quarks. On an atomic level, increasing amplitudes is indistinguishable from increasing force. The precession increases whereby releasing radiation jets. In the atom these radiation jets are gluons. On a cosmic scale, these are the same jets of radiation leaving some binary star systems. Precession related radiation is localized and has particle like characteristics. This radiation exists between all force couplings such that it is considered a force mediator. In the case of atoms, gluons are considered color force mediators. Logic, mechanics, mathematical form and experimental confirmation strongly suggest wave functions are physical structures with physical meaning. These structures are natural solutions to Hilbert’s Sixth Problem or the relationship between statistics and mechanics.

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