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Chapter 7



The study of the form, function, and composition of matter has been, and continues to be, one of the greatest intellectual challenges of all time. In ancient times the Greek Empedocles (495-435 B.C.) came up with the idea that matter is composed of earth, air, fire, and water. In 430 B.C. the idea of Empedocles was rejected by, the Greek, Democritus of Abdera. Democritus believed that the substances of the creation are composed of atoms. These atoms are the smallest bits into which a substance can be divided. Any additional subdivision would change the essence of the substance. He called these bits of substance "atomos" from the Greek word meaning "indivisible". Democritus was, of course, correct in his supposition, however, at the time, no evidence was available to confirm this idea. Ancient technology was primitive and could not to confirm or contest any of these ideas. Various speculations of this sort continued to be offered and rejected over the next 2,000 years.

The scientific revolution began in the seventeenth century. With this revolution came the tools to test the theories of matter. By the eighteenth century these tools included methods of producing gases through the use of chemical reactions, and the means to weigh the resultant gases. From his studies of the gaseous by-products of chemical reactions, French chemist Antoine Lavoisier (1743-1794) discovered that the weight of the products of a chemical reaction equals the weight of the of the original compound. The principle of "the conservation of mass" was born. For his achievements Lavoisier is today known as the father of modern chemistry.

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Late in the eighteenth century, the first use of another new tool began to be applied to test the theories of matter. This tool is electrical technology. The first electrical technology to be applied to the study of matter, electrolysis, involves the passing of an electrical current through a conductive solution. If an electrical current is passed through a conductive solution, the solution tends to decompose into its elements. For example, if an electric current is passed through water, the water decomposes, producing the element hydrogen at the negative electrode and the element oxygen at the positive electrode. With the knowledge obtained from the use of these new technologies, English schoolteacher, John Dalton (1766-1844) was able to lay down the principles of modern chemistry. Dalton's theory was based on the concept that, matter is made of atoms, all atoms of the same element are identical, and atoms combine in whole number ratios to form compounds.

Electrical technology became increasingly more sophisticated during the nineteenth century. Inventions such as the cathode ray tube (a television picture tube is a cathode ray tube) allowed atoms to be broken apart and studied. The first subatomic particle to be discovered was the electron. In 1897, J.J. Thomson demonstrated that the beams seen in cathode ray tubes were composed of electrons. In 1909, Robert Millican measured the charge of the electron in his, now famous, oil drop experiment. Two years later, Ernest Rutherford ascertained the properties of the atomic nucleus by observing the angle at which alpha particles bounce off of the nucleus. Niels Bohr combined these ideas and in 1913, placed the newly discovered electron in discrete planetary orbits around the newly discovered nucleus. The planetary model of the atom was born. With the appearance of the Bohr model of the atom, the concept of the quantum nature of the atom was established.

Pick the icon to view the Bohr model of the atom.

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As the temperature of matter is increased, it emits correspondingly shorter wavelengths of electromagnetic energy. For example, if a metal poker is heated it will become warm and emit long wavelength infrared heat energy. If the heating is continued the poker will eventually become red hot. The red color is due to the emission of shorter wavelength red light. If heated hotter still, the poker will become white hot emitting even shorter wavelengths of light. An astute observer will notice that there is an inverse relationship between the temperature of the emitter and the wavelength of the emission. This relationship extends across the entire electromagnetic spectrum. If the poker could be heated hot enough it would emit ultra-violet light or X-rays.

The German physicist Max Karl Ludwig Planck studied the light emitted from matter and came to a profound conclusion. In 1900, Planck announced that light waves were given off in discrete particle-like packets of energy called quanta. Today Planck's quanta are know known as photons. The energy in each photon of light varies inversely with the wavelength of the emitted light. Ultraviolet, for example, has a shorter wavelength than red light and, correspondingly, more energy per photon than red light. The poker, in our example, while only red hot cannot emit ultraviolet light because its' atoms do not possess enough energy to produce ultraviolet light. The sun, however, is hot enough to produce ultraviolet photons. The ultraviolet photons emitted by the sun contain enough energy to break chemical bonds and can "sun" burn the skin. The radiation spectrum cannot be explained by any wave theory. This spectrum can, however, be accounted for by the emission of a particle of light or photon. In 1803, Thomas Young discovered interference patterns in light. Interference patterns cannot be explained by any particle theory. These patterns can, however, be accounted for by the interaction of waves. How can light be both a particle and a wave?

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In 1924, Prince Louis de Broglie proposed that matter possess wave-like properties. 1 According to de Broglie's hypothesis, all moving matter should have an associated wavelength. De Broglie's hypothesis was confirmed by an experiment conducted at Bell Labs by Clinton J. Davisson and Lester H. Germer. In this experiment an electron beam was bounced off of a diffraction grating. The reflected beam produced a wave like interference pattern on a phosphor screen. The mystery deepened; not only does light possess particle-like properties but matter possesses wave-like properties. How can matter be both a particle and a wave?

Pick the icon to view one form of the Germar Davisson experiment.

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Throw a stone in a lake and watch he waves propagate away from the point of impact. Listen to a distant sound that has traveled to you from its source. Shake a rope and watch the waves travel down the rope. Tune in a distant radio station, the radio waves have traveled outward from the station to you. Watch the waves in the ocean as they travel into the shore. In short, waves propagate, its their nature to do so, and that is what they invariably do. Maxwell's equations unequivocally demonstrate the fields propagate at light speed. Matter waves, however, remain "stuck" in the matter. Why do they not propagate? What "sticks" them? An answer to this question was presented by Erwin Schrödinger and Werner Heisenberg at the Copenhagen conventions. The Copenhagen interpretation states that elementary particles are composed of particle-like bundles of waves. These bundles are know as a wave packets. The wave packets move at velocity V. These wave packets are localized (held is place) by the addition of an infinite number of component waves. Each of these component waves has a different wavelength or wave number. An infinite number of waves each with a different wave number is required to hold a wave packet fixed in space. This argument has two major flaws. It does not describe the path of the quantum transition and an infinite number of real waves cannot exist within a finite universe.

Max Born attempted to side step these problems by stating that the wave packets of matter are only mathematical functions of probability. Only real waves can exist in the real world, therefore an imaginary place of residence, called configuration space, was created for the probability waves. Configuration space contains only functions of kinetic and potential energy. Forces are ignored in configuration space.
"Forces of constraint are not an issue. Indeed, the standard Lagrangian formulation ignores them...In such systems, energies reign supreme, and it is no accident that the Hamiltonian and Lagrangian functions assume fundamental roles in a formulation of the theory of quantum mechanics.."

Grant R. Fowles University of Utah
Ordinary rules, including the rules of wave propagation, do not apply in configuration space. The propagation mystery was supposedly solved. This solution sounds like and has much in common with those of the ancient philosophers. It is dead wrong!
"Schrödinger never accepted this view, but registered his concern and disappointment that this transcendental, almost psychical interpretation had become universally accepted dogma."

Modern Physics Serway, Moses, Moyer; 1997
Einstein also believed that something was amiss with the whole idea. His remark,
"God does not play dice"
indicates that he placed little confidence in these waves of probability. For the most part, the error made little difference, modern science advanced, and bigger things were discovered. It did, however, make at least one difference; it forestalled the development of gravitational and low level nuclear technologies for an entire Century.



Matter is composed of energy and fields of force. Matter can be mathematically modeled but a mathematical model does not make matter. Matter waves are real, they contain energy, are the essence of mass, and convey momentum.
" This result is rather surprising... since electrons are observed in practice to have velocities considerably less than the velocity of light it would seem that we have a contradiction with experiment.

Paul Dirac, his equations suggested that the electron propagates at light speed. 11
[Force PA.] Matter does not disperse because it is held together by forces. These forces generate the gravitational field of matter, establish the inertial properties of matter, and set matter's dynamic attributes. The remainder of this chapter will be spent qualifying these forces and the relationship that they share with matter. The ideas to follow are central to this author's work. Reader's who have no interest in math may skip to the conclusion without missing the essential details of this chapter. Essentially the math shows that forces within matter are responsible for many of the properties of matter.

This concept will be extended in Chapter 10. The various fields that compose matter have radically different ranges and strengths. The force, that pins the various fields within matter, will be explored. An understanding of the structure of the restraining forces has revealed the path of the quantum transition.

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A version of this section was published in "Infinite Energy" Vol 4, #22 1998

The matter wave function is composed of various fields. Photons were employed to represent these various fields. Photons exhibit the underlying relationship between momentum and energy of a field (static or dynamic) in which disturbances propagate at luminal velocities. Consider photons trapped in a massless perfectly reflecting box. The photon in a box is a simplistic representation of matter. Light has two transverse modes of vibration and carries momentum in the direction of its travel. All three modes need to be employed in a three dimensional model. For the sake of simplicity this analysis considers only a single dimension. The photons in this model represents the matter wave function and the box represents the potential well of matter. As the photons bounce off of the walls of the box momentum "p" is transferred to the walls of the box. Each time a photon strikes a wall of the box it produces a force. This force generates the gravitational mass associated with the photon in the box. The general formula of gravitational induction, as presented in the General Theory of Relativity 3, 4 (this equation was derived in Chapter 6) is given by Equation #2.

Eg = G / (c2r) (dp / dt)

Equation # 2 The gravitational field produced by a force

Eg = the gravitational field in newtons / kg

G = the gravitational constant

r = the gravitational radius

dp/dt = force

Each time the photon strikes the wall of the box it produces a gravitation field according to equation #2. The gravitational field produced by an impact varies with the reciprocal of distance "1/r". The gravitational field produced by matter varies as the reciprocal of distance squared "1/r2". This author has ascertained how the "1/r2" gravitational field of is produced by a force. It will now be shown that the superposition of a positive field that varies with an "1/r" rate over a negative field that varies with an "1/r" rate, produces the "1/r2" gravitational field of matter. An exact mathematical analysis of the gravitational field produced by the photon in the box will now be undertaken.

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Pick the bouncing ball for a message from the author.
energy in a box L = The dimensions of the box

p = momentum

t = the time required for the photon to traverse the box= 2L/c

r = the distance to point X

The far gravitational field at point X is the vector sum of the fields produced by the impacts on walls A and B.

This field is given by below.

Eg at x = 1/r field from wall A - 1/r field from wall B

Equation 3 Showing the super-position of two fields.

Eg at x = (G / [ c2 (r+L) ] ) ( dp / dt ) - (G / [c2r] ) (dp / dt)

Equation 4 Simplifying.

Eg at x = - (G / c2) (dp / dt) [ L / (r2 + r L) ]

Equation 5 Taking the limit to obtain the far field.

Eg at x = limas r>>L - (G / c2) (dp / dt) [ L / (r2 + r L) ]

The result , Equation #7, is the far gravitational field of matter. Far, in this example, means greater than the wavelength of an elementary particle. In the case of a superconductor far means longer than the length of the superconductor.

Eg at x = - (G / c2) (dp / dt) L / r2        Equation #7

This momentum of an energy field that propagates at light speed is given by the equation below 2 .

p = E / c

E = the energy of the photon

c = light speed

p = momentum (radiation pressure)

The amount of force (dp / dt) that is imparted to the walls of the box depends on the round trip travel time of the photon. Equation 8 gives the force on the walls of the box.

dp / dt = Dp / Dt = (2E / c) / (2L / c) = E / L        Equation #8   Note: This force is 29.05 Newtons at the classical radius of the electron.

Equation #8 was substituted into Equation #7. Equation 9 is the far gravitational field produced by energy bouncing in a box

Eg at x = - (G / c2) (E / L) (L / r2)        Equation #9

Equation 10 is Einstein's relationship between matter and energy.

M = E / c2        Equation #10

Substituting mass for energy yileds Equation #11, Newton's formula for gravity 5 .

Eg at x = - GM / r2        Equation #11

Forces are produced as energy is restrained. These forces induce the gravitational field of matter. 6, 10

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Note: a version of this section was published in "The Journal of New Energy" Vol 5, September 2000

In 1924 Prince Louis DeBroglie proposed that matter has a wavelength associated with it. 1 Schrödinger incorporated deBroglie's idea into his his famous wave equation. The Davission and Germer experiment demonstrated the wave nature of the the electron. The electron was described as both a particle and a wave. The construct left many lingering questions. How can the electron be both a particle and a wave? Nick Herbert writes in his book "Quantum Reality" Pg. 46;

The manner in which an electron acquires and possesses its dynamic attributes is the subject of the quantum reality question. The fact of the matter is that nobody really these days knows how an electron, or any other quantum entity, actually possesses its dynamic attributes."

Louis deBroglie suggested that the electron may be a beat note. 7 The formation of such a beat note requires disturbances to propagate at light speed. Matter propagages at velocity V. DeBroglie could not demonstrate how the beat note formed. This author's model demonstrates that matter vibrates naturally at its Compton frequency . The luminal Compton wave is pinned in place by restraining forces ( ref to Chapter 10 ). The reflected wave doppler shifts as it is restrained. The disturbances combine to produce the dynamic DeBroglie wavelength of matter. The animation shows the DeBroglie wave as the superposition of the original and the Doppler shifted waves.

Java animation Pick the icon to view an animation on the DeBroglie wavelength of matter.

The harmonic vibration of a quantum particle is expressed by its Compton wavelength. Equation #1A expresses the Compton wavelength.

lc = h / Mc

Equation #2A gives the relationship between frequency f and wavelength l. Please note that the phase velocity of the wave is c.

c = f l

Substituting Eq #2A into Eq. #1A yields Eq #3A the Compton frequency of matter.

fc = Mc2 / h

A doppler shifted component of the original frequency is produced by the restraint of the wavefunction. Classical doppler shift is given by Eq #4A.

f2 = f1 ( 1 +- v / c)

A beat note is formed by the mixing of the doppler shifted and original components. This beat note is the deBroglie wave of matter.


Equation #5A and the above express a function "F" involving the sum of two sin waves.

          F(L,t)| = amplitude orig. wave + amplitude reflected wave                
                | L held constant

         F(t) = sin(2p fc t + p) +   sin[ 2p fc (1 +- v/c) t ]                             

Substituting Eq #4A into Eq #5A yields Eq #6A.

         F(t) = sin[2p t(Mc2/h)+ p] + sin[2p t(Mc2/h)(1 +- v/c)]  

Refer to the figure above. A minimum in the beat note envelope occurs when the component waves are opposed in phase. At time zero the angles differ by p radians. Time zero is a minimum in the beat note envelope. A maximum in the beat envelope occurs when the component waves are aligned in phase. The phases were set equal, in Equation #7A, to determine the time at which the aligned phase q condition occurs.

q 1 = q2

2 p t ( M c2 / h ) + p = 2 p t ( M c2 / h )( 1 +- v / c )

ct = (+ -) h / 2 Mv

ld = h / Mv

The result, Equation #10A, is the deBroglie wavelength of matter. Reflections contain a luminal Comption wave. The superposition of these reflections is the deBroglie wave of matter.

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An analysis was done that described inertial mass in terms of a restraining force. This force restrains disturbances that propagate at luminal velocities. Consider energy trapped in a perfectly reflecting containment. (see fig. below)

               |           \  |
               |       P1   \ |
               |             \|
              A|             / B            
               |        P2  / |
               |           /  |
               |<---- L------>|

              Matter wave in a box 

This energy in a containment model is a simplistic representation of matter. In this analysis no distinction will be made between baryonic, leptonic, and electromagnetic waves.

The wavelength of the energy represents the Compton wavelength of matter. The containment represents the surface of matter. The field propagates at light speed. Its momentum is equal to E/c. The containment is at rest. The energy is ejected from wall "A" of the containment, its momentum is p1. The energy now travels to wall "B". It hits wall B and immediately bounces off. Its momentum is p2 . The energy now travels back to wall "A", immediately bounces off, its momentum is again p1. This process repeats continuously. If the energy in the containment is evenly distributed throughout the containment, the momentum carried by this energy will be distributed evenly between the forward and backward traveling components. The total momentum of this system is given in equation #1C.

pt =(p1 /2 - p2 /2)

The momentum of a flow of energy is given by equation #2C
         p = E/c                                           Eq #2

                            E = energy

                            c = light speed

                            p = momentum

         Substituting Eq. #2C into Eq. #1 yields Eq. #3C.


         pt  = [E1 /2c - E2/2c]         Eq. #3C         

Given the containment is at rest. The amount of energy in the containment remains fixed, the quantity of energy traveling in the forward direction equals the quantity of energy traveling in the reverse direction. This is shown in equation #4.

         E1  = E2                                          Eq #4C          


Substituting Eq. #4C into Eq. #3C yields Eq #5C.

pt = (E/2c)(1 - 1) Eq #5C
Equation #5 is the total momentum of the system at rest. If an external force is applied to the system its velocity will change. The forward and the reverse components of the energy will then doppler shift after bouncing off of the moving containment walls. The momentum of a an energy flow varies directly with its frequency. Given that the number of quantums of energy is conserved, the energy of the reflected quantums varies directly with their frequency. This is demonstrated by equation #6C.

E2  = E(1) [ff / fi]                                   Eq. #6C         

Substituting Eq. #6C into Eq #5C. yields eq. #7C.


pt  = (E/2c)[(ff1/fi1) - (ff2 /fi2)]                      Eq #7C          

Equation #7C is the momentum of the system after all of its energy bounces once off of the containment walls. Equation #7 shows a net flow of energy in one direction. Equation #7C is the momentum of a moving system. The reader may desire to analyze the system after successive bounces of its energy. This analysis is quite involved and unnecessary. Momentum is always conserved. Given that no external force is applied to the system after the first bounce of its energy, its momentum will remain constant.

Relativistic doppler shift is given by equation #8C.

(ff / fi) = [1 - v2 / c2]1/2 / (1 +- v/c), Eq #8C
                                    v = velocity with respect to the observer

                                    c = light speed

                                ff/fi = frequency ratio 
                         + or - depends on the direction of motion

Substituting equation #8 into equation #7C yields equation 9C

                  -        .5                  .5  _                          
           =  E  | (1-v2/c2)          (1-v2/c2)      |
             --- |  -----------   -  ----------     |                Eq #9C                                 
              2c | (1-v/c)            (1+v/c)       |
                  -                                _

                 _                                        -                                                                                                 _              .5                     .5_  
          =   E | (1+v/c)(1-v2/c2)       (1-v/c)(1-v2/c2)   |
            --- |  ----------------  -  -----------------  |
             2c | (1+v/c)(1-v/c)         (1-v/c)(1+v/c)    |
                 -                                        -

                 -         .5             -
             E  | (1-v2/c2)  (1+v/c-1+v/c) |
             -- |  ----------------------- |
             2C |     (1-v2/c2)            |
                 -                        - 

                     ___ Ev________


Substituting mass for energy, M = E/c2

          =        ___Mv______

The result, equation #14C is the relativistic momentum of moving matter. This first analysis graphically demonstrates that inertial mass is produced by a containment force at the surface of matter. A fundamental change in the frame of reference is produced by the force of containment. This containment force converts energy, which can only travel at light speed, into mass, which can travel at any speed less than light speed. 8

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Note: A version of this analysis has been published in this author's book Elementary Antigravity , Vantage Press 1989, ISBN 0-533-08334-6

A version of this analysis has been published in The General Science Journal at

According to existing theory the matter wave emerges from the Fourier addition of component waves. This method requires an infinite number of component waves. Natural infinities do not exist within a finite universe. The potential and kinetic components of a wave retain their phase during a Fourier localization. The aligned phase condition is a property of a traveling wave. The Fourier process cannot pin a field or stop a traveling wave.

Texts in quantum physics commonly employ the Euler formula in their analysis. The late Richard Feynman said, "The Euler formula is the most remarkable formula in mathematics. This is our jewel." The Euler formula is given below:

e i q = cosq + i sinq

The Euler formula describes the simple harmonic motion of a standing wave. The cos component represents the potential energy of a standing wave. The sin component represents the kinetic energy of a standing wave. The kinetic component is displaced by 90 degrees and has a i associated with it. The localization of a traveling wave through a Fourier addition of component waves is in error. To employ this method of localization and then to describe the standing wave with the Euler formula is inconsistent. This author corrected this error through the introduction of restraining forces. The discontinuity produced at the elastic limit of space restrains the matter wave. The potential and kinetic components of the restrained wave are displaced by 90 degrees. A mass bouncing on the end of a spring is a good example of this type of harmonic motion. At the end of it travel the mass has no motion ( kinetic energy = zero) and the spring is drawn up tight ( potential energy = maximum ). One quarter of the way into the cycle the spring is relaxed and the mass is moving at its highest velocity (kinetic energy = maximum). A similar harmonic motion is exhibited by the force fields. The energy of a force field oscillates between its static and magnetic components.

The Standing Wave

Mass energy ( Em ) is a standing wave. A standing wave is represented on the j axis of a complex plane.

         | Em = Mc2

The phase of a standing wave is 90 degrees. All standing waves are localized by restraining forces.

A traveling wave has its kinetic and potential components aligned in phase. An ocean wave is a good example of this type of harmonic motion. The wave's height ( potential energy ) progresses with the kinetic energy of the wave.


The Traveling Wave

The energy "E" contained by a wave carrying momentum "P" is expressed below.

E = Pc

The traveling wave expresses itself through its relativistic momentum "P".

P = Mv / (1- v2 / c 2 )1/2

Substituting yields the amount of energy that is in motion "Eq". Energy flows are represented on the X axis of a complex plane.

     Eq = Mvc / (1- v2 / c 2 )1/2


The vector sum of the standing ( Em ) and traveling ( Eq ) components equals the relativistic energy ( Er ) of moving matter.

[ Er ] 2 = [ Em ]2 + [ Eq ] 2

[ Er ] 2 = [ Mc2 ] 2 + [ Mvc / (1- v2 / c 2 )1/2 ] 2

Er = Mc2 / (1- v2 / c 2 )1/2

The relativistic energy is represented by the length of the hypotenuse on a complex plain

[ Phase diagram ]

The ratio of standing energy to the relativistic energy [ Em / Er ] reduces to (1- v2 / c 2 )1/2. This function express the properties of special relativity. The arc sin of this ratio is the phase;

g = arc sin (1- v2 / c 2 )1/2

The phase g expresses the angular separation of the potential and kinetic energy of matter. The physical length of a standing wave is determined by the spatial displacement of its potential and kinetic energy. This displacement varies directly with the phase g. The phase g varies inversely with the group velocity of the wave. This effect produces the length contraction associated with special relativity.

Time is represented on the Z (out of the plain) axis on a complex diagram. The rotation of a vector around the X axis into the Z axis represents the change in potential energy with respect to time. The rotation of a vector around the Y axis into the Z axis represents a change in potential energy with respect to position. Relativistic energy is reflected on both axes. The loss in time by the relativistic component Er is compensated for by gain in position.

The phase g of a wave expresses the displacement of its potential and the kinetic energy. When placed on a complex diagram the phase directly determines the relativistic momentum, mass, time, and length. These effects reconcile special relativity and quantum physics.

The analysis reveals information not provided by special relativity. The ratio of traveling energy to the relativistic energy ( Eq / Er ) reduces to v/c. The simplicity of the ratio suggests that it represents a fundamental property of matter. In an electrical transmission line this ratio is known as the power factor. The power factor is a ratio of the flowing energy to the total energy. The construct of Special Relativity may be derived a-priori from the premice that the group velocity of the matter wave is V and the phase velocity of the matter wave is c. The difference between these two velocities is produced by reflections. Reflections result from restraining forces. The same principles apply to all waves in harmonic motion.

This model requires a restrained luminal wave. What is the nature of this restraing force? Are forces beyond the four known forces required? This author will show that no addtional forces are required. The restraining force is produced throgh the action of the known forces. The nature of this restraining force will be presented in Chapter 10. An analysis of this restraining force, in Chapter 12, revealed the path of the quantum transition.

Kinetic and potential energy were represented as vectors on a two dimensional complex plain. The rotation of this complex plain through a third dimension added the element of time to the construct. The inclusion of additional dimensions should enable this model to be extended into the realm of high energy physics. The extended model would contain, at its core, a unification of relativity and quantum physics.

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Einstein's principle of equivalence states that gravitational and inertial mass are always in proportion. The photon has no rest mass and a fixed inertial mass. What is its gravitational mass of the photon? General relativity states that gravity warps space. Photons take the quickest path through this warped space. The path of a photon is affected by gravity. The effect has been measured. The light of star was bent as it passed near the sun. The momentum of the light was altered. The principle of the conservation of momentum requires that the sun experiences an equivalent change in momentum. The bending light must generate a gravitational field that pulls back on the sun. Bending light exerts a gravitational influence.

Photons from the extremes of the universe have traveled side by side for billions of years. These photons do not agglomerate. The slightest agglomeration would result in a decrease in entropy. This decrease would be in violation of the laws of thermodynamics. Photons traveling in straight lines extert no gravitational influence.

Matter gives up energy during the process of photon ejection. The principle of the conservation of energy requires that the negative gravitational potential and the positive energy of the universe remain in balance. The ejected photon must carry a gravitational influence that is equivalent to the gravitational mass lost by the particle.

These conditions are satisfied by a photon with a variable gravitational mass. This mass varies directly with the force (dp/dt) it experiences.

Hubbles' constant expresses the expansion space in units of (1/time). Ordinarily, the effects resulting for the Hubble expansion are quite tiny. At great distances and at high velocities significant effects do, however, take place. As a photon travels through space at the high velocity of light it red shifts. This red shift may be considered to be the result of an applied force. This force is produced by the acceleration given in equation #1D.

              Acceleration = Hc                                                           Eq #1D

                                        H = Hubble's constant, given in units of (1/sec)
c = light speed
To demonstrate the gravitational relationships of a photon the principle of the conservation of momentum will be employed. According to this principle exploding bodies conserve there center of gravitational mass. Mass M ejects a photon while over the pivot I. The gravitational center of mass must remain balanced over the pivot point I. Mass M1 is propelled to the left velocity at v1 and the energy of the photon E2 travels to the right at velocity c. The product of the velocity and time is the displacement S.

            <---S1---> <---------S2--------->

            Mass                           photon

            Matter and a energy on a balance beam 

The center of mass of an exploding body is qualified by equation #2D.  This center is both inertial and gravitational.

             M1 S1  = M2 S2                Eq #2D           

The gravitational field of the particle was descirbed with Newton's formula of gravity, see 2C below left.

The general formula of gravitational indiction, as presented in the General Theory of 

Relativity 3, 4  is given below.

           Induced grav. field = G/(c2r) dp/dt 

This equation (as derived in Chapter 6 ) was describes the gravitational influence of the photon, see 2C below right.

        (Newton's grav. field)(displacement) = (Einstein's grav. influence)(displacement)   Eq 2C

        (GM1 /r2) S1   = G/(c2r) force S2           

        (GM1 /r2)S1   = G/(c2r) dp/dt S2
         Substitutinhg vt and ct for displacement S and multiplying by r squared

         GM1(v1 t) = (G/c2) dp/dt (ct)r           Eq #5D             

         Substituting for force, dp/dt = Ma = MHc = (E2/c2)Hc = E2H/c    Eq #6D

         G(M1v1)t = (G/c2 )( E2 / c) Hctr             

         Substituting momentum p for M1v1 and E2/c                                   

         Gp1 t = (G/c2)p2 Hctr   

         Setting the momentums equal. p1  = p2       

         c = Hr                                             Eq #11D

Force induces the gravitational field of matter and energy. The confinement of mass energy produces a field that drops off at a one over r squared rate. A gravitational field is also produced by the acceleration of energy through Hubble’s constant. This gravitational field drops off at a one over r rate. Both mechanisms produce a equivalent effect at the edge of the visible universe. The equivalance conserves the negative gravitational potential of the universe. The speed of light depends upon the product of Hubble's constant and the radius of the visible universe. This qualification is essentially consistent with the measured cosmological constants. The gravitational constant G is an identity within these equations and is not dependent upon the condtion.

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Schrödinger's Wave Equation Revisited

Schrödinger's wave equation is a basic tenement of low energy physics. It embodies all of chemistry and most of physics. The equation is considered to be fundamental and not derivable from more basic principles. The equation will be produced (not derived) using an accepted approach. Several assumptions are fundamental to this approach. The flaws within these assumptions will be exposed.

This author will derive Schrödinger's wave equation from a set of fundamental classical parameters. The Schrödinger's wave equation was fundamentally derived from the premice that the speed of sound with the nucleus is 1.094 million meters per second and that restraining forces confine a luminal wave. This author's approach is classical.

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The accepted approach

The classical wave equation is given by equation #A1.

Ñ2 Y = (1/v2) (d2Y / d t2)

Ñ2 is the second derivative of the function in three dimensions.

Ñ2 = (d2 /dx2 + d2 /dy2 + d2 /dz2 )

The wave equation describes a classical relationship between velocity, time, and position. The velocity of the wave packet is v. It is an error to assume that the natural velocity of a matter wave is velocity v. Disturbances within the force fields propagate at light speed c. The matter wave Y is also a force field. Like all fields it must propagate at velocity c. Equation #3 below expresses the wave equation as of function of position and time.

Ñ2 Y(x, t) = (1/v2) (d2Y(x, t) / d t2)

Pg_15 UP

The exponential form of the sin function (e jwt ) is introduced. This function describes a sin wave. The j in the exponent states that the wave contains real and imaginary components. The potential energy of wave is represented by the real component and the kinetic energy of a wave is represented by the imaginary component. In a standing waves these components are 90 degrees out of phase. Standing waves are produced by reflections. The required reflections are not incorporated within current models. Current models include (e jwt ) ad-hoc.

Y(x, t) = Y(x) e jwt


Ñ2 Y(x) e jwt = (1/v2) (d2Y(x) e jwt / d t2)

A solution is obtained by integrating twice with respect to time.

Ñ2 Y (x) e jwt /( jw) = (1/v2) d (Y(x) e jwt ) / d t

Ñ2 Y(x) e jwt / -w2 = (1/v2) Y(x) e jwt

After the double integration the equation is reduced.

Ñ2 Y(x) = ( -w2/v2) Y(x)

Replacing angular velocity w with frequency f.

Ñ2 Y(x) = ( -4p2f2/v2) Y(x)

Frequency squared f divided by velocity squared v equals (1/l) squared.

Ñ2 Y(x) = ( -4p2/ l2 ) Y(x)

The Schrödinger equation describes the deBroglie wave of matter. The deBroglie wave and Planck's constant were introduced ad-hoc. The introduction of the deBroglie wave was questioned by Professors Einstein and Langevin. The introduction became accepted because it matched with the experimental facts.
"Schrödinger also had to explain how wave packets could hold together, elaborate the meaning of the wave function, and demonstrate how the discontinuities of quantum phenomena arise from a continuous wave processes."
The Great Equations, Robert P. Creese, Pg. 248

l2 = h/Mv

Ñ2 Y(x) = ( -4p2M2v2/ h 2 ) Y(x)

Ñ2 Y(x) = ( -4p2M (Mv2)/ h 2 ) Y(x)

The relationship between kinetic, total, and potential energy is expressed below.

(Mv2) = 2 (Total - Potential)


Ñ2 Y(x) = -4p2 M 2(E - U) Y(x) / h 2


- h 2 Ñ2 Y(x) / (-4p2 2 M) = (E - U) Y(x)

Substituting ħ for h / 2p

- ħ 2 Ñ2 Y(x) / ( 2 M) = (E - U) Y(x)

The result below is the time independent Schrödinger equation. The Schrödinger equation states that the total energy of the system equals the sum of its kinetic and potential energy. Energy is a scalar quantity. Scalar quantities do not have direction. This type of equation is known as a Hamiltonian. The Hamiltonian ignores restraining forces. The unrestrained matter wave propagates at velocity v. It is an error to assume that an unrestrained wave propagates without dispersion.

[ (- ħ 2 Ñ2 /2M ) + U ] Y(x) = E Y(x)

Pg_16 UP

A New Approach

This author discovered the velocity of sound within the nucleus Vt from his observation of cold fusion and antigravitational experiments.

Vt = wc rp

The electron's elastic constant was classically extracted from this velocity.

K-e = 29.05 /rx

The electron's Compton angular velocity was produced as a condition of the elastic constant K-e at a radius rx equal to the ground state radius of the hydrogen atom (.529 angstroms).

Vt = [K-e/M-e]1/2 rp

The frequency is known as the Compton frequency of the electron.

The simple harmonic motion is of a restrained wave is given by. Please not the Compton angular velocity, not the deBroglie wavelength, was is used in the formulation.

d2Y(x) / dt2 = -wc2 Y(x)

In order to give a result in conventional units the classical value for the Compton angular velocity, below, was placed into the formulation.

wc = M-e c / ħ

Squaring and factoring.

wc2 = M (Mc4)/ ħ 2

The Compton angular velocity squared was substituted for w2 below.

d2Y(x) / dt2 = - [M (Mc4)/ ħ 2] Y(x)

Dividing by light speed squared.

d2Y(x) /dt2 (1/c2) = - M (Mc2)/ ħ 2 Y(x)

H. Ziegler pointed out in a 1909 discussion with Einstein, Planck, and Stark that relativity would be a natural result if all of the most basic components of mass moved at the constant speed of light. 13 This author's work is based on the idea that local fields are restrained at elastic discontinuities. Disturbances withn the restrained fields propagate at light speed. Refer to the wave equation A1. Substituting Ñ2 for acceleration divided by light speed squared. This step embodies the idea the disturbances in the matter wave propagate at luminal velocities. This empodument provides for a unification of Special Relativity and quantum physics.

Ñ2 Y(x) = - M (Mc2)/ ħ 2 Y(x)

Mass energy is expressed as the difference between the total energy and the potential energy of the matter wave. U always equals 1/2 E there for a factor of two was employed to get the total positive energy.

Mc2 = 2(E - U)


Ñ2 Y(x) =-[2M ( E - U ) / ħ 2 ] Y(x)

The result below is the time independent Schrödinger equation.

[ (- ħ 2 Ñ2 /2M )+ U ] Y(x) = E Y(x)

The time independent Schrödinger equation has been derived from a simple technique. The deBrobie wave was not incorporated ad-hoc into the solution. Disturbances within the matter wave propagate at luminal velocities. Restraining forces prevent dispersion. The deBroglie wave arose naturally from the restraint of the Compton wave.

Pg_17 UP


The movement of ordinary matter does not produce a net magnetic field. The movement of charged matter does produce a net magnetic field. Charged matter is produced by the separation of positive and negative charges. The derivation used to develop Newton's formula of gravity (Equation #3) shows that matter may harbor positive and negative near field gravitational components.

The wavefunctions of superconductors are collimated. The collimated wave functions act in unison like a single macroscopic elementary particle. The near field gravitational components of a superconductor are macroscopic in size. The rotation of these local gravitational fields is responsible for the gravitational anomaly observed at Tampere University. 9

Pg_18 UP


Natural flux is pinned into the structure of matter. Forces are produced through the action of the containment. These forces induce the gravitational mass of matter, generate matter's relativistic properties, and determine matter's dynamic properties.

The nature of the bundling force will be presented in Chapters 10 & 11. An analysis of the bundling force revealed the path of the quantum transition.

Pg_19 UP


 1.  French aristocrat Louis de Broglie described the electrons wavelength in his Ph. D. 
     thesis in 1924.   De Broglie's hypothesis was verified by C. J. Davisson and L. H. Germer 
     at Bell Labs.  

 2.  Gilbert N. Lewis demonstrated the relationship between external radiation 
     pressure and momentum.  Gilbert N. Lewis.  Philosophical Magazine, Nov 1908.

 3.  A. Einstein,   Ann d. Physics 49,  1916.

 4.  Einstein's principle of equivalence was experimentally  confirmed by  R.v.  Eötös 
     in the 1920's.

     R.v.  Eötös,  D. Pekar,  and Feteke,  Ann. d. Phys 1922.

     Roll, Krotkov and Dicke followed up on the Eötvös experiment and confirmed the 
     principle of equivalence to and accuracy of one part in 10 11  in the 1960's.

     R.G. Roll,  R.  Krokow  &  Dicke,  Ann.  of Physics 26,  1964.

     MATHENATICA (1687).


 6. Jennison, R.C. "What is an Electron?"  Wireless World, June 1979. p. 43.

         "Jennison became drawn to this model after having experimentally 
         demonstrated the previously unestablished fact that a trapped           
         electromagnetic standing wave has rest mass and inertia."

      Jennison & Drinkwater Journal of Physics A, vol 10,
      pp.(167-179)   1977

      Jennison & Drinkwater Journal of Physics A, vol 13,
      pp.(2247-2250)  1980

      Jennison & Drinkwater Journal of Physics A, vol 16,
      pp.(3635-3638)  1983

 7.  B. Haisch & A. Rueda of The California Institute for Physics and Astrophysics
     have also developed the deBroblie wave as a beat note.  Refer to:  

 8.  Znidarsic F. "The Constants of the Motion"   The Journal of New Energy
     Vol. 5, No. 2 September 2000

 9.  "A Possibility of Gravitational Force Shielding by Bulk YBa2Cu307-x",
     E. Podkletnov and R. Nieminen,  Physica C, vol 203
    (1992), pp 441-444.

 10.  Puthoff has shown that the gravitational field results from the cancellation of
      waves.  This author's model is an extension version this idea.

             H.E. Puthoff, "Ground State Hydrogen as a
             Zero-Point-Fluctuation-Determined State"

             Physical Review D, vol 35, Number 3260, 1987

             FORCE", Physical Review A, vol 39, Number 5, March 1989

 11.  Ezzat G. Bakhoum "Fundamental disagreement of Wave Mechanics with Relativity" 
       Physics Essays  Volume 15, number 1, 2002

 12.  John D. Barrow and John K. Webb  "Inconstant Constants" 
       Scientific American  June 2005

 13.  Albert Einstein, "Development of our Conception of Nature and Constitution of Radiation," 
        Physikalische Zeitschrift 22 , 1909.

// end of chapter 7 .............................................................................