THE DYNAMIC THEORY A New View of Space-Time-Matter by Pharis E. Williams

Dedication I dedicate this work to my family; to my father and uncle

who encouraged my thinking and individualism, to my mother for her steady love, to my brothers and sister for their confidence in my ability, to my children for growing up with 'Dad's theory', and to my wife Jeri for she bore the brunt of my mental absence.

Copyright � 1993 by Pharis E. Williams

PREFACE Present books, such as "The Arrow of Time" by Roger Highfield and Peter Coveney and "The Big Bang Never Happened" by Eric Lerner, talk of a new revolution is science. The first points to work by Ilya Prigogine and others with regard to the flow of time and the dichotomy between the time flow in the universe and physical theories wherein time may flow forward and backward. The "Unended Quest" in " The Arrow of Time" is to find how a foundation of science might be laid that describes dynamic systems showing this one-way aspect in time. In "The Big Bang Never Happened" Lerner also points out the need to find physical theories which correspond to the directivity of nature's time. The main discussion though concerns explanations of cosmological phenomena in terms of plasmas and Maxwellian electromagnetic concepts. I am in agreement with the authors of both these books with regard to the majority of their points. I disagree with Highfield and Coveney in that a foundation for physical theories restricted by a flow of time has been found and reported starting in 1976. My disagreement with Lerner is very limited, but may point out an important difference in our thinking. Let me quote from Lerner's introduction where he states; "Today we again hear renowned scientists, such as Stephen Hawking, claiming that a 'Theory of Everything' is within our grasp, that they have almost arrived at a single set of equations that will explain all the phenomena of nature --gravitation, electricity and magnetism, radioactivity, and nuclear energy --from the realm of the atoms to the realm of the galaxies and from the beginning of the universe to the end of time. And once again they are wrong. For quietly, without much fanfare, a new revolution is beginning which is likely to overthrow many of the dominant ideas of today's science, while incorporating what is valid into a new and wider synthesis." I believe Lerner is correct. But only in the sense that I do not believe it possible to know all of the phenomena of nature "from the beginning of the universe to the end of time." What I put forth in this book is my research which shows that one can start with a small, simple set of equations and derive the basis for the currently accepted branches of physics by imposing restrictive assumptions. The search for a unifying field theory began in the early 1800's when scientists began searching for a way of unifying the electromagnetic and gravitation fields. When the proton-proton scattering results showed a deviation from Coulombic scattering, once again scientists began trying to find a way of unifying the fields, or forces, of nature. This was done immediately upon the heels of assuming that the deviation from Coulombic scattering must come, not from changes in Maxwellian electromagnetism, but from an independent strong nuclear force. It has always appeared to me that one should go back and address this assumption of independence before seeking a means of unification.

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One doesn't need to read too much of the scientific literature from the 1930's to the present to see how much has been devoted to the notion of unifying the forces, and/or fields, of nature. Within this body of work lies the basis for Hawking's "Theory of Everything." I believe this work misses the point of unification. For instance, if we wish to approach a unification, what should we unify? Should we unify the fields, or should we unify the various branches of physics? It seems rather difficult to believe that nature is divided into the different branches of physics, such as thermodynamics, Newtonian mechanics, relativistic mechanics, and quantum mechanics, just because we learned how to formulate the basis for each branch at different times in our scientific advancement. Further, given a variational principle and a metric we know how to derive field equations and force laws. Therefore, shouldn't we be seeking to unify the various branches of physics and deriving the necessary fields from that unification rather than trying to unify the fields and not reconciling the difference between the foundations of the different branches? In my research I chose to seek a way of unifying the various branches of physics. This entailed seeking a simple set of physical laws from which one may derive the foundations of the different accepted branches of physics as subsets of this more general set of laws. What has emerged from this work is that there is a logical necessity for the branches of physics that comes from the imposition of different restrictive assumptions. The type of geometry need not be assumed as Newton and Einstein did, but is dictated by the fundamental laws. The laws produce, not one, but two variational principles from which we may derive the field equations and force laws. What resulted from the attempt to unify the branches of physics produced not only the desired result, but, also that of unifying the fields and forces of nature also. The fundamental laws, which could be written on a T-shirt, produce field equations and force laws which accurately describe phenomena intended to be included in Hawking's "Theory of Everything." It does not, however, allow for the existence of a Big Bang or beginning or end of time. Furthermore, since the fundamental laws are based upon generalizations of classical thermodynamics, the equations of motion derived from them come complete with an Arrow of Time built in. I first reported this predicted flow of time in 1981. If I were asked to explain why the research reported in this book has not gained any wider distribution than it currently enjoys, I would have to offer up our system of refereed journals as the most important reason. But hand-in-hand with this must go the notion that "everyone knows that one may derive classical thermodynamics from any number of different force laws by using statistical mechanics." This notion was refuted by Peter G. Bergmann in 1979, yet it persists today.

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On the other hand, if one were to accept the potential of having equations of motion derived from generalizations of classical thermodynamics, then it is not difficult to imagine an Arrow of Time accompanying them. But this is small incentive to a referee. Neither is the ability to derive the field equations and the force laws for the different branches of physics much more incentive for the referee to give a thumbs-up for such a theory which 'everyone' knows is doomed before it gets started. The many attempts to get portions, or all, of this research published in the refereed journals have produced many interesting comments. These comments are interesting from the point of view that they expose the human side of referees, not that they are based upon scientific evaluation. Let me offer three excerpts as examples: from the physics department of a name university, "While the equations you've derived are not wrong, we somehow like it better the old way," from a scientist at a government laboratory, "If you ask me to shoot you down, I can't. If you ask me to help you, I won't. I suggest that you learn to play the game and then someone may listen to you," and from a journal dedicated to speculation, "We no longer have the time to consider articles which look into the foundations of physics." What I sought to do was to answer some personal questions about science using all of the rigor contained in the logic of mathematics. What I found was a methodology by which we may see how the various physical phenomena from the nuclear realm to the cosmos come from a single, simple set of three fundamental assumptions. Many current interpretations concerning fundamental aspects of several existing theories are shown to be wrong, misleading, or too restrictive. Notice that I said many current interpretations are wrong, not many current theories are wrong. What I found is that there is a much more general theory available in which the current theories are subsets or first, or second, order approximations. That doesn't mean these theories are wrong any more than the validity of the Special Theory of Relativity means that Newton's equations of motion are wrong. It only means that Newton's dynamics applies only to a limited range of velocities . If we then use Newton's equations of motion for velocities approaching the speed of light our interpretations will of necessity be wrong. However, we didn't know these interpretations were in error until Einstein put forth his more general theory. The same is found to be true of many interpretations based upon the current theories which the Dynamic Theory shows to be wrong when viewed in its more general light. Also, the reported research shows how the various branches of physics fit together into a unified picture of a nature built upon the dimensions of space, time, and mass.

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TABLE OF CONTENTS Preface i Chapter 1 1 Overview 1.1 Questions concerning the theoretical basis 1 1.2 Possible new theoretical approach 2 1.3 A new view of space, time, and matter 6 Chapter 2 36 New Theoretical Fundamentals A. General Laws 2.1 First Law 36 2.2 Second Law 39 2.3 Absolute Velocity and Einstein's postulate 45 2.4 The concept of Entropy 49 2.5 Third Law 51 B. General Relations 2.6 Energy and Maxwell's Relations 52 2.7 Equilibrium conditions 55 2.8 Stability conditions 56 C. Geometry 2.9 Geometry required by fundamental laws 58 D. Mechanical systems near equilibrium 2.10 Special relativistic and classical mechanics 71 2.11 Energy concepts 77 2.12 Non-isolated systems 81 E. Quantum mechanics 2.13 Quantum Mechanics derived 82 2.14 On the derivation of thermodynamics from statistical mechanics 85 F. Summary 2.15 Summary of new theoretical fundamentals 87 Chapter 3 90 Five-Dimensional Systems A. Systems near an equilibrium state 3.1 Equations of motion 91 3.2 Energy equation 95 B. Systems with non-Euclidean manifold 3.3 General variational principle 97

3.4 Gauge function field equations 99 3.5 Energy-momentum tensor

100 3.6 Force density vector 104 3.7 Equation of energy flow 106 3.8 Momentum conservation 106 3.9 Gauge field pressure 108 Chapter 4 110 Five-Dimensional Quantization A. Quantization in five dimensions 4.1 Quantization 110 4.2 Five-dimensional Hamiltonian 111 4.3 Five-dimensional Dirac equation 112 4.4 "Lorentz" covarience 113 4.5 Spin 114 4.6 Dirac equation with fields 115 4.7 Allowed fundamental spin states 116 B. Quantized fields 4.8 Quantum condition applied to particles 119 4.9 Radial field dependence 121 4.10 Self-energy of charged particles 127 4.11 Nuclear phenomena 132 4.12 Hiesenberg's Uncertainty Principle and geometry 138 4.13 Nuclear masses 144 Chapter 5 157 Five-Dimensional Gravitation 5.1 Charge-to-Mass ratio and magnetic moments 157 5.2 Perihelion advance 167 5.3 Redshifts 171 5.4 "Fifth" force 183 5.5 Inertial and Gravitational mass equivalence 192 5.6 Cosmology 194 Chapter 6 198 Electromagnetogravitic Waves 6.1 Wave equations 198 6.2 Wave solutions 198 6.3 Non-thermal transmission 205 6.4 Wave boundary conditions 208 6.5 Reflection and refraction 217 6.6 Complex refraction angles 224

6.7 Assumptions and wave solutions 227 Chapter 7 239 Hydrodynamic Systems 7.1 First fundamental quadratic form 241 7.2 Second fundamental quadratic form 245 7.3 Tensor derivatives 249 7.4 Relativistic hydrodynamics 256 7.5 Classical hydrodynamics 257 7.6 Shock waves 259 7.7 Mass conservative electrodynamics 262 Chapter 8 267 Experimental Tests 8.1 Speed-of-light 268 8.2 Index of refraction 269 8.3 Neutron interferometer 270 8.4 Nuclear masses 271 8.5 Gravitational rotor 271 8.6 Nuclear Lamb shift 276 Chapter 9 277 Epilogue 9.1 Only three basic assumptions 277 9.2 Geometry is specified 279 9.3 The Arrow of Time 279 9.4 Mass as a coordinate 280 9.5 Non-singular gauge potential 280 9.6 Unification of the branches of physics 281 9.7 The pedagogical aspect of the Dynamic Theory 281 9.8 Where to from here? 282

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CHAPTER 1 OVERVIEW 1.1 Questions concerning the theoretical basis It seems that throughout my working career I have been a trouble-shooter. This started when I entered the Navy as an Electrician's Mate working on the power electrical equipment on Navy ships. Troubleshooting was the main job, whether it was finding some electrical malfunction or the presence of saltwater in an electrical box. Later, as a Naval Officer with an Electrical Engineering degree, I was constantly required to ferret out some sort of trouble. This at times would involve missile systems, gun systems, radars, sonar systems, boilers, or other systems. It seemed only natural then to employ this same procedure to investigate what appeared to me as problems in the foundations of physics. Though I had often asked "Why?" when confronted with some new assumption or adopted postulate, the first really puzzling facet of current physics I encountered was the concept of relativistic kinetic energy from Einstein's Special Theory of Relativity. The puzzling part was that it depended upon the speed of light independent of the mechanism by which this energy might be transferred. To better illustrate what puzzled me, consider the transfer of energy between two charged particles on collision courses. If the particles have near-miss trajectories, then the energy is primarily transferred by the electrical forces between the charges. From the view of retarded potentials, or the concept of a limiting speed of electromagnetic signal transmission, it is rather easy to accept the energy transferred being dependent upon this limiting velocity. But suppose the particles are uncharged and the interaction is strictly a gravitational one. Again the concept of a limiting signal speed would imply that the energy exchanged between the particles depend upon this limiting velocity. But is it the same as the limiting signal velocity for the electromagnetic case? Do gravitational waves travel at the same speed as electromagnetic waves? Einstein, in the Special Theory of Relativity, adopted the position that the constancy of the speed of light forces a modification of Newton's dynamic law. This modification implies that all forces have the same limiting velocity, namely, the speed of light. There exists an abundance of theoretical and experimental evidence that the speed of light becomes the limiting velocity whenever electromagnetic forces are involved. The point that bothered me was whether other forces, such as gravitational, should also have the same limiting velocity. Though we have had reports of the detection of gravitational waves, we have no experimental determination of the speed of a gravitational wave. Therefore, I object to the viewpoint that the modification to Newton's law should be applied to all forces without some additional justification. Let me describe an analogy which may not hold in the strictest sense yet may serve to illustrate my point of view. A river, flowing toward the sea,

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carries energy with it. The speed with which this energy can move from one point to another is the velocity of the river's current. The river produces a force on a boat tied up to a pier on the river. When the boat is set adrift, this force accelerates the boat. However, the maximum velocity to which the river can accelerate the boat is the current velocity; this is the velocity with which the energy of the river can propagate. From this point of view the speed of light, being the propagation velocity of electromagnetic energy must be the limiting velocity associated with electromagnetic forces. Certainly nature would be much simpler if all forces have the same limiting velocity. Yet without some experimental evidence of the propagation of gravitational energy, I find it difficult to feel comfortable with Einstein's modification of Newton's law justified by electromagnetic experimental evidence and arguments of simplicity. The fundamental philosophical viewpoint that the force depends upon velocity and vanishes as the velocity approaches the limiting velocity raises another question concerning Einstein's modification of classical mechanics. Under Einstein's modification Hamilton's principle is written with a relativistic mass which depends upon the velocity and a velocity independent force. Does this represent a different philosophy or are both views equivalent? More specifically, are the "real" concepts to be taken as a mass independent of velocity together with a velocity dependent force or should we associate the velocity dependent relativistic mass and velocity independent forces with "real" world? Or does it make any difference which we chose? At this point I faced the first major decision. If I adopted Einstein's postulates, then it appeared that I would be required to change my intuitive beliefs concerning certain physical phenomena. I found this extremely difficult to do. On the other hand, if I did not embrace these postulates, I would have to replace them with something that would say essentially the same thing in all cases where the Special Theory of Relativity has been found to be very accurate. Not only this but if a new point of view were adopted, then virtually the entire sphere of physics may need to be reviewed in order to ensure that the new point of view did not conflict with currently used theories where they have experimental verification. 1.2 Possible new theoretical approach History records the advancements in physics which came from the efforts of people new to the field. Therefore my lack of training in physics might be turned into an advantage if I sought to determine a philosophical basis unhampered by the directed philosophy that comes from a study of physics as currently taught. This is in contradistinction with current practices and procedures of academicism where mastery of current theories generally precedes the development of a new one. To deliberately choose this deviation risks accusations of arrogance and naivete. On the other hand such a choice

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seemed the best way of avoiding the danger of becoming so familiar with current ways of thinking as to make it improbable of giving due attention to other ways. Having decided to look for a new foundation for physics I was faced with the question of how to begin. I recalled some Ozark hill philosophy I overheard as a youngster. A native Ozarkian was giving directions to a stranger who was trying to find a certain fishing hole. The directions went something like this: "See yonder road going down that holler? Well, go down thar 'bout five mile and you'll come to a fork in the road. Take the right hand fork. Now that's the wrong one but you take it anyways. After you've gone a piece, you'll come to a log across the road. Now you know you're on the wrong road. So go back and take the left hand fork. You can't miss it." A quick review of physics reveals that there are different branches with different sets of fundamental laws or postulates. Though it is easy to see how the distinction between these branches came about, it was difficult for me to believe that nature shared the same divisions. I felt that all natural phenomena should be explained by a single set of fundamental laws. This belief is somewhat like a grove of redwood trees or bamboo forest. Above the ground each tree appears as a distinct plant. Yet we know that below the ground they may be found to grow from the same root system. Thus, I felt that a more fundamental approach might display the unity in nature and that prior attempts at unification in the search for a unified field theory could be likened to attempts to tie the trees together at the tree top level rather than down at the root level. Is nature symmetrical in time? Does everything run backward in time as well as forward? Obviously, not every process in nature will run backwards, yet the equations of motion in Newtonian and relativistic mechanics are time symmetrical. I believe in an asymmetrical nature and this belief played a role in the eventual selection of fundamental laws. How then did I use this philosophy to determine a set of generalized laws on which to base an attempt to construct a new approach to physics? Before proceeding let me offer a word of caution. During any theorization the philosophy of the theorist plays such an important role that an attempt to understand the theory is aided by a knowledge of this philosophy. Therefore the following includes not only the philosophical basis upon which the theory is based and the mathematical development but also ideas and beliefs which played a part in the various decisions. Because of the individualistic nature of philosophy the following will deviate occasionally from a strict third person presentation, risking a loss of professional appearance, to the clearly personal first person. Newtonian mechanics fails to describe events involving high velocities, relativistic mechanics fails to describe the atom, and gravitational effects have resisted quantization. If these are viewed as logs and the Ozarkian's directions are followed, then we must retrace our steps and seek another approach rather

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than attempting to chop up the log and continue to push forward up one of these roads. The branch of thermodynamics, however, does not appear to have a log somewhere along the way. Here the classical thermodynamic laws are very general, particularly Caratheodory's statement of the second law. Thus the thermodynamic laws appeared to be the fork in the road where a new route might be chosen. However, in mechanics we talk of equations of motion, field equations, and geometry while in thermodynamics we speak of equations of state and equilibrium. If a generalization of the classical thermodynamic laws is adopted, how might we obtain the equations with which we are familiar in mechanics? More particularly, how could this type of general law yield geometry and a variational principle? The second law of thermodynamics can produce a variational principle through principles such as increasing entropy and minimizing free energy, but can it also produce a geometry? This seemed to be a crucial point. If the laws could not produce a geometry, then a geometry would have to be assumed, thus necessitating an additional assumption. The belief that a simple fundamental set of laws should lead to the fundamental principles of the different branches of physics made the thought of additional assumptions abhorrent. The notion that the adopted laws should specify the type of geometry that must be used seemed very satisfying. Newton found that the absolute nature of Euclidean geometry brought undesirable features. Einstein, in his General Theory, displayed the benefits that might be gained by going to a more general geometry. He showed that physical phenomena might be displayed as elements determined by certain physical laws. This is essentially the question here. Can a set of laws, which are generalizations of the classical thermodynamic laws, determine the metric elements and hence the geometry? By appealing to the mathematics of functions of more than one variable we find that a quadratic form becomes involved when a maximum or minimum is sought. Further, this quadratic form generates a natural geometry for that function. In thermodynamics the stability conditions provide a similar quadratic form and therefore the quadratic form which specifies the stability conditions should form a natural geometry for a physical system governed by laws such as the thermodynamic laws. Thus the foundations of the theory have been outlined, namely the belief that all physical phenomena should be derivable from a single set of physical laws which are generalizations of the classical thermodynamic laws. Such a theory should be capable of describing all the dynamic events in nature. Therefore it seems appropriate to call it the "Dynamic Theory". Obviously, for such a theory to be tenable it must reproduce, or be consistent with, the various fundamental postulates and/or laws currently used in the various branches of physics. Indeed it should do even more. It should also reduce the number of necessary assumptions and provide an unprecedented unification of physics.

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Further, there is the possibility that the theory might produce an experimentally verifiable prediction. The first requirement that should be placed upon the Dynamic Theory is that it reproduce, or be consistent with, current theories. In order to show that the Dynamic Theory satisfies this requirement, Section A of Chapter 2 states the adopted laws and then sections of the remainder of the book show how appropriate restrictions upon the system do yield the fundamental principles for the various current theories. Though a theory which has the capability of displaying a unification of physical theories might have significant value based solely upon this capability, it would become more attractive if it could explain phenomena for which no explanation exists or make some new prediction which might lead to an experimental test of the theory. Since restrictions were placed upon the system in order to show how current theories may be obtained, the easiest way to see the expanded coverage of the theory is to relax one or more of the restrictions and consider a more general system. In Chapters 3, 4, 5, 6, and 7 some of the previously imposed restrictions are relaxed and the results are worked out for several types of systems. Chapter 8 presents some experiments which might test the Dynamic Theory. A theory, such as the Dynamic Theory, immediately poses several problems which are not associated with its validity or applicability. First, there is a new point of view to be dealt with. Initially it would appear to be inconsistent with all past concepts of system energy or relativistic concepts. Yet in the end it is completely consistent with current theories and sheds an entirely new light upon physical phenomena. Another imposing difficulty with the Dynamic Theory stems from its generality. The scope of the theory includes all physical phenomena while in the past half century the vast amount of scientific knowledge that has been accumulated has demanded specialists. Increasing expansion of mankind's knowledge demands further specialization. Such a progression produces no demand for a generalist. The result is that the greater portion of this theory will be outside the field of many readers. Closely associated with this problem is another. Throughout science symbols and words are used to denote concepts and quantities. The limited number of available symbols and words together with the expanded scope of scientific knowledge requires duplication. For the specialists this duplication can be somewhat minimized. However, in the case of a general theory touching virtually all areas of specialization the problem becomes very significant. In particular, if a certain symbol or set of words is used, a particular notion or concept may be associated with them by the reader. This association will likely depend upon the reader's specialty and therefore will vary with the reader. Any attempt to choose symbology or word usage aimed at a particular specialty risks increased confusion for readers in other fields. Therefore, the reader is cautioned to keep in mind that conceptualizations and symbology familiar

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because of its use in one branch of physics may now take on an entirely new meaning. 1.3 A New View of Space-Time-Matter The history of mankind's attempts to unify electromagnetic and gravitational fields, or interactions, began when man begin to learn of electric and magnetic fields. The first formal theory attempting to unify the two fields of science was presented in 18361. However, no theory has yet been suggested that has gained undeniable experimental verification. Theoretical physicists are still at work trying to find a theory that will ultimately unify the forces of nature. Such is the strength of the belief in the unity of nature. The theory developed below adopts the premise that a description of physical phenomena should be based upon a simple set of fundamental postulates and that the current physical theories should be found to be subsets of this more general theory by applying restrictive assumptions. The selection of the three following fundamental laws reflect this premise. Generalized Laws In looking for a choice of fundamental basis for a theory to unify the various branches of science, consider the following. Newtonian mechanics fails to describe events involving high velocities, relativistic mechanics fails to describe the atom, and gravitational effects have resisted quantization. On the other hand, one finds that thermodynamics is the one branch of science which has always been found to hold. Here one finds the classical thermodynamic laws to be very general, particularly Caratheodory's statement of the Second Law. In mechanics the basic equations discussed are equations of motion, field equations, and geometry, while in thermodynamics the basic equations are equations of state and equilibrium. If a generalization of the classical thermodynamic laws are adopted as a fundamental basis for a unifying theory, how may the familiar equations from mechanics be obtained? The crucial point is how to obtain geometry and a variational principle from these general laws. Given a geometrical description and a variational principle, established procedures may be used to obtain equations of motion and field equations. Geometry may be obtained from a quadratic form. Therefore, stability conditions should yield a natural geometry based upon laws generalized from the classical thermodynamic laws. Further, in thermodynamics we find two variational principles; one in the maximum entropy principle for isolated systems, the other is the minimum free energy principle for non-isolated systems.

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First Law The First Law is taken as the statement equating the energy exchanged between the system and its surroundings to the change in the system energy plus any work that the system does. The form for expressing this law is

In Eqn. (1), dU represents the differential change in the system's energy, ðE represents any and all energy exchanged between the system and its surroundings that cannot be expressed by a work term. There is no restriction in this law concerning the number of independent variables. The dimensionality depends only upon the applicable, independent work terms. However, in this presentation it is beneficial to initially place some restrictions upon the type and number of allowed work terms. Therefore, a system with only one work term which is the pdv expansion work of thermodynamics will be called a "thermodynamic" system. A system with three mechanical fdx work terms will be called a "mechanical" system. An important aspect of this law is that, while the energy of the system is a function that is independent of the path, both the energy exchanged with the surroundings and the work done depend upon the path by which the system goes from one state to another. The path dependence of these terms places severe limitations upon the utility of this law and will become important when viewing relativistic and Newtonian mechanics using the new theory. Second Law Caratheodory's statement of the Second Law of Thermodynamics is very abstract and does not depend upon the type or number of variables used and, therefore, is already in very general form. The law simply says that there exist states to which the system may not go and then be able to return to its original state. Though Caratheodory formed this statement in terms of neighborhoods, it is known from thermodynamics that it contains the aspects of prohibiting perpetual motion; to be exact, perpetual motion of the second kind. The point is that this law seems intuitively to apply to mechanical systems as well as thermodynamic systems. The Second Law is stated as: In the neighborhood (however close) of any state of a system of

any number of independent variables, there exist states that cannot be reached by reversible E-conservative (ðE=0) processes.

n).1,..., =(j ; qdf - dU = E jj_

1

8

Obviously, if attention is restricted to purely thermodynamic systems with only a pdv work term, these laws produce classical thermodynamics. Therefore, the important question is whether or not the laws contain the existing mechanical theories when only mechanical fdx work terms are considered. In thermodynamics, Caratheodory used his statement of the Second Law to show that the Second Law guarantees the existence of an integrating factor for the First Law. One important feature of such a result is that the integrating factor converts the path dependent First Law into a path independent statement. Two other features resulting from Caratheodory's work have increased significance when applied to a mechanical system. Caratheodory showed, in classical thermodynamics, that the integrating factor is a function of temperature only and that it is independent of the system. When a mechanical system is considered, the integrating factor can be shown2 not only to exist but also to be a function of the velocity only and independent of the type of force considered. Since the integrating factor is strictly a function of velocity, an absolute velocity may be defined as in thermodynamics where an absolute zero temperature is defined. Thus, the absolute velocity is defined as that constant velocity at which a system may undergo a process from one solution curve to another without exchanging energy with its surroundings. Mechanical Entropy The integrating factor may be used to define a mechanical entropy just as we do for a thermodynamic system. Here the definition becomes

where S is the mechanical entropy and the process is a reversible one. Thus, the path independent function obtained by using the mechanical integrating factor is the function defined as the mechanical entropy. The Second Law may be used, as done in Section 2.4, to show that an isolated mechanical system, which cannot exchange energy with its surroundings, undergoing a spontaneous, or irreversible, process must experience an increase in its mechanical entropy. Third Law Just as in thermodynamics, where a Third Law was needed in order to associate the entropy of one system to the entropy of another, so also a Third Law is needed here. The Third Law may be stated:

,)q(

E = dS&φ

_

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The entropy of a system, when the integrating factor becomes infinite, is a constant, and this constant may be taken to be zero.

Some of the immediate results of these adopted laws may now be presented. In particular, the definition of the absolute velocity says the integrating factor goes to zero for this unique velocity. The Third Law combines with the Second to say that the absolute velocity may not be obtained in a finite number of steps. Thus, the absolute velocity becomes a unique limiting velocity. Also, the Second Law showed that the integrating factor was independent of the type of force considered. Therefore, the limiting velocity does not depend upon the force and, hence, must be the same regardless of the type of force. Thus, not only must all forces have the same limiting velocity, but since the absolute velocity is unique and the only velocity found in Nature that exhibits this characteristic is the speed of light, then the speed of light must be the absolute velocity. Further, since the definition of the absolute velocity is made for a constant-velocity process, Einstein's assumption concerning the constancy of the speed of light comes directly from the adopted laws (see Section 2.3). Geometry In order to find the equations of motion for a mechanical system, the geometry required by the adopted laws must first be determined. Since the mechanical system was considered to have three fdx work terms, the energy of the system becomes a function of four independent variables: three space variables and the mechanical entropy. Thus, the quadratic form obtained from the stability conditions may be expressed in terms of the variables of space and mechanical entropy and is

Adopting this quadratic form as the metric of a general system whose thermodynamic variables are held fixed, the metric may be written as

where the summation convention is used and

1,2,3).=,(

; 0>)dq)(dq(qq

U+)dq(dS)(qS

U2+)(dSSU 22

22

2

βα

βα

βα

α

α ∂∂∂

∂∂∂

∂∂

(2)

0,1,2,3)=j(i, ; dqdqh=)(ds jiij

2

(3)

qq

U=h ji

2

ij∂∂

∂ 5

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where q0 is the entropy. Thus, the stability conditions provide a metric in the four-dimensional manifold of space-entropy. The arc length s, in the space-entropy manifold, may be parameterized by choosing

where c is the unique velocity appearing in the integrating factor of the Second Law. The metric may now be written as

But Einstein's relativistic theories are in space-time manifolds. In order to show that the proposed theory contains Einstein's theories, a space-time manifold must come from the adopted laws. This is indeed the case if the mechanical system is restricted by requiring that it be isolated (ðE=0). This restriction establishes the condition necessary for the principle of increasing mechanical entropy which becomes a variational principle

In order to use this variational principle, Eqn. (4) may be expanded, solved for dq0 and squared to arrive at the quadratic form

where

with /dt.dq = q nn& 11 By defining x0 = ct and xn = qn then Eqn. (6) may be written as

cdt,=dtq=ds 0& 6

0,1,2,3).=j(i, ; dqdqh = )(dtc jiij

22

(4)

0. = )dq( = )S(d 202 ∫∫ δδ

(5)

],dqdqh-Adtdqh2+)(dtc[h1 = )dq( 0

22

00

20 βααβ

αα

(6)

h

)q(h+h

qqh-hc +

hqh = A

00

20

0000

2

00

0 &&&& αα

βααβ

αα _

11

where f = h00. This metric obviously reduces, in the Euclidean limit of constant coefficients, to the metric of Minkowski's space-time manifold of Special Relativity. Thus, the stability conditions and the principle of increasing entropy combine to require that the equations of motion for an isolated system be the equations of geodesics in a space-time manifold. However, this manifold, whose arc length is the entropy, is related to another space-time manifold by a gauge function so that a discussion of geometry involves two space-time manifolds. Recalling Eqn.(7) we have

The path independence of the entropy fully specifies the geometry of both manifolds3. For the entropy manifold, the geometry is required to be Riemannian with a vector curvature. The other manifold, which may be called the "sigma" manifold, is required to have a Weyl geometry with both a vector curvature and a distance curvature. The distance curvature refers to the changes of the length of a vector under parallel displacements in the sigma manifold and is found to be given by

The requirement of two manifolds for an isolated system and the fact that the adopted laws fully determine the geometry of each are two of the most significant aspects of the proposed theory. The requirement that there be two manifolds coupled by a gauge function gives rise ultimately to Maxwell's electromagnetic theory as well as quantization. The fact that the laws specify the geometry removes the necessity of assuming a particular geometry and leads to the removal of objections to Weyl's unified field theory of 19184 and to London's quantization of Weyl's work in 19275. It is these aspects of the theory which allow the unification of the different branches of physics. In 1918, the German mathematician Weyl proposed a unified field theory based upon his extension of geometry. However, this theory has not gained acceptance, partly because his theory produced only Einstein's General Theory of Relativity and Maxwell's Electromagnetism. Weyl's theory said nothing in addition to these theories. Another reason Weyl's theory failed to

0,1,2,3)=j(i, ; dxdxgf1 = )dq( ji

ij20 ˆ

(7)

.dxdxg = )(df1 = dxdxg

f1 = )dq( ji

ij2ji

ij20 σˆ

(8)

)l.dx( = lx

f21 = dl i

ii φ

∂

∂ log

(9)

12

gain acceptance is that Einstein produced an argument that, using Weyl's theory, the spectral lines produced from an atom must be dependent upon the history of the atom, which contradicts experience. The proposed theory removes both these objections. The first objection is quickly removed by the fact that, from the point of view of this theory, the system has been restricted to be, first, a strictly mechanical system, and secondly, an isolated system. The removal of either of these restrictions allows the theory to discuss events that cannot be addressed by Einstein's General Theory of Relativity or Maxwell's Electromagnetism as will later be shown. The second objection is removed by the fact that the Second Law requires that the entropy and, in addition, the change in entropy to be independent of the path and hence, Einstein's argument of path dependence is nullified. Later it will be shown how the theory arrives at the predictions of phenomena predicted by Einstein's General Theory and Maxwell's Electromagnetic Theory from a different approach in which the geometry is required to be that of Weyl. SPACE-TIME-MASS In 1918 Weyl published his book titled "Space-Time-Matter" in which he discussed Einstein's theories and his own unified theory of gravitation and electromagnetism. Yet, within the text, matter did not share the same role as space and time, though an equivalence was implied by the title. Space and time were coordinates in the relativistic manifold while matter was not. The theory proposed here does for mass what Einstein's Special Theory of Relativity did for time; mass also becomes a coordinate on an equal footing with space and time. Just as relativity created considerable conceptual difficulty, so also can the proposed theory be expected to create conceptual difficulty. However, if the unification provided by the theory is considered as justification for attempting to see what the theory might produce, then a fifth dimension with physical interpretation follows rather quickly. The first restriction placed upon any system discussed thus far has been that of restricting the system to be either a thermodynamic system, with only a pdv work term, or a mechanical system with three fdx work terms. By removing this restriction, a system must be considered, which may be called a thermo-mechanical system, that experiences four types of work. Thus, the first law includes four work terms and, therefore, involves five dimensions. Since the specific volume is the reciprocal of the mass density, the First Law may also be written in terms of the mass density. For such a system the coordinates become the specific entropy, mass density, and the three space variables. If five dimensions, which include the mechanical and thermodynamic variables, seem odd consider how thermodynamics is taught. The First Law of Thermodynamics is written on the blackboard, equating the differential heat exchanged between the system and its surroundings with the differential change in the internal energy plus the differential work terms. In the work

13

terms are the three mechanical work terms in addition to the thermodynamic work. It is then pointed out to the students that the right hand side of the equation has five independent variables and it is stated that five equations are needed which relate these variables in order to have a solvable system. Usually the first statement made at this point is that conservation of mass guarantees that the mass density may be written as a function of space and time and, therefore, only four additional equations are needed, which are stated as being the Equation of State and the three mechanical equations of motion, such as Newton's. But what about the case when mass isn't conserved? Can mass density be written as a function of space and time for this case also? If it may not then the fundamental dimensionality of nature must be five dimensions. Where does this lead? It has already been shown that the stability conditions lead to metrics upon which the Entropy Principle works to provide equations of motion when the metric coefficients are assumed known and field equations for these coefficients when they are not known. Thus, it is necessary only to work out what the implications of the five fundamental dimensions would be, compare them to the existing theories in those regions of physical phenomena where the existing theories are known to work and see if there is some predictable critical experiment that may be conducted to test the new theory. To begin the investigation of the implications of this five-dimensional system first consider the system to be isolated. The principle of increasing entropy becomes effective and the equations of motion are the equations of geodesics in a five-dimensional manifold of space, time, and mass density. The First Law for five dimensions may be written as

where the tilde denotes specific quantities. The entropy variational principle, as stated in Eqn. (5), becomes

where now q0 is the specific entropy. The system's specific energy is now given in terms of the five variables specific entropy, space, and mass density. The stability condition, and hence the metric, is then stated in terms of these same variables. The stability condition is stated as

1,2,3) = ( ,dqF - dP - Ud = E 2 αλλ

αα

~~_

(10)

0 = )dq( = )dq( = )(dS 20202 γδγδδ ∫∫∫

(11)

14

where q4 = ã/a0. The metric may then be written as Eqn. (3) with the indices running from 0 through 4. Eqn.s (7) and (8) give the five-dimensional geometry when the indices also take on the value 4. Equations of Motion The equations of motion are obtained when it is assumed that the coefficients of the metric are given and one looks at the Euler equations giving the variations of the coordinates which satisfy the variational condition. By using the variational principle from Eqn. (11) one finds the force densities to be given by

with

where ui are the components of the five-dimensional velocity vector, the

are the Christoffel symbols, and fi are the components of the five-dimensional acceleration vector. Obviously, if the mass density is considered to be conserved such that u4 = 0 and the system is near equilibrium so that a flat metric makes a good approximation, then the volume integral of Eqn. (13) becomes the force-mass-acceleration relation of Special Relativity. Therefore, Einstein's Special Theory is obtained within this theory by employing the restrictions of an isolated system near equilibrium, with conservation of mass. It is interesting to note that the inertial mass density comes from the fact that the stability conditions are given in terms of specific quantities while the Entropy Principle is stated in terms of the entropy. This fact will take on an even more interesting character when we consider the comparison between inertial and gravitating mass. Another interesting fact is that if the First Law is considered for an isolated system, one obtains

0,1,2,3,4) = (i ; 0 > dqdqqq

U = dqdqh iiii

2ii

ii∂∂

∂~

(12)

f = F ii γ

(13)

uulk

i +

dqdu = f kl

0

ii

lk

i

15

so that

When the energy integral of Eqn. (14) is evaluated one finds

where u4 = /a0 is used and it is assumed that the system is near rest. The energy density in Eqn. (15) includes the rest energy term because the integral requires it; not because of a constant of integration as in the Special Theory of Relativity. Further, because the system was considered to be isolated, pE=0, then the appearance of the rest energy term in the expression for the system specific energy brings with it some sublities of interpretation not found in Einstein's Special Theory where energy and mass are equated one-for-one. For instance, the one-to-one correspondence between energy and mass exists only for resting mass when mass is conserved. Also notice that the Special Theory of Relativity energy equivalence may exist only for isolated systems. Also, if we require the usual conservation of mass then dã/dt=0 and Eqn. (15) reduces to the rest energy plus the classical kinetic energy. Gauge Fields When the standard variational techniques are used on the metric for the isolated, five-dimensional system, it is found6 that the gauge function yields a gauge field with ten components as

−−−−

−−

−−

−−

=

00

00

0

3214

3123

2132

1231

4321

VVViVVBBiEVBBiEVBBiEiViEiEiE

Fij (16)

and eight partial differential equations, Eqn. (17),

1,2,3) = ( ; dxF - dp - Ud = 0 = E 2 αγγ

αα

~~_

1,2,3,4) = ( ; dxF = Ud ααα

~~

(14)

)()a(2

1 + v21 + c = U 2

2o

22 γγ

γγ &~

(15)

16

which replace Maxwell's four equations and the equation of charge continuity. However, there are four new field components appearing in these eight field equations. When these are assumed to be zero the system of equations collapses back to the Maxwell's equations of electromagnetism. It is no surprise that the collapse of the eight equations produces the Maxwell system; this has been shown by many researchers. The objective becomes one of how are these new field components to be interpreted? Initial investigations led into the five-dimensional quantum mechanics and to a predicted magnetic moment for neutrally charged particles7 (discussed later). Current theories ascribe these anomalous magnetic moments to the strong nuclear force. This led to the erroneous interpretation that these new field components must be related to the nuclear forces. This turned out to be wrong when later research was conducted in which a closer look was taken at the concept of fundamental particles. Fundamental Particle Fields The concept of fundamental particles might be rather loosely stated as something like "smallest possible" or "cannot be further divided". But one generations' fundamental particles have been divided by the next generation until there now exists a plethora of "fundamental" particles and the search for more continues. But how can the concept of "fundamental particle" be stated with mathematical rigor? If a mathematical statement for this "state" can be put forth, then the logic of mathematics may be used upon the field equations and it should then be possible to determine what fields these "particles" or "states" might have.

Jc

4- = t

Vc1 + V

Ea =

tV

c1 + V

0 = Ba + Vx

0 = Ja + J + t

4 = Va + E

cJ4 = V

a + tE

c1 - Bx

0 = Ex + tB

c1

0 = B

44

04

0

40

40

0

π

γ

γ

γρ

πργ

πγ

∂

∂∆

∂∂

∂∂

∆

∂

∂∆

∂∂

∆∂∂

∂

∂∆

∂∂

∂∂

∆

∆∂∂

∆

_

_

_

_

(17)

17

Consider the concept of a fundamental particle and look for a mathematical definition for it. First, consider the realm of thermodynamics where the very stable states are isentropic states and, therefore, suppose that the fundamental particles are isentropic states. When one looks at the metric for an isentropic state of an isolated system one finds that the condition which the German physicist London imposed upon Weyl's theory in 1927 is required. Namely, one finds that in order to satisfy the isentropic condition the line integral formed by the gauge potentials and the differentials of the metric variables must be quantized, or since ðE=0, then, from Eqn. (8), (dσ)=(dσ)0 so that

which is satisfied only if

where N is an integer and i is the square root of minus one. When a line integral is encountered in the class room the students are generally asked to find the value of the line integral given a certain path. Here though, one has a line integral that already has a value. There are then two questions that might be asked. First, if the gauge potentials are given, what are the paths allowed? London's work answered this question5. The only paths possible are those given by the solutions to the quantum mechanical equations of motion. Further consequences of this result will be discussed later. The second question that might be asked of the line integral is; what gauge potentials are allowed by the line integral if the value of the integral is independent of the path? This is asking what potentials may be used in the integral which will produce a quantized value for the integral independent of the path considered? This is the same as asking "What fields may a particle have if these fields are to be independent of the path?" If the value of the integral is to be independent of the path, then Eqn. (19) must be true even when all dxj are zero but one. Thus, the quantum condition requires that

where there is no summation on k. Eqn. (20) must be true for all k, and because one is free to choose the path, the ϕk must reflect the quantization represented by the integer N. Therefore,

1 = e dxijφ∫

(18)

iN2 = dx jj πφ∫

(19)

iN,2 = dxkk πφ∫

(20)

18

where there is no sum on j and the may not be quantized. Thus, Eqn. (21) represents the first response to the question concerning what ϕj are allowed for fundamental particles; the gauge potentials must be quantized. This is the first known quantization of the gauge potentials for particles which is required by some fundamental condition, such as the isentropic state requirement. Restating; this is the first display of a logical necessity for quantization of electric charge based upon fundamental principles and obtained by restrictive assumptions. By using the mathematical approach of assuming a solution in the form of a product of functions of independent variables and setting

the trial solution was run through the eight field equations of Eqn. (17)8. The result produced for the radial function is

Here ë depends upon the particle and the potential displays some familiar attributes of the Maxwellian gauge potential and some that are, at first, surprising. The potential corresponding to the classical electromagnetic potential

where Z is the quantum number required by the quantum condition, depends only upon the radial distance from the particle, not just the usual 1/r dependence. At first glance one is prompted to state that this is the Yukawa potential. However, the exponent in the Yukawa potential goes as r rather than 1/r. One may also note that this potential has no singularities for any value of the radial distance r. At distances much greater than ë this potential (herein called the Neo-Coulombic potential) has the familiar 1/r form from electrostatics and Newtonian gravitation. When the radial distance equals lambda the potential has its maximum absolute value. Because of the overriding effect of the exponential the potential returns to zero as r tends to zero. The Neo-Coulombic potential is so well behaved that all of its derivatives

φφ ~jjj N =

(21)

,fffff = f rt21

γφθlog

.erk = f r

-r

λ

(22)

erZk = r

-r

λφ

(23)

19

are also non-singular. This property will prove to be of extreme value when considering such a potential in quantum mechanic systems since no renormalization is required. Therefore, the usual problems arising with renormalization do not appear with this potential. The Neo-Coulombic potential gives the electric field radial component a long range 1/r2 dependence that we know for the electric field,

It also requires that the electric field rise to a maximum absolute value as r decreases from infinity, go to zero as r approaches lambda, reverse sign as r becomes smaller than lambda, go to another maximum absolute value and then approach zero as r tends to zero. This short range behavior is drastically different from that of the usual electrostatic field and will have enormous consequences for the nuclear phenomena wherein the radial separations are of the order of the lambdas of the fundamental particles. The next thing noticed about the gauge potentials arrived at by the above method is that the new three dimensional vector field has two multiplicative factors, for

The first factor has the same radial dependence of the electric field and hence the long range 1/r2 dependence. If this is to represent a physical field other than the electric field then it must be the gravitational field. To further confuse the issue, the second multiplicative factor involves a dependence upon time. At first this may seem to run counter to all knowledge of gravitational effects; however, later it shall be shown that this time dependence is all important in gravitational phenomena. Is it possible then that the ten gauge field components may be made up of the three electric field components, three magnetic field components, three gravitational field components, and the gravitational potential? Only by working through the predictions of the theory in the various areas of physical phenomena can it be determined whether the predictions can be supported by the experimental evidence or if the predictions run counter to the evidence. If there exists experimental evidence that is in measurable direct conflict with the predictions of the theory then the theory must be wrong. On the other hand, if the predictions are supported by the evidence and predictions exist which may

.er-1

rZk = E r

-2r

λλ

(24)

rr e

rrbtWV

λλ −

−+= 11)1( 2

(25)

20

be used to test the validity of the theory then the theory deserves more than a offhand dismissal just because it disagrees with existing theories or beliefs. Quantization Derived The strength of the quantum-theoretical structure is such that it has swept aside virtually every attack upon it. However, using classical definitions of commutivity it may be shown9 that the anti-commutivity of the position and momentum is dependent upon the metric approximating a flat metric. If a realm of conditions exists that does not allow a flat metric approximation then the commutators must depend upon the geometry. One finds that

where the

are the Christoffel symbols. This much does not depend upon any theory whatsoever, but only upon the mathematics of differentiation. Since the quantum Poisson brackets must correspond to the classical Poisson brackets, then they also depend upon the geometry in the same fashion. In the past it has been possible to argue that if the only physical field that affects the geometry is the Einsteinian gravitational field, then it is possible to ignore this geometrical effect upon the commutivity of space and momentum in nuclear phenomena. If, however, the gravitational field is described by a gauge field then this argument is nullified because the gauge fields do play a large role in the realm of nuclear physics. The German physicist London produce a quantization of Weyl's theory in 1927. In his work, London showed that if the arc length of the metric was required to return to its original value, a quantization was produced and that the wave function was proportional to this arc length. However, there was a difficulty with his work; it required an imaginary distance. The proposed theory not only removes the difficulty of the imaginary distance but further, logically produces the quantization conditions when the system is placed under an additional restriction. The quantum condition, as stated before, comes from restricting one's attention to systems which are isentropic. The requirement that the system have a constant entropy is the simplest restriction that produces London's quantization. The imaginary distance appearing in London's work also appears here in the entropy manifold. However, the attractive electromagnetic force comes from a negative gauge function which couples the "distance" in the manifold with the Weyl geometry to the entropy manifold. In the entropy manifold the change in

ψδ

Ψ x

sl

j+gi = ]p,x[ s

jlklkj h

sl

j

21

entropy is the distance and, therefore, distance must always be real and non-negative for an isolated system because of the principle of increasing entropy. The proposed theory then logically produces London's assumption and removes the difficulty with imaginary distances. Further, it is found that the quantization conditions are limited to a system with a distance curvature, or gauge function. Thus, the interpretation of universal application of a non-varying, least unit of action coming from Heisenberg's Uncertainty Principle rests with the existence, or lack, of a distance curvature and not with the existence of a vector curvature. Equivalently, only forces that may be expressed in terms of a gauge function, or distance curvature, may exhibit quantization, while forces describable by only a vector curvature cannot be quantized. If the above interpretation of the new field components as gravitational field components holds up as gauge field components then gravitational effects may be quantized as well as the electromagnetic effects. This description of the derivation of quantum mechanics from generalizations of the classical thermodynamics runs counter to the commonly held belief that one may derive classical thermodynamics using statistical methods and a variety of force laws. This contention is, however, without rigorous support, as may be seen when one considers the development of statistical thermodynamics. For instance, in order to talk of a statistical temperature one must start by assuming Newtonian physics (this constitutes three fundamental assumptions). Given Newtonian, or other physics, one can talk of an energy distribution, canonical ensembles and statistical temperature; however, one must make an additional fundamental assumption (the Equipartition Law) before the statistical heat capacities may be obtained. In order to obtain thermodynamics two more assumptions are required. It was pointed out by Peter G. Bergmann10 that using the statistical approach one may obtain an expression for the difference in the heat exchanged between the system and the surroundings and the element of work done. In classical thermodynamics this difference is the change in internal energy which is path independent. In the statistical approach the difference is obtained without reference to the internal energy. To claim that the statistically derived expression is an exact differential is a logically new assertion; it constitutes the First Law of Thermodynamics. In addition, the assumptions of statistical thermodynamics allow the derivation of the fact that the differential of heat exchanged must be greater than, or equal to, the multiplication of the statistical temperature by the differential change in the statistical entropy. This product of statistical thermodynamic properties is similar to an identical product of thermodynamic properties. In statistical thermodynamics it is asserted that the ratio of the statistical temperature and the classical temperature is Boltzman's constant. Once this assertion is made, the statistical entropy may be related to the classical entropy. However, there is no logical necessity that the ratio of temperatures be a constant from the

22

statistical approach; only if it is a constant can there be a one-to-one correspondence between the statistical entropy and the classical entropy. The above quantum condition establishes the conditions assumed by London and, therefore, one may follow his work in deriving the Schrodinger quantum mechanics. London's work establishes how quantum mechanics may be derived within the framework of a larger theory and will not be repeated here. Rather, a sketch of the five-dimensional quantum mechanics will be presented11. The variational principle required by the entropy principle is given by Eqn. (11). Because multiplication by a constant does not change the problem, one may write

By defining the velocity vector as uj = dxj/dq0 and the momentum as pj = ãcgjkuk, where the fact that gjkujuk = 1 has been used, one may show that pjpj = ã2c2, which is the five-dimensional "momentum-energy" relation. Because of the benefits of a first-order differential wave equation, Dirac sought to find a first-order operator equation that also satisfied the second 0-order Klein-Gordon equation (the operator equivalent to the momentum-energy relation). This can also be done in five dimensions by taking the specific Hamiltonian operator to be

By taking the four partial derivatives in Eqn. (27) as the components of the four-vector specific momentum operator, one may write

where natural units, h = c = 1, have been used. If one takes the p0|> = h|> and requires that the alphas and beta are to be chosen such that solutions of this equation are also solutions of Eqn. (28), one finds the restrictions imposed upon the choice of the alphas and beta to be

0. = )dq(c 202γδ ∫

(26)

.-x

+x

+x

+x

i = h 44332211 βαααα

∂

∂

∂

∂

∂

∂

∂

∂

(27)

) + P(- = h βα_

(28)

23

The set of 4 x 4 matrices satisfying the requirements of Eqn. (29) is given as

where I is the 2 by 2 identity matrix and the sigmas are the 2 by 2 Pauli spin matrices. Then the five-dimensional Dirac equation may be taken to be

where we have used the four-dimensional vector operator. By defining

then Eqn. (31) may be written as

Taking into consideration Eqn. (32) with the gauge fields of Eqn. (16), one arrives at

0. = +

1, =

,P = )P(2

22

αββα

β

α_

(29)

−

−=

=

=

=

2

24 0

0

,3,2,1;0

0

,0

0

σ

σα

σ

σα

β

j

II

j

jj ,

(30)

(x)) - i( = (t)t

i ψβαψ ∆∂

∂ _

(31)

,4,3,2,1;;0 =−== jjj βαγβγ

0. = (x)1) + (i jj ψγ∂

(32)

0 = Fi21-1-)-)(i-(i jk

jkkk

jj ψσφφ

∂∂

(33)

24

where

−−−−

−−

−−

−−

=

022221022122022220

0

3214

3123

2132

1231

4321

iuiuiuxuissx

iuisisxiuisisxxxxx

jk

&

&

&

&

&&&&

σ (34)

and s is the usual intrinsic spin while u is a new spin appearing because of the added dimension. By expanding, one finds that Eqn. (33) becomes

Recalling the field equations of Eqn. (17), even a particle without an electric charge (that is an electrically neutral particle) may have a magnetic moment because, for ñ = J = 0, one finds

If these new fields are to be interpreted as the gravitational fields then Eqn. (36) may be interpreted as requiring a magnetic moment for spinning, gravitating particles. An interesting result occurs when one looks at the allowed fundamental spin states. In the five-dimensional quantization of the space-time-mass manifold, three spin vectors appear. One of these is the familiar three-component spin vector of relativistic quantum mechanics; the second of the three is a new three-component spin vector; the remaining is a four-component spin vector defined below. Using the theorem, if α satisfies α2 = a2 where a is a number, then the eigenvalues of α are +a, it is not difficult to show that the component eigenvalues are:

If, in analogy with the eigenvalues for the total angular momentum, one writes

0. = ]xV-ivEi+uV2+sB2+1-)-)(i-[(i 44

kkjj ψφφ &•••∂∂

(35)

.Va- =

tE

c1-Bx Va- = E 0

40

γγ ∂∂

∂∂

∆∂∂

•∆

(36)

1,2,3,4. =j 1,2,3 = ; 43 = S ,

21 = u ,

21 = s j

2 ααα ±±

(37)

25

then the possible eigenvalues become

However, the following relations, which specify the components of a four-dimensional spin vector which, when added to the angular four-momentum, commutes with the specific Hamiltonian, restrict the number of possible combinations of these eigenvalues.

The question to be asked now seems to be, how many combinations of the above eigenvalues are allowed? The answer may be shown to be octets. This predicted result compares with the experimental findings of Gel Mann. By deriving the quantization conditions and using London's derivation of the quantum mechanics from this condition one obtains classical atomic physics by assuming that the effects of the gravitational gauge field components may be neglected. Thus, there appears to be no effect of the proposed theory upon the atomic physics that is now known. There is an astonishing effect of the Neo-Coulombic potential upon how one might describe nuclear phenomena. One of the first features noted about the potential was its return to a zero value as the radial value approaches zero. This has the effect of producing a force given by,

If this force is repulsive when r is infinite for like particles, it becomes zero when the separation is at the distance lambda and becomes a strongly attractive force when the separation becomes less than lambda. This is just the sort of behavior found when proton-proton scattering was first done at high enough energies to see a deviation from Coulombic scattering. The expression for the Neo-Coulombic scattering cross-section was found to be

1) + S(S = 43 = S jjj

2

.23,-

21 = S ,

21+ = u ,

21+ = s j__ αα

.s+s-s = S ,u+u+s = S

u-u+s = S ,u-u-s = S32142133

31223211

.11

12

1 rerr

kqF

λλ −

−

=

(38)

,)

2(

d2V2mqq = d

402

21 δθθθπ

σ

sin

sin

26

where

This scattering cross section for like-particle interaction appears to have the right dependencies to explain the scattering data. It remains to compare prediction with existing experimental data to determine the validity of the predictions and the ability of the Neo-Coulombic potential to explain the Strong Nuclear Force with that portion of its radial dependence that causes the value of the potential to return to zero. When unlike particles are considered care must be taken to keep the lambdas in the forces straight. The force on any charged particle due to the presence of another, second particle, is the product of the charge of the first particle and the field of the second particle. Thus, the force on the first particle goes to zero at the lambda of the second. For the force on a proton due to the field of an electron

while the force on an electron due to the field of a proton is

By looking at proton and electron like-particle scattering data it would appear that the lambda of the proton must be much larger than that of the electron. If this is the case then the force on the electron due to the near presence of the proton goes to zero while the proton is still attracted to the electron. Any further decrease in the separation causes the electron to experience a repulsive force; although the proton is still attracted to the electron. This immediately raises the eyebrows. Can it be that Newton's Third Law, concerning the equal and opposite forces, does not hold in Nature? The answer is, certainly. Newton's Third Law does not hold in high-speed electromagnetic interactions when viewed by the retarded potentials; it was found to be violated during beta decay until the hypothesis of the neutrino reinstated the summation of particle spins. Should one then throw out the unlike-particle forces because they violate Newton's Third Law without seeing what predictions these forces might lead to?

.

)-()2

(kE4

23+1

)2

()-(21+1)

2(

kE46+1

= 2

2 4

42

θπθθλ

θθπ

θλ

δ

sinsin

tansin

er-1

rk- = Eq = F r

-e2epp

eλλ

(39)

.er-1

rk = Eq = F r

-p2pee

pλλ

(40)

27

If one proceeds with the unlike-particle forces, he finds very quickly that it appears possible that the proton might find a very close orbit, at a separation from the electron by a distance lambda, in which it could settle down into a Bohr orbit around the electron. On the other hand the electron would experience no force from the orbiting proton. Such a state might cause one to think of the neutron. Here one runs into the question of particle spins that beta decay brought out and which led to the hypothesis of the neutrino. Also, the argument is offered that Heisenberg's Uncertainty Principle requires that the electron could never be in an orbit so tightly bound that the orbit is less than nuclear separations. This argument hinges upon the unit of action being Planck's constant. But remember the dependence of the Poisson brackets upon the geometry? Another argument against the neutron being an electron and proton in nuclear-sized orbits is based on an argument that the principle of angular momentum cannot be conserved. The neo-coulombic forces, which require that the force between the electron and proton be directed on a line between them, also requires that the angular momentum be conserved. However, the unit of action depends upon the gauge function and this requires that, when Bohr-type orbits are considered, there is an effective unit of action for the electron orbit and a different effective unit of action for the proton orbit. Thus, the effective unit of action for the electron orbit requires that in the neutron the orbital angular momentum would be given by he and its intrinsic spin angular momentum would be (1/2)he. Similarly, for the proton the orbital angular momentum would be hp and the spin (1/2) hp. After the neutron decays, the angular momentum is the sum of the two particles' intrinsic spin angular momenta, which is given by because both particles are free and therefore, each has an intrinsic spin angular momentum of (1/2) h. Thus, the conservation of angular momentum is expressed as

Experimental evidence of orbital and/or spin angular momentum is contained in the experimental magnetic moments. If one equates the intrinsic and orbital magnetic moments of the electron and proton while they are in the orbital configuration to the experimental value of the neutron's magnetic moment they have

. = ++)+(+21

pepe hhhhh __

(41)

-+21+ n

pB

e µµ

h

h

h

h __

(42)

28

where ìB is the Bohr magneton and ìn is the nuclear magneton. Eqn.s (41) and (42) require that he =8.0517 x 10-4h and hp = 0.66585h. Thus, within the proposed theory the neutron appears to be a proton in orbit around an electron. Not surprisingly then, it is possible to build a nuclear model of the protons-around-electrons, and electrons-around-positrons, states that allow one to predict the masses of the nuclei which have a mass number less than 10 amu with better RMS error than the best of the semi-empirical mass formulas have for mass numbers greater than oxygen13. This should possibly be considered all the more significant since the semi-empirical mass formulas have ever increasing errors for the low mass numbers and are not even used below an amu of 16. It remains for this nuclear model to be extended to the higher mass numbers, but it appears from the work done thus far that one can only expect that the correspondence with experiment will improve with increasing mass numbers. Is it possible that the Neo-Coulombic forces can explain the phenomena associated with the weak forces? Certainly the nuclear mass predictions argues that a nuclear model based upon these orbits does not miss far and is a much cleaner model than currently used. Initial looks at the neutrino experiments using the proposed theory offer other explanations for these experimental results but are too lengthy to include here. It should be remembered that these experiments must be explained by the proposed theory if the unlike forces are to fully account for phenomena that the weak nuclear forces are now thought to explain. The long range 1/r2 dependence of the new three-dimensional vector gauge field component suggests that these components are the components of the gravitational field. If this is to be the case the proposed theory must then explain the same phenomena that the General Theory of Relativity predicts. First, note that the gravitational field components in the gauge field tensor must have units equivalent to the electric field components. Following up on this, one finds that a charge-to-mass ratio is needed to convert the gravitational field units from the familiar units of acceleration to the volts/meter units used in the gauge field tensor. By considering the new fields and comparing them to the currently used fields one finds that this ratio is given by the square root of the product of the gravitational constant and the dielectric constant, or

An interesting result follows immediately. If the fundamental charge-to-mass ratio works as it appears to, and electrically neutral spinning bodies have magnetic moments, then the predictions of magnetic moments for electrically neutral bodies may be made by determining the effective charge density of the rotating gravitating body using the charge-to-mass ratio and the spin of the body. A simple calculation of the earth's magnetic moment, assuming uniform

coul/kg. 102.4296x = G = -11εβ

(43)

29

mass distribution, by this method produced a prediction of the magnetic moment 1.06 times the actual value14. This prediction seems surprisingly close considering the uncertainties in the density measurements of the mass distribution of the earth. One of the predictions of Einstein's General Theory of Relativity concerns the tendency of light from stars and other objects in the heavens to be shifted towards the red color end of the spectrum. Looking at the emission and reception of light within the framework of the proposed theory one finds that the unit of action, which establishes the energy of any state, depends upon both the relative time and the gravitational field at the time and place of the emission and reception. This is so because the theory holds the gravitational field to be a gauge field and it is the gauge function that determines the applicable unit of action. It is not difficult to show that

Thus, for a metric with only a gauge function the effective unit of action would be given by

By recalling the gauge gravitational field of Eqn. (25), one may use Eqn. (45) to find the expression for the unit of action for emission of a photon to be

where the subscript, e, denotes emission. Similarly, the unit of action for the reception of a photon can be found to be

If photon energy is conserved between emission and reception then

.xsl

j +gi = ]p,x[ s

jlklkj ψδψ

h

(44)

].fff[2 = rt γexphh′

(45)

e

R)bt+(1W = R

-

e

eee e

eλexphh

(46)

.eR

)bt+(1W = R-

r

rrr r

r

λexphh

(47)

. = rree νν hh

(48)

30

If one sets te = 0, tr = L/c, W = (-GM/c2), and b = -H, then they find the shift in frequency given by By looking at the first order approximations of this prediction one finds that the time dependence of the gravitational field produces the linear dependence and is given by Hubble's constant while the gravitational potential produces the same prediction that comes from Einstein's theory.

Looking a little closer one finds that the time dependence of the red shift produces an experimental number, H-1 = (5.6+0.6) x 1017 sec. (1.61 x 10-18 sec-1 < H < 2.0 x 10-18 sec-1), that corresponds to the same time dependence that has been measured and reported for the moon's orbit15 (b=1.9 x 10-18 sec-1), well within experimental error. It is somewhat pleasing that a prediction coming from the same time dependence originating in the gauge function leads to a comparison of phenomena involving cosmological distances agrees with phenomena involving the much shorter distance involved in the moon's orbit. Another possible plus to this prediction is that, because the prediction involves an exponential dependence upon time and gravitational potential between the emission and reception of the light, then the distances that are currently ascribed to distant bodies by their red shifts may be much greater than the actual distances. Also, the possible red shifts from dense gravitating bodies may be much greater than is now believed possible thereby removing the mystery from many objects. The time dependence of the gravitational field stems from the principle increasing entropy and is a direct result of this inflation-like effect imposed upon the universe by the denial of perpetual motion. An additional implication follows for the use of dating processes which depend upon radioactive processes in that the unit of action changes with time in accordance with that same time dependence. The results would be that all of the dates would have to be adjusted downward. The prediction of the advance of the perihelion of the planetary orbits is the one prediction of Einstein's General Theory of Relativity that requires the entire formal theory. Within the proposed theory one obtains an advance to the planetary orbital perihelion by simply using the low velocity Newtonian equations of motion with the Neo-Coulombic gravitational potential, which is

.1exp 2 −

+

−

−=∆ −

−−

r

re

e

r

r

R

e

Re

r

Rr

e

ecHL

ReM

ReM

cG

λλλ

λ

λ (49)

31

The perihelion advance predicted by the General Theory of Relativity is given by16

Thus, the lambda of the sun would have to be given by

if the proposed theory is to be identical in its prediction of planetary perihelion advance to Einstein's General Theory of Relativity. Currently there is much discussion of experimental evidence of the need for a fifth, and even a sixth, force in Nature. The evidence points to a decreased gravitational strength when compared with Newtonian gravitation. Consider the Neo-Coulombic gravitational force which must go to zero at some value of distance that is representative of the body in question. The obvious conclusion is that the gravitational force in the proposed theory must become less than the Newtonian value as distance is decreased. Thus, a new independent force may not be necessary at this time. There are numerous implications of this feature of the Neo-Coulombic force which will have large effects upon the concept of the universe presented by the proposed theory. For example, a gravitational force which becomes repulsive with decreasing distance denies the type of gravitational collapse now discussed by cosmologists. Neither can it support the singularities now called Blackholes. The possibility of the existance of distant bodies so massive that light cannot escape their gravitational pull has not yet been investigated. A number of possible experimental tests have been considered. A few of these will be presented here. The proposed theory presents a picture of the universe in which the electromagnetic and gravitational fields are components of a single gauge field tensor and, therefore, are fields on equal footing and also, more importantly, inductively coupled. This implies that manipulation of one field will inductively produce another of the fields. It is this type of inductive coupling which causes a magnetic field to be created by the flow of current. The electric field which is the source of the voltage in the alternator providing the power for home use was

.L

GMm32 2

2

≈

λπδ θ

(50)

.Lc

mMG32 = 22

222

GTR

πδ θ

(51)

c

GM = 2sunλ

(52)

32

inductively created by passing a conductor through a magnetic field. Is it not then possible to create a gravitational field by the manipulation of the electromagnetic fields if the inductive coupling presented in the proposed theory exists? Where might this inductive coupling most likely show up? One area of phenomena is in wave properties such as electromagnetic or, in this case, electromagnetogravitic waves. The five-dimensional wave solutions have an additional transverse field component17 which is opposite in direction to the electric field component. This additional component is the gravitational field component. One of the results of the possible existence of this gravitational component is that while the wave energy density depends upon the sum of the squares of all wave components, the radiation pressure depends upon the sum of the squares of the electric and magnetic components, but the square of the gravitational component is subtracted from the sum of the others. This implies that the radiation pressure would always be a little less than the energy density rather than always equal to it. The initial experiments on radiation pressure and energy density showed just this difference, however, the difference was within experimental error. To date the known experimental techniques do not appear to have sufficient accuracy to measure the expected difference in these quantities. Another experimental technique which has a much better chance of detecting the new wave component is the neutron interferometer device. Here the gravitational component is directed opposite the electric and a polarized laser beam may be used to deflect one leg of the neutron's path without causing an interaction between the magnetic component and the neutron's magnetic moment. The sensitivity of the interferometer is such that even an extremely small amount of energy in this component might be discernable. The phase velocity of the five-dimensional waves are found to be dependent upon any divergence in the flow of a medium through which the wave is passing. This allows the possibility of slowing down the wave significantly by causing a divergent flow. The divergence possible from nozzles in continuous flow is too small to allow for other difficult factors affecting the speed of light, such as the index of refraction, to be accounted for with sufficient effect left over for clean measurement of the slow down. On the other hand, if the divergent flow were created using explosives, the one might be able to slow down gamma rays to about half the speed of light. This would involve all the usual difficulties of one-shot testing plus some other possible problems. The very nature of the five-dimensional manifold places restrictions upon some phenomena. For instance, when looking at shockwaves in material using the proposed theory it is found that the phenomena is predicted on a four-dimensional hypersurface embedded by the conservation of mass within the five-dimensional space. This has an effect that appears like a viscosity and puts a very distinctive anti-symmetrical profile into the shock front. The recent advances in our abilities to measures differential times and distances make it

33

appear possible to measure the rise of a shock by using a Laser Velocity Interferometer with an electronic streak camera. The predicted asymmetry is such that it would be easy to discern it from the classical symmetrical Newtonian viscosity. The preceding has presented the fundamental laws of the proposed theory and how each of the existing theories may be shown to be either within the scope of the new theory or superseded by it. All of the results of the assumption of the laws presented here have, of course, been arrived at through rigorous mathematical logic using these laws as the starting point. The mathematical derivations have been left to later chapters in order to provide for a better flow of the overview discussion and to limit the length of the overview. The single most important concept hoped to have been conveyed in the preceding is that the classical laws of thermodynamics contain within them the generality, applicability, and strength to allow them to provide the basis for a description of a nature much more general than the sum of all the currently known theories and contain within it the current theories as subsets. Starting from its general five-dimensional form, the theory provides a metric in the form of the stability conditions. Two variational principles are given by the basic laws. The first, and the more general, is a Principle of Minimum Free Energy, and has not been pursued in the above discussion. Secondly, the Entropy Principle has been used throughout the preceding discussion to limit the realm of phenomena to those for which current theories are used. The basic laws determined the type of geometry of the metric. The variational principle stemming from the Entropy Principle was used to obtain the equations of motion and the field equations when the appropriate restrictions were imposed. It was the restrictions employed that allowed the concentration upon phenomena related to certain current theories. To help interpret the five-dimensional field components, the isentropic restriction was imposed. This restriction required a quantization from which it was found that one could derive quantum mechanical equations of motion, following London's work. When the characteristics of fundamental particle gauge potentials were sought which satisfied this quantization condition it was found that the Neo-Coulombic potential appeared, requiring that the gravitational field and potential be components of a gauge field on equal footing with the electric and magnetic fields. Thus, the isentropic restriction produces the subset in which Quantum Mechanics and the fields of the fundamental particles are found. If the system is restricted to be an isolated one and one looks at the trajectories required by the Entropy Principle, he finds that they are given by equations of motion in five dimensions. By saying that mass density may be written as a function of space and time one finds that the trajectories lie on a four-dimensional, space-time, hyper-surface embedded within the five-dimensional manifold. If one further restricts his attention to those events

34

near equilibrium states, the metric may be approximated by a flat metric, and one finds the equations of motion to be those of Einstein's Special Theory of Relativity. A further restriction to slow moving things brings about the reduction to Newtonian equations of motion. Turning from equations of motion to the forces of Nature, the proposed theory presents only one type of force, the gauge force, which shows up in three, three-component vector fields plus a scalar field. These fields correspond to the fields now known as the electric, magnetic, gravitational fields and the gravitational potential. Because the proposed theory displays the three forces together in a single five-dimensional field one probably should refer to all three as components of the electromagnetogravitic (EMG) force. The theory appears to describe the phenomena currently described by the Strong Nuclear Force by the Neo-Coulombic electrostatic force which reverses its sign as the separation of like particles is reduced. For the Weak Nuclear Force the theory offers the asymmetrical unlike-particle force. The Neo-Coulombic gravitational force not only provides the classical gravitational predictions plus the planetary perihelion advance prediction, but includes a prediction which appears to correspond to the recently observed experimental results which have brought forth talk of a fifth force in nature. Currently used cosmological and gravitational red shifts were found to be the first-order approximations to the red shift predictions from the proposed theory. The full exponential character of the time and gravitational potential dependence of the red shifts may find usefulness in helping to describe the universe by helping to clear up some of the mysteries of the cosmos.

35

References: 1 O.F. Mosotti, 'Sur les forces qui regissent la constitution interieur des corps',

Turin (1836). Mossotti's essay was translated into English and published in Taylor's Scientific Memoirs, 1, 448 (1839).

2 P. E. Williams, 'The Dynamic Theory: A New View of Space, Time, and Matter',

Los Alamos National Laboratory Report, LA-8370-MS, pp. 11-18 (1980). (see also Section 2.2)

3 pp. 39-49 of Ref.[2]. (see also Section 2.2) 4 H. Weyl, 'Space-Time-Matter', (1922). 5 F. London, 'Quantum Mechanische Deulung der Theorie, von Weyl', z. Physik,

{\bf 42\/}, 375-389 (1927). 6 pp. 87-91 of Ref.[2]. (see also Section 3.4) 7 pp. 111-119 of Ref.[2]. (see also Section 4.6) 8 P. E. Williams, 'The Possible Unifying Effect of the Dynamic Theory', Los

Alamos National Laboratory Report, LA-9623-MS pp.57-61 (1983). (see also Section 4.9)

9 pp. 76-77 of Ref.[8]. (see also Section 4.13) 10 P.G. Bergmann, 'Basic Theories of Physics: Heat and Quanta', Dover (1950). 11 pp. 111-124 of Ref.[2]. (see also Sections 4.1-4.7) 12 pp. 119-124 of Ref.[2]. 13 pp. 83-99 of Ref.[8]. (see also Section 4.13) 14 D. Halliday and R. Resnick, 'Physics', Third Edition, Wiley (1978). (see also

Section 5.1) 15 T. Van Flandern, 'Is Gravity Getting Weaker?', Sci. Am. (1976). 16 R. Adler, M. Bazin, and M. Schiffer, 'Introduction to General Relativity', Second

Edition, McGraw-Hill (1965). 17 pp. 38-54 of Ref.[8]. (see also Chapter 6)

36

CHAPTER 2 NEW THEORETICAL FUNDAMENTALS The Dynamic Theory uses a different viewpoint, or approach, to present a description of physical phenomena. Therefore the first criterion that it must meet is that it must not be in conflict with existing theories in a field of physics where the existing theory gives an adequate and accurate description. To show that the Dynamic Theory meets this criterion, this section will present the adopted laws and then proceed to show how the fundamental principles of existing theories may be obtained from these laws. A. General Laws In the following development physical concepts are necessary, as are symbols for these concepts. Because this development will merge certain thermodynamic conceptualizations into mechanics, a notational dilemma must be faced. On the one hand, it is desired to preserve the thermodynamic conceptualization by using familiar symbols from that theory. On the other hand, descriptions of mechanical systems are also sought. The formulism then looks either like thermodynamics with familiar thermodynamic quantities replaced by mechanical quantities, or it looks like mechanics into which thermodynamic quantities intruded. In either case there is danger of confusion. One could avoid the dilemma by choosing entirely different symbols for the variables of the theory. But then the whole takes on an artificially abstract character. Since the purpose of this formulation is to bring out the power of the thermodynamic conceptualization, it was decided to use the suggestiveness of the thermodynamic or mechanical symbols whenever convenient; the reader is asked to keep an open mind and not make premature association with the symbols used. 2.1 First Law. The concept of conservation of energy is fundamental to all branches of physics and therefore represents a logical beginning for a generalized theory. Therefore, in terms of generalized coordinates or independent variables, the notion of work, or mechanical energy, is considered linear forms of the type

where the forces Fi may be functions of the velocities (dqi/dt = ui) as well as the coordinates qi and the summation convention is used. The inclusion of velocities in forces reflects the belief that forces should depend upon the velocities. This will become clearer when these work terms are included in the first law.

n),1,2,..., = (i dq )u,...,u,q,...,q(F = W in1n1i_

37

The line integral ³c Fi dqi then represents the work done along the path C by the generalized forces. A system may acquire energy by other means in addition to the work terms; such energy acquisition is denoted dE. The system energy, which represents the energy possessed by the system, is considered to be

dU will be assumed to be a perfect differential. With these concepts, then the generalized Law of Conservation of Energy, which is adopted as the first law of the Dynamic Theory, has the form

Positive dE is taken as energy added to the system by means other than through the work terms and Fi is taken as the component of the generalized force acting on the system which caused displacement dqi. In the First Law the dimensionality is n + 1 and is determined by the system considered. There is no limitation on the quantity or type of variables that may be used. However, in this presentation and in practice, it will be beneficial to place restrictions upon the type and number of allowed work terms. A system with only one work term, which is the pdv expansion work of classical thermodynamics, will be called a "thermodynamic" system and the dimensionality will be two. A system with three or less fdx work terms will be called a "mechanical" system with the appropriate dimensionality. Obviously, if there are three mechanical work terms, the dimensionality will be four. A system with a combination of thermodynamic and mechanical work terms will be considered later. In an infinitesimal transformation, the First Law is equivalent to the statement that the differential

is exact. That is, there exists a function U whose differential is dU; or the integral ³dU is independent of the path of the integration and depends only on the limits of integration. This condition is not shared by ³dE or ³ÿW . The path dependence of ³ÿW is another reason that the generalized forces are assumed to be functions of velocity as well as position. In Newtonian mechanics forces are usually assumed to be dependent on position only so that the simplicity of path independence may be used. Though even in Newtonian mechanics certain forces are taken as velocity dependent. Friction forces are an example.

).u,...,u ,q,...,qU( n1n1

n).1,..., = (i dq F - dU =

W - dU = Ei

i

__

(2.1)

dqF + E = dU ii_

38

This statement of the generalized First Law is consistent with the

First Law of thermodynamics in that if there is only one generalized force,

which is taken to be the pressure, and one generalized coordinate, the

volume, then Eqn. (2.1) becomes

where F = -P with the convention that work of expansion is work done by

the system on its surroundings. Here the system energy, U, is the

thermodynamical internal energy. There should then be no confusion

when Cartheodory's statement of the second law is applied to this

thermodynamic system. However, when considering the application of

generalizations of the classical thermodynamic laws to mechanical systems

some confusion may be expected. During the initial portion of this

development, it is desired to demonstrate the applicability of the

generalized laws to mechanical systems. Therefore, it may help avoid

confusion to think of the generalized coordinates of a mechanical system

as the space coordinates of a mass point. Obviously, there exists systems

in nature that may be considered to consist of a continuous distribution of

mass points. Such a system may be thought of as a composite system of

an infinite number of subsystems and, therefore, involve an infinite

number of "generalized coordinates," or "degrees of freedom." However,

just as in classical mechanics, we may later make the transition from mass

points to matter in bulk; then the generalized coordinates, qi, used here

may better be termed independent variables.

To explore some of the consequences of the exactness of dU,

consider a system whose variables are F, q and u. The existence of the

state function U, or an equation of state, means that any pair of these

three parameters may be chosen to be the independent variables that

completely specify the system. For example consider U = U(F,q) then

The requirement that dU be exact immediately leads to the result

The "energy capacity" of a system at the position q with dq = 0 may be

defined as

and the "energy capacity" of a system under a constant force is defined as

Pdv + dU = E = Q __

dq.qU + dF

FU = dU

Fq»¼

º«¬

ªww

»¼º

«¬ªww

.qU

F =

FU

q F qq F »»¼

º

««¬

ª»¼

º«¬

ªww

ww

»»¼

º

««¬

ª»¼º

«¬ªww

ww

»¼º

«¬ªww

»¼

º«¬

ª

cuU =

_u_E = C

qqq

39

2.2 Second Law.

There are processes that satisfy the First Law but are not observed

in nature. The purpose of the dynamic second law is to incorporate such

experimental facts into the model of dynamics.

The statement of the Second Law is made using the axiomatic

statement provided by the Greek mathematician Caratheodory, who

presented an axiomatic development of the Second Law of thermodynamics

that may be applied to a system of any number of variables. The Second

Law may then be stated as follows:

In the neighborhood (however close) of any equilibrium state of

a system of any number of dynamic coordinates, there exist

states that cannot be reached by reversible E - conservative

(dE = 0) processes.

When the variables are thermodynamic variables, the E-conservative

processes are known as adiabatic processes.

A reversible process is one that is performed in such a way that, at

the conclusion of the process, both the system and the local surroundings

may be restored to their initial states without producing any change in the

rest of the universe.

Consider a system whose independent coordinates are a generalized

displacement denoted q, a generalized velocity u (with u = dq/dt), and a

generalized force F. It can be shown that the E-conservative curve

comprising all equilibrium states accessible from the initial state, i, may be

expressed by V(u,q) = constant, where V represents some as yet

undetermined function. Curves corresponding to other initial states would

be represented by different values of the constant.

.uU =

_u_E = C

FFF »¼

º«¬ªww

»¼

º«¬

ª

c

40

Reversible E-conservative curves cannot intersect, for if they did, it would be possible, as shown in Figure 1, to proceed from an initial equilibrium state i, at the point of intersection, to two different final states f1 and f2, having the same q, along reversible E-conservative paths, which is not allowed by the Second Law. When the system can be described with only two independent variables, such as on the E-conservative curve, then if these variables are q and u and F is a generalized force,

Regarding U = U(q,u), then

where all quantities on the right-hand side are functions of u and q. An E-conservative process for this system is

Figure 1. If two reversible E-conservative curves could intersect it would be possible to violate the Second Law by performing the cycle i, f1, f2, i.

Fdq. - dU = E_

dqF - qU +du

uU = E

uq »»¼

º

««¬

ª»¼

º«¬

ªww

»¼º

«¬ªww_

0. = dqF - qU +du

uU

uq »»¼

º

««¬

ª»¼

º«¬

ªww

»¼º

«¬ªww

41

Solving for du/dq yields

The right hand member is a function of u and q, and therefore, the derivative du/dq, representing the slope of a E-conservative curve on a (u,q) diagram, is known at all points. Equation (2.2) has therefore a solution consisting of a family of curves, see Figure 2, and the curve through any one point may be written

A set of curves is obtained when different values are assigned to the constant. The existence of the family of curves V(u,q) = constant, generated by Eqn. (2.2) representing reversible E-conservative processes, follows from the fact that there are only two independent variables and not from any law of physics. Thus it can be seen that the First Law may be satisfied by any of these V = constant curves. The Second Law requires that these curves do not intersect. Therefore the Second Law, together with the First law, leads to the conclusion that through any arbitrary initial-state point, all reversible E-conservative processes lie on a curve, and E-conservative curves through other initial states determine a family of non-intersecting curves. To see the results of this conclusion consider a system whose coordinates are the generalized velocity u, the generalized displacement q and the generalized force F. The First Law is

where U and F are functions of u and q. Since the (u,q) surface is subdivided into a family of non-intersecting E-conservative curves V(u,q) = constant where the constant can take on various values V1, V2, ..., and points on the surface may be determined by specifying the value of V along with q, in all regions where the Jacobian of the transformation does not

Figure 2. The First Law fills the (u,q) space with slopes. The V curves represent the solution curves whose tangents are the required slopes. The Second Law requires that these curves do not intersect.

.

uU

F - qU-

= dqdu

q

u

»¼º

«¬ªww

»»¼

º

««¬

ª»¼

º«¬

ªww

(2.2)

constant. = q)(u, = VV

Fdq - dU = E_

42

vanish, so that U, as well as F, may be regarded as functions of V and q. Then

dqqU + dU = dU

q»¼

º«¬

ªww

»¼º

«¬ªww

V

VV

and

dq.F - qU + dU = E

q »»¼

º

««¬

ª»¼

º«¬

ªww

»¼º

«¬ªww

V

VV

_

Since V and q are independent variables this equation must be true for all values of dV and dq. Suppose dV = 0 and dq z 0. The provision that dV = 0 is the provision for an E-conservative process in which dE = 0. Therefore, the coefficient of dq must vanish. Then, in order for V and q to be independent and for dE to be zero when dV is zero, the equation for dE must reduce to

,dU = Eq

VV »¼º

«¬ªww_

with

F. = qU»¼

º«¬

ªww

V

Defining a function O by

»¼º

«¬ªww

{cV

OU

q

then dE = OdV where O= O(V,q). Now, in general, an infinitesimal of the type Pdx + Qdy + Rdz + ..., known as a linear differential form, or a Pfaffian expression, when it involves three or more independent variables, does not admit of an integrating factor. It is only because of the existence of the Second Law

that the differential form for dE referring to a physical system of any number of independent coordinates possesses an integrating factor. Two infinitesimally neighboring reversible E-conservative curves are shown in Figure 3. One curve is characterized by a constant value of the function VA, and the other by a slightly different value VA + dV = VB. In any

Figure 3. Two reversible E-conservative curves, infinitesimally close. When the process is represented by a curve connecting the E-conservative curves, energy dE = OdV is transferred.

43

process represented by a displacement along either of the two E-conservative curves dE = 0. When a reversible process connects the two E-conservative curves, energy dE = OdV is transferred. The various infinitesimal processes that may be chosen to connect the two neighboring reversible E-conservative curves, shown in Figure 3, involve the same change of V but take place at different O. In general O is a function of u and q. However, it is obvious that O may be expressed as a function of V and u. To find the velocity dependence of O consider two systems, one and two, such that in the first system there are two independent coordinates u and q and the E-conservative curves are specified by different values of the functions V of u and q. When dE is transferred, V changes by dV and dE = OdV where O is a function of V and q. The second system has two independent coordinates u, and q' and the E-conservative curves are specified by different values of the function V' of u and q'. When dE is transferred, V' changes by dV' and dE = O'dV' where O' is a function of V' and u. The two systems are related through the coordinate u in that both systems make up a composite system in which there are three independent coordinates u, q, and q' and the E-conservative curves are specified by different values of the function Vc of these independent variables. To help visualize the situation it may be noted that the composite system is, in essence, two particles joined together and traveling with the same velocity but not sharing the same location. Since V = V(u,q) and V' = V'(u,'), using the equations for V and V', Vc may be regarded as a function of u, V and V'. For an infinitesimal process between two neighboring E-conservative surfaces specified by Vc and Vc + dVc, the energy transferred is SEc = OcdVc where Oc is also a function of u, V and V'. Then

.d + d +du u

= d cccc V

VVV

VVV

V c»¼º

«¬ª

cww

»¼º

«¬ªww

»¼º

«¬ªww (2.3)

Now suppose that in a process there is a transfer of energy dEc between the composite system and an external reservoir with energies dE and dE' being transferred, respectively, to the first and second systems, then

E + E = Ec c___ and

,d + d = d cc VOVOVO cc or

.d + d = dcc

c VO

OV

O

OV c»

¼

º«¬

ª c»¼

º«¬

ª (2.4)

Comparing Eqns. (2.3) and (2.4) for dVc then

0. = u

c

wwV

44

Therefore Vc does not depend on u, but only on V and V'. That is Vc = Vc(V,V'). Again comparing the two expressions for dVc we find

. = also = c

c

c

c VV

O

OVV

O

Ocw

wcww

Therefore the two ratios O/Oc and /Oc are also independent of u, q and q'. These two ratios depend only on the V's, but each separate O must depend on the velocity as well (for example, if O depended only on V and on nothing else, the dE = OdV would equal f(V)dV which is an exact differential). In order for each O to depend on the velocity and at the same time for the ratios of the O's to depend only on the V's, the O's must have the following structure:

).,g( (u) = with),(f (u) = ),f( (u) =

c VVIO

VIOVIO

cccc (2.5)

(The quantity O cannot contain q, nor can contain q', since and /Oc must be functions of the V's only.) Referring now only to the first system as representative of any system of any number of independent coordinates, the transferred energy is, from Eqns. (2.5),

.d )f( (u) = E VVI_ Since f(V)dV is an exact differential, the quantity 1/M(u) is an integrating factor for dE. It is an extraordinary circumstance that, not only does an integrating factor exist for the dE of any system, but this integrating factor is a function of velocity only and is the same function for all systems. It would be nice if there were a simple way of deriving the functional form of M(u). In thermodynamics we opted to take the easy way out by assuming that the integrating factor was simply the reciprocal of the temperature. However, for mechanical systems we will find the functional form of the integrating factor when we determine the equations of motion. The fact that a system of two independent variables has a dE that always admits an integrating factor regardless of the axiom is interesting, but its importance in physics is not established until it is shown that the integrating factor is a function of velocity only and that it is the same function for all systems. 2.3 The Absolute Velocity and Einstein's Postulate.

45

The universal character of M(u) makes it possible to define an

absolute velocity. Consider a system of two independent variables q and u,

for which two constant velocity curves and E-conservative curves are

shown in Figure 4. Suppose there is a constant velocity transfer of energy

E between the system and its surroundings at the velocity u, from a state

b, on a E-conservative curve characterized by the value V1, to another state

c, on another E-conservative curve specified by V2. Then since it is seen

that

u. constant at )df( (u) = _E 2

1VVI V

V³

For any constant velocity process between two other points a to d,

at a velocity u3 between the same E-conservative curves the energy

transferred is

.u constant at )df( )u( = E_ = )u_E( 33332

1VVI V

V³

Taking the ratio of

'E = M(u) = a function of the vel. at which 'E is transferred.

'E3 M(u3) same function of vel. at which 'E3 is transferred

Then the ratio of these two functions is defined by

M(u) = 'E (between V1 and V2 at u)

M(u3) 'E3 (between V1 and V2 at u3)

or

(u).)u(

E_ = _E3

3 II »

¼

º«¬

ª

By choosing some appropriate velocity u3 it follows that the energy

transferred at constant velocity between two given E-conservative curves

decreases as M(u) decreases, or the smaller the value of 'E the lower the

corresponding value of M(u). When 'E is zero M(u) is also zero. The

corresponding velocity u0 such that M(u0) is zero is the "absolute velocity".

Therefore, if a system undergoes a constant velocity process between two

E-conservative curves without an exchange of energy, the velocity at which

this takes place is called the absolute velocity.

The definition of the absolute velocity requires constant velocity

processes be considered. All Galilean frames of reference will display this

process as one of constant velocity. Further, if all reference frames are to

be of equal status then observers in all Galilean reference frames must

share the dE = 0 constant velocity process equivalently. Furthermore,

Figure 4. Two constant velocity energy transfers, E3 at u from b to c and E3 at u3 from a to d, between the same two conservative curves M1 and M2.

46

each observer will have the same value for the absolute velocity or else one of the frame will enjoy a privileged nature. Just as the absolute temperature in classical thermodynamics is a limiting quantity we may suspect that the absolute velocity will also turn out to be a limiting quantity. Because of our experimental evidence that the speed of light behaves as a limiting velocity when electromagnetic forces are involved and the absolute velocity is independent of the force or type of system and is therefore unique, it must be the speed of light. Thus, the first two laws of the Dynamic Theory require Einstein's postulate concerning the speed of light. To be more specific, the absolute velocity is unique for all Galilean frames of reference. There is one such velocity already known and that velocity is the speed of light, c. Therefore, the absolute velocity must be the speed of light and the same for all Galilean observers. This is Einstein's postulate. The above may be put on a more rigorous basis by observing that for the E-conservative process

.dqF - qU + du

uU = 0 = E D

DDD

D »¼

º«¬

ª

ww

ww_

If dE = 0 is to be invariant for all points qD then we must have

0 = F - qU

DDww

and thus duD= 0, for all D = 1,2,3. Thus the allowed transformations are those with constant velocities. This, of course, was just what was required by the statement of, or restriction to, a constant velocity process. Then all Galilean observers will agree upon the identification of an E-conservative system in absence of any work on the system. Now let us suppose that at the time t, a system is at point p(q1, q2, q3) in Q. If our system is E-conservative and traveling at the absolute velocity, c, then in dt seconds it will be at the point qD + dqD where dqD = uD cdt. Now the speed is given by

c, = uug = uu EDDE

DD

where gDß is the metric for the space and the metric is parameterized using the absolute velocity, c, which is the only velocity with an adequate definition thus far. Now an observer in another frame Q' sees the system at the point P' given by (q1, q2, q3) at the time t. In dt seconds the system will have moved to a point given by qD+ dqD and the speed will be given by

c = uug = uu ccccc EDDE

DD

or dtc = qdqdg 22ccc ED

DE

47

since the process must specify the E-conservative process at the absolute velocity, c. But, since the Q observer must be Galilean then

.a + qa = t

ta + qa = q44

4

4

EE

DEDE

D

c

c

If we specialize so that gDß = GDE, g'DE = GDE, (ie. Euclidean) and we specify the relative motion between q and q' to be only in the q' direction, then our transformation is of the form

t.a + qa = tta + qa = q

44

41

14

11

cc

cc (2.9)

Substituting equation (2.9) into (2.7) we find that

.)a( + c = )a(c

)a(c + 1 = )a(aac = aa

214

2244

2

241

2211

44

41

211

14

These are three equations in four unknowns. We need one further relation. But for the moment we have

1] - )a[(c = c - )a(c = )a( 24

42224

4221

4 and

.)a(c + 1 = )a( 241

2211

If the q' moves with constant velocity v with respect to q then

.a +7u a

a +u a = dta + dqadta + dqa =

dtqd

44

41

14

11

44

41

14

11

»¼

º«¬

ª»¼

º«¬

ªc

For u = 0,

v.- = aa =

dtqd

44

14c

Thus a41 = -va44 which implies 1], - )a[(c = )a(v = )a( 24

4224

4221

4 or

.+

cv - 1

1+ = a

2

244 J__

{

Then a41 = +Jv, and since )a(c + 1 = a 24

121

1 we have

,cv+ = a 2

41 »¼

º«¬

ªJ_

and .+ = a1

1 J_ We now have

48

t. + ucv+ = t

q = q

q = q

tv-q+ = q

2

33

22

11

JJ

J

c»¼

º«¬

ªc

c

c

c

_

_

Now if (dq'/dt) = v, this implies

0. = 1 +

cv+

v + v+ = tdqd

2

2

»»»»»

¼

º

«««««

¬

ª

¸¹

ᬩ

§cc

__

__

This means we must take the + sign for a11. If (dq'/dt) = 0 we find

if we take the - sign for a41. If (dq'/dt) = c then

if the sign of a14 is taken as - and the sign of a44 is taken as +. Thus, we have

or

v- = 1 +

cv+

v+ = tdqd

2

2

»»»»»

¼

º

«««««

¬

ª

¸¹

ᬩ

§cc

__

_

c = 1

ccv

v - c = tdgd

2

2

»»»»»

¼

º

«««««

¬

ª

r¸¹

ᬩ

§r

cc

J

J

J

J

= ac

v- = a

v- = a

= a

44

241

14

11

¸¹

ᬩ

§

49

Equations (2.10) are the transformations of Einstein's Special Theory of Relativity, which, in Einstein's derivation needed only his postulate concerning the speed of light and the requirement that physics be the same for all Galilean observers. Here, in the Dynamic Theory we have shown that the Second Law requires Einstein's postulate and the transformations of Special Relativity for Galilean observers. It should be noted that since the absolute velocity (or the speed of light) is unique the answer to whether there may be a different limiting velocity for different fundamental forces is answered by the Second Law. The Second Law states that there is only one limiting velocity independent of the type of force considered. Note that the function defined above as J goes to zero as v tends to c. This is a property required of the integrating factor M(u) and raises suspicions concerning he functional form we will ultimately determine for M. 2.4 The Concept of Entropy. In a system of two independent variables, all states accessible from a given initial state by reversible E-conservative processes lie on a V(u,q) curve. The entire (u,q) space may be conceived as being filled by many non-intersecting curves of this kind, each corresponding to a different value of V. In a reversible non-E-conservative process involving a transfer of energy dE, a system in a state represented by a point lying on a surface V will change until its state point lies on another surface V + dV. Then dE = OdV, where 1/O, the integrating factor of dE, is given by O = M(u) f(V), and therefore dE = M(u)f(V)dV or

Since V is an actual function of u and q, the right-hand member is an exact differential, which may be denoted by dS; and

.qcv - t = t

q = q

q = q

vt) - q( = q

12

33

22

11

»¼

º«¬

ªc¸

¹

ᬩ

§c

c

c

c

J

J

(2.10)

.)df( = (u)E

VVI_

(u)E = dS

I_

50

where S is the mechanical entropy of the system and the process is a reversible one. The Dynamic Theory's Second Law may be used to prove the equivalent of Clausius's theorem, which is stated here without proof. Theorem: In any cyclic transformation throughout which

the velocity is defined, the following inequality holds:

where the integral extends over one cycle of the transformation. The equality holds if the cyclic transformation is reversible. Then for an arbitrary transformation

with the equality holding if the transformation is reversible. The proof of this statement may be seen by letting R and I denote respectively any reversible and any irreversible path joining A to B, as shown in Figure 5. For path R the assertion holds by definition of S. Now consider the cyclic transformation made up of I plus the reverse of R. From Clausius' theorem

0, E - ERI d³³II__

or

S(A).- S(B) Ed - EdRI {³³II

Another result of the Second Law is that the mechanical entropy of an isolated (dE = 0) system never decreases. This can be seen since an isolated system cannot exchange energy with the external world because dE = 0 for any transformation. Then by the previous property of the entropy,

0 S(A)- S(B) d where the equality holds if the transformation is reversible. One consequence of the Second Law is that of all the possible transformations from one state A to another state B the one defined as the change in the entropy is the one for which the integral

IE I B

A_

³{

is a maximum. Thus

,ddE1 = S(A)- S(B) B

A WWI »¼

º«¬

ª³

_max

0, (u)E

d³I_

S(A),- S(B) (u)EB

A d³I_

51

where W is a parameter that indicates position along the path from A to B, or

If U = U(W,q,u,du/dW), then the change in the entropy is given by the integral

The u and q which maximize 'S will be denoted as v and x then, with U = U(x,v), F = F(x,v), and M(v) the v and are given by the solution of the system of equations

where

x' = dx/dW and v' = dv/dW. Thus, the Dynamic Theory's Second Law provides an answer to the question that is not contained within the scope of the First Law: In what direction does a process take place? The answer is that a process always takes place in such a direction as to cause an increase of the mechanical entropy in the universe. In the case of an isolated system, it is the entropy of the system that tends to increase. To find out, therefore, the equilibrium state of an isolated one-dimensional system, it is necessary merely to express the entropy as a function of q and u and to apply the usual rules of calculus to render the function a maximum. The equations, which describe the path the system takes toward the maximum of entropy, are the equations of motion for the isolated system. When the system is not isolated, there are other entropy changes to be taken into account. 2.5 Third Law. The Second Law enables the mechanical entropy of a system to be defined up to an arbitrary additive constant. The definition depends on the existence of a reversible transformation connecting an arbitrarily chosen

.dddqF -

ddU1 = S(A)- S(B) B

A WWIWI »

¼

º«¬

ª¸¹·

¨©§

¸¹·

¨©§

³max

.dddqF -

ddU1 = _S B

A WWIWI »

¼

º«¬

ª¸¹·

¨©§

¸¹·

¨©§

³

0 =

vG -

vG

dd

0 = xG -

xG

dd

ww

»¼º

«¬ª

cww

ww

»¼º

«¬ª

cww

W

W

,ddxF - U1 =G »¼

º«¬ªww

WWI

52

reference state 0 to the state under consideration. Such a reversible transformation always exists if both O and A lie on one sheet of the state surface. If two different systems are considered, the equation of the state surface may consist of several disjoint sheets. In such cases the kind of reversible path previously mentioned may not exist. Therefore, the Second Law does not uniquely determine the difference in entropy of two states A and B, if A defines a state of one system and B the state of another. For this determination a Third Law is needed. The Third Law may be stated, "The Mechanical Entropy of a system at the absolute velocity is a universal constant, which may be taken to zero." In the case of a purely thermodynamic system the absolute quantity is the absolute zero temperature, while for a mechanical system the absolute quantity is the absolute velocity. The Third Law implies that any energy capacity of a system must vanish at the absolute velocity. To see this, let R be any reversible path connecting a state of the system at the absolute velocity u0 to the state A, whose entropy is to be found. Let CR (u) be the energy capacity of the system along the path R. Then, by the Second Law,

But according to the Third Law, S(A) o 0 as uA o u0. Hence it follows that CR(u) o 0 as u o u0. In particular, CR may be Cq or CF. The statement of the Third Law above reflects the restriction to mechanical work terms. A general statement of third law that is independent of the number or type of variables is "The generalized entropy of the system, when the integrating factor vanishes, is a universal constant, which may be taken to be zero." B. General Relations 2.6 Energy and Maxwell's Relations. In thermodynamics a discussion of equilibrium and stability conditions is best done if the enthalpy, Helmholtz's and Gibb's functions are defined first. Therefore, the mechanical analogues of these functions are defined here. Each branch of physics such as thermodynamics and particle dynamics has its own developed procedures. If both branches can be described by the same basic laws, then the procedures developed in thermodynamics may prove to be useful in particle dynamics and vice versa. Once the mechanical enthalpy, mechanical Helmholtz's and mechanical Gibbs' functions are defined, it is then easy to write down the resulting mechanical Maxwell and mechanical energy capacity relations. To begin the development of the Maxwell relations, the mechanical entropy was defined as

.(u)du(u)C = S(A) R

uu

A

O »¼

º«¬

ª³

I

53

Then, since dE = dU -Fdq,

where

Define the mechanical enthalpy as H = U - Fq. Then

Therefore

The mechanical Helmholtz's function can be defined as K = U - M(u)S, and

or, with M'(u) = dM/du,

This leads to

The mechanical Gibb's function may be defined as G = H - M(u)S then

so that

.(u)E dS

I_

{

dq,F - dU = dSII

Fdq + (u)dS = dU I

(2.11)

qdF. - (u)dS = dH I

(2.12)

q.- = FH ; (u) =

SH

SF»¼º

«¬ªww

»¼º

«¬ªww

I

(u)dS -Sdu du(u)]d[ - dU = dK I

I

Fdq. -(u)du S- = dK Ic

(2.13)

(u)F. = qK ; (u)S- =

uK

uq

II »¼

º«¬

ªwwc»¼

º«¬ªww

qdF, +(u)Sdu - =dG Ic

(2.14)

54

From the differential Eqns. (2.11), (2.12), (2.13), and (2.14) the Maxwell relations for a mechanical system may be written:

The energy capacity at the position q can be defined as

Define the energy capacity with a constant force as

then

and

The three generalized laws have been formulated and a few results of these laws have been seen. The next step is to derive the stability conditions to obtain the quadratic forms necessary for a metric. The derivation of the equilibrium and stability conditions is identical to the derivation of the thermodynamic equilibrium and stability conditions with the variables changed to represent the mechanical variables q, u, S and F instead of the thermodynamic variables T, V, S and P.

q. = FG ; (u)S- =

uG

uF»¼º

«¬ªwwc»¼

º«¬ªww

I

.uq =

FS(u)

uF- =

qS(u)

Sq- =

Fu(u)

SF =

qu(u)

Fu

qu

Fq

qS

»¼º

«¬ªww

»¼º

«¬ªwwc

»¼º

«¬ªww

»¼

º«¬

ªwwc

»¼º

«¬ªww

»¼º

«¬ªwwc

»¼º

«¬ªww

»¼

º«¬

ªwwc

I

I

I

I

(2.15)

.uS(u) =

uE C

qqq »¼

º«¬ªww

»¼º

«¬ªww

{ I

»¼º

«¬ªww

»¼º

«¬ªww

{uS(u) =

uE C

FFF I

,uF

uq

(u)(u) = )C - C(

qFFq »¼

º«¬ªww

»¼º

«¬ªww

cII

.

qFqF

= CC

u

S

q

F

»¼

º«¬

ªww

»¼

º«¬

ªww

55

2.7 Equilibrium Conditions.

To establish the criteria for equilibrium, consider, Clausius's

theorem

or

For an E-conservative system dE = 0, then 'S t 0, or S(B) t S(A). Therefore

the mechanical entropy tends toward a maximum so that spontaneous

changes in an E-conservative system will always be in the direction of

increasing mechanical entropy.

Now by First Law 'E = 'U – F'q. Therefore \M'S t 'U - F'q, which

is analogous to the Clausius inequality in thermodynamics. Now consider

a virtual displacement (U,q) o (U + GU, q + Gq), which implies a variation S

o S + GS away from equilibrium. The restoration of equilibrium from the

varied state (U + GU, q + Gq) o f(U,q) will then certainly be a spontaneous

process, and by the Clausius inequality M(-GS) > -(GU - FGq). Hence, for

variations away from equilibrium, the general inequality

must hold. The inequality sign is reversed from the sign in Clausius'

inequality because hypothetical variations G away from equilibrium are

considered rather than real changes toward equilibrium.

In a spontaneous process,

M'S t 'Erev = 'U + work done by the system.

The "work" consists of two parts. One part is the work done by the

negative of the force F. It may be positive or negative, but it is inevitable.

Only the rest is free energy, which is available for some useful work. This

latter part may be written as

The maximum of A is

0, E

- E

R

BA

I

BA d³³ II

__

S(A).- S(B) E

E

R

BA

I

BA {³d³ II

__

0 > S - qF - U IGGG

(2.16)

q.F + U - E = A rev ''''

56

which is obtained when the process is conducted reversibly.

The least work, GAmin, required for a displacement from equilibrium

must be exactly equal to the maximum work in the converse process

whereby the system proceeds spontaneously from the 'displaced' state to

equilibrium (otherwise a perpetual motion machine may be constructed.

Corresponding to Eqn. (2.17) then,

The equilibrium criteria may then be expressed as GAmin t 0. In words: At

equilibrium the mechanical free energy is a minimum. Any displacement

from this state required work.

2.8 Stability Conditions.

To decide whether or not an equilibrium is stable, the inequality

sign in Eqn. (2.16) must be ensured. The conditions for stability may take

different forms depending upon which variables are taken as the

independent variables.

To derive the stability conditions when q and S are taken as the

independent variables consider the terms of second order in small

displacements beginning with the general condition

Choose U = U(q,S), which, because of the identity

is a natural choice for the independent variables, and expand GU in powers

of the Gq and GS

The inequality (2.16) then shows that in Eqn. (2.18),

second order terms + third order terms + . . . > 0.

q,F + U - S = A '''' Imax

(2.17)

S. - qF - U = A IGGGG min

0. > S - qF - U IGGG

Fdq - dU = dSI

order... third of terms +

SSU + Sq

SqU2 + q

qU

21 +

qF + S = U

22

222

2

2

GGGG

GIGG

»¼

º«¬

ªww

»¼

º«¬

ªww

w»¼

º«¬

ª

ww

(2.18)

57

Retaining only the second order terms, the criterion of stability is that a quadratic differential form be positive definite;

If this is to hold true for arbitrary variations in Gq and GS, the coefficients must satisfy the following:

An alternate approach is seen when u and q are considered to be the independent variables, a quadratic form in Gu and Gq may be found by using K = U - MS so that

The terms GSGu cannot be neglected because in Clausius's inequality, which is the actual stability condition, the variations are finite, and therefore, from Eqn. (2.16) the following is obtained:

Expanding in powers of Gu and Gq,

and

But

and

0. > SSU + Sq

SqU2 + q

qU 2

2

222

2

2

GGGGww

www

ww

(2.19)

0. > Sq

U - qU

SU ; 0 >

qU 2

2

2

2

2

2

2

»¼

º«¬

ªww

w»¼

º«¬

ª

ww

»¼

º«¬

ªww

ww

-u Sdud - S - U = K GI

IGGG

... + qqK

21 +u q

uqK + q

qK

21 +u S

dud - qF = K 2

2

222

2

2

GGGGGI

GGww

www

ww

u.qF - qU1 + uu

U1 =u S 2 GGI

GI

GG »¼

º«¬

ªww

ww

- uU =

uK

ww

ww

F. = qKww

58

Therefore

and

Then

and the quadratic form in Gu and Gq is

Since

then

Other quadratic forms may be derived by using different independent variables; however, these two quadratic forms will suffice for this development. C. Geometry 2.9 Geometry Required by the Fundamental Laws. There is nothing that specifies which of the quadratic forms coming from the stability conditions should be adopted as the metric. Thus the choice may be based upon simplicity and/or applicability. However, it becomes obvious that if we choose one of the forms using the velocity as our metric and then obtain equations of motion, then the equations of motion will become third order differential equations in time since the velocity is itself first order and the equations of motion are second order differential equations.

u.qF - qU1

u- =

uF =

quK2

GGI

I»¼

º«¬

ªww

»¼

º«¬

ªww

ww

www

.uU1

dud - S

u- =

uK

2

2

2

2

ww

»¼

º«¬

ªww

ww

III

ququ

K - )u(uK + S

u- =u S

dud 2

22

2

2

2

GGGI

GGI

www

»¼

º«¬

ªww

ww

»¼º

«¬ª

0. > )u(Su

2 + uK - )q(

qK 2

2

2

2

22

2

2

GI

G »¼

º«¬

ªww

ww

ww

F, = qK

u»¼

º«¬

ªww

0. > qF =

qK

u2

2

»¼

º«¬

ªww

ww

59

The fact that these equations of motion will become third order differential equations in time displays a time asymmetry that appears to correspond to nature. However, third order equations are difficult or impossible to solve. To avoid the difficulty of third order equations of motion, suppose we adopt the quadratic form of Eqn. (2.19) as the metric for our system. Thus we are adopting a manifold with coordinates of space and mechanical entropy. This choice is not totally arbitrary because we wish to choose a metric that will display the metric of Einstein's Special and General Relativity as subsets of our metric. Looking toward this objective guides us in the choice of metric. It now becomes desirable to extend our system beyond the dimensionality used thus far. Such an extension brings up a question concerning the integrating factor. With one work term the differential of the entropy was written as

Then if for each dimension the exchange of energy is denoted by dEj, then

where there is no summation intended for fi dVi. Since each dSi is a perfect differential, then the total change in mechanical entropy may be written as

However, the question which arises is whether there exists a single integrating factor M such that

To see this consider the element of work considered before as

Since each dUi is in itself a perfect differential, then dU = ¦i dUi so that

or

.df = E = dS ii VI_

,df = E = dS iii

ii V

I_

.df = E = dS = dS iiii

iiii VI

¦¦¦_

.df = E = E = dS iiii

ii VII

¦¦__

n.1,..., = i ; dqF = W iii¦_

)dqF - dU( = dqF - dU = E iiii

iiiii ¦¦¦_

.E = E ii __ ¦

60

If the system is total E-conservative in the sense that

then dE = 0 is a Pfaffian differential equation. This equation is integrable and has an integrating factor M. The integrability is guaranteed by the Second Law since it is impossible to go from one initial state to any neighboring state. Then, just as in the one-dimensional case, the perfect differential follows:

But since

then

Now following the same argument presented in Section 2.2 concerning the composite system, dE = OdV where V is a function of all the Vi and the ui. Therefore, since dEi =OidVi, then

Now

so that dE = ¦idEi or OdV = ¦iOidVi and

It follows that the wV/wui = 0 and that the ratios Oi/O are also independent of the qi. Therefore the O's have the form Oi = M fi and O = M F(Vi, Vi, ..., Vn) and also

The right hand side is a perfect differential and therefore so is the left.

0, = E = E ii__ ¦

.E = E = dS ii II__

¦

,df = E iiii VI6_

.df

= dS ii

ii VI

I¦

.d + d = E ii

ii »¼

º«¬

ªww

¦ VVVI

O_

»¼

º«¬

ªww

ww

¦ VV

OIV i

i

iii d + du

u = d

.d = d ii

i VOOV »¼º

«¬ª

¦

.df = d = dF = fd = S iiiii

iii

i VVIO

VOOV ¦»

¼

º«¬

ª¦»¼

º«¬ª

¦_

61

Since each Oi/fi is an integrating factor and O/F is also an integrating

factor, it follows that M(u1, u2, ..., un) is an integrating factor for the dE as

well as for dEi = ¦idEi. Therefore

The importance of this question may be seen in terms of the

difficulty that would be created if a universal integrating factor could not

be found. For then each additional work term would require its own

integrating factor to be determined individually.

Thus assured that an overall integrating factor exists, then the

existence of an overall entropy function is guaranteed so that

for any i and the quadratic form may be extended to include three spatial

work terms and thus becomes

Adopting this quadratic form as the metric of a general system whose

thermodynamic variables are held fixed, we may then write this metric as

where the summation convention is used and

with q0 = S/F0, the scaled mechanical entropy for dimensional

correctness.

Thus, the stability conditions provide a metric in the

four-dimensional manifold of space-mechanical entropy. However, the

existing relativistic theories are theories in a space-time manifold.

Therefore, if these theories are to be contained within the Dynamic Theory,

then the space-time manifold must be found within the Dynamic Theory.

The arc length s in the space-mechanical entropy manifold may be

parameterized by choosing ds = u0dt = cdt, where u0 = c is the unique

velocity appearing in the integrating factor of the second postulate. There

are two reasons for choosing the unique velocity. First, it is the only well-

defined velocity we have thus far. Secondly, we may look ahead to the

.E = E = dS ii II__

¦

dqF - dU = E = dS 1i

III_

1,2,3. = ,

; 0 > )dq)(dq(qq

U + )dq(dS)(qS

U + )(dSSU 22

22

2

ED

EDED

DD ww

www

www

0,1,2,3, =j i, ; dqdqh = )(dS jiij

2

(2.20)

,qq

U = h ji

2

ijww

w

62

metric of the Special Theory of Relativity. The metric may now be written as

Now suppose the systems considered are restricted to only E-conservative systems. Then the principle of increasing mechanical entropy may be imposed in the form of the variational principle

In order to use this variation principle, Eqn. (2.21) may be expanded, solved for (dq0) and squared to arrive at the quadratic form

where

with uD = dqD/dt. By defining x0 = ct, xD = qD ; D= 1, 2, 3, then Eqn. (2.22) may be written as

where f = h00. This metric obviously reduces, in the Euclidean limit of constant coefficients, to the metric of Minkowski's space-time manifold of Special Relativity. It is interesting to note that in the metric of Eqn. (2.22) the difference in the sign on the time and space elements of the metric come from stability conditions given in terms of space and mechanical entropy while the variational principle was taken to be the Entropy Principle. In this fashion the Second Law guarantees the limiting aspect found in Einstein's Special Theory of Relativity. In his General Theory of Relativity, Einstein assumed the space-time manifold to be Riemannian. However, this assumption involves the a priori assumption that the scalar product be invariant. This assumption was later questioned by Weyl in his generalization of geometry. From the viewpoint that the adopted postulates of the Dynamic Theory should

0,1,2,3. =j i, ; dqdqh = )(dtc jiij

22

(2.21)

0. = )dq( 20³G

> @,dqdqh - Adtdqh2 + )(dtch1 = )dq( 0

22

00

20 EDDE

DD

(2.22)

)u(hh + uu

hh +

hc + u

hh = A 2

00

0

0000

2

00

0 DDEDDEDD _

0,1,2,3 =j i, ; dxdxgf1 = )dq( ji

ij20 c

(2.23)

63

contain the other theories it then becomes desirable to determine whether or not these postulates specify the geometry of the (dq0)2 space-time manifold. More particularly do the adopted postulates lead to a geometry that includes the geometry of current theories? To arrive at a more general geometry would not be a limitation for it would certainly include the others. Recalling Eqn. (2.23), we can define

Now the Second Law guarantees the existence of the function mechanical entropy and that dq0 be a perfect differential; therefore

where q0i = wq0/wxi. Then the exactness of dq0 is stated by

By defining the parallel displacement of a vector to be

and using Eqns. (2.26) and (2.27) it may be seen that the connections must be symmetrical, or

This result should not be taken to mean that only symmetric connections need to be considered. Rather it means that given the ij's that maximize (dq0)2, then the connections are symmetrical. However, since a variational principle must be used to determine the ij 's, then both symmetric and antisymmetrical connections will have to be considered. In Weyl's generalization of geometry he found it necessary to assume the symmetry of the connections. He proved a theorem showing that the symmetry of the connections guaranteed the existence of a local

.dxdxg )(df1 dxdxg

f1 = )dq( ji

ij2ji

ij20 {{c V

(2.24)

,dxq = dq i0i

0

(2.25)

0. = q - q 0ij|

0j|i

(2.26)

]] vsv

isi dx = d *

(2.27)

. = vki

vik **

(2.28)

64

Euclidean limiting manifold and used this theorem in support of the symmetry assumption. Here we find that the Second Law requires that the connections formed by the solution coefficients must be symmetrical thus guaranteeing, through Weyl's theorem, the existence of a local Euclidean geometry within the Dynamic Theory. Suppose now we consider whether the order of differentiating the change in entropy makes any difference. This means that we must use symmetric connections since the actual change in entropy will be determined by the metric coefficients that generate a maximum. Therefore, consider the difference

Since (dq0)2 = q0iq0jdxidxj from Eqn. (2.25), using Eqn. (2.24) we find q0iq0j = gij. Then

Thus

Likewise

Therefore the difference must be

Using the definition Eqn. (2.27) we see that

Now

- xx

)dq( = )dq_( ji

20220

www

.)q(d + dxdx]qq + qq[ = x

)dq( 20k

ji0j

0k|i

0i

0kj|k

20

ww

.qq2 + qq2 +

dxdx]qq + qq + qq + qq[ = xx)dq(

0k

0l|k

0l

0k|l

ji0lj|

0k|i

0j

0l|k|i

0l|i

0kj|

0i

0l|kj|lk

202

www

.qq + qq2 +

dxdx]qq + qq + qq + qq[ = xx

)dq(

0l

0k|l

0k

0l|k

ji0kj|

0l|i

0j

0k|l|i

0k|i

0lj|

0i

0k|lj|kl

202

www

.dxdx]q)q - q( + q)q - q[( = )dq_( ji0j

0k|l|i

0l|k|i

0i

0k|lj|

0l|kj|

20

.q = q

also ,q = q

,qdx = qd

rki

0r

0i|k

rik

0r

0k|i

0r

sris

0i

*

*

*

65

Similarly

Therefore

Then defining the vector curvature as

the difference may be written as

However, recall that q0iq0j = ij; then

But gri =gir and Rijkl = girRrjkl, so that

So the difference will vanish if Rjilk = -Rijlk. Now since

differentiation will result in

or

.x

+ q =

xq + q =

xq + q =

]q[x

= q

l

riks

ikr

sk0

r

l

rik0

rrik

srl

0s

l

rik0

rrik

0l|r

rik

0rl

0l|k|i

»¼

º«¬

ªw*w

**

w*w

**

w*w

*

*ww

.x

+ q = q k

rils

ilr

sk0

r0

k|l|i »¼

º«¬

ªw*w

**

.-+x

-x

qq =] q-q[q sil

rsk

sik

rslk

ril

l

rik0

r0

i0

k|l|i0

l|k|i0

j »¼

º«¬

ª****w

*ww*w

- x

R l

rikr

ilk w*w{

.dxdx]Rqq + Rqq[ = )dq_( jirjlk

0r

0i

rilk

0r

0j

20

.dxdx]Rg + Rg[ = )dq_( jirjlkir

rilkjr

20

.dxdx]R + R[ = )dq_( jiijlkjilk

20

,dxdxg = dxdxqq = )dq( jiij

ji0j

0i

20

)dxdxgd( = )dxdxqqd( = )dqd( jiij

ji0j

0i

20

66

which can be written as

But gij = q0iq0j. Therefore

or

and

Now interchange jis to sij to get

Then interchange jis to isj so that

Add Eqns. (2.31) and (2.32) and subtract Eqn. (2.30).

or

,)dxdxd(g + dxdxdg =

)dxdxd(qq + dxdxdqq + dxdxqqdji

ijji

ij

ji0j

0i

ji0j

0i

ji0j

0i

).dxdxd(g + dxdxdg =

)dxdxd(qq + dxdxqdxq + dxdxqqdxji

ijji

ij

ji0j

0i

ji0r

srjs

0i

ji0j

0r

sris **

dg = gdx + gdx ijirsr

jsrjsr

is **

x

g = g + g s

ijrjsri

risrj w

w**

.x

g = + s

ijijsjis w

w**

(2.30)

.xg

= + jis

jsisij ww

**

(2.31)

.xg

= + jsi

sijisj ww

**

(2.32)

- - + + + jissijisjjsisij *****

67

and

Now by using the symmetry of gij it can be shown that

and therefore '(dq0)2 = 0. This is the necessary and sufficient condition that the differential entropy change may be transferred from an initial point to all points of the space in a manner that is independent of the path. The distinguishing feature of Riemannian geometry is the invariance of the scalar product under a vector transplantation. Therefore to determine whether the (dq0)2 space is a Riemannian space, consider the vector i and i. Now since [i = gij[j and

then

Or, since gijgij = Gii = 1 and

then

Thus the change in the covariant and the contravariant vectors are given by

Now consider the change in the scalar product [iKi. Then

»¼

º«¬

ª

w

w

w

w

ww

*x

g +

x

g +

xg

21 = s

ijisj

jsi

sij

(2.33)

.x

g +

x

g +

xg

2g = g = s

ijisj

jsi

rs

sijrsr

ij »¼

º«¬

ª

w

w

w

w

ww

**

R - = R ijlkjilk

,dg + dxx

g = gdx = dx = d j

ijsj

sijk

rksr

isrsr

isi [[[[[w

w**

.dxx

g - gdx = dg sj

sijk

rksr

isj

ij [[[w

w*

, + = x

gijsjiss

ij**

w

w

68

Renaming the indices in the second term yields

Thus the geometry of the (dq0)2 manifold is Riemannian. Next consider the question of what is the geometry of the (dV)2 space? Equation (2.24) shows that we may write (dV)2 = f(dq0)2, which is reminiscent of Weyl's generalized geometry. Further we have

Then in the sigma space an arbitrary vector [i would have a length given by the self-scalar product

where l is the length of the vector in the entropy space. If we differentiate Eqn. (2.34), we have

However, in the entropy space the length of the vector is unchanged under parallel displacement so that

Comparing Eqn. (2.35) with the definition of the parallel displacement of a vector, Eqn. (2.27), we find that

plays a role similar to that of the connections *ijk in the definition of parallel displacement of a vector. Therefore we shall define the change in the length of a vector under displacement to be

.dx - dx =

)dx(- + dx =

d + d = )d(

ri

sirs

ir

sris

rsirsi

ir

sris

iii

ii

i

K[K[

K[K[

K[K[K[

**

**

.dx) - ( = )d( sir

ris

ir

ris

ii K[K[K[ **

.g f = g ijijc

- g = |||| = l jiij

2 [[[ cc

(2.34)

2fldl. + dxxf

l = ldl2 ii

2

ww

ccc

.ldxx

lnf21 =

fl

dxxf

21 = ld i

ii

ic

wwc

wwc

(2.35)

x

lnf = ii w

wI

69

This is the same definition Weyl made in his generalization of geometry. However, there is a difference in the way it was obtained. Weyl chose this definition in analogy with the connections * and the definition then led to the second more general metric. In this theory the fundamental laws lead us to two metrics and Eqn. (2.35) for the change in the length of a vector under displacement. Therefore, we have no choice. Thus within the Dynamic Theory Eqn. (2.35) is a derived equation and Eqn. (2.36) only renames the logarithmic derivative. Using Eqn. (2.36) we may obtain, in general,

Renaming the various summation indices, rearranging terms, and using the length of a vector, we obtain

Since this must hold for arbitrary choice of [i and dxk, we conclude that

This is the same system of linear equations for the connections *ijk as Eqn. (2.30) except that the inhomogeneous term ijk has now to be replaced by gijk - 2gij Mk. Therefore the same linear algebra as before leads to

where (ijk) is the usual Christoffel symbol of the second kind. Now, since the entropy space is Riemannian, then in the entropy space we have M'i=0 and *'ijk = -(ijk) and the length l of a vector is unchanged under parallel displacement. However, the same displacement law in the sigma space, with metric gij, leads to the relation

)l.dx( = dl iiI

(2.36)

.dxg + dxg + dxg =

)gd( = )dx(l2 = dlklij

lkijkjli

lkijkji

kij|

jiij

ii

22

[[[[[[

[[I

**

.dxg2 = dx]g +g + g[ kjikij

kjiljkil

likljkij| [[I[[**

0. = g + g + )g2 - g( ljkli

likljkijkij| **I

- g + g[g + jk

i - = jlkklj

liijk II¸

¹

ᬩ

§*

(2.37)

70

Thus r (1/2)(wlnf/wxk) plays the role of Mk in Eqn. (2.36). It follows then that the ordinary connections -(ijk) constructed from ij are equal to the more general connections ijk constructed according to Eqn. (2.37) from gij and Mk = (1/2)(wlnf/wxk): This can also be seen by direct computation from Eqn. (2.36)

.fg = g ijijc (2.39) and

We may interpret the change of metric from ij to gij by Eqn. (2.40) as a change of scale for the length at every point of the Riemannian manifold by the variable gauge factor f. This transformation is called a gauge transformation, and Mk is called a gauge vector field. The generalized geometry thus separates the problem of measure-ment of angles from that of measurement of length. For instance, the angle between the two vectors [i and Ki at a given point of the space is measured by the ratio

This ratio does not change under the gauge transformation Eqn. (2.40). The gauge transformation is therefore an angle-preserving, or conformal, change of metric. On the other hand, the length of vectors will change under Eqn. (2.40) according to Eqn. (2.35). Thus the metric tensor ij determines angles, while one needs also the gauge vector Mk to measure length. Considering the sigma space, which is characterized by the tensor field ij and gauge vector k. The same argument as before shows that we may, without changing the intrinsic geometric properties of vector fields, replace the geometric quantities by use of a scalar field f as follows:

.ldxx

lnf21+ =

dxxf

l =

fgd+ = gd+ = ld

kk

kk

jiij

jiij

cwwww

cc

_

__ [[[[

(2.38)

. = ijk

ijk **c

(2.40)

.)g)(g(

g =

|||| |||| jiij

jiij

jiiji

i

KK[[

[[

K[K[

71

That is, in the new metric, vectors will have the same law of affine

transplantation and the angle between different vectors at the same point

of the manifold will be preserved, but the local lengths of a vector will be

changed according to

Thus the general Weyl geometry of the sigma space admits also a

conformal gauge transformation.

D. Mechanical Systems Near Equilibrium

2.10 Special Relativistic and Classical Mechanics

Classical mechanics describes the motion of a system, which could

be a particle, for which the energy of the system is a constant. The

equations of motion yield trajectories resulting from the action of forces;

they may also be obtained from the Principle of Least Action. When the

action integral is treated as a variational problem with variable end points,

the method of Lagrangian multipliers yields the same equations as does

Hamilton's Principle. However, if the variational problem is transformed to

a new space in which the new variational problem has fixed end points,

then the metric for this space is displayed, and the equations of motion are

geodesics in this space.

In classical mechanics the Principle of Least Action as formulated by

Lagrange has the integral form

In curvilinear coordinates the integral assumes the form

where D,E = 1, 2, 3.

Defining

. = ,x

lnf21+ = ,fg = g i

jki

jkkkkijij **cwwcc II

(2.41)

.fl = l 22c

.sdvm = A PP

2

1x³

(2.42)

dt,dt

dxdt

dxmg = dxdtdxmg = A )Pt(

)Pt(PP

2

1

2

1

ED

DEE

D

DE ³³

,dt

dxdt

dxg2m = T

ED

DE

72

the integral becomes

Then the principle of least action may be stated as: Of all curves C' passing through P1 and P in the neighborhood

of the trajectory C, which are traversed at a rate such that, for each C', for every value of t, T+V=F, that one for which the action integral A is stationary is the trajectory of the particle.

The transformation of variables may be carried out to display the metric

where hDE =2m(E0-V)gDE. Here different particles in the same field and with different energies E0 would appear to have different geometries, a situation which has been previously taken to be impossible and therefore precluded the geometrization of dynamics(see page 6 of ref. 46). However, in view of Weyl's generalization of geometry, treating the variational problem in the Principle of Least Action as transformed to a new space in which the varia-tional problem has fixed end points, in effect, is a transformation into a space with Weyl geometry where the gauge function is 2m(E0-V). Thus changing the energies does not change the geometry since it will still be a Weyl space. Suppose now that the concepts of classical mechanics are compared with the concepts from the point of view of the Dynamic Theory. The energy of the system in classical mechanics is a constant of the motion and therefore the change in kinetic energy is the negative of the change in potential energy, which may be written as

However, for classically conservative forces dH is a perfect differential. Therefore for this system with only one work term the force is a function of position only. This suggests the association of the classical energy of the system, H, with the system energy, U, which is also a perfect differential. Now if the system is isolated, or E-conservative, then

But if dU=dH=0 then F must be zero. This points out an important difference between classical physics and the Dynamic Theory. A classically

2Tdt. = A )Pt()Pt(

2

1³

dxdxh = )(ds 2 EDDE

(2.43)

0. = dV + dT = dH

dq.F - Ud = E = 0 _

73

conservative system is one for which the system's energy is a constant of the motion. However, the E-conservative system, within the Dynamic Theory, is one for which dE = 0. Thus an E-conservative system which is also conservative in the classical sense must have no forces F which may depend upon velocity as well as position but may have forces which arise from -(wU/wq) = F and must be functions of positions only. Suppose we now turn our attention to the mechanics of Special Relativity. In the Special Theory of Relativity Einstein sought to put Newtonian mechanics into a form that would leave the speed of light invariant. The resulting dynamics exhibits the notion of a unique velocity in a similar sense to the previously defined absolute velocity. Within the Dynamic Theory we may display the appearance of the Special Theory's foundations by using the generalized entropy principle rather than being required to assume the existence of Newton's equations of motion on an a priori basis. Newtonian mechanics is displayed in its simplest form for particles, so we shall make the restrictive assumption that the mass density, J0, such that

We will also assume that the gij are constants, thus

and gDE=GDE. Our variational problem depends upon the integral

where we have used the definition

Because we have assumed that the mass, m0, is a constant we can write our integral as

We can make a change of variables by letting

so that the integral becomes

m = d(vol) 00J³

1,2,3, = , ,dxdxg - )(dtc = )(d 222 EDV EDDEˆ

,duugfm = )dq(m = )(dS = dS = S kjjk

20

200

2mq0

2S0

S0

2

1

0

VVV

ˆ³³³³

.ddx = u

jj

V

.duugf = mS = q kj

jk0

0 2

1VV

Vˆ³

,d uug = ds kjjk Vˆ

(2.44)

74

We can now define a new function

and then consider a further change in variables such that

If we substitute this new variable into our integral we find

with the auxiliary selection that 2T - m0c2f = 0. The problem of determining geodesics has now been converted into a statement of the principle of maximum generalized entropy: Of all curves C', passing through P1 and P2 in the

neighborhood of the trajectory C, which are traversed at a rate such that, for each C', and for every value of W, 2T = m0c2f, that one for which the generalized entropy integral, q0, is maximum (stationary) is the trajectory of the particle.

When stated in the form of a variational equation, this principle reads

with the auxiliary condition 2T-m0c2f=0, on C'. The dynamics is unaffected by the addition of a constant to the gauge function; therefore, let

where h is a constant. The auxiliary condition now reads

dS. f = q SS

0 2

1³

,cm2f = T 2

0

.d m2T = d ,fc = dS

0WW

,d 2T = qcm )P()P(

00

2

1WW

W³

(2.45)

0, = d 2T )P()P(

2

1WG W

W³

,2f

cm - h = V 20

.C on 0, = h - V + T c

75

We can solve this variational problem by making use of the Lagrangian multiplier method for a problem with constraints. We construct a function G=2T+OM, where M=T+V-h=0, and determine the solution of the system of equations

This system has a solution for which O=-1Sokol, and it follows that the trajectory C is determined by the solution of the system

We assumed that the gjk were constants; therefore, if we make the definition

then the equations in Eqn. (2.46) become

because dV=cdW and W=m0gjkujuk. If we multiply these equations by gli and sum them, we obtain

From the metric with constant coefficients we get

or

0. = h- V + T

withn,,0,1,2, =j 0, = uG

dd -

xG

jjxxx»¼

º«¬

ªww

ww

W

(2.45)

n.,0,1,2, =j ,x

V - = x T -

u T

dt d

jjj xxxww

ww

»¼

º«¬

ªww

(2.46)

,x

V - = F jj ww

> @ ,F = ugcmd d

jk

jk2

0 ˆV

> @ n.,0,1,2, = l ,F = Fg = umd d l

jljl

0 xxxˆV

(2.47)

1,2,3, = , ,xxg - c = dtd 2 EDV ED

DE &&ˆ

.v - c = dtd 22V

(2.48)

76

Substituting Eqn. 2.48 into Eqn. 2.47 we find that

where vl=dxl/dt. Thus we have

where E=v/c. Now Eqn. (2.49) can be rewritten as

Because E=0 in a local coordinate system x,

where, in a local reference frame x, al=(1/c2)(d2xl/dt2). In Eqn. (2.51) we have the form of Newton's second law in classical mechanics. We may rewrite Eqn. (2.50) in the form

These equations are the equations of motion of the Special Theory of Relativity and come from the geodesic equations of the variational problem,

> @

l

l

lll

vvc

cmdtd

vc

ddt

dtdxcm

dtd

vc

ddt

ddxcm

dtducm

ddF

»¼

º«¬

ª

��

»¼

º«¬

ª¸¹·

¨©§

�

»¼

º«¬

ª

22

20

22

2022

20

20

1

,1V

VVV

, - 1vm

dt d

- 1

1 = F2

l0

2l

»»

¼

º

««

¬

ª

EE

(2.49)

.1

12

02

»»¼

º

««¬

ª

� �

EE

ll vm

dtdF

(2.50)

,am = dtxdm = F l

02

l2

0l

(2.51)

. - 1vm

dt d = f F - 1

2

l0ll2

»»

¼

º

««

¬

ª{

EE

(2.52)

77

in Eqn. (2.42), based upon the generalized entropy principle with the restrictive assumptions that the mass density, J, be a constant and that the metric coefficients, gjk, are independent of the mass distribution. Thus, we have shown that the special relativistic equations of motion and the Newtonian equations of motion are required by the generalized entropy principle, but that they represent a limited subset of the entropy principle. 2.11 Energy Concepts Newtonian and relativistic mechanics talks of potential and kinetic energy while classical thermodynamics, which forms the basis of the Dynamic Theory, contains concepts with units of energy such as entropy, enthalpy, and free energy. We may use these common fundamental principles within the Dynamic Theory to explore how the mechanical energy concepts fit among the general thermodynamic energy functions. It seems that this will be of more than a little benefit when trying to keep all of the energy-based concepts in proper perspective. First, let us recall the First Law, with n=1,

whereas the differential change in the generalized entropy is

where dS is a total differential. If we suppose that the system energy, U, may be a function of position, x, and the velocity, v=dx/dt, then we may write

Because dS is a total differential, then

This requires that

Fdx, - dU = E_

(2.53)

,Fdx - dU = dSII

(2.54)

dx.F - dxxU1 + dv

vU1 = dS

III ww

ww

.vU1

x = F -

xU1

v

¿¾½

¯®­

ww

ww

¿¾½

¯®­

»¼º

«¬ªww

ww

II

78

because M=M(v) from the second law. Further, dU is a total differential so Eqn. (2.55) becomes

Now consider the functional form of the force from the equations of motion in Eqn. (2.52),

where F(x) is strictly a function of x because it came from the gauge function. Then

Thus, Eqn. (2.55) may be written as

In order to satisfy Eqn. (2.57) we find )=M and U=U(v). By substituting these results into the differential expression for the entropy, Eqn. (2.54) we find

- vx U1 2

www

¸¹

ᬩ

§I

(2.55)

only.velocity of function = v- = F -

xU

vF

I

Iww

»¼º

«¬ªwwww

(x),F (x)F - 1 = F 2 ˆˆ ){E

(2.56)

(x).Fdvd =

vF ˆ)ww

.)(ˆ

)(ˆ

I

Idvd

xFxU

dvdxF

�

»¼º

«¬ª )�ww

)

(2.57)

dx, (x)F - - 1

dvdvdU

= dS2

ˆE

¸¹·

¨©§

(2.58)

79

which is a perfect differential whereby we have found that U is strictly a function of velocity. Now consider the First Law for an isolated system, or

but, using Eqn. (2.56) this may be written as

Then by integrating we find that

This is Einstein's energy integral which, because of the equations of motion, becomes

In his Special Theory of Relativity Einstein interpreted the right-hand side of Eqn. (2.59) to be the kinetic energy; therefore, he chose the integration constant to be -m0c2 in order that T=0 when v=0. Here, Eqn. (2.59) is the energy of the system and, therefore, will not be zero when v=0. Thus, the constant of integration should be taken as zero, giving the energy by

If we differentiate Eqn. (2.60) with respect to the velocity, we find

Substituting this result into Eqn. (2.58), the change in entropy becomes

This expression may now be integrated because

Fdx, - dU = 0 = E_

dx. F(x) - 1 = dU 2E

dx. F = U - U PP0 0³

constant. + - 1cm = U

2

20

E

(2.59)

. - 1cm = U

2

20

E

(2.60)

.)1()1(

221

TT

K

TT

TT

K

EII

B

vmcvcm

vU

�

�

ww

(x)dx.F - ) - (1

vdvm = dS 220 ˆE

80

³

³³

��

��

�

K TT

TT

K

K TTK

K

EE

E

),()1()(

21

)(ˆ)1(

2

1

xVdcm

dxxFdvvm

SS

and

By setting the constant of integration at 1/2 m0c2, we get

Thus, the generalized entropy for a purely mechanical system has two parts. One, depending entirely upon the velocity, and which we may call kinetic entropy, is given by

The second term in the mechanical entropy is a function of position only and may be called the entropy potential, V(x). We may look at the kinetic entropy differently if we go back to the variable changes during the presentation of the maximum entropy principle, because there we had

but c/v - 1cd = d 22WV 199 therefore,

But, by Eqn. (2.48) this becomes

constant. + V(x) + ) - (1

cm21

= S 2

20

E

V(x). + ) - (1

vm21

= Sor V(x) + ) - (1

cm21

= S 2

20

2

220

EE

E

(2.61)

.) - (1

vm21

= T 2

20

E

(2.62)

,d m2t = d uug = ds

0

kjjk WVˆ

.

ddt

xxgcm21 =

ddu

ddugcm2

1 = uugcm21 = T

2kj

jk2

0

kj

jk2

0kj

jk2

0

¸¹·

¨©§V

VV

&&ˆ

ˆˆ

81

which is the kinetic entropy of Eqn. (2.62). Thus, we find that it is the mechanical entropy, S, that must have a constant value along any trajectory for an isolated system, because

for the trajectory, and therefore, S=h=constant. Thus we have established the following for the trajectories of an isolated system: Mechanical Entropy: S = h = constant

j

j

j

j

jj

x

x

j

j

dxFvdvm

or

dxFdUEdLawFirst

cmvUsystemtheofenergyst

xFFForce

cmTsystemtheofenergyKinetic

dxFWsystemthebydoneWork

cmUsystemtheofEnergy

xFxVEntropyPotential

cmTEntropyKinetic

ˆ1)1(

0

,0:

)0(:Re

)(ˆ1:

11

1:

:

1:

)(ˆ)(:

)1(:

2

23

2

0

20

2

2

20

2

20

2

20

2

1

EE

E

E

E

E

���

�

�

»»¼

º

««¬

ª�

�

�

�

�

³

³

2.12 Non-Isolated System Thus far we have consistently required the system to be isolated. Obviously there are a large number of physical phenomena for which this restriction may not be used, even as an approximation. Therefore, relaxation of this restriction should provide description of a large and important class of systems. The remainder of this book involves the investigation of the predictions of the Dynamic Theory assuming the system is isolated. This may give the implication the non-isolated system is less important or less interesting. This is not the intention of the

,) - (1

vm2

1

= T2

20

E

0 = h - V + T

82

presentation. Rather, the presentation is aimed at displaying the fact that existing theories are subsets of the Dynamic Theory. In order to do this we must stay with the assumption of the isolated system. One of the benefits of the Dynamic Theory is the capability of using procedures currently used in one branch of physics in another where prior to the unification displayed here would have been thought impossible. A system in which this procedure should produce significant results is a nonequilibrium thermodynamic system. Thermodynamics tells us that we must minimize the free energy, but the ability to use this as a variational principle to obtain equations of motion is a procedure that the Dynamic Theory now makes possible for this thermodynamic system. E. Quantum Mechanics 2.13 Quantum Mechanics Derived In 1927 F. London derived quantum principles from Weyl's geometry.(47) However, the results of his work made it difficult to define length as a real number and because of this Weyl later interpreted the mathematical formalism of his unified theory as connected with transplanting a state vector of a quantum theoretical system. Suppose that we consider an isolated, or E-conservative, system so that dE = 0. Then, because of the Second Law dq0t0 which is the principle of increasing mechanical entropy. Then certainly (dq0)2t0 and also, since (dq0)2=f(dV)2, then f(dV)2t0. However, if f<0, then (dV)2<0 since it is the product that must remain greater than, or equal to, zero. In this case

But

and

which implies that the element of arc (dV) is given by

where (dV)0 is some initial value of the element of arc. Now suppose an equilibrium, or reversible, state is desired so that dq0=0. Thus, the desired condition is a null trajectory of the (dq0)2

.)(d-f- = dq 20 V

)(d dx = )d(d kk VIV

,dx = )(d)d(d k

kIVV

³³

,e )(d = )(d dx0

kkIVV ³

83

manifold. Then, if fz0, the desired condition is also a null trajectory of the (dV)2 manifold. This implies that d(dV)=0 or (dV)=(dV)0, so that

which is satisfied only if

where N is an integer. This is the quantum condition London introduced. To illustrate how this condition arises from the Dynamic Theory's approach, suppose a description of a hydrogen atom is desired. A hydrogen atom is in a stable condition and, if isolated, satisfies the conditions dE = 0 and dq0=0. These conditions along with fz0 establish the quantization of the integral in Eqn. (2.63). To show how the Dynamic Theory removes from London`s work the difficulty of defining length as a real number, consider an elementary presentation of London's. Suppose the field of a proton to be given by

Equality of forces for the simple case of circular motion requires that

Thus the period is given by T=2Sr/v and the velocity by

Now

so that

Solving for the radius shows that the allowed radii are

By choosing

1, = e dxkkI³

iN,2 = dx kk SI³

(2.63)

0. i ; 0 ; r

= i

1

0 z{IDI

.re =

rmv

2

22

.mre = v

iN,2 = cT = dx 0k

k SII ³³

iN.2 = e

mrc2 = rcT 1

1

SSDD

.cm)(

eN- = r221

22

D

84

where h is Planck's constant, then the possible radii become

which are the Bohr radii.

The imaginary D1 presented the difficulty, in London's work, of

defining length as a real number. In the Dynamic Theory real distance, or

length, may be defined, and properly should be, in the (dq0) manifold.

Recalling that the definition of the potentials is

it may easily be seen that if f<0, then Mk becomes imaginary as does the

length of arc in the (dV)2 manifold since the length of arc is given by

However, the arc length in the (dq0)2 manifold is real since dq0t0 by the

Second Law.

It should be noted that the conditions for quantization are not

restricted to dE=0, dq0=0, and f<0 as used here. Any set of conditions

which results in the final element of arc (dV) being equal to the initial

element of arc (dV)0 results in quantum conditions. It is particularly

significant to note that the quantization involves only forces, which may be

described in terms of the "distance curvature" and does not involve forces

describable by a vector curvature. Thus interpreting the gauge potentials

Mk to be electromagnetic potentials provides quantum effects for

electromagnetic forces.

Here, again, is a distinction between curved and Euclidean

manifolds, though here it appears slightly different. The Dynamic Theory

requires a quantization. However, this quantization depends upon the

existence of a gauge function and appropriate restrictive conditions. Thus

a curved space may exhibit quantum effects but only if the curvature is

accompanied by a gauge function or a distance curvature.

Thus the Dynamic Theory, through London's quantization, not only

supports the contention that "God does not play with dice all the time" but,

further, may supply the answer to the question, "What is waving in the

wave function?" London showed that the wave function is directly related

to the element of the arc length in the sigma manifold. Therefore the

"waving" is the tendency of this element of arc length to increase and

,i 137

i hc

ei2 = 2

1 DS

D {|

,me4

hN = r 22

22

S

,xf + = k

21

k ww ln_I

.)(d = 2VV ³

85

decrease around a closed path. Using the calculus of complex variables, the quantum number becomes the order, or multiplicity of the zero of (dV). This may also be stated in terms of null trajectories. Einstein's null trajectory was the path light travels and remains so here. However, this is the zeroth order null trajectory. The remaining null trajectories for the complex arc length are given, as London showed, by the equations of Quantum Mechanics. 2.14 On The Derivation of Thermodynamics from Statistical Mechanics It is commonly believed that one can "derive" thermodynamics from a variety of force laws using the techniques of statistical mechanics. This belief is not supported when one considers the development of statistical thermodynamics. For instance, in order to talk of a statistical temperature, W, one must start by assuming Newtonian physics (this constitutes three fundamental assumptions). Given Newtonian physics one can talk of an energy distribution, canonical ensembles and statistical temperature; however, one must make an additional fundamental as-sumption (the Equipartition Law) before the statistical heat capacities may be obtained. In order to obtain thermodynamics we need two more assumptions. To display the assumption necessary for the first law of thermodynamics let me quote from page 85 of "Basic Theories of Physics: Heat and Quanta" by Peter G. Bergmann. "The difference between the heat transferred to the system and the work performed by it,

is, according to our previous discussions, the increase in u. But in a systematically thermodynamic approach (that is, using only macroscopic observations and concepts), we get the differential expression, Eqn. (2.64) without reference to u. From that point of view, to claim that this expression is an exact differential is a logically new assertion; and this assertion constitutes the First Law of Thermodynamics." The assumptions of statistical thermodynamics allow us to derive the differential of the heat exchanged in the form

where W is the statistical temperature and dV is the statistical entropy. Further, it may be shown that

,R)dY - J( + dC = W - Q iiin

1i=¦TQ__

(2.64)

,d = Q VW_

(2.65)

86

By comparing Eqn. (2.66) with the classical thermodynamic statement

and

we find that the statistical expression, Eqn. (2.66), is analogous to the classical expression, Eqn. (2.68), for the second law of thermodynamics. Also we may equate Eqns. (2.65) and (2.67) to obtain

and Eqns. (2.66) and (2.68) are simultaneously satisfied provided that W/T>0. In statistical thermodynamics it is asserted that W/T=kB, where kB is Boltzman's constant. Once this assertion is made then we have kBdV=dS, hence S=kBV. However, there is no logical necessity that the ratio W/T be a constant from the statistical approach, and only if it is a constant can we have a one-to-one correspondence between the statistical entropy and the classical thermodynamic entropy. The misconception that classical thermodynamics may be derived from Newtonian mechanics without the necessity of making additional assumptions is further entrenched by authors, such as Kittel, who in his text Thermal Physics says the following on p. 49, "We show in Chapter 8 that W is proportional to the conventional absolute temperature which is measured in degrees Kelvin"; (This implies a logical necessity) On p. 427 the author states, "By analogy with the relation dQ = WdV we `assume' that the Kelvin temperature T has the property dQ = kBTdV for a reversible process; here kB is a constant to be determined and V is the entropy." (The implied logical necessity is reduced to an assumption.) We are so familiar with Newtonian mechanics and its basic validity that it is difficult for us to consider that it might be derivable from some other physical concept and its associated fundamental assumptions. Further, classical thermodynamics, even before statistical mechanics gave

.Q dW

V_

t

(2.66)

TdS, = Q_

(2.67)

,TQ dS _

t

(2.68)

TdS, = dVW

(2.69)

87

rise to the distinction between microscopic and macroscopic views, never talked of anything resembling equations of motion. Added to these factors is the somewhat long logical progression from the adopted laws of the Dynamic Theory to Newtonian mechanics. On the one hand the First and Second laws adopted by the Dynamic Theory give rise to a generalized entropy principle that requires that any dynamics for an isolated system must occur so as to seek a maximum of the generalized entropy. Thus we have a variational principle based upon maximizing the entropy. On the other hand the laws produce, through the stability conditions, a metric to be used in this variational principle in which the type of geometry is specified and need not be assumed as in Newtonian and relativistic mechanics. The Euler equations resulting from this variational principle taken in the non-relativistic, Euclidean, three-dimensional limit, for particles become

or inertial mass, m, times the acceleration, d2xi/dt2, must equal the force, Fi. Thus the adopted laws, through restrictive assumptions, do lead to Newton's Second Law. Newton's First Law comes from considering the motion in the absence of any force. To arrive at Newton's Third Law one must show that all of the forces, allowed by the adopted laws of the Dynamic Theory, must be symmetrical. A violation of this symmetry of forces that has recently been found will be shown in Section 4.11 for forces within the nucleus; and, therefore, the Dynamic Theory does not require Newton's Third Law to hold within the nucleus, but does for atomic forces and macroscopic matter. F. Summary 2.15 Summary of new theoretical fundamentals When this investigation was initiated, it was concluded that Einstein's postulate of the constancy of the velocity of light could not be adopted since it was felt that experimental evidence in electromagnetism alone did not justify applying it as a limiting velocity to all types of forces. However, we find that Einstein's postulate is required by the Dynamic Theory which approaches physical phenomena from a different way. The new viewpoint indeed supports Einstein's every contention with regards to Special Relativity and his uneasiness concerning quantization. Further, the Dynamic Theory supports Einstein in such a way that it seems only the early successes of his theories kept Einstein himself from coming to the same realization. This is, of course, speculation, but it was Einstein who returned to very fundamental concepts in order to establish a basis for his relativity

,F = dt

xdm i2

i2

88

theory. He was also known to be aware of the tremendous strength of classical thermodynamics since he wrote, "A theory is the more impressive the greater the simplicity of its premises are, the more different kinds of things it relates, and the more extended is its area of applicability. Therefore the deep impression that classical thermodynamics made upon me. It is the only physical theory of universal content concerning which I am convinced that, within the framework of applicability of its basic concepts, it will never be overthrown." Thus it seems only the fact that Caratheodory's statement of the second law, which is the key to the development of the Dynamic Theory, did not make its appearance before the relativistic theory had achieved such stupendous successes kept Einstein from eventually investigating its possible extended application. The key points in the development of the Dynamic Theory seem to be the recognition of the generality of the thermodynamic laws and their independence upon the number or type of variables considered and the recognition that the quadratic forms associated with the stability conditions from natural metrics leading to a geometrical description of the dynamics of the system independent of the variables used in the description. There are numerous conclusions and implications that could be reiterated here; however, only a few of the seemingly more significant ones will be discussed. The first one is the existence of an integrating factor for any system describable by the First Law, particularly an integrating factor independent of the type of force considered. It is this fact that ultimately leads to a unique limiting velocity for all forces. However, in speaking of the absolute velocity for mechanical systems, care must be taken to point out that, as far as the three laws are concerned, it does not represent an absolute barrier. Rather the laws only state that, for a mechanical system with only three work terms representing the work done by three spatial forces the absolute velocity represents an upper and lower limit. Thus solutions with velocities greater than the speed of light are also allowed. However, so long as the system is subjected to only these three forces, then its velocity may never cross this barrier. This absolute barrier effect may be expected to change if another force term representing an additional dimension is found necessary. The reduction in the number of fundamental laws or postulates is significant. This together with the unifying effect of the three laws promises to simplify the study of physical phenomena by founding the entire realm of physics upon a common set of conceptualizations. In this chapter it was shown that the Special Theory of Relativity was a special case obtained from the fundamental laws adopted by the Dynamic Theory. Einstein’s postulate concerning the speed of light was an immediate result of the Second Law. Further, it was shown that Quantum Mechanics resulted when stable isentropic states were required. This displays a different relationship between these theories than has previously

89

been envisioned. Here there is no question concerning one being more fundamental than the other. The determining factor is not which is more fundamental, but what restrictions are placed upon the system. For example, if the system is in an "isentropic state" that state is to be determined by the equations of Quantum Mechanics. If the system is not required to be isentropic or otherwise restricted so that the entropy must return to its original value after completing a loop the quantum Mechanics does not describe the system. Rather, in this case one must turn to the equations of Einstein's Special Theory of Relativity. A further note should be considered here. The equations of motion that have been derived here and the Quantum Mechanic equations of motion which London derived from the isentropic condition describe the system as if "tends" toward an equilibrium. This is the origin of the motion. That is, the tendency to seek a maximum of entropy for the isolated system. In this chapter there is no clear way to improve our understanding of nuclear phenomena, nor is there any clarification of gravitational effects. Further, ramifications of the theory will be pursued in the next three chapters, which will bear on these points.

90

CHAPTER 3 - FIVE-DIMENSIONAL SYSTEMS During the preceding development of the Dynamic Theory, there did not appear to be anything that approached a description of nuclear effects. Of course quantum theorists may respond that the nuclear effects lie within the realm of quantum theory. This, however, does not seem to be a strong argument since current nuclear theory appears to depend upon a number of ad hoc postulates. If it is supposed that nuclear theory cannot be extracted from some aspect of the preceding four-dimensional world view, then how might the Dynamic Theory produce a foundation for nuclear theory? At this point there may appear to be no obvious way. Therefore, let us proceed on a different tack. Thus far we have constantly adhered to the policy of dividing systems into two types: thermodynamic systems with only a work term of the pdv type and mechanical systems with three mechanical, or spatial, work terms. The generality of the adopted laws places no restrictions upon the number or type of variables used. In particular, there is no restriction coming from the laws themselves which says we cannot use four work terms, one the thermodynamic pdv term and three mechanical Fdq terms. Obviously pdv itself is just another Fdq type term with the pressure as the generalized force and the volume as the generalized displacement. When we teach thermodynamics we write the First Law with the right hand side of the equation being the change in internal energy (system energy), the thermodynamic work term, and three spatial work terms. We then tell the students that since the right hand side of the equation involves five unknowns we must have five independent equations in order to have a solvable system. The first equation offered is the conservation of mass which we state guarantees that we may write mass density as a function of space and time. But is this really true for all space and for all time? The rub comes in attempting to visualize a world description in five dimensions. Many arguments may be envisioned which tend to imply only a four-dimensional manifold is needed. The kinetic theory of gases relates the pressure to the average velocities of the particles contained. Does that not imply that thermodynamics ultimately rests on a four-dimensional manifold? Recall that the system in the kinetic theory is basically in equilibrium. Statistical thermodynamists may claim that thermodynamics is basically statistical in nature and is fundamentally tied to order and disorder and hence to the four-dimensional world of quantum theory. But remember that the overall system, to which the statistical approach is applicable, is a composite system made up of many subsystems each in an equilibrium state. What happens to this argument when the number of individual particles is not infinitely large?

91

Still there seems to be no substantial support for a five-dimensional world from the point of view of current theories. This is to be expected though in view of the difficulties experienced in the transition from the classical three-dimensional world to the four-dimensional space-time of Einstein's theories. Obviously had the extension of the universe been restricted on a priori grounds to three-dimensional Euclidean space, Einstein's theory would have been rejected on first principles. On the other hand, as soon as we recognize that the fundamental continuum of the universe and its geometry cannot be posited a priori and can only be disclosed to us from place to place by experiment and measurement, a vast number of possibilities are thrown open. Among these the four-dimensional space-time of relativity, with its varying degrees of non-Euclideanism, has found a place. So also may the five-dimensional view of the Dynamic Theory be found within the possibilities. Ultimate judgment upon its necessity, or applicability, should rest upon a comparison of the theory's predictions with reality. A. Systems Near an Equilibrium State The metric coefficients are made up of the second partial derivatives of the system energy function and, therefore, if the system remains near an equilibrium state, then the value of these derivatives evaluated at the equilibrium state may be used as a first approximation for the metric coefficients. In this case the geometry will be Euclidean and, from the preceding four-dimensional development, the Euclidean manifold produced by applying the E-conservative restriction was Minkowski's space-time continuum of Special Relativity. Therefore, suppose we begin an investigation of the five-dimensional world by staying very near an equilibrium state so as to simplify the description to a five-dimensional generalization of Minkowski's space-time manifold. 3.1 Equations of Motion Suppose that we consider some sort of system requiring four work terms and for the moment not concern ourselves as to exactly what this system might be. Thus, for our system we will have thermodynamic as well as mechanical variables and the First Law becomes

Where the v and are considered as specific quantities. That is, these quantities are related to a unit of mass such as is customary in thermodynamics. The specific volume is the reciprocal of the mass density J. Using the mass density instead of the specific volume the First Law becomes

1,2,3. = ; dqF - Pdv + Ud = E DDD

~~~_

92

This law now requires that the system's specific energy U be a function of five independent variables so that U = U(S,q1,q2,q3). Thus, the First Law requires a five-dimensional manifold of specific entropy, space, and mass density for a general system. Since the system under consideration needs both thermodynamic and mechanical variables, we can no longer refer to the entropy as mechanical or thermodynamic; however, the limiting case where the mass is held fixed must produce the mechanical entropy. The procedure established by the Dynamic Theory is to take the stability condition quadratic form as the metric for a stable system. Thus, the coefficients of the metric become the second partial derivatives of the energy function. In order to simplify the metric, suppose for the present that we restrict our system to be very near an equilibrium state so that we may consider the second partial derivatives to be constants. This is in essence considering a local Euclidean manifold; the symmetry of the geometric connections guarantees that we may do this. Since the metric coefficients are constants, a transformation may be found such that the cross terms are zero. Then in this coordinate system and when

the metric becomes

If we again consider the restriction d = 0 so that we are talking of an E-conservative system for which the principle of increasing entropy holds, then we have the variational principle given by

Solving Eqn. (3.1) for dq0 and squaring we get

or

The entropy manifold given by Eqn. (3.3) is a five-dimensional Minkowski-type manifold with coordinates of space-time-mass. We may,

1,2,3. = ; dqF - dP - Ud = Ed 2 DJJ

DD

~~~¸¹

ᬩ

§

,a

q and FS q

0

4

0

0 J{{

1,2,3. = ; )dq( + dqdq + )dq( = )(dtc 242022 DDD (3.1)

0. = )(dS 2³G (3.2)

1,2,3, = ; )dq( - dqdq - )(dtc = )dq( 242220 DDD (3.3)

1,2,3,4. = , ; dt

dq dt

dqg - c = dt

dq 20 2

EDED

DE ¸¹

ᬩ

§¸¹

ᬩ

§¸¹

ᬩ

§

93

therefore, follow the procedure Minkowski and Einstein used in the Special Theory of Relativity. First, to avoid confusion, let us rename the coordinates as

Then define the five-dimensional velocity vector as

and define the five-dimensional acceleration vector as

Now the specific entropy is the arc length and the variational principle is based upon the entropy. Therefore, if we multiply the specific entropy by the mass density, we have the entropy density. The variational problem becomes

Notice how the mass has entered our variational problem. It has entered because our metric was in terms of the "specific entropy", or entropy per unit mass. The variational problem is based upon the entropy, not the specific entropy. Thus, the mass density is required in the variational problem to correct this difference. The importance of this lies in the fact that this is the origin of the "inertia" which appears in the following equations of motion. The Euler equations for this problem are

or

Using the fact that gijuiuj = 1, the Euler equations become

.q x and ,q x ,q x ,q x ct, x 443322110 {{{{{

0,1,2,3,4 = i ; dqdx u 0

ii {

. dqdx

dqdx

jk

i +

dqxd

qu f 0

k

0

j

02

i2

0

ii

¸¹

ᬩ

§{{

GG

0. = )dq( = )dq( 20202 JGJG ³³ (3.4)

0 = uug

uuxg

- uugx

- uug

ug

dq d

jiij

jikik

jiijkji

ij

jij

0

»»»»

¼

º

««««

¬

ªww

ww

»»

¼

º

««

¬

ª JJJ

0. = uug

uuxg

- uug

ug

dq d + uug

x -

uug

ugua

jiij

jiiik

jiij

jij

0ji

ijijiij

jij40

°°¿

°°¾

½

°°¯

°°®

­ww

»»

¼

º

««

¬

ª

ww

»»

¼

º

««

¬

ª J

94

where the Fi are force densities. Obviously if we hold the mass density fixed, u4=0, then the volume integral of this equation becomes the force-mass- acceleration relationship of Special Relativity. Now Since

then

Then

where ß = v/c with v being the four-dimensional speed. The force density equation may now be written as

Consider

but GJ/Gt=a0v4, so that the force density equations may be written as

F ugua - x

= f ijij

40i

i {wwJ

J (3.5)

1,2,3,4 = ;uu - c = dt

dq and qu = f 2

0 2

0

ii D

GG DD

¸¹

ᬩ

§

1,2,3,4. = :uu = v where

dqdx

t

v-c =

dqdt

tu =

qu = F

20

i

22

0

i

0

ii

DGGJ

GG

JGG

J

DD¸¹

ᬩ

§

,

dtdx

- 1

1t

- 1c =

dtdx

v - c

1t

v - c = F

222

2222

¸¸

¹

·

¨¨

©

§

¸¸

¹

·

¨¨

©

§

D

DD

EGG

E

J

GGJ

.dt

dx - 1

1t

c = F - 1

222

¸¸

¹

·

¨¨

©

§ DD

EGGJ

E

;dt

dx - 1

1t +

dtdx

- 1

1dt

= dt

dx - 1t

222 »

»

¼

º

««

¬

ª

»»

¼

º

««

¬

ª

»»

¼

º

««

¬

ª DDD

EGG

JE

GJ

E

JGG

95

We may define

as the effective mass density or "relativistic" mass density; then

By defining F - 1c F 22 ˆDD E{ 23 so that

we see that this force density becomes Einstein's special relativistic force density when v4 = 0, or for constant "rest mass." Thus, the equations of motion, Eqn.s (3.6), reduce to Einstein's special relativistic equations of motion when dJ/dt=0. 3.2 Energy Equation Now for our system the restriction that

requires that

or if p/J2 is considered as another generalized force density, then

Thus, by integrating the expression for the system's specific energy change, we should arrive at the Einstein energy equation if we hold dx4/dt � 0. Therefore, we shall perform the integration using the force densities given by Eqn. (3.6) to get the system's energy, or

.dt

dx - 1

1cua -

dtdx

- 1t

c1 = F - 1

22

40

222

»»

¼

º

««

¬

ª

»»

¼

º

««

¬

ª DDD

EE

JGG

E

E

JJ

2

1

- 1 {

. - 1c

vva - dt

dxt

c1 = F - 1

22

401

22

EJ

GG

EDD

D»¼

º«¬

ª

E

JGG DD

D

2

401

- 1vva -

dtdx

t = F »

¼

º«¬

ªˆ (3.6)

1,2,3 = ,dxF - dP - Ud = 0 = E 2 DJJ

DD

~~~_

1,2,3, = ,dxF + dp = Ud 2 DJJ

DD

~~

1,2,3,4. = ,dxF = Ud DDD

~~

96

But c2ß2 = uĮuĮ and c2ß(d/dt) = uĮ duĮ/dt; therefore,

Now ß depends upon u and not upon x4 or J; therefore

or

If the internal energy is considered to be the system's energy when the spatial velocities uD; D = 1, 2, 3 are taken as zero, then the internal energy density given by

At the condition where u4 is also zero the internal energy density is then

By taking the constant of integration to be zero, this internal energy density then is seen to correspond to Einstein's "rest energy" where here the "rest energy" is in terms of a four-dimensional "at rest" state. If we make the usual approximation of allowing ß2<<1, then the system's energy density is approximately given by

dt. - 1dt

duu + uu

- 1

1dtd =

dx - 1uua -

dtdx

- 1dtd = dxF = U - U

22

tt

2

40

2

pp

pp0

0

00

°°¿

°°¾

½

°°¯

°°®

­

»»»»

¼

º

««««

¬

ª

¸¸

¹

·

¨¨

©

§³

°¿

°¾½

°

°®­

»»

¼

º

««

¬

ª³³

EEJ

EE

J

DD

DD

DDD

DD

~~~

.

) - (1

dc =

dt - 1

c + c - 1

1dtd = U - U

2 23

2

2

222

2

tt0

0

0

E

EEJ

E

EEE

EJ

EE³

°¿

°¾½

°

°®­

»»

¼

º

««

¬

ª

¸¸

¹

·

¨¨

©

§³

&~~

E

J2

2

0 - 1c = U - U ~~

constant. + - 1c = U

2

2

E

J~

constant. +

cu - 1

c = U4

2

¸¹

ᬩ

§

J~

constant. + c = U 2J~

97

where here u4 = dJ/dt is used. This displays the classical limit system energy density for an E-conservative system very near equilibrium. B. Systems With Non-Euclidean Manifold Suppose now we relax the assumption that the system is very near an equilibrium point so that the second partial derivatives are no longer constants but are functions. This is essentially the same transition as Einstein made going from his Special to General theory; however, the logic of the transition is much simpler here. The only change in the logic appears in the relaxation of the assumption of nearness. There is, of course, a drastic increase in mathematical difficulty since the metric com-ponents are no longer constants. 3.3 General Variational Principle We shall consider a system that must be described by both thermodynamic and mechanical variables. When written in terms of the mass density, the First Law for this system may be written as

where the tilde denotes specific quantities. Following the prescribed procedures of the Dynamic Theory we shall take the stability condition quadratic form as the metric for our system. Thus, the metric coefficients will be given by the second partial derivatives

where q4=J/a0. The metric may then be written as

where D,ß=1,2,3,4. Imposing the restriction that the system be E-conservative, SE=0, results in the principle of increasing entropy, so that

In terms of the specific entropy the variational principle may be written as

,)()a(2

1 + v21 + c = U 2

20

22 JJ

JJ &~

1,2,3, = ,dqF - dP - Ud = E 2 DJJ

DD

~~~_

0,1,2,3,4, =j i, ,qq

U = h ji

2

ijww

w

,dqdqh + dqdqh2 + )dq(h = )(dtc 00

2000

22 EDDE

DD (3.3A)

0. = )(dS 2³G

98

Solving the metric given by Eqn. (3.3A) and squaring yields the expression

with

This metric in a five-dimensional manifold of space-time-mass may be rewritten as

where

and

with x0=ct, x1=q1, x2=q2, x3=q3, and x4=J/a0. Thus we may write

Having established the metrics in Eqn. (3.8) in the manner prescribed by the Dynamic Theory, the geometry must be Weyl geometry; wherein the potential five-vector is defined as

and the field tensor is given by

0. = )dq( = )dq( 2020 JGJG ³³ (3.4)

1,2,3,4, = , },dqdqh - Adtdqh2 + )(dtc{h1 = )dq( 0

22

00

20 EDEDDE

DD»

¼

º«¬

ª (3.7)

.)h(

)qh( + qq

hh -

hc + q

hh = A 2

00

20

0000

2

00

0 &&&&

KKQKKQKK _

,)(dh1 = )dq( 2

00

20 V»¼

º«¬

ª

0,1,2,3,4, =j i, ,dxdxg )dq( jiij

20 ˆ{

0,1,2,3,4, =j i, ,dxdxq )(d jiij

2 c{V

.dxdxgf1 = )(d

f1 = dxdxq = )dq( ji

ij2ji

ij20 c¸

¹

ᬩ

§¸¹

ᬩ

§Vˆ (3.8)

x

lnf+ i

21

i ww

{ _I (3.9)

. - F ij,ji,ij II{ (3.10)

99

3.4 Gauge Function Field Equations

In order to isolate the field equations resulting from a gauge function

from the field equations produced by a vector curvature, let us consider a

local Euclidean manifold for (dV)2.

Now the field tensor given by Eqn. (3.10) has 25 components. We

would like to determine the field equations for these components. The

quickest, though not the only, way is to consider the five dimensions to be

x0=ict, xD=x

D, D=1,2,3,4. The field tensor is then defined to be

Using Bianchi's identities

and the various combinations of the indices 0, 1, 2, 3, 4 we obtain the field

equations

The definition of the five-vector current density

yields the equations

.

0V-V-V-iV-

V0B-BiE-

VB0B-iE-

VB-B0iE-

iViEiEiE0

= F

3214

3123

2132

1231

4321

ij

0 = xF +

xF +

xF

jki

ijk

kij

ww

ww

ww

0. = Ea +

tV

c1 + V

0 = Ba + Vx

0 = tB

c1 + Ex

0 = B

04

0

J

J

ww

ww

'

ww

'

ww

'

x'

(3.11)

Jc4

xF

iiij S{

ww (3.12)

.cJ4- =

tV

c1 + V

cJ4 = V

a + tE

c1 - Bx

4 = Va + E

44

0

40

S

SJ

SUJ

ww

x'

ww

ww

'

ww

x'

(3.13)

100

In addition to these field equations there is the statement of conservation of charge where

so that

For ease in future reference to these eight field equations they may be rewritten as

3.5 Energy-Momentum Tensor If we follow the approach of relativistic electrodynamics, we may define the tensor {T} in terms of the field tensor {F} according to

Using the field tensor to calculate the components of the energy-momentum tensor we find that the components are given by

0,1,2,3,4, = i 0, = xJ

ii

ww

0. = Ja + J + t

40 J

Uww

x'ww (3.14)

[h] cJ4- =

tV

c1 + V

[g] Ea =

tV

c1 + V

[f] 0 = Ba + Vx

[e] 0 = Ja + J + t

[d] 4 = Va + E

[c] c

J4 = Va +

tE

c1 - Bx

[b] 0 = Ex + tB

c1

[a] 0 = B

44

00

0

40

40

0

SJ

J

JU

SUJ

SJ

ww

x'

ww

ww

'

ww

'

ww

x'ww

ww

x'

ww

ww

'

'ww

x'

(3.15)

»¼º

«¬ª

¸¹·

¨©§{ FF4

1 + FF 41 T ststjk

kjjk G

Sλ

λ

101

and

where

Equations (3.15) form a set of eight Maxwell-type equations which obviously reduce to Maxwell's four equations. The wave equations for the new field quantities may be derived using standard assumptions.

and

Therefore,

For the vector field we have:

and

,3 2, 1, = ] ,VV + )B x E[( 4-1 = T 40 DS DDD

] ,V + V + B + E[

81 = T 22

422

00 S

] ,V E[

4i = T 04 xS

,3 2, 1, = ] ,)B x V( + EV[

41 = T 44 DS DDD

] ,V - E - B + V[

81 = T 2222

444 S

,]}V - V + B + E[ 21 - VV - BB + EE{

41 = T 22

422GDE

S EDEDEDDE

3. 2, 1, = ,ED

tV

c1 +

dtV =

dtJ

c4 =

tV

c1 + )V(

t 24

24

24

2

www

x'w

wwx'

ww S

.V + dtV

c1 = E a - =

tV

c1 + )V( 04 'x'

wx'

ww

x'ww

x''x'J

. E a - tJ

c4 =

tV

c1 - V 0

422

42

242

JS

ww

x'ww

ww

'

)J c4( - =

tV

c + )V( 4

4 S'

ww'

x''

; J c

4 = V tc1 + V + )Vxx( 44

2 ''ww

'''S

102

therefore

ButJ

SUww

x' V a - 4 = E 40 70, so that

andJ

Sww

ww

'V a - J

c4

tE

c1 - Bx 0 72 , so that

Now the wave equations for the usual vector and scalar potentials are

and

We may differentiate these with respect to the mass density and substitute them into our wave equations and get

and

where

The field energy density may be defined by

. )B x ( a + E ca + J

c4 =

tV

c1 - V 0

t

20

42

2

22

JJS

ww

'ww

w'ww

'

t

V a - 4 a - tJ

c4 =

tV

c1 - V 4

004

224

2

242

ww

ww

ww

ww

' SUJ

S

. V a - tE 2 + J

c4 a + J

c4 =

tV

c1 - V 0042

2

22

JS

JS

ww

ww

ww

'ww

'

J c

4 - = tA

c1 - A 2

2

22 S

ww

'

. 4- = t

c1 - 2

2

22 SU

II

ww

'

J

S24

220

4242

2

242 V a +

tJ

c4 = V

t

c1 - V

ww

ww

ww

'

, V a - tE2 a + J

c4 =

tV

c1 - V 0042

2

22

»»¼

º

««¬

ª

ww

ww

ww

'ww

'JJ

S

. A a - V = V and a + V V 0044 JJI

ww

ww

{

] , V + V V + B B + E E [ 81 2

4xxx{S

[

103

and the electrical Poynting vector may be defined by

Now the electrical Poynting vector represents the outward flow of the electromagnetic field energy through a surface. If we take the total vector, whose components are T0D, to be the total flow of energy, then the vector

with components VV4c

4 DS81must be the outward flow of energy due to changes of the

mass density within the surface. Let us designate the mass energy vector as

so that the total energy vector is

whose components are

The Dynamic stress tensor may be defined as the three-dimensional tensor whose elements are

The Maxwell stress tensor is defined in electrodynamics as the three-dimensional tensor with elements

In terms of the Maxwell stress tensor, the Dynamic stress tensor may be written as

Then in terms of the above defined quantities

. )B x E( 4c S E S

{

,)VV( 4c S 4m S

{

S + S = S mE

. T )ic( - =] VV + )B x E( [

4c = S 04 DDDD S

]}V2 - [ 21 - VV - BB + EE{

41 = T 2D [GS DEEDEDEDDE

]}B + E[ 21 - BB + EE{

41 = T 22M GS DEEDEDDE

. ]}V - V[ 21 - VV{ - T = T 2

42MD G DEEDDEDE

104

Suppose we calculate the trace of the energy-momentum tensor:

3.6 Force Density Vector. The force density vector may be defined in terms of the divergence of the energy-momentum tensor. Therefore, suppose we calculate the five-dimensional divergence of the tensor {T}, or

Because of the antisymmetry of Fjk, the first term may be written as

By interchanging the indices k and l

Using the Bianchi identity

the terms contained within the parentheses may be written as

.

]V - E - B + V[ 81)]BxV( + EV[ )VE(

4i

)]BxV( + EV 4i}T{S

ci -

)VE( 4iS

c1 -

= {T}

22224

4

4

D

¸¸¸¸¸¸¸¸¸

¹

·

¨¨¨¨¨¨¨¨¨

©

§

x

x

SS

S

S[

.] V - E - V + B[ 81 =

]V21 - V 2

1 - B 21[

41 =

)]V - V + B + E(23 - V - B + E[

41 +] V + B[

41 =

T +] V + B[ 41 =

]V - E - B + V[ 81 + + T = T = {T}t

24

222

24

22

224

2222224

2

424

2

22224

DjjjjR

S

S

SS

S

S[

DD

.] FF 41 + FF[

x

41 =

xT

ststjkkjkkjk

GS λλw

www

. FxF = F

xF

kkj

kkj

λλ

λλ

ww

ww

. F x

F + xF

21 = F

xF = F

xF

kj

kj

kkj

kkj

λλ

λλλλλ

λ

¸¹

ᬩ

§ww

ww

ww

ww

,0 = x

F + xF +

xF

jkkj

kj

ww

ww

ww λ

λ

λ

105

Substituting this back into the expression for the divergence, the last term will be canceled because l, k, s, and t are dummy indices. Then the divergence becomes

By interchanging the indices k and l on the right-hand side we obtain

The Dynamic force density five-vector may now be defined as

Therefore, the components of K are given by

But the five-vector current density is given by

The components of the five-vector force density become

Now, since Jk = (icU, J, J4), then

and

where J = Uu. These then are the components of the force density five-vector resulting from a gauge field in the Dynamic Theory. These components reduce to the four components of the Lorentz force density should V4 = V = 0.

. x

)FF(41 - = F

xF

21 = F

xF

ikk

kjk

kkj

ww

ww

ww λλ

λλ

λλ

. x

FF41 =

xT

kk

jkjk

ww

ww λ

λS

. x

F F41 =

xT k

jkk

jkλ

λ

ww

ww

S (3.16)

. {T} Div K 5{

. x

FF41 = K k

jkj λ

λ

ww

S

. Jc-4 =

xF

kk Sλ

λ

ww

. FJc1- = Jc

4 F41- = K kjkkkjj ¸

¹·

¨©§ S

S

,V

cJ +)] B x u(

c1 + E[ = K

] ,VJ + E J[ ci = K

4

440

U

x (3.17)

,c

V J - V = K 44x

U

106

With the interpretation that the four force density components with subscript 1 through 4 are the force density vectors which appear in the First Law as FD, then the force density vector provides the connection between the First Law and the geometry of the sigma manifold discussed in section 2.9. Thus, the existence of the vector field Mi is also demanded by the Dynamic Theory and need not exist as a separate assumption. 3.7 Equation of Energy Flow. Consider the zeroth component of the Dynamic force density five-vector

Then

or

or, since x4 = J/a0,

Rearranging the terms

and separating out the electrical Poynting vector leads to

This then is the five-dimensional energy flow equation. 3.8 Momentum Conservation The expression for the conservation of momentum may be obtained from the space portion of the force density five-vector

. x

T + x

T + x

T = x

T = K 4040

000

k0k

0 ww

ww

ww

ww

DD

x

)VE( 4i +

xS

ci -

(ict) =] VJ + EJ[

ci

444 wxw

ww

ww

xS

[DD

,x

)VE(41 -

xS

c1 +

tc1 =] VJ + EJ[

c1

444 wxw

ww

ww

xS

[DD

. )VE(4

ca - xS +

t = VJ - EJ- 0

44 JS[

DD

wxw

ww

ww

x

JS

[wxw

xww )VE(

4ca + VJ - EJ - =

t + S div 0

44

. )VE(4

ca + S div - VJ - EJ - = t

+ S div 0m44E JS

[wxw

xww

107

Butx

TE

DE

ww 110 is the three-dimensional divergence of the Dynamic stress tensor {TD},

therefore,

If we consider a volume in which all the material is contained and outside of which the field vanishes, then integrating over this volume yields

The integral of K gives the total force (i.e., the time derivative of the mechanical momentum p less the vector ¸

¹·¨

©§ EJ D 2 - 1/v& 113. Now define the vector

Then define

so that

Using the divergence theorem the volume integral may be converted to a surface integral so that

If the field vanishes outside of V, it must do so also on the boundary surface s, hence

Therefore, it is not the mechanical momentum p but the quantity p + G which is conserved. Therefore, we must interpret G as the momentum of the field and

. 3 2, 1, = , ,x

T + x

T + x

T = x

T = K 44

00

kk EDD

E

DEDD

ww

ww

ww

ww

.)] B x V( + EV[ 4a + }T{ div +

tS

c1 - = K 4

0D2 JS w

www

. dv }T{ div = dv )]}BxV( + EV[ 4a -

tS

c1 + K{ D

v4

02

v³³ w

www

JS

. dt } - 1v + )B x V( + EV[{

4a -

cS g

20

02

E

JJS

D&

ww

³{

,G d g v

{³ V

. dv }T{ div = )G + p( dtd D

v³

. da n }T{ = )G + p( dtd D

s

x³

. 0 = )G + p( dtd

108

as the momentum density of the field. 3.9 Gauge Field Pressure The Dynamic stress tensor is given by

Now separate the three-dimensional dynamic stress tensor into a traceless and an isotropic tensor.

where

and

Now

and

> @ dt - 1

v4 + )BxV( + EV 4a -

cS = g

2

02 ¸

¸

¹

·

¨¨

©

§x

ww

³E

JSJS

&

. V8

3 -] V - B + E[81 - =

]}V - V + B + E21 - VV - BB + EE{

41 = T

24

222

224

22D

SS

GS DEEDEDEDDE

W

GS

GSS

GSS

JS

DEDE

DE

DEEDEDED

DEEDEDED

DEEDEDEDDE

c{ + t

]V - V3 + B + E[ )81)(

31( -

]V - B + E[)81)(

32( -] VV - BB + EE[

41 =

]}V - V + B + E[ 81 - VV - BB + EE{

41 =

}]V - V + B + E[ 21 - VV - BB + EE{

41 = T

224

22

222

224

22

224

22D

]V - B + E[)31( - VV + BB + EE{

41 t 222G

SS DEEDEDEDDE {

.] V - V3 + B + E[ )24

1( - 224

22GS

W DEDE {c

0 )] V - B + E( - V - B + E[ )41( = }t{t 222222

r {SDE

.] V - V3 + B + E[ )81( - = }{t 22

422

r SW DEc

109

Consider the definition

Then

and

The isotropic part of the stress tensor is usually called the "pressure." Therefore, define 3p = t in accordance with customary notation, so that

With the exception of the factor of 3 this reduces to the "radiation pressure" for an electromagnetic field when V = V0 =0. Note that this pressure may be zero since it is the sum and difference of squares, or p = 0, when

This may prove to be an important point when considering boundary conditions in cosmology or the study of elementary particles.

.] V - V3 + B + E[ )24

1( - = t 31 22

422G

SWG DEDE c{

]V - V3 + B + E[ )81( - = t 22

422

S

¸¸¸

¹

·

¨¨¨

©

§

c

t00

0t0

00t

= W DE

.] V - V3 + B + E[ )24

1( - = p 224

22

S

. V3 + B + E = V 24

222

110

Chapter 4. QUANTIZATION IN FIVE DIMENSIONS

The preceding development provides a tremendous wealth of mathe-matical abstractions. However, there seems within it no readily apparent method of interpreting the new fields. If there appears to be no physical entity which may be associated with the new field quantities, then the development will have gone for naught. On the other hand, with the notion of nuclear fields in mind it seems that if the new field quantities are included in a quantized picture, then perhaps the relation to nuclear fields may be made. In the following the requirement for quantization is provided by appropriate restrictions upon a system whose description is taken from the Dynamic Theory. However, the use of the five-dimensional Dirac equation has not yet been shown to result from the Dynamic Theory. Schrodinger's quantum mechanics may be obtained using London's work, but I am not aware of a procedure to arrive logically at Dirac's equation even though I feel that the method exists. As it now stands, the use of the generalized Dirac equation must be accepted as an independent fundamental as-sumption. 4.1 Quantization. The system under consideration now is a five-dimensional system with arc element

Now since our system is an E-conservative, dE = 0, system the principle of increasing entropy requires that (dq0)2 > 0 so that f(dV)2 t 0. Introducing the quantization conditions results in

where

If we restrict ourselves to a (dV)2 space which is the local Euclidean space, then (dV)2 is the five-dimensional Minkowski-type manifold; using London's work we would produce a five-dimensional quantum dynamical system.

. )f(d = )dq( 220 V

,4 3, 2, 1, 0, =j ,in2 = dx jj SI³

. a

x ,x ,q x ,q x ,ct x and x

nf 0

4322110j

1/2

jJ

I {{{{w

wr{

"

111

4.2 Five-Dimensional Hamiltonian. We previously showed that the principle of increasing entropy resulted in

as the variational principle for a local Euclidean manifold. Since multiplication by a constant does not change the problem we may take our variational problem to be

Defining the velocity vector as uj=dxj/dq0 and the momentum as pj=wL/wuj=Jgjkuk, where we have used the fact that gjkujuk = 1, then we may form the contravariant momentum as

so that

since Jc2 = Jgjkujuk. Equation (4.1) is the five-dimensional "momentum-energy" equation. We may now follow London's procedure to obtain our wave function for the five-dimensional system. However, a quicker way to investigate the effect of the Dynamic Theory upon quantum mechanics would seem to be that of adopting Dirac's equation in a five-dimensional form and following a development analogous to standard four-dimensional relativistic quantum mechanics. With this in mind, then we shall adopt the form

to be the five-dimensional specific Hamiltonian operator. The partial derivative operators are specific operators and hence are dimensionless in natural units. In Eqn. (4.1) the D's and E do not involve derivatives and must be Hermitian in order that h be Hermitian. By taking the four partial derivatives in Eqn. (4.1) as the four-vector specific momentum operator we may write

0 = )dq( 20JG ³

. 0 = )dq(c 202JG ³

,u g g = pg = p kjk

kjk

j"

"J

,c =

)uu g ( = u g u = )u g g( )ug ( = pp

22

kjik

kjkj

2k

jkjj

jj

J

JJ

GJJJ ""

"""

(4.1)

EDDDD~~~~~ -

x +

x +

x +

x i = h 44332211 ¸

¹

ᬩ

§ww

ww

ww

ww (4.2)

. ) + p ( - = h ED~~ x (4.3)

112

4.3 Five-Dimensional Dirac Equation. If we take |>h = |>p0 10 and require that the D's and E are chosen such that solutions of this equation are also solutions of Eqn. (4.3), we find the restrictions imposed upon the choice of the Į's and E to be:

and

where natural units, c = 1, are used. A set of 8 x 8 matrices satisfying the requirements of Eqn. (4.4) is

where

and

Then the five-dimensional Dirac equation may be taken to be

where the' 17is a four-dimensional operator. By defining

then Eqn. (4.6) may be written as

By virtue of the properties of the s’D~ 20 and E~ 21 plus the fact that

,1 =

,p = )p (2

2

E

D~

~ x

,0 = + DEED ~~~~ (4.4)

,A0

0A = ,3, 2, 1, = i

0

0 = ,

0

0 = 4

i

ii ¸

¹

ᬩ

§¸¸¹

·¨¨©

§¸¸¹

·¨¨©

§D

D

DD

E

EE ~~~ (4.5)

¸¹

ᬩ

§¸¸¹

·¨¨©

§¸¹

ᬩ

§

OI

I0 = A and ,3 2, 1, = i

0

0 = ,

I-0

OI =

i

ii

V

VDE

. 1-0

01 = and ,

0i

i-0 = ,

01

10 = ,

10

01 = I 321 ¸

¹

ᬩ

§¸¹

ᬩ

§¸¹

ᬩ

§¸¹

ᬩ

§VVV

,(x) ) - (i = (x) t

i <'x<ww ED

~~ (4.6)

,4) 3, 2, 1, = ( - ; 0 PDEJEJ PP ~~~{{

. 0 = (x) 1) + (i jj <w J

k j for 0

,4 3, 2, 1, = k =j for 1-0 = k =j for 1

{ = g jk

z

113

the anticommutator of the J-matrices must satisfy

4.4 "Lorentz" Covariance. Under a five-dimensional Lorentz transformation

we shall suppose each component of the wave function <(x) transforms into a linear combination of all four components:

where S is a Dirac spinor satisfying

By using an infinitesimal Lorentz transformation given by

whereH jk 28 are a set of 16 numbers, then S (T) may be shown to be given by

where the matrix T is given by

Equations (4.7), (4.8), and (4.9) suffice to guarantee the Lorentz covariance of the five-dimensional Dirac equation. Following standard quantum mechanical procedure we shall adopt the probability current density to be

with the requirements that: wk ji = 0 , jktransforms as a contravariant vector, and jk must be real.

. g2 = },{ jiij JJ

x L = x kkj

jc

,(x) S= )x( TL (x) <c<c<&

. L = S S kjk

j-1 JJ

(4.7)

HT jk

jk

jk d + g = L (4.8)

,)d (T exp = )( S0

TTT

³ (4.9)

. 41 = T kj

jk JJH

(x) (x) = (x) j kk \J\

114

4.5 Spin. In the three-dimensional space the angular momentum is given by the vector L as the cross product of the coordinates and momenta. We shall then define the angular four-momentum to be the four-dimensional cross product

where x4 is the mass density and

Hijk = ^ 0 if any two indices are alike,

1 for even permutation to align indices in ascending order,

-1 for odd permutation to align indices in ascending order.

Then the commutator of the components of the angular four-momentum with the specific Hamiltonian is not zero; for instance

Now suppose there exists a four-spin vector S such that the sum of the angular four-momentum and the four-spin vector commutes with the specific Hamiltonian; then if we define a new three-spin vector u given by the components ui = iJ4J1, u2=iJ4J2/2, and u3=iJ4J3/2, and take the usual spin vector s, given by s1=iJ2J3/2, s2=iJ1J3/2, and s3=iJ1J2/2, the components of the four-spin vector may be shown to be

and

In analogy with standard relativistic quantum mechanics the

eigenvalues of the four-spin components can be shown to be 43 + _ 36. It

may also be shown that the set of observables P, h, and S where P is the four-momentum and S is the four-spin, form a complete set of commuting observables.

px L kjijkH{

. pi - pi + pi - pi + pi - pi =h] ,L[ 4202404101401202103 JJJJJJJJJJJJ

,u + u + s = S ,u - u + s = S ,u - u - s = S

2133

3122

3211

. s + s - s = S 3214

115

4.6. Dirac Equation with Fields. In analogy with relativistic quantum mechanics we take the five-dimensional Dirac equation to be

where Mj is five-vector potential. By operating on the left with 1] - ) - [(i i

jj JIw 38 and separating JJ kj 39 into symmetric and antisymmetric parts as

then Eqn. (4.10) becomes

Separating I kjw 42 into symmetric and antisymmetric parts as

and defining the field tensor as

Eqn. (4.11) becomes

Now since

,

0n-n-n-x-

n0s2is2i-x-

ns2i-0s2ix-

ns2is2i-0x-

xxxx0

=

3214

3323

2312

1211

4321

jk

�

�

�

�

����

V

where

,0 = 1] + ) - [(i jjj <w JI (4.10)

, + g ] ,[ 21 + },{

21 = jkikkjkiki VJJJJJJ {

0. = ] )i-i-+(-+1-)-)(i-[(i jkkjkjkjkj

jjjj <wwwwww VIIIIII (4.11)

) - ( 21 + )(

21 = jkkjkjkj IIII wwww

, - = F jkkjjk II ww

. 0 = ] iF 21 - 1 - ) - )(i - [(i jk

jkkk

jj <ww VII (3.28)

and 3, 2, 1, = n 4 jj {V

116

0V-V-V-V-

V0BB-E

VB-0BE-

VBB0E-

VEEE0

= F

3210

3123

2132

1231

0321

jk

plus recalling the seven Maxwell-type equations from Eqn. (3.15)

,Jc4 - =

tV + V 0, = E

a + tV

c1 + V 4

404

SJ w

wx'

ww

ww

'

0, = Ba + V x ,V

a - cJ4 =

tE

c1 - B x 00 JJ

Sww

'ww

ww

'

then Eqn. (4.12) may be written as

and thus becomes the Dirac equation with fields E, B, V4, and V. Suppose we consider a system without an electric charge so that p = J = 0, then by Eqn. (4.13) we still have

and, therefore, there will still be a magnetic moment. 4.7. Allowed Fundamental Spin States. In the five-dimensional quantization of the space-time-mass manifold three spin vectors appear. One of these is the familiar three-component spin vector of relativistic quantum mechanics. The second of the three is a new three-component spin vector while the remaining one is a four-component spin vector. Using the theorem: If D satisfies D2 = a2 where a is a number, then the eigenvalues

of D are ra. Then it is not difficult to show that the component eigenvalues are

,V a - 4 = E 0, = tB

c1 + E x 0, = B 4

0 JSU

ww

x'ww

'x'

(4.13)

0. = ] V ni - xVi - x Ei - s B2 + 1 - ) - )(i - [(i 44

kkjj <xxxww �II

JJ w

www

'ww

x'V

a - = tE

c1 - B x and Va - = E 0

40

. 4 3, 2, 1, =j and 3 2, 1, = ,43 = S ,

21 + = u ,2

1 + = s 2j DDD __

117

If, in analogy with the eigenvalues for the total angular momentum, we write

then the possible eigenvalues becomes

However, the following relations, which were shown to be required for S to commute with the specific Hamiltonian, restrict the number of possible combinations of these eigenvalues.

and

The question to be asked now is, how many combinations of the above eigenvalues are allowed? For S1 = 1/2 the combination s1 = -1/2 and u2 = 1/2 is impossible. For S1 = 1/2 the combination s2 = -1/2 and u1 = -1/2 is impossible. For S3 = 1/2 the combination s3 = -1/2 and u1 = -1/2 is impossible. For S4 = 1/2 the combination s1 = -1/2 and s2 = 1/2 is impossible. For S1 = -3/2 only one combination is possible: s1 = -1/2, u2 = 1/2,

and u3 = 1/2. For S2 = -3/2 only one combination is possible: s2 = -1/2, u1 = -1/2,

and u3 = 1/2. For S3 = -3/2 only one combination is possible: s3 = -1/2, u1 = -1/2,

and u2 = -1/2. For S4 = -3/2 only one combination is possible: s1 = -1/2, s2 = 1/2,

and s3 = -1/2. Now because S4 is a combination of the first terms of each of the components S1, S2, S3, not all of the above listed 16 combinations are possible. For S4 = 1/2 the following combinations of (s1, s2, s3, u1, u2, u3) are possible.

,1) + S(S = 43 = S jj

2j

. 23- ,

21 = S ,

21 + = u ,2

1 + = s j__ DD

,u + u + s = S ,u - u + s = S ,u - u - s = S

2133

3122

3211

.s + s - s = S 3214

118

(1) (�, �, � ; - �, �, - � for S1 = S2 = S3 = � (2) (�, �, � ; �, - �, �) for S1 = S2 = S3 = � (3) (�, - �, - � ; �, �, - �) for S1 = S2 = S3 = � (4) (- �, - �, � ; �, - �, - �) for S1 = S2 = S3 = � The remaining combinations are: (5) (- �, �, - � ; - �, - �, - �) for S4 = S3 - E; S1 = S2 = � (6) (-�, �, - � ; �, �, �) for S4 = S1 - E; S2 = S3 = � (7) (�, - �, - � ; - � - �, �) for S2 = S3 = - E; S1 = S4 = � (8) (- �, - �, � ; - �, �, �) for S1 = S3 = - E; S2 = S4 = �. Thus there is an octet of possible combinations. There are also some obvious symmetries in these combinations. An aid in seeing these symmetries is the vector defined as t where

Then for each of the eight combinations above we find (t1, t2, t3) given by (1) t= (0,0,0) (5) t= (1, 0, -1) (2) t=(0,0,0) (6) t= (-1, 0, 1) (3) t= (0,1,1) (7) t= (0, -1, -1) (4) t= (1,1,0) (8) t= (-1, -1, 0) Thus, the eight combinations correspond to four distinct t vectors which carry a r sign. Or t1=(0, 0,0) ;t2 = (0, 1, 1) ; t3 = (1, 1, 0) ; t4 = (1, 0, -1) For + tD we have: t1 o (s;u) = (�, �, � ; - �, �, - �) t2 o(s;u) = (�, - �, - � ; �, �, - �) t3 o (s;u) = (- �, - �, � ; �, - �, - �) t4 o (s;u) = (- �, �, - � ; - �, - �, - �) . For -tD we have: -t1 o (s;u) = (s,u) = (�, �, � ; �, - �, �) -t2 o (s;u) = (s,u) = (�, - �, - � ; - �, - �, �) -t3 o (s;u) = (s,u) = (- �, - �, � ; - �, �, �) -t4 o (s;a) = (s,u) = (- �, �, - � ; �, �, �) Now by defining the vectors: a = (�, �, �) ; b= (- �, �, - �) c = (�, - �, - �) ; d =(�, �, - �)

u + u t ; u - u t ; )u + u(- t 123312321 {{{

119

We may write t1 o (s;u) = (a;b) -t1 o (s;u) = (a;-b) t2 o (s;u) = (c;d) -t2 o (s;u) = (c;-d) t3 o (s;u) = (-d;c) -t3 o (s;u) = (-d;c) t4 o (s;u) = (b;-a) -t4 o (s;u) = (b;a) The octet is then made up of the combinations: (a;rb); (c;rd); (b;ra); (-d;rc) . The appearance of octets for basic quantum numbers is reminiscent of elementary particle theory. Thus, the Dynamic Theory seems to give promise to the hope of tying elementary particles to fundamental principles in a new way. B. Quantized Fields Much difficulty was encountered in trying to find a solution to the wave equations. This stimulated a return to thoughts of fundamental particles. The motivation for this change was primarily the feeling that it would be more productive to get away from the wave solutions for a while, but also there was the haunting feeling, retained for some five years, that the new fields played a role in particle structure. This feeling was based primarily on the role the new fields appear to play in the five-dimensional quantization and their role in the self-energy of charged particles. 4.8. Quantum Condition Applied to Particles. The quantum condition

was required when generalized isentropic states were considered. Given the thermodynamic basis for the three fundamental laws, it seems natural to think that if the Dynamic Theory were to say anything about fundamental particles then it should probably come from considering generalized isentropic states. Thus, the quantum condition, Eqn. (4.14), should play a crucial role. This also was the condition from which London began his work, which showed that this condition produces quantum mechanics, but quantum mechanics describes interactions between particles such as electrons and nuclei. It does not specify what types of particles are allowed.

n, , 1, 0, =j iN,2 = dx jj xxx³ SI (4.14)

120

That the three adopted laws must apply to individual fundamental particles is tantamount to the notion that these three laws must specify what particles are allowed and, thereby, must specify their allowed fields. If we again look at the quantum condition, Eqn. (4.14), we see that it is given as a line integral that must have a quantized value. Our usual first encounter with a line integral involves the evaluation of a given line integral when the path is specified. Because the quantum condition represents a line integral that may only have certain values, London asked a legitimate question when he asked what paths would be allowed given the electrostatic potential in advance8. This points out that there are three parts to any line integral, the integrand, the path, and the integral value. Another question that may be asked of the quantum condition is given that the integral value may only be 2SiN, what are the possible M allowed for a particle that must retain its identity along any path? This is equivalent to asking what fields are fundamental particles allowed to have if we are free to move them anywhere in the manifold? To be more specific, we are asking what T are allowed by the quantum condition if the dxj are to be independent? If the dxj are to be independent, then we may choose all dxj to be zero except dxk. Then the quantum condition requires

Equation (4.15) must be true for all k, and because we are free to set the path, then the Mk must reflect the quantization represented by the integer N. Therefore,

where the may not be quantized. Thus, Eqn. (4.16) represents the first response of the quantum condition to the question concerning what Mj are allowed for fundamental particles; the gauge potentials must be quantized. The definition of the gauge potentials is

where f was the gauge function. The field tensor was defined by

where covariant differentiation is required. There are restrictions placed upon these fields, for they must obey the set of eight differential equations given by Eqn. (3.15).

k). on sum(no iN2 = dxkk SI³ (4.15)

, sum)(no N = jjj II ~ (4.16)

,x

lnf = j

21

j ww

I (4.17)

, - = F jk,kj,jk II (4.18)

121

4.9. Radial Field Dependence. Any potential II ~

jjj N = 62 allowed by the quantum condition must also satisfy

Eqn.s. (3.15). Even so, Eqn.s. (3.15), (4.17) and (4.18) represent three stages of differentiation, starting with the gauge function, f. In looking for the restrictions Eqn.s. (3.15) place upon the quantized potentials, we may employ a technique of mathematics in the solution of differential equations. We try to find a solution in the form of the product of functions of the separate variables. However, our trial solution must produce potentials of the form in Eqn. (4.16). Therefore, suppose we try to find a solution of the form

where

with ft = function of time only, fr = function of spherical radius only, etc., and the function G is defined by the system of partial differential equations,

The definition of the gauge potential, using the trial solution of Eqn. (4.19), now produces

but by the defining relations for G this becomes

If we define

then we have the proper form

We may now use our trial solution to write the potentials in spherical coordinates:

FG = lnf 21

(4.19)

,fffff = F rt JIT

. sum)(no xF

FG) - N(

= xG

jj

j ww

ww

,xG F +

xFG =

x(FG) =

xlnf = jjjj

21

j ww

ww

ww

ww

I

sum).(no xF N =

xF G) - N( +

xFG = jjjjjj w

www

ww

I

,xF = jj ww

I~

sum).(no N = jjj II ~ (4.20)

122

,FNa = fffff Na = 21lnf

a =

,F rsinN = fffff

rsinN = lnf

rsin1 =

,F rN = fffff

rN = 2

1lnf

r1 =

,FN = fffffN = r

21lnf

=

440rt4004

33

rt32

1

3

22

rt2

2

11rt11

JIT

JIT

JIT

JIT

II

TTITI

TI

I

c

c

c

c

w

w

¸¹·

¨©§

¸¹·

¨©§

ww

¸¹·

¨©§

¸¹·

¨©§

¸¹·

¨©§

w

w

w

w

where the notation f t c 72 denotes dft/dt and Fj denoted

,ddF = F

,ddF = F

,drdF = F

,dtdF = F

3

2

1

0

I

T

and

. ddF = F 4 J

Substituting the potentials given by Eqn. (4.21) into the definition of the field tensor and using the required covariant differentiation, we obtain the field components

,F icN = ffff f

icN =

(ilc)lnf = 0

0rt

021

0 ¸¹·

¨©§

¸¹·

¨©§

ww

JITI

(4.21)

123

,Fa)N - N( = V ,Fa )N - N( = V

,rFN - F

r)N - N( = B

,rFN - F

rsin)N - N( = B

,F cotN - rF

rsin)N - N( = B

,c

F )N - N( a = V

,Fcrsin)N - N( = E

,Fcr)N - N( = E

,Fc)N - N( = E

24042

14041r

2212

21

3313

13

332332

r

040404

0303

0202

0101

r

T

I

T

I

T

T

TT

T

»¼º

«¬ª

»¼º

«¬ª

»¼º

«¬ª (4.22)

and Fa )N - N( = V 34043I

These field components reflect the quantization of the potentials. However, the quantization of the fields is not a simple quantization because each component depends upon the difference of quantum numbers. The field components given by Eqn. (4.22) must satisfy the differen-tial equations of Eqn. (3.15). Therefore, if we substitute Eqn. (4.22) into Eqn. (3.15), we will obtain the restrictions upon the quantum numbers, Nj, and the functions ft, fr, fT, fM, and fJ required for these fields to be the fields of a fundamental particle. We begin with the equation

This equation becomes

in spherical coordinates. Substituting from Eqn. (4.22) and simplifying, we finally arrive at

This requires that

Substituting from the definition of the Fj, we find

. 0 = B x' (3.15a)

,0 = B rsin

1 + )B (sin rsin

1 + rrB

r1 2r

2¸¹·

¨©§ w

ww

¸¹·

¨©§

ww

¸¹

ᬩ

§I

T

TTT

T

> @^ ` 0 = N)N-N( + N)N-N(F + )fr+f( rN)N-N( r

133121223313323

3Tcot

sin ¸¹

ᬩ

§4

.] N)N-N( + )N-N[(F = Fr + F( rN)N - N( 3311223313332 Tcot

124

If 0 fff t zc JI 78, this may be rewritten as

However, we may divide by frf cot and separate the equation into

where

The left-hand side of Eqn. (4.23) is a function of r only, while the right-hand side is a function of T only. Therefore, Eqn. (4.23) must be a constant. We can then write

where the constant On depends upon the set of quantum numbers, N1, N2, and N3, for the particle. Thus, On depends on the particle under consideration. The radial equation in Eqn. (4.24) may be integrated immediately with the result

The appearance of this exponential functional form for the radial dependence is surprising and, at first, pleasing. The surprise is that this functional form comes only from the gauge function, f, playing the guiding role in the gauge fields and the field equation

which is a purely classical equation. Thus, the exponential neo-coulombic radial function does not appear at first to depend upon the fifth dimensionality, but only upon the quantum condition so that even a four-dimensional approach would have produced this same radial function.

> @ > @ . N)N-N( + N)N-N( fffff = fffffr + fFfffr N)N-N( 331212rtrt2

rt332 JITJITJITT cccccot

� � > @N)N-N( + N)N-N( ff = ffr+rf N)N-N( 331212rr2r332 TTT cccot

,f

f =

f k]fr + rf[

r

r2

r

T

T

Tcotcc (4.23)

. N)N-N(

N)N-N( + N)N-N( = K332

331212

, = k = f

kf =

f]fr+rf[

n0r

r2

r OOT T

T

cotcc (4.24)

. e rk = f r

-R

n ¸¹·

¨©§

¸¹·

¨©§ O (4.25)

,0 = B x'

125

It is pleasing to see the appearance of the exponential neo-coulombic radial functional of Eqn. (4.25) because the electron catastrophe has haunted theoreticians since the inverse radial dependence of the columbic potential was first seen. The radial function in Eqn. (4.25) is well behaved everywhere; as r o f , fr o 0 and as r o 0, fr o 0. A quick glance at the function might cause one to think it is the Yukawa potential, but a closer look will show that the exponent is the inverse of the exponent in the Yukawa potential. The value for fT may be obtained by integrating the remaining portion of Eqn. (4.24). This integration produces

. )( k = f 03

OT Tsin

If the exponential neo-coulombic function corresponds to reality, then On must be small, less than 10-17m for electrons, and of the order of magnitude of 10-15m for protons. Therefore, O0 must be very small, which in turn implies fT is very close to a constant. The equations resulting from substituting the quantized potentials into the remaining non-source equations [3.15b], [3.15e], and [3.15f], produce the following restrictions

°°¯

°°®

­

o

cc

cc

cc

0 = ff N)N-N(

0 = ff N)N-N(

0 = ff N)N-N(

[3.15b]

221

313

332

JT

JI

JI

°¯

°®

­

o

cc

cc

cc

0 = ff N)N-N(

0 = ff N)N-N(

0 = ff N)N-N(

[3.15e]

t221

t313

t332

T

I

I

[3.15f] (satisfied identically). When the potentials are substituted into the equations with source terms, Eqn.s [3.15c-e] and [3.15h], the resulting equations are very complex. To reduce the complexity of the equations, the assumption was made that all source terms were zero; that is,

This assumption reduced the complexity somewhat but still left a system of equations that, thus far, is unsolved. However, an interesting aspect of this assumption is the possible existence of a radial electric field without the presence of any electric charge within, or upon, the particle. This possibility rests upon the pressure of the term Jww /V 4 86 in the Eqn. [3.15d]. Much was learned about the interaction of charged particles by considering only the radial dependence of the electric field while temporarily neglecting the magnetic field or any potential variation of the electric field with azimuthal angles. This latter is the spherically symmetric field assumption. Having not yet obtained a complete solution to the system of equations that is the result of substituting the quantized

. 0= J ;0 = J 0; = 4U

126

fields into the eight field equations, it proved beneficial to make the

assumption of spherically symmetric fields in which the only variation of

the fields is the radial dependence specified by the neo-coulombic radial

function.

Then, if we want to explore the radial dependence of static forces

between the fundamental particles allowed by the quantum condition, we

must consider the force law,

whose spatial components may be written

By restricting our concern to static forces we can concentrate on the force

density,

. VcJ + E = K 4U (4.26)

Thus, the radial dependence of the electric field, E, and the V field are all

that need to be considered at the moment. Substituting the radial

function, Eqn. (4.25), into the field expressions, Eqn. (4.22), we find

er - 1

rZk = E ,

r-n

2rn ¸¹·

¨©§

¸¹·

¨©§¸¹

ᬩ

§ OO (4.27)

and

er-1

rWg = V ,

r-n

2rn ¸¹·

¨©§

¸¹·

¨©§¸¹

ᬩ

§ OO

where Z = (N1-N0) and W = (N1-N4) so that the quantum number Z appears

in the radial electric field the same as it does classically.

From Eqn. (4.27), the electric field of fundamental particles allowed

by the Dynamic Theory is quantized by the quantum condition, and the

quantum steps may only be integer steps. This would necessarily preclude

any particles with fractional charge steps.

Substituting the radial fields of Eqn. (4.27) into the force law, Eqn.

(4.26), and integrating the charge density over the physical extent of the

particle, we find the radial force between two particles is

er - 1

rWgg

+ er - 1

rZkq

= F ,r

-n

2

2r

-n2

1 nn ¸¹·

¨©§

¸¹·

¨©§

¸¹·

¨©§¸¹

ᬩ

§¸¹·

¨©§¸¹

ᬩ

§ OO OO (4.28)

where

. d(Vol) cJ = g ,d(Vol) = q 4

21 ¸¹·

¨©§³³ U

If we consider the electric force in Eqn. (4.28) and restrict our attention to r

such that r >> On, then the electric force becomes the columbic force

,JF c1 = K kjkj ¸¹·

¨©§

. VcJ + )B x J(

c1 + E = K 4 ¸

¹·

¨©§U

127

Further, the other force term, based upon the V field, may be seen to also have the same long-range form,

Thus, this force is also an inverse square long range force. Therefore, the nonelectric force in Eqn. (4.28) cannot be a nuclear force. What then must be the interpretation of this force, or must its appearance be interpreted to mean that nature cannot have a five-dimensional character? The only force known in nature that has a long-range character, in addition to the electrostatic force, is the gravitational force. But how can we interpret the V field as the gravitational field when Einstein showed that the gravitational field could be explained by a vector curvature in a four-dimensional manifold and the V field is a gauge field in a five-dimensional manifold? If the V field were to be considered as gravitational, then the bending of light around the sun, predicted by Einstein's General Theory of Relativity, must have another explanation. Is this possible? If the V field is gravitational, then is there room in the Dynamic Theory for an explanation of nuclear phenomena or must it also follow the current approach to nuclear physics, thereby requiring similar additional assumptions? These and other questions occur when we see the long-range character of the V field force. It is this theoretical quandary, presented by the V field, that spoils the pleasant surprise of seeing a non-singular electrostatic field emerge from the quantum condition. A number of possibilities appear dependent upon the answers obtained to the previous questions and/or others. Primarily, the possibilities pertain to the validity of the five-dimensional view, for it is from this five-dimensionality that the V field comes. One reasonable approach, in attempting to find a possible way out of this quandary, seems to be to suppose the gravitational interpretation is a possibility and then see how the Dynamic Theory compares with measurable experimental evidence. 4.10 Self-Energy of Charged Particles. One of the difficulties in Maxwellian electromagnetism is the infinite self-energy that is predicted for a charged particle. This "electron catastrophe," or singularity, does not exist with the non-singular neo-coulombic field, and the self-energy of a charged particle may be found. In classical electromagnetic theory the self-energy of a charged particle is discussed but its value has not been established. This is

. rZkq = F 2

1E

. rWgg = F 2

2V

128

because the expression for the self-energy is a function of the radius associated with the physical extent of the charge distribution. Thus, the radius of the charged particle must be known before the self-energy can be determined. Currently the self-energy of a charged particle is equated with the energy associated with its inertial mass by E = mc2. Then the radius associated with its energy is taken as the "radius" of the particle. There is no intention that this radius be the physical radius of the particle though it compares favorably with experimental values. The question arises here of whether or not the Dynamic Theory, with the five-dimensional viewpoint, can theoretically predict the self-energy and/or the radius of the physical extent of the mass or charge distribution of the particle. One of the beneficial aspects of the generalization of physical theory as done in the Dynamic Theory is the possibility of using conceptualizations and procedures developed in one branch of physics in another branch. This aspect of the theory appears applicable here. The self-energy of a charged particle is the notion that a certain amount of energy be associated with the existence of the particle and its charge. This notion may be associated with the notion of free energy used in thermodynamics, for, if the self-energy of the charged particle is its free energy, then it represents the energy which may be "freed" upon converting the particle into energy. Conversely, this would represent the energy required to assemble the charged particle. With the conceptualization of free energy the second law provides the condition for a stable equilibrium state, namely that a charged particle in an equilibrium state must exist at a minimum of its free energy. Thus, if the self-energy, or free energy, of a charged particle is sought, then minimizing its free energy will yield the desired result. The free energy was defined, in analogy with the thermodynamic case, as

where D depends upon the applicable work terms which here will be taken as the three spatial dimensions, so that D = 1, 2, 3. The first law is given by

while the second law yields

for a quasi-static, reversible process. Therefore, the differential change in the system energy is

,Fx - S - U G DDI{ (4.29)

dxF - dU = Ed DD

dxF - dU = dS DDI

.dxF + dS = dU DDI (4.30)

129

Differentiating Eqn (4.29) gives the differential change in the free energy as

Substituting Eqn (4.30) into (4.31) yields

The force in Eqn (4.32) is considered to be the Lorentz force

so that Eqn (3.48) becomes

If we wish to consider the change in free energy with respect to a change in the charge at a constant velocity, we find that, since U is a function of velocity only, dU= 0. The specification of constant velocity stems from the desire to obtain the self-energy of a charged particle; therefore, the particle should be considered as sitting still, so that it will have no kinetic energy. The differential change of free energy for a stationary particle is then

so that for U = constant

but

is independent of the charge q and, therefore,

If the charge is not in motion, then

.dFx - dxF - Sd- dS - dU =dG DDD

DII (4.31)

.dFx - Sd- =dG DDI (4.32)

])Bxv( + Eq[ = F DD

}.])Bxv( + Eq[d{x - Sd- =dG DDI

}.])Bxv[( + Eqd[ + ])Bxv( + E{dq[x - Sd- =

},])Bxv( + Ed{q[x - Sd- =dG

DDD

DD

I

I

.q

])Bxv( + E[qx - ])Bxv( + E[x- =

qG

¸¸¹

·¨¨©

§

ww

¸¹

ᬩ

§ww D

I

DD

D

I

)Bxv( + E

.])Bxv( + E[x- = qG

DD

I¸¹

ᬩ

§ww

Ex- = qG

DD

I¸¹

ᬩ

§ww (4.33)

130

since v = 0. If G is the self-energy of a charged particle, then by Eqn. (4.32)

the change in the self-energy is given with respect to a change in the charge q. If we assume a spherically symmetric charge density, U, then

We may then find the free energy by the integration

where R represents the radius within which the charge density U is contained. The field Eqn. [3.15d] is

The entire right-hand side of this field equation behaves as a charge density; therefore, we may equally write

where it is understood that either JH ww )/V( 4 109 is zero or U is considered to be a total effective charge density. In either event, Eqn. (4.34) gives us

when we consider only a radially symmetric field Er. Thus,

Substituting Eqn. (4.35) into Eqn. (4.33) the self-energy is then found to be

If we now use the neo-coulombic electric field given by

,Er - = qG

rww

. drr4 = dv = dq 2 USU

,drEr4 - =

dqEr - =dG

r3

R

0

r

q

q

G

G 00

US³

³³

. )V(a - 4 = )E( 4

0 JHSUHw

wx'

,4 = )E( SUHx' (4.34)

,4 = r

)Er(r

r2

2SUH

ww

¸¹

ᬩ

§

. )Erd( rr

= dr r2

2 ¸¹

ᬩ

§SHU (4.35)

. G + )Er)d(E(r - =G 0r2

r³H (4.36)

131

in the integral of Eqn. (4.36), we find

which may be integrated so that

To find the specific value of the self-energy, we must find the R that minimizes G. Therefore, set

After carrying out the required differentiation and simplifying, this is satisfied if

Equation (4.38) only has one positive root, which is

Substituting this result into Eqn. (4.37), the self-energy becomes

or

when O is given in units of fermi. An example may be a proton for which O is approximately 1 fermi if the proton-proton scattering data is considered. Then if O~fermi,

Gp = -7.26235 x 10-3 MeV + Gop , so that

Gop = 938.263 MeV (4.40)

er - 1

r4e = E r

-2r

¸¹·

¨©§

¸¹·

¨©§ OO

SH

,G + er - 1

4eder

- 1r4

e e - =G 0r-

r-

R

0»¼

º«¬

ª¸¹·

¨©§¸¹·

¨©§

¸¹·

¨©§ ¸

¹·

¨©§

¸¹·

¨©§

³OO O

SHO

SH

. G + e41 -

R21 -

R43 +

R

)2(4e- =G 0R

2-23

2

2¸¹·

¨©§

»»¼

º

««¬

ª¸¹·

¨©§

¸¹·

¨©§

¸¹·

¨©§

»¼

º«¬

ª OOOOHOS

(4.37)

. 0 = RGww

. 0 = 52 - R

53 + R2 ¸

¹·

¨©§

¸¹·

¨©§ OO (4.38)

. 0.4 = R O

,G + (0.063379) )(4e- =G 02

2

»¼

º«¬

ª

HOS

,G + (fermi)

fermi- MeV10 x 7.26235- =G 0

-3

O (4.39)

132

is the part of the proton rest energy independent of its charge. The charge energy of the proton would then be

Gcp = -7.26235 keV , which is negligibly small compared to the non-charge energy Gop. What is the nature of the energy Gop? It is not energy caused by the presence of electric charge on the protons. Also the self-energy, G, was found for a resting particle. If we associate the resting self-energy, G, with the rest mass as

Gp = mopc2 = Gcp + Gop , and Gcp is the portion of the proton's rest energy that is due to its charge, then Gop must be that portion of the rest energy that is due to the proton mass above. In this case, the proton mass energy, Gop, is given by Eqn. (4.40). Suppose we consider an electron and assume that Oe - ~ 10-3 fermi. Then

Ge- = 0.511 MeV = -7.26235 MeV + G0e-, or the mass energy of the electron would then be

Goe- = 7.773 MeV , whereas its charge energy is

Gce- = -7.26235 MeV 4.11 Nuclear Phenomena. The electrostatic force, appearing in Eqn. (4.27), differs significantly from the columbic force only when r becomes small enough to be of the order of magnitude of the On. The first experimental evidence that the scattering of charged particles by other charged particles was not always columbic was the Rutherford scattering data. The appearance of the exponential multiplier in the neo-coulombic force of Eqn. (4.27) prompts us to ask whether or not the difference between this force and the columbic force suffices to explain nuclear phenomena without resorting to the postulation of a new short-range force such as the nuclear force. An obvious starting point to explore the possibility that the neo-coulombic force might apply to nuclear phenomena would probably be the

133

Rutherford scattering formula. This may be done; however, the appearance of the exponential term makes an analytical expression difficult, if not impossible, to obtain. We may arrive at a solution of limited usefulness if we assume that r >> O. Further, in considering particle scattering, we shall restrict our consideration to scattering of like particles, only, so we are guaranteed that only one O is involved. The best way to investigate the scattering cross sections is to start with the solutions of the equations of motion for planetary orbits in which the force is given by the neo-coulombic force instead of the simple inverse r2 force from Newton's gravitational force. We find the radial equation becomes

where u = 1/r, k is the gravitational constant, and L is the orbital angular momentum. The exponential function may be expressed in terms of a power series, and our radial equation becomes

by assuming r>>O so that O/r = Ou <<1, then we may neglect the terms with nt3 in our radial equation. The result of this assumption is

with

This equation may be compared with the classical equation,

or with the general relativistic equation6,

which has the identical form of our equation. Thus, the same method of perturbations may be used to obtain a solution as was used for the relativistic case. The result of this calculation is the solution,

,eu)- (1LMk =u +

dud u-

22

2OO

T

.

)u(-n!

1)+(n+1LMk =

...+n!

)u(-+...+2!

)u(-+u)(-+1u)-(1LMk =u +

dud

n1)=(n2

n2

22

2

»¼º

«¬ª

¦

»¼

º«¬

ª

f O

OOOOT

uL2Mk3 +

LMk =u +

dud 2

2

2

22

2

2

¸¹

ᬩ

§ ODT

.LMk2 + 1 =

22 O

D

,LMk =u +

dud

22

2

T

,Mku3 + LMk =u +

dud 2

22

2

T

134

where

is the increase in the perihelion. Notice we have shown that the neo-coulombic force will predict an advance in the perihelion of planetary orbits with the solutions to our planetary orbits equation. We will discuss this further at a later time. We needed the solution given by Eqn (4.41) in order to obtain an expression for the scattering cross section of like particles. If we now consider the solution, Eqn. (4.41), obtained with the as-sumption that r >> O, then the scattering cross section may be expressed as

where

The appearance of the factor G expresses the first-order deviation of the scattering cross section of the neo-coulombic force from that of the columbic force. However, the assumption that r >> O implies a limit on the minimum impact parameter for which this cross section retains validity. Therefore, a computer solution is probably necessary to really investigate the scattering of charged particles using the neo-coulombic force. Figure 6. Comparison of coulomb and neo-coulomb forces at short range. Another way of visualizing the neo-coulombic force is to make a plot of it and compare it with a plot of the columbic force. Figure 6 compares these two forces plotted with the separation variable in fermions and normalized so that the columbic force at one-fermion separation is unity. Note that this plot compares the forces for like particles to ensure that O is the same for both particles. Figure 6 shows that the neo-coulombic force

> @,) - - ( e + 1LMk =

r1 =u 002 TGTDTcos (4.41)

L2

Mk3 = 2

2

0TO

TG

,

2

d2 mv2qq = d

420

21 GTTTSV

»»»»

¼

º

««««

¬

ª

¸¹·

¨©§¸

¹

ᬩ

§

sin

sin (4.42)

.

)-( 2

kE4

23 + 1

22 + 12

kE6 + 1

2

2 4

)-(42

°°°

¿

°°°

¾

½

°°°

¯

°°°

®

­

»»¼

º

««¬

ª¸¹·

¨©§

¸¹·

¨©§¸¹·

¨©§

»¼

º«¬

ª¸¹·

¨©§

¸¹·

¨©§

¸¹·

¨©§

{

TSTTO

TTO

G

TS

sinsin

tansin

135

is virtually indistinguishable from the coulomb force for separations greater than approximately 10O. However, at a separation of exactly O, the force is identically zero. In terms of the classical notion of nuclear forces, we would say that at separations greater than 10O, the nuclear force is negligible, whereas at a separation of O the magnitude of the nuclear force was equal to the magnitude of the coulomb force. The neo-coulombic force becomes an attractive force for separations less than O. This is exactly the behavior to be expected of a non-singular potential. For a potential to be non-singular it must tend to zero as r goes to zero. Such a potential which tends to zero for r tending to zero and for r tending to f must have a maximum absolute value in between. At that maximum the force, being determined by the slope of the potential, will go to zero and will be of the opposite signs on each side of the zero. Now let us look at the force between unlike particles, say a proton and an electron. Consider the electron and proton to be placed on a horizontal surface separated by a distance, r, with the proton to the right of the electron. Thus, the long-range attractive forces between these two particles will cause the proton to experience a force to the left while the electron will experience a force to the right. We may than write the force on the proton that is due to the positive charge of the proton being in the electron field as

where the electron field involving the electron lambda has been accounted for. The electron force owing to the electron charge being in the proton field is given by

Figure 7 plots both these forces as a function of the separation, r, where, Op = 10-15m, or Op = 1 fermi has been assumed. The electron-electron scattering data show that the electron-electron interaction behaves in a coulombic manner even when separations are approximately 0.01-0.1 fm. To be consistent with this data, we have assumed Oe = 10-3 fermi. From this plot of the force on the proton and the force on the electron, we see that for separations less than about 10 fermis the forces become extremely unsymmetrical. This immediately and visually demonstrates that the neo-coulombic exponential force violates Newton's

,)u( e r

- 1rk- =

Eq = F

xr-e

2

epp

e

ˆ¸¹·

¨©§

¸¹·

¨©§ OO (4.43)

. )u(e r

- 1rk =

Eq = F

xrp-p

2

pee

ˆ¸¹·

¨©§

¸¹

ᬩ

§ OO (4.44)

136

third law requiring that the force on the proton be equal in magnitude and opposite indirection to the force on the electron. The question arises whether or not a violation of Newton's third law has ever been seen as the result of an interaction between an electron and a proton? The answer, based on a neutron disintegration from which a proton and electron emerge, is definitely yes; Newton's third law was seen to be violated. To reinstate Newton's third law in neutron disintegration and all other beta decay, Pauli postulated the existence of the neutrino. Fermi later developed his theory of weak interactions,11 from which appeared the necessity to talk of a fourth force in nature. Can it be that the neo-coulombic force, which requires distinct O for distinct fundamental particles, accounts for the action of the weak forces also? The possibility that it might opens the theoretical flood gates and a virtual tidal wave of questions surges forth. Does this mean the neutrino does not exist? What about the experimental evidence submitted in support of the capture of a free neutrino?12 Could this mean that the neutron might be bound states of an electron and proton? This question should be followed by, what about conservation of angular momentum in neutron decay (i.e., spin), conservation of linear momentum, and Heisenberg's uncertainty principle? Figure 7. Neo-coulombic forces between unlike particles at short range. The preceding questions do not begin to scratch the surface of the theoretical questions that need to be answered as the result of considering the possibility that the force law of Eqn. (4.27) with only a gravitational force plus the neo-coulombic force might explain the phenomena now thought to require four distinct forces in their explanation. However, the appearance of a non-singular force with the apparent range of the neo-coulombic form cannot be thrown out offhand. Therefore, it seems that the only reasonable choice is to systematically and thoroughly explore the pos-sibilities. If we again consider the plots of the proton and electron forces in Fig. 7. we see that, at atomic separations and greater distances, the forces obey Newton's third law and the difference between the neo-coulombic and columbic forces is so small that it could not be detected in atomic or macroscopic phenomena. But as the separation becomes smaller, the picture begins to change. When the r approaches Op, the electron is no longer attracted to the proton as strongly as the proton is attracted to the electron. If the separation is exactly Op, then the electron is indifferent to the proton's presence. The proton, on the other hand, is still very much attracted to the electron. If for the moment, we ignore the interpretation of Heisenberg's uncertainty principle that would say it cannot be, then we could easily imagine a circular proton orbit around a stationary electron, during which the proton stays at a radius of Op from the electron. The

137

electron should be stationary during such motion because it would experience no force. We now consider a separation between the electron and proton, which is some simple fraction of Op. Here, we find the electron repulsed by the proton, but the proton is still attracted to the electron. Notice that the force on both particles, from our initial positioning of the proton on the right, is to the left. If both particles were given an angular momentum such that they were placed into synchronized circular orbits, then because their synchronous motion always results in the force on both particles being directed along the line separating them and from the proton toward the electron or from the electron away from the proton then, again ignoring arguments from the uncertainty principle, circular orbits in which the electron is in a small orbit about a space point could be imagined, where the proton is in a much larger orbit about the same space point. Let us follow this picture a little farther and write simple Newtonian-like force laws for this situation. The situation envisioned is presented in Fig. 3. The electron position is given by re from the origin, and the position of the proton is given by rp. The separation between them is r = rp - re . (4.45) Because the force is always directed along the line separating the two particles, we may write the radial equation of motion for the proton as

where the assumed circular motion has been taken into account and vp is the tangential proton velocity. The electron equation of motion is given by

. e r - 1

rk =

rvm r

-p2

e

ee

p¸¹

ᬩ

§

¸¹

ᬩ

§ OO (4.47)

Figure 8. Electron and proton orbits. In both equations k < 0. The right-hand side of Eqn. (4.46) and (4.47) are both functions of the separation, r, whereas the two left-hand sides are individually functions of re and rp. A solution is possible only when the three equations, Eqn.s (4.45)-(4.47), are solved simultaneously. An alternative approach is to add Eqn. (4.46) and (4.47) to obtain the equation of motion for the center of mass,

,er - 1

rk- =

rv

m re-e

2p

pp

¸¹·

¨©§

¸¹·

¨©§ OO (4.46)

,er - 1 - er

- 1 rk =

r2vm +

r2v

m r-e

r-p

2e

ee

p

pp

pp¸¹

ᬩ

§¸¹

ᬩ

§¸¹·

¨©§

¸¹

ᬩ

§ OO OO (4.48)

138

or

where R = (mprp + mere)/(mp + me) and M = mp + me. From Eqn. (4.48) we see that bound states, where the center of mass is in motion as the result of the asymmetrical force, may only occur when the separation is less than Op. All of these equations of motion exhibit a feature not usually found in equations of motion. That is, because the force depends on the separation, r, between the particles and not strictly on the position, re, for the election, then the usual integration of the force over a change of position, which produces the potential energy, cannot be readily done because,

However, Eqn. (4.45) and Eqn. (4.47) may be used to obtain re as a function of r, or vice versa, so the integration of Eqn. (4.49) may be completed. The transcendental function in the force law prohibits an analytical solution of re as a function of r. Therefore, only numerical or graphic solutions of these equations are possible. 4.12 Heisenberg's Uncertainty Principle and Geometry. The suggestion that bound states of electrons and protons might exist where the orbits are of the approximate order of magnitude of nuclear dimensions, is essentially a return to the notion that a neutron might be such a state. This idea gave way under arguments of conservation of momentum and Heisenberg's Uncertainty Principle to the view that electrons are forbidden to be found within the nucleus. Therefore, let us take another look at those fundamental tenets of quantum mechanics, the Poisson brackets. The classical Poisson bracket is defined by

where F and G are any two functions of the canonically conjugate variables qj and pj. The special relations that occur when F and G are qj and pj, respectively, are especially important in quantum mechanics; these are, classically:

,er - 1 - er

- 1 rk =

RMV

r-e

r-p

2

2p

p

¸¹

ᬩ

§¸¹

ᬩ

§¸¹·

¨©§

¸¹

ᬩ

§ OO OO

. dr er

- 1 rK - =

drF - = )rV(

er-p

2

eee

p¸¹

ᬩ

§

¸¹

ᬩ

§¸¹

ᬩ

§³

³

OO (4.49)

^ ` ,qG

pF -

pG

qF = GF,

jjjjj¸¸¹

·¨¨©

§

ww

ww

ww

ww¦

139

{qj, qk} = 0 {pj, pk} = 0 (4.50) {qj, pk} = 0 , where Gjk is the Kronecker delta. The classical Poisson brackets of Eqn. (4.50) are obtained when Euclidean spaces are assumed. However, the definition of Poisson brackets remains valid for general metric spaces, when the notion of covariant differentiation is used. If we now consider the momenta, expressed in a general coordinate system, the covariant components,

and (4.51)

are the contravariant components. Covariant differentiation must be carried out with respect to contravariant vector components. There, in a general space the canonically conjugate variables to be considered are xj and pk, and the Poisson bracket of the position and momenta becomes

or

Quantum mechanics adopts the operator,

for the momentum. This, in general case, becomes the covariant operator

The operator for the contravariant momentum components is then

xmg = p kijj �

xm = pg = p jl

jlj �

^ `

. xl sj

+ =

pl n

k +

xp

px -

pp xl s

j +

xx = p ,x

kls

jl

nl

k

l

j

l

k2

l

jkj

GG»»¼

º

««¬

ª

¿¾½

¯®­

»»¼

º

««¬

ª

¿¾½

¯®­

ww

ww

ww

»»¼

º

««¬

ª

¿¾½

¯®­

ww

(4.52)

^ ` . xk sj

+ = p ,x sjk

kj

¿¾½

¯®­

G

,p xi

_jj ow

w¸¹·

¨©§

� � . p , i_

jj o¸¹·

¨©§ (4.53)

140

Now if we look at the quantum Poisson bracket, where the operators are operating on a scalar <, then

This may be written in terms of the classical Poisson bracket, Eqn. (4.52), as

If the space is Euclidean, then the gkl become the Kronecker delta and the Christoffel symbols vanish and the quantum Poisson bracket of Eqn. (4.55) becomes

because pk = gkl pl = Gklpl = pk. However, from Eqns. (4.53) and (4.54), we see that the metric does play a role in the quantum operators. This should also be seen in the use of the operators in the Schrodinger Hamiltonian operator, because

becomes the operator to be used in a general space and, of course, is the operator used in applying Schrodinger's equation to the hydrogen atom.

� � . p , g i_ j

ljl o¸

¹·

¨©§ (4.54)

. x l sj

+ g i =

xx g

i_ -

xl sj

+xx

i_g-

xg

i_

x=

l ),x( g i_ -

xg

i_

x = ] p ,x[

sjl

kl

ljjl

sl

jkl

lklj

jkll

kljkj

\G

\

\\

\I\

»»¼

º

««¬

ª

¿¾½

¯®­

ww

¸¹·

¨©§

»»¼

º

««¬

ª

¿¾½

¯®­

ww

¸¹·

¨©§

ww

¸¹·

¨©§

»¼

º«¬

ª¸¹·

¨©§

ww

¸¹·

¨©§

!

(4.55)

> @ ^ ` . p,xg_ i = p ,x ljklkj \\

> @ , i_ = p ,x jkkj \G\

� �

¸¹

ᬩ

§ww

ww

¸¹

ᬩ

§ww

¸¹·

¨©§

ww

ww

x g g

x

g_ =

x g

i_ G

x

g1

i_ =

p g x

g

1 i_ = pp

jjl

j

2

ljl

j

jj

jj

(4.56)

141

The geometrical effect may be seen also in Dirac's equation by considering that the restrictions,

must be met in order for solutions of

where

is also a solution of pjpj = m2 in natural units. The first restriction may be rewritten as

by the definition of the momenta. Then, if we expand the left-hand side and equate coefficients of the pjpk, we find that (D)2 = -g11 , (D)2 = -g22 , (D)2 = -g33 , (3.77)

and

From Eqn. (3.77), for a Euclidian metric where gjk = Gjk, these restrictions reduce to the usual restrictions. Any metric properties will affect these restrictions and will therefore feed into the solutions. Now, of what benefit it this discussion of geometrical effect upon quantum mechanics in considering the neo-coulombic force? Recall that the neo-coulombic force came from a gauge function in a Weyl space. A gauge function has a geometrical effect that could be thought of as effectively changing the unit of action in quantum mechanics. To see the basis for this statement, let us recall the quantum Poisson bracket operations on a scalar,

,0 = +

and ,1 =

,pp = )p (2

jj

2

DEED

E

D x

(4.57)

,|>H = |>po

� � ,m + p - = H ED x

pp g = xxg m = pp = )p( kjjkkjjk

2ji

2j��D

^ `^ ` ,g2 - = = +

,g2 - = = +

13311331

12213221

DDDDDDDDDDDD

^ ` . g2 - = + + 23322332 DDDDDD

142

and let us define

then we can write

which has the same form now used but the effective unit of action _c 158 depends on the geometry as seen by Eqn. (3.78). We may look at the effective unit of action in yet another way. Recall, from the principle of maximum entropy, that the generalized entropy is the action. Thus, quantization of the action is a quantization of the generalized entropy. But, because the entropy space is tied to the sigma space, we have

The gauge function is a function of the space point; therefore, the gauge function varies continuously from point to point in the space. Thus, if the generalized entropy is quantized, so must be V. We may write

where the difference between _c 161 and _ 162 contains the geometrical difference between q0 and V. But how can we determine the relationship between _c 163 and _ 164? From the principle of maximum entropy, the potential energy function was defined as the negative integral of the force through a distance just as it is in classical mechanics, because

but the potential energy plays the role of the gauge function. The equations of motion for the asymmetrical forces between unlike forces showed that an analytic form for the potential energy may be unobtainable owing to the transcendental nature of the forces and because the forces depend upon the separation between the particles, not their positions. Thus, there appears no way, at the moment to obtain an analytical expression for _c 166, and we must resort to a numerical solution for the unlike particle case.

> @ , xl sj

+ g_ i = p ,x sjl

klkj \G\»»¼

º

««¬

ª

¿¾½

¯®­ (3.74)

,xl sj

+ g_ = _ sjl

kljk

»»¼

º

««¬

ª

¿¾½

¯®­

c GG (3.78)

> @ , _ i = p ,x jkkj G\ c (3.79)

� � . )f(d = dq 20 2 V

n_ , = = _n = q0 Vc

,dxdV - = F jj

143

The absence of an analytical expression for the effective unit of action, _c 313, does not completely stop us from considering the possibility that a neutron may be a proton in a large orbit about an electron in a small orbit. We may, for the moment, acknowledge the difficulty of obtaining an analytical expression for _c 314 by allowing the _c 315, or unit of action, for the proton and electron to be a function of their orbit, and we may designate _e 316 to be the effective unit of action for the electron orbit in a neutron and _ p 317 to be the unit of action for the neutron's proton orbit. If the effective unit of action depends upon the orbit, as it appears here that it must, then the interpretation that Heisenberg's Uncertainty Principle rules out the possibility of an electron being contained within nuclear discussions is inapplicable. Another argument against the neutron being an electron and proton in nuclear-sized orbits is based on an argument that the principle of angular momentum cannot be conserved. The neo-coulombic forces, which require that the force between the electron and proton be directed on a line between them, requires that angular momentum be conserved. However, the effective unit of action for the electron orbit requires that, in the neutron the orbital angular momentum would be given by _e 318 and its

intrinsic spin angular momentum would be _21

e 319. Similarly, for the proton the orbital

angular momentum would be _ p 320 and the spin _)21( p 321.

After the neutron decays, the angular momentum is the sum of the two particles' intrinsic spin angular momenta, which is given by _ 322 because both particles are free and, therefore, each has an intrinsic spin angular momentum

of )_21( 323. Therefore, the conservation of angular momentum is expressed as

Experimental evidence of orbital and/or spin angular momentum is contained in the experimental magnetic moments. If we equate the intrinsic and orbital magnetic moments of the electron and proton while they are in the orbital configuration to the experimental value of the neutron's magnetic moment we have

where P E 326 is a Bohr magneton and P n 327 is a nuclear magneton. Equations (3.80) and (3.81) represent two equations in the two unknowns, ne 328 and _ p 329, which may be solved to obtain the effective units of action for the electron and proton orbits making up a neutron such that angular momentum in conserved during neutrons' decay and that the correct magnetic moment of the neutron is ensured. Substituting the experimentally measured values of intrinsic magnetic moments

� � ._ = _ + _ + _+ _+ 21

pepe __ (3.80)

. 1.91315 - = __

+ __

- __

22 +

__

22 + nn

pen

pe PPPPP EE ¸¹

ᬩ

§¸¹

ᬩ

§¸¹

ᬩ

§¸¹·

¨©§

¸¹

ᬩ

§¸¹·

¨©§ __ (3

144

for the electron and proton into Eqn. (3.81) produces a more accurate solution because this contains the anomalous magnetic moments. Then we would have

The only simultaneous solution of Eqns. (3.80) and (3.82), for which _e 331 and _ p 332 are both positive, are

The values of the effective units of action for the proton and electron given in Eqn. (3.83) show that angular momentum is conserved during the decay of a neutron when the neutron is considered to be a proton in orbit around an electron under the neo-coulombic force. The third major argument against a neutron being a state of electron and a proton orbits stems from the experimental evidence on the violation of Newton's Third Law during decay. That is, the energy of the electron emerging after decay is inconsistent with the equal and opposite columbic forces between an electron and a proton. Here, we find that the neo-coulombic forces are unequal in magnitude and opposite in direction; thus the energy of an electron emerging as the result of crossing from such an orbit cannot be consistent with Newton's third law. There now exists a fourth argument against this picture of a neutron: the possible existence of the neutrino. The above picture of the neutron produces no need to postulate the existence of neutrinos. What then can be said about the experimental evidence that has been put forward in support of the capture of free neutrinos?12 A conclusive answer will need to await further investigation. 4.13 Nuclear Masses The difficulty produced by the asymmetry of forces that arises in the interaction of an electron with a proton may be avoided if two protons are considered to be in orbit about the single electron. If we think of a snapshot of such a case we would find that the situation depicted in Fig. 9 allows us to visualize the forces. Figure 9. Two protons in orbit about a single electron. The force on the electron would be zero because it has a proton on each side diametrically opposed to one another. The force on each proton will be made up of two parts; one, the force that is due to the presence of the electron, and the other, owing to the other proton. The symmetry

. 1.91315 - = __

+ __

- h_

2.79275 + __

2.002319 + nne

pe

npe PPPPP E ¸

¹

ᬩ

§¸¹

ᬩ

§¸¹

ᬩ

§¸¹

ᬩ

§__ (3.82)

._ 0.66586 = _

_ ,10 x 8.0517 = _p

-4e (3.83)

145

guarantees that each proton will experience an identical force, if circular orbits are assumed, toward the center of rotation. The force on the proton on the left would be

To be sure, quantum mechanical procedure should be used; however, it may be beneficial to begin by assuming circular orbits similar to Bohr's initial approach to atomic structure. This may indicate the potential utility of the force in Eqn. (3.83), as well as perhaps identifying procedures to be used later. Any nuclear orbits should probably be relativistic; therefore, in cylindrical coordinates, where the velocity for motion in a plane is given by

then we have

For circular orbits, this becomes

Thus, the relativistic equations of motion for the proton become

Equation (3.85) separates into two equations

and

The second of these equations says that the angular momentum is given by

e(2r) - 1

)(2r|k| + er

- 1 r

|k|- = F 2r-e

2r-e

2ppe

1¸¹

ᬩ

§¸¹·

¨©§

»¼

º«¬

ª¸¹·

¨©§ OO OO (3.83)

, r + rr = v TT ˆˆ ��

c

) r + r( = 1 = cv - 1 = 2

222

2

2 TJ��

c r - 1 = 2

22TJ� (3.84)

rer-1-e2r

-141

r|k|=

)r+rr(mdtd

r-e

2r-p

2p ep

���

»¼

º«¬

ª¸¹·

¨©§

¸¹

ᬩ

§¸¹·

¨©§

¸¹

ᬩ

§

»»¼

º

««¬

ª¸¹·

¨©§

¸¹

ᬩ

§ OO OOJ

TT ˆˆ (3.8

. rer - 1 - e2r

- 141

r|k| =

rm dtd

r-e

2r-p

2p ep ˆ»

¼

º«¬

ª¸¹·

¨©§

¸¹

ᬩ

§¸¹·

¨©§

»¼

º«¬

ª ¸¹·

¨©§

¸¹

ᬩ

§ OO OOJ�

(3.86)

. 0 = rm

dtd p

»»¼

º

««¬

ª

JT�

146

where _c 342 indicates that whereas the unit of angular momentum will be a constant for a given orbit, it may be different for different orbits. The first of Eqn. (3.86) is

but for circular motion 0 = r� 344, therefore,

Substituting from Eqn. (3.87) into Eqn. (3.88) we have

The potential energy for one of the protons can be found by integrating the force and is

Then the total energy of the three-body system, including rest energy, would be

However, by substituting Eqn. (3.87) into Eqn. (3.84) and solving for J, we find

,_n = L = rmp

2pc

JT� (3.87)

,er - 1 - e2r

- 141

r|k| = dt

drm -

)rM-rm( =

rmdtd

r-e

2r-p

22

p2ppp ep

»¼

º«¬

ª¸¹·

¨©§

¸¹

ᬩ

§¸¹·

¨©§

¸¹

ᬩ

§»¼

º«¬

ª ¸¹·

¨©§

¸¹

ᬩ

§ OO OOJ

J

JT

J

�����

. er - 1 - e2r

- 141

r|k|- =

rmr

-e2r

-p2

2p ep

»¼

º«¬

ª¸¹·

¨©§

¸¹

ᬩ

§¸¹·

¨©§

¸¹

ᬩ

§ ¸¹·

¨©§

¸¹

ᬩ

§ OO OOJT� (3.88)

. er - 1 - e2r

- 141

r|k|- =

rm)(n

r-e

2r-p

23p

22ep

»¼

º«¬

ª¸¹·

¨©§

¸¹

ᬩ

§¸¹·

¨©§

¸¹

ᬩ

§c ¸¹·

¨©§

¸¹

ᬩ

§ OO OOJ! (3.89)

. e - e4

1r

|k| =

drer - 1 - e2r

- 141

r1|k| - = dr F(r) . - = V(r)

r-

2r-

r-e

2r-p

2

ep

ep

»¼

º«¬

ª¸¹·

¨©§

»¼

º«¬

ª¸¹·

¨©§

¸¹

ᬩ

§¸¹·

¨©§

¸¹

ᬩ

§³³

¸¹·

¨©§

¸¹

ᬩ

§

¸¹·

¨©§

¸¹

ᬩ

§

OO

OO OO

(3.90)

. cm + cm2 + e - e4

1 r

|k|2 = E 2e

2p

r-

2r-

Tee

JOO»¼

º«¬

ª¸¹·

¨©§ ¸

¹·

¨©§

¸¹·

¨©§ (3.91)

147

Thus, substituting Eqn. (3.92) into Eqn. (3.89) produces a transcendental equation whose solution gives r(n), which may then be used in Eqn. (3.90) to obtain the total energy of the system. The mass of the system should then be found from

Because this system has one electron and two protons, it has a total electric charge of +1 and would have a mass of approximately 2 amu. This is the same characteristic exhibited by the deuterium nucleus. If this is the structure of the H2 nucleus, then the mass given by Eqn. (3.93) should correspond to the mass of the ground-state nuclear mass for n = 1. If the 1e-, 2p+ case existing where n = 1 is the ground-state H2 nucleus, then is the excited state represented by two protons in the n = 2 state or can it be represented by one proton in an n = 1 orbit and one in an n = 2 orbit? The equations developed here consider only the case when both protons are in the same orbit. Any consideration of the protons being in different orbits introduces an asymmetry in the forces and a similar difficulty faced in the neutron case. Therefore, for the moment we will consider only the simpler cases, where symmetry reduces the complexity of the solution. Notice, though, that even in the simpler symmetric case, no analytical solution exists of Eqn. (3.89) for r(n) because the force contains a transcendental function. By allowing !c351to be different for each n, then for the ground and first excited states of the H2 there are four quantities to be determined: (1)_ , , ep cOO 352, and _c 353. Of course, in theory, we could determine both O p 354 and O e 355 from scattering experiments: then we would only have two, (1)_c 356and (2)_c 357. But if we discover exactly how the _c 358 depends upon the orbit, then a solution of Eqn. (3.89) would represent a pure theoretical prediction of both the ground state mass, m(n = 1), and the excited state mass, m(n = 2), for then we would be able to express (r)_ = _ cc 359. If we think about the possibility of adding an additional proton to the 1e-, 3p+ case, we are faced with a question. Can the additional proton be placed in the n = 2 orbit without considering the asymmetry thus introduced into the system, or must we consider a single orbit with three protons symmetrically spaced? The answer lies partly in the solution of the appropriate Schrodinger or Dirac equation, because this would inform us of the number of protons that are allowed in a given orbit. This, however, would not answer the question concerning how the asymmetry introduced by a single proton in the n = 2 orbit affects the problem. To

.

rcm_n + 1

1 =

p

2

¸¹

ᬩ

§ cJ (3.92)

. cE = M 2

T ¸¹

ᬩ

§ (3.93)

148

obtain an answer to this question the situation should be addressed by both methods and both results should be compared with the experimental evidence. Whichever approach proves to be correct, the result would be a nucleus with a total charge of +2 and a mass number of 3, or Z = 2, and A = 3. This corresponds to the He3 nuclei. If the third proton can be placed in the n = 2 orbit of the solution for a single-electron case, and determining O p 360 and O e 361 by scattering experiments, plus determining (2)_ and (1)_ cc 362 by the bound and excited states of the H2 nucleus, then the mass of the He3 is a pure theoretical prediction. We shall consider this a little later. The fact that the excited state of the H2 nucleus is a virtual state implies that adding two protons may itself be nearly a virtual state and thus a cutoff would occur in the number of protons that can exist in a bound state for a one-electron core. To possibly look at other nuclei, suppose first that we go back and consider the like particle force between two electrons,

This force is repulsive until the separation between the electrons becomes less than Oe, and then it becomes attractive. But can there be bound states of two electrons? The answer lies in the effect of the gauge on _c 364. If the _c 365 for unlike particles is less than _c 366, then the sign reversal between the force of unlike particles and the force for like particles should mean that the _c 367 for like particles should be larger than _ 368. Thus, bound states of like particles would be forbidden. As an example of the sign reversal effect on _c 369, suppose that the gauge for the unlike electron, proton case is given by

or

Then,

or

er - 1

r|k| = F r

-e2ee

e ¸¹·

¨©§

¸¹·

¨©§ OO (3.94)

,e - e rk- = lnf r

-r

-21 ep

¸¹·

¨©§¸¹·

¨©§ ¸

¹·

¨©§

¸¹

ᬩ

§ OO

. e - erk- = f r

-r

- ep

»¼

º«¬

ª¸¹·

¨©§¸¹·

¨©§ ¸

¹·

¨©§

¸¹

ᬩ

§ OOexp (3.95)

,dx dx gf)f(d = ddx g = )dq( kjjk

2xkjjk

20 ˆV

. gf = g jkjk ˆ

149

Suppose g jkˆ 374 = G jk 375; then Eqn. (3.78) would become

Then, from Eqn. (3.96), we find

Now OO pe < < 378; therefore, when r = Op, the gauge function, f, from Eqn. (3.95)

is less than unity because k < 0. Thus the _c 379 given by this function is always less than or equal to . On the other hand, for like particles where only one O is involved, the gauge function would be given by

Then, because k > 0, f t 1, requiring _ _ tc 382. Although like-particle bound states may be forbidden by the uncertainty principle for large _c 383, bound states of unlike particles of nuclear dimensions are allowed by an _c 384that may be much, much less than _ 385. Next, we might consider possible bound states between electrons and positrons with subnuclear dimensions. If the O for the positron, Oe+, is less than the O for the electron, Oe-

, then we could have bound states with two electrons in orbit about a single positron that would be given by equations exactly like the equations for the 1e-, 2p2+ states, where the positron replaces the electron in the 1e-, 2p+ equations and the two electrons in the 1e+, 2e- case replace the two protons in the 1e-, 2p+ shell. It is possible now to consider a 1e+, 3e- core, thus introducing questions concerning asymmetry aspects and other possible questions. However, owing to the Oe + < Oe - << Op, the core structure is one where in electrons orbit about positrons and the electron orbits are << Op. For the protons in a shell orbit the interior core structure may be negligible, just as the internal structure of the nucleus has almost no effect on the atomic electron orbits. We shall not explore the core structure here but shall consider only the effect of different-core excess electron charge upon allowed proton shell orbits. If we denote the excess electron charge of the core by the integer Y, by which we mean the total number of core electrons less the number or core positrons, then by denoting the number of shell protons in orbit around this nuclear core, we find that the charge on the nucleus, Z, is given by

> @. f_ =

0 + g_f = _

jk

jlkl

jk

GGG ˆc

(3.96)

. f = __c

_

. erk = f r

-»¼

º«¬

ª¸¹·

¨©§ ¸

¹·

¨©§ O

exp (3.97)

Y - A = Z (3.98)

150

Equation (3.98) indicates that the excess core electron number behaves identically with the neutron numbers in current nuclear theory, although there are no neutrons as such in this nuclear model. Indeed, the neutron, in this picture, is simply another state, namely Y = 1 and A =1. This suggests a picture of the nucleus in which there are protons in orbits about a nuclear core. The number of protons are given by the current mass number, A. The radii of the proton shell orbits are approximately the value of Op; that is, about 1 fermi. The core may be made up of electrons in orbit about positrons and is sized approximately the same as Oe-, which is much, much less than Op. This view of the nucleus is similar to that of the atomic view, but here the nuclear core plays the role of the atomic electrons. The force law for the shell proton or-bits would then be given, from Eqn. (3.85), by

The equations specifying the proton shell orbits are

and

The total energy of the nuclei would be given by

where A(n) is the number of protons with the quantum number, n; R(n) is the radius of the proton orbit with the number n; Ec(Y) is the energy of the nuclear core for which Y is the excess electron charge; and J[R(n)] is the relativistic J evaluated for R(n). The mass of the nuclei with energies given by Eqn. (3.101) would then be

. er - 1 Y - e2r

- 141

r|k| = F r

-e2r

-p2

ep

»¼

º«¬

ª¸¹·

¨©§

¸¹

ᬩ

§¸¹·

¨©§ ¸

¹·

¨©§

¸¹

ᬩ

§ OO OO (3.99)

»¼

º«¬

ª¸¹·

¨©§

¸¹

ᬩ

§¸¹·

¨©§

¸¹

ᬩ

§ ¸¹·

¨©§

¸¹

ᬩ

§

er - 1 Y - e2r

- 141

r|k|- =

rm)(hn

r-e

2r-p

23p

22ep OO OOJ (3.100)

¸¹

ᬩ

§ crcm

_n + 1

1 =

p

2J

(Y)E +

[R(n)mA(n)

+ Ye

- e 41

R(n)|k|A(n) = Y) E(A,

c

2p

R(n)-

2R(n)-

n

e

p

¿¾½

»¼º

¯®­

«¬ª

»¼

º«¬

ª

»¼

º«¬

ª

¦

JO

O

(3.101)

151

. Y) E(A, c1 = Y) M(A, 2 ¸¹

ᬩ

§ (3.102)

This approach has a simple look to it. For instance, if Y = 1 then Ec =

0.511 MeV, the rest energy of the electron. Then the ground state for 2H

would be

whereas the excited state is

Using Ec = 0.511 Me V we find that the energy of a single proton in the n =

2 orbit would be

Table I. Experimental and predicted nuclear masses.

Experimental Predicted 'M = Ep - EE Predicted

MassMass 'M/A BE/A

Y Z A (MeV) (MeV) (MeV) (MeV) (MeV)

1 1 2 1875.0 1873.7 -1.3 -0.7 2.1

1 1 2 1877.9 1879.7 1.8 0.9 1.2

1 2 3 2808.3 2819.9 2.6 0.9 1.7

1 3 4 3749.5 3748.2 -1.3 -0.3 1.5

2 1 3 2808.9 2804.2 -4.7 -1.2 4.4

2 2 4 3727.3 3736.1 8.8 2.2 4.9

2 3 5 4667.5 4668.1 -4.8 0.1 5.2

2 4 6 5604.7 5600.0 -4.7 -0.8 5.4

3 2 5 4667.8 4668.3 0.5 0.1 5.4

3 3 6 5601.4 5600 -1.0 -0.2 5.5

3 4 7 6531.8 6532.3 -1.6 0.1 5.6

4 1 5 4691.8 4689.2 -1.6 -0.3 1.5

4 2 6 5605.5 5610.4 4.9 0.8 4.1

4 3 7 6533.3 6531.6 -1.6 -0.2 5.9

4 4 8 7454.3 7452.7 -1.6 -0.2 7.3

5 2 7 6545.7 6545.7 0.6 0.1 4.6

5 3 8 7471.2 7471.2 -1.5 -0.2 5.9

5 4 9 8392.2 8395.5 0.8 0.1 --

6 2 8 7482.5 7482.5 1.0 0.1 3.8

6 3 9 8406.7 8404.7 -2.0 -0.2 5.3

6 4 10 9325.0 9326.0 1.0 0.1 6.5

,E + E2 = 1) E(2, c1

. E + E2 = )1 E(2, c2*

> @2

E - )1 E(2, = E c*

2

152

Thus, the 3He nuclei energy would be

Now using the tabulated experimental data14 we find that the predicted nuclear mass of the 3He should be

compared to the tabulated value of 3.014848 amu. Similarly, the predicted mass for 4Li should then be 4.028288 compared to the tabulated 4.025231. The difference between the predicted and the tabulated values are 1.8 and 0.7 MeV/nuclear, respectively, for the 3He and 4Li nuclei. Because the core energy and orbital energy levels should change

when the excess electron number of the core changes, we may construct Table I, where selected nuclei are used to establish the core and shell energy levels for different Y. Predictions of the mass of other nuclei are made using the energy Eqn. (3.101) and assuming that the number of protons in a full shell corresponds to the number of electrons in the atomic shells, i.e., 2, 8, 18, .... For each Y, some of the experimental masses are used to establish an energy value; therefore, the predicted value appears the same as the experimental. In each case, the energy value established by this data point appears in the appropriate column. The RMS error in the predicted values of all 21 nuclei was 4.3 MeV, with an arbitrary selection of which nuclei were used to establish an energy level. A better way of approaching the establishment of the energies would be to take the average value of all possible ways to find a particular energy. Table II lists the energy-level average values needed in the total energy equation for the same 21 nuclei. By using the average values from this

> @ . 2

E - )1 E(2, + 1) E(2, = 1) E(3, c*

> @

amu , 3.020562 =

amu 2

10 x 5.49 - 2.016000 +amu 2.012836 = 1) E(3,-4

Table II. Energy-level average values.

Y 2E1+Ec 2E1+6E2+Ec E1 E2 (amu) (amu) (amu) (amu) 1 2.011441 8.048629 1.002976 1.006198 2 2.009952 8.012742 -- 1.000465 3 2.009979 8.013231 -- 1.000542 4 2.067271 8.000773 -- 0.988917 5 2.071474 8.018938 -- 0.991244 6 2.100007 8.032752 -- 0.988969

153

table in Eqn. (3.101) we may construct another table, which compares the predicted masses with the experimental masses and also tabulates the pre-dicted binding energy per nucleon (i.e., BE/A). The RMS error in the predicted masses in Table III was 2.9 MeV. A comparison between the M/A and BE/A will readily display the predicted error in the binding energy per nucleon, because M/A is the error in mass per nucleon; therefore, the sum of M/A and the predicted BE/A is the experimental binding energy per nucleon. In the development of the energy, Eqn. (3.101), we assumed that an extra proton could be added in an orbit and the interaction between that new proton and the other protons could be ignored. This assumption was made even after we saw that any odd proton sets up asymmetrical forces. Thus, errors could have been expected. Still, the RMS errors from this crude averaging procedure do not appear too inaccurate when even the best of the semiempirical nuclear mass formulas does not address nuclei below a Z of 16 because of the large errors that arise.15 To avoid the errors resulting from ignoring proton-proton interaction, we must reconsider the simplest case, a single electron and three protons in orbit around this electron. Proton-proton interaction suggests that the protons will arrange themselves in a plane spaced on the points of an equilateral triangle, as in Fig. 10. For this case, the force on a single proton would be

where R = 3r 397. Then the force becomes

This force differs from the two-proton force of Eqn. (3.99) by the coefficient of the separation, r, in the proton-proton portion of the force. The relativistic circular orbit equations of motion would be

which differs from Eqn. (3.100) only be replacing R = 2r by R = 3r 400. Then the energy is

,er - 1

6

R|k|2 + er

- 1 r

Y|k|- = F R-p

2r-e

23pe¸¹

ᬩ

§¸¹·

¨©§

¸¹

ᬩ

§¸¹·

¨©§

¸¹·

¨©§ OO OSO cos

. er - 1 Y - e

3r - 1

31

r|k| = F r

-e3r

-e23

ee

»»¼

º

««¬

ª¸¹·

¨©§

¸¹

ᬩ

§¸¹·

¨©§ ¸

¹·

¨©§

¸¹·

¨©§ OO OO (3.103)

,er-1 Y-e

3r-1

31

r|k|- =

rm)n(n

r-e

r-p

23p

22ep¸¹·

¨©§

¸¹

ᬩ

§¸¹·

¨©§

¸¹

ᬩ

§¸¹·

¨©§

¸¹

ᬩ

§c OO OOJ (3.10

154

. 4 E+[R(n)]

cm3+Ye-e3

1R(n)

|k|3 = Y) (n, E c

2p

r-

3r-

3er

°¿

°¾½

°°®­

»¼

º«¬

ª¸¹·

¨©§ ¸

¹·

¨©§

¸¹

ᬩ

§

JOO (3.105)

If we consider a possible symmetric orbit of four protons, as pictured in Fig. 11, then the force on each proton would be

where R = 2r 402. Figure 10. Four protons in symmetric orbit.

Table III. Mass predictions for selected nuclei.

Expermental Predicted kM Mass 2E1 + Ec E2 Mass EP - EE Y Z A (amu) (amu) (amu) (amu) (MeV)/A 1 1 2 2.012836 2.012836(2) - 2.012836 0 1 1 2 2.016000 Excited 2H 1.00773(2) 2.016000 0 1 2 3 3.014848 (2) 1.001484(1) 3.020562 1.8 1 3 4 4.025231 (2) 1.010179(2) 4.028288 2 1 3 3.015484 (2) (1) 2.998268 -5.3 2 2 4 4.001422 (2) (2) 4.004472 0.7 2 3 5 5.010676 1.992064(2) (3) 5.010676 0 (2) 2 4 6 6.016880 1.006204(4) 6.016880 0 3 2 5 5.011046 2.004404(2) (3) 5.011046 0 3 3 6 6.013260 (2) 1.002214(4) 6.013260 0 3 4 7 7.012129 (2) (5) 7.015474 0.4 4 1 5 5.035709 (2) (3) 5.036203 0.1 4 2 6 6.017709 (2) (4) 6.024955 1.1 4 3 7 7.013707 2.069947(2) (5) 7.013707 0 4 4 8 8.002459 2.056639(2) 0.988752(6) 8.002459 0 5 2 7 7.026850 2.057980(2) (5) 7.026850 0 5 3 8 8.020624 (2) 0.993774(6) 8.020624 0 5 4 9 9.009337 (2) (7) 9.014398 0.5 6 2 8 8.032752 2.079702(2) (6) 8.032752 0 6 3 9 9.024927 0.992175(7) 9.024245 -0.1 6 4 10 10.010689 (2) (8) 10.017102 0.6

. e2r - 1

)(2r|k| + eR

- 1

R

4|k|2

+ er-1

r|k|Y- = F

2r-p

2r-p

2r-e

24

pp

e

¸¹

ᬩ

§¸¹

ᬩ

§

¸¹·

¨©§

¸¹

ᬩ

§¸¹

ᬩ

§

¸¹·

¨©§

¸¹·

¨©§

OO

O

OO

SO

cos

(3.106)

155

Thus Eqn. (3.106) becomes

where F2 is the two-proton force of Eqn. (3.99). We can also write the circular motion equation of motion as

and the energy equation as

The foregoing discussion on a possible method of accounting for proton-proton orbit interaction seems to imply that we must do every possible nuclear configuration differently. The final answer to this question must await further research. The discussion was presented to point out that ignoring the proton-proton interactions may be an error source in the predicted masses. For instance, consider the force for the symmetric four-proton orbit, F4, given by Eqn. (3.107). This force may be a stronger attractive force for a given Y and r than F2, so long as 2/ < r pO 406, but it is weaker for 2/ > r pO 407. Thus, for orbits where 2/ < r pO 408, four protons with n = 1 can produce less ground-state energy from E4 than the energy expression in Eqn. (3.99) does with two protons in the n = 1 orbit and two in the n = 2 orbit. The large error would thus diminish in the prediction of the 4He nuclei mass. The appearance of the exponential term requires the use of numerical solutions, whether one tries the circular orbit approximation or uses the quantum mechanical approach. This will be the subject of future research. However, the masses predicted by the very simple assumptions and the comparison of experimental and predicted binding energy per mass number, plotted in Fig. 7, seem to imply that a more detailed solution may prove very useful.

,(Y)F + e

r2 - 1

r2|k|2 =

er - 1 Y - e2r

- 141 + e

2r - 1

22

r|k|- = F

2r2-p

2

r-e

2r-p

2r-p

24

p

epp

¸¹

ᬩ

§

¸¹·

¨©§

¸¹

ᬩ

§¸¹

ᬩ

§

¸¹

ᬩ

§

»»¼

º

««¬

ª¸¹·

¨©§

¸¹

ᬩ

§¸¹·

¨©§

¸¹

ᬩ

§

O

OOO

O

OOO

(3.107)

,(Y)F - er3

- 1 r2

|k|2- = rm)h(n

2r3-p

23p

22p¸¹

ᬩ

§

¸¹

ᬩ

§c OOJ (3.108)

. (Y)E - y) (n,E 2 + e R(n)

|k|22 =

(Y)E+[R(n)]

cm4+Ye+e4

1+e22

R(n)|k|4 = Y) (n,E

c2r2-

c

2p

r--

2r-

r2-

4

p

epp

¸¹

ᬩ

§

¸¹·

¨©§

¸¹

ᬩ

§¸¹

ᬩ

§

»»¼

º

««¬

ª¸¹·

¨©§

¸¸¹

·¨¨©

§

O

OOO

J (3.109)

156

Figure 11. Binding energy per mass number versus mass number.

157

CHAPTER 5 - GRAVITATION 5.1 Charge-to-Mass Ratio and Magnetic Moments In Chapter 1 a brief overview of the Dynamic Theory was presented. The fundamental principles of the Dynamic Theory were presented in Chapter 2. From these fundamental laws the constancy of the speed of light was derived, the required geometry was obtained, classical and special relativistic equations of motion were derived, and the conditions requiring quantum mechanics were displayed. The requirements of the fundamental laws were carried further in Chapter 3 by looking at the gauge fields of the resulting five-dimensional geometry when mass is considered as an independent variable. When quantization conditions are considered in five-dimensions we found experimental features of particle physics required by these new laws. For example, we saw that octets are required fundamental states reminiscent of Gell-Man's eight-fold way; the allowed fields for fundamental particles were shown to be quantitized in electric charge; and the radial field dependence to display a short-range non-singular behavior which allowed it to predict nuclear masses from it's deviation from the Coulombic radial dependence and nuclear decay (beta decay) from the asymmetry of while particle forces. Thus, in chapters 2, 3, and 4 we have shown how the Dynamic Theory reproduces, by using the appropriate restrictive assumptions, the fundamentals of all the current branches of physics except gravitation. In Chapter 4 the radial field dependence was derived and the long-range dependence required that the new field components be interpreted as the gravitational field and the gravitational potential. In this chapter we will explore a few aspects of this interpretation. In particular, we will look at some of the predictions of the Dynamic Theory in comparison with Einstein's General Theory of Relativity. Before we plunge into the derivation of a prediction to compare with the General Theory of Relativity let us first consider a question which arises from the necessity of keeping the units straight among the field quantities E, B, V, and V4 when they are all to be considered as components of the five-dimensional gauge field. By considering the units of these field components it is soon found that a charge-to-mass ratio is needed in order that the units of the gravitational field components may be compared, or put in the same equation, with the electric and magnetic components. Let us first see if we can determine this ratio. In the Chapter 4 the derivation of the fields allowed for fundamental particles was presented. These field expressions give rise to the specification of a charge-to-mass ratio which allows conversion of classical gravitation field units to electromagnetic field units. The gravitational field component in the system of field equations with the electric and magnetic components brings up the requirement for a

158

gravitation-to-electromagnetic unit conversion. This need may be seen by looking at the different field quantities. First consider the electric case. The field units are given by [E] = volt/meter, while the expression for the electric force density is Fe = UE with units of newton/meter3. For the gravitational field, in the Dynamic Theory, the units are [V] = webers/meter squared, while the gravitational "current" density has units given by [J4] = ampere/meter2. Thus, the gravitational force density is given by Fa = (J4/c)V where again the units are newton/meter3. In order to compare this system to the classical gravitational system we need to be able to go from a gravitational field with units of newton/kilogram to units of volt/meter. Now

Thus, if ß is a quantity with units of coul/kg, then (1/ß) is the conversion factor we seek. Similarly, we need to convert the gravitational mass density, (J4/c), with units of coul/m3 to units of kg/m3. Obviously ß will also be the conversion factor for this also. The question is; How do we determine this charge-to-mass ratio and is it unique? If we consider the fields the Dynamic Theory gives for fundamental particles we may determine ß. Thus, let us look at the solution of the gauge function for fundamental particles given previously, that is

We showed that this became

for fundamental particles. The functional dependence of the gauge function upon time or mass density was not determined then. Recalling from past reading that measurements of a time dependence of the earth's gravitational field have been reported(). We may proceed to make the simplest possible assumption about the functional form for fJ and ft, namely linear dependence. If the functions have only weak dependence upon time or mass density then this dependence could easily be masked in experimentation not specifically designed to look for it. Thus, lets' consider the form

¸¹

ᬩ

§¸¹·

¨©§

kgcoul

mvolt =

kg - mcoul - volt =

kgnt

. fffff = F ln tr21

JIT

»»

¼

º

««

¬

ª ¸¹·

¨©§

reff = f ln r

-

t21

O

J (5.1)

»»

¼

º

««

¬

ª ¸¹·

¨©§

re ) w+ (s bt) + (a = f ln r

-

21

O

J (5.2)

159

where a, b, s, and w are constants to be evaluated using known information about the proton and time dependence measurements of the gravitational field. The gauge potentials are then

and

Using these potentials the field quantities become

and

In Eqns. (5.4) we may see the effect of the time and mass density dependence of the fields. First, notice that the electric field, Er, vanishes as the quantity b vanishes. From the expression for the radial component of the gravitational field, Vr. we see that b is the time dependence of the gravitational field. Thus, in order for an electric field to exist there must be a time dependence of the gravitational field. Similarly, one may see that the electric field must depend upon the mass density in order for there to be a gravitational field. For a gravitational potential, V4, to exist not only must the gravitational field depend upon time but also the electric field must depend upon the mass density. This is a rather extraordinary revelation!

0 = =

re

r - 1) w+ (s bt) + (aN - =

dr)f (ln =

re ) w+ (s b

icN =

(ict))f (ln =

32

2r

-

121

1

r-

021

0

II

OJI

JI

O

O

¸¸

¹

·

¨¨

©

§¸¹·

¨©§w

¸¸

¹

·

¨¨

©

§¸¹·

¨©§

ww

(5.3)

. r

ebt) + w(aa = )f (lna = r

-

021

04 ¸¸

¹

·

¨¨

©

§

ww

O

JI

,0 = V = V

,re

r - 1 wbt) + (a a )N - N( = V

,0 = B = B = B

,0 = E = E

,re

r - 1 ) w+ b(s

c)N - N( = E

2r

-

041r

r

2r

-01

r

IT

O

IT

IT

O

O

OJ

¸¸

¹

·

¨¨

©

§¸¹·

¨©§

¸¸

¹

·

¨¨

©

§¸¹·

¨©§

(4.4)

. r

e(bw) c

)N - N(a = Vr

-040

4 ¸¸

¹

·

¨¨

©

§ O

160

Now we must check these field quantities in the eight field equations. Because B = 0, then

is satisfied. Next

implies

which is satisfied by the spherical symmetry of the E field. Then

is satisfied if the current density vector, J, vanishes as it should for particles. Looking at

we find that this is satisfied by a spatial charge distribution of

The continuity equation

requires

Then

implies

0 = B x'

0 = E x + tB

c1

'ww

0 = E x '

J

SPPHww

¸¹·

¨©§

ww

¸¹·

¨©§'

Va - J

c4 =

tE

c - B x 0

JH

SUww

x' Va - 4 = E 4

0

cr

er - 2) w+ )b(sN - N(

4 = 4

r-

01OO

OJ

SH

U¸¹·

¨©§

¸¹·

¨©§ (5.5)

J

Uww

x'ww J

a + J + t

= 0 0

. 0 = J 4

Jww (5.6)

0 = Ba + V x 0 Jww

'

161

which is also satisfied by the spherical symmetry of V. When we consider the radial component of

we find that it is satisfied if

while the components in the and directions are satisfied identically. The last equation is

This equation requires that

Note that the expression for J4 satisfies Equation (5.6). We also find that the radial dependence of J4 is identical to the radial dependence of U. Thus, we may rewrite Eqn (5.8) as

T.C. Van Flandern, of the Naval Observatory, has reported a measured very small time rate of decrease in the gravitational field given by him to be approximately 6-parts in 1011 per year. If we designate this rate of decrease by dG/dt, then

From the experimental measurement

then

Thus, the non-zero field quantities are

0 = V x '

Jw

w'

ww E

a = V + tV

c1

04

,N = N 04 (5.7)

. Jc4 - =

tV

c + V 4

4 ¸¹·

¨©§

ww

¸¹·

¨©§x'

PPH S

. er - 2bt) + )(aN - N(

r4cwa- = J r

-014

04

OOO

SP¸¹·

¨©§

¸¹

ᬩ

§ (5.8)

. ) w+ b(s

bt)w + (aca- = J2

04 U

JPH »¼

º«¬

ª

. G = ab � (5.9)

, 10 x 1.9- /yr10 x .6 G -1-18-11 sec##�

. a bt + a #

162

where Z = N1 - N0. We should note in Eqns. (5.10) that the gravitational field depends upon the universal constant, ao, being non-zero. This implies a linkage between the maximum mass conversion rate and the gravitational field in somewhat a similar way as the electric field depends upon the speed of light. Further, it should be noted that if the electric field does not depend upon the mass then there could be no gravitational field. Now the total charge is given by

Therefore, using Eqn. (5.5) and the spherical element of volume dv = r2sinT drdTdM, we have

where R is the radius of physical extent of the particle. If the mass density in the particle is a constant, J0, then the charge is given by

UJPH

S

OOJH

U

O

O

O

J

O

O

O

O

) w+ b(s

awca- = J

cr4

er-2) w+ a(sGZ-

=

,r

er - 1

waa Z

r

er - 1

bt) + w(aa Z= V

,r

er - 1

) w+ z(sGa = E

20

4

4

r-

2

r-

0

2

r-

0r

2

r-

r

»¼

º«¬

ª

¸¹·

¨©§

»»»»

¼

º

««««

¬

ª¸¹·

¨©§

#

»»»»

¼

º

««««

¬

ª¸¹·

¨©§

»»»»

¼

º

««««

¬

ª¸¹·

¨©§

�

�

(5.10)

. ds = qvol

U³

°¿

°¾½

°

°®­

»»¼

º

««¬

ª¸¹·

¨©§

¸¹

ᬩ

§

¸¹·

¨©§

¸¹

ᬩ

§

³

³

ITTO

JS

OH

ITTO

ZJOSH

O

O

ddrdre

r-2)w+(s

c4GaZ- =

. )ddrdr( re

r - 2) + (s

c4GaZ- = q

2r

-R=r

0=r

24r

-

vol

sin

sin

�

�

163

Similarly, we may denote the gravitational "charge" as

Then, using Eqn. (5.8), we have

Now the electrons' force is given by

or

since s >> HJ0. If we compare this with the classical expression for r >> J, then we must have

or

Thus,

. er 1) w+ (sc

GaZ- =

drre

r-2)w+(s

cGaZ- = q

R-

o

2r

-R=r

0=r

O

O

OJ

H

OOJ

OH

¸¹·

¨©§

°¿

°¾½

°

°®­

¸¹·

¨©§

¸¹

ᬩ

§³

�

�

(5.11)

. dvolcJ =

cM

24

vol³

.er - 1

cwaaZ

ds r

e 4

r - 2waaZ-

c

M

r-0

r-0

vol

O

O

OP

SP

OO

¸¹·

¨©§#

»»¼

º

««¬

ª¸¹·

¨©§

# ³ (5.12)

� � � �»»»»

¼

º

««««

¬

ª¸¹·

¨©§

x¸¹

ᬩ

§»¼

º«¬

ª¸¹·

¨©§

¸¹

ᬩ

§

r

er - 1

w+ scGZa er

- 1 w+ sc

GaZ=Eq=F 2

r-

0r-

0re

O

O

O

JO

JH ��

er - 1

r

er - 1

s c

GaGZ F r-

2

r-

22

222

eO

O

OO

¸¹·

¨©§

¸¹·

¨©§

¸¹

ᬩ

§#

�

SH

OH

O

4eZ er

- 1c

sGZa 22

r-

2

#¸¹·

¨©§

¸¹

ᬩ

§ �

,G4

ce er - 1sa 22

22

r-22

�HSO O

#¸¹·

¨©§ (5.13)

164

which gives the product of a and s in terms of the experimental quantities H, c, e, and dG/dt. Let us now turn to the gravitational force, which is given by

By comparing this with the classical expression for the gravitational force we find we must have

or

If we consider the particle to be a proton then Z = 1, and the ratio of the magnitude of the electric force to the gravitational force is given by

thus

and

from Eqn. (5.14). Then, we have

,

er-1G4

ec asr

2-2

1OO

SH ¸¹·

¨©§

r#

�

. r

er

- 1)waa(Z =

re

r - 1wa)a(Zer

-1waaZ =

V M= F

2r

2-220

2r

-

0r-0

rg

O

OO

OP

OOP

¸¹·

¨©§

¸¹·

¨©§

¸¹·

¨©§¸¹

ᬩ

§

MG = er - 1)waa(Z 2

r-2

0 PO O¸¹·

¨©§

er

- 1

G

aZm = aw

r-0 OO

P

¸¹·

¨©§¸¹

ᬩ

§ (4.14)

,mG4

e = F 2

2

R SH

F4

e = mGR

22

SH

F4

e = )waa(R

22

0 SHP

165

Now choose

so that, by Eqn. (5.11),

and, from Eqn. (5.15),

From Eqn. (5.13) we find

These values point out the extremely weak time dependence of the gravitational field and the very weak mass dependence of the electric field. This mass dependence may be better seen if we write

then substitute for the field parameters to arrive at

. F4ca

e = F4a

e = awR0r0 SHSH

P¸¹

ᬩ

§ (5.15)

He = a (5.16)

,Ge = b �¸¹·

¨©§H

(5.17)

.

er-1ceZa

Gm = w

2r-2

1

oOO

H

¸¹·

¨©§

(5.18)

er-1G4

c = s

2r-2

1OO

S ¸¹·

¨©§�

(5.19)

»»»»

¼

º

««««

¬

ª¸¹·

¨©§

r

er - 1

c

) w+ (sGaZ = E 2

r-

r

OOJ�

166

Now let us return to the search for the charge-to-mass ratio since we have all the necessary information. The quantity we defined as the gravitational "charge" is given by Eqn. (5.12). If we divide this by c we have

but, by Eqn. (5.14), this becomes

or, rewriting

Thus, the charge-to-mass ratio we seek is given by

or

.r

er - 1

GeamG4 + 1

4Ze =

r

er - 1

e

mGa4G + 1

4Ze =

r

er - 1

F4

1ac +

G4c

cGZe = E

2

r-

0

2

r-

2

2

0

2

r-

r0r

»»»»

¼

º

««««

¬

ª¸¹·

¨©§

»»¼

º

««¬

ª¸¹

ᬩ

§¸¹·

¨©§

»»»»

¼

º

««««

¬

ª¸¹·

¨©§

»»¼

º

««¬

ª¸¹·

¨©§

»»»»

¼

º

««««

¬

ª¸¹·

¨©§

»»¼

º

««¬

ª¸¹

ᬩ

§

O

O

O

O

HSJ

SH

OHSJ

SH

O

SJ

SH

�

�

�

�

,er-1

cwaaZ =

cM

r-0 OO

P¸¹·

¨©§

er-1Gm =

er-1G

cm =

er-1G

aZm

caZ =

cM

r-

r-

r-

0

0

O

O

O

OH

OP

P

OP

P

¸¹·

¨©§

¸¹·

¨©§

¸¹

ᬩ

§

¸¹·

¨©§

¸¹

ᬩ

§¸¹

ᬩ

§

.er-1Gm =

cM

r-OO

H ¸¹·

¨©§ (4.20)

,G = m

M/c = HE ¸¹·

¨©§ (5.21)

. coul/kg 10 x 2.4296 = G = -11HE

167

The surprising, and pleasing, thing about this result is that it is

formed as the product of two known physical quantities rather than

depending upon new quantities, such as a0 and , whose values are not well

known. Further, in retrospect, it appears to be the simpliest, if not the

only, combination of an electromagnetic parameter and a gravitational

parameter whose units are coul/kg.

It is worthwhile to point out that the dependence of the fields upon

time and mass density (ie, b and w) is extremely small but is essential in

establishing ß, and the inductive coupling between the electromagnetic

and gravitational fields.

The charge-to-mass ratio brings up the notion that a rotating

gravitational, electrically neutral body should have a magnetic moment

stemming from the effective electric charge associated with the

gravitational mass. Given the charge-to-mass ratio we may quickly look at

its prediction for the earth's magnetic moment.

Using ß, the "effective" charge associated with a gravitational mass is

given by . M = qeff E

For the earth this effective charge would be . coul 10 x 1.454 = q 14

eff

Thus, if the magnetic moment of the earth is given by � � ,A/2Mq = effP

where A is the earth's angular momentum then we have

)10 x 2(5.983)10 x )(7.2910 x )(9.7110 x (1.454 =

I2Mq

=

24

5-3714

eff ZP ¸¹

ᬩ

§

or

m - amp 10 x 8.6 = 222P

This predicted value of the earth's magnetic moment compares very

well with the experimental value of 8.1 x 1022 amp- m2.

5.2 Perihelion Advance

No serious suggestion that the additional vector field in the five-

dimensional gauge equations of the Dynamic Theory be the gravitational

field can be made without giving due consideration to the explanation of

the planetary perihelion advance provided by Einstein's General Theory of

Relativity. Though several attempts have been made to explain the

perihelion advance by other means none has succeeded in casting much

doubt on Einstein's explanation.

Let us recall some of the main features of the classical problem of

planetary orbits. Kepler's first law states that a planet describes a closed

elliptical orbit with the sun at a focal point. However, the presence of such

168

small influences as other planets moving in the suns' field causes a perturbation in the motion of a given planet, and the resulting orbit is not precisely elliptic. Indeed, one may think of the actual orbit as a slightly bumpy ellipse which may precess in the plane of motion; that is, the perihelion shifts about and does not always occur at the same angular position. The fact that the idealized classical orbit is a closed ellipse is a result peculiar to the Newtonian inverse-square law; in fact, Newton himself found that, if the force of gravity were proportional to 1/r(2+G) instead of 1/r2, then a planetary orbit would not be closed and a perihelic shift of order G would occur. Indeed, this result was taken to indicate that, since planetary orbits are very nearly closed, the Newtonian inverse-square law must be very accurate, as in fact it is. Let us now ask were may there be room for differences between the predictions of classical celestal mechanics and the celestial mechanics of the General Theory of Relativity or the Dynamic Theory presented here. Since Kepler's first law is experimentally verified to be correct to a high accuracy, we might expect that non-Newtonian Theories may merely add a few bumps to the nearly elliptic orbits and contribute somewhat to perihelic motion. Since angles are much more conveniently measured in astronomy than are distances, it is natural to concentrate on perihelic motion. Conveniently enough, there is, in fact, a well-known discrepancy in classical mechanics concerning the perihelic motion of the planet Mercury. Because of Mercury's high velocity and eccentric orbit, the perihelion position can be accurately determined by observation. The difference between the classically predicted perihelic shift (due to perturbations by other planets) and the observed perihelic shift is 43 seconds of arc per century. Even though this is a very small difference, it is about a hundred times the probable observational error and represents a true discrepancy from the very precise predictions of celestial mechanics which has bothered astronomers since the middle of the last century. The first attempt to explain this discrepancy consisted in hypothesizing the existence of a new planet, Vulcan, inside the orbit of Mercury, and much theoretical work was done to predict the position of Vulcan, using the known perturbation on Mercury's orbit. However, careful observation failed to discover the hypothetical planet, and the hypothesis was finally abandoned in 1915 when Einstein used general relativity theory to explain the observed effect. Now let us look at what the Dynamic Theory offers as an explanation for the perihelic advance and then compare it to the predictions of the general relativity theory. The classical equations of motion are

. 0 = R2m + mrF(r) = mr - rm 2

TTT

����

��� (5.22)

The second of Eqns. (5.22) has the solution

169

where L is the angular momentum. Using Eqn. (5.23) the first of Eqn.s (5.22) may be written

or

where

We are seeking the prediction of the Dynamic Theory with respect to the perihelion advance. This may be found by comparing the frequency of small radial oscillations about steady circular motion for the effective potential given by Eqn. (5.25) for the non-singular potential of the Dynamic Theory with the frequency of revolution. By considering the non-singular potential of the Dynamic Theory, Eqn. (5.25) becomes

with K = -GMm, where G is the gravitation constant, M is the mass of the sun, and m is the mass of the planet of interest. Equation (5.23) gives the frequency of revolution. To determine the frequency of small radial oscillations about steady circular motion we need to evaluate the second derivative of the effective potential, v, the radius for which the first derivative is zero. The first derivative of the effective potential is obtained by differentiating Eqn. (5.26) with respect to r. This may be found to be

The second order derivative of the effective potential may be found to be approximately

,mrL =

2T� (5.23)

rM

L + F(r) = rM3

2��

> @ ,(r)v r

- rM cww

{�� (5.24)

. Mr2L + v(r) = (r)v

2

2

¸¹

ᬩ

§c (5.25)

,Mr2L + er

K = (r)v2

2

r-

¸¹

ᬩ

§¸¹·

¨©§c �

O (5.26)

> @ . mrL - er

- 1 rK- = (r)v

r 3

2

r-

2 ¸¹

ᬩ

§¸¹·

¨©§

¸¹

ᬩ

§cww OO (5.27)

,mr

L3 + er2 - 1

r2K

r(r)v

4

2

r-

32

2

¸¹

ᬩ

§¸¹·

¨©§¸¹

ᬩ

§#wcw OO (5.28)

170

when terms involving O2/r2 are considered negligible with respect to terms involving O/r. We may determine r0 from the condition

The radius, r0 is the radius of near circular orbit and the effect of the exponential factor and (1-O/r) factor will be negligible for O<<r. Thus, we may approximate Eqn. (5.29) by

so that

If we approximate the exponential factor in Eqn. (5.28) by its power series expansion and retaining only those terms whose dependence upon O/r are linear or less, then Eqn. (5.28) is approximated by

Now the frequency of small radial oscillations about steady circular motion may be found from

Thus, we have

An approximation for the frequency of small radial oscillations about steady circular motion may now be made by taking the square root of Eqn.

. mrL - e

r - 1

rGMm = 0 =

r(r)]v[

30

2

r-

020

0 ¸¹

ᬩ

§¸¹

ᬩ

§¸¹

ᬩ

§wcw OO (5.29)

¸¹

ᬩ

§¸¹

ᬩ

§

mrL -

rGMm = 0 3

0

2

20

. GMm

L r 2

2

0 ¸¹

ᬩ

§# (5.30)

.

mrL3 +

r3 - 1

r2K =

mrL3 +

r - 1

r2 - 1

r2K

r(r)v

4

2

3

4

2

32

2

¸¹

ᬩ

§¸¹·

¨©§¸¹

ᬩ

§

¸¹

ᬩ

§¸¹·

¨©§¸¹·

¨©§¸¹

ᬩ

§#wcw

O

OO

(5.31)

,r = r r(r)v

m1 02

22

»¼

º«¬

ªwcw¸

¹·

¨©§{Z

. L

mGM6 + 1L

mMG =

LmMG3 +

LmGM3 - 1

LmMG2- =

LGMm

mL3 +

LmGM3 - 1

LGMm2GMm-

m1

2

2

6

644

6

644

2

2

6

644

2

2 42

2

2

2

2 32

»¼

º«¬

ª¸¹

ᬩ

§¸¹

ᬩ

§

»¼

º«¬

ª¸¹

ᬩ

§¸¹

ᬩ

§

°¿

°¾½

°

°®­

¸¹

ᬩ

§¸¹

ᬩ

§»¼

º«¬

ª¸¹

ᬩ

§¸¹

ᬩ

§¸¹·

¨©§#

O

O

OZ

(5.32)

171

(5.32) and considering the second terms of the second factor as small compared to one. Thus, we have

The perihelion advance per revolution may now be found as the difference between Eqns. (5.33) and Eqn. (5.23) evaluated at r0, divided by the orbital frequency, or

so that

The perihelion advance predicted by Einstein's General Theory of Relativity is given by

If O were to be such as to provide an identical prediction as the General Theory then O would have to satisfy

For G = 6.7 x 10-8 gr-1cm3/sec2, M = 1.98 x 1033gr, and c = 3 x 1010 cm/sec,

or

This is an extremely small value compared to the radius of the sun but is sufficient within the Dynamic Theory to provide the same prediction of perihelion advance as the General Theory of Relativity.

. L

mGM3 + 1 L

mMG 2

2

3

322

»¼

º«¬

ª¸¹

ᬩ

§¸¹

ᬩ

§#

OZ (5.33)

¸¹

ᬩ

§

°¿

°¾½

°°®­

»¼

º«¬

ª¸¹

ᬩ

§¸¹

ᬩ

§

¸¹

ᬩ

§

mMGL

LmMG -

LmGM3 + 1

LmMG 2 =

- 2 =

322

3

3

322

2

2

3

322 OS

TTZ

SGT�

�

. L

GMm3 2 2

2

¸¹

ᬩ

§#

OSGT

. Lc

mMG3 2 = 22

222

GiR ¸¹

ᬩ

§STG

c

GM = 2O

cm 10 x 1.47 = cm)10 x (3

)10 x )(1.9810 x (6.7 = 15210

33-8

O

. m10 x 1.47 = 3O

172

5.3 Redshifts

Einstein's General Theory of Relativity predicted the advance of the

perihelion of planetary orbits by using the full effect of the geometrical

equations. We saw in the previous section that the Dynamic Theory

predicts a similar planet orbit perihelion advance. Another of the

predictions of Einstein's Theory concerns the redshifts associated with

light received from distant light emitting objects or when light travels

through a changing gravitational field.

The Dynamic Theory should also predict frequency shifts that are

experimentally measurable. If it does not then it doesn't have the same

strength of predictability as Einstein's General Theory.

There are two types of redshifts resulting from the theoretical

approach of the Dynamic Theory. First, there is an expansion red shift due

to the increasing "entropy" of the universe. Secondly, there is a frequency

shift caused by a difference in the gravitational strength between the point

of emission of the light and the point of its reception. Both of these types

of frequency shifts are the result of a difference in the effective unit of

action at the emission point and the reception point.

Both of the above types of frequency shifts may be referred to as of

geometrical in origin in that they both come from the gauge function.

However, each originates from a different variable change in the gauge

function. For instance the expansion shift involves considering the

universe as an isolated system resulting in the entropy principle requiring

small frequency shifts toward the red. This comes from the gauge function

being dependant upon time. The second type of frequency shift comes

from the gauge functions dependence upon space and mass.

We may first consider these types of frequency shifts to be

independent and look at each in turn. Then we shall consider them

together. First, we will need to consider the local systems where a photon

is first emitted and then where it is received. In both systems the energy of

a photon is given by hv, where v is the frequency and h is the effective unit

of action. The effective unit of action is the product of the gauge function

and Planck's constant if a locally flat metric is considered.

At the heart of both types of frequency shifts is the gauge function

which has previously been given as

and found to be

fffff = fln tr21

JIT (5.34)

,r

eff = fln r-

t21

¸¸

¹

·

¨¨

©

§ O

J

173

where ft and fJ indicates functions of time and mass density. We need to determine more about the gauge function than we previously have. The square of the arc lengths differ by the multiplicative gauge function as

or

From this we see that the differential change in entropy is given by

Recalling that there can be no decrease in the entropy for an isolated system we must then consider the possible effect of the entropy principle upon the universe as an isolated system. We can see from this line of thinking that ft in Eqn. (5.35) is the one to focus on for the moment. The simplest function is of course the linear function and this linear dependence appeared previously in section 5.1. Suppose then, we consider the effective unit of action for an isolated universe at some time which we will set at t = 0. We find

which can be written

at t = 0. Here the value of the effective unit of action corresponds identically with Planck's constant, h. At some later time t = T the effective unit of action would be given by

where A is a constant. Let us further consider a change of variables using the distance light will travel in free space instead of the time, T. Since T = L/C, L is the distance variable we seek. We now have

If a photon were emitted at t = 0 it would have been emitted with an effective unit of action given by Eqn. (5.36). If that photon is received at

)f(d = )dq( 220 V

. )(d r

eff2 = )dq( 2r-

20 VO

JH »»

¼

º

««

¬

ª

¸¸

¹

·

¨¨

©

§exp

. d r

eff = dq r-

t0 V

O

J »»

¼

º

««

¬

ª

¸¸

¹

·

¨¨

©

§exp (5.35)

,fh = h 010

[0] h = h10 exp (5.36)

[AT] , H = h1T exp

. c

AL H = h1T »¼

º«¬ªexp (5.37)

174

the later time, T, at a distance of L, then the universe's effective unit of

action would be given by Eqn. (5.37) at reception. If the energy of the

photon when emitted is given by Q e10 h 85, then no loss of energy by the photon until

reception would require that

Substituting from Eqn. (5.36) and Eqn. (5.37) into Eqn. (5.38) we find

The frequency shift would be given by

or

The question arises whether or not the frequency shift given by Eqn.

(5.40) is red or blue? From Eqn. (5.40) it may be seen that the frequency

shift is negative if A > 0, thus the shift is red or blue as A is positive or

negative. Going back to Eqn. (5.35) and using the gauge function of Eqn.

(5.37) we see that

This indicates that a given element of arc length, dı, yields a larger change

in entropy, dqo, at the time T = L/C than before at time t = 0. Thus, the

entropy change is increasing and our universe is expanding.

The expansion red shift given by Eqn. (5.40) may also be expressed

in terms of wavelength as

Equation (5.41) may be expanded as

which may be approximated by

QQ r1Te

10 h = h (5.38)

. c

AL = re »¼º

«¬ªexpQQ (5.39)

e

e - 1 = - =

cAL

cAL

e

er

e

¸¹·¨

©§

'

QQQ

Q

Q

. 1 - e = cAL-

eQ

Q' (5.40)

Vde = dq cAL0 ¸¹·¨

©§

. 1 - e = 1 - = cAL

r

e

e QQ

O

O' (5.41)

... + c

AL 3!1 +

cAL

2!1 +

cAL =

32

e¸¹·

¨©§

¸¹·

¨©§

¸¹·

¨©§'

O

O (5.42)

175

when AL << c. Experimentally it has been determined that

where H is the Hubble constant which is given by

Then we find the predicted frequency shift given by Eqn. (5.42) is seen in nature when the constant, A, is taken as the Hubble constant. From Eqn. (5.41) we see that experimentally found red shifts can be used as astronomical markers from the expression

For small experimental red shifts, compared to one, the astronomical distances L, given by Eqn. (5.44), are not significantly different from the linear markers given by the approximation of Eqn. (5.43). However, as the red shift begins approaching unity the difference becomes significant. Turning to the second type of frequency shift, and returning to Eqn. (5.34) to again write

The effective unit of action is expressed by

But when in a locally Euclidean geometry the gkl = Gkl then Eqn (5.46) becomes

From the expressions for the gauge potentials

c

AL e#

'

O

O (5.43)

,c

HL e#

'

O

O

. 10 x 0.6) + (5.6 = H 17-1 sec_

. + 1 Hc = L

e»¼

º«¬

ª¸¹

ᬩ

§ '¸¹·

¨©§ expln

O

O (5.44)

. r

e ff = f r-

21

¸¸

¹

·

¨¨

©

§ O

JHln (5.45)

. x l sj

+ gfh = h s

ie

kl

jk

1

»»¼

º

««¬

ª

¿¾½

¯®­

³³ ˆ (5.46)

. r

eff2 h = fh = hr

-

t1

»»

¼

º

««

¬

ª

¸¸

¹

·

¨¨

©

§ O

Jexp (5.47)

176

and

Using Eqn. (5.49), F14 gives the radial component of the gravitational field. From Eqn. (5.48) we find

and

The expression for the gravitational field given in Eqn. (5.50) is in terms of a field density. By integration over the volume occupied by the gravitational mass density we have the gravitational field

where M is the total gravitating mass. For any weak time variation in the field we can ignore the time dependence. Thus, we have

to be compared with the experimental field

Certainly for r>>O Vr in Eqn. (5.52) is approximated by the experimental expression of Eqn. (5.53) if A = -G and

From Eqn. (5.54) we find that

x

)f(ln = x

(lnf) 21 = 3

21

jj ww

ww

I (5.48)

II ij,ji,ij - = F (5.49)

¸¹·

¨©§¸¸

¹

·

¨¨

©

§

ww

r - 1

reff- =

r)f(ln =

2r

-

t21

1O

IO

J

. r

e r

- 1 ddf

fa - = Fr

-

to14 ¸¸

¹

·

¨¨

©

§¸¹·

¨©§¸¹

ᬩ

§O

J OJ

(5.50)

,r

e r

- 1 dMdf

fa - = Vr

-M

tor ¸¸

¹

·

¨¨

©

§¸¹·

¨©§¸¹

ᬩ

§O

O (5.51)

¸¸

¹

·

¨¨

©

§¸¹·

¨©§

¸¹

ᬩ

§r

e r

- 1 dMdf

aa - = Vr

-M

or

OO (5.52)

. r

GM = V 2r exp (5.53)

. M= dMdf M

177

The conversion to a field density may be done by dividing by the mass, M, so that the gauge function in Eqn (5.47) becomes

or

where c2 has been used to obtain a unitless quantity, which must be the case for f. Now, using the unit of action given by Eqn. (5.55), suppose a photon is emitted from one body with a gravitational field

and is received in another gravitational field

the conservation of photon energy would then require

or

so that

The frequency shift would then be

constant. + M21 = f 2

M

»»»»

¼

º

««««

¬

ª

¸¸

¹

·

¨¨

©

§¸¹·

¨©§

re

cM

constant + M212G -

H = hr

-

2

2

1O

exp

,r

e cGM- h = h

r-

21

»»

¼

º

««

¬

ª

¸¸

¹

·

¨¨

©

§¸¹

ᬩ

§O

exp (5.55)

,eRc

GMR

-

12

11

1O

¸¹

ᬩ

§

eRcMG

R-

22

22

2O

¸¹

ᬩ

§

QQ rree h = h cc

»¼

º«¬

ª¸¹·¨

©§¸¹

ᬩ

§»¼

º«¬

ª¸¹·¨

©§

¸¹

ᬩ

§e

RcGM- = e

RcGM- R

-

22

2rR

-

12

1e 2

2

1

1 OOQQ expexp

. eRc

GM + eRc

GM- = R-

22

2R

-

12

1er 2

2

1

1

»¼

º«¬

ª¸¹

ᬩ

§¸¹

ᬩ

§ OOQQ exp (5.56)

178

For R2 >>O2 and R1 >>O1, then Eqn. (5.57) may be approximated by

The approximation in Eqn. (5.58) shows that if M1/R1 > M2/R2 then this

frequency shift given by Eqn. (5.57) is negative, or towards the red end of

the spectrum.

We can make the further simplification of assuming that both

then we would have the approximation that

In terms of wavelengths we have

with the above assumptions Eqn. (5.60) is approximated by

Suppose we look at this red shift for a photon emitted from the

surface of the sun and received at the earth's surface. The needed

numbers are:

G = 6.67x10-11nt-m2/kg2 Msun = 329,390(5.983x1024kg)

Mearth = 5.983x1024kg Rsun = 6.953x108m

Rearth = 6.371x106 c = 3x108 m/sec

Thus, from Eqn. (5.61),

»¼

º«¬

ª¸¹

ᬩ

§¸¹

ᬩ

§'e

RcGM - e

RcGM = - = R

-

12

1R

-

22

2

e

er

e1

1

2

2 OO

QQQ

Q

Q exp (5.57)

. 1 - e cG

e

2¿¾½

¯®­

¸¹

ᬩ

§#

'

Q

Q (5.58)

,1 < < Rc

GM and 1 < < Rc

GM2

22

12

1

. Rc

GM - Rc

GM 1

21

22

2

e_

Q

Q' (5.59)

»¼

º«¬

ª¸¹

ᬩ

§¸¹

ᬩ

§

'

eRc

GM - eRc

GM =

1 - =

R-

22

2R

-

12

1

r

e

e

2

2

1

1 OO

QQ

O

O

exp (5.60)

. Rc

GM - Rc

GM 2

22

12

1

e_

O

O' (5.61)

179

or 0.63 in km/sec. This is the same as predicted by Einstein's General Theory of Relativity. For a terrestrial test of the red shift the prediction would be

If 'R = 72 ft = 21.95m, then 'O/Oe | 2.4x10-15. Since the approximation given by Eqn. (5.59) may be expressed as

where 'M = G[(Me/Re)-(Mr/Re)], then it may be seen that the red shift given by Eqns. (5.57) and (5.60) produce the red shifts predicted by Einstein's General Theory of Relativity if R1>>O1, R2>>O2, GM1<<c2R1, and GM2<< c2R2. However, if these conditions of approximation are not met then one must resort back to Eqn.'s (5.57) and (5.60) for the predicted red shifts. Suppose one considered a photon which may have been emitted on a dense star such that GM1/C2R1 is too large to allow a simplification of the exponential expression. If this photon were received on earth then, by Eqn. (5.60), we would have

Because of the small value of (GMearth)/(C2Rearth) 7 x 10-10, then the approximation becomes

where GM/c2R is the gravitational field at photon emission. From Eqn. (5.62) we may learn something of a stars' density by the red shift in the light received from it. For instance, Eqn. (5.62) has

Notice that even the large red shifts displayed by quasars are allowed by Eqn. (5.63) without requiring them to be at the far reaches of our universe.

,10 x 2.1 6-

e#

'

O

O

. R Rc

GM = R + R

M - RM

cG

222e

'¸¹

ᬩ

§»¼

º«¬

ª¸¹·

¨©§

'¸¹·

¨©§¸

¹

ᬩ

§' _O

O

,cd 2

e

'' _Q

Q

. 1 - earth Rearth M -

RM

cG

e

e2

e»¼

º«¬

ª¸¹

ᬩ

§¸¹

ᬩ

§#¸¹

ᬩ

§ ' expexpO

O

,1 - e rcGM

e 22

2¸¹

ᬩ

§¸¹

ᬩ

§ ' _expO

O (5.62)

. + 1 ln Rc

GMe

2 »»¼

º

««¬

ª¸¹

ᬩ

§ '

O

O

exp

_ (5.63)

180

We have looked at the predicted frequency shifts as if they were independent phenomena. In the sense that they stem from the same gauge function, then both types of redshifts may be present in any light received. Thus, we should look at the expression for the entire red shift. Suppose we let

where s, w, z, a, b, and k are to be evaluated. The effective unit of action becomes

for t = 0. From the above, let us take s = w = 0, then Eqn. (5.65) becomes

when integrated over the entire gravitating mass as before and the subscript, e, refers to the unit of action at the place and time of emission of the photon. A similar expression is found at the place and time of reception, or

Equations (5.65) and (5.66) may be used to express the frequency shift required to conserve photon energy, since

Thus, we have

then

erk bt) + (a )z + w+ (s = fff = fln R

-2tr2

1 O

J JJ ¸¹·

¨©§

,bt) + )(az + w+ (serk h = h

2r

-10 »

¼

º«¬

ª¸¹·

¨©§ JJ

Oexp (5.64)

,eR

M)bt + kz(a h = h R-

e

ee1e e

e

»¼

º«¬

ª»¼

º«¬

ª Oexp (5.65)

. eR

M)bt + kz(a h = h R-

r

rr1r r

r

¿¾½

¯®­

»¼

º«¬

ª Oexp (5.66)

. h = h r1re

1e QQ

,eR

)bt + (azkM - eR

)bt + (azkM = R-

r

rrR

-

e

eeer r

r

e

e

¿¾½

¯®­

»¼

º«¬

ª»¼

º«¬

ª OOQQ exp

181

Similarly, the expression for the wavelength shift becomes

If we write Eqn. (5.67) as a power series and make the approximation of keeping only the first term, we find

where we've also let te = 0. By letting tr = L/c Eqn. (5.68) becomes

We want to evaluate the unknowns a, k, z, and b in terms of the previously determined quantities such as G, c, and H. Therefore, suppose that the gravitational field at the time of emission of the photon is the same as here on earth at its reception, then we find Eqn. (5.69) becomes

Experimentally, we have found the expansion red shift is given by

thus, we should have

¿¾½

¯®­

»¼

º«¬

ª'e

Rbt + (aM - e

R)bt + (aM zk = R

-

r

rrR

-

e

ee

er

r

e

e OO

Q

Q exp

. 1 - R

)bt + (aM - eR

)bt + (aM zk = e

eeR

-

r

rr

er

r

¿¾½

¯®­

»¼

º«¬

ª' O

O

O exp (5.67)

,R

e(a)M - R

e)bt + (aM zk e

R-

e

r

R-

rr

e

e

e

r

r

»»¼

º

««¬

ª#

'OO

O

O (5.68)

. R

eaM - R

ecbL + aM

zk e

R-

e

r

R-

r

e

e

er

r

»»»»

¼

º

««««

¬

ª¸¹·

¨©§

#'

OO

O

O (5.69)

. CR

zkbLM earth

earth

e#

'

O

O (5.70)

,c

HL = e¸¹

ᬩ

§ '

O

O

exp

. R

zkbM = Hearth

earth (5.71)

182

On the other hand if the time between photon emission and reception is sufficiently close then our approximation to Eqn. (5.69) can be written as

where Rr>>Or and Re>>Oe. Thus, in order to compare with experiment, we set

If we now substitute Eqns. (5.71) and (5.73) into Eqn. (5.68), we have

where R/M is the radius and mass of the earth as in Eqn. (5.71). While the full expression, Eqn. (5.67) becomes

From Eqn. (5.75) it may be seen that, if the receiving location is the earth, then the time dependence in Eqn. (5.75) is given by

so that b, of Eqn. (5.67) is given by -H. Thus, the gauge function dependence upon time is also given by b= -H. Recall that a time dependence of the gravitational field has been reported by T. Van Flandern with a reported value of b = -1.9 x 10-18sec-1. The measured Hubble constant is H-1 = (5.6 r 0.6) x 1017 sec so that 2.00 x 10-18 sec t H t 1.61 x 10-18 sec. Thus, we see that the Dynamic Theory predicts that the expansion redshift and a time decrease in the gravitational field strength are both the result of the time variation of the gauge function. Further, it seems amazing that two such different and difficult type measurements have such a good agreement! Returning to the wavelength shift given by Eqn. (5.75), it may be seen that the contribution of expansion, or gravitational potential, to the total red shift is contained within this equation. Equation (5.75) has three unknowns: the astronomical distance L, the mass of the emitting star (or object), Me and the size of the emitting star, Re. Given only two pieces of

,RM -

RM zka

e

e

r

r

e»¼

º«¬

ª' _O

O (5.72)

. cG - = zka 2 (5.73)

,eMR

RM

cHL +

ReM -

ReM

cG- R

-

r

r

e

R-

e

r

R-

r2

er

re

e

r

rO

OO

O

O¸¹·

¨©§¸¹

ᬩ

§¸¹·

¨©§

»»¼

º

««¬

ª¸¹

ᬩ

§#' (5.74)

1.-eMR

RM

cHL+

ReM-

ReM

cG-= R

-

r

r

e

R-

e

r

R-

r2

eR

re

e

r

r

°¿

°¾½

°

°®­

¸¹·

¨©§¸¹

ᬩ

§¸¹·

¨©§

»»¼

º

««¬

ª¸¹

ᬩ

§' OOO

O

O exp (5.75)

Ht = c

HL

183

experimental data such as the redshift and the apparent luminosity, we

can determine the astronomical distance, L, and the gravitational density,

Me/Re by assuming a mechanism for the light production (ie, sun-like).

Given another data point such as light fluctuation periodicity then a

limiting size might be obtained.

5.4 "Fifth" Force

Is a "fifth" force really necessary? Obviously, a new force is much

more exciting than finding an explanation for the measured effect within

another force description. A new force may be more tenable because it

may appear not to compete with existing forces. A correction to existing

forces may lack excitement and must certainly be shown to be compatible

with existing forces where they are measured to great accuracy. A

correction to an existing force is usually difficult to find and may go against

the preference of many. But to assume there is an additional force is to

assume its independence and would then necessitate yet another force to

be "unified".

Accurate measurements show that the gravitational force of the

Earth differs from Newton's Law at close range. More specifically, the

difference in the Earth's gravitational field over a difference of height in a

deep well is not the same as predicted by Newton's Law. This leads to a

simple choice; either Newton's Law of gravity needs to have a correction or

an independent fifth force is needed to explain the difference. In the past

when we found that the proton-proton scattering data differed from the

coulombic predictions we opted for an independent force and have had the

fun of searching for a method of unifying electromagnetism and the strong

force every since. We could make that same choice here, or we could

investigate the difference in the prediction of the non-singular potential

and Newton's law of gravity. To do this we need to look at the gravitational

attraction on a mass in a well deep down from the surface of the Earth.

Freshman texts on physics typically show how to calculate the

gravitational influence of a thin spherical shell on a mass both inside and

outside of the shell. This is the procedure we need here because a mass in

a deep well feel the influence of both the mass of the Earth interior to it

and in the shell of the Earth exterior to it where the shell thickness is the

depth of the well. If we recall the procedure for this using the Newtonian

potential, then we remember that, for the 1/r potential, all of the mass

interior to the test mass attracts it as if the mass were located at the center

of the Earth. On the other hand we recall that there is no gravitational

influence due to the mass in the outer shell which is exterior to the test

mass in the deep well.

For a potential which differs from the 1/r Newtonian potential these

conclusions may not be true. Indeed, ones first suspicion is that they are

184

not the correct conclusions. What we now need to do is to calculate the influence on a test mass both outside and inside of gravitating mass. First, suppose we calculate the gravitational influence on a mass exterior to a thin spherical shell. Using the neo-Newtonian potential we find the integral to be

dx exf(x)

dxd = F

dxex - 1 1 +

xr - R

RrtmG- = F

x-

r+R

r-Rx=

x-

2

22

2

r+R

r-Rx=

»¼º

«¬ª

¸¹·

¨©§

»¼

º«¬

ª»¼

º«¬

ª

³

³O

OOUS

Figure 12. Neo-Newtonian Potential. Figure 13. Neo-Newtonian Force. Figure 14. Gravitational attraction of a section dS of a spherical shell of matter on m. where by making use of the integral tables and a lot of algebra we arrive at the solution F is an improper integral for R=r. This means that terms in the series have denominators which tend to zero as R tends to r. We must show that the series converges because this is the case when our test mass is at the bottom of a deep well. It would then be at the outer surface of the inner mass. It is easy to see that as O tends to zero the solution tends to the classical solution. If we now consider our test mass to be inside a thin shell and look at the force.

as O o 0, F o 0 which is the classical result. Suppose now we look at a couple of approximations which may give some insight into the influence of the exponential term in the potential. First, let us consider a mass outside another mass for which R-r>>O, then we would have the approximation

°¿

°¾½»¼

º«¬

ª

¯®­

»¼º

«¬ª

»¼º

«¬ª¸

¹

ᬩ

§

»¼º

«¬ª

¸¹

ᬩ

§

¦

³

f

)R-(r1 -

)r+(R1

N.N!)(-2 +

r+RR-r 2 + e - e 2r

RrmG- =

dx exf(x)

dxd

RrtmG- = F

NN

N

1=N

r)-(R--

r+R-

2

x-

r+R

r)--(Rx=2

OO

OUS

US

OO

O

log

. > > r-R if; r-Rr-2R - 1

RGMm- F

222OO »

¼

º«¬

ª¸¹

ᬩ

§¸¹

ᬩ

§#

185

This result shows that the force of attraction on a test mass outside another mass is reduced by the second term in the square brackets. This is, of course what we should have expected from a potential which deviates from the Newtonian potential by turning around and going back to zero. The first deviation from Newtonian-like character would be to become weaker. The other approximation to consider is that for the expression for the force an the test mass inside the shell. For this we find that inside the shell if r-R>>O, then

This is a force away from the center of the shell and toward the inside of the shell. The Big Question is: What is the force when the test mass is on the immediate exterior, or interior, of a shell? That is, do we have convergence of the infinite series in the solutions for both the inside and outside forces on the test mass. To address this consider the absolute value of the ratio of the n+1 and the nth terms in the force for a mass outside of a shell of finite thickness t, then from our previous results we found that

The question arises whether or not F is finite for t > 0? If R=r+t/2 then R+r=2r+ t/2 and R-r = t/2, or r = R - t/2 ? R+r = 2R - t/2 ; R-r = t/2

if we have 2R>>t/2 and 2R>>O then we may make the following approximation,

. > > R-r if; )R-rr(

R-rR + r )(- R

GMm- = F22

22

2OO »

¼

º«¬

ª¸¹

ᬩ

§#

»¼

º«¬

ª¯®­

¸¹·

¨©§

»¼º

«¬ª

»¼

º«¬

ª

¦f

)r+(R1 -

)r+(R1

N.N!)(-2 +

r+Rr-R 2 + e + e 2r

RtrmG- = F

NN

N

1=N

r-R-

r+R-

2

OO

OUS OO

log

°°

¿

°°

¾

½

»»»»»

¼

º

«««««

¬

ª

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¹·

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¼

º

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¬

ª

¸¸¸¸

¹

·

¨¨¨¨

©

§

»¼º

«¬ª

»¼

º«¬

ª

¦f

2t1 -

2t-2R

1 N.N!

)(-

2t-R

2 +

2t-2R

2t

2t-R

2 + e + e 2 R

mtrG- = F

NN

N

1=N

2t

-

2t-2R

-2

2

OO

OUS OOlog

186

Now look at the ratio of two successive terms in the series

But R>>t/2 so that

Therefore, the series converges absolutely! Now how about an expression for the force for a large number of shells of thickness t sufficient to make up a sphere? If there are M shells making up the sphere then the thickness t = Re/M. While if we want to monitor the force on a test mass at the surface of the sphere then the limits on x in the integration varies with each shell. For instance for the outer shell the limits on x would be

MR = t and R = R but

2t - 2R = r+R x

2t = r-R e

edd 154 or the limits on x would be

For the next shell 23t - 2R = r+R x t

23 = r-R dd 156 or

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ª¸¹·

¨©§

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¹

·

¨¨¨¨

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»¼º

«¬ª¸

¹

ᬩ

§#

¦f

)(t

2 - )(2R

1 N.N!

)(-2R

+

2R2t

2R

+ e + e 21

RGMm- F

N

N

N

N

1=N

2t

-2R

-2

OO

OOOlog

»»»»»

¼

º

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ª

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ª

2t - )(2R

2t - )(2R

tR)1+(N

=

t2 -

)(2R1

N.N!)(-

t2 -

)(2R1

1)!+1)(N+(N)(-

= aa

NN

1+N1+N

2

N

N

N

N

1+N

1+N

1+N

1+N

N

1n+

O

O

O

»¼

º«¬

ª

fo#

fo t)1+(N2

N

aa

N 2N

1+N Olimlim

»¼º

«¬ªdd

2M1 - 2 R =

2MR - R2 x

2MR

ee

ee

187

Thus, for the Nth shell we have limits given by

Then we have

Now, looking at

on the other hand for

Further, since »¼º

«¬ª

2M1)-(2N - 1 R = r e 162

If we choose M such that O > > MRe 164 then we may write

»¼º

«¬ªdd

2M3 - 2 R =

MR

23 - R2 x

MR

23

ee

ee

»¼º

«¬ªdd¸

¹·

¨©§¸¹·

¨©§

2N1-(2N - 2 R x

NR

21 - 2N

ee

> @ °¿

°¾½

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»¼

º«¬

ª

°°®­

»¼

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ª

»¼º

«¬ª

°°®­

»¼

º«¬

ª»¼

º«¬

ª

¦f

¸¹·

¨©§

»¼º

«¬ª

)1-(2N

)(2N - 1+2N - 4N

)(2N RN!-N

)(- +

1)-(2N-4N1-2N

N1)-(2N-2 R

+

e + e 21

Rm(N)GM- = (N)F

N

N

N

N

Ne

N

1=N

e

2N1-2N R

-

N1)-(2N-2 R

-2e

ee

O

O

OO

log

»¼º

«¬ª

»¼º

«¬ª

»¼º

«¬ª

N1 R

- = N = N if

N1-2 R

- = 0 = N if

N1)-(2N-2 R

- =

e

1

e

1

e

1

O

OO

exp_

exp_exp

¸¹·

¨©§

¸¹·

¨©§

2N1-1 R

- = N = N if

MR-2 = 1 = N if

21-2N

NR

- =

e

2

e2

e2

O

OO

exp_

exp_exp

»¼º

«¬ª

¸¹

ᬩ

§2M

1)-(2N - 1 MR 4 = M

23e

(m) SU

188

Suppose we look at three shells such that M = 3 then we find

or

Now we can do the same sort of thing for the "inside the shell" force. This case has the lower limits on x changed so that

with

> @ °¿

°¾½

°¿

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»¼

º«¬

ª

°°®­

»¼

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§

¦f

1-2m - 4M)1-(2m

1 - )1-(2m RN.N!

)(2M)(- +

1)-(2m-4M1-2m

M1)-(2m-2 R

+

M1-2m R

-

M1)-(2m-2 R

- 2 21

2M1)-(2m - 1

MR 4

RGm- = F

NN

N

Ne

NN

1=N

e

ee

23e

2e

(m)

O

O

OOSU

log

¸¹·

¨©§¸¹

ᬩ

§

¿¾½

¯®­

»¼

º«¬

ª

¿¾½

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xx»¼

º«¬

ª

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½

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§

¦f

65

3R 4 = M if

R4.198 - 1

RmGM- = F

52.398)3 -

R 23 -

R5 23 - 1

RmGM- =

][111 - 1

RN.N!6)(- +

1-121

35 R

+

31 R2

-

31 - 2 R2

- 1 61 - 1

3R 4

RGM- = F

23e

1e

2e

11

ee2e

1

N

0

Ne

NN

1=Ne

ee

2e

2e

1

SUO

OOO

OO

OOSU

log

¿¾½

¯®­

¸¹

ᬩ

§

R3.548 - 1

RGMm- = F

e2e

outO

2t R-r = x where ,dx

xef(x)

dxd

RrtmG- = F

x-r+R

R-rx=2in t

»»¼

º

««¬

ª³

OUS

189

Then for a shell with Re as the inner radius and a thickness of t then r=Re=t/2 and we desire to know what the gravitational effort for R=Re. Then we have R+r=2Re+ t/2 and r-R = t/2 then

if O>>2Re, Re>>t/2, then we have the approximation

Example: Compare the change in gravitational field as one goes from the Earth's surface down a deep well to a depth of d. The Newtonian gravitational strength at the earth's surface is

For the neo-Newtonian force we must see the same force, thus, we must set

¿¾½

¯®­

¸¹·

¨©§

»¼

º«¬

ª ¦f

xN.N!)(- +

x1 2 + e x +

xr1R(- = ex

f(x)N

N

1=Nx

-22

x- O

OOO

log

°°

¿

°°

¾

½

°°

¿

°°

¾

½

°°

¯

°°

®

­

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º

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ª

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º

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ª¸¹·

¨©§

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¼

º

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¬

ª¸¹·

¨©§

2t1 -

2t + R2

1 N.N!

)(- +

2t + R2

2t

2 +

e2t - e 2

t + R2 + e

2t

2t + R - R

-

e

2t + R2

2t + R-R-

R

tm2t + R G-

= F

N

e

N

e

2t

-

2t + R2

-e2

te

22e

2t + R2

-

e

e

22e

2e

e

in

e

e

OO

US

OOO

O

� � »¼

º«¬

ª¸¹

ᬩ

§¸¹

ᬩ

§

°°¯

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¼

º

«««««

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§

¸¹·

¨©§

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§#

¦f

t2 -

R21

N.N!)(-

R2 +

R4t

R2 +

te 2 +

R2R2

- 1

4t

RGMm- F

N

N

eN

N

1=Neee

t2-

e

e2e

in

OOO

OO

log

R

mM-G = F 2e

eN

¸¹

ᬩ

§#

R3.55 - 1

RGMm-

RmMG- = F

e2e

2e

eNN

O

190

or

Now the Newtonian gravitation at a depth of d.

or

The variation in force from the surface to the depth is then

or

On the other hand, the gravitation force in the well with the neo-Newtonian potential would be given by

¸¹

ᬩ

§

R3.55 - 1

M = M

e

e

O

� � � �d-R2d-R

mGM- = d-R

mGM- = (d)F 2e

2e

e

e2

eN

¸¹

ᬩ

§#

R2d-1R

mGM- (-d)F

e

2e

eN

»¼

º«¬

ª¸¹

ᬩ

§

»»»»»

¼

º

«««««

¬

ª

¸¹

ᬩ

§¸¹

ᬩ

§#'

RmGM

RmGM =

R2d-1

R2d+1-1

R

mGM =

R2d-1R

mMG + R

mGM- F

2e

e2e

e

e

e2e

e

e

2e

e2e

eN

.R

mGM R2d F 2

e

e

eN »

¼

º«¬

ª¸¹

ᬩ

§#'

� � °¿

°¾½

»¼

º«¬

ª

¯®­

¸¹

ᬩ

§»¼º

«¬ª

¦f

t1 -

R41

N.N!)(-2

RM2M -

R4t

RM2M - e - 1

M2M -

R4MM + 1

RmGM- = (-d)F

Ne

N

N

1=Nee

eeet

2-

eee2e

eNN

OO

OO Olog

191

then 'FNN = FNN(Re) - FNN (Re-d)

but ¸¹

ᬩ

§

R3.55 - 1

1 = MM

e

e O181 so that for d = 5,500 ft = 1,676m and Re = 6.4 x 106m then

Now

Thus, we have

� � � �

� � � � °¿

°¾½

°¿

°¾½

»¼

º«¬

ª

°°®­

¯®­

¸¹

ᬩ

§»¼º

«¬ª'

¦f

d1 -

R41

N.N!)(-2

d-R2 -

R4d

R2 - e - 1

21 -

d-R4

MM + 1

4-RmGM +

RmGM- = F

Ne

N

N

1=Ne

eed

2-

eee2

e2e

eNN

OO

OO Olog

� � °¿

°¾½

»»¼

º

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ª

¯®­

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«¬ª

¸¹

ᬩ

§»¼

º«¬

ª¸¹

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§#'

¦f

)(1676

1 - 102.56x

1 N.N!

)(-210 x 1.492 -

10 x 1.512 + e - 1 954- MM + 1

RmGM

R2d F

N7 N

N

1=N

4-

3-d

2-

e2e

e

eNN

OO

OO

� �

OO

OO

OO

OO

O

414-13-4

311-10-3

27-7-2

3-4-

N7 N

N

1=N

10 x 2.11 10 x (-1.2674324

16+

10x 9.44+ 10 x (-2.124373

8-

10 x 3.56- 10 x (-3.562.24+

10 x 1.193 10x 5.9661

+2

= )(1676

1 - 102.56x

1 N.N!

)(-2

oxxx

oxx

o

o

»»¼

º

««¬

ª¦f

¸¹

ᬩ

§

¸¹·

¨©§¸¹

ᬩ

§¸¹

ᬩ

§#'

¸¹·

¨©§¸¹

ᬩ

§¸¹

ᬩ

§#'

R2d-1 R

mGM- = (-d)F

d3.548 - 1

RmGM

R2d F

16763.548 - 1

RmGM

R2d F

e

2e

eN

2e

e

eNN

2e

e

eNN

O

O

192

But the neo-Newtonian gravitational force at the bottom of a well of depth d is given by

which may be written

This gives the first order approximation of the deviation from Newtonian gravitation predicted by the neo-Newtonian potential and shows that the predicted gravitational force of the Earth decreases more rapidly than Newtonian gravitation does. 5.5 Inertial and Gravitational Mass and their Equivalence There are three ways in which mass appears in Newton's Second Law when gravitational forces are considered. Consider his gravitational force law which may be written

and his Second Law which is

In these equations there are the inertial mass, m, and two gravitational masses, m1 and m2. The force equation comes from considering the force on m2 due to the the gravitational field of m1. In this case m1 is usually referred to as the active gravitational mass while m2 is the passive gravitational mass. Classically Newton's Third Law is imposed in order to show that the ratios of active and passive gravitational masses must be equal. Consider

»¼

º«¬

ª

¸¹

ᬩ

§

»¼

º«¬

ª

¸¹

ᬩ

§

»¼

º«¬

ª

¸¹

ᬩ

§

R7.096-1

R2d-1 R

mGM- = (-d)F

R3.548-1

R2d-1 R

mGM- +

R3.548 - 1

R2d-1 R

GMm- = (-d)F

e

e

2e

eNN

e

2

e

2e

e

e

e

2e

NN

O

O

O

¸¹

ᬩ

§

R7.096-1 (-d)F = (-d)F

eNNN

O

rr

mGm- = F3

21

. dt

rdm = )v(mdtd = F 2

2

193

so that

where the subscripts a and p refer to the mass's role as either an active or passive gravitational mass. This leads us to the equation

which means that since the ratios must be equal the ma and mp may be made equal. The equality of inertial and gravitational mass is not predictable by Newton's laws. Rather, it is taken as an assumption. This assumption has been subjected to increasingly accurate experimentation by Eotvos in the 1880's, by Dicke in 1964, and by Braginski in 1971. The present limit of comparision between gravitational and inertial mass in about one part in 1012. Now let's consider these same three mass concepts in the context of the Dynamic Theory. First, there is the inertial mass density. It makes its appearance in Section 3.1 when we impose the principle of increasing entropy as a variational principle. The metric element is given in terms of the specific entropy while the entropy principle is in terms of the entropy density. The effect of this is to introduce the mass density as a product of the acceleration into the equations of the force densities (see Eqn. (3.5)). The same inertial mass concept leads to the Einstein energy and mass relation in Section 3.2. The other two mass concepts enter first through the field equations given by Eqn.s (3.15) and from them the force densities in Eqn.s (3.17). In Section 5.1 we went through the field equations to determine the charge-to-mass conversion needed to keep the units consistant. Here we found that the passive gravitational mass given by Eqn. (5.12) was

while the gravitational field associated with a gravitational mass is given by Eqn. (5.10), when the evaluated parameters are used, as

F- = F 21

mGm = mGm 1p2a2p1a

mm =

mm

2p

1p

2a

1a

Gm M p H#

(5.76)

194

where the mass in the gravitational field equation is to be considered the acitve mass and therefore we've used the subscript a to denote this. The gravitational force due to the passive mass M being in the gravitational field Vr is then

By looking at Eqn. (5.78) we may see that, first, it is the active gravitational mass that has the time dependence and not the passive gravitational mass. Further, only when the active gravitational masses are identical with their O's the same will the active and passive gravitational masses be equal. We've used the subscripts 12 in the force equation to denote the force on mass 1 in the pressence of the field of mass 2. We may consider the force on mass 1 when in the field of mass 2 and we find

If we form the ratio of Eqn.s (5.78) and (5.79) we find that

Further, only for identical gravitational masses will Newton's Third Law be satisfied within the Dynamic Theory. 5.6 Cosmology The hot big bang model of the Universe is the model which is in vogue now. Virtually all the journals print numerous articles relating to some aspect of the hot big bang model. The model is based upon the

> @»»»»

¼

º

««««

¬

ª¸¹·

¨©§

r

er - 1

tG + 1 Gm = V 2

r-a

ar

aOO

H�

(5.77)

. rr

er - 1

t)G + (1mGm- = F 123

r-2a

2a1p12

2a

»»»»

¼

º

««««

¬

ª¸¹·

¨©§ OO

� (5.78)

. rr

er - 1

t)G + (1mGm = F 123

r-1a

1a2p21

1a

»»»»

¼

º

««««

¬

ª¸¹·

¨©§ OO

� (5.79)

. er

- 1mm

er - 1mm

= F-F

r-2a

1a2p

r-1a

2a1p

21

12

2a

1a

»¼

º«¬

ª¸¹·

¨©§

»¼

º«¬

ª¸¹·

¨©§

O

O

O

O

(5.80)

195

Newtonian Gravitational and the notion of a scale of the universe that is

changing with time. This notion is borrowed from Einstein's General

Theory of Relativity, however Einstein's theory is not used in the hot big

bang model itself. It would seem a shame to discuss a new gravitational

potential such as presented in this book without some discussion of its

possible impact upon the hot big bang model. It would have been

preferable to wait until the entire solution could be presented. However,

this is not possible now so this presentation will include a discussion of

how one might expect the new potential to impact the hot big bang model

and the problems that render the solution illusive.

The development of the standard big bang model begins with

considering a spherical piece of the universe with an observer at the

center. This sphere is considered to be filled with "dust" of density U(t) with

a galaxy of interest placed at the outer boundary of the sphere which has a

radius denoted by x. When Gauss's law and Newton's laws of motion and

gravitation is used one arrives at

But consider what happens if one wishes to compare this with the non-

singular potential of the Dynamic Theory. Then Eqn. (5.81) becomes

Now let us replace x with the comoving coordinate x=R(t)r where R(t) is the

scale factor of the universe and r is the comoving distance coordinate as is

done in the standard model. When we also normalize the density to its

value at the present epoch, Uo , by U(t)=UoR-3(t) we obtain

We can begin to see the trend to be expected from the universe from

Eqn. (5.83) by noting that should we look back in time to the point when

R=r/O then we would have a point in time, say T1 when the acceleration of

the universe would have been zero. At times before T1 there would have

been an acceleration outward while for times after T1,such as the current

epoch, the rate of the expansion of the universe is slowing down. This is a

very different story than is told from by the standard model. But how is it

different? It is the same as the standard model in that from Eqn. (5.83)

one sees that the universe was forced into expansion at early times and is

now slowing down its rate of expansion. One big difference between the

story to be told by Eqn. (5.83) and the standard model is that Eqn. (5.83)

gives the reason for the initial expansion and it denies that the universe

x.m(t)G3

4- = x

Gm(t)x3

4- = dt

xdm g2g

3

2

2

g USUS (5.81)

.ex-1(t)xG

34- = ex

-1x

m(t)Gx3

4- = dt

xdm xx-

2g

3

2

2

gOO O

USOUS

¸¹·

¨©§

¸¹·

¨©§

. R

eRr

-13G4

- = dt

Rd2

Rr-

o2

2 OOUS¸¹·

¨©§ (5.83)

196

was ever collapsed to a singular point as supposed by the standard model. To better see the first contention we should proceed a little further. If we multiply Eqn. (5.83) by dR/dt and integrate with respect to

time we find In Eqn. (5.84) we have included the term for the radiation for completeness. Now let us evaluate the constant of integration, k, by setting the values of R, dR/dt, U, and H at their present day values of 1, Ho, Uo, and Ho. Then Eqn. (5.84) becomes

If we now make the definitions

Eqn. (5.85) may be put into Eqn. (5.84) to obtain

We may now take a look at some of the implied dynamics from Eqn. (5.86). First look at the dynamics as R tends to infinity and there is no radiation. For this case we would have

which is the same as in the standard model. Now suppose we look backward in time to the time when dR/dt was zero? Then Eqn. (5.86) becomes

This is a transcendental equation which could be solved for R if we knew O, r and the density of the dust and radiation at the current epoch. It may be seen from Eqn. (5.87) that if there is no radiation and R does not equal zero then

.kc - c2R(t) + eR(t)

3G8 = R 2

2

2

Rr-22

»¼

º«¬

ª HU

S O� (5.84)

.kc - c3

G4 + e3G8

= H 22

or

-o2o

HSUS O (5.85)

,e and G8

H3 c

r-

o2o

c UU

SU

O

{:{

.1 - R1

c2H + - 1 +

Re

eH = R 2c

2o

2o

r-

Rr-

2o

2 ¸¹

ᬩ

§¸¸

¹

·

¨¨

©

§:

:UH

O

O

� (5.86)

), - (1H = R 2o

2 :f�

� �.R - 1c2

+ R + eR - eR = 0 22

o2cr

-2oRr

-o

HUUUOO (5.87)

197

There is also a trivial solution at R=0 in Eqn. (5.87) but for this case the acceleration is also zero and therefore no dynamics are allowed. While at first glance it may appear that we have as good a developed solution as is arrived at in the standard model, consider the following points. Our galaxy was considered to be on the outside limit of a sphere of dust. For the current epoch the density of the dust is very small locally to the galaxy and Gauss's law for considering the total mass of the sphere of dust to be placed at the center should hold very well. But what about when we are looking back in time when the density was a lot greater. At some density we are no longer able to approximate result used in Eqn. (5.82) but must use the solution developed in the discussion of the Fifth Force. Then our conclusions arrived at above are only good in a general sense and are not quantitatively accurate. A second point concerns the fact that we have developed the gauge function in prior sections. If this is the scale of the universe as the gauge function is supposed to be, then why are we again trying to solve for it here? If the scale of the universe is given by the gauge function then the dynamics may be over specified if we put the radiation into the equation for the acceleration such as Eqn. (5.83). On the other hand, what is the source for the radiation? If there is no hot big bang for the radiation to come from where might it originate? The Dynamic Theory displays an inductive coupling between the electromagnetic and the gravitational fields. could the radiation be due to the expansion of the gravitating mass of the universe? If so then a knowledge of the gauge function might turn the equation of motion for our galaxy into a prediction of the radiation required at the present time. This prediction might then be compared to the measured radiation. However, there is the necessity to have a O for the universe. It may be obtained from the gauge function also as GM/c2. But how is M determined? Perhaps this is sufficient to point out that the overall picture of cosmology to be given by the Dynamic Theory is not yet complete but in any event will likely be very different from the hot big bang model of the universe. Will it allow for high temperatures needed for accounting for the abundances of the elements? Since it allow for the universe to be much smaller in the past it would have the associated high temperatures. Yet it should not have the infinite temperatures associated with a singular universe.

. - e

e = R

o

cr

-

Rr-

»¼

º«¬

ª

UUO

O

(5.88)

198

Chapter 6 Electromagnetogravitic Waves

Given the system of equations, Eqn. (3.15), and the interpretations

that; E is the electric field, B is the magnetic field, V is the gravitational

field, and V4 is the gravitational potential, then the question arises as to

how the electromagnetogravitic waves may propagate.

6.1 Wave Equations

The usual assumptions such as wP/wJ = wH/wJ = 0 and U = 0 may be

used to derive the wave equations for the four field quantities. Other

assumptions used are that the media is isotropic and that J = VE and J4 =

V4V4. The resulting wave equations are

,E-)E(c4a=V

a+tV

c-

tV

c4-V 4

02

2202

2

2242

»»¼

º

««¬

ª

ww

ww

ww

ww¸

¹

ᬩ

§ww

¸¹

ᬩ

§'

JV

JVSP

JPHVSP (6.1)

,0 = Ea +

tE

c-

tE

c4- E 2

2202

2

222

JPVSPV

ww

ww¸

¹

ᬩ

§ww

¸¹

ᬩ

§' (6.2)

,0 = Ba +

tB

c-

tB

c4- B 2

2202

2

222

JPHSPV

ww

ww¸

¹

ᬩ

§ww

¸¹

ᬩ

§' (6.3)

and

. 0 = Va + tV

c -

tV

c4- V 2

42

202

42

24

24

42

JPHVSP

ww

ww¸

¹

ᬩ

§ww

¸¹

ᬩ

§' (6.4)

The inhomogeneous term in Eqn. (6.3) displays an interconnection

between the electric and gravitational waves. Further, this term produces

the question of whether the propagation vector for the gravitational wave

can be the same as the propagation vector for the electric wave. In

Maxwellian electromagnetism it may be shown that the propagation vector

for the electric wave must be the same as the propagation vector for the

magnetic wave. However, this is not true, in general, for this system of

equations.

6.2 Wave Solutions

Given that the propagation vectors may be different a trial solution

may be sought during which the conditions for identical propagation

vectors may be exposed.

If the waves are considered to be propagating in the positive x-

direction, then the trial solutions may be taken to be of the form

199

� �> @� �> @� �> @ , k - xk - ti- V = V

, k - xk - ti- B = B

, k - xk - ti- E = E

4vvO

4bbO

4ee0

JZ

JZ

JZ

exp

exp

exp

and � �> @ , k - xk - ti- V = V 444404 JZexp (6.5)

By making the definitions

,cka = A ,cka = A ,cka = A ,cka = A

4404

4v0v

4b0b

4e0e

(6.6)

and substituting the trial solutions, Eqn. (6.5) into the wave equations, Eqn.s (6.1)- (6.4), we obtain the indicial relations:

� � � � ¸¹

ᬩ

§»¼

º«¬

ª

ww

¸¹

ᬩ

§

VE

ki + - Aei4 + A - i4 + = ck4e

42v4

2v0

2

D

D

JV

VVSPVSPZZPH

� �� � ,A - )(kc = ck

,A - )(kc = ck2b

2b

2

2e

2e

2

(6.7)

and � � ,A - i4 + = ck 2

442

42

VSPZZPH where D = 1, 2, 3, and (kc)2 = PHZ2 + i4SPZV. Substituting the trial solutions into the continuity equation of Eqns. (3.14), we find

� � . k

i- = - 4

444 J

VVV

ww

¸¹

ᬩ

§ (6.8)

The ratio (ED/VD) appearing in Eqn. (6.7) indicates that we need to know the relationship between the components of the V field and the components of the E field. These relationships may be found by substituting the trial solution into each of Eqn. (3.15). In this substitution, the further limiting assumption that the electric field may be polarized so the Ez = 0 and Ex = 0 is made in order to simplify the solution. It should be pointed out though that, in contradistinction with Maxwellian electrodynamics, Ex is not required to be zero by the differential equations. The differential equations require the following relationship among the non-zero components, given the trial solutions and the imposed restrictive assumptions:

,E ck = B y

ez ¸

¹·

¨©§Z

(6.9)

and

. E kk A = E A- = V y

v

eby

ey ¸

¹

ᬩ

§¸¹·

¨©§

¸¹·

¨©§

ZZ

Thus, the imposed assumptions reduce the solution to only three non-zero components, Ey, Bz, and Vy. If we consider the different expressions, from Eqn. (6.9), for Bz, and take the partial derivative with respect to the mass density we find

200

JZ

J w

¸¹·

¨©§w

ww E

ck = B

ye

z

requires that

� � . A - A ca

i = kk1

eb0

e

e¸¹

ᬩ

§ww

¸¹

ᬩ

§J

(6.10)

From Eqn. (6.9) we also find ,kA = kA veeb (6.11)

while differentiating with respect to J-produces

� � . A - A ca

i = AA1

ev0

e

e¸¹

ᬩ

§ww

¸¹

ᬩ

§J

(6.12)

With the assumption that there is no longitudinal field component the surviving system of equations is

� �� �

� � ,A

ca4 + A - )(kc = ck

,A - )(kc = ck

,A - )(kc = ck

e

02v

2v

2

2b

2b

2

2e

2e

2

JVSPZww

¸¹

ᬩ

§

(6.13)

� �

� �A - Acai = A

A1

A - Acai = k

k1

,kAA = k

ev0

e

e

eb0

e

e

ee

bv

¸¹

ᬩ

§ww

¸¹

ᬩ

§

¸¹

ᬩ

§ww

¸¹

ᬩ

§

¸¹

ᬩ

§

J

J (6.14)

where the definition

SPZVZPH i4 + = )(kc 22 (6.15) has been used. The partial derivative, wV/wJ, appearing in Eqn. (6.14) may be found from experiment in the following manner; because

¸¹

ᬩ

§ww

¸¹·

¨©§ww

ww

JV

JV T

T =

where T is the temperature. The conductivity, V, is the reciprocal of the resistivity, T, whose linear dependence upon the temperature is given by

� �> @T - T + 1 r = r 00 D Then

. r- = T

= T 0

2r1

DVV

ww

ww ¸

¹·

¨©§

On the other hand, the coefficient of volume expansion is defined as

,T

(vol.) (vol.)

1 = w

wE

but

201

,vol.

mass = J

Therefore,

. = T

JEJww

Thus,

. r = 2

0

EJVD

JVww (6.15)

The solution of Eqn. (6.15) is

¸¹

ᬩ

§

¸¹

ᬩ

§

JJDE

J

V0

n

0

l

r = (6.16)

where J0 is the mass density evaluated at the temperature T0. Given the expression for wV/wJ, it may be seen that the system of Eqns. (6.13) and (6.14) represent six-equations in the six unknowns, Ae, Ab, Av, ke, kb, and kv. The system may be solved as shown in the following. The unknown, kb, appears only in one equation; therefore, this equation may be considered to determine kb once the solution for the other unknowns are determined. Eqn. (6.11) may be used to eliminate Ab, in Eqn. (6.10) leaving us with four equations in four unknowns. Eqn. (6.10) now looks like

Jww

¸¹

ᬩ

§ kA

cia- k = k e

e

0ev (6.17)

Eqn. (6.13) may be used to determine Av by rewriting it as

. AA

cia- A = A e

e

0ev Jw

w¸¹

ᬩ

§ (6.18)

This may be used to eliminate Av from the third of Eqn.s (6.7) leaving three equations in three unknowns. By differentiating the first of Eqn. (6.7) with respect to the mass density, it becomes

. AA2 - i4 = kck2 ee

e2e JJ

VSPZ

J ww

ww

ww (6.19)

Substituting Eqns. (6.17) and (6.18) into the third Eqn. (6.7) and using Eqn. (6.19) results, after some manipulation, in

0. = A1i2+A

A1i2 2-A

A1 )(kc

e

22e

e

e

e

22

¸¹

ᬩ

§¸¹

ᬩ

§ww

»¼

º«¬

ª

ww

¸¹

ᬩ

§¸¹

ᬩ

§ww

»¼

º«¬

ª

ww

¸¹

ᬩ

§JV

SPZJJ

VSPZ

J (6.20)

This is a quadratic equation in wAe/wJ whose solutions are

»¼º

«¬ªw

w

ww )(kc - A + A

)(kc

i2 = A 22

ee2e _J

VSPZ

J

or

202

. )(kc - A + A )(kc

i2 = A 22ee2

e

»¼º

«¬ª»

¼

º«¬

ª

ww _SPZV

Therefore we have

But from the definition of (kc)2 we find that

Thus,

Eqn. (6.21) now becomes

By using the method of substitution, recognizing that it may be put into a homogeneous form, and realizing the solution may be complex, we arrive at the solution of Eqn. (6.22) as

where c2 is a constant of integration such that

that is, c2 may depend upon P, H, and Z but not V. Eqn. (6.23) is a quadratic equation in c2 and may be solved yielding

. (kc)d(kc) =

)(kc - A + A

dA22

ee

e

»¼º

«¬ª _

(6.21)

VSPZdi4 = )d(kc 2

. ki

d(kc) = )2(kc

2(kc)d(kc) = )2(ki)d(kc =

)(kcdi2

22

2

2WSPZ

. (kc)d(kc) =

)(kc - A + A

Ad22

ee

e

»¼º

«¬ª _

(6.22)

> @1 - c)(kc c21 = A 2

22

2e ¸

¹

ᬩ

§

(6.23)

,0 = c2

Vww

203

Because c2 does not depend upon V it is unaffected by setting V = 0, so if Aeo = Ae (V = 0) then

By substituting Eqn. (6.25) back into Eqn. (6.23) we find the sign before the radical must be taken to be positive. Then the expression for Ae becomes

where

Using Eqn. (6.26) in the system of equations, Eqn. (6.14), we may, after a great deal of algebra, write the total solution as:

where

Now we may write

with

. )(kc

)(kc + A + A = c 2

22Ee

2_

(6.24)

� � . + A + A = c 2

22eoeo

2ZPH

ZPH_

(6.25)

. hi4 + A = A eoe SPZV

(6.26)

> @ . + A + A 2

1 + A h22

eoeo2

eo

°¿

°¾½

°

°®­

{ZPHZPH

,hi4 + A = A eoe SPZV

(6.27)

> @ . + A + A 2

1 + A h2

eoeo2

eo

°¿

°¾½

°

°®­

{ZPHZPH

,i + = k eee ED

204

Now we have

where

The propagation vector for the v component may be written as

with

� � . hA2 - 1 =

,)h(4 + A - 1 =

,1 - 4 + 12

c

=

1 + 4 + 1 2

=

eo

2

2

2eo

2 21

e

2 21

e

VV

ZPHSPZV

ZPHHH

ZHVSHPZ

E

ZHVSHP

D

~

~

~~~

~~~

»¼

º«¬

ª

°¿

°¾½

°

°®­

¸¹·

¨©§

¸¹·

¨©§

°¿

°¾½

°

°®­

¸¹·

¨©§

(6.28)

ED avavv i + = A

> @ . )h(4 + A

h4 ca[ = f

,f) - (1 h4 = ,f) + (1 A =

22eo

0

av

eoav

SPZ

JV

SPZ

SPZVED

»¼

º¸¹

ᬩ

§ww

(6.29)

, + = k vvv ED

� �

. f - 4

a2 - 1 =

,h

fA + -

- 1 =

,4 + 1 2

c

=

,1 + 4 + 1 2

=

avav

2eo

2

2av

2av

2 21

v

2 21

v

»»¼

º

««¬

ª¸¹

ᬩ

§c

»»¼

º

««¬

ª¸¹

ᬩ

§c

°¿

°¾½

°

°®­

¸¹·

¨©§

ccc

¸¹·

¨©§

°¿

°¾½

°

°®­

¸¹·

¨©§

ccc

SPZVE

VV

ZPHZPHEDHH

ZHVSHPZ

E

ZHVSHP

D

(6.30)

205

Now we have

with

Then

where

The system of equations, Eqns. (6.26) through (6.31), representing the solution is an extremely complicated system and should be put on a computer in order to fully explore all the ramifications of this solution. 6.3 Non-thermal Transmission through Media It is the intent of this section to briefly show the effect of the solution given above and discuss how this solution may be useful in modeling electromagnetic interactions with biological systems. Therefore, consider the question of component attenuation, or how the different components of the electromagnetogravitic wave may be attenuated? A simpler question would be, "For what frequencies will the components not be attenuated at all?"

ED ababb i + = A

� �� �

� �� � .

+ +

= F

, + +

= D

,hD4 + FA = ,hF4 - DA =

2e

2e

evve

e2e

evev

eoab

eoab

2

ED

EDED

ED

EEDD

SPZVESZVD

ED bbb i + = k

(6.31)

. f - 42

- 1 =

, -

- 1 =

,1 - 4 + 1 2

c

=

,1 + 4 + 1 2

c

=

avav

2

2av

2av

2 21

b

2

21

b

»¼

º«¬

ªc

»»¼

º

««¬

ªc

°¿

°¾½

°

°®­

¸¹·

¨©§

¸¹·

¨©§

°¿

°¾½

°°®­

c¸¹·

¨©§

SPZVEDVV

ZPHEDHH

ZHVSHPZ

E

ZH

VSHPZD

ˆ

ˆ

ˆˆˆ

ˆˆ

206

From Eqn. (6.28) we find that the electric component will pass unattenuated if V=0. This is satisfied by two conditions. The first condition is that V = 0 which is the classical condition of a perfect dielectric. The other condition is that

Substituting the definition for h into Eqn. (6.32) we find, after some manipulation, that this is satisfied if

which has only one real solution

The complex solutions are:

Considering the real solution and assuming Aae2 to be real, we find that

We do not yet know the dependence of Aeo upon P, H, Z. The assumption that Aeo is linear in Z would mean that the relative strength of the gravitational component compared with the electric component, given by

does not depend upon frequency in free space. In Eqn. (6.33), the assumption that

¸¹

ᬩ

§

A21 = h

eo

(6.32)

. 0 = 3 - A

- A

- A 2

eo

2

2eo

2 2

2eo

2

¸¹

ᬩ

§¸¹

ᬩ

§¸¹

ᬩ

§ ZPHZPHZPH

(6.33)

. A 1.7971 = 2ae

2cZPH

. A 1.0434) i + (-0.8985 = eo2 _ZPH

. A 1.7971 = eoc PH

Z

,E A- = V ye

y ¸¹·

¨©§Z

, = Aeo KZ

(6.34)

207

implies that there are no frequencies for which Ee = 0, and this would be

consistent with classical theory.

If now we look at the frequencies for which Ev = 0, we find, from Eqn.

(6.29), that Ev = 0 when V' = 0, or

Substituting for the defined quantities in Eqn. (6.35), assuming K2<<PH, and disregarding negative frequencies, we find two possible frequencies for

which Ev = 0, or

and

The magnetic component is unattenuated when Eb = 0, or when

The condition specified by Eqn. (6.37) represents a seventh order

polynomial in Z, therefore, the roots of this polynomial have not been

sought. It may be noted though that there are up to seven possible

frequencies for which the magnetic component is unattenuated.

Thus, for frequencies satisfying the conditions of Eqns. (6.35) and

(6.37), the gravitational or the magnetic component respectively will

experience no attenuation. Because these conditions result in polynomials

in Z, then there must be frequency regions where either Ev < 0 or Eb < 0, or

both Ev and Eb are negative. In these regions the gravitational and/or the

magnetic component will experience an amplitude growth.

On the other hand, from Eqn. (6.32), we found that there were no

frequencies for which Ee < 0 for V > 0. This then leads to the possibility

. f = 42

- 1 avav¸¹

ᬩ

§SPZV

ED

(6.35)

,4 - 2

ca16- 1

2

2

0

21

c1

»»»»»

¼

º

«««««

¬

ª

¸¹·

¨©§¸

¹

ᬩ

§ww

¸¹

ᬩ

§#

HS

HPJV

SK

KZ

(6.36)

°°¿

°°¾

½

°°¯

°°®

­

¸¹·

¨©§

»»¼

º

««¬

ª¸¹

ᬩ

§¸¹

ᬩ

§

¸¹

ᬩ

§

¸¹

ᬩ

§ww

¸¹

ᬩ

§#

HS

PHK

PHS

PHK

H

JV

S

KZ

4 - 2 - 1 4 + 2 2 - 1 2

ca4 1

2o

21

c2

. 2 = 4 abab EDSPZV

(6.37)

208

that the growth in the gravitational and/or magnetic component is at the expense of the electric component. If then, non-thermal transmission is defined to be transmission during which none of the wave energy is deposited in the media, we find that our simple solution will support non-thermal transmission for frequencies satisfying Eqn. (6.33). For this type of transmission the energy originally in the electric component experiences an attenuation and is transferred to the gravitational and/or the magnetic components which experience a gain. The net result is the transmission of energy through the media without loss, only a change in form. We have seen that the three fundamental postulates of the Dynamic Theory have led to the use of mass density as a fifth dimension, fundamentally independent of space and time. The five-dimensionality of the theory produced the eight differential equations describing the allowed interrelationships between the five dimensional gauge fields. Other investigations in fields allowed for fundamental particles produces the interpretation of the V field as the gravitational field. With the interpretation of the V field came the question of how these waves might propagate given their specified interrelationships. In answer to this question we've shown that for simplified, continuous media there exist frequencies for which the electric component is attenuated while gain in the gravitational and/or magnetic components is experienced. This gives rise to the possibility of transmitting energy through the conductors where no such energy transmission is allowed by Maxwellian electromagnetism. Even though biological systems are complex structures, is it not possible that the five-dimensional fields of the Dynamic Theory have applicability in describing radiation interaction with these systems? Is it possible that a description of non-thermal effects of radiation on biological systems may be aided through the use of the non-thermal transmission effects discussed here? A great deal of discussion these days concentrates on nonlinear approaches. The five dimensional waves provide a linear description of effects which in four dimensions would appear as nonlinear. Thus, it would seem possible to replace nonlinear four-dimensional problems with five dimensional linear ones. 6.4 Boundary Conditions Classical work on boundary conditions of field vectors generally starts with Coulombs' Law. Here polarization of materials enter the picture. Polarization can be caused by either alignment of molecules or induced asymmetry. From these considerations the electric dipole moment is defined as

lq = p

209

where p is the electric dipole moment and l is a vector from -q to +q. The net dipole moment per unit volume is the polarization, P, of the medium. From this we get

where q is the net polarization charge within the volume. If the density of the polarization charge is U'l then we have

where the minus sign arises since by definition, the direction of the polarization vector is from negative to positive, whereas the electric field is directed from positive to negative. Thus, we arrive at

Now in order to write an equation like

that is valid in a dielectric medium and account for both free charges and polarization, U and U' respectively we must write

or using Equation (3.53)

Maxwell named the quantity in parenthesis the dielectric displacement, or

Therefore, the equation for a dielectric media becomes

From experiment it is found that a large class of media exhibit P proportional to E, for field strengths not too great. Thus,

q = dvol P vol

cx'³

dvol - = qvol

Ucc ³

Ucx' - = P

SU4 = E x'

,) + ( 4 = E UU cx'

(6.38)

. 4 = )P4 + E( SUSx'

. P4 + E = D S

. 4 = D SUx'

E = P eF

(6.40)

210

where xe is the electric susceptibility of the medium. Then

The proportionality factor between D and E is called the dielectric constant and

Therefore,

Thus, Equation (6.39) may be written as

Consider now that in the Dynamic Theory we derived the equation

Thus it may be seen by comparing Equation (6.41) with Equation (6.42), that the second term on the right hand side plays a role of gravitational polarization charge density, or

Then we would have

or

or

. E)4 + (1 = D eFS

. 4 + 1 = eFSH

. E = D H

. 4 = )E( SUHx'

(6.41)

. )V(ao - 4 = )E( 4

JH

SUHw

wx'

(6.42)

. P 4 = V4)(ao gx'w

wHS

JH

(6.43)

,4 = )P 4 + E( g SUSHx'

> @ ,4 = P 4 + E)4 + (1 ge SUSFSx'

> @ . 4 = gP 4 + P 4 + E SUSSx'

(6.44)

211

Thus, in order to include the gravitational polarization the dielectric displacement must be given by

Now the dielectric polarization, P, is an averaging over a finite volume, thus when using the Gaussian pillbox in looking at boundary conditions it is assumed that the pillbox may contain free charges but not polarization charges. Thus, when considering boundary conditions we must look at the displacement vector D not the field strength E. Therefore, consider the usual "Gaussian pillbox" of cross-sectional area S and thickness 't. Let n be the unit normal to the surface S. The pillbox volume is V = S't and is assumed to contain free charge but no polarization charge, nor gravitational polarization (we may want to rethink this assumption concerning gravitational polarization when it is better understood.) If we integrate over the volume V we have

or, by using the divergence theorem,

The left hand side may be integrated by noting that since the normal component of D is involved there is no contribution from the sides. Thus, since the volume V can be made sufficiently small, we have

If

s = t)( 0t

UU'o'

lim 57; or free surface charge density, then

relates the charge in the normal component of D across a boundary to the surface density of free charge on that boundary. If Us = 0 then the normal component D is continuous across the boundary. Equation (6.41) may also be written as

. P4 + P4 + E = D gSS

dv 4 = dv D vv

US ³³ x'

. dv 4 = ad D vs

US ³³ x

. t S 4 = )Sn D - n D( = n D + n D 11221122 'xxxx USˆˆˆˆ

s 4 = n )D - D( 12 USˆx

(6.45)

. s 4 = n - )P 4 - E - P 4 + E( g111g222 USSHSH

212

This points out the need to consider the physical meaning of the gravitational polarization but we won't go into that at this time. The next condition that must be fulfilled at the boundary comes from

for static fields so that wB/wt = 0. (This may safely be assumed since even for non-magnetostatic field the contribution by the wB/wt term vanishes in using Stokes theorem). Now construct a rectangular path which has sides '1 width 't, and for which the sides parallel a segment of the bounding surface. Then by Stokes' theorem

Thus,

Now, ,n- = n 21 ˆˆ 63 therefore

Since the contribution from the ends is proportional to 't, the second term vanishes in the limit as 't o 0. Thus we have

Now

so that

or

O = Ex'

. 0 = dan).E.( = l.dE os

ˆ'³ ³

ends. from oncontributi + 1)n.E( + 1)n.E( = l.dE 2211 '' ˆˆ

. 0 = (ends) + 1n).E - E( 212 'ˆ

,0 = n - )E - E( 212 ˆ . 0 = n x )EE.(n 120 ˆ

n x n = n 02 ˆˆˆ

0 = )n x n).(E - E( 112 ˆˆ

. 0 = n)xEE.(n 120 ˆ

213

But the orientation of the rectangle is arbitrary so that

Eqn. (6.42) implies that the tangential component of E must be continuous. For the magnetic induction field, since here

which is the same as in the classical case we would have

or the normal component of B is continuous across the boundary. For the tangential component we must consider the possibility of another source term as we did for .Ex' 73 For an Amperian loop current I, a directed loop area S, the magnetic dipole moment is defined by

Averaging over a volume we obtain the magnetization, M, which is the net dipole moment per unit volume,

From this the Amperian current density becomes

Now, in the classical case, we have, for electrostatic fields,

where J is the Amperian current density. Thus, we could write

We can then define the magnetic intensity vector as

. 0 = n x )E - E( 12 ˆ

(6.46)

0 = Bx'

0 = n )B - B( 12 ˆx

. cSI = mc

. dvmd = M

. M x c = J '

)J + J(c

4 = B x c'S

. Jc

4 = )M 4 - B( x SS'

. M 4 - B = H S

214

Again, experimentally, numerous materials are found such that

is a good approximation for the magnetization for small fields. xm is called the magnetic susceptibility. Thus we would have

where P is the permeability of the material. Now in the Dynamic Theory we have, when 0�/0t = 0,

If we define a gravitational magnetization by

then Eqn. (6.43) becomes

and the magnetic intensity vector should be defined by

or

It may be seen from Eqn. (6.44) that the gravitational magnetization adds to the Amperian magnetization and could lead to misinterpretations. Now, in an analogous fashion to the dielectric displacement vector, the boundary condition becomes

where K is the surface current density according to

H = M mF

H = H )4 + (1 = B m PFS

. )/V(a- J

c4 = )/B( x 0 J

PSP

w'

(6.47)

,V a- = M x 4 0g

JPP'

Jc

4 = M 4 - B x g SP

SP »

»¼

º

««¬

ª'

M 4 - B = H gSP

. )M + M( 4 - B = H gS

(6.48)

K c

4 = n x )H - H( 12 ¸¹·

¨©§ Sˆ

215

Thus, if there is no surface current density, then

or the tangential component of H must be continuous across the boundary. Now we need to do a similar thing for V and V4. Starting with the equation

we will look at the gravitational field defined at

where E is the gravitational charge-to-mass ratio. Thus, we have

where the quantity (J4/Ec) is the free gravitational mass density. Now by using the divergence theorem we have

By using a Gaussian pillbox again, then for a sufficiently small box

For free gravitational mass density to exist on the boundary, the product J4 ǻt must remain finite as t o 0. Therefore, in the limit

fo

o''

J

0 tt)J( = K lim

0 = n x )H - H( 12 ˆ

Jc4 - =

t)V(

c1 + V. 4

4 SHP w

w'

EPV = G

Jc4 - =

t

V

c1 + G. 4

4

ESE

H

w

¸¹

ᬩ

§w

'

. dv c

J 4- = dv t

V

c1 + ad.G 4

v

4

vs¸¹

ᬩ

§¸¹

ᬩ

§w

³³³ ES

EH

. t Sc

J 4- = t

Vc

t S + )n.G + n.G( 441122 '¸

¹

ᬩ

§ww'

ES

EH

(6.49)

. m = c

t J 0 t s

4»¼

º«¬

ª 'o' E

lim

216

Then Eqn. (6.45) becomes

Thus, if there is no free gravitational mass on the surface, the normal component of the gravitational field must be continuous. Question: The relationship between G and V is similar to that between H and B. Is it fair then to consider a similar behavior between them? By this I mean since

then can

where Ȥg may be called the gravitational susceptibility and N is yet unnamed? Now lets look at

For this we construct the closed rectangular path across the boundary. Then we would have

Where S is the rectangular area ǻ1 ǻT and no is the unit vector normal to the rectangle and lies along the boundary of the surface. Performing the line integral, we obtain

But since n- = n 21 ˆˆ 102 and the contribution from the ends is proportional to ǻt we have, as ǻt o 0.

or

. m 4- = n).G - G( s12 Sˆ

M 4 + H = H 4 + H = H = B m SFSP

N 4 + G = G 4 + G = G = Vg SFSP

E

Jww

'B ao- = V x

)dan.B( ao = adn).v x ( = l.dV 0s

0s Jw

wcc'³ ³³

t 1 n.B ao- =

ends) from ion(contribut + 1 )n.V( + 1 )n.V( = )l.dV(

0

2211

''¸¸¹

·¨¨©

§

ww

''³

ˆ

ˆˆ

J

0 = )n x n).(V - V( 1022 ˆˆ

. 0 = n x )V - V( .n 120 ˆˆ

217

But, since the orientation of the rectangle is arbitrary, then

This states that the tangential component of V, the gravitational induction,

must be continuous.

The boundary condition for the scalar, V4 is that V4 must be

continuous because it is a scalar.

It perhaps should be noted that the new physical notions that

appear in the foregoing could prove extremely important should one

consider going into materials development.

6.5 Reflection and Refraction

First we shall consider normal incidence as shown in Figure (15)

Figure 1. EMG wave propagating in the z-direction.

Applying the boundary conditions on the tangential components we

have:

and

or by using J1=Jo we have

and

But

and

0 = n x )V - V( 12 ˆ

e H = E H + e H

e E = e E - e Eik

2ik0

1ik0

0

ik02

ik01

ik00

b24 2

0b14

14bo o

e24 2E14 1

4eo o

JJJ

JJJ

e V = e V - e V 24v214v1o4vo ik02

ik01

ik00

JJJ

� �

� �eH = H - He E = E - E

14b124b2

14e124e2

k - ki02

01

00

k - ki02

01

00

JJ

JJ

(6.50)

� �e V = V - V 14v124v2 k - ki02

01

00

JJ

,M4 - B = H g

P

S

P

218

Therefore, if we assume for the moment that gravitational magnetization is zero, or 0, = M g 112 then

Note: This assumption places some, perhaps severe, restrictions upon Jww /V 114 and we will have to come back and look at these, but for now it seems like a

reasonable assumption to allow us to proceed with reasonable simplicity. Thus, from the first two of Eqn. (6.50) we have

and

Where

By adding Eqn. (6.51) we find, after rearranging the terms,

On the other, by subtracting Eqn. (6.51) and rearranging, we have

For the solution sought in the non-thermal biological section Ez=V4=Vz=0. Thus, if we stay with that solution we need only look at Vx, but

Defining

. E ck = B x

ey ¸

¹·

¨©§Z

. E ck = H 0

0e10

0 ¸¹·

¨©§Z

e E = E - E ei02

01

00

J'

(6.51)

.eE 1k2k = E + E eio

2e

eo1

oo

J'¸¹

ᬩ

§

. k - k 14e124e2e JJJ {'

> @ . E k + ke k2 = E 0

0e2e

e-ie10

2

J'

(6.52)

� �> @ . E

k + kk - k = E 0

0e2e1

e1e201

. E 4cka- + V xe0

x ¸¹·

¨©§

Z

219

then

Thus, the last of Eqn. (6.50) becomes

Using Eqn. (6.52) this becomes

Comparing Eqn. (6.54) with Eqn. (6.53) we find

which is only satisfied only if

Equation (6.55) implies that the dependence of the electric field upon mass density is not influenced by the type of material there. This is a result that is a direct consequence of the assumption previously made and is further evidence that we must return to that assumption soon. For now we shall forge ahead. We shall now consider the case where the incident wave impinges upon the boundary interface at an oblique angle To. The wave is polarized so that the electric component is parallel with the interface. For the incident wave we have

,cka = A 4e0e

. EA- = V xe

x ¸¹·

¨©§Z

. E e AA- = E - E 0

2ei

e

e00

01

J'¸¹

ᬩ

§

(6.53)

E )k + k(k2

AA - 1 = E 0

0e2e1

e1

e1

e201 »

¼

º«¬

ª

(6.54)

� �k2 AA - k = k- e1

e1

e2e1e1 ¸

¹

ᬩ

§

. A = A e2e1

(6.55)

220

For the reflected waves

The refracted waves are given by

The tangential components of �, H, and V, can be continuous across the boundary only if the phases of the field vectors are all equal at the interface.

For each component the propagation vectors keo, ke1, and ke2 are coplanar, so if r is chosen to lie in the interface and in the plane of the propagation vectors, then we have,

� �

°°°°°°

¿

°°°°°°

¾

½

¸¹·

¨©§

¸¹

ᬩ

§

x

x

x

. E A- =

e V = V and

Exk c =

e H = H

e E = E

0eo

k - rk-i(wt-oo0

0eo1

)k - rk-i(wt-000

)k - rk-i(wt-000

14vovo

14bob0

14eoeo

Z

ZPJ

J

J

� �

°°°°

¿

°°°°

¾

½

¸¹·

¨©§

¸¹

ᬩ

§

x

E A- = V and

Exk c = H

e E = E

1e1

1e11

)k - rk-i(wt-011

14e1e1

Z

ZP

J

(6.56)

� �

°°°°

¿

°°°°

¾

½

¸¹·

¨©§

¸¹

ᬩ

§

E A- = V and

Exk c = H

e E = E

2e2

2e22

2

)k - r.k-i(wt-012

24e2e2

Z

ZP

J

(6.57)

.k + r k = k + r k = k + r k

k + r k = k + r k = k + r k

k + r k = k + r k = k + r k

24v2v214v1v114vovo

24b2b214b1b114bobo

24e2e214e1e114eoeo

JJJ

JJJ

JJJ

xxx

xxx

xxx

(6.58)

221

For keo-ke1 we find

From this we find that sinTeo = sinTe1 if J1 = 0 or if .k = k 4eo4e1 133 Not wanting to restrict ourselves un-necessarily by assumptions, lets continue. For other components we have

and

Again using Eqn. (6.58) we find

and

However, for keo = ke1, subtracting Eqn. (6.60) from Eqn. (6.59) yields

so that we have as a required result

In a similar fashion we have

and

.k+ k = k+ k = k+ k 24e2e2e214e1e1e114eoeoeo JTJTJT sinsinsin

� �.k - k k

- = 4eo4e1eo

1eoe1 ¸

¹

ᬩ

§ JTT sinsin

� �k - k k

- = 4bo4b1bo

1bob1 ¸

¹

ᬩ

§ JTT sinsin

� �.k - k k

- = v4v4vo

1V1v 21¸

¹

ᬩ

§ JTT sinsin

k - k = k - k 11222 e4e4ee2e1e1 JJTT sinsin

(6.59)

k-k = k-k 1222 4eoe4ee2eoeo JJTT sinsin

(6.60)

0 = )k - k( 4e14eo1J

.k = k 4e14eo

(6.61)

k = k 4b14bo

k = k 4v14vo

222

With this result Eqn. (6.59) becomes

Similarly, for the other components we have

and

Because of Eqn. (6.57), we must have

Now the tangential components of �, H, and V must be continuous at the interface. Therefore,

and

Eqn. (6.65) may be written in terms of the electric component then we would have

and

k

)k-k(-

kk =

e2

14e124e2e1

e2

e1e2

JJTT sinsin ¸

¹

ᬩ

§

(6.62)

k

k - k(-

kk =

b2

14b124b2b1

b2

b1b2

JJTT sinsin ¸

¹

ᬩ

§

(6.63)

k

)k-k(-

kk =

V2

14v124v2v1

v2

v1v2

JJTT sinsin ¸

¹

ᬩ

§

(6.64)

. = ; = ; = v1vob1b1e1eo TTTTTT

,n x H = n x )H + H(

,nxE = n x )E + E(

21o

21o

ˆˆˆˆ

(6.65)

. nx V = n x )V + V( 21o ˆˆ

> @ ,n)xExk( = nx)Exk( + )Exk(

,nxE = n)xE + E(

2e2

11e1oeo

21o

2ˆˆ

ˆˆ

¸¹

ᬩ

§

PP

(6.66)

).nxE(A = n)xEA + EA( 2e21e1oeo ˆˆ

223

From the first and last of Eqn. (6.66) we have

or

Since both �o and �1 are perpendicular to the plane of incidence then Eqn. (6.66) requires that

But, since Aeo = Ae1, this can be satisfied only if

We've seen this result before. Now suppose we expand the triple cross products in Eqn. (6.62), then

All the products );E n( xˆ 155 vanish, and ;k(-1)j = k n ejejej Tcosˆ x 156 j = 0, 1, 2, so that

If we use the fact that Teo = Te1, and rearrange the terms we arrive at

Since the electric vectors are all parallel to the boundary surface, we must have

n)xE + E(A = n)xE1A - EA( 1oe21eoeo ˆˆ

> @ 0. = n x E)A - A( + E)A - A( 1e2e1oe2eo ˆ

(6.67)

.E)A - A(- = E)A - A( 1e2e1oe2eo

.A = A e2e1

> @ > @

> @.k)E n( - E)k n(

= k)E (n - E)k n( + k)E n( - E)k n(

e222e22

1

e111e1eoooeo

xx¸¹

ᬩ

§

xxxx

ˆˆ

ˆˆˆ

PP

.Ek = E k - Ek 2e22

11e1e1oeoeo ¸

¹

ᬩ

§

PP

TT coscos

e kk = )E - E( )k-ki(

e2e1

eo

2

1eo1o

oo 14e124e2 JJT

PP

T coscos ¸¹

ᬩ

§¸¹

ᬩ

§

(6.68)

eE = E + E )k-ki(O2

O1

OO 14e124e2 JJ

(6.69)

224

We may combine Eqn. (6.68) and (6.69) by subtraction to eliminate E2o and obtain

Eliminating EO we have

Equations (6.62) and (6.63) may be used to determine the refracted angles for each component while Eqns. (6.69) and (6.70) determine the magnitude of the reflected and refracted electric field components. The magnitudes of the reflected and refracted magnetic and gravitational components may be found using Eqns. (6.56) and (6.57). 6.6 Complex Refraction Angles In order to solve the refraction problem when one, or both, mediums at an interface are conductors then we must have a computer code capable of solving the complex angle problem. Thus, we must learn how to interpret the complex angle of refraction then how to compute it. We may start with the equations already derived for the sine of the refraction angles, which are

and

� �> @. E

kk+

k- = E 0

0

e2e12

e21eo

e2e21eo01

»»¼

º

««¬

ª¸¹

ᬩ

§T

PP

T

TPT

coscos

coscos

(6.70)

»»¼

º

««¬

ª¸¹

ᬩ

§T

PP

T

T JJ

e2e12

e21eo

)k-ki(eo0

2

kk+

e 2 = E24e214e1

coscos

cos

(6.71)

,k

)k-k( -

kk = 2

e2

14e124e2e1

e2

e1e

JJTT sinsin ¸

¹

ᬩ

§

(6.62)

,k

)k-k( -

kk =

b2

14b124b2b1

b2

b1b2

JJTT sinsin ¸

¹

ᬩ

§

(6.63)

. k

)k-k( -

kk =

v2

14v124v2v1

v2

v1v2

JJTT sinsin ¸

¹

ᬩ

§

(6.64)

225

With the realization that one or more of the k's in Eqns. (6.62), (6.63), or (6.64) may be complex then one must consider the right-hand-side to be complex. Thus, we have the situation that sinT2 for each component, would be complex and, therefore, we must consider T2 to be complex. Thus consider the case where

with

Then, by trigometric identity, we may write

Equating Eqns. (6.72) and (6.73), we find

From Eqn. (6.74) we find that

and

Now Į2 is a real angle and the expression for sinĮ2 in Eqn. (6.75) reduces to the usual expression for the sine of the refraction angle. Thus, we shall take Į2 to be the angle of refraction and it is given by Eqn. (6.75). We must now learn how to find Į2 and E2 given x and y. We may start by rewriting Eqn. (6.75) as

and

iy + x = 2Tsin

(6.72)

. i + = 222 JJT

. h cosa i + = 22222 EDEDT sincoshsinsin

(6.73)

.iy + x = h i + 2222 EDED sincoscoshsin

(6.74)

E

D2

2x =

coshsin

. =y 22 ED sinhcos

(6.75)

E

D2

2 hx =

cossin

(6.76)

226

Now using Eqn. (6.76) the trigometric identity

becomes

or

when coshE2 and sinhE2 are written in terms of exponentials. Now suppose we define

where w � 1 always. Then Eqn. (6.77) becomes

or

This may be rewritten as

Equation (6.79) is a quadratic equation in w which has the solutions

. hy =

22

ED

sincos

TT cossin 22 + = 1

¸¹

ᬩ

§¸¹

ᬩ

§

EE 2

2

2

2

hy + x = 1

sincosh

»¼

º«¬

ª»¼

º«¬

ª

e-e2y +

e+e2x = 1 2222 -

2

-

2

EEEE

(6.77)

2

e + e = )(2 = w2-222

2EE

Ecosh

(6.78)

2)-e + e(

y4 + 2)+e+ e(

x4 = 1 2222 2-2

2

2-2

2

EEEE

¸¹

ᬩ

§¸¹

ᬩ

§1-w

y2 + 1+w

x2 = 122

. 0 = 1 - )y - x2( + )y + x( 2w - w 22222

(6.79)

227

The expression under the radical is non-negative (as may be shown by a lengthy procedure) so w is real, which must be the case from Eqn. (6.78). Consider three cases: Case A: x2 + y2 > 1 Let x2 + y2 = 1 + į where į � 0, then Eqn. (6.80) becomes

If y = 0, then because w � 1, the positive sign must be used. If y = 0, then there is no need for Eqn. (6.80). Case B: x2 + y2 � 1 Let x2 + y2 - 1 + į where į � 0, then Eqn. (6.80) becomes

Again, because w � 1, clearly the upper sign must be chosen. Case C: x2 + y2 = 1 Equation (6.76) now becomes

If y � 0, the upper sign must be used. If y � 0, there is no need for Eqn. (6.80). From Eqn. (6.80) and the above logic on how to choose the proper sign we may obtain w from x and y. Then from w we find E2 using Eqn. (6.78) or

then by Eqn. (6.72) we may find Į2 from

. y4+)1-y+x( + )y + x( = w 222222 _

(6.80)

. y4 + + - 1 = w 22HH _

. y4 + + - 1 = w 22GH _

2y + 1 = w _

2

coshw = 2arc

E

(6.81)

. x = 2

2 ¸¹

ᬩ

§

ED

coshsinarc

(6.82)

228

Thus Eqn.s (6.80), (6.81) and (6.82) give us Į2 and E2 for any x and y. The Į2 and E2 must be checked against Eqn.s (6.75) and (6.76) to resolve any ambiguities. 6.7 Assumptions and Wave Solutions In order to try to bring to light the significance of the variation of material properties with respect to changes in the mass density in the attempt to obtain wave solutions let us consider each assumption that must be made. To do this let us begin with the form of the trial solution. The classically assumed form for a trial solution is

for plane wave propagation in the x-direction. In as far as it may be possible we would like to stay with a similar form. Thus suppose we try the form

Once the form of the trial solution is chosen then one can look at the eigenvalues of the differential operations since the trial form is the exponential form. Consider the partial derivative with respect to x

if we assume that

Since we desire to consider how waves of a certain frequency propagate it seems appropriate to adopt this assumption. Next, consider the eigenvalue of the differention

There is nothing thus far in our considerations that forces us to consider only those cases for which the phase is strictly linear in x, or where the density constant cannot depend upon space. In the classical case I believe one can use the Maxwell equations, or the wave equations coming from them, to show that the phase must be linear in x. The same may prove true for these five-dimensional waves but it has not yet been

> @kx)-i(wt- exp

> @)k-kx-i(wt- 4Jexp

)] ,k-kx-[-i(wt)] k-(kx

xi[ =

)]}k-kx-[-i(wt{x

44

4

JJ

J

exp

exp

ww

ww

0 = xwww

}

xk + xk + k +

xki{x =

)k+(kxx

i x

44

4

ww

ww

ww

ww

oww

JJ

J

229

proven. For the sake of simplicity let's make the same assumption here and hold in abeyance any attempt to prove a linear relationship. Therefore, lets assume

With regard to k4 we have no precedence set by classical theory and with no real feeling for the physical interpretation of k4 we are left to our own devices. For the moment suppose we make no assumptions regarding the dependence of k4 upon space, but we might assume isotropy is mass density. Thus, our eigenvalue for the space differential operator we have

Now let's look at the mass density differential operator, or

if we assume

The assumption that the frequency should not depend upon the mass density seems justifiable since we want to determine how a wave of a certain frequency will propagate. Therefore, we want to control the frequency. We should not, however, allow this desire to lock us into this assumption. Given the assumption Eqn. (6.83) we have

By analogy with the classical result that the phase of the wave depends only linearly upon x it seems a fair assumption that we may simplify some by assuming that

We have no real justification for this assumption at the moment though. However, with this assumption our mass density operator becomes

0 = xkww

)xk+i(k

x4

ww

oww

J

)]k-kx-t[-i( )k+(kx i =

)]}k-kx-t[-i( {

44

4

JZJJ

JZJ

exp

exp

»¼

º«¬

ªww

ww

. 0 = JZww

(6.83)

. k + k + kx = )k+(kx 44

4 JJ

JJ

J ww

ww

ww

0 = k4

Jww

230

If we consider the potential variation of k with respect to mass density then we run into the dependence of k upon P and V and whether these material properties depend upon mass density. If they do, as experiment tends to show then

Since we desire to retain the correspondence to experiment we shall make no assumptions concerning the dependency of k upon mass density. The remaining operator is the time operator. For this we have

if we assume that

and

From the point of view that both k and k4 are determined by material properties then these assumptions appear appropriate for static materials. Thus the eigenvalue of the time differential operator is

We have now chosen a general form for the solution we will seek. But there are several potential components to the wave. For example, there are the electric transverse, magnetic transverse, gravitational transverse, electric longitudinal, and the gravitational longitudinal components. In addition there is the scalar wave component, the gravitational potential. In the classical case it may be shown that the propagation constant may be the same for both the electric and magnetic components. That is not so with these more complex five-dimensional waves. Thus, we should allow for the possibility that each component may have a different propagator. With this in mind we will try to find wave solutions with the following trial forms.

. )k + ki(x 4JJ ww

oww

. k + k = k¸¹

ᬩ

§ww

¸¹·

¨©§ww

¸¹

ᬩ

§ww

¸¹·

¨©§ww

ww

JV

VJH

HJ

|)k-kx-t-i(| i- = |})k-kx-t-i(| { t 44 JZZJZ expexpww

0 = tkww

. 0 = tk4

ww

. i- t

Zoww

231

By using these four forms which constitute our trial solution we may find that two, or all, of the k's must be the same but we aren't forcing them to equality prematurely. Now lets put our trial solutions into the field equations. Lets start with the field equation

The x and y components of this equation require that

For the z component we have, after simplifying,

The second field equation is

The x and y components require that

However, the z component gives us

Now look at the field equation

.)] k-xk-[-i(wt V = V

)]k-xk-[-i(wt V = V

)]k-xk-[-i(wt B = B

)]k-xk-[-i(wt E = E

4444O4

4vvO

4bb0

4eeO

J

J

J

J

expexp

exp

exp

. 0 = Ex + tB

c1

'ww

¸¹·

¨©§

(3.15b)

. 0 = B = B yx

. E xk + k

wc = B y

eez ¸

¹·

¨©§

ww

¸¹·

¨©§ J

(6.84)

. 0 = Ba + Vx O Jw

w'

(3.15f)

. 0 = V z

. E

xk + k

xk + k

kx + k w

ca- = V y4V

4ee

b4b

0y

»»»»

¼

º

««««

¬

ª

wwww

¸¹·

¨©§

*ww

¸¹·

¨©§

J

J

Q

(6.85)

232

We see that we face more assumptions. The first one concerns whether or not H varies with space. The classical assumption seems appropriate here also. That is, if the medium is isotropic then

The second assumption is that there are no free charges so that

The third place for a possible assumption resides in the possible dependence of H upon mass density. However, experiment indicates that H does vary with changes in mass density. Therefore, it seems inappropriate to assume differently. Thus, Eqn. (3.14d) requires that

From Eqn. (6.82) we see that the gravitational potential, V4, depends only upon the longitudinal electric field component. The fourth field equation is

The x component of this vector equation requires

. )V(a - 4 = )E( 4

O JH

SUHw

wx'

(3.15d)

. 0 = xwwH

. 0 = U

. ]k+k+

i1-

)Exxk+k(

a1- = V4

444

4ee

O

JJH

H

J

ww

ww

¸¹·

¨©§

ww

¸¹

ᬩ

§ (6.86)

. E a = V + tV

c1

04 Jww

'ww

¸¹·

¨©§

(3.15g)

. Ex }kx+k( a +

|i-kk|a

k

)xk+k)(

xk+k(

{wc- = V

e4eO

444O

4

4ee

444

x

x

J

JH

HJ

JJ

ww

ww

¸¹·

¨©§

ww

www

ww

¸¹·

¨©§

(6.88)

233

From the y component of Eqn. (3.15g) we find

If we compare Eqn. (6.88) with Eqn. (6.85) we must have

The z component of Eqn. (3.15g) is an identity, thus we can turn to the fifth field equation, which is

The usual conductivity assumptions seem appropriate here and are taken as

Therefore, Eqn. (3.15h) becomes the indicial relation

when Eqns. (6.86) and (6.88) are used. The sixth field equation is

.Ey )kx+k( w

ca- = V e4e

0y Jw

w¸¹·

¨©§

(6.88)

. k+kxk+k = kx+kx

k+k b4b

4ee

e4e

4vv ¸

¹

ᬩ

§ww

¸¹·

¨©§

ww

¸¹

ᬩ

§ww

¸¹·

¨©§

ww

JJJ

(6.89)

. J c

4- = t

Vc

+ V 44 SPPH

ww

¸¹·

¨©§x'

(3.15h)

V = J and ,E = J 444 VV

(6.90)

.

xk+k4)i4+( =

ei-kx+k

xkx+ka+

xk+kx

k+kxk+kc

4ee

2444

e4e

20

4ee

444

4vv

2

¸¹·

¨©§

ww

°¿

°¾½

»»¼

º

««¬

ª¸¹

ᬩ

§ww

ww

°°®­

»¼

º«¬

ª¸¹·

¨©§

ww

ww

¸¹·

¨©§

ww

¸¹·

¨©§

ww

JSPZVZPHJH

J

JJJ

(6.91)

. Va- J

c4 =

tE

c - Bx O J

SPPHww

¸¹·

¨©§

ww

¸¹·

¨©§'

(3.15c)

234

The x component of this equation produces the indicial relation

The z component of Eqn. (3.15c) is an identity but the y component gives us another indicial relation in

The seventh field equation

is an identity since Bx = 0. However, the last remaining field equation

is not satisfied identically. Rather, it requires

. kx+ka +

i-kx+k

xk+kx

k+kkx+k c = i4+

e4e

2O

444

4ee

444

v4v

22

°¿

°¾½¸¹

ᬩ

§ww

°°

¯

°°

®

­

»¼

º«¬

ª¸¹

ᬩ

§ww

ww

¸¹·

¨©§

ww

¸¹·

¨©§

ww

¸¹

ᬩ

§ww

J

JH

HJ

JJ

JSPZVZPH

(6.92)

)kx+k)(kx+k(ca +

)x

k+k)(xk+k( = wi4+w

e4e4

22O

4bb

4ee

2

JJ

JJVSPPH

QQ w

www

ww

ww

(6.93)

. 0 = B.'

(3.15a)

J

Uww

x'ww Ja + J +

t = 0 4

O

(3.15e)

235

if we assume

We are once more in a dilemma created by our ignorance. We don't know what V4 is other than a "gravitational conductivity". Thus, it would appear that we cannot assume that it is independent of the mass density. Thus Eqn. (6.94) may be put into the form

This is an extremely curious equation. It relates the electrical conductivity to the gravitational conductivity and includes in this relation the variation of the dielectric constant with respect to changes in the mass density. In the absence of any knowledge about V4, the gravitational conductivity, let us assume that there is a linear relationship between it and the mass density. The only justification for this assumption comes from the association of gravitational mass to gravitational conductivity and also to inertial mass and hence mass density. Regardless of our lack of knowledge lets assume

Further, in attempting further simplification lets assume

Using these assumptions we may rewrite Eqn. (6.95) as

J

VVw

www )V(

a- = xE 44

Ox

(6.94)

. 0 = xw

wV

. 4 - i = kx+k ) - ( 4444

»»¼

º

««¬

ª

ww

¸¹

ᬩ

§ww

¸¹·

¨©§

¸¹

ᬩ

§ww

JV

JH

HV

JVV

(6.95)

. = 4 KJV

. = 4VQV

°¿

°¾½

°°®­

»¼

º«¬

ª¸¹·

¨©§

ww

¸¹·

¨©§

¸¹

ᬩ

§ww

VQH

JQJ

-li = k x+k 4

44 ln

(6.96)

236

The right hand side of Eqn. (6.96) is in terms of quantities which should be available through experimentation, thereby giving us a differential equation in the two unknowns k44 and k4. We now have a system of equations given by the following numbered equations: (6.84), (6.86), (6.87), (6.88), (6.89), (6.91), (6.92), (6.93), and (6.96). These equations give the field components in terms of Ey and Ex. However, the eight field equations cannot determine a relationship between the longitudinal and transverse components. This is left for the energy-momentum tensor to determine. In order to attempt a reduction of these equations we might try the simplifying assumptions:

and

Using these assumptions let us now look at the requirements that come from the interchangeability of substitution and differentiation. For instance,

when differentiation is taken first. On the other hand if we take the substitution first then we get

By comparing these two we see that

When we compare the two expressions for the partial derivative of Bz with respect to the mass density we find

Turning next to the transverse gravitational component, the partial derivative with respect to x requires that

,0 =

xk

,0 = xk

4b

4e

wwww

. 0 = x

k4v

ww

,E ck ik =

xB

ye

bz ¸

¹·

¨©§

ww

Z

. E ck ki =

xB

ye

ez ¸

¹·

¨©§

ww

Z

. k = k eb

. )k-ki( = )lnk(4e4b

e

Jww

237

This sets up something of a dilemma since ke should depend upon both H and V. These in turn depend upon J. Therefore, Eqn. (6.97) is a result that does not seem to correspond to experiment. Thus, our choice of simplifying assumptions appears too restrictive. But which assumption is the one that must be relaxed? An investigation into the necessity of the assumptions made seems required prior to making advancement toward the solutions of the electromagnetogravitic wave equations.

. 0 = ke

Jww

(6.97)

238

References W.R. Adey, Tissue Interactions with Nonionizing Electromagnetic Fields, Physiological Reviews, Vol. 61, No. 2, April 1981. D. ter Haar and H. Wergelande, Elements of Thermodynamics, 1966. H. Weyl, Space, Time, and Matter, Dover, 1922. R. Adler, M. Bazin, and M. Schiffer, Introduction to General Relativity, 1965. Th. Kaluza, Sitzungsber. d. Preuss. Akad. d. Wiss., 1921, p. 966. O. Veblen, Projektive Relativitatstheorie, Berlin, springer, 1933. W. Pauli, Ann D. Physik, 19, 305 (1933); 18, 337 (1933). A. Einstein and Mayer, Berl. Ber., 1931, p. 541; Berl. Ber. 1932, p. 130. A. Einstein and P. Bergmann, Ann. of. Math., 39, 683 (1938). Einstein, V. Bargmann, and P.G. Bergmann, Theodore von Karmaan Anniversary Volume, Pasadena, 1941, p. 212.

239

Chapter 7 Hydrodynamic Systems The equation of motion for the fifth dimension, mass density, appears as a generalization of the principle of the conservation of mass. Further, in classical hydrodynamic systems five equations in five unknowns are used. It seems logical then to expect the five equations of motion appearing in the five-dimensional Dynamic Theory to be generalizations of the classical equations. An added incentive to investigate the possibilities of this generalization is gained when electromagnetically contained ionized plasmas with mass conversion are considered. For if the five equations are generalizations of the classical hydrodynamic equations, then the use of the five-dimensional fields allowing mass conversion should provide an entirely new viewpoint of a controlled fusion reactor. Since it is suspected that the five equations of motion resulting from the application of the principle of increasing entropy to a thermo-mechanical system are generalizations of the classical equations, it then becomes necessary to show that this is indeed the case. This seems possible by restricting the system so that it corresponds to the usual system considered. First, from the Dynamic Theory approach, the manifold required for a description of the system is the five-dimensional manifold of space, time, and mass density. Within this manifold the continuity equation of mass no longer holds for the general system. We can, however, restrict our system by first requiring that the system remain on a hypersurface within the five-dimensional manifold. For a system so restricted, any of the five dimensions may be considered as functions of the other four. In particular, since by custom in hydrodynamics the mass density is considered to be a function of space and time, we may consider the mass density to be the variable chosen to be function of the others or

so that

Such a system will be constrained to be on a hypersurface embedded within the five-dimensional manifold of space, time, and mass density as shown and upon this hypersurface will be described in a four-dimensional manifold of space and time. If we further restrict our system by requiring that the total derivative of the mass density to be zero or

� �x ,x ,x ,x = 3210JJ

. dx x

= d DD

JJ ¸

¹

ᬩ

§ww

240

then

or

which is the usual continuity equation. Thus, by restricting the system to this particular hypersurface we have constrained the system to obey the continuity equation as does a usual hydrodynamic system. Not only does this restriction place our system within the space-time manifold where we may compare the resulting four equations of motion with the equations of motion in relativistic theories but, since the seven gauge field equations must hold in the five-dimensional manifold they must also hold on the hypersurface. This allows the new field quantities to be expressed as functions of the �, B fields and the partial derivatives of the mass densities. Further, it appears that the additional B field equations may be used to determine a dependence of the E and B fields upon the mass density and/or its changes. Then by comparing the equations of motion obtained here for the system restricted to the mass conservation hypersurface with the relativistic Navier-Stokes equations it should be possible to identify the viscous coefficients with the field quantities and perhaps see how the viscosity depends upon these fields as I feel it does. Since we have restricted the system to a hypersurface where the mass density is a function of space and time, then the surface is defined by five equations of the type

Further, since x4 = Ȗ/a0 and x4 = x4(x0, x1, x2, x3), then Eqn. (7.1) becomes

and

dx dxd = 0 = d D

D

JJ

vdx

+ vx

+ vx

+ t

= 0 = dtd 3

32

21

1

JJJJJ www

ww

ww

,0 = v grad + t

xww

JJ

. )u ,u ,u ,u(x = x 3210ii

(7.1)

u = x,u = x,u = x,u = x 33221100

. )u ,u ,u ,u( f = x 32104

241

Since u0, u1, u2, and u3 are independent variables, the locus defined

by Eqn. (7.1) is four-dimensional, and these equations give the coordinates

xi of a point on the hypersurface when u0, u1, u2, and u3 are assigned

particular values. This point of view leads one to consider the surface as a

four-dimensional manifold S embedded in a five-dimensional enveloping

space. We can also study surfaces without reference to the surrounding

space, and consider parameters u0, u1, u2, and u3 as coordinates of points

in the surface.

If we assign to u0 in Eqn. (7.1) some fixed value u0 = u0, we obtain a

three-dimensional manifold

which is a three-dimensional manifold lying on the hypersurface S defined

by Eqn. (7.1). By assigning fixed values for any three of the four

hypersurface variables we obtain a net of curves, on the hypersurface,

which may be called coordinate curves.

Obviously the parametric representation of a hypersurface in the

form of Eqn. (7.1) is not unique, and there are infinitely many curvilinear

coordinate systems which can be used to locate points on a given

hypersurface S. Thus, if one introduces a transformation

and

where the uĮ (u-0, u-1, u-2, u-3) are of class C1 and are such that the

Jacobian

does not vanish in some region of the variables u, then one can insert the

values from Eqn. (7.2) in Eqn. (7.1) and obtain a different set of parametric

equations

defining the hypersurface S. Equation (7.2) can be looked upon as

representing a transformation of coordinates in the hypersurface.

4) 3, 2, 1, 0, = (i ),u ,u ,u ,u( x = x 3210ii

,)u ,u ,u ,u( u = u

,)u ,u ,u ,u( u = u

,)u ,u ,u ,u( u = u

3-2-1-0-22

3-2-1-0-11

-3-2-1-000

,)u ,u ,u ,u( u = u -3-2-1-033

(7.2)

)u ,u ,u ,u(

)u ,u ,u ,u( = J 3-2-1-0-

321O

ww

)u ,u ,u ,u( f = x -3-2-1-0ii

(7.3)

242

7.1 First Fundamental Quadratic Form The properties of hypersurfaces that can be described without reference to the space in which the hypersurface is embedded are termed "intrinsic" properties. A study of intrinsic properties is made to depend on a certain quadratic differential form describing the metric character of the hypersurface. We proceed to derive this quadratic form for our restricted system. It will be convenient to adopt certain conventions concerning the meaning of indices to be used. We will be dealing with two distinct sets of variables: those referring to the five-dimensional space in which the hypersurface is embedded (these are five in number) and with four coordinates u0, u1, u2, and u3 referring to the four-dimensional manifold S. In order not to confuse these sets of variables we shall use Latin letters for the indices referring to the space variables and Greek letters for the hypersurface variables. Thus, Latin indices will assume values 0, 1, 2, 3, 4 and Greek indices will have the range of values 0, 1, 2, 3. A transformation T of space coordinates from one system X to another X will be written as

a transformation of Gaussian hypersurface coordinates, such as described by Eqn. (7.2) will be denoted by

A repeated Greek index in any term denotes the summation from 0 to 3; a repeated Latin index represents the sum from 0 to 4. Unless a statement to the contrary is made, we shall suppose that all functions appearing in the discussion are of class C2 in the regions of their definitions. Consider the hypersurface S defined by

where the xi are coordinates covering the five-dimensional space in which the hypersurface S is embedded, and a curve C on S defined by

where the uĮ's are the Gaussian coordinates covering S. Viewed from the surrounding space, the curve defined by Eqn. (7.4) is a curve in a five-dimensional manifold, which we shall assume, for the present, is

,)u ,u ,u ,u( x = x 3210ii

(7.4)

WWWWDD 21 ,)( u = u dd

(7.5)

243

Riemannian entropy manifold of the Dynamic Theory, and its element of arc is given by the formula

From Eqn. (7.4) we have

where, as is clear from (7.5),

Substituting from Eqn. (7.6) and Eqn. (7.7), we get

where

The expression for (dq0)2, namely

is the square of the linear element of C lying on the hypersurface S, and the right hand member of (7.8) can be called the First Fundamental quadratic form of the hypersurface. The length of arc of the curve is given by

where

W

DD

ddu = u� 25 and q0 is the specific entropy. The total change in the entropy

along the curve C would then be

dxdx g = )dq( jiij

20

(7.6)

du ux = dx

ii D

Dww

(7.7)

. d ddu = du WW

DD

� � ,duduA =

duduux

uxg = dq

ji

ij0 2

EDDE

EDED w

wwwˆ

. ux

uxg A

ji

ij EDDE ww

ww

{ ˆ

(7.8)

� � ,duduA = dq0 2 EDDE

,d buA_ = q - q2

1

01

02 WJ ED

DE

W

W

��³

244

Consider a transformation of surface coordinates

with a non-vanishing Jacobian

It follows from Eqn. (7.11) that

and hence Eqn. (7.9) yields

If we set

we see that the set of quantities AĮȕ represents a symmetric covariant tensor of rank two with respect to the admissible transformations Eqn. (7.11) of hypersurface coordinates. The fact that the AĮȕ are components of a tensor is also evident from Eqn. (7.9), since (dq0)2 is an invariant and the quantities AĮȕ are symmetric. The tensor AĮȕ is called the covariant metric tensor of the hypersurface. Since the form Eqn. (7.9) is positive definite, the determinant

and we can define the reciprocal tensor AĮȕ by the formula AĮȕ AȕȖ = The properties of surfaces concerning the study of the first fundamental quadratic form

� � . d uuA_ = q - q2

1

01

02 WJJ ED

DE

W

W³

(7.10)

)u ,u ,u ,u( u = u 3210DD

(7.11)

uu = J E

D

ww

,ud uu = du EE

DD

ww

� � . ud ud uu

uuA = dq0 2 GJ

G

E

J

D

DEww

ww

,uu

uuA = A G

E

J

D

DEJGww

ww

0 > A = A DE

� � duduA = dq0 2 EDDE

245

constitute a body of what is known as the 'intrinsic geometry of surfaces.' They take no account of the distinguishing characteristics of surfaces as they might appear to observer located in the surrounding space. Two surfaces, a cylinder and a cone, for example, appear to be entirely different when viewed from the enveloping space, and yet their intrinsic geometries are completely indistinguishable since the metric properties of cylinders and cones can be described by the identical expressions for square of the element of arc. If a coordinate system exists on each of the two surfaces such that the linear elements on them are characterized by the same metric coefficients AĮȕ, the surfaces are called "isometric." Thus, if our description of the restricted system is done only in terms of the intrinsic geometry of the hypersurface we may lose sight of features which may characterize our system when viewed from the enveloping space. Therefore, in order to characterize the shape of the surface we must develop a view which involves the enveloping space. 7.2 Second Fundamental Quadratic Form An entity that provides a characteristic of the shape of the surface as it appears from the enveloping space is the normal line to the surface. The behavior of the normal line as its foot is displaced along the surface depends on the shape of the surface, and it occurred to Gauss to describe certain properties of surfaces with the aid of a quadratic form that depends in a fundamental way on the behavior of the normal line. Before we in-troduce this new quadratic form let us recall the definition Eqn. (7.8),

We note that the foregoing formulas depend on both the Latin and Greek indices, and we recall that the Latin indices run from 0 to 4 and refer to the surrounding space, whereas the Greek indices assume values 0, 1, 2, and 3 and are associated with the embedded hypersurface. Furthermore, the dxi and gij's are tensors with respect to the transformations induced on the space variables xi, whereas such quantities duĮ and AĮȕ are tensors with respect to the transformation of Gaussian surface coordinates uĮ. Equation (7.8) is a curious one since it contains partial derivatives

uxi

Dww 35 depending on both Latin and Greek indices. Since both AĮȕ and gij in Eqn (7.8)

are tensors, this formula suggests that

uxi

Dww 36 can be regarded either as a contravariant space vector or as a covariant surface

vector. Let us investigate this set of quantities more closely.

. 3) 2, 1, 0, = ,( 4) 3, 2, 1, 0, =j (i, ux

uxg A

ji

ij EDEDDE w

www

{ ˆ

246

Let us take a small displacement on the hypersurface S, specified by the surface vector duĮ. The same displacement, as is clear from Eqn. (7.7), is described by the space vector with components

The left-hand member of this expression is independent of the Greek indices, and hence it is invariant relative to a change of the surface coordinates uĮ. Since duĮ is an arbitrary surface vector, we conclude that

is a covariant surface vector. On the other hand, if we change the space coordinates, the duĮ, being a surface vector, is invariant relative to this change, so the Eqn. (7.13) must be a contravariant space vector. Hence we can write Eqn. (7.13) as

where the indices properly describe the tensor character of this set of quantities. Let A and B be a pair of surface vectors drawn from one point P of S.

FIG HERE Then using Eqn. (7.14) they can be represented in the form

The five-dimensional vector product, defined by

. duux = dx

ii D

Dww

(7.12)

uxi

Dww

(7.13)

ux x

ii

DD ww

{

(7.14)

Bx = B and Ax = A iiii DD

DD

(7.15)

,BA = N jikijk H

(7.16)

247

is the vector normal to the tangent plane determined by the vectors A and B, and the unit vector n perpendicular to the tangent plane, so oriented that A, B, and n form a right-handed system, is

We call the vector n the unit normal vector to the hypersurface S at P. Clearly, n is a function of coordinates (u0, u1, u2, u3), and as the point P(u0, u1, u2, u3) is displaced to a new position P(u0 + du0, u1 + du1, u2 + du2, u3 + du3), the vector n undergoes a change

whereas the position vector r is changed by the amount

Let us form the scalar product

If we define

so that Eqn. (7.19) reads

the left-hand member of Eqn. (7.20), being the scalar product of two vectors in a Riemannian space by being in the entropy manifold, is an invariant; moreover, from symmetry with respect to Į and ȕ , it is clear that the coefficients duĮ duȕ in the right-hand member of Eqn. (7.20) define a covariant tensor of rank two. The quadratic form

BA

BA = n jikij

EDDEH

H

(7.17)

duun = nd DDw

w

(7.18)

. duur = rd DDw

w

. duduur

un = rd nd ED

ED ww

xww

x

(7.19)

¸¸¹

·¨¨©

§

ww

xww

ww

xww

ur

un +

ur

un

21 = b DEEDDE

,dudub - = rd nd EDDEx

(6.20)

248

called the second fundamental quadratic form of the hypersurface, will be

shown to play an essential part in the study of hypersurfaces when they

are viewed from the surrounding space, just as the first fundamental

quadratic form

rd rd A x{ 49 or

did in the study of intrinsic properties of a hypersurface.

We can rewrite the formula Eqn. (7.17) in terms of the components

xĮi of the base vectors aĮ. We denote the covariant components of n by ni

and observe that its covariant components ni are given by

and

Substituting in Eqn. (7.22) from Eqn. (7.15) and Eqn. (7.23), we get

and, since this relation is valid for all surface vectors, we conclude that

Multiplying Eqn. (7.24) by İĮȕ, and noting that İĮȕİĮȕ=2, we get the desired

result

,dudub B EDDE{

(7.21)

,duduA = A EDDE

T

H BABA = n

kjijk

isin

(7.22)

. AA = B A EDDEHTsin

(7.23)

� � 0 = BA xx - n kjijki

EDEDDE HH

. xx = n kjijki EDDE HH

(7.24)

. xx21 = n kj

ijki EDDE HH

(7.25)

249

It is clear from the structure of this formula that ni is a space vector which does not depend on the choice of surface coordinates. This fact is also obvious from purely geometric considerations. 7.3. Tensor Derivatives We wish to reduce the second fundamental quadratic form eqn. (7.21) analytically by the operation of tensor differentiation of tensor fields which are functions of both surface and space coordinates. To do this we shall first present the concept of tensor differentiation introduced by A. J. McConnell*. Let us consider a curve C lying on a given hypersurface S and a vector Ai defined along C. If IJ is a parameter along C, we can compute the intrinsic derivative

GWG Ai

56 of Ai, namely,

In formula eqn. (7.26) the Christoffel symbols

¿¾½

¯®­

jki

g 58 refer to the space coordinates xi and are formed from the metric

coefficients gij. This is indicated by the prefix g 59 on the symbol. On the other hand, if we consider a surface vector A defined along the same curve C, we can form the intrinsic derivative with respect to the surface variables, namely,

In this expression the Christoffel symbols

¿¾½

¯®­EJD

a 61 are formed from the metric coefficients a Įȕ associated with the Gaussian

hypersurface coordinates uĮ. A geometric interpretation of these formulas is at hand when the fields Ai and AĮ are such that

0 = Ai

GWG 62 and

0 = AGWG D

63. In the first equation the vectors Ai form a parallel field with respect to

C, considered as a space curve, whereas the equation

,ddxA jk

i g +

dtdA = A k

jii

WGWG

¿¾½

¯®­ˆ

(7.26)

. dduA

a +

ddA = S

WEJD

WGWG J

EDD

¿¾½

¯®­

(7.27)

250

0 = AGWG D

64 defines a parallel field with respect to C regarded as a surface curve. The

corresponding formulas for the intrinsic derivatives of the covariant vectors Ai and AĮ are

and

Consider next a tensor field T i

D 67, which is a contravariant vector with respect to a transformation of space coordinate xi and a covariant vector relative to a transformation of surface coordinates uĮ . An example of a field of this type is the tensor

nx = x

iI

DD ww 68 introduced earlier. If

T iD 69 is defined over a surface curve C, and the parameter along C is IJ, then

T iD 70 is a function of IJ. We introduce a parallel vector field Ai along C,

regarded as a space curve, and a parallel vector field BĮ along C, viewed as a surface curve, and form an invariant

The derivative of ij(IJ) with respect to the parameter IJ is given by the expression

which is obviously an invariant relative to both the space and surface coordinates. But, since the fields Ai(IJ) and BĮ(IJ) are parallel,

and eqn. (7.30) becomes

WWGW

GddxA ij

k g -

dAd = A j

kii

¿¾½

¯®­ˆ

(7.28)

. dduA

a -

dAd = A

WDEJ

WGWG E

JDD

¿¾½

¯®­

(7.29)

. BAT = )( ii DDWI

,ddBAT + Bd

dAT + BAddT =

dd

ii

ii

i

i

WWWWI D

DD

DDD

(7.30)

,dduB

a =

ddB and

ddxA ij

k g =

dAd j

ki

WEJD

WWW

JE

D

¿¾½

¯®­

¿¾½

¯®­ˆ

251

Since this is invariant for an arbitrary choice of parallel fields Ai and BĮ, the quotient law guarantees that the expression in the brackets of Eqn. (7.31) is a tensor of the same character as T i

D 75. We call this tensor the intrinsic tensor derivative of T i

D 76 with respect to the parameter IJ, and write

If the field T i

D 78 is defined over the entire hypersurface S, we can argue that, since

is a tensor field and

W

J

ddu 80 is an arbitrary surface vector (for C is arbitrary), the expression in the bracket is a

tensor of the type T i

DJ 81. We write

and call T i

,JD 83 the tensor derivative of T i

D 84 with respect to uȖ. The extension of this definition to more complicated tensors is obvious from the structure of Eqn. (7.32). Thus the tensor derivative of T i

DE 85 with respect to uȖ is given by

If the surface coordinates at any point P or S are geodesic, and the space coordinates are orthogonal Cartesian, we see that at that point the tensor derivatives reduce to the ordinary derivatives. This leads us to

. BA dduT

a -

ddxT jk

i g +

ddT =

dd

ii

kj

iD

J

GDD

WEJG

WWWI

»»¼

º

««¬

ª

¿¾½

¯®­

¿¾½

¯®­ˆ

(7.31)

. dduT

a =

ddxT jk

i g +

dTd =

tT i

kj

ii

WDJG

WWGG J

GDDD

¿¾½

¯®­

¿¾½

¯®­ˆ

WJD

GGWG J

GJDJDD

ddu T

a - xT jk

i g +

uT T ikj

ii

»»¼

º

««¬

ª

¿¾½

¯®­

¿¾½

¯®­

ww

{

T

a - xT jki

g + uT T ikj

ii

GJDJD

DJ DJG

¿¾½

¯®­

¿¾½

¯®­

ww

{ ˆ

(7.32)

. T

a - T

a - xT jki

g + u

T = T iikii

i, DGGEJDEJ

DEJDE EJ

GDJG

¿¾½

¯®­

¿¾½

¯®­

¿¾½

¯®­

ww

(7.33)

252

conclude that the operations of tensor differentiations of products and sums follow the usual rules and that the tensor derivatives of gij, AĮȕ, Gijk, İĮȕ and their associated tensors vanish. Accordingly, they behave as constants in the tensor differentiation. The apparatus developed in the preceding section permits us to obtain easily and in the most general form an important set of formulas due to Gauss. We will also deduce with its aid the second fundamental quadratic form of a surface already encountered. We begin by calculating the tensor derivative of the tensor xi

D 87, representing the components of the surface base vectors aĮ. We have

from which we deduce that

Since the tensor derivative of aĮȕ vanishes, we obtain, upon differentiating the relation

Interchanging Į, ȕ, Ȗ cyclically leads to two formulas:

and

If we add Eqn. (7.36) and Eqn. (7.37), subtract Eqn. (7.35), and take into account the symmetry relation Eqn. (7.34), we obtain

This is the orthogonality relation which states that xi

,ED 94 is a space vector normal to the surface, and hence it is directed along the unit normal ni. Consequently, there exists a set of functions bĮȕ such that

,x }

a{ - xx}jki

{ g + uu

x = x ikji2

iGEDEDDE DE

Gˆ

www

. x = x i,

i, EDED

(7.34)

. 0 = xx g + xx g

,xx g = Aj

,i

ijji

,ij

jiij

JEDEJD

EDDE

ˆ

ˆ

(7.35)

0 = xx g + xx g j,

iij

ji,ij DJEJDE ˆˆ

(7.36)

. 0 = xxg + xx g J,

iij

ji,ij EDJDEJ ˆˆ

(7.37)

. 0 = xx g ji,ij JED

253

The quantities bĮȕ are the components of a symmetric surface tensor, and the differential quadratic form

is the desired second fundamental form. Now since G i

ji

j = nn 97, and ng = n j

iji ˆ 98, then

but since

xx 21 = n kj

ijki EDDE HH 100, then

We now have, in Eqn.s (7.8) and (7.38), the formulas necessary to determine the first and second fundamental quadratic forms for our system constrained to a four-dimensional hypersurface. Our objective is to show that by appropriately constraining our system we arrive at the Navier-Stokes equations. Let us determine the first fundamental quadratic form. First recall that our system was restricted so that x4 = x4(xo, x1, x2, x3) or the mass density is a function of space and time; then we have the relations

and

Since Eqn. (7.8) is

. nb = x ii, DEED

du du b B EDDE{

; nx g = b ji,ij EDDE ˆ

. xxx 21 = b kji

,ijk GJEDJG

DE HH

(7.38)

,u = x

,u = x

,u = x

,u = x

33

22

11

00

. )u ,u ,u ,u( f = )x ,x ,x ,x( f = x 321032104

,xx g = ux

uxg = A ji

ij

ji

ij EDE

DDE ˆˆww

ww

254

then

where

In a similar fashion we may determine the remaining coefficients and find that

where

where the hĮȕ are functions of the partial derivatives of the mass density with respect to space and time in addition to space and time from the gi4ˆ 109 where i = 0, 1, 2, 3, 4.

Though we may use Eqn. (7.38) to determine the metric coefficients for the second fundamental quadratic form, it is not necessary for the current presentation. The hypersurface which is embedded in the five-dimensional space is a four-dimensional curvilinear space-time manifold. Thus the relativistic hydrodynamic equations are applicable here so long as the metric coefficients are determined as coefficients of the hypersurface quadratic form. The complete energy-momentum tensor for a fluid in a flat Riemannian space-time manifold is given by

where

ds

du uD

D {� 111, s is the arc length. Then based upon this energy momentum tensor the flow

of a fluid under the effect of its own internal pressure force is given by setting the divergence of Eqn. (7.41) equal to zero, or

,)f(g + f g2 = g = A 20440040000

uf f 00 ww

{

h + g = A DEDEDE ˆ

(7.39)

,3 2, 1, 0, = , ; ffg + fg2 = h 444 EDEDEDDE ˆˆ

(7.40)

)g - uu(cP + uu = T 2

DEEDEDDE J ����

(7.41)

255

If we reduce Eqn. (7.41) to the non-relativistic limit, the use of Eqn. (7.42) gives us

where IJĮȕ = PgĮȕ is the three-dimensional stress tensor of an ideal fluid. If in Eqn. (7.41) we use the fact that the metric coefficients for the hypersurface may be written as the sum of Eqn. (7.40), then we have

where it must be remembered that the uu �� ED 115 are also dependent upon this same sum. In the non-relativistic limit the effects of this sum of metric tensors appear as a sum in the stress tensor

Recall that the gDE 117 refer to the three-dimensional space viewed from the five-dimensional manifold. The hĮȕ, however, contain the information about the surface embedded in the five-dimensional space. If we then associate the tensor

with the viscous stresses, we are saying that the viscous stresses depend upon the geometric character of the hypersurface. In the limit of small displacements we write the strain velocity tensor as

Then the first order coefficients of viscosity are related to the strain velocity tensor and viscous stresses according to

. 0 = ,T EDE

(7.42)

,3 2, 1, = ,, , = g a, GEDJW DGDE

EG

� � ,h - g - UucP + uu = T 2

DEDEEDEDDE J ˆ����

(7.43)

. 3 2, 1, = , ,Ph - gP - = EDW DEDEDE ˆ

(7.44)

Ph - t DEDE {

(7.45)

� � . v + v 21 = e ,, DEEDDE�

256

If we then use Eqn. (7.45) in Eqn. (7.46), we find that the relationship be-tween the geometric character of the hypersurface and the viscous coefficients is given by

Equation (7.47) then expresses the functional dependence of the viscous coefficients upon the strain velocities, pressure, mass density, and their derivatives. 7.4. Relativistic Hydrodynamics. By viewing classical hydrodynamics to be given by the embedding of a four-dimensional hypersurface within a five-dimensional manifold, the association Eqn. (7.47) between the geometric properties of the hypersurface and the viscous coefficients could be tentatively made. We may now go back and develop this relationship more completely. The hypersurface, which becomes embedded in the five-dimensional manifold by the restriction that x4 = x4(x0, x1, x2, x3) is a four-dimensional relativistic manifold. Thus, for the surface we may use the relativistic energy-momentum tensor, which is

where

dxdx u 0

PP { 123 and µv = 0, 1, 2, 3. The divergence of Eqn. (7.48) yields the flow

equations for a fluid under the effects of its own internal pressure. However, from the viewpoint of the Dynamic Theory, the surface metric coefficients may be written in terms of the metric coefficients of the first four space coordinates as given by Eqn.s (7.40) and (7.41), or

� � . v + v 2

c = t n,n,

n

GG

DEGDE

(7.46)

� � . v + v 2

c = Ph - n,n,

n

GG

DEGDE

(7.47)

� � ,g - uu cP +u u = T 2

PQQPPQPQ J

(7.48)

. 3 2, 1, 0, = , ,g = A EDDEDE ˆ

257

Thus, the square of the arc length for the entropy manifold may be written as

or, if

dqdx u 0

DD { 126, then

Then on the hypersurface the energy momentum tensor would become

or

Since the surface coordinates, xĮ, are the same as the first four coordinates of the surrounding space, the velocities uĮ are the same whether considered as surface or space vectors. The difference between the surface view and a four-dimensional space view appears in the metric coefficients. Thus, while the square of the arc element on the surface is unity, the square of the arc element in the surrounding space is not, or

but

7.5. Classical Hydrodynamics. Suppose we consider the metric given by gDE 132 to be a flat space then because of Eqn. (7.49) we may write

If we then form the space divergence

� � dxdxh + dxdxg = dxdxA = dq0 2 EDDE

EDDE

EDDE ˆ

. Uuh + uug = uuA = 1 EDDE

EDDE

EDDE ˆ

� �A - uucP + uu = T 2

DEEDEDDE J

� � . h - g - uucP + uu = T 2

DEDE

EDEDDE J ˆ

(7.49)

uuA = 1 EDDE

. uuh - 1 = uug EDDE

EDDEˆ

� � . hcP - g - uu

cP + uu = T 22

DEDEEDEDDE J ˆ

258

this may be written as

where h0 has components h0Į, = 1, 2, 3. Therefore,

so that if hv is a four-vector with components h h0 QQ { 137, then

The remaining components of the divergence are given by

which may be rearranged to read

If we look at the non-relativistic limit, then, by neglecting the terms P(v/c), we get

The multiplicative factor 1/c2 on the right-hand side suggests that

which is the assumption we chose to place our system on a particular surface. This corresponds to a classical system where conservation of mass is assumed. Therefore, on the surface of a curve specified by

,0 = H - cu

cP +

cu

x + h

cP -

tc1 = ,T 0

200

20

¿¾½

¯®­

¸¹

ᬩ

§ww

¿¾½

¯®­

ww D

DD

DQ JJQ

,0 = )Ph( c1 - )v (P

c1 +

t)Ph(

c1 - )v ( +

t

c1 0-

22

00

2 x'xw

w»¼º

«¬ª x'ww

JJ

,)Ph( + t

Ph( c1 + )v (P

c1 = )v ( +

t 0

00

2 »¼

º«¬

ªx'

ww

x'x'ww

JJ

. ),Ph(c1 + ,vc

P - v Pc1- = )v ( +

t 2 QDJJ QDx'x'ww

,0 =

cPh - +

cvv

cP +

cvv

x +

hcP -

cPv +

cu

tc1 = ,T

2222

023

DEDE

EDED

E

DDD

DQ

GJ

JQ

¯®­

¸¹

ᬩ

§ww

¿¾½

¯®­

ww

. )v v (P +

t)Pv(

c1 -

x)Ph( +

t)Ph(

c1 + )v ( +

t v -

xP- = v v +

tv

2

0

»¼

º«¬

ªx'

ww

ww

ww

»¼º

«¬ª x'ww

ww

»¼

º«¬

ª'x

ww

DD

E

DEDD

DD

D

JJ

J

. )Ph( + t

)Ph(c1

c1 = )v ( +

t0-

00

2 »¼

º«¬

ªx'

ww

x'ww

JJ

,0 )v ( + t

#x'ww

JJ

259

we must then have

or

Thus, we may write

where

The term

t)Ph(

c1 0

ww D

148 has been neglected in Eqn. (7.50).

Thus we see that the geometric character of the hypersurface, contained in the term PhĮȕ, behaves as if it were a viscous effect to be added to the normal viscous effects. Recalling Eqn. (7.41), it may be seen that the viscous-like effects of the geometry of the hypersurface depend upon the density gradient. If these terms exist, they must be very small in everyday phenomena. Yet if we consider phenomenon which involve very large density gradients, these terms could become large enough to see. 7.6. Shock Waves. One field of physical phenomena that displays large density gradients is shock waves. Therefore let us take a quick look at the effect of these additional terms on the description of a shock front for a steady, one-dimensional shock. The total stress in a steady, one-dimensional shock would be given by

,0 = ,T QPQ

x

)Ph( + t

)Ph(c1 +

xP- = v v +

tv

)

E

DED

DD

D

Jw

ww

www

»¼

º«¬

ª'x

ww

. 3 2, 1, = ,),Ph( +

t)Ph(

c1 +

xP- =

x)Ph( +

t)Ph(

c1 +

xP- = a

0

0

EE

J

DED

D

E

DED

DD

ww

ww

ww

ww

ww

EWJ DED , = a

. )h - gP(- = Ph + gP- = DEDEDEDEDEW ˆˆ

(7.50)

,dxdu +

x

a1 - 1 P =

2

2O

¸¹·

¨©§

°¿

°¾½

°°®­

¸¹·

¨©§ww

¸¹

ᬩ

§K

JV

260

when g11 = 1 and h11 is evaluated using Eqn. (7.41). However, for a steady shock we also have the jump conditions

and

These equations represent the conservation of mass, momentum, and energy. By using the conservation of mass relation we may write the total stress as

where

may be called the effective viscous coefficient. Since within the shock front the velocity gradient du/dx is negative, we see that the effective viscous coefficient acts so as to thicken the shock front when compared to the classical viscous coefficient . Using the second jump condition, an expression for the velocity gradient is

which may be approximated by

The effect of the correction term on the velocity gradient is seen in Eqn. (7.53), because the multiplicative factor outside the brackets is the classical expression for the negative velocity gradient. The effect of the

,k = +u k

,k =u

21

1

VJ

. k = 2

- Ek 3

221

V

,dxdu + P = eff ¸

¹·

¨©§KV

¸¹·

¨©§{

dxdu

uakP - 42

0

21

eff KK

(7.51)

,u]k + k - [PuaPk4 + 1 - 1

Pk2ua =

dxdu

122420

21

21

420

°¿

°¾½

°

°®­

KK (7.52)

. u]k + k - [PuakP - 1u]k + k - [P1- =

dxdu

122420

21

12¿¾½

¯®­

¸¹

ᬩ

§

KK

(7.53)

261

correction term lessens the negative velocity gradient and extends the shock front. The effect of the correction term in Eqn. (7.51) is estimated by considering the strong shock dependence of pressure upon shock velocities. For instance, the shock pressure, from the jump conditions, is

If the shock velocity is related linearly to the particle velocity as the assumed solid equation of state, U = co + sup, then Eqn. (7.54) becomes

Thus, for strong shocks, P varies approximately as the square of the shock velocity. Consider Eqn. (7.52) or (7.53). From either of these equations, the velocity gradient varies as the square of the shock velcoity. Using these two conclusions in Eqn. (7.51) for the total viscosity Șeff and remembering that the integration contant k1 is given by -ȖoU, the effective viscosity varies approximately as the square of the shock velocity or, essently , as the pressure. The conclusion is that if the effective viscosity varies with the pressure, an increase by the same factor of 103 must be accompanied by a viscosity increase by the same factor of 103. This explains the apparent discrepancy between the low and high pressure aluminum viscous effects. For instance, the Asay-Bertholf limits are: P = 25 GPa Ș > 40 poise P = 36 GPa Ș < 2,500 poise . Another experiment places an upper limit of 103 poise for a shock pressure of 40 GPa. If 102 poise is considered representative of the viscosity when P � 10 GPa, then from Eqn. (7.51), a pressure of 103-104 GPa must be accompanied by a viscous effect of 104-105 poise. This total viscosity estimate is supported by numerical integration across the shock front using the Tillotson equation of state for aluminum. The classically predicted rise times for shocks of 40 GPa with Ș = 575 p and 5x103 GPa for Ș = 5x104p are duplicated by using the effective viscosity experssion in Eqn. (7.51) with Ș = 1.0 p and a0 � 365 g/cm4. Thus, the Dynamic Theory correlates these data points that appear contradictory by classical theory. Further, these data points provide an estimate of the new universal constant appearing in the Dynamic Theory.

. uU = P poJ

(7.54)

. )c - (UsU

= P o0J

262

This value of a0 provides an estimate of other predictions of the theory in

fields other than shock waves.

7.7. Mass Conservative Electrodynamics

One of the incentives for seeking to determine whether the five

equations of motion were generalizations of the classical hydrodynamic

equations was the possibility of shedding new light upon fusion plasmas.

Now before mass conversion is accomplished the plasma must reach

certain conditions. The attainment of these conditions involve

electromagnetic fields not encountered in usual circumstances on earth. If

the Dynamic Theory is to be believed, then perhaps it may provide new

insight into the attainment of the appropriate conditions before mass

conversion begins.

The following development still assumes conservation of mass in

order to see the geometry of the hypersurface for a system under the

influence of electromagnetic fields.

Suppose we now describe the behavior of charged matter under the

influence of an electromagnetic field from the viewpoint of the Dynamic

Theory. From this viewpoint the conservation of mass has the effect of

restricting our system to a four-dimensional hypersurface which is

embedded in the five-dimensional manifold of space, time, and mass

density.

Since we desire to consider the effects of an electromagnetic field we

must consider a gauge function. When a gauge function exists, the square

of the arc length in the entropy space is related to the square of the arc

length in the sigma space by

When the system is restricted to a hypersurface by the relation x4 = x4(x0,

x1, x2, x3), then the entropy surface may be written as

where

Likewise for the sigma surface

� � � � . d h1 = dx dx g

h1 = dx dx g = dq 2

00

jiij

00

jiij

0 2V¸

¹

ᬩ

§¸¹

ᬩ

§ ˆˆ

� � ,du du a = dq0 2 EDDEˆ

. x x g = ux

uxg = a ji

iji

i

ij EDEDDE ww

wwˆˆ

du du a = )(d 2 EDDEV ˆ

263

where

Thus, we have

The principle of increasing entropy requires that the equations of motion be geodesics in the entropy space but they will appear as equations involving forces in the sigma space. We desire to expose these forces and therefore should work in the sigma space. Our objective then is to determine the effect of embedding a four-dimensional surface given by x4 = x4(x0, x1, x2, x3) in the sigma space and thus obtain a sigma surface describing a system subjected to the classical conservation of mass restric-tion. Having previously determined the metric coefficients for the entropy space by Eqns. (7.39) and (7.40) we may write the coefficients for the sigma surface as

However by considering the effects of the electromagnetic field as a force we must first consider the space field tensor:

If we restrict ourselves to the classical field quantities E and B and for the moment assume that the field quantities V4 and V are zero or negligible, then we obtain only the effects of the hypersurface viewpoint. This assumption seems reasonable considering the interpretation of the new field quantities are gravitational effects. Under this assumption our field tensor becomes

. x x g = a jiij EDDE ˆˆ

. a h1 = a00

ˆˆ DEDE ¸¹

ᬩ

§

> @ . h + g h = a h = a 0000 DEDEDEDE ˆˆˆ

0 V- V- V- V-V 0 B- B E-V B 0 B- E-V B- B 0 E-

V E E E 0

= F

3210

3123

2132

1231

0321

ij

0 0 0 0 00 0 B- B E-0 B0 B- E-0 B- B 0 E-

0 E E E 0

= F123

132

231

321

ij

264

We can now use this space field tensor to determine the appearance of the fields when viewed from the surface. The surface field tensor will be given by

But since xĮi = įĮi for i, = 0, 1, 2, 3 and xĮ4 = fĮ, the surface field tensor of a purely electromagnetic space field tensor is only the four-dimensional portion of the space field tensor since Fi4 = 0 for i = 0, 1, 2, 3, 4. Thus when we use the relativistic energy-momentum tensor for the surface, we have

which is the relativistic energy-momentum tensor for matter under the influence of electromagnetic fields. But since h + g = a DEDEDE ˆˆ 169, then Eqn. (7.55) becomes

or

where

is the four-dimensional space relativistic energy momentum tensor and

is the portion of the energy-momentum tensor which contains the geometrical properties of the hypersurface. From Eqn. (7.56) we can say that the Dynamic Theory has the appearance of adding a term to the relativistic energy-momentum tensor. This term contains the geometrical character of the surface and represents the difference between the appearance of the energy-momentum tensor when viewed from the surrounding space as compared to the view from the hypersurface.

. x x F = F jiij EDDE

,FFa41 + F F

c1 + u u = T 2 »¼

º«¬ª

DEDEPQDQP

DQPPQ J ˆ

(7.55)

� � »¼º

«¬ª

F F h + g 41 + F F

c1 + uu = T 2 DE

DEPQPQDQPQPPQ DJ ˆ

T + T = T georelPQPQPQ

(7.56)

»¼º

«¬ª{ FF g

41 + F F

c1 + uu T 2rel DE

DEPQDQPQPPQ DJ ˆ

F F h c41 T 2geo DE

DEPQPQ ¸¹

ᬩ

§{

265

If we take the divergence of the energy-momentum tensor Eqn. (7.56), we have

The additional force terms from the surface geometry are given by

But if we define

as the electromagnetic energy density, where

then the geometric energy-momentum tensor becomes

and the additional forces are given by

We may also look at the radiation pressure predicted by the Dynamic Theory to see how the surface restriction affects the relativistic prediction of radiation pressure. The relativistic radiation pressure is taken as one third of the three-dimensional Maxwell stress tensor which is the space portion of the energy-momentum tensor, or

where Į, ȕ = 1, 2, 3. To get the equivalent stress tensor for the Dynamic radiation pressure we must add the space portion of Eqn. (7.57) so that the total stress tensor becomes

. ,T + ,T = ,T georel QPQ

QPQ

QPQ

� � . F = ,F F h c41

2P

DEDEPQ Q¸

¹

ᬩ

§

W[DEDE 16- FF {

(7.57)

� �B + E 81 = 22

S[

ch 4- = T 2geo

[S PQPQ

� �Q[S PQP ,h

c4- = F 2

� � [GS DEEDEDDE - BB + EE

41 = T M

� �

� � � � . h + - BB + EE 41 =

h 4 - - BB + EE 41 = T

DEDEEDED

DEDEEDEDDE

G[S

[[GS

266

We can then obtain the negative of the trace by

The radiation pressure is then given by

The first term in Eqn. (7.58) is the classical radiation pressure in electrodynamics. The remaining three terms give the difference between the pressure predicted by the Dynamic Theory and the classical prediction. To determine what this difference is let us restrict our system to again be very near equilibrium so that the gĮ4 = 0 for Į = 0, 1, 2, 3 and g44 = constant. Thus we have a flat space. For this space the

from Eqn. (7.58) and g44 = -1. Thus

By substituting Eqn. (7.59) into Eqn. (7.58) the pressure becomes

However, since the classical pressure is given by

3

= Pc[ 187, then the pressure predicted by the Dynamic Theory becomes

We see then that the Dynamic Theory predicts a decrease in the radiation pressure as a result of viewing the system to be restricted to a four-dimensional hypersurface embedded in a five-dimensional space. The amount of this decrease in pressure depends upon the gradient of the

� � � �

� �> @ . h + h + h - - - =

h + h + h - 3 - B + E 41 - = {T}-

332211

33221122

[[

[[S »¼

º«¬ª

.] h + h + h + [13

= P 332211[

(7.58)

¸¹

ᬩ

§ww

x

ag

= h2

20

44DDDJˆ

. x

+ x

+ x

a1 - = h + h + h 3

2

3

2

1

2

20

332211»»¼

º

««¬

ª¸¹

ᬩ

§ww

¸¹

ᬩ

§ww

¸¹

ᬩ

§ww JJJ

(7.59)

. x

+ x

+ x

a1 - 1

3 = P 3

2

21

2

20 °¿

°¾½

°

°®­

»»¼

º

««¬

ª¸¹

ᬩ

§ww

¸¹

ᬩ

§ww

¸¹

ᬩ

§ww JJJ[

. x

+ x

+ x

a1 - 1 P = P 3

2

2

2

1

2

20

cD°¿

°¾½

°

°®­

»»¼

º

««¬

ª¸¹

ᬩ

§ww

¸¹

ᬩ

§ww

¸¹

ᬩ

§ww

¸¹

ᬩ

§ JJJ

267

mass density and the constant a0. Once the constant a0 is determined, then the deviation in predicted pressures can be specified. This prediction should appear in attempts to use electrodynamic forces to control ionized plasmas and perhaps there are large enough density gradients for these predictions to show up in cosmological events. References: *A. J. McConnell, Absolute Differential Calculus, London, 1931, Chapters IV - XVI.

268

Chapter 8 Experimental Tests Every new theory should possess some feature that can be checked experimentally, for the objective in the creation of a new physical theory should be a better understanding of physical phenomena. The following suggested experiments represent but a few possible tests of the five-dimensionality of the Dynamic Theory, for each depends upon the five-dimensional fields. 8.1 Speed-of-Light Measurements The various speeds associated with the five-dimensional plane wave have been studied in detail.17 Here, for simplicity, we will limit our discussion to phase velocity, defined as that velocity at which the wave phase remains constant. The trial solution used in the wave equation was

)];(exp[ 4JZ Kkxti ��� (8.1) therefore, the wave phase

.4JZI kkxt �� (8.2) If we set

,0 dtdZ

we find

kdtkd

kv p

)( 4JZ�

Substituting for k4 from Chapter 6, the phase velocity becomes

,20

3 JJ

Z�

kcaC

kvp � (8.3)

where it is assumed that k and k4 are independent of time. Now if mass is conserved, then

,, ju jJJ � � (8.4) where the uj are the components of the medium flow where vp is defined. By substituting Eqn. (8.4) into Eqn. (8.5), the phase velocity becomes

.,0

3 jukca

Ck

v jp ¸

¹

ᬩ

§�¸

¹·

¨©§

JZ (8.5)

In classical electromagnetism any uniform motion of the medium is not reflected in the speed-of-light measurements. The same thing is true of the phase velocity given by Eqn. (8.5). On the other hand, a divergence in the flow of the medium will affect the phase velocity. The velocity change, owing to a divergence in the flow, is inversely proportional to the density of the medium. Because of the change also is proportional to parameter C3, which has not yet been completely determined by the wave solution, it cannot be seen yet whether any envisioned experiment could measure the predicted change in velocity. To do so would require completing the wave solution to determine C3.

One suggested experiment might be measuring the phase velocity in the divergent flow coming out of a nozzle in a hypervelocity wind tunnel. Although such an

269

experiment might not be sensitive enough to detect the predicted velocity change particularly, µ and would change because of the change in mass density.

Another possibility, which was suggested by Bobby G. Craig, is to measure the travel time of a strong beam of gamma rays through a divergent flow of gas created by explosives. This may create the largest divergent flow possible, but whether or not other experimental difficulties could be surmounted to make reliable measurements is unknown. 8.2 Index of Refraction The change in the parameters µ and was mentioned in the speed-of-light experiment discussion. From the plane wave solutions, we found that

JHH H )( 310

0 CCcai

�

and

»¼

º«¬

ª��

)(1 032

20

0

HHZP

PP

CC

Classically, the index of refraction for dielectrics is given by (µ)1/2. However, given the wave solution, we must consider the boundary conditions as the wave passes through a boundary between two media, determine the energy transmission and reflection coefficients, and then find the index of refraction from a modification of Snell's law. That is, the index of refraction should indicate the angle of the refracted wave with respect to the incident wave.

A cursory look at a five-dimensional wave incidence upon a boundary produces the relation

.sin

)(sinsin

20

422400

0

2

2

0

TJJ

TT

kkk

kk �

� (8.6)

But from Chapter 6 we find

,)(121

212

3 ¸¹·

¨©§

»¼

º«¬

ª�¸

¹

ᬩ

§�

cCC

CC

k ZPH

Z

if y=0. Also,

.0

344 caCCk J

J �

Then we have

,sin

)]()([(sinsin

200

23203042400

0

2

2

0

TZEJJ

EE

TT

aCCCCca ���

� (8.7)

where

.)(1 312

3 CCCC

�� Z

PHE

Then, if the frequency is high enough,

270

200

23203042400

0

2

sin

)]()([

TZEJJ

EE

aCCCCca ���

!!

We would be tempted to define as the index of refraction. On the other hand, the classical

notion of the index of refraction involves the ratio of the sin of the incident and refracted

waves. In Eqn. (8.7), the appearance of sin 2 in the right-hand side makes matters more

difficult. However, C45 is a phase angel we may set at zero, and we may choose the

reference medium, 0, to be free space for which 0 = 0; then Eqn. (8.7) can be written as

.

][sin

1

sin

sin

0

420232

20

2

2

0

E

JTZ

E

TT

cCaCa

�»¼

º«¬

ª�

Then we may define

,sin

)(

0

403

TZJ

EKa

cCaC �� (8.8)

so that Eqn. (8.7) becomes

,sin

sin

0

2

2

0

KK

TT

so long as the reference medium is free space. If we call the index of refraction, we find

that

,sin

)()(1

0

403

31

2

3

TZJ

ZPHK

acCaCCC

CC �

��¸¹

ᬩ

§�

depends upon both the frequency and the mass density.

A possible experimental test may be obtained by applying rigorous boundary

conditions to the five-dimensional wave incident upon a boundary. This would verify or

correct the modified Snell's law given by Eqn. (8.7). The frequency dependence of the

refracted wave angle that was determined experimentally may be compared with the

predicted angle. Another comparison may be done by considering the density dependence

of the refracted wave.

8.3 Neutron Interferometer

A neutron interferometer can detect extremely small differences in forces upon each of

two neutron beans by using interference techniques. It can detect the difference in the

earth's gravitational field that is due to a height change of only 2 cm near sea level with

some thirty fringe shifts.

If the long-range character of the V field, as seen from the radial dependence

required for a fundamental particle,

¸¹

ᬩ

§�

¸¹·

¨©§ �� r

r err

WgVO

O1

2

requires that the V field is to be interpreted as the gravitational field, then the force law,

,4 V

cJEK � U

271

would require that J4/c be interpreted as the gravitational mass density. This would require

³ v

g dVcJm 4

to be the gravitational mass of a particle contained within the volume V. The gravitational force on a particle in a gravitational, or V, field would be given by .VmF g This implies that the transverse V field accompanying the and B component in the electromagnetic wave would apply a force on a neutron through an interaction with its gravitational mass. Therefore, a beam of neutrons passing through a polarized layer beam should be slightly deflected owing to the gravitational field component. This effect would be most easily detected if, through the use of some appropriate mirror, a standing optical wave could be created using a polarized laser beam. Then a neutron beam passing through an appropriate part of the gravitational component of this standing wave would have all the neutrons deflected in the same direction.

The sensitivity of the neutron interferometer may be such that, if one neutron beam passes through a standing optical wave created by a laser of appropriate frequency, very minuscule deflections could be detected. The appropriate laser frequency should be chosen to maximize the predicted deflection. This, of course, requires that the wave solution be completed so that the relative strength of the transverse gravitational component is known; that is, because

,3yy ECV ¸

¹·

¨©§ � Z

we must know C3 before we can choose the best laser frequency and power. If a deflection is detected and has the predicted dependence upon laser frequency

and power, then the electromagnetic wave must be accompanied by a gravitational component. 8.4 Nuclear Mass We infer from the neo-coulombic electrostatic force that the nucleus may be made up of complex orbits of electrons and protons, plus possibly positrons, as discussed in Chapter 4. The transcendental nature of the forces involved requires the use of a computer in solving the equations of motion. Computer solutions may be obtained and the masses predicted; then these predicted masses may be compared with the existing experimental masses. A good comparison between the predicted and experimental masses, accounting for possible errors introduced by any assumptions made to obtain a solution, would increase the theory believability. 8.5 Gravitational Rotor The continuity equation in the Dynamic Theory is

J

Uw

����ww

400 JaJ

t

272

If we consider only steady state conditions such that

0 ww

tU

then we have

.40 JJa ��� wwJ

(8.9)

Equation (8.9) states that if one can create a non-zero divergence in the current density then one creates a particular variation between the gravitational mass density and the inertial mass density. This is in violation of both the classical conservation of charge and Einstein's assumed equivalence principle.

Suppose we consider what happens when we pass a current into the apex of a cone, as shown in Figure (16). FIGURE 1: Current into the apex of a cone Any position on the exterior of the cone is given by fxdy � where the height of the cone is

,fdh

therefore f = d/h. If a steady current, I, is flowing into the apex of the cone, then at any x the current

density is given by

.ˆˆcos1 yx

areaJ TT �¸

¹·

¨©§

But the area is given by ,)2( tyArea S while the

22

coshd

h�

T

thus our current density vector becomes

.ˆˆ2 22

yoxhd

hyt

IJ �»¼

º«¬

ª

�¸¹

ᬩ

§

S (8.10)

Now we may form the divergence of this current density, noting that the above assumes that the current density is a constant throughout the thickness t.

273

2)(1

2cos

)(1

2cos

)/([2cos

xhdtIh

xhxtdhI

xhddtI

x

xJx

yJy

xJxJ

�¸¹·

¨©§ �

»¼

º«¬

ª�w

w¸¹·

¨©§

¿¾½

¯®­

�ww

ww

ww

�ww

��

ST

ST

ST

(8.11)

Substituting the result, Eqn. (8.11), into equation (8.9), we have

24

0 )(2cos

xhdtIhJa

�

ww

ST

J (8.12)

This is a differential equation whose solution is

020

4 )(2cos J

xhdtaIhJ �

�

STJ (8.13)

Thus, the gravitational charge density is given by

,)(2

cos 402

0

4

cJ

xhcdtaIh

cJ

��

S

TJ (8.14)

Suppose we now consider two cones joined at their bases as shown in Figure (17).

FIGURE The element of torque about the point A experienced due to the presence of the earth's gravitational field V is found from the relation, torque = force x distance. Therefore,

)(2

)()(2

cos2 402

0

tyVxcJVxdxty

xhcdtaIhd SS

STJ

W �»¼

º«¬

ª�

� (8.15)

The effect of the constant of integration term with J40 is to predict a constant torque without current flow. Since this should have been noticed we shall take J40 = 0. Thus

T

JW

TJ

W

cos)(

cos)(

)/1(

0

20

dxxhca

vxId

xdxxhcda

VhxdIhd

»¼

º«¬

ª�

�

»¼

º«¬

ª���

(8.16)

We obtain the total torque by integrating from x given by

hdxdD �

or )1/(0 � dDhx to .0 x Thus, we have

274

³ ³� �»

¼

º«¬

ª�

0

)1/( 0

.)(dDhx xh

xdxcavIdtorque J

W

Then the torque is determined by

^ `

TJ

TJ

TJ

cos]/ln[)(

cos)]1/(ln[)()ln(

cos|)ln(

0

0

0)(

0

¿¾½

¯®­ ��¸¹

ᬩ

§ �

¿¾½

¯®­ ������¸¹

ᬩ

§ �

��¸¹

ᬩ

§ �

�

dDhDddh

cavI

dDhhhdDdhhh

cavI

xhhxcavItorque x

dDhx

or

TJ

cos)]/ln()/1[(0

dDdDcavhItorque ��¸

¹

ᬩ

§ � (8.17)

Suppose we pick some parameters; such as I = 10 amps, h = d = 0.1 m, D = 0.01 m. With these parameters

.sec103059.3

)1.0()1.0(sec)/(103

1.0

)01.0(ln

1.

01.01)1.0(

)]/ln()/1[(cos

110

228

2

��u�

�u

»¼º

«¬ª ��

��¸¹·

¨©§

m

mdDdD

ch T

Thus, we have

.)sec103059.3( 19

0

ampsa

vtorque ��u¸

¸¹

·¨¨©

§�

J

We now need to choose a material to obtain the mass density, determine the gravitational field V, and obtain a value for a0. Let's do them in the reverse order. Shock wave physics investigations produced an extremely rough estimate of a0 which was

./104~ 470 mkga u

The earth’s gravitational field strength, at sea level, is given by

./100336.4

)/(104296.2

8.9

18.9

11

11

mvoltkgcoul

kgmcoulvolt

kgmcoulvoltv

u�

u

¸¹

ᬩ

§��

�

¸¹

ᬩ

§��

� E

If we choose aluminum as our material, then = 2.7 x 103 kg/m3 and our torque becomes

275

¸¹

ᬩ

§

�

��u�

�u�

uu

u�u

�

�

mnt

lbftmnt

coulvolt

coulmkg

mvoltmkgtorque

.73757.0100.9

100.9

)103059.3()/104(

)/100336.4)(/107.2(

2

2

9

47

1133

so that

..10638.6 2 lbfttorque �u �

This is not a very large torque, however, different cone parameters could be chosen to

optimize the torque.

There is another aspect which I don't yet know how to approach. In electric

motors there is a phenomena known as armature reaction which tends to limit armature

current far more than the armature resistance does. I suspect there is a

Figure 2. Upright cone, powered from within.

somewhat analogous reaction here that may further reduce the torque but it would take

time to investigate this possibility.

One final point on the creation of a gravitational field. A long held desire of

mankind is to be able to exert some control over the grip the earth's gravitational field has

over him. A slight variation of the above torque device might allow the generation of this

control.

Suppose we look at a single cone set upright as shown in Figure (18). From the

equations generated before for the torque we see that the lift force generated by this

simple device would be given by integrating the element of force

dxxhca

VIdF)(

cos

0 ��

TJ

or

)/(cos

]}1/1{ln[cos

|){ln(cos

0

0

0

)1/(

0

Ddnca

vI

dDca

vI

xhca

vIForce dDhx

l¸¹

ᬩ

§ �

���

�¸¹

ᬩ

§ � �

TJ

TJ

TJ

Thus, since V = 4.0336 x 1011 volt/m, the force becomes

)/100336.4()/ln( 11

22

0

mvolthdca

hDdIForce u�

J

Obviously, other physical shapes may achieve similar results; perhaps with an even

greater levitation force than the simple cone.

276

8.6 Nuclear Lamb Shifts The atomic Lamb shift experiments were some of the best experiments in science because of the comparison between predictions and experimental data. We now should have the capability of doing gamma ray spectroscopy. Then preditions of the nuclear energy levels as given by the nuclear model and potential given in Chapter 4 may be checked experimentally.

277

Chapter 9 Epilogue There are brief summaries at the end of Chapters 1 and 2, which give some measure of summation of the Dynamic Theory. Here I wish to provide a little further discussion in three areas. First, what is really new in the Dynamic Theory? Secondly, how might it help us teach science and physics? The third topic is where might the theory lead? This does not mean that there are not other new things presented in the preceding chapters. For instance, it is not new to state that the Unit of Action appearing in the derivation of Heisenberg's Uncertainty Principle depends upon the geometry. Anyone who has gone through the derivation considering covariant differentiation has ended up with this conclusion. Indeed, this is necessary in order to get the correct predictions of atomic states. However, when one considered Einstein's vector curvature and the atom or nucleus it was easy to argue that the vector curvature was so small that on the order of the nucleus or the atom the curvature could not influence the unit of action to any meaningful extent. This statement is perfectly true and easily supportable so long as your description of the phenomenon does not involve a gauge function. What is new in the Dynamic Theory is the appearance of a full gauge function. Now we can no longer assume that geometry can be assumed away on the scale of the nucleus. Indeed we have shown that all of the observation laws and experimental data are satisfied if the neutron is a proton in nuclear orbit around and electron. What is really new? 9.1 Only three basic assumptions. The Dynamic Theory is based upon only three fundamental assumptions as stated in Chapter 2. When I tried to count the needed fundamental assumptions in all of our current branches of physics I came up with something more than twenty. I am aware that one can generate considerable discussion about whether or not specific one of the twenty plus assumptions were really fundamental or not. However, the criteria I used was whether or not I knew of a method by which it could be derived from another assumption. If the assumption could not be derived, then it must be fundamental.

Though I've been told many times that you can't do it, I see very little logic restraining one from deriving Quantum Mechanics from a continuum theory. For example, if one jiggles a guitar string that is tied down on only one end, there is a continuum of solutions possible. On the other hand, if both the ends of the string are tied down, and this represents an additional restriction, then only certain, and quantified solutions are possible. Why not the same thing in the more broader sense of physics? Table IV shows the necessary restrictive assumptions that must be made in order to start with the

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Table I. Restrictive Assumptions to Reach the Branches of Physics. Branch of Physics Restrictive Assumptions Classical Thermodynamics i. only a pdv work term Special Theory of Relativity Group A assumptions

i. isolated system, dE=0 ii. only 3 spatial work terms iii. near equilibrium

(Uij=constant) iv. variation of paths

Newtonian Mechanics Group A assumptions, plus (4-dimensional) v. only three spatial work terms Electromagneto-Gravitic Fields Group C assumptions

i. isolated system, dE=0 ii. gauge field equations

Maxwellian EM Fields Group C assumptions, plus iii. only three spatial work

terms Quantized Gauge Potentials [e(-O/r)/r] Group C assumptions, plus

iv. Ij independent of path v. isentropic states

Strong Nuclear Force i. like particle forces, O1=O2

Weak Nuclear Force i. unlike particle forces, O1zO2

Atom Physics (Classical) i. 4-D Quantum Mechanics ii. r >> Omax

Perihelion Advance Group A assumptions, plus vi. quantized gauge potentials

Redshifts i. quantized gauge potentials ii. geometrical unit of action

fundamental assumptions of the Dynamic Theory to the foundations of various branches of physics and theories.

One example of the ability of the Dynamic Theory to derive the fundamental assumptions of the various branches of physics which is not specifically pointed out in Table IV is that of deriving Einstein's postulate concerning the limiting aspect of the speed of light. We saw, in Chapter 2, that this is a direct result of the Second Law and has, therefore, the same place in the theory as the limiting temperature in thermodynamics.

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9.2 Geometry is specified. Scientists have searched for Lagrangians in their attempts to find a way of unifying the fields of nature. One reason for doing so is because given a variational principle, such as the Principle of Least Action, they can arrive at equations of motion and then field equations. One problem with this approach is that they must make an assumption with respect to the geometry of their space when they employ the Principle of Least Action. This was the same necessity that Newton faced. Newton had no choice of geometries at his disposal. There was only Euclidean geometry to be had. Einstein chose the Riemannian geometry for use in his relativistic theories but rejected the more general Weyl geometry when it was proposed. The Dynamic Theory leaves us no choice. The fundamental laws specify what the geometry must be. To me this is extremely satisfying from the sense that I feel that properly chosen laws should do just that; they should leave no choice as to geometry.

A good many physicists know that one may choose a Lagrangian and use the Principle of Least Action to arrive at equations of motion. Few, if any, know that by doing so they must assume a type of geometry in this process and, thereby, have interjected a restriction into the process. Therefore, the more power to a process that does not allow this unwitting interjection of a restriction.

Another extremely important point with regards to the geometry is that within the Dynamic Theory there are two geometries to be considered. This is as fundamental within the Dynamic Theory as the fact that the concepts of heat and entropy are two different things is fundamental in thermodynamics. The temperature is the integrating factor in thermodynamics and, as such, plays a pivotal role between the heat and the entropy. Within the Dynamic Theory the gauge function was shown to be a geometrical integrating factor between the two geometries. 9.3 The Arrow of Time The notion of irreversibility is embodied in the laws of thermodynamics, but is not in the Newton and Einstein laws of motion. Yet mankind has been constantly aware of the relentless march of history that Omar Khayyám expressed in his:

The moving finger writes and having writ, moves on; nor all your Piety nor Wit, Shall lure it back to cancel half a Line, Nor all your tears wash out a Word of it.

Many articles and books have been written on the subject of time symmetry, but still questions remain. I first wrote on the Arrow of Time in 1981. The recent book, "The Arrow of Time" by Peter Coveney and Roger Highfield is an excellent one. It sets forth the problem of time symmetry in the mechanical theories in a very clear fashion.

The Dynamic Theory adopts generalizations of the classical laws of thermodynamics as the basis for a new view of all physical phenomena. It was shown in Chapter 2 how the adoption of these laws led to an integrating factor for purely mechanical systems and that this integrating factor was strictly a function of velocity. In Chapter 3 we saw the scope of the entropy increase further, though it still retained the connection to the energy exchange by the integrating factor. Further, we saw that for the isolated systems the Second Law produced an Entropy Principle that the entropy could never decrease, or dS0. From this we derived equations of motion using the

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expression for the square of the differential of the entropy. The square masks the fact from the Second Law we obtained dS0. Further, in Chapter 2 we discussed the fact that we could have obtained our equations of motion as third order equations in time, but chose not to in order to have second order equations.

All of this means that the Arrow of Time is part of the Dynamic Theory from the point of adoption of the Second Law. 9.4 Mass as a coordinate Hermann Weyl titled his 1918 book "Space-Time-Matter" which somewhat implies that matter is considered on the same footing as space and time. This, of course, is not supported by the contents of the book. In the book he treats space and time as coordinates while he leaves matter in its usual place in mechanics as either being the inertial or gravitating mass. Further, the fact that the Dynamic Theory goes into five-dimensions presents no new factor on that basis alone. Many other researchers have looked into five dimensions in order to try to obtain the necessary degrees of freedom with which to build into their theory the various fields thought to be needed to describe the universe. What is different about the Dynamic Theory is that it treats mass on an equal footing as space and time. This means that it is treated as a coordinate the same as space and time. No other researcher, looking into five-dimensional systems, allowed any physical significance to the fifth dimension. That is to say that they wished to have the added freedom of the five dimensions but did not wish to allow the fifth dimension to play a physical role as space and time were allowed to do. 9.5 Non-singular gauge potential There are two aspects of the non-singular potential which makes it's appearance something that is really different. The first is the fact that the maximum absolute value of the potential is different for different particles. This is the extraordinary feature, which leads to a description of phenomena usually reserved for the "Weak Force." The second new aspect of the non-singular potential is that when applied to the planetary orbits the potential produces the correct variation of perihelion advance as a function of orbit size by itself. Numerous gravitational potentials were guessed and tried in attempts to obtain an alternative to Einstein's General Theory of Relativity. One of the most severe tests for these candidate potentials is the variation of the predicted advance as a function of the orbit size. None of them could pass this test. The prediction of the perihelion advance within the Dynamic Theory depends upon the orbital parameters in the same fashion as the General Theory of Relativity.

A further utility of having a non-singular potential is that there is no need to renormalize any functions including the gauge potential or any of its derivatives. In Quantum Mechanics renormalization has always created problems and/or discussions. With nothing to renormalize the problems and the need for discussions go away.

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9.6 Unification of the Branches of Physics Numerous researchers have worked on the problem and an untold amount of time has gone into the attempts to find a unified field theory. Still a unified field theory is but a promise of the future. The promises still hold out hope of a theory in the "near" future. These promises remind one of the carrot on the pole. The more the poor horse tries to reach the carrot the more the pole moves the carrot ahead. Nowhere in my reading of books written on physics and unified field theories do I recall reading anything about an attempt to unify the various branches of physics except in my own hand or word processor.

The unification of the various branches of physics is the result of the generalizations of the laws of classical thermodynamics and seeking how to obtain equations of motion from them. Once the Entropy Principle was seen to provide a variational principle from which equations of motion could be obtained the method of unifying the branches of physics became visible.

While it may seem rather anticlimactic to be so brief about a point which has as much significance as this point, there seems to be little that needs to be added to the seven preceding chapters. 9.7 The pedagogical aspect of the Dynamic Theory The precept that all current branches of physics (i.e. classical thermodynamics, Newtonian mechanics, Special Relativistic mechanics, and Quantum mechanics) plus all the forces in nature (i.e. electromagnetic, gravitational, weak, and strong nuclear) stem from a single, simple set of fundamental laws may now be used to teach each of these branches and forces by the application of a different set of restrictive assumptions. The logic and rigor underlying this ability comes from the Dynamic Theory, which shows that three fundamental laws may be used with restrictive assumptions to derive the fundamentals of the various branches and forces. This not only displays the interrelationships of the different branches and forces, but also sets up the excellent teaching situation where different restrictive assumptions are the only difference between the very dissimilar branches.

Students may be excused for some level of confusion during their advancement through a school system wherein they are confronted with additional fundamental assumptions as they encounter new branches of physics. This confusion may be somewhat enhanced by the necessity of learning increasing skills in mathematics. However, it is the sheer number of fundamental assumptions currently perceived necessary for providing the basis for the existing branches of physics that is the source of the confusion.

The notion that the different forces in nature might somehow be tied together is the impetus behind the unified field theory hunt. Indeed this notion was behind the first formal theory, which attempted to unify the electromagnetic and gravitational fields as far back as 1836. Since then innumerable scientists have conducted investigations into the unification of the forces, or fields, in nature. However, no theory has yet been suggested that has gained undeniable experimental verification. Theoretical physicists

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are still at work trying to find a theory that will ultimately unify the forces of nature. Such

is the belief in the unity of nature.

On the other hand the unification of the branches of physics has not enjoyed the

same level of attention. Indeed, relatively speaking, there is very little discussion in the

literature of this concept. This concept was the motivation for the development of the

Dynamic Theory. The unification of the forces comes as an additional feature.

To based a study of science and physics upon the three fundamental laws and then

lead to the various branches of physics by restricting our attention by specific

assumptions would seem to be a very logical way to learn about our universe.

9.8 Where to from here?

This might be one of my favorite topics. I seldom get the chance to do much work in this

area anymore and certainly have few with which to discuss the topic. One of the things

that I disliked about the course of instruction that I received was that the prevailing

attitude from virtually all of my professors was that "We now knew where advancement

could be made." If this were true then where was there any room for new work? Why

should I study a subject for which there was nothing new to be learned? This was a

terrible turn-off.

However, I didn't believe them then and don't believe them now.

From the Dynamic Theory's point of view there is a great deal of things to be

learned! For example, almost everything in the preceding chapters refers to systems

which are isolated and for which the Entropy Principle was employed. What do the non-

isolated equations of motion look like? Wouldn't they describe a particle's transition from

one stable state to another?

We know that when one has a non-isolated thermodynamic system and are

pumping heat energy in or out of the system we must then minimize the free energy to

determine what happens. The same logic would apply here. If we wish to seek solutions

for non-isolated systems we need to minimize he free energy to obtain the non-isolated

equations of motion. This should give us the ability to better describe what happens in

these systems.

Notice, though, what this implies. We need to be vary careful when we are

considering mechanical systems to classify them as isolated or non-isolated. This is

something we never had to worry about before. It is also a way of seeing that there may

be a lot more to be learned if we do things differently.

The new things to be learned are not limited to non-isolated systems. Consider the

simpler question, can something go faster than the speed of light? From the relativistic

point of view one must answer that something going slower than the speed of light now

must forever remain slower than the speed of light. Similarly, things faster must remain

faster. But is the same conclusion true in the Dynamic Theory? The answer is no. From

the five-dimensional point of view the limiting aspect comes from

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0

2

44

2

2

0

cag

cv

dtdq J�

��

when time rate of change in entropy goes to zero. Should d/dt0 and g44<0 then v may be

allowed to be greater than c. What does this mean?

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In nuclear weapons and reactors, mass is converted into energy. However, Einstein's theory, which predicts the energy released in this conversion, says nothing about the rate at which this conversion can or does proceed. On the other hand, the Dynamic Theory not only provides an additional equation of motion that can be solved to find the mass conversion rate as a function of time, but it predicts a limiting rate of mass conversion. The limiting mass conversion rate comes from Eqn. (9.1) when v=0 and g44=1, then .0max ca J� Where does this lead us?