## A Unified Field Theory by N.F.J. Matthews

Dr. N.F.J. "Sy" Matthews

In April 1999, Dr. Matthews published the results of his 31-year research project. The results include a unified field theory. The field equations of Dr. Matthews' theory ("Electromagnetic - Kinematic - Gravitational (

*EKG*) field equations") are a super-set of Einstein's gravitational equations, Maxwell's electromagnetic equations, and a set of kinematic field equations.

From the Preface:

**"One day as a college sophomore, I asked my physics professor what an electron looked like. With a knowing, good-natured smile he responded that no one knew the structure of an electron and, in his opinion, no one ever would. His answer was a great surprise to me, for I thought physicists should know such things."**

From the last chapter:

**"The electron can be viewed as a structure containing four clouds of moving particles. Particles of one cloud carry charge while particles of the other three clouds carry angular momentum. No particle has mass or linear momentum. All four clouds are related to one another because all of their particle current densities are linear combinations of derivatives of ... and .... The first two clouds are further related to one another because each of their particle volume densities is constructed from a linear combination of derivatives of ... and .... The last two clouds are further related to one another because each of their particle volume densities is constructed from a derivative of ...."**

## Download the book here!

With copyright permission from the copyright owner, the entire, original, unedited, unmodified book (except for the copyright notice added to every page) can be downloaded here:http://www.nofreewill.com/uft/Unification_of_Electromagnetism,_Kinematics,_and_Gravitation_by_N.F.J._Matthews-ev1.00.pdf.zip

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Many thanks to Betty Matthews, Tildon H. Glisson, Sherman Yeargan, and Chip Matthews for allowing me to make this book available to the world. And, most importantly, thanks to Dr. Matthews for figuring all of this out and writing this book.

## Related Research, Citations, and Researchers

Dr. Matthews' work is published in his book entitled "Unification of Electromagnetism, Kinematics, and Gravitation" by N.F.J. Matthews, Copyright 1999 by N.F.J. Matthews, ISBN: 0-9671473-0-1

To obtain a copy of the monograph by N. F. J. Matthews entitled, "Unification of Electromagnetism, Kinematics, and Gravitation," send a check for $48.00 (forty-eight US dollars) ("to cover printing and shipping", $45.00 for walk-in pickup) with the check made out to:

"NC State Engineering Foundation, Inc."

with "Matthews book" written on the "for" line.

If this is too costly, you might check out a library copy. Numerous university libraries in the US have received a copy, as well as the Smithsonian Library and Library of Congress.

The ISBN number is 0-9671473-0-1. Quantities are limited.

Note that this book is not a glossy, professionally bound publication. It is a photocopy bound in a simple black cover, much like the way dissertations are printed.

Mail the check, along with your complete mailing address, to the publisher:

Engineering Communications

7 Page Hall, Box 7901

College of Engineering

North Carolina State University

Raleigh, NC 27695-7901

Attention: Nate DeGraff

Engineering Communications

http://www.engr.ncsu.edu/communications/

Phone: (919) 919-515-3394

engr-communications@ncsu.edu

Nate DeGraff can be contacted by:

e-mail: nate_degraff@ncsu.edu

phone: (919) 515.3848

On Amazon.com:

"Unification of Electromagnetism, Kinematics, and Gravitation" by N.F.J. Matthews

http://www.amazon.com/gp/product/0967147301/ref=oh_o02_s00_i00_details

#### About Dr. Matthews:

A native of Clinton, North Carolina, Dr. Matthews joined the NC State University faculty in 1964. He earned the admiration and respect of his colleagues and students through his dedication to teaching, research, and service to the university and the community. He received several teaching awards and was named an Alumni Distinguished Professor in 1992. He retired from the university in 1998.

Prior to his retirement from NC State University, Dr. Matthews held several administrative positions, including director of graduate programs for the Department of Electrical and Computer Engineering and chair of the Courses and Curricula Committee for the department. He served as a mentor and friend to young faculty members and provided counsel to his peers.

Dr. Matthews received his BSE and MSE in mathematics from George Washington University in 1957 and 1959. He received his masters and doctoral degrees in electrical engineering from Princeton University in 1961 and 1964.

### Below are excerpts from Dr. Matthews' book.

### From the Preface:

**One day as a college sophomore, I asked my physics professor what an electron looked like. With a knowing, good-natured smile he responded that no one knew the structure of an electron and, in his opinion, no one ever would. His answer was a great surprise to me, for I thought physicists should know such things.**

Some 20 years later, I began a serious search for a set of equations that might lead to an acceptable answer to my question. After much consideration, I concluded that such a set of equations should satisfy the following conditions:

***** The equations should be classical field equations. They should have no probabilistic or quantum-mechanical feature.

***** The equations should be generally covariant. Consequences of the equations should not depend on the coordinate system in which the equations might be written.

***** The equations should be gauge invariant. Variables of the equations should lead to observable, measurable quantities.

***** The equations should be inherently free of singularities. Source terms of the equations should appear as volume densities and current densities analogous to charge density and electric current density appearing in Maxwell's equations.

***** The equations should have a nonlinear aspect. Either the differential equations themselves should be nonlinear or algebraic constraints on field and source variables of the equations should be nonlinear. Superposition should not be a general property of the theory.

***** The equations should yield conservation laws of charge, energy, linear momentum, and angular momentum in the same sense that Maxwell's equations yield the law of charge conservation.

***** The equations should be consistent with Maxwell's equations of electromagnetism.

***** The equations should be consistent with Einstein's equations of general relativity.

**This book describes a field theory that satisfies the eight requirements described above. At the heart of the theory is a set of 24 linear, first-order, gauge-invariant, covariant field equations, eight of which are Maxwell's equations and 16 of which contain the essence of Einstein's second-order equations of general relativity. All 24 field equations are identities in affine connections in the same sense that Bianchi's equations of Riemannian geometry are identities. Local conservation laws of charge, angular momentum, velocity of center of gravity, energy, and linear momentum follow directly from the field equations. Four coordinate conditions and a gauge condition are imposed on the 24 equations for the purpose of removing the five ambiguities inherent in gauge-invariant, covariant equations. The imposed conditions take the form of five nonlinear algebraic constraints on the field and source variables.**

### From Chapter 1 - Strength of Maxwell's Equations:

**In his investigation of field theories, Einstein[1] used the concept of**

*strength*of a system of field equations to compare the worth of one system with that of another. He assumed that each field variable could be written as a Taylor series in the neighborhood of a point*p*. By repeated differentiation and evaluation of the field equations at*p*, he could obtain relations among the Taylor coefficients. In effect, the field equations would specify some coefficients in terms of others so that, in principle, many coefficients could be eliminated from the series. The remaining Taylor coefficients would be free data; they could be chosen freely provided they remained consistent with the field equations. Einstein called the amount of this free data the strength of the system of field equations.Several authors have examined strengths of field theories. Penney [2], Hoenselaers [3], Burman [4], and Murphy [5], presented strengths of various field theories including those of Einstein, Maxwell, Weyl, Klein-Gordon, and Dirac. Mariwalla [6] showed that approximations of the strengths of Maxwell's potential and field-strength formulations were equal. Schutz [7] suggested that the strength of a field theory was related in a well-defined manner to the number and character of arbitrary functions required to specify a solution locally.

As a supplement to the work of Mariwalla and Schutz, this chapter presents verification that the strength of each of the two formulations of Maxwell's theory in empty space can be interpreted as initial data required to yield a solution to the field equations of that formulation. Also, the amount of initial data required for a solution of the field-strength formulation is shown to be equal, without approximation, to that required for a solution of the potential formulation [8].

**Exact equivalence of the strengths of the two formulations leads to a useful concept. An investigator familiar with the second-order potential formulation, but ignorant of the first-order field-strength formulation, could reasonably decide to examine the possibility of expressing the content of Maxwell's theory by a set of first-order field equations simply because the character of the initial data predicted by the strength of the potential formulation is precisely that of a set of first-order differential equations. And, of course, the investigation would meet with success.**

Equivalence of the strengths of the two formulations of Maxwell's theory suggests that ** the concept of strength may be useful in predicting possible unknown formulations of established field theories.** The strength concept is used in Chapter 2 to show that in empty space Einstein's second-order theory of general relativity has a first-order formulation.

### From Chapter 1, Section 1.1 - Strength and Functional Order:

The*strength*of a system of classical field equations is defined as the amount of free data that can be specified arbitrarily for the system of equations. In order to describe this amount of free data, we assume that each field variable of the system can be written as a Taylor series in the neighborhood of an arbitrary point. In calculating the strength of the system of field equations, we must identify the order of each variable of the system and we must determine the number of

*n*th-order Taylor coefficients for each variable. The strength of the system of equations is described in terms of the number of

*n*th-order Taylor coefficients that can be chosen freely.

The *order* of a function is an integer describing the dimension of the function relative to a reference function whose order is an integer chosen arbitrarily.

...

Taylor coefficients themselves and even the forms of the Taylor coefficients have no significance in strength calculations. However, keeping track of the number of *n*th-order coefficients is of utmost importance.

...

The strength concept permits us to compare the number of free *n*th-order Taylor coefficients of one set of functions with that of another set. Here, the functions of one set are taken to be linear combinations of derivatives of the functions of the other set. Keeping track of the *order* of each function permits us to accurately compare the numbers of free *n*th-order Taylor coefficients of the two sets. Our ability to make such comparisons allows us to draw important conclusions about the differential equations satisfied by the two sets of functions.

In subsequent sections, Maxwell's field-strength components ... are taken to be functions of second order. Maxwell's potentials ... are then functions of first order. The reasons for taking the components of ... here to be functions of second order are discussed in Sec. 1.3.

### From Chapter 1, Section 1.2 - Strength of Maxwell's First-Order Equations:

In this section, we derive the strength of the first-order formulation of Maxwell's theory in empty space and show that the strength is equal to the amount of initial data required for a solution of the equations....

These functions can be chosen freely provided they are consistent with the field equations. The selected functions can be viewed as initial data which can be correlated with the Cauchy initial-value problem as applied to the field equations. Verification that this amount of initial data is required for a solution of Maxwell's equations is given in Appendix B.

The fact that the strength of Maxwell's first-order equations represents the amount of initial data required for a solution of the equations suggests that strengths of differential equations describing other theories can be used for determining amounts of initial data required for solutions of those differential equations. The technique may be used as a supplement to the standard method of determining required amounts of initial data.

### From Chapter 1, Section 1.3 - Strength of Maxwell's Second-Order Equations:

If the concept of strength of a field theory is to be of value, then the strength of the theory should be independent of the manner in which the theory is presented. One would expect the theory to have the same strength when considered from different points of view....

This agreement between the strengths of the potential formulation and the field-strength formulation means that each formulation requires exactly the same amount of initial data for a solution to the respective set of field equations. Of course, the form of the initial data for each formulation is expected to be different, but this doesn't detract from the significance of (1.26).

...

### From Chapter 1, Section 1.4 - Conclusion:

Equation (1.26) shows that the amount of initial data required for the set of second-order field equations (1.20), relating functions ... of first order, is precisely the same as the amount of initial data required for a set of first-order differential equations, (1.6) and (1.7), relating functions ... of second order.If an investigator, unaware of the existence of the first-order field-strength formulation, expressed the strength of the second-order potential formulation in the form of the right-hand side of (1.18), he might feel compelled to seek a reformulation of the theory in the form of eight first-order differential equations having two identities of second order. He would expect the equations to relate six dependent variables, each of which could be written as a linear combination of the first derivatives of .... He would also expect the equations to exhibit conformal invariance. He could not be certain that the first-order equations would exist, but he could be sure that they would have the predicted form if they did exist. Knowing what to look for, the investigator could easily determine the first-order field-strength formulation we know to be valid.

These results for Maxwell's theory suggest that the concept of strength of a field theory as described here may be useful in obtaining unknown formulations, together with their invariance properties, of established field theories. We use this application of the strength concept in Chapter 2 to obtain a first-order formulation of Einstein's second-order, empty-space, gravitational field equations.

### From Chapter 2 - First-Order Formulation of Einstein's Equations in Empty Space:

In this chapter, the strength of Einstein's second-order gravitational field equations for empty space is computed and shown to be equal to the exact amount of initial data required for a local solution of the equations.While the strength derived here agrees with the original strength obtained by Einstein [1], the derivation differs from Einstein's in two respects. First, 40 Christoffel symbols are not treated here as independent variables as was done by Einstein and later by Mariwalla [2]. Hoenselaers [3] pointed out that such treatment was unnecessary and undesirable. Second, coordinate conditions are imposed here, which permits the strength to be interpreted correctly as an amount of initial data. **Even though Einstein and Mariwalla interpreted strength as the amount of free data, they did not display the free data in a form that could be interpreted specifically as an amount of initial data necessary for a solution of the field equations.**

...

The suggestion that the unknown first-order equations may exist is supported by the analogy in Maxwell's theory described in Chapter 1. A similar argument is used here to suggest the possible existence of a set of first-order partial differential equations which contain the essence of Einstein's second-order equations locally.

A set of field equations having the properties of the unknown equations is shown here to be the vanishing local divergence of Weyl's conformal tensor.

### From Chapter 2, Section 2.1 - Strength of Einstein's Equations:

...It is not essential that coordinate conditions be imposed in the strength derivation. However, if coordinate conditions are not imposed, the four derivatives ... remain indeterminate, and the four metric coefficients ... remain ambiguous [6].

**Imposing appropriate coordinate conditions eliminates the four arbitrary functions existing in Einstein's equations and yields a well-defined, initial-value problem.**This permits the strength of Einstein's equations to be interpreted as the amount of initial data required for a solution. Harmonic coordinate conditions are used here because they are simple and because they are commonly used by others. Imposition of coordinate conditions does not change the strength of the field theory, a result verified in (2.9) below.

...

### From Chapter 2, Section 2.3 - Local Coordinates:

...As a policy in following chapters, we express field equations and conservation laws in covariant form, and then, if desired, express the covariant equations in local coordinates. This serial policy is established so we do not inadvertently discard partial derivatives of Christoffel symbols that are imbedded in second and higher-order covariant derivatives.

### From Chapter 2, Section 2.4 - First-Order Formulation of Einstein's Equations:

...The possible existence of field equations having properties like those of (2.38) is deduced from examination of the amount of initial data required to yield a local solution of Einstein's equations in empty space. The unknown equations described by Zn of (2.14d) require initial data in the form of four functions of three variables and six functions of two variables. It is not immediately clear that this amount of initial data is required for a solution of (2.38). Verification that this amount of initial data is required is given in Appendix E.

### From Chapter 2, Section 2.5 - Chapter 2 Conclusion:

**Einstein [1] and other investigators [2,3] used strength primarily to calculate a single number, the coefficient of freedom, which gave only a vague idea of how much data was required for a solution. The treatment of strength in this chapter demonstrates that strength, as it pertains to Einstein's equations, has far greater utility. Strength not only gives the amount of initial data required for a solution, but it displays the amount of data in a meaningful form.**

In order to establish a well-defined initial-value problem, we must impose four coordinate conditions on ... <Einstein's gravitational> field equations <for empty space> (2.1). The harmonic coordinate conditions (2.4) imposed in Sec. 2.1 limit permissible coordinate transformations to those satisfying (2.5). This limitation implies that the initial-value problem described by (2.1) and (2.4) taken together pertains only to a local region [14]. Thus, any use of the strength concept for examining the initial-value problem should be restricted to a local region. This restriction is demonstrated clearly by the fact that the amount of initial data required for (2.1) and (2.4) is the same as that required for (2.38).

The analyses of this chapter give strong evidence that (2.38) is a satisfactory first-order formulation of Einstein's empty-space field equations (2.1) locally. This evidence lends credence to speculations that (2.36) may constitute meaningful gravitational wave equations when combined with Einstein's field equations in both empty and non-empty space. Indeed, Hawking [15] uses (2.35) in considering small perturbations in an otherwise flat universe. Campbell et al [16, 17, 18] express local versions of (2.35) in forms analogous to Maxwell's equations and examine their physical meanings. Other contributions include those of Obata et al [19, 20], Drew and Gegenberg [21], Dymnikova [22], Novello [23], Tchrakian [24], Lesche and Som [25], and Lichnerowicz [26].

Clearly, the local vanishing of the divergence of the conformal tensor in (2.38) is a necessary consequence of Einstein's empty-space field equations (2.1) locally. Conversely, as we prove in Appendix F, if (2.38) is valid in a local region of space-time, then a Ricci tensor satisfying (2.1) in the region can always be found.

This final result adds compelling evidence to the argument that (2.38) is a satisfactory first-order formulation of (2.1) locally.

Having justified (2.38) as a local first-order formulation of Einstein's equations (2.1) insofar as initial data is concerned, we examine in Chapter 3 the consequences of generalizing the result to the covariant, non-empty space identity of (2.35).

### From Chapter 3 - First-Order Formulation of Einstein's Equations in Non-Empty Space:

We presented evidence in Chapter 2 that (2.38) constituted a first-order formulation of Einstein's second-order field equations in a local region of empty space. In this chapter, we generalize (2.38) to yield generally covariant first-order equations in non-empty space.After using Einstein's equations for non-empty space to eliminate the Ricci tensor ... and the Riemann scalar *R* from the contracted Bianchi identities (2.35), we show that the resulting equations, commonly called *Bianchi-type field equations*, constitute 24 generally covariant equations, only 16 of which are algebraically independent. A vanishing divergence of the source tensor of the Bianchi-type field equations represents a set of six equations that we interpret locally as conservation laws of angular momentum and velocity of center of gravity. A different generally covariant divergence of the Bianchi-type field equations yields a set of 10 higher-order field equations, one of which we interpret as a generalization of Newton's law of gravitation. A vanishing divergence of the source tensor of these last 10 equations yields four equations interpreted locally as conservation laws of energy and liner momentum.

Conservation laws permit us to identify volume densities of conserved quantities and to recognize current densities associated with conserved quantities. Volume densities and current densities appear as source terms in the contracted Bianchi identities making identification of associated components of the conformal tensor ... as physically meaningful field quantities feasible. In the next section, we examine general properties of conservation laws and point out certain characteristics of conservation laws that require careful consideration.

### From Chapter 3, Section 3.1 - Conservation Laws:

**Development of conservation laws of charge, angular momentum, energy, linear momentum, and velocity of the center of gravity in a closed system, all derivable from a set of field equations, is one of the primary objectives of this book.**Since conservation laws play prominent roles in the theory presented in this and subsequent chapters, it is appropriate to describe here briefly what we mean when we refer to an equation as a conservation law.

...

Every conservation law treated in this and subsequent chapters has three important properties:

*****The conservation law is derivable from a set of field equations.

*****In curved space, the conservation law takes the form of a generally covariant vanishing divergence of a global source tensor.

*****In flat space, the conservation law takes the form of a vanishing local 4-divergence of a local source tensor whose components can be computed throughout a region of the flat space.

This last property prohibits a conservation law from containing isolated partial derivatives of global Christoffel symbols since derivatives are expected to be unknown and undeterminable.

...

**Identifying the covariant divergence of the Einstein source tensor with degenerate electromagnetic field equations in general and with the vanishing electromagnetic source tensor ... in particular raises serious question about interpreting the vanishing covariant divergence of Einstein's source tensor as conservation laws of energy and linear momentum, as is commonly done in the literature [1].**Indeed, because of the analytical results presented in Chapter 4, we are compelled to do two things:

* Interpret the vanishing of the divergence of Einstein's source tensor as a statement that the electromagnetic source tensor is zero in a system whose electromagnetic potentials are zero.

* Seek new conservation laws of energy and linear momentum.

**Conservation laws of charge, angular momentum, energy, linear momentum, and velocity of the center of gravity in a closed system should be an integral part of any successful theory unifying electromagnetism and gravitation.** We consider the conservation laws of this chapter to be an abbreviated set of interim conservation laws which serve as stepping stones to a complete set of conservation laws together with their parent field equations developed in Chapters 4 and 5.

### From Chapter 3, Section 3.3 - Energy-Momentum Tensor:

...Indeed, we show in Chapter 5 that this can be done so that (3.15) for ... = ... = 0 can be viewed as a generalization of Newton's law.

### From Chapter 3, Section 3.4 - Conclusion:

**The first-order formulation (3.6) of Einstein's gravitational theory derives its credibility from the fact that the amount of initial data required to provide a solution of Einstein's equations (2.1) in empty space for the metric tensor ... is precisely the same as the amount of initial data required to provide a solution of (2.38) for the components of Weyl's conformal tensor ....**Generalizing this result to arrive at the field equations of (3.6) for non-empty space is a natural step.

Local conservation laws follow in a straightforward manner upon taking appropriate covariant divergences of the source terms appearing in the field equations (3.6) and noting that, after differentiation, terms linear in Christoffel symbols can be neglected in local Minkowski coordinates. In order for energy or linear momentum to be computed for a given physical system, the field equations must be solved for Weyl's conformal tensor. These ideas are analogous to those that apply to relationships existing between the field-strength and potential formulations of Maxwell's field theory.

The first-order formulation of Einstein's theory has several interesting features:

* The field equations (3.6) are linear and of first order in the field variables ..., suggesting that solutions of a large class of meaningful physical problems can be addressed.

...

* Local conservation laws for energy, linear momentum, angular momentum, and velocity of the center of gravity follow directly from the field equations (3.6).

...

**The vanishing divergence ... of (3.18) is interpreted here locally as conservation laws of energy and linear momentum. If this interpretation is correct, then the four equations of (3.5), which follow from Einstein's equations (3.1) and the contraction (2.7) of Bianchi's identities, cannot be interpreted as conservation laws; they must play some other role.** Indeed, we show in Chapter 4 that, in a gauge invariant system, a degenerate form of electromagnetic field equations reduces to (3.5). In particular, the four equations of (3.5) are interpreted in Chapter 4 as corresponding to vanishing electric charge density and vanishing electric current density. This suggests that the contracted Bianchi identities (2.7) cannot lead to conservation laws of energy and linear momentum.

**While the field equations of (3.6) appear to constitute a satisfactory first-order formulation of Einstein's second-order field equations (3.1), two features militate against acceptance of either (3.1) or (3.6) as an ultimate set of field equations that can describe the behavior of an arbitrary physical system. The two shortcomings are:**

*** Neither set is gauge invariant.**

*** Neither set embraces Maxwell's electromagnetic field equations.**

**Both of these deficiencies, as they pertain to (3.6), are addressed in Chapter 4.**

### From Chapter 4 - Field Equations and Conservation Laws in Weyl's Space:

The 16 first-order field equations (3.6) and their associated conservation laws, (3.13) and (3.18), are generally covariant so that their dependent variables have relativistic meaning. However, since the field equations and conservation laws of the formulation are not gauge invariant, the conserved quantities associated with (3.13) and (3.18) are not observables [1]. This deficiency obscures physical interpretation of the field variables and conserved quantities.With a view toward expressing the first-order formulation of Einstein's theory described in Chapter 3 in gauge-invariant form, we begin this chapter with some useful relations among gauge-invariant tensor densities ....

...

The gauge-invariant Bianchi identities constitute 24 independent, linear, first-order, gauge-invariant, field equations. Eight equations are interpreted as electromagnetic field equations. The remaining 16 equations are viewed as kinematic (angular-momentum and velocity) field equations. A divergence of the 16 kinematic equations yields 10 additional, first-order field equations having higher-order dependent field variables. We call these last 10 equations the gravitational field equations. **One of these 10 equations is viewed as a gauge-invariant generalization of Newton's law of gravitation.**

Generally covariant, gauge-invariant, conservation laws of energy, linear momentum, angular momentum, velocity of the center of gravity, and charge follow directly from the field equations.

The *Electromagnetic-Kinematic-Gravitational (EKG)* field equations and conservation laws of this chapter are identities in Weyl affine connections in the same sense that the Bianchi relations are identities in Christoffel symbols. **Recognition of these field equations and conservation laws as identities suggests that man's search for accurate statements of physical laws may be in reality a search for identities exhibiting appropriate invariances.**

### From Chapter 4, Section 4.1 - Intensors:

...Here, adopting the essence of Eddington's convention [4], we give the special name

*intensor*to a tensor density that is gauge invariant. Associated with every intensor is a weight which describes the density character of the intensor.

...

We refrain from using the phrase

*intensor density*since the word

*density*in this context is unnecessary, cumbersome, and redundant.

### From Chapter 4, Section 4.4 - Bianchi Intensor Identities:

Bianchi's tensor identities (2.26) of Riemannian geometry can be generalized to yield intensor identities in Weyl's geometry. Taking the incovariant derivative of both sides of (4.1) with respect to ..., twice permuting the last three indices (...) of the resulting intensor equation, and adding the resulting three equations yield equations we call the*Bianchi intensor identities*:

... + ... + ... = 0. (4.30)

These equations are identities in the Weyl affine connections .... The validity of (4.30) is easily verified by direct substitution using the definition of ... given in (4.1) and the definition of the incovariant-differentiation operation given in Sec. 4.3.

**The Bianchi intensor identities of (4.30) contain a wealth of information about the physical world. Extracting this information and interpreting it properly are challenges because of algebraic complexities of the intensor equations. Care and fortitude, however, lead to enlightening results.**

**The identities of (4.30) form the foundation from which electromagnetic, kinematic, and gravitational field equations are developed in Sec. 4.5.** It is important to note that all field equations that result from contractions of (4.30) are identities in the Weyl affine connections. No assumption about conditions under which the equations are valid is required except the assumption that the algebraic rules and incovariant-differentiation rules of the previous sections are acceptable.

### From Chapter 4, Section 4.5 - Field Equations:

**In this section, we derive field equations of electromagnetism, kinematics, and gravitation from contractions of the Bianchi intensor identities of (4.30). We derive conservation laws of charge, angular momentum, velocity of center of gravity, energy, and linear momentum from the field equations. In effect, we obtain all field equations and all conservation laws from the Bianchi intensor identities.**

Since the Bianchi equations of (4.30) are identities in the Weyl affine connections, all field equations and conservation laws derived from (4.30) are also identities in the Weyl affine connections. **Hence, neither the field equations nor the conservation laws originate from empirical laws. However, comparison of our results with empirical laws (e.g., Maxwell's field equations) permits us to give meaningful names to the field equations and conservation laws (identities) we obtain. The matching of empirical laws with identities in this manner leads us to conjecture that accepted empirical laws have the form they do because they can have no other form - i.e., empirical laws are identities in disguise.**

All intensors in this section are expressed in contravariant form. Expressing intensor equations in contravariant form permits us to easily combine equations and to accurately describe the several different incovariant divergences we present in subsequent sections.

### From Chapter 4, Section 4.5a - Electromagnetic Field Equations:

**The form of the electromagnetic field equations of (4.33) suggests a contradiction to the argument that the divergence of Einstein's tensor being zero leads to conservation laws of energy and linear momentum.**For ... = 0, the intensor ... of (4.34) reduces to

... (4.36)

where ... is ..., the contravariant form of Einstein's tensor of (4.19). Clearly, ... is associated here with .... This observation leads us to interpret the vanishing of ... as the vanishing of ... in a physical system in which ... is zero.

**The appearance of ... here casts doubt on the interpretation of (3.5) as conservation laws of energy and linear momentum. Equation (3.5) arises because ... vanishes. The divergence ... is unlikely to perform double duty. We cannot expect ... to correspond to zero electric charge and current densities and, at the same time, lead to conservation laws of energy and linear momentum.**

**This result, considered together with other results presented subsequently, suggests that specifying a gauge-invariant form of Einstein's tensor, namely ..., gives the inherent character of a system.** Source terms of field equations contain .... Solutions of the field equations then supplement the investigator's knowledge of the system under consideration.

### From Chapter 4, Section 4.5b - Kinematic Field Equations:

...Thus the kinematic field equations of (4.37) exhibit a feature that the electromagnetic field equations do not have. The kinematic field equations contain source terms ..., none of which appears in a conservation law, while all source terms ... in the electromagnetic field equations appear in a conservation law.

**This contrast is one of the features that makes the kinematic field equations of (4.37) more intricate and complicated than the electromagnetic field equations of (4.32) and (4.33).**

### From Chapter 4, Section 4.5c - Gravitational Field Equations:

Identifying ... as an energy-momentum intensor, believing Eq. (4.49) can be written as a set of first-order set of equations, and recognizing (4.49) for ... = ... = 0 as a generalization of Newton's law of gravitation justifies identifying (4.49) as a set of gravitational field equations. Thus, we call the combination of the electromagnetic field equations (4.32) and (4.33), the kinematic field equations (4.37), and the gravitational field equations (4.49) the*Electromagnetic-Kinematic-Gravitational (EKG)*field equations.

### From Chapter 4, Section 4.6 - *EKG* Conservation Laws:

**Conservation laws for charge, angular momentum, velocity of the center of gravity, linear momentum, and energy follow directly from**

*EKG*field equations (4.32), (4.33), (4.37), and (4.49). Our ability to derive conservation laws from the*EKG*field equations is one of the most important properties of the field equations. This ability permits us to make meaningful correspondences between the observed physical world and its mathematical representation....

If we were to attempt to perform the tasks described above, we would encounter difficulty because of the structure of the conservation laws. Since the conservation laws are expressed here in intensor form, they are valid in any coordinate system and in any gauge system. However, in order for the conservation laws to have practical value, they should be expressed in a specific, local, coordinate system and in a specific gauge system. Simply replacing incovariant differentiation locally with partial differentiation in order to accomplish this task is inappropriate since partial derivatives of affine connections ... appear in every conservation law. In general, these derivatives of affine connections are not zero even in a local Minkowski coordinate system. Writing the

*EKG*conservation laws in a local coordinate system and in a specific gauge system without the explicit appearance of partial derivatives of the affine connection is a realistic possibility and a worthwhile endeavor. We pursue the endeavor in Chapter 5.

### From Chapter 4, Section 4.7 - Conclusion:

If the*EKG*field equations and the conservation laws of this chapter can be expressed in a local coordinate system as described above, specific physical systems can be examined in a straightforward manner. We expect that the behavior of a physical system can be found from appropriate initial conditions and from ten equations which are generalizations of Einstein's equations (3.1), namely,

... (4.60)

where ... is a set of 10 selected functions which satisfy the charge conservation law (4.35) and which have the same algebraic properties as the source intensor ... describing the system. The total system charge, angular momentum, and velocity of the center of gravity can be computed directly from .... After the electromagnetic field equations are solved for ..., source terms of the kinematic field equations can be computed and the kinematic field equations can be solved for .... Finally, the total linear momentum and energy of the system can be computed.

**From (4.13), (4.14), and (4.17) for ... = 0, Eq. (4.60) reduces to Einstein's equations of gravitation (3.1) so that (4.60) can be interpreted as a gauge-invariant form of Einstein's equations in non-empty space. The thrust of our effort in Chapter 5 is not toward solving the generalized Einstein equations (4.60) for the metric coefficients ... because we have no particular use for them. Rather, we seek coordinate conditions and a gauge condition that will permit us to express the conservation laws of charge, angular momentum, velocity of the center of gravity, linear momentum, and energy in a local coordinate system without partial derivatives of affine connections appearing in the EKG conservation laws.**

**Our effort in Chapter 5 is rewarded with the determination of a gauge condition and four coordinate conditions, all of which, in combination, serve to eliminate the appearance of stand-alone, affine-connection derivatives in conservation laws expressed in an arbitrary, local coordinate system.**

### From Chapter 5 - *EKG* Field Equations and Conservation Laws in a Locally Flat Weyl Space:

**The**While the incovariant form of the

*Electromagnetic-Kinematic-Gravitational (EKG)*field equations and conservation laws of Chapter 4 constitute a unification of Maxwell's theory of electromagnetism and Einstein's theory of gravitation.*EKG*equations gives physical meaning to the field variables and conserved quantities, such a form in not appropriate for solving specific physical problems since affine connections, which are unknown quantities, appear in the equations separate from their appearance within field and source variables, ..., ..., and .... Also, the gravitational field equations and all of the

*EKG*conservation laws contain partial derivatives of the affine connections. These derivatives are unknown quantities even in local coordinate systems.

This chapter treats the problem of expressing the *EKG* field equations and conservation laws in a local coordinate system and in a limited gauge system. The local systems selected here are those for a flat Weyl space, i.e., those for a four-space in which Weyl's curvature intensor ... is zero. The resulting *EKG* field equations and conservation laws are expressed locally in intensor form so that they exhibit Lorentz invariance and a limited gauge invariance.

Choice of a gauge condition, permitted by the gauge invariance of the general theory, is made in Sec. 5.3a where the law of charge conservation is examined in the locally flat Weyl space. Selection of an acceptable gauge condition is found to be quite naturally limited to imposition of a condition on the local source intensor ....

Choice of four coordinate conditions, permitted by the covariance of the general theory, is made in Sec. 5.3c where conservation laws of energy and linear momentum are examined in the locally flat Weyl space. Appropriate coordinate conditions are found to be conditions relating source intensors, their local incovariant derivatives, and dependent field variables.

### From Chapter 5, Section 5.2 - Combined Intensors and Their Incovariant Derivatives:

In any attempt to express the*EKG*field equations and conservation laws of Chapter 4 in a local system, incovariant derivatives of intensors must be expanded in terms of affine connections ... and partial derivatives of the intensors. Several of the resulting equations contain partial derivatives of affine connections, which are unknown quantities even at .... These affine-connection derivatives stand alone; i.e., they do not appear within intensors such as ..., ..., ..., and

*S*. This complicates the task of writing the

*EKG*field equations and conservation laws in a local system. All derivatives of affine connections somehow must be eliminated or joined together in order to form intensors at ....

...

Incovariant derivatives of combined intensors play a central role in the development of local

*EKG*field equations and conservation laws. We describe here how incovariant differentiation of a combined intensor is performed. In particular, we describe a procedure for writing expressions for global incovariant derivatives of the product of a local metric intensor and a global intensor.

### From Chapter 5, Section 5.3 - Local *EKG* Field Equations and Conservation Laws:

The *EKG*field equations and conservation laws of Chapter 4 are expressed in arbitrary coordinate and gauge systems, the ambiguities of which make any search for meaningful solutions of the field equations difficult or impossible. In this section, the

*EKG*field equations and conservation laws are expressed in local coordinate and gauge systems corresponding to a locally flat Weyl space.

**Use of local systems permits an investigator to impose symmetry conditions on physical systems of interest and to seek meaningful solutions of the field equations for the physical systems.**

### From Chapter 5, Section 5.3a - Electromagnetic System:

...Condition (5.59) corresponds to an equation commonly called a gauge condition in electromagnetism. We prefer to call (5.59) a

*Charge-Ambiguity-Removal (CAR)*condition since this name better describes the

*purpose*of the condition. The

*CAR*condition serves to guarantee the validity of the charge conservation law (5.58) and to make the charge density determinate and unambiguous.

One may be tempted to conclude from (5.57), (5.58), and (5.59) that the law of charge conservation itself depends on the act of choosing the *CAR* condition. Such a conclusion is incorrect. Equation (5.57) states that charge is conserved, but indeterminate. The *CAR* condition (5.59) simply makes the conserved charge determinate.

### From Chapter 5, Section 5.3c - Gravitational System:

...Imposing the four conditions of (5.81) is justified because the Bianchi intensor identities (4.30) contain, on account of their general covariance, four arbitrary functions that can be selected freely. Equation (5.81) is a natural choice for conditions on the four arbitrary functions.

Conditions (5.81) correspond to equations commonly called coordinate conditions in general relativity. We prefer to call (5.81) *Energy-Linear-Momentum-Ambiguity-Removal (ELMAR)* conditions since the name is more descriptive of the *purpose* of the conditions. The *ELMAR* conditions serve to guarantee the validity of the conservation laws of energy and linear momentum (5.80) and to make the energy and linear-momentum densities determinate and unambiguous.

The validity of (5.80) depends only on the validity of the electromagnetic field equations, (5.51) and (5.56), of the kinematic field equations (5.61), and of the *ELMAR* conditions (5.81). In effect, we can interpret the information contained in the four *ELMAR* conditions as being the same as the information contained in the four conservation laws of energy and linear momentum.

### From Chapter 5, Section 5.4 - Conclusion:

The fact that the choice of ... depends on ... and ...**suggests that the**

*EKG*field equations of (5.51), (5.56), and (5.61), the*EKG*conservation laws of (5.58), (5.66a), and (5.80), and the ambiguity-removal conditions of (5.59) and (5.83) may provide a basis for theoretically explaining the existence of certain entities of nature which exhibit properties of cohesiveness such as the photon, the neutrino, and elementary particles having non-zero rest masses.### From Chapter 6 - Strength of the *EKG* Field Equations:

In this chapter, we use the strength concept developed in Chapter 1 to examine the amount of initial data required for a solution of the local *EKG*field equations of Chapter 5. The procedure used here is the same as that used for calculating the strengths of Maxwell's equations and Einstein's equations presented in Chapters 1 and 2, respectively. The amount of initial data computed from the strength calculations is shown to be equal to that required to yield a solution of the local

*EKG*field equations.

### From Chapter 6, Section 6.5 - Conclusion:

...If the theory of Sec. 6.2 is sufficiently rich in content, then selection of appropriate functions for these quantities may constitute a significant step in an attempt to construct an acceptable model of a stable elementary particle.

The *CAR* and *ELMAR* conditions of (6.15) and (6.16) culminating in (6.26), (6.34b), (6.34c), and (6.34d) can play an important role in such an endeavor. The *CAR* and *ELMAR* conditions are nonlinear in ..., ..., and .... The nonlinear character of these conditions indicates that different solutions of the field equations cannot be superposed and that evolution of the source intensor ... in space-time depends on the field intensors ... and .... **This feature supports the assertion that the theory of Sec. 6.2 may contain ingredients required for construction of mathematical models of elementary particles.**

### From Chapter 7 - *EKG* Field Equations and Conservation Laws in Curvilinear Coordinates:

In Chapter 5, the *EKG*field equations and conservation laws are expressed in a locally flat Weyl space in which the equations exhibit Lorentz invariance and a limited gauge invariance. The forms of the equations clearly reveal their Lorentz and limited-gauge invariances, but the physical meanings of many of the equations are hidden by the somewhat obscure intensor notation in which relations are expressed. Further, for physical systems having spatial symmetries different from that appropriate for Minkowski space, the compact notation of Chapter 5 promises to make investigation cumbersome at best.

This chapter rectifies these shortcomings to an extent. **The EKG field equations and conservation laws are written here in standard vector notation using three-dimensional divergence and curl operators.** Meaningful names and symbols together with conventional dimensions are assigned to source terms and field variables by multiplying the flat-space

*EKG*field equations and conservation laws of Chapter 5 by appropriate universal constants. Employed constants include the charge of the positron

*e*, the speed of light in free space

*c*, the permittivity of free space ..., and Planck's constant

*h*. The fine structure constant ... appears in the equations in a natural way.

### From Chapter 7, Section 7.3 - Electromagnetic System:

...The procedure used here to obtain the equations above demonstrates the approach used for dealing with more complicated, less familiar, field equations and conservation laws of the kinematic system in the next section.

### From Chapter 7, Section 7.4 - Kinematic System:

The kinematic field equations of (5.61) constitute 24 equations, eight of which can be derived from the other 16. Six identities of second order exist among the 16 so that 10 independent equations remain to yield solutions for the 10 independent components of the field variables .... The 24 field equations fall into three sets of equations, each set containing eight equations. Any set can be obtained from the other two.Writing the three sets of field equations in standard, three-dimensional notation in local curvilinear coordinates is a formidable task, not because the equations are particularly complicated, but because many field and source variables must be defined and expressions for them must be derived. The task reduces mainly to establishing a consistent notation and developing a satisfactory bookkeeping system.

The three sets of field equations are obtained from (5.61) in a manner similar to that used for obtaining the electromagnetic field equations (7.16) - (7.19) from (5.51) and (5.56). Each of the 24 equations of (5.61) is multiplied by appropriate universal constants, and substitutions for the source terms and field variables are made. The equations are then grouped together in meaningful forms.

### From Chapter 7, Section 7.4a - Kinematic Field Equations:

...In general, the principle of superposition cannot be applied to the field equations (7.32a) - (7.32d) and (7.16) - (7.19). Different solutions of the field equations can be superposed only in a source-free region, i.e., where components of the source intensor ... are all zero. In a region that is not source free, source terms for different systems cannot be superposed since components of the source intensor must satisfy the

*CAR*condition (7.31), which is a nonlinear relationship. Also, as shown in Sec. 7.5, the source components ... must satisfy the four

*ELMAR*conditions, which are also nonlinear in ....

### From Chapter 7, Section 7.4b - Conservation of Angular Momentum and Velocity of Center of Gravity:

Conservation laws of angular momentum and velocity of the center of gravity of a physical system in local curvilinear coordinates are obtained by combining (5.66a) with (7.33a), (7.33b), (7.36a) - (7.37c), and their permutations. Results are ... (7.56a) ... (7.56b)**The conserved quantities, charge, angular momentum, and velocity of the center of gravity, do not depend on solutions for E and H of the electromagnetic field equations (7.16) - (7.19) or on solutions for ... and ... of the kinematic field equations (7.32a) - (7.32d). However, as shown in Sec. 7.5, the conserved quantities, energy and linear momentum, do depend on E, H, ..., and ..., so that the field equations must be solved before total energy and linear momentum of a system can be computed.**

### From Chapter 7, Section 7.5 - Gravitational System:

**The gravitational field equations (5.72) can be viewed as five independent first-order equations relating 36 linearly independent field variables. Since the number of field variables far exceeds the number of equations, unambiguous solutions of (5.72) for ... cannot be obtained in general. At first glance, insolubility of the equations is troubling, but closer examination suggests a logical reason for this circumstance. The field variables ... take no part in conservation laws. Therefore, ambiguities in ... in no way affect conserved quantities. A question then arises as to whether ... plays any**

*fundamental role*in describing physical systems. Some of the components of ... have clear physical meanings from practical points of view, but whether the meanings are necessary for describing the behavior of a system is questionable.Equations (5.72) are not written here in local coordinates since a general solution of them is not contemplated. However, the equation for ... = ... = 0 is of particular interest, and we express it here in local coordinates ....

...

**Equation (7.62) is viewed as a generalization of Newton's law of gravitation.** The vector ** G** is interpreted as gravitational flux density (energy/area) and the source scalar ... is interpreted as energy density (energy/volume). The anomalous field term ... is a curiosity. For ... independent of time, Eq. (7.62) is a statement of Newton's law of gravitation. However, since ... plays an essential role in the Lorentz invariance of (7.62), one would expect ... to be necessarily time dependent in general.

Equations (7.62) and the field equations (5.72) as a whole deserve further investigation. We do not examine them here because conserved quantities (constants) of physical systems depend only on their source terms. Values of conserved quantities of physical systems do not depend on field terms of (5.72).

Substituting for ... in (5.80) using (7.59) and (7.64) - (7.66) yields the law of energy conservation, ... (7.67a) and the conservation laws for the three components of linear momentum, ...

...

As stated previously in this section, conserved quantities do not depend on the fields ..., but they do depend on solutions of the electromagnetic field equations and on solutions of the kinematic field equations.

...

Since the first two terms on the left-hand side of (7.72b) are recognized as the ... component of Lorentz force density (force/volume), Eq. (7.72b) and its permutations can be interpreted as a statement requiring that force densities be balanced at every point in a closed physical system.

### From Chapter 7, Section 7.6 - Indexed-Vector Operations:

...One may wonder why in orthogonal curvilinear coordinates we use indexed vectors in describing the

*EKG*theory rather than describing the theory using the well established formalism of dyadics [4,5,6]. There are several reasons for using indexed vectors rather than dyadics.

First, indexed vectors are simple, dyadics are complicated. An indexed vector represents only one vector in space; dyadics represents three vectors in space. An indexed vector has three components; a dyadic has nine components. The index on an indexed vector clearly identifies the coordinate with which the indexed vector is associated; a dyadic carries no such identification because the dyadic represents three vectors, each associated with a different coordinate.

Second, the notation associated with indexed vectors is less abstract (less compact) than the notation associated with dyadics, making the indexed-vector formulation easier to interpret. The indexed-vector notation permits us to write the kinematic field equations as three separate sets of equations, one set associated with each coordinate. Indexed vectors within the equations have clear, simple, physical meanings. Dyadic notation would obscure meanings of the equations because the meaningful indexed vectors would not appear explicitly in the equations. We favor making the transition from the abstract intensor notation of Chapter 5 to the less abstract indexed-vector notation here rather than to dyadic notation, another abstract notation.

Finally, in rectangular coordinates, the hyperdivergence, hypercurl, and hypergradient operations reduce to the familiar divergence, curl, and gradient operations, respectively, so that meaningful analogies can be drawn between electromagnetic field equations and the kinematic field equations and between conservation laws of charge, angular momentum, velocity, energy, and linear momentum. Such analogies are obscured in a dyadic notation.

### From Chapter 7, Section 7.8 - Plane Waves In Rectangular Coordinates:

The form of the kinematic field equations of (7.32a) - (7.32d) resembles the form of the electromagnetic field equations of (7.16) - (7.19), leading us to suspect that solutions for the two sets of equations may have similar characteristics. In order to examine some of these expected similarities, we briefly treat here a simple physical system, namely, that of plane waves traveling in the positive*z*direction in free space in a rectangular coordinate system.

**We find that electromagnetic waves cannot propagate unless kinematic waves accompany them. Kinematic waves, however, can propagate alone, i.e., without being accompanied by electromagnetic waves.**

...

**These two simple cases imply that radio waves must be accompanied by kinematic waves, but that kinematic waves can propagate alone. If this is true, then we may wonder why kinematic fields have never been predicted and detected experimentally. The answer is contained in volume-density terms appearing in conservation laws. Angular momentum volume density ... described by (7.52a) and velocity volume density ... described by (7.54a) do not depend on components of ... or ... at all. Examination of (7.70a) and (7.70b) reveals that kinematic fields can make no contribution to energy density ... or linear momentum density ... of a system if all components of ... of the system are zero. Therefore, kinematic fields cannot directly affect either angular momentum or velocity of any system. Kinematic fields cannot directly affect either energy or linear momentum of a system if all components of ... of the system are zero.**

**The plane-wave system described here should bear little resemblance to a system describing a photon. A photon carries angular momentum as well as energy so that a photon system must embrace non-zero components of ....**

### From Chapter 7, Section 7.9 - Conclusion:

A physical system is specified when particular functions of space and time are specified for the 10 components of .... If these functions are properly selected and if relative affine connections ... are negligibly small everywhere in local space-time, then the conserved quantities, charge, angular momentum, and velocity of the center of gravity, of the system can be computed. Source terms of the electromagnetic field equations can be computed and the electromagnetic field equations can be solved for the electromagnetic field strengths,**and**

*E***, after which the source terms of the kinematic field equations can be computed. The kinematic field equations can then be solved for kinematic field strengths, ... and ..., after which the conserved quantities, energy and linear momentum, can be computed.**

*H*...

**Satisfactory resolution of the problem may produce techniques for constructing theoretical models of elementary particles. Such models should depend strongly on the**

*CAR*and*ELMAR*conditions because the nonlinear characteristics of these conditions have the responsibility of holding the particles together. The theory is expressed in local curvilinear coordinates here in order to facilitate construction of models having different spatial symmetries.The *CAR* and *ELMAR* conditions are automatically satisfied in systems for which all ... are zero. Such systems include that of traveling electromagnetic and kinematic waves. Evidence indicates that electromagnetic waves must be accompanied by kinematic waves while kinematic waves can propagate alone.

### From Chapter 8 - *EKG* Field Equations and Conservation Laws in Cylindrical Coordinates:

**In this chapter we use the**

*EKG*field equations and conservation laws of Chapter 7 to obtain a set of differential equations that we expect to lead to mathematical models of physical systems described in cylindrical coordinates.We write the equations in cylindrical coordinates rather than in spherical coordinates or some other curvilinear coordinate system because angular momentum can be computed more easily in a cylindrical coordinate system. Using cylindrical coordinates in which to model a photon, electron, or other elementary particle having angular momentum in the z direction, we can compute the angular momentum of the system from only the z component of angular momentum density while still taking advantage of possible ... symmetry of the system. In spherical coordinates, for example, the z component of angular momentum density would have to be computed from superposition of two components of angular momentum density, namely, the radial component and the copolar-angle component, each of which would have to be determined separately.

We limit our efforts in this chapter to setting up a satisfactory notation and expressing all equations and field variables of the previous chapter in a cylindrical coordinate system. No attempt is made here to solve the field equations. An approach to solving the equations for a specific physical system is presented in Chapter 9.

### From Chapter 8, Section 8.2 - Electromagnetic System:

...The condition indicates that different solutions of the

*EKG*field equations cannot be added together safely to yield a different solution of the equations. Superposition of solutions is not a general property of the theory.

**Thus the**

*CAR*condition constitutes a property of the theory that shows promise in providing a mechanism for holding elementary particles like the photon and electron together, i.e., for keeping the particles from losing their identities and flying apart.### From Chapter 8, Section 8.4 - Gravitational System:

As described in Sec. 7.5, the ten gravitational field equations of (5.72) constitute only five independent equations while the gravitational field equations themselves contain 36 independent vector components. This fact makes an unambiguous solution of the equations for a system of low spacial symmetry impossible to obtain. An argument is presented in Sec. 7.5 that a solution of the equations is unnecessary. None of the 36 field variables appears in the density of any conserved quantity.**Therefore, all gravitational field variables can remain ambiguous and unknown without affecting our knowledge of the**

*fundamental parameters*(the conserved quantities) of the physical system.**One of the gravitational field equations of interest, however, is the generalization (7.62) of Newton's law of gravitation, which in our cylindrical coordinate system takes the form, ... (8.51)**

...

**We do not seek solutions for ... and the three components of G because we do not have enough equations from which to determine them. Even if the functions, ..., ..., ..., and ..., could be found unambiguously, they would contribute nothing to determination of conserved quantities. The decisive factor militating against their significance is the fact that they appear nowhere in densities of conserved quantities. Likewise, field variables of the other nine gravitational field equations of (5.72) must remain ambiguous and they are not examined here.**

...

**The conservation laws of energy and linear momentum state that energy or linear momentum inside any infinitesimal volume of space can change only if non-zero energy or linear momentum, respectively, moves through the surface bounding the volume. We can be confident that our interpretation of (8.56) - (8.57c) (as conservation laws of energy and linear momentum) is correct because we recognize the term, ..., in (8.53) as stored electromagnetic energy density (energy/volume) and because we recognize the components of**

*D*x*B*in (8.55a), (8.55b), and (8.55c) as components of electromagnetic linear-momentum density (linear momentum/volume). Other terms in the expressions for ..., ..., ..., and ... (even though such terms are unfamiliar to us) have dimensions of densities of energy and linear momentum, respectively, thus making the general meaning of each term clear.

**The four ELMAR conditions state that net power (generated power minus dissipated power) and net force (active force minus reactive force) inside any infinitesimal volume of space is zero.** All terms of the

*ELMAR*condition (8.58a) have the dimension of ..., which we recognize as electric power density (dissipated power/volume). Likewise, all terms of the three

*ELMAR*conditions (8.58b), (8.58c), and (8.58d) have the dimension of ..., which we recognize as electromagnetic force density (active force/volume) and which has the same direction as the time derivative of linear-momentum density (linear momentum/time/volume or reactive force/volume). Thus, the

*ELMAR*conditions imply that energy and linear momentum are conserved even though, strictly speaking, the

*ELMAR*conditions cannot be called conservation laws. That the set of

*ELMAR*conditions is closely related to the set of conservation laws is no surprise since either set can be derived from the other set with use of the electromagnetic and kinematic field equations.

### From Chapter 8, Section 8.5 - Wave Equations:

...**The 16 scalar wave equations above, the charge conservation law (8.13), the**

*CAR*condition (8.14), and the four*ELMAR*conditions of (8.58a) - (8.58d) must be satisfied if we wish to describe the detailed structure and behavior of a physical system in cylindrical coordinates.### From Chapter 8, Section 8.6 - Conclusion:

**Having expressed the**

*EKG*equations in cylindrical coordinates, we are now in a position to seek solutions of the field equations and/or the wave equations subject to cylindrical symmetry conditions and boundary conditions yet to be established.The effort expended in obtaining explicit forms of the kinematic wave equations has been justified. We have succeeded in decoupling components of velocity field strengths ... from components of angular-momentum field strengths ..., irrespective of any spatial or temporal symmetry imposed on the system. A close examination of the kinematic field equations of (8.69a) - (8.70e) reveals that, in effect, total decoupling of each individual component of ... and ... from all other components can be accomplished if cylindrical symmetry is imposed on the system, that is, if no field or source component depends on the coordinate ....

**In the next chapter, we review results obtained in Chapter 6 pertaining to the strength of the theory and formulate a mathematical description of an isolated electron spinning on its own axis.**

### From Chapter 9 - Formulation of the Electron Problem:

**In this chapter we proceed with development of a mathematical model for the internal structure of a free electron. We treat the electron as a spinning top with its axis of rotation lying along the z axis in a cylindrical coordinate system. We assume the electron has rotational symmetry and, though spinning, the electron exhibits no translational motion. Thus, all source and field variables describing the electron are taken to be independent of polar angle ... and time**

*t*so that field and source variables depend only on the displacement coordinates,*r*and*z*.We first reexamine the strength of the *EKG* equations of Chapter 6 and determine the exact amount of free data required for a solution of the equations under the symmetry constraints described above.

Next, we specify data for certain components of the source indexed vectors .... The specified data are selected to satisfy the charge conservation law and to simplify the *CAR* condition. Using free data to satisfy the conservation laws of velocity and angular momentum is not required since these laws are identities in components of .... Thus the conservation laws are automatically satisfied.

We then proceed with examination of the eight independent electromagnetic component field equations in conjunction with 16 independent kinematic component field equations. These 24 component field equations fall naturally into four groups, each group consisting of six field equations and containing its own subset of field and source variables. For each of these four groups, the pertinent six field equations are examined, free data is specified, and solutions for the participating components of ..., ** D**,

**, ..., and ... are sought. Source and field components must satisfy not only the electromagnetic and kinematic field equations but they must also satisfy the**

*H**CAR*condition and the four

*ELMAR*conditions.

### From Chapter 9, Section 9.3 - Electromagnetic and Kinematic Field Equations:

The electromagnetic field equations (8.10a) - (8.10d), constitute eight independent 1st-order component differential equations. In our stationary, symmetric system, the four equations containing components of electric field vectors (**and**

*E***) are decoupled completely from the four equations containing components of magnetic field vectors (**

*D***and**

*H***).**

*B*The kinematic field equations, (8.15a) - (8.17d), constitute 24 1st-order component differential equations, only 16 of which are independent. The remaining eight equations can be obtained from the 16 independent ones. We have a certain latitude in selecting the particular 16 equations to identify as independent. Here, we select the 16 independent component equations as those belonging to (8.16a) - (8.17d). A cursory examination of these 16 equations in our stationary, symmetric system reveals that the eight equations containing components of velocity field vectors (... and ...) are decoupled completely from the eight equations containing components of the angular-momentum field vectors (... and ...). In subsequent sections, this decoupling is used to great advantage in our attempt to solve the equations.

The eight electromagnetic field equations and the 16 independent kinematic field equations are intimately coupled. Source terms of the electromagnetic field equations contain components of the source vectors ... while the source terms of the kinematic field equations contain components of both the source vectors ... and the electromagnetic field vectors, ** D** and

**. However, a close examination of the equations reveals that the 8 + 16 = 24 equations fall naturally into four groups, each group consisting of six component field equations. No group contains a field or source component of any other group so that the groups are completely decoupled. In order to distinguish one group from another, we give each group a name that pertains to a significant source quantity appearing in that group. Names of the groups are the**

*H**charge group*, the

*angular momentum group*, the

*velocity group*, and the

*electric current group*. We examine each group separately in sections immediately following.

### From Chapter 9, Section 9.4 - *ELMAR* Conditions:

... This form of the four

*ELMAR*equations, together with the four groups of component field equations and their partial solutions, can be used to derive the conservation laws of energy (8.56) and linear momentum, (8.57a), (8.57b), (8.57c), for the stationary symmetric system. Our ability to obtain the four conservation laws in this manner serves as a check on the accuracy of this form of the

*ELMAR*conditions.

From Chapter 9, Section 9.5 - Conclusion:

In order to make the electron problem reasonably tractable, we require the field and source variables of the free electron model to be independent of polar coordinate ... and time *t*.

...

Analysis presented in Sec. 9.3 reduces the electron problem to finding solutions of six 2nd-order differential equations, (9.28a), (9.28b), (9.43), (9.74), (9.75), and (9.76a), whose source terms contain the variables, ..., ..., ..., ..., ... and ..., respectively. The source variable ... in (9.75) can be found from the solution of (9.76a). We propose to use the four

*ELMAR*conditions and the

*CAR*condition to find the five remaining five source variables, ..., ..., ..., ..., and ....

...

... Satisfaction of the

*CAR*and

*ELMAR*conditions guarantees that ..., ..., ..., ..., and ... are correctly specified.

**A detailed procedure for solving the electron problem with this approach is given in Appendix J.**

**Nonlinearity of the CAR and ELMAR conditions in components of ... provides the glue that holds the electron together.** Nonlinearity of the conditions indicates that superposition does not hold, thereby indicating that multiple solutions of the field equations cannot be added together to yield other solutions.

**We expect only one solution to exist for a given set of boundary conditions.**

Clearly, the energy of the system depends, not only on the electric and magnetic fields as expected, but on the velocity flux density ... and the source vectors, ... and ..., as well.

**The electron can be viewed as a structure containing four clouds of moving particles. Particles of one cloud carry charge while particles of the other three clouds carry angular momentum. No particle has mass or linear momentum. All four clouds are related to one another because all of their particle current densities are linear combinations of derivatives of ... and .... The first two clouds are further related to one another because each of their particle volume densities is constructed from a linear combination of derivatives of ... and .... The last two clouds are further related to one another because each of their particle volume densities is constructed from a derivative of ....**

The first cloud is made up of a set of infinitesimal electric charges, each of which moves on a circle in the plane perpendicular to the *z* axis.

...

The second cloud is made up of a set of infinitesimal spinning tops, each top having its angular momentum directed in the ... direction and each top moving on a circle in a plane perpendicular to the *z* axis.

...

The third cloud is made up of another set of infinitesimal spinning tops, each top having its angular momentum directed in the ... direction and each top moving along a closed contour in the plane ... = ....

...

The fourth cloud is made up of yet another set of infinitesimal spinning tops, each top having its angular momentum directed in the ... direction and each top moving along a closed contour in the plane ... = ....

...

The appearance of ... as a source term in (9.43) supports once again the argument that **spatial curvature produces electromagnetic and kinematic fields in the same sense that spatial curvature produces gravitational fields.**

### From Appendix J - A Procedure for Solving the Electron Problem:

**We briefly describe here a procedure for solving the electron problem formulated in Chapter 9.**

...

**If the computed values, < charge of electron (J.1) >, < rest mass of electron (J.2) >, < angular momentum of electron (J.3) >, and < magnetic moment of electron (J.4) > above agree with experimentally accepted values for the charge of an electron, mass of the electron, Planck's constant, and the magnetic moment of the electron, then we have strong evidence that ..., ..., ..., and ... describe the internal structure of a free electron.**

### Above are excerpts from Dr. Matthews' book.

There are 2 "papers" that I know of that Dr. Matthews published on this topic:

1. "** On the strength of Maxwell's equations**" by N.F.J. Matthews, North Carolina State University, Raleigh, North Carolina 27695-7911

Published in "J. Math. Phys.

**28**(4), April 1987 0022-2488/87/040810-05$02.50 (c)1987 American Institute of Physics"

2. "** The Strength of Einstein's Equations**" by N.F.J. Matthews, North Carolina State University, Raleigh, North Carolina 27695-7911, USA

Published in "

*General Relativity and Gravitation, Vol. 24, No. 1, 1992*0001-7701/92/0100-0017$06.50/0 (c)1992 Plenum Publishing Corporation"

First hand quotes and information directly from Matthews:

1. While teaching an electromagnetics class, "Raise your hands if you do not like details... Shame on you."

2. When I asked him what he thought about Quantum Mechanics, his reply was, "Thou shalt not know." I should have asked a follow up question on this but I took it as meaning that this is not one of the Ten Commandments.

3. When I asked him if he had found the equations that Einstein was looking for, he said, "yes."

4. To his electromagnetics class two days before the first test, "failing to prepare is preparing to fail." I do not know where Matthews got this quote, but I have seen it elsewhere attributed to Benjamin Franklin and John Wooden, but I am not certain of its origin.

5. Matthews said he got the idea of this research topic from the Appendix II of Einstein's book, "The Meaning of Relativity"* where he first learned of Einstein's idea of the "strength" of a set of equations. Matthews turned Einstein's intuition of the "strength" of a set of equations into a rigorous mathematical interpretation, applying it first to Maxwell's Equations, second to Einstein's gravitational equations, and finally to determine a consistent form (a consistent pattern) and set of equations that would be a superset of Maxwell's Equations and Einstein's gravitational equations at the same time. The results being his theory above.

6. To his electromagnetics class, "Symmetry is a powerful argument."

7. To his electromagnetics class, "Science is in a sorry state."

8. To his electromagnetics class, "Geometry is important." Matthews was talking about how changing the geometry of the situation changes the results. For example, kinks in network cables alter the electrical properties in these transmission lines and cause reflections where they otherwise would not be. Twist ties being too tight when bunching cables together cause similar problems.

* Appendix II of "The Meaning of Relativity" by Albert Einstein

ISBN-10: 1567311369

ISBN-13: 978-1567311365

Publisher: Mjf Books; Publication Date: July 1997; Edition: 5

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