Maxwell_equations_from_four-vectors.pdf

A Proposal of an Algebra for Vectors and anApplication to Electromagnetism

Diego Saa 1

Abstract. A new mathematical structure intended to formalize the clas-sical 3D and 4D vectors is briefly described. This structure is offered tothe investigators as a tool that bears the potential of being more ap-propriate, for its use in Physics and science in general, than any of theother mathematical structures of geometric origin, such as the Hamilton(or Pauli or Dirac) quaternions, geometric algebra (GA) and space-timealgebra (STA). The application of this algebra in electromagnetism isdemonstrated, where current concepts are reproduced, in some cases,and modified, in other cases. Several physical variables are proved tosatisfy the wave equation. It is suggested the need of an electromagneticfield scalar, with which Maxwells equations are derived as the resultof a simple four-vector product. As a byproduct, new values and unitsfor the dielectric permittivity and magnetic permeability of vacuum areproposed.

Mathematics Subject Classification (2010). 02.10.De, 03.50.De, 06.20.fa.

Keywords. four-vectors, quaternions, four-vector derivative, electromag-netic theory.

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Contents

1. Introduction 32. Four-vectors 82.1. Advantages of four-vectors 102.2. Complex four-vectors 103. Four-vector algebra 113.1. Sum, difference and conjugates 113.2. Four-vector product 123.3. The Magnitude (Absolute Value) and the Norm 133.4. Unit four-vector 133.5. Identity four-vector 133.6. Multiplicative inverse 144. Four-vector calculus 144.1. Derivatives 154.2. Differential of interval 164.3. Four-Velocity 164.4. Four-gradient 165. Four-vectors in electromagnetism 175.1. Maxwells equations from electromagnetic four-vector 185.2. Electromagnetic four-vector from potential four-vector 225.3. Charge-current four-vector 245.4. Generalized charge-current continuity equations 255.5. Some electromagnetism in classical Physics 255.6. Solution of wave equation 265.7. Covariance of physical laws 275.8. Electromagnetic forces 276. Discussion 29Acknowledgment 30References 31

Algebra for Vectors and an Application 3

1. Introduction

Four-vectors are regarded as the most proper mathematical structure for thehandling of the pervasive four-dimensional variables identified in the Physicsof the twentieth-century.In this paper, a new product is defined for four-vectors, which implies a newalgebra for the handling of four-vectors. It is a new non-associative algebrathat embraces vectors of up to four dimensions that can be extended with-out undue effort to further dimensions. If you are familiarized with vectors,you should find it very easy to work with this new mathematical structure,because it is a rather obvious formalization of vectors.Nevertheless, to the knowledge of the present writer, this algebraic structurehas not been discovered before, despite the utmost and acknowledged impor-tance of vectors. The scalar and vector products (or dot and cross products)have not been defined and operated before with a single integrated and co-herent vector structure comparable to the one proposed here.By endowing vectors with the new product, vectors acquire more proper-ties and extend their use for the easier and correct handling of the four-dimensional physical variables. The new four-vector product reveals both theclassical dot and cross products, when the first element of their operands,called the scalar in quaternionic terminology, is zero. This will be explainedlater.The author had to decide whether the new mathematical structure shouldbe given a new name, or to maintain the classical one, despite the fact thata new mathematical form for the product is attributed to them.The decision of this author has been to preserve the name for the structure,which also preserves the functionality of three-dimensional vectors.The reader will be able to judge that this decision is justified by the factthat the remaining operations and interpretations continue being the sameas what he is used to call simply four-vectors or vectors.In the present paper, the terms vector product, four-vector product orsimply product will be used to refer to the new product. The classical dotand cross products will be named as such. Later it will be shown that thetensor product of a covariant by its corresponding contravariant four-vectorcan be reproduced by carrying out the product of the four-vector by itself,since no distinction is found between the covariant and contravariant formsof a four-vector in orthonormal coordinates.

The rest of this section is essentially a recount of vector history, so thebored reader might wish to skim to the next section to confront our newproposal.The most similar, to four-vectors, mathematical structure proposed to rep-resent the physical variables has been the quaternions.Quaternions were known by Gauss by 1819 or 1820, but unpublished. Theirofficial discovery is attributed to the Irish mathematician William RowanHamilton in 1843 and they have been used for the study of several areas of

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Physics, such as mechanics, electromagnetism, rotations and relativity [1],[2], [3], [4], [5], [6]. James Clerk Maxwell used the quaternion calculus in hisTreatise on Electricity and Magnetism, published in 1873 [7]. An extensivebibliography of more than one thousand references about Quaternions inmathematical physics has been compiled by Gsponer and Hurni [8].

The Americans Gibbs and Heaviside discovered the modern vectors be-tween 1888 and 1894. Their work may be considered a sort of combinationof quaternions and ideas developed around 1840 by the German physicistHermann Grassman. The notation was primarily borrowed from quaternionsbut the geometric interpretation was borrowed from Grassmans system.Hamilton and his followers, such as Tait, considered quaternions as a math-ematical structure with great potential to represent physical variables. Nev-ertheless, they have not lived up to the expectations of physicists.

By the end of the nineteenth century both mathematicians and physi-cists were having difficulty applying quaternions to Physics.The authors of reference [9] explain that quaternions constituted an interme-diate step between a plane geometric calculus (represented by the complexnumbers) and the contemporary vector analysis. They allowed to simplifythe writing of a problem and, under certain conditions, allowed the geomet-ric interpretation of the problem. Their multiplication has two physicallymeaningful products, but, supposedly, the presence of two parts in the samenumber complicated the direct handling of the quaternions.Maxwell, Heaviside, Gibbs and others noted the problems with the quater-nions and began a heated debate, with Tait and other advocates of quater-nions, which by 1894 had largely been settled in favor of modern vectors.Gibbs was acutely aware that quaternionic methods contained the most im-portant pieces of his vector methods. However, in the preface of Gibbs book,Dr. Edwin Wilson affirms that Notwithstanding the efforts which have beenmade during more than a half century to introduce Quaternions into physicsthe fact remains that they have not found wide favor. [10]

Crowe comments that Maxwell in general disliked quaternion methods(as opposed to quaternion ideas); thus for example he was troubled by thenon-homogeneity of the quaternion or full vector product and by the factthat the square of a vector was negative which in the case of velocity vec-tor made the kinetic energy negative. The aspects that Maxwell liked wereclearly brought in his great work on electricity; the aspects he did not likewere indicated only by the fact that Maxwell did not include them. [11]Heaviside was aware of the several difficulties caused by quaternions. Hewrote, for example, Another difficulty is in the scalar product of Quater-nions being always the negative of the quantity practically concerned. Yetanother is the unreal nature of quaternionic formulae [12]. The difficultywas a purely pragmatic one, which Heaviside expressed saying that there


is much more thinking to be done [to set up quaternion equations], for themind has to do what in scalar algebra is done almost mechanically [12].There is great advantage in most practical work in ignoring the quaternionaltogether, and also the double signification of a vector above referred to,and in abolishing the quaternionic minus sign. (Heavisides emphasis) [12].In principle, most everything done with the new system of vectors could bedone with quaternions, but the operations required to make quaternions be-have like vectors added difficulty to using them and provided little benefit tothe physicist. Precisely Crowe quotes the following paragraph attributed toHeaviside: But on proceeding to apply quaternionics to the development ofelectrical theory, I found it very inconvenient. Quaternionics was in its vecto-rial aspects antiphysical and unnatural, and did not harmonise with commonscalar mathematics. So I dropped out the quaternion altogether, and kept topure scalars and vectors, using a very simple vectorial algebra in my papersfrom 1883 onward. [11]

Alexander MacFarlane was one of the debaters and seems to have beenanother of the few in realizing what the real problem with the quaternionswas. MacFarlanes attitude was intermediate - between the position of thedefenders of the GibbsHeaviside system and that of the quaternionists. Hesupported the use of the complete quaternionic product of two vectors, buthe accepted that the scalar part of this product should have a positive sign.According to MacFarlane the equation j k = i was a convention that shouldbe interpreted in a geometrical way, but he did not accept that it implied thenegative sign of the scalar product. [13] (The emphases are mine).He incorrectly attributed the problem to a secondary and superficial mat-ter of representation of symbols, instead of blaming to the more profounddefinition of the quaternion product. MacFarlane credited the controversyconcerning the sign of the scalar product to the conceptual mixture doneby Hamilton and Tait. He made clear that the negative sign came from theuse of the same symbol to represent both a quadrantal versor and a unitaryvector. His view was that different symbols should be used to represent thosedifferent entities. [13] (The emphasis is mine).

At the beginning of the twentieth century, Physics in general, and rel-ativity theory in particular, was lacking the appropriate mathematical for-malism to represent the new physical quantities that were being discovered.But, despite the fact that it was recognized that all physical variables such asspace-time points, velocities, potentials, currents, etc., must be representedwith four values, quaternions were not used to represent and manipulatethem. It was necessary to develop some new mathematical tools in orderto manipulate such variables. Besides vectors, other systems such as tensors,spinors and matrices were developed or used to handle the physical variables.In the course of the twentieth century we have witnessed further efforts toovercome the remaining difficulties, with the development of other algebras,

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which recast several of the ideas of Grassman, Hamilton and Clifford in aslightly different framework. Examples in this direction are Hestenes Geo-metric Algebra in three dimensions and Space Time Algebra in four dimen-sions. [14], [15], [16], [17] [18]

The commutativity of the product was abandoned in all the previousquaternions and in some algebras, such as the one of Clifford. According toGaston Casanova [19] It was the English Clifford who carried out the deci-sive path of abandoning all the commutativity for the vectors but conservingtheir associativity. [19]. Also the Hestenes geometric product conservesassociativity [18]. In this sense, the associativity of the product is finally aban-doned in the four-vector product defined later in the present paper. This is acollateral effect of the proposed algebra, and constitutes a hint about the formthe new four-vectors handle, for example, a sequence of rotations. Besides,the complex numbers are not handled as in the Hamilton quaternions, wherethe real number is situated in the scalar part and the imaginary number inthe vector part. Rather, four-vectors allow that a whole complex number beplaced in each component, so it is possible to have up to four complex num-bers. But, what is more important, it is known that in quantum mechanics,observables do not form an associative algebra, so the present one seems tobe the natural algebra for Physics.

Our intent is to raise the interest in this algebra and try to convince thereader that the presented here is one of the most important mathematicaltools for Physics.

Paraphrasing Martin Erik Horn [2] about quaternions, Having impor-tant consequences for the learning process, the analysis of four-vector rep-resentations of other relativistic relationships should be a further theme ofphysics education research. . . Due to its structural density, the four-vectorrepresentation is without a doubt a more unified theory in comparison to thematrix representation.

I would add that the use of four-vectors allows discerning constants,variables and relations, previously unknown to Physics, which are needed tocomplete and make coherent the theory.

In summary, it has been an old dream to express the laws of Physicswith the use of quaternions. But this attempt has been plagued with recurringpitfalls for reasons until now unknown to both physicists and mathematicians.Quaternions have not been making problem solving easier or simplifying theequations.

I believe that this has been due to an internal problem in the defini-tion of the product of the Hamilton quaternions. With the vector algebraproposed in a previous paper [20] and briefly revised here, the author hopes


that the interest and use will reverse in favor of four-vectors, instead of theHamilton, Pauli or Dirac quaternions, tensors, geometric algebra, spacetimealgebra and other formalisms.

Despite the fact that the original developers of vector theory had iden-tified the difficulties, it is a fact that, after more than one hundred years ofits inception, vector theory has not yet been endowed with the needed four-vector product, comparable in characteristics to the one of quaternions. Thisdeficiency is overcome in the present paper.

In the present paper, the synthesis of all the Maxwell equations whichare equivalent to a simple four-vector product is performed through the de-rivative (four-gradient) of a new electromagnetic four-vector.This is the application of four-vectors to electromagnetism studied in thispaper. It consists in reproducing some known formulas, in particular the fourMaxwell equations, by taking a single four-vector product and in develop-ing other expressions that describe the interaction between charges, currents,potentials and electromagnetic fields. Relevant examples are the derivationsthat show several physical variables satisfying the homogeneous wave equa-tion. In particular, the potentials and the charge-current four-vectors satisfycorresponding dAlembert equations.

Several derivations, mainly based on classical vector algebra, have beenabridged in order to maintain the length of this paper within reasonablelimits. The reader should have no problem in reproducing them with thesuggestions provided.

Take attention of number 3 in section 3.2, where an example of thenon-associative nature of the classical vector cross product is exposed, via anexample.This is well-known, but most of the mathematical tools used in Physics usu-ally handle only associative products. Therefore, the non-associative mathe-matical structure here proposed could be more appropriate for the handlingof vectors.

The investigators might wish to delve into some possible applicationsfor evaluating the real potential of this structure and to help to further itsdevelopment. The present author suggests Euclidean, projective and confor-mal geometries and, within Physics, it would be interesting to explore theLorentz invariance of Maxwells equations, as well as applications in classicalMechanics and even in Relativity, Quantum Mechanics and Diracs theory.

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2. Four-vectors

The present author, in a former paper [20] proposed the mathematical struc-ture used in the present paper. A revision of the basic algebra is performedin this and following sections, in order to maintain this paper self-contained.

The proposal is that four-vectors are four-dimensional numbers of theform:

A = e at + i ax + j ay + k az (1)

or, assuming that the order of the basis elements e, i, j and k is theindicated, then those basis elements can be suppressed and included implicitlyin a notation similar to a vector or 4D point:

A = (at, ax, ay, az) (2)

Four-vectors will be denoted in general with a bold upper-case letter. Three-vectors will be denoted in general with bold lower-case letters.The t, x, y and z, as sub- or super-indexes of the elements, should be inter-preted as the space-time coordinate associated to the respective element ofthe four-vector.

The classical three-dimensional vectors are represented by just the threespatial elements of a four-vector. We will represent them also by using theabbreviated form from expression 2, namely with comma-separated elementsand implicit basis elements i, j and k.Physicists are accustomed to referring to the first element of a four-vector asthe scalar, and to the remaining three elements as the vector part of thefour-vector. For example, the electromagnetic four-potential is conceived ofas constituted of the scalar potential and the vector potential [21]. Wewill also follow such terminology.

Four-vector elements can be any integer, real, imaginary or complexnumbers.

The four basis elements e, i, j and k satisfy the following relations.Theserelations define the four-vector product. For simplicity, the operator for theproduct will not be shown in print, it is represented implicitly by the space be-tween the pair of four-vectors to be multiplied, and must be assumed presentwhenever two four-vectors are separated by a space. The square representsthe product of the element by itself:

e2 = i2 = j2 = k2 = e (3)


Also, the following rules are satisfied by the basis elements:

e i = i e = i, (4)e j = j e = j, (5)

e k = k e = k, (6)i j = j i = k, (7)j k = k j = i, (8)k i = i k = j. (9)

The relations 3 to 9 give an important operational mechanism to reduceany combination of two or more indexes to just one.We will usually make use of this algebra. However, there exists a second,or alternative algebra, where the right-hand side values of the first threerelations 4-6 are positive. That is:

e i = i e = i, (10)e j = j e = j, (11)e k = k e = k. (12)

These properties of the e, i, j, k bases characterize the four-vector prod-uct as noncommutative but, what is more important and different with re-spect to the previous Hamilton and Pauli quaternions as well as to the CliffordAlgebra (see [19], p. 5 axiom 3), the product is in general nonassociative.This means that the order of the products must be given explicitly, by group-ing them with parentheses.As an example where the order is relevant, consider the following productof the four basis elements: ((i e) j) k. With the use of 4, reduce i e to ithen by 7 i j to k and finally, by the last relation 3, k k to e. This isone result. Now consider the same ordering of symbols but with a differentgrouping: (i (e j)) k. First reduce the two middle basis elements with theuse of 5 e j to j, then i j to k and then k k to e, we get thesame result but with the sign changed.

If we put these rules into a multiplication table, for four-vectors theylook like this:

** e i j ke e i j ki i e k jj j k e ik k j i e

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2.1. Advantages of four-vectors

The four-vector operations have extensive applications in electrodynamicsand relativity. Some of the advantages proposed for the Hamilton quaternions,Geometric Algebra and Space-Time Algebra, which should be extended toour new four-vectors, but are not explored here, are:

1. Four-vectors express rotation as a rotation angle about a rotation axis.This is a more natural way to perceive rotation than Euler angles [22].

2. Non singular representation (compared with Euler angles, for example)3. More compact (and faster) than matrices. For computation with rota-

tions, four-vectors offer the advantage of requiring only 4 numbers ofstorage, compared with 9 numbers for orthogonal matrices [23]. Com-position of rotations requires 16 multiplications and 12 additions infour-vector representation, but 27 multiplications and 18 additions inmatrix representation...The four-vector representation is more immuneto accumulated computational error. [23].

4. The real quaternion units defined by Hamilton together with the scalar1 (or rather e in our notation) have the advantage to form a closedfour element group, which is not the case with the Pauli-units [24].

5. Every four-vector formula is a proposition in spherical (sometimes de-grading to plane) trigonometry, and has the full advantage of the sym-metry of the method [25].

6. Unit four-vectors can represent a rotation in 4D space.7. Four-vectors have been introduced because of their all-attitude capa-

bility and numerical advantages in simulation and control [26].

Quaternions have been often used in computer graphics (and associatedgeometric analysis) to represent rotations and orientations of objects in 3Dspace. This chores should be now undertaken by the four-vectors, which aremore natural, and more compact than other representations such as matri-ces. Besides, the operations on them, such as composition, can be computedmore efficiently. Four-vectors, as the previous quaternions, will see uses incontrol theory, signal processing, attitude control, physics, and orbital me-chanics, mainly for representing rotations/orientations in three dimensions.The spacecraft attitude-control systems should be commanded in terms offour-vectors, which should also be used to telemeter their current attitude.The rationale is that combining many four-vector transformations is morenumerically stable than combining many matrix transformations.

2.2. Complex four-vectors

Regularly, four-vectors contain real elements, for example for applications ingeometry. However, the elements handled by complex four-vectors are com-plex numbers.The collection of all complex four-vectors forms a vector space of four com-plex dimensions or eight real dimensions. Combined with the operations ofaddition and multiplication, this collection forms a non-commutative andnon-associative algebra. There is no difficulty in obtaining the multiplicative


inverse of a complex four-vector, when it exists, within four-vector algebrasuggested below. However, there are complex four-vectors whose elements aredifferent from zero but whose norm is zero. Therefore, complex four-vectorsdo not constitute a division algebra.

However, complex four-vectors are very important in the study of elec-tromagnetic fields, as will be seen in the following.

3. Four-vector algebra

A cursory revision of four-vector algebra is performed next. For a more ex-tended analysis of this algebra the reader should refer to a previous paper ofthe present author [20].Let us define two four vectors A and B :

A = eat + iax + jay + kaz

B = ebt + ibx + jby + kbz

3.1. Sum, difference and conjugates

The sum of two four-vectors is another four-vector, where each componenthas the sum of the corresponding argument components.

A + B = e(at + bt) + i(ax + bx) + j(ay + by) + k(az + bz) (13)

The difference of two four-vectors is defined similarly:

A B = e(at bt) + i(ax bx) + j(ay by) + k(az bz). (14)The conjugate of a four-vector changes the signs of the vector part:

A = eat iax jay kaz (15)From this definition it is obvious that the result of summing a four-vector

with its conjugate is another four-vector with only the scalar component dif-ferent from zero. Dividing by two such result, the scalar component is isolated.The previous operation defines the operator named the anti-commutator orthe Hamiltons scalar operator S : (A + A)/2 = SA. Similarly, the result ofsubtracting the conjugate of a four-vector from itself is a pure four-vector(that is, one whose scalar component is equal to zero), when divided by twodefines the commutator or the Hamiltons vector operator V : (AA)/2 = VA

The complex conjugate or Hermitian conjugate of a four-vector changesthe signs of the imaginary parts. Given the complex four-vector:

A = e(at + ibt) + i(ax + ibx) + j(ay + iby) + k(az + iby) (16)

Then its complex conjugate is:

A = e(at ibt) + i(ax ibx) + j(ay iby) + k(az iby) (17)

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3.2. Four-vector product

Using relations 3 to 9, the four-vector product is given by:

A B = e(atbt + axbx + ayby + azbz) + (18)

i (atbx + axbt + aybz azby) +j (atby axbz + aybt + azbx) +k(atbz + axby aybx + azbt).

With the notation of three-dimensional vector analysis it is possible toget a shorthand for the product. Regarding i, j, k as unit vectors in a Cartesiancoordinate system, we interpret a generic four-vector A as comprising thescalar part a and the vector part a = i ax + j ay + k az. Then we writeit in the simplified form A = (a, a). With this notation, the product 18 isexpressed in the compact form:

A B = (a b + a b,a b + a b + a b) (19)The product for the alternative algebra, which uses relations 10-12 in-

stead of 4-6, isA B = (a b + a b, a b a b + a b) (20)

where the usual rules for vector sum and dot and cross products arebeing invoked.Then, the alternative algebra simply switches the signs of the first and sec-ond terms in the vector side of the product and can be computed using theregular product 19, using conjugates: A B.

The following properties for the product are easily established:

1. If the scalar terms of both argument four-vectors of the product arezero then the resulting four-vector contains the classical scalar and vec-tor products in its respective components.

2. The product is non-commutative. So, in general, there exist P and Qsuch that P Q 6= Q P.

3. Four-vector multiplication is non-associative so, in general, for threegiven four-vectors P, Q and R, P (Q R) 6= (P Q) R.Note that this is different from the Hamilton quaternions and the so-called Clifford Algebras, see for example [27]. It reflects the well knownfact that the associative law does not hold for the vector triple product,for which: p (q r) 6= (p q) r. Just to provide an example, forthe case of classical vectors, let us assume the three vectors p=(1,5,2),q=(0,1,0) and r=(1,2,3). Then, the product p (q r) gives (-5,7,-15)whereas the product (p q) r gives (-2,7,-4).

In order to reproduce this result with the use of four-vectors, thescalar terms, if any, must be set to zero before performing the products.


The non-associativity of the product, to account for this property, can-not be found in the quaternions, in geometric or Clifford algebras or inthe standard tensor algebra for four-vectors.

4. The product of a four-vector by itself produces a result different fromzero only in the first or scalar component, which is identified as thenorm of the four-vector. In this sense it is similar to the dot product invector calculus:

A A = (a2t + a2x + a

2y + a

2z, 0, 0, 0) (21)

Note that this expression is substantially different with respect tothe Hamilton quaternions, in which the square of a quaternion is givenby

A A = (a2t v v, 2 atv), (22)where v represents the three-vector terms of the quaternion. Not

only the scalar component has terms with the sign changed, but non-zero term appears in the vector part of the quaternion. This has been asource of difficulty to apply Hamilton quaternions in Physics, which isovercome by our four-vectors.

5. The multiplicative inverse of a four-vector is simply the same four-vectordivided by its norm.

3.3. The Magnitude (Absolute Value) and the Norm

The magnitude, or absolute value, of a four-vector is defined as the squareroot of the sum of squares of its elements:

|A| =a2t + a

2x + a

2y + a

2z (23)

It can be computed as the square root of the scalar component of the productA A.

The norm is defined as the square of the absolute value. It can becomputed as the scalar component of the product A A.

3.4. Unit four-vector

A unit four-vector has the magnitude equal to 1. It is obtained by dividingthe original four-vector by its magnitude or absolute value.

3.5. Identity four-vector

The identity four-vector is a unit four-vector that has the scalar part equal tounity and the vector part equal to zero. Let us denote it with 1 = (1, 0, 0, 0).It has the following properties, where A is any four-vector:

1 A = A, A 1 = A

As you can see, the 1 is the right identity. The alternative product mentionedin section 3.2 makes 1 the left identity.

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3.6. Multiplicative inverse

The multiplicative inverse or simply inverse of a four-vector A is denoted byA1, and evaluated as the vector divided by its norm:

A1 = A/|A|2 (24)The product of the vector by its inverse is the identity four-vector:

A A1 = A1 A = 1

4. Four-vector calculus

We define a four-vector as a set of four quantities, which transform like thecoordinates t, x, y and z. This representation has been enormously successfulin Physics, although, at first sight, it would seem that the time does notmixes with the spatial coordinates, the electrical charges with the currentsor the energy with the momenta.The following Subsections describe several four-vectors that resemble andwork as the corresponding ones in Classical Physics. When differences ap-pear they are duly noticed.In particular, any scientist has to wonder what the effect would be if theremaining physical variables that still do not have the four-vector form, suchas the electric and magnetic fields, were represented as four-vectors. In Sub-sections 5.1 and following, such a proposal is explored. In particular, theelectromagnetic four-vector is proposed, which includes the new scalar fields in combination with the electric and magnetic fields in the vector part ofthe four-vector.This proposal results in a coherent theory that allows deriving the Maxwellsequations, and produces a set of formulas compatible with most of the cor-responding classical ones. When some difference appears, such as in the caseof the so-called Lorenz gauge, where the same Classical Physics has had diffi-culty in proposing just one gauge [28], [29], there appears the electromagneticscalar field in a surprising position. In addition, Classical Physics has not beenable to provide a definition for electrical charge in terms of potentials, and,again, there appears the electromagnetic scalar as a brick for its constructionbut without destroying the known relations, such as the equation for charge-current continuity.In the following Subsections, we will first attempt to reproduce some of thecalculus-based four-vectors. Throughout we try to stick to the (-,+,+,+) sig-nature convention, with a Minkowski metric. In order to verify that all theequations are satisfied, you can begin, for example, with a potential of theform of equation 107. As a practical recommendation, if all the operators,such as gradient, divergence and curl are maintained as shown below, thenthe time derivative of your B function needs to be multiplied by the squareroot of 3 when it appears isolated, such as in Faradays law, 56.


4.1. Derivatives

The time derivative of a four-vector is defined, as is usual for vectors, derivingeach component separately.

The time derivative, d/dt, of a product of two four-vectors has a formsimilar to the conventional derivative of a product, but maintaining the order(in the following formulae it is assumed that the example four-vectors A andB are functions of the variable t):

d

dt(A B) = A

dB

dt+dA

dtB (25)

Derivative of the square of a four-vectorIf in the previous expression we replace B by the A four-vector:

d

dt(A A) = A

dA

dt+dA

dtA (26)

Now if we swap the order of the factors in the last product, we get theconjugate of the other so, adding both, we note that the vector component isset to zero. There remains only the scalar component different from zero. Thesame can be achieved if we derive the (scalar) obtained by first multiplyingA by itself. This proves that the result of the derivative of the square of afour-vector is the same either if the four-vector is first multiplied by itself andthen derived or if the derivative rule of a product is applied before derivingits components. The resulting scalar component is of the form:

AdA

dt+dA

dtA = (2(a a+ b b+ c c+ d d), 0, 0, 0) (27)

Derivative of the product of a four-vector by its inverse:We know that the product of a four-vector A by its inverse A1 is the identityfour-vector, which is a constant. Therefore, in the right-hand side of thederivative we acquire the null four-vector (zero in all components):

d

dt(A A1) =

d

dt(1, 0, 0, 0) = (0, 0, 0, 0) (28)

Or, expanding the derivative of the product:

d

dt(A A1) =

dA

dtA1 + A

dA1

dt= (0, 0, 0, 0) (29)

Example: Given the four-vector

A = (cos(a t), Log(b t2), c/ sinh(t3), d)

The time derivative of A is

dA

dt= (a sin(at), 2/t,3 c t2 coth(t3) csch(t3), 0)

The inverse of A and its derivative are much longer and by reasons of spacecannot be included here. However, the reader can verify equivalence 29 by im-

plementing the product 18 in some system such as Maple or MathematicaR

.

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4.2. Differential of interval

An arbitrary interval differential is expressed as a four-vector in which eachcomponent is the projection of the interval over each coordinate axis. As anexample, in Cartesian coordinates, we define the four-vector dS or intervalfour-vector as follows:

dS = (i c dt, dx, dy, dz) (30)

Where i is the imaginary unit,1, here and in the following equations.

The square of the interval is a relativistic invariant, which appears from theproduct of the interval four-vector by itself:

dS2 = dS dS = c2 dt2 + dx2 + dy2 + dz2 (31)Four-vectors offer this great advantage since there is no difference betweenthe contravariant and covariant forms of a four-vector. The fact is that, whenwe use orthonormal sets of coordinates, both sets of basis vectors, that iscovariant and contravariant, coincide and there is no difference in the repre-sentation of a vector in each of these basis.

4.3. Four-Velocity

In order to obtain the velocity four-vector, just factor out the time coordinatedifferential in the interval four-vector and divide everything by the propertime differential:

U = (i c, x, y, z) (32)

or concisely

U = (i c, v) (33)

Where the factor is the quotient of the coordinate time differential dividedby the proper time differential and in practice can be disregarded for smallvelocities:

=dt

d=

11 v2/c2 (34)

4.4. Four-gradient

We know that the total differential (magnitude df of an arbitrary scalarfield, given as a function of the time and space coordinates) is

df =f

tdt +

f

xdx +

f

ydy +

f

zdz (35)

From this relation we extract the partial derivatives and separate themfrom the interval differential, defined in Subsection 4.2, so that their prod-uct restores the magnitude df . In this way we discover the four-vector appropriate for electromagnetism (later we will conclude that 0 is equal to1/c): { i0 f

t,f

x,f

y,f

z

}= f (36)

we recognize this as the four-gradient of the scalar field f . Notice that thescalar field f is not a four-vector.


In general, if we suppress the scalar field and leave the rest as an emptyoperator, we obtain the four-gradient :

=( i0

t,

x,

y,

z

)(37)

and simplifying:

=( i0

t,) = (i t,) (38)

where the three-dimensional vector del is the important Hamilton operator

= ( x,

y,

z

)(39)

The product of the four-gradient by itself gives the dAlembert operator inthe scalar component of the resulting four-vector. For simplicity, let us writeonly the component different from zero:

= 2 = 202

t2+

2

x2+

2

y2+

2

z2(40)

Or more concisely, using the del () operator:

2 = 202

t2+2 (41)

This dAlembert operator generates wave equations when operating on ascalar field such as the electromagnetic scalar potential, or on a vector field,such as the electric field.

5. Four-vectors in electromagnetism

Vlaenderen and Waser [30] have proposed a scalar component for the elec-tromagnetic field, and exhibit some reasons to justify its experimental need.Note however that they include new terms in Maxwells equations, just asLyttleton and Bondi or the extended Proca equations do [31]. In all thesecases the equations are different from the classical Maxwells ones and theorigin of such terms is not clear.Since the twenty century, it is known that every physical variable should berepresented with four components, one time component and three spatialcomponents. For example, the energy and momentum, the scalar and vectorpotentials, or the charge and the current, constitute and are represented asfour-vectors. However, the physicists have not been able to discover the cor-responding four-vectors for the the electric and magnetic fields, E and B.Therefore, let us try to complete the four-vector by assuming that the elec-tromagnetic four-vector, M, includes the scalar component s, in the followingform:

M = (s ,m) (42)

which, expanding the spatial components, is:

M = (s ,mx,my,mz) (43)

18 Diego Saa 1

where the elements of the vector, or spatial, component m are of the form

mi = i Ei +1

0Bi (44)

Or, by using the magnetic field H, we can make disappear from Maxwellsequations all manifestations of the magnetic permeability 0:

mi = i Ei + Hi (45)

5.1. Maxwells equations from electromagnetic four-vector

We intend to derive the Maxwells equations from the simple four-vectorproduct (four-gradient):

M = 0 (46)

This represents the product of the four-gradient 38 by the electromagneticfour-vector 42.Then, expanding with the four-vector product schema 19:

M =( 0 s

t+ m,s+ i 0 m

t+ m) (47)

or, expanding with 44:

M =( i 0 s

t+ i E + 1

0 B, (48)

0 Et

+ i00

B

t+s+ i E + 1

0 B) (49)

To reach to Maxwells equations let us equate this four-vector to zero. Eachof the real and imaginary components must be equated to zero independently(so it is possible to simplify the imaginary units):

E = 0 st (50)10 B = 0 (51)

E = 00 Bt (52)10 B = 0 Et s (53)

Let us compare these with the well-known Maxwells equations:

Gauss electric field law:

E = 0 , (54)Gauss magnetic field law:

B = 0, (55)Faradays law:

E = Bt , (56)Amperes law:

1 0 B = 0 Et + J (57)

Our intention of generating the Maxwells equations by taking the four-gradient of the electromagnetic four-vector has been almost fulfilled. We


would be done if the set of equations 50-53 had become identical to theset 54-57.Let us try to identify the differences that appeared in the first, third and lastequations, and how to overcome such differences, if possible.

First, in order to unify the last equation of each set, the gradient of thescalar of the electromagnetic field must be equal to the current density, thatis

s = J (58)Not every vector field has a scalar potential; those which do are called conser-vative. And conversely, it is known that if J is any conservative or potentialvector field, and its components have continuous partial derivatives, then ithas a potential with respect to a reference point, of the form J = s.

Therefore, the current density vector J satisfies the conditions of a con-servative field. Besides, from classical vector analysis we know that the curlof any gradient is zero, so the curl of J is zero, J = 0.

Consequently, applying Stokes theorem to this curl, the line integral ofJ around any closed loop is zero:

J dl = 0 (59)

Also, following a similar reasoning as in section 2.3 of Griffiths [32], we canconclude that the electromagnetic scalar between a reference point O and apoint r is independent of the path and given by the line integral:

s(r) = rO

J dl (60)

Second, we want that the first equations of both sets be identical witheach other. This is achieved when the time derivative of the scalar componentof the electromagnetic field, s/t, is made identical to the electric chargedensity, , divided by the square of the permittivity, 0:

s

t= /20 (61)

As will be seen later, all physical variables considered in the presentpaper satisfy the dAlembert wave equation with constant coefficient 20. Thismeans that the propagation speed of the electromagnetic waves, which is thespeed of light, is some function of the absolute permittivity of vacuum, andvice versa.

The dAlembert wave equation with coefficient 20 requires that 0 beinterpreted as equal to the inverse of the speed of light. Now, since in ourtheory 0 is independent from 0, we might preserve, or not, the knownrelation:

c =10 0

= 2.998 108m/s (62)

20 Diego Saa 1

Let us assume that 0 is defined also as equal to the inverse of the speed oflight, so the previous relation is satisfied:

0 = 0 =1

c. (63)

Physicists are aware that the choice of units of many universal constants,such as 0 and 0, is completely arbitrary in current Physics. For example,Prof. Littlejohn of University of California at Berkeley expresses the followingin his lecture notes on Quantum Mechanics: In Gaussian units, the unit ofcharge is defined to make Coulombs law look simple, that is, with a forceconstant equal to 1 (instead of the 1/4pi0 that appears everywhere in SIunits). This leads to a simple rule for translating formulas of electrostatics(without D) from SI to Gaussian units: just replace 1/4pi0 by 1. Thus, thereare no 0s in Gaussian units. There are no 0s either, since these can beexpressed in terms of the speed of light by the relation 00 = 1/c

2. Insteadof 0 and 0s, one sees only factors of c in Gaussian units. [33].The definition of 0, as the inverse of the speed of light, is strictly necessaryfor 0, since the electromagnetic waves displace at the speed of light, but 0does not appear to have such requirement.

Therefore, assuming that 0 = 1/c, let us replace it in the known rela-tion: q2 = 2h 0 c, [34]. This imposes the requirement that the elementarycharge, q, be redefined as equal to the square root of Plancks constant, h,multiplied, for macroscopic applications, by double the fine structure constant:

q =

2 h (64)

Therefore, the dimensions of charge become [M12LT

12 ].

From this relation we can compute the conversion constant, let us name it C,used to convert the electrical units to mechanical units. When this conversionconstant is used, the electrical units, such as the coulomb and ampere, are notanymore indispensable, except for compatibility with previous knowledge:

C =

2 h

q[M

12LT

12 coul1] (65)

The square of C divided by 2 is the von Klitzing constant (about 25812.808ohm). CODATA 2002 defines this constant as independent and has aboutseven digits precision (in 1990 the CIPM adopted exact values for the vonKlitzing constant). With the above mentioned proposal the von Klitzing con-stant is a function of Planck constant and of the value of the elementarycharge. The number of correct digits can be duplicated. Several other con-stants such as the elementary charge, Plancks constant, fine structure con-stant and electron mass, can also be obtained with several additional digitsof precision by making use of the quantum Hall conductance measurements.This paper is not about physical constants so I cannot go farther on this.However, the above hints should be enough for the experts to use profitablyto improve the precision of several constants.


Using equations 50-53 let us prove that the electric and magnetic fieldssatisfy the dAlembert wave equation. In both cases it is necessary to use thefollowing vector identity, for any vector X:

(X) = ( X)2X (66)First, let us take the time derivative of the proposed Faraday equation 52:

2B

t2= 0

0 E

t(67)

Replacing here the definition of E/t obtained from the proposed Ampereequation 53:

2B

t2= 0

0 ( 1

0 0 B + 1

0s) (68)

By vector calculus, the curl of any gradient is always zero, so the second termat the right-hand side vanishes. Applying vector equivalence, 66, and usingGauss equation for the magnetic field, 51, the result 2B = 0 is immediate:

202B

t2+2B = 0 (69)

In a similar form, for the electric field, let us take the curl of the proposedFaradays equation, 52, and use the vector equivalence 66:

( E)2E = 00 B

t(70)

Replacing the divergence of E by its equivalent from the proposed Gaussequation for the electric field, 50, we find the equation:

(0 st

)2E = 00 B

t(71)

Equating this to the time derivative of the proposed Ampere equation, 53,multiplied by 0, the result 2E = 0 is immediate:

202E

t2+2E = 0 (72)

After some standard and very well known operations within the study ofelectrodynamics, we have arrived to the corresponding dAlembert equation(or four-dimensional Laplace equation) for the spatial components, E andB, of the electromagnetic field.

In the following Subsections, similar wave equations are inferred forother physical variables.

22 Diego Saa 1

5.2. Electromagnetic four-vector from potential four-vector

Let us define the potential four-vector as:

A = (i,A) (73)To obtain the rank 2 electromagnetic tensor (Faradays tensor) (here

multiplied by the speed of light), in current Physics it is necessary to carryout the following tensor operations [32]:

F = A A (74)All the signs of the elements of this rank 2 tensor appear in the same or-

der as the one provided in the product for four-vectors, defined in section 3.2:

F =

0 Ex Ey Ez

Ex 0 cBz cByEy cBz 0 cBxEz cBy cBx 0

(75)With four-vectors we defined the simple electromagnetic four-vector 42,

which contains more information than this rank 2 tensor (the four-vector in-cludes information about the scalar component, which is new to Physics, andthe four-vector product 18 generates the correct signs in identical positionsas in Faradays tensor 75).

Therefore, in covariant notation, Faradays tensor should be written as:

F = A A + gs (76)The scalar field s and the electromagnetic fields E and B can be defined

by performing the following four-vector product:

M = A (77)

Replacing the four-gradient by its definition 38, where the inverse of thespeed of light is assumed from now on for both 0 and 0, by relation 63, andexpanding this product with 19 we get

M = (s, i E + cB) =( 1

c

t A,i 1

c

A

t i+A) (78)

Equating the corresponding real and imaginary components of this identity,we find the definition of the electromagnetic scalar in terms of potentials as:

s = 1c

t A (79)

Its form is identical to the Lorenz gauge. This gauge is normally assumedequal to zero in current Physics. However, if this were the case, the electro-magnetic scalar, s, would be zero and, according to the present theory, wewould get the homogeneous Maxwells equations. This means that the Lorenzgauge amounts to assuming that the charge and current densities are zero.Precisely some studies conclude that the Lorenz gauge is unphysical. For ex-ample, in [35], the authors say we see that the Lorenz gauge is in conflict


with the physical phenomena. This is a crash of the Lorenz gauge. Now it isessential to find a new transformation of the equations.The spatial components contain the well known definitions for the electricfield in terms of potentials:

E = 1c

A

t (80)

and of the magnetic field:

c B = A (81)Substituting these definitions into 50-53, they simplify into identities

and wave equations for the potentials and A. Thus, replacing first in Gausselectric field law, 50, we derive the wave equation for the scalar potential :

E = 1c

s

t(82)

( 1c

A

t) = 1

c

t

( 1c

t A) (83)

Both terms containing A are identical since the time and space derivativescan be interchanged. Then

1c22

t2+2 = 0 (84)

Which is the wave equation for the scalar potential, 2 = 0.Now, by substituting the definition of the magnetic field into 51, produces:

(A) = 0 (85)Which is an identity because the divergence of a curl is always zero.Next, let us take Faradays law, 52,

E = Bt

(86)

and replace in it the definitions of the electric and magnetic fields in termsof potentials:

( 1c

A

t) = 1

c

t

(A) (87)The time and space derivatives of the A potential from the first term canbe interchanged and in this way becomes identical to the first term in theright-hand side, being simplified. The curl of a gradient is always zero, so weobtain an identity.Finally, from Amperes law, 53,

c B = 1c

E

ts (88)

by replacing the definitions of the fields in terms of potentials we derive thewave equation for the vector potential :

(A) = 1c

t

( 1c

A

t)( 1

c

t A) (89)

24 Diego Saa 1

Apply the vector identity 66 to the left-hand side term and simplify the twoterms containing :

( A)2A = 1c22A

t2+( A) (90)

Then simplify the terms at the margins, obtaining 2A = 0:

1c22A

t2+2A = 0 (91)

5.3. Charge-current four-vector

Equation 58 defines the current as the gradient of the electromagnetic scalar.We can take the curl of both sides of such equation, with which the left-handside becomes zero because the curl of any gradient is zero. This proves thatthe (local) circulation of current is zero. This is new to Physics.Next we can derive the equation of conservation of charge by obtaining thedivergence of the classical Amperes law, 57, and replacing the divergence ofE by its definition in 54. But, instead of that, let us take the divergence ofour new equation 53. The left-hand side becomes zero because the divergenceof any curl is zero. We obtain:

0 =1

c

t( E) + (s) (92)

Using the Gauss electric field law, 50:

0 =1

c

t(1

c

s

t) + (s) (93)

From here and using equations 61 and 58 it is easy to recover the conservationof charge equation:

J = t

(94)

However, equation (93) is equivalent to this one and constitutes the waveequation for the electromagnetic scalar, 2s = 0:

1

c22s

t22s = 0 (95)

There is no need to continue with these operations. It is more profitable toquestion whether four-vectors can generate this kind of equations. The answeris, of course, in the positive. First, we have to apply the gradient four-vector,38, to the negative of the electromagnetic scalar, s, with which we obtain thecurrent four-vector :

J = (s) = (i 1c

s

t,s) (96)

In other words, again by equations 61 and 58, we arrive at the classic charge-current four-vector

J =(i c, J

)(97)


5.4. Generalized charge-current continuity equations

Next, apply (of course, this means apply the four-vector product) the gra-dient four-vector to the current four-vector just defined:

J =( i 1

c

t,)(i c, J) (98)

The resulting four-vector, equated to zero, produces three equations, whichconstitute the generalized charge-current continuity equations. In particular,the first one is the well known equation for conservation of charge:

t+ J = 0 (99)

1

c

J

t+ c = 0 (100) J = 0 (101)

The second equation can also be derived by taking the gradient of Gausselectric field law, equating with the time derivative of Amperes law, apply-ing the vector equivalence 66 and simplifying the emerging wave equation forthe electric field, which is known to be zero.Similarly, the last expression is a completely reasonable vector equationwithin our theory, since in equations 96 and 97 (or in 58) we had definedthe current (3D) vector as the gradient of the electromagnetic scalar. Re-membering a theorem (see Feynman [21]) of differential calculus of vectorsthat says that the curl of a gradient is zero, the identity is proved.

Finally, taking the time derivative of 99 and subtracting the divergenceof 100 we find that the charge density satisfies the wave equation 2 = 0:

1c22

t2+2 = 0 (102)

Also, take the time derivative of 100, subtract the gradient of 99, applythe vector equivalence 66 and simplify with 101. With which we concludethat the current density satisfies the wave equation 2J = 0:

1c22J

t2+2J = 0 (103)

5.5. Some electromagnetism in classical Physics

Classical electromagnetic theory finds inhomogeneous wave equations forboth potentials and electromagnetic fields, whereas we found homogeneousequations for all physical variables. Jackson, [38] p.246, shows the equationfor the B field:

2B 1c22B

t2= 0

( J) (104)The curl at the right-hand side of this equation should be zero if the

current density, J, constituted a conservative field; since, in such situation, itwould be equal to the gradient of some scalar potential. With this, the curlof a gradient is automatically zero.

26 Diego Saa 1

Jackson, [38] p.246, also shows the equation for the E field:

2E 1c22E

t2= 1

0

( 1c2J

t

)(105)

Whose right-hand side is, again, considered as different from zero inclassical Physics, whereas in our theory it is immediately zero after replac-ing the definitions of charge and current densities in terms of our proposedelectromagnetic scalar.

On the other hand, for the case of the scalar and vector potentials,Jackson, [38] p.240, shows how to uncouple the equations for these variables,through the use of the so called Lorenz condition, which arbitrarily equatesto zero the definition of the electromagnetic scalar of our theory, effectivelyforcing to zero the charge and current densities, which conform the right-handsides of the inhomogeneous equations for potentials.

5.6. Solution of wave equation

Suppose that a function of space and time u(t, x, y, z) satisfies the partialdifferential equation 2u = 0:

1c22u

t2+2u

x2+2u

y2+2u

z2= 0 (106)

where c is a constant with the dimensions of speed. The classical solution ofthis equation is a periodic function. On the other hand, Coulombs law andthe potentials, in the solutions obtained by Lienard and Wiechert, are notperiodic. One might wonder how are we going to reproduce such results. Aslong as the present writer knows, Coulombs law has never been derived fromfirst principles, and even Newtons law of gravitation has the same form and,therefore, it is very probable, and rather obvious, that its origin is also in thewave equation.The answer is that we have to pick the new form as the ansatz of the solution:

u(r, t) =a

t k r (107)This form (for electromagnetic and gravitational potentials), or some

small integer power of it (square for gravitational, electric and magneticforces), avoids the infinities for radius close to zero, and is a promising non-harmonic solution also for problems in other areas of Physics. In particular,the present author proposes this form to describe the law of gravitation.Direct substitution in the wave equation shows that an arbitrary functionu(r, t) = f( tk r), such as the suggested above, or any linear combinationof such solutions, satisfies the wave equation, where is angular velocity, kis a wave vector pointing in any direction, and r is a position vector from anarbitrary origin. The a is an appropriate constant of charge, charge density,amplitude of electric field, etc., depending on the problem being solved.After replacing the proposed solution in the wave equation, the dispersionrelation is obtained:

= c k (108)


5.7. Covariance of physical laws

In section 4.2 it was shown the form of the differential of interval. Now wehave to question whether such form is preserved when a relativistic boost isapplied.Let us assume the usual Lorentz transformations, which define a boost in thex direction:

dt = (dt vdxc2 )/

1 v2c2 , (109)

dx = (dx vdt)/

1 v2c2 , (110)dy = dy, (111)dz = dz. (112)

Then, let us replace these values into the space-time four-vector

dS = (i c dt, dx, dy, dz) (113)and let us compute its square, via the standard four-vector product,

dSdS. The reader can verify that this four-vector product preserves the squareof the interval:

(c2 dt2 + dx2 + dy2 + dz2, 0, 0, 0) (114)This is important for guaranteeing that the four-vector product pre-

serves the covariance of physical laws. As Feynman explains, The fact thatthe Maxwell equations are simple in this particular [four-vector] notation isnot a miracle, because the notation was invented with them in mind. But theinteresting physical thing is that every law of physics[...] must have this sameinvariance under the same transformation. Then when you are moving at auniform velocity in a spaceship, all of the laws of nature transform togetherin such a way that no new phenomenon will show up. It is because the prin-ciple of relativity is a fact of nature that in the notation of four-dimensionalvectors the equations of the world will look simple. [21]

This is the reason why it is of paramount importance to represent allphysical variables with four-vectors and, besides, that the operations overthem should preserve their form. As we have seen, current Physics does notintegrate the electromagnetic fields E and B into a four-vector, as was ex-plained before, because it lacks the equivalent of the time component, whichis represented by our scalar field, s. In the following section we will revealother formula where current Physics has incorrectly dismissed four-vectors.

5.8. Electromagnetic forces

Engelhardt [36] explains that either the Lorentz force, or the field equations,or both must be suitably modified to account for the force on a particle inits rest-frame. It is, of course, well known that the Lorentz force must bemodified anyway to include the effect of radiation damping, when a chargeproduces electromagnetic waves due to strong acceleration. Whether a mod-ification of the Lorentz force alone leaves [Maxwell] equations intact, is an

28 Diego Saa 1

open question. In 1890 Hertz was aware of the fact that the final forms of theforces are not yet found (emphasis in the original).In the following, a new formula is suggested for completing the computationof electromagnetic forces, without affecting the Maxwell equations.Quantum phenomena such as the Aharonov-Bohm effect has received expla-nations appealing to the potentials but not to the electromagnetic fields orthe Lorentz forces: in an ideal experiment, the electron sees no B or E fields,though it does traverse different potentials A and V. [37]In the present paper, both aspects have been improved, with the additionof the scalar electromagnetic field and the additional term for force that isproposed below, as a function of the current (flow of electrons in the A-Beffect).

The forces over a test charge are computed in classical Physics by meansof the classical Lorentz force equation, by multiplying the charge, q, by anexpression reminiscent of the electromagnetic four-vector:

F = q(E + v B

)(115)

As should be clear, the charge does not constitute a four-vector, and the ap-pearance of the velocity vector v is not very natural.

We propose the following formula to compute the forces associated withcurrents in an electromagnetic field, as an extrapolation and correction of theprevious one. It is reached by multiplying the inverse of the speed of light bythe current four-vector and by the electromagnetic four-vector (see Jackson[38], p. 611):

F =1

cJM =

(i c, J

)(s , i E + c B

)(116)

By expanding the indicated product we get:

F =(i( s+

1

cJ E) + J B, i (1

cJ E cB) + (s

cJ + E + JB)) (117)

Whittaker [39], proposed a scalar force. Jackson [38], in page 611, ex-cept for the minor appearance of the speed of light in the last term, showsour second scalar term, that is 1c J E, together with the two classical vectorterms, which appear at the end of (117).Prykarpatsky and Bogolubov [40] and Martins and Pinheiro [41] have ob-tained part of our first real vector term, (s J/c), since our definition of theelectromagnetic scalar in terms of potentials includes the divergence of thevector potential (refer to expression (79) ), which those authors multiply bythe charge and velocity. This is simply recognized as current, J.


6. Discussion

Four-vectors, in the form proposed by the present author, emerge in thispaper as a possibly more appropriate mathematical tool for the handling ofvectors and the study of fundamental physical variables and their describingequations.

This new mathematical structure seems to be a formalization of the clas-sical vectors. Its simplicity contributes to the possibility of more extended andfruitful uses in all branches of science.

As an illustration of such applications, four-vectors have allowed, in thispaper, to identify a new component of the electromagnetic field, which is theelectromagnetic scalar. At the difference of [30, eqs. (57)-(60)], our electro-magnetic scalar does not require to be artificially appended to Maxwellsequations, but constitutes intrinsic part of their derivation and structure.

All the classical physical variables mentioned in the present paper, suchas charge and current densities, scalar and vector potentials, electric, mag-netic and the new electromagnetic scalar fields have been proved here tosatisfy the homogeneous wave equation, which gives a strong argument toconclude that our universe is of a single wave-like constitution.In this paper, the author proposed several wave equations, where some ofthem are new or different to the ones of standard Physics. This is important,but also quite risky for the present theory, since current Physics is quite welltested. However, a new scientific theory, with the aim of being worthy andsuccessful, should really understand and describe how nature works. More-over, it should be falsifiable, and make testable predictions of experimentaloutcomes not yet put to test.In the reader should remain the questioning whether the theory that has beenexposed represents reality or not. To answer this, numerous differences havebeen proposed in this paper with respect to the current accepted models,which should make it easy for the physicists to locate the discrepancies, andreject the theory if points are found where it is inconsistent with experiment.As with every theory, it is up to the practitioners, in this case the physicistsand mathematicians, to locate the failures or problems, if any.The reproduction of many known equations, some of them rather complex,such as all the Maxwells equations, originated on simple four-vector prod-ucts, provides a very strong reassurance in favor of the proposed vector alge-bra as one of the most correct and powerful mathematical tools to apply inPhysics.

The periodic solutions of the wave equation are well known, but thenon-periodic solutions are not known, or have been ignored, despite theirimportance to Physics. The classical Coulomb and Newtonian equations forthe electrostatic and gravitational forces and potentials seem to be traceable

30 Diego Saa 1

to the wave equation, which is rather natural after acceptance of equation64 for the conversion between mass and charge. The present author has pos-tulated the non-periodic solutions of the dAlembert equation as the correctexpressions for these phenomena.

The reader should have noticed that the gradient of several four-vectors,such as the electromagnetic and the current four-vectors, as well as all thewave equations, were equated to zero in order to generate the Maxwells equa-tions and others. Why should it be so? The present author does not have theanswer. The question should be posited to Nature and is left to the readerto discover. This problem has the signature of the situations mentioned inSubsection 2.2, according to which some complex four-vectors may have zeromagnitude despite the fact that they are non-zero. Feynman [11] in chapter25 concludes that All of the laws of physics can be contained in one equa-tion. That equation is U = 0 [his emphasis]. Our statement is stricter, in thesense that the relations satisfied by the physical variables are not arbitrarybut all of them are just standard linear wave equations (dAlembert equation).

The proposal of changing the existing definitions of the dielectric permit-tivity and magnetic permeability of vacuum appears as too radical. However,with the proviso that the theory proposed in this paper is correct and evenwithout it, the new definitions allow to simplify and simultaneously preservethe coherence of all Physics.

A proposal was given to dismiss the classical electromagnetic units calledcoulomb and ampere. Therefore, the mechanical units, kilogram, meterand second, are the only ones that remain as required to study Physics ingeneral and electromagnetism in particular.

Acknowledgment

The author wishes to express his gratitude to Dr. Delbert Larson for perform-ing a comprehensive and exhaustive revision of the paper and for providingmultiple suggestions for improvement. Also, thanks to Dr. Cesar Costa ofEscuela Politecnica Nacional, who provided some important observations. Ofcourse, any remaining errors are full responsibility of the present author.Finally my deep thanks to Dr. Bertfried Fauser, who kindly provided therespective endorsement to a slightly different version of the present paper, inorder that it be published in ArXiv, no matter if it was later suppressed pub-lication by anonymous managers of such Cornell web site, without providingany explanation.


References

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[2] Martin Erik Horn. Quaternions in university-level physics considering specialrelativity. Arxiv preprint physics/0308017, 2003.

[3] S. de Leo. Quaternions and special relativity. Journal of Mathematical Physics,37:295568, June 1996.

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[41] Alexandre A. Martins and Mario J. Pinheiro. On the electromagnetic ori-gin of inertia and inertial mass. International Journal of Theoretical Physics,47(10):27062715, 2008.

Diego Saa 11 Emeritus. Departamento de Ciencias de Informacion y Computacion, EscuelaPolitecnica Nacional, Ladron de Guevara E11-253, Quito Ecuador. Tel. (593-2)2567-849.e-mail: [email protected]

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