A Proposal of an Algebra for Vectors and an Application to Electromagnetism

Diego Sa´a 1

Abstract. A new mathematical structure intended to formalize the classical 3D and 4D vectors is briefly described. This structure is offered to the investigators as a tool that bears the potential of being more appropriate, for its use in Physics and science in general, than any of the other mathematical structures of geometric origin, such as the Hamilton (or Pauli or Dirac) quaternions, geometric algebra (GA) and space-time algebra (STA). The application of this algebra in electromagnetism is demonstrated, where current concepts are reproduced, in some cases, and modified, in other cases. Several physical variables are proved to satisfy the wave equation. It is suggested the need of an electromagnetic field scalar, with which Maxwell’s equations are derived as the result of a simple four-vector product. As a byproduct, new values and units for the dielectric permittivity and magnetic permeability of vacuum are proposed. Mathematics Subject Classification (2010). 02.10.De, 03.50.De, 06.20.fa. Keywords. four-vectors, quaternions, four-vector derivative, electromagnetic theory. 2 Diego Saá 1

Contents 1. Introduction 3 2. Four-vectors 8 2.1. Advantages of four-vectors 10 2.2. Complex four-vectors 10 3. Four-vector algebra 11 3.1. Sum, diﬀerence and conjugates 11 3.2. Four-vector product 12 3.3. The Magnitude (Absolute Value) and the Norm 13 3.4. Unit four-vector 13 3.5. Identity four-vector 13 3.6. Multiplicative inverse 14 4. Four-vector calculus 14 4.1. Derivatives 15 4.2. Diﬀerential of interval 16 4.3. Four-Velocity 16 4.4. Four-gradient 16 5. Four-vectors in electromagnetism 17 5.1. Maxwell’s equations from electromagnetic four-vector 18 5.2. Electromagnetic four-vector from potential four-vector 22 5.3. Charge-current four-vector 24 5.4. Generalized charge-current continuity equations 25 5.5. Some electromagnetism in classical Physics 25 5.6. Solution of wave equation 26 5.7. Covariance of physical laws 27 5.8. Electromagnetic forces 27 6. Discussion 29 Acknowledgment 30 References 31 Algebra for Vectors and an Application 3

1. Introduction

Four-vectors are regarded as the most proper mathematical structure for the handling of the pervasive four-dimensional variables identified in the Physics of the twentieth-century. In this paper, a new product is defined for four-vectors, which implies a new algebra for the handling of four-vectors. It is a new non-associative algebra that embraces vectors of up to four dimensions that can be extended without 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 structure has not been discovered before, despite the utmost and acknowledged importance of vectors. The scalar and vector products (or dot and cross products) have not been defined and operated before with a single integrated and coherent vector structure comparable to the one proposed here. By endowing vectors with the new product, vectors acquire more properties and extend their use for the easier and correct handling of the four- dimensional physical variables. The new four-vector product reveals both the classical dot and cross products, when the first element of their operands, called the “scalar” in quaternionic terminology, is zero. This will be explained later. The author had to decide whether the new mathematical structure should be given a new name, or to maintain the classical one, despite the fact that a 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 fact that the remaining operations and interpretations continue being the same as what he is used to call simply “four-vectors” or “vectors”. In the present paper, the terms “vector product”, “four-vector product” or simply “product” will be used to refer to the new product. The classical dot and cross products will be named as such. Later it will be shown that the tensor product of a covariant by its corresponding contravariant four-vector can be reproduced by carrying out the product of the four-vector by itself, since no distinction is found between the covariant and contravariant forms of a four-vector in orthonormal coordinates.

The rest of this section is essentially a recount of vector history, so the bored reader might wish to skim to the next section to confront our new proposal. The most similar, to four-vectors, mathematical structure proposed to represent the physical variables has been the quaternions. Quaternions were known by Gauss by 1819 or 1820, but unpublished. Their oﬃcial discovery is attributed to the Irish mathematician William Rowan Hamilton in 1843 and they have been used for the study of several areas of 4 Diego Sa´a 1

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

The Americans Gibbs and Heaviside discovered the modern vectors between 1888 and 1894. Their work may be considered a sort of combination of quaternions and ideas developed around 1840 by the German physicist Hermann Grassman. The notation was primarily borrowed from quaternions but the geometric interpretation was borrowed from Grassman’s system. Hamilton and his followers, such as Tait, considered quaternions as a mathematical 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 physicists were having difficulty applying quaternions to Physics. The authors of reference [9] explain that quaternions constituted an intermediate step between a plane geometric calculus (represented by the complex numbers) and the contemporary vector analysis. They allowed to simplify the writing of a problem and, under certain conditions, allowed the geometric interpretation of the problem. Their multiplication has two physically meaningful products, but, supposedly, the presence of two parts in the same number complicated the direct handling of the quaternions. Maxwell, Heaviside, Gibbs and others noted the problems with the quaternions and began a heated debate, with Tait and other advocates of quaternions, which by 1894 had largely been settled in favor of modern vectors. Gibbs was acutely aware that quaternionic methods contained the most important pieces of his vector methods. However, in the preface of Gibbs’ book, Dr. Edwin Wilson affirms that “Notwithstanding the efforts which have been made during more than a half century to introduce Quaternions into physics the 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 the non-homogeneity of the quaternion or full vector product and by the fact that the square of a vector was negative which in the case of velocity vector made the kinetic energy negative. The aspects that Maxwell liked were clearly brought in his great work on electricity; the aspects he did not like were indicated only by the fact that Maxwell did not include them.” [11] Heaviside was aware of the several difficulties caused by quaternions. He wrote, for example, “Another difficulty is in the scalar product of Quater- nions being always the negative of the quantity practically concerned. Yet another is the unreal nature of quaternionic formulae” [12]. The difficulty was a purely pragmatic one, which Heaviside expressed saying that “there Algebra for Vectors and an Application 5 is much more thinking to be done [to set up quaternion equations], for the mind has to do what in scalar algebra is done almost mechanically” [12]. “There is great advantage in most practical work in ignoring the quaternion altogether, and also the double signification of a vector above referred to, and in abolishing the quaternionic minus sign.” (Heaviside’s emphasis) [12]. In principle, most everything done with the new system of vectors could be done with quaternions, but the operations required to make quaternions be- have like vectors added difficulty to using them and provided little benefit to the physicist. Precisely Crowe quotes the following paragraph attributed to Heaviside: “But on proceeding to apply quaternionics to the development of electrical theory, I found it very inconvenient. Quaternionics was in its vectorial aspects antiphysical and unnatural, and did not harmonise with common scalar mathematics. So I dropped out the quaternion altogether, and kept to pure scalars and vectors, using a very simple vectorial algebra in my papers from 1883 onward.” [11]

Alexander MacFarlane was one of the debaters and seems to have been another of the few in realizing what the real problem with the quaternions was. “MacFarlane’s attitude was intermediate - between the position of the defenders of the Gibbs–Heaviside system and that of the quaternionists. He supported the use of the complete quaternionic product of two vectors, but he 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 should be interpreted in a geometrical way, but he did not accept that it implied the negative sign of the scalar product”. [13] (The emphases are mine). He incorrectly attributed the problem to a secondary and superficial matter of representation of symbols, instead of blaming to the more profound definition of the quaternion product. “MacFarlane credited the controversy concerning the sign of the scalar product to the conceptual mixture done by Hamilton and Tait. He made clear that the negative sign came from the use of the same symbol to represent both a quadrantal versor and a unitary vector. His view was that different symbols should be used to represent those different entities.” [13] (The emphasis is mine).

At the beginning of the twentieth century, Physics in general, and relativity 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 as space-time points, velocities, potentials, currents, etc., must be represented with four values, quaternions were not used to represent and manipulate them. It was necessary to develop some new mathematical tools in order to 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 to overcome the remaining difficulties, with the development of other algebras, 6 Diego Saá 1 which recast several of the ideas of Grassman, Hamilton and Clifford in a slightly different framework. Examples in this direction are Hestenes’ Geo- metric Algebra in three dimensions and Space Time Algebra in four dimensions. [14], [15], [16], [17] [18]

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

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

Paraphrasing Martin Erik Horn [2] about quaternions, “Having important consequences for the learning process, the analysis of four-vector representations of other relativistic relationships should be a further theme of physics education research. . . Due to its structural density, the four-vector representation is without a doubt a more uniﬁed theory in comparison to the matrix representation.”

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

In summary, it has been an old dream to express the laws of Physics with the use of quaternions. But this attempt has been plagued with recurring pitfalls for reasons until now unknown to both physicists and mathematicians. Quaternions have not been making problem solving easier or simplifying the equations. 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 algebra proposed in a previous paper [20] and brieﬂy revised here, the author hopes Algebra for Vectors and an Application 7 that the interest and use will reverse in favor of four-vectors, instead of the Hamilton, Pauli or Dirac quaternions, tensors, geometric algebra, space–time algebra and other formalisms.

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

In the present paper, the synthesis of all the Maxwell equations –which are equivalent to a simple four-vector product– is performed through the derivative (four-gradient) of a new electromagnetic four-vector. This is the application of four-vectors to electromagnetism studied in this paper. It consists in reproducing some known formulas, in particular the four Maxwell 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 derivations that show several physical variables satisfying the homogeneous wave equation. In particular, the potentials and the charge-current four-vectors satisfy corresponding d’Alembert equations.

Several derivations, mainly based on classical vector algebra, have been abridged in order to maintain the length of this paper within reasonable limits. The reader should have no problem in reproducing them with the suggestions provided.

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

The investigators might wish to delve into some possible applications for evaluating the real potential of this structure and to help to further its development. The present author suggests Euclidean, projective and confor- mal geometries and, within Physics, it would be interesting to explore the Lorentz invariance of Maxwell’s equations, as well as applications in classical Mechanics and even in Relativity, Quantum Mechanics and Dirac’s theory. 8 Diego Sa´a 1

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

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

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 the indicated, then those basis elements can be suppressed and included implicitly in 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 interpreted as the space-time coordinate associated to the respective element of the four-vector.

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

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

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

e2 = i2 = j2 = k2 = e (3) Algebra for Vectors and an Application 9

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 reduce any 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 three relations 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 product as non–commutative but, what is more important and different with respect to the previous Hamilton and Pauli quaternions as well as to the Clifford Algebra (see [19], p. 5 axiom 3), the product is in general non–associative. This means that the order of the products must be given explicitly, by grouping them with parentheses. As an example where the order is relevant, consider the following product of the four basis elements: “((i e) j) k”. With the use of 4, reduce “i e” to i then by 7 “i j” to k and finally, by the last relation 3, “k k” to e. This is one result. Now consider the same ordering of symbols but with a different grouping: “(i (e j)) k”. First reduce the two middle basis elements with the use of 5 “e j” to −j, then “−i j” to −k and then “−k k” to −e, we get the same result but with the sign changed.

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

** e i j k e e –i –j –k i i e k –j j j –k e i k k j –i e 10 Diego Sa´a 1

2.1. Advantages of four-vectors The four-vector operations have extensive applications in electrodynamics and relativity. Some of the advantages proposed for the Hamilton quaternions, Geometric Algebra and Space-Time Algebra, which should be extended to our 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 rotations, four-vectors offer the advantage of requiring only 4 numbers of storage, compared with 9 numbers for orthogonal matrices [23]. Com- position of rotations requires 16 multiplications and 12 additions in four-vector representation, but 27 multiplications and 18 additions in matrix representation...The four-vector representation is more immune to accumulated computational error. [23]. 4. The real quaternion units defined by Hamilton together with the scalar “1” (or rather “e” in our notation) have the advantage to form a closed four 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 associated geometric analysis) to represent rotations and orientations of objects in 3D space. This chores should be now undertaken by the four-vectors, which are more natural, and more compact than other representations such as matrices. Besides, the operations on them, such as composition, can be computed more efficiently. Four-vectors, as the previous quaternions, will see uses in control theory, signal processing, attitude control, physics, and orbital mechanics, mainly for representing rotations/orientations in three dimensions. The spacecraft attitude-control systems should be commanded in terms of four-vectors, which should also be used to telemeter their current attitude. The rationale is that combining many four-vector transformations is more numerically stable than combining many matrix transformations.

2.2. Complex four-vectors Regularly, four-vectors contain real elements, for example for applications in geometry. However, the elements handled by complex four-vectors are complex numbers. The collection of all complex four-vectors forms a vector space of four complex dimensions or eight real dimensions. Combined with the operations of addition and multiplication, this collection forms a non-commutative and non-associative algebra. There is no diﬃculty in obtaining the multiplicative Algebra for Vectors and an Application 11 inverse of a complex four-vector, when it exists, within four-vector algebra suggested below. However, there are complex four-vectors whose elements are diﬀerent from zero but whose norm is zero. Therefore, complex four-vectors do not constitute a division algebra.

However, complex four-vectors are very important in the study of electromagnetic 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 extended analysis of this algebra the reader should refer to a previous paper of the 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 component has 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 different from zero. Dividing by two such result, the scalar component is isolated. The previous operation defines the operator named the anti-commutator or the Hamilton’s scalar operator S:(A + A)/2 = SA. Similarly, the result of subtracting 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 two defines the commutator or the Hamilton’s vector operator V :(A−A)/2 = V A

The complex conjugate or Hermitian conjugate of a four-vector changes the 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) 12 Diego Sa´a 1

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

AB = 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 to get a shorthand for the product. Regarding i, j, k as unit vectors in a Cartesian coordinate system, we interpret a generic four-vector A as comprising the scalar part a and the vector part a = i ax + j ay + k az. Then we write it in the simpliﬁed form A = (a, a). With this notation, the product 18 is expressed in the compact form: AB = (a b + a · b, −a b + a b + a × b) (19) The product for the alternative algebra, which uses relations 10-12 instead of 4-6, is AB = (a b + a · b, a b − a b + a × b) (20) where the usual rules for vector sum and dot and cross products are being invoked. Then, the alternative algebra simply switches the signs of the ﬁrst and second terms in the vector side of the product and can be computed using the regular 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 are zero then the resulting four-vector contains the classical scalar and vector products in its respective components.

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

3. Four-vector multiplication is non-associative so, in general, for three given four-vectors P, Q and R, P (QR) 6= (PQ) R. Note that this is different from the Hamilton quaternions and the so- called Clifford Algebras, see for example [27]. It reflects the well known fact 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, for the 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, the scalar terms, if any, must be set to zero before performing the products. Algebra for Vectors and an Application 13

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

4. The product of a four-vector by itself produces a result different from zero only in the first or “scalar” component, which is identified as the norm of the four-vector. In this sense it is similar to the dot product in vector calculus: 2 2 2 2 AA = (at + ax + ay + az, 0, 0, 0) (21) Note that this expression is substantially different with respect to the Hamilton quaternions, in which the square of a quaternion is given by 2 AA = (at − 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 a source of difficulty to apply Hamilton quaternions in Physics, which is overcome by our four-vectors.

5. The multiplicative inverse of a four-vector is simply the same four-vector divided 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 square root of the sum of squares of its elements: q 2 2 2 2 |A| = at + ax + ay + az (23) It can be computed as the square root of the scalar component of the product AA.

The norm is deﬁned as the square of the absolute value. It can be computed as the scalar component of the product AA. 3.4. Unit four-vector A unit four-vector has the magnitude equal to 1. It is obtained by dividing the 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 to unity 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 mentioned in section 3.2 makes 1 the ‘left identity’. 14 Diego Sa´a 1

3.6. Multiplicative inverse The multiplicative inverse or simply inverse of a four-vector A is denoted by A−1, and evaluated as the vector divided by its norm:

A−1 = A/|A|2 (24)

The product of the vector by its inverse is the identity four-vector:

AA−1 = A−1 A = 1

4. Four-vector calculus We define a four-vector as a set of four quantities, which transform like the coordinates t, x, y and z. This representation has been enormously successful in Physics, although, at first sight, it would seem that the time does not mixes with the spatial coordinates, the electrical charges with the currents or the energy with the momenta. The following Subsections describe several four-vectors that resemble and work as the corresponding ones in Classical Physics. When differences appear they are duly noticed. In particular, any scientist has to wonder what the effect would be if the remaining physical variables that still do not have the four-vector form, such as the electric and magnetic fields, were represented as four-vectors. In Sub- sections 5.1 and following, such a proposal is explored. In particular, the electromagnetic four-vector is proposed, which includes the new scalar field s in combination with the electric and magnetic fields in the vector part of the four-vector. This proposal results in a coherent theory that allows deriving the Maxwell’s equations, and produces a set of formulas compatible with most of the corresponding classical ones. When some difference appears, such as in the case of the so-called Lorenz gauge, where the same Classical Physics has had difficulty in proposing just one gauge [28], [29], there appears the electromagnetic scalar field in a surprising position. In addition, Classical Physics has not been able to provide a definition for electrical charge in terms of potentials, and, again, there appears the electromagnetic scalar as a brick for its construction but 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 the calculus-based four-vectors. Throughout we try to stick to the (-,+,+,+) signature convention, with a Minkowski metric. In order to verify that all the equations are satisfied, you can begin, for example, with a potential of the form of equation 107. As a practical recommendation, if all the operators, such as gradient, divergence and curl are maintained as shown below, then the time derivative of your B function needs to be multiplied by the square root of 3 when it appears isolated, such as in Faraday’s law, 56. Algebra for Vectors and an Application 15

4.1. Derivatives The time derivative of a four-vector is deﬁned, as is usual for vectors, deriving each component separately.

The time derivative, d/dt, of a product of two four-vectors has a form similar to the conventional derivative of a product, but maintaining the order (in the following formulae it is assumed that the example four-vectors A and B are functions of the variable t): d dB dA (AB) = A + B (25) dt dt dt Derivative of the square of a four-vector If in the previous expression we replace B by the A four-vector: d dA dA (AA) = A + A (26) dt dt dt Now if we swap the order of the factors in the last product, we get the conjugate of the other so, adding both, we note that the vector component is set to zero. There remains only the scalar component different from zero. The same can be achieved if we derive the (scalar) obtained by first multiplying A by itself. This proves that the result of the derivative of the square of a four-vector is the same either if the four-vector is first multiplied by itself and then derived or if the derivative rule of a product is applied before deriving its components. The resulting scalar component is of the form: dA dA A + A = (2(a a˙ + b b˙ + c c˙ + d d˙), 0, 0, 0) (27) dt dt Derivative of the product of a four-vector by its inverse: We know that the product of a four-vector A by its inverse A−1 is the identity four-vector, which is a constant. Therefore, in the right-hand side of the derivative we acquire the null four-vector (zero in all components): d d (AA−1) = (1, 0, 0, 0) = (0, 0, 0, 0) (28) dt dt Or, expanding the derivative of the product: d dA dA−1 (AA−1) = A−1 + A = (0, 0, 0, 0) (29) dt dt dt Example: Given the four-vector A = (cos(a t), Log(b t2), c/ sinh(t3), d) The time derivative of A is dA = (−a sin(at), 2/t, −3 c t2 coth(t3) csch(t3), 0) dt The inverse of A and its derivative are much longer and by reasons of space cannot be included here. However, the reader can verify equivalence 29 by im- R plementing the product 18 in some system such as Maple or Mathematica . 16 Diego Saá 1

4.2. Differential of interval An arbitrary interval differential is expressed as a four-vector in which each component is the projection of the interval over each coordinate axis. As an example, in Cartesian coordinates, we define the four-vector dS or interval four-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 the product 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 between the contravariant and covariant forms of a four-vector. The fact is that, when we use orthonormal sets of coordinates, both sets of basis vectors, that is covariant and contravariant, coincide and there is no difference in the representation 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 coordinate differential in the interval four-vector and divide everything by the proper time differential: U = γ(i c, x,˙ y,˙ z˙) (32) or concisely U = γ(i c, v) (33) Where the γ factor is the quotient of the coordinate time differential divided by the proper time differential and in practice can be disregarded for small velocities: dt 1 γ = = (34) dτ p1 − v2/c2 4.4. Four-gradient We know that the total differential (magnitude “df ” of an arbitrary scalar field, given as a function of the time and space coordinates) is

∂f ∂f ∂f ∂f df = dt + dx + dy + dz (35) ∂t ∂x ∂y ∂z From this relation we extract the partial derivatives and separate them from the interval differential, defined in Subsection 4.2, so that their product restores the magnitude df . In this way we discover the four-vector ∂ appropriate for electromagnetism (later we will conclude that 0 is equal to 1/c): ∂f ∂f ∂f ∂f − i , , , = ∂f (36) 0 ∂t ∂x ∂y ∂z we recognize this as the four-gradient of the scalar field “f ”. Notice that the scalar field f is not a four-vector. Algebra for Vectors and an Application 17

In general, if we suppress the scalar ﬁeld and leave the rest as an empty operator, we obtain the four-gradient: ∂ ∂ ∂ ∂ ∂ = − i , , , (37) 0 ∂t ∂x ∂y ∂z and simplifying: ∂ ∂ = − i , ∇ = (−i ∂ , ∇) (38) 0 ∂t t where the three-dimensional vector “del” is the important Hamilton operator ∂ ∂ ∂ ∇ = , , (39) ∂x ∂y ∂z The product of the four-gradient by itself gives the d’Alembert operator in the scalar component of the resulting four-vector. For simplicity, let us write only the component diﬀerent from zero:

2 2 2 2 2 2 ∂ ∂ ∂ ∂ ∂∂ = = −0 + + + (40) ∂t2 ∂x2 ∂y2 ∂z2 Or more concisely, using the “del” (∇) operator: 2 2 2 ∂ 2 = −0 + ∇ (41) ∂t2 This d’Alembert operator generates wave equations when operating on a scalar 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 electromagnetic field, and exhibit some reasons to justify its experimental need. Note however that they include new terms in Maxwell’s equations, just as Lyttleton and Bondi or the extended Proca equations do [31]. In all these cases the equations are different from the classical Maxwell’s ones and the origin of such terms is not clear. Since the twenty century, it is known that every physical variable should be represented with four components, one time component and three spatial components. For example, the energy and momentum, the scalar and vector potentials, or the charge and the current, constitute and are represented as four-vectors. However, the physicists have not been able to discover the corresponding 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 electromagnetic four-vector, M, includes the scalar component s, in the following form: M = (s , m) (42) which, expanding the spatial components, is: M = (s , mx, my, mz) (43) 18 Diego Saá 1 where the elements of the vector, or spatial, component m are of the form 1 mi = i Ei + Bi (44) µ0 Or, by using the magnetic field H, we can make disappear from Maxwell’s equations all manifestations of the magnetic permeability µ0: mi = i Ei + Hi (45) 5.1. Maxwell’s equations from electromagnetic four-vector We intend to derive the Maxwell’s equations from the simple four-vector product (four-gradient): ∂ M = 0 (46) This represents the product of the four-gradient 38 by the electromagnetic four-vector 42. Then, expanding with the four-vector product schema 19: ∂s ∂m ∂ M = − + ∇ · m, ∇s + i + ∇ × m (47) 0 ∂t 0 ∂t or, expanding with 44: ∂s 1 ∂ M = − i 0 + i ∇ · E + ∇ · B, (48) ∂t µ0 ∂E 0 ∂B 1 − 0 + i + ∇s + i ∇ × E + ∇ × B (49) ∂t µ0 ∂t µ0 To reach to Maxwell’s equations let us equate this four-vector to zero. Each of the real and imaginary components must be equated to zero independently (so it is possible to simplify the imaginary units): ∂s ∇ · E = 0 ∂t (50) 1 ∇ · B = 0 (51) µ0 ∇ × E = − 0 ∂B (52) µ0 ∂t 1 ∂E ∇ × B = 0 − ∇s (53) µ0 ∂t Let us compare these with the well-known Maxwell’s equations: Gauss’ electric field law: ρ ∇ · E = 0 , (54) Gauss’ magnetic field law: ∇ · B = 0, (55) Faraday’s law: ∂B ∇ × E = − ∂t , (56) Ampere’s law: 1 ∂E ∇ × B = 0 + J (57) µ 0 ∂t Our intention of generating the Maxwell’s equations by taking the four- gradient of the electromagnetic four-vector has been almost fulfilled. We Algebra for Vectors and an Application 19 would be done if the set of equations 50-53 had become identical to the set 54-57. Let us try to identify the differences that appeared in the first, third and last equations, and how to overcome such differences, if possible.

First, in order to unify the last equation of each set, the gradient of the scalar of the electromagnetic field must be equal to the current density, that is ∇s = −J (58) Not every vector field has a scalar potential; those which do are called conservative. And conversely, it is known that if J is any conservative or potential vector field, and its components have continuous partial derivatives, then it has a potential with respect to a reference point, of the form J = −∇s.

Therefore, the current density vector J satisfies the conditions of a conservative field. Besides, from classical vector analysis we know that the curl of any gradient is zero, so the curl of J is zero, ∇ × J = 0. Consequently, applying Stokes’ theorem to this curl, the line integral of J around any closed loop Γ is zero: I J · dl = 0 (59) Γ Also, following a similar reasoning as in section 2.3 of Griffiths [32], we can conclude that the electromagnetic scalar between a reference point O and a point r is independent of the path and given by the line integral: Z r s(r) = − J · dl (60) O Second, we want that the first equations of both sets be identical with each other. This is achieved when the time derivative of the scalar component of the electromagnetic field, ∂s/∂t, is made identical to the electric charge density, ρ, divided by the square of the permittivity, 0: ∂s = ρ /2 (61) ∂t 0 As will be seen later, all physical variables considered in the present 2 paper satisfy the d’Alembert wave equation with constant coefficient 0. This means that the propagation speed of the electromagnetic waves, which is the speed of light, is some function of the absolute permittivity of vacuum, and vice versa. 2 The d’Alembert wave equation with coefficient 0 requires that 0 be interpreted as equal to the inverse of the speed of light. Now, since in our theory µ0 is independent from 0, we might preserve, or not, the known relation: 1 c = √ = 2.998 × 108m/s (62) 0 µ0 20 Diego Saá 1

Let us assume that µ0 is defined also as equal to the inverse of the speed of light, so the previous relation is satisfied: 1 = µ = . (63) 0 0 c Physicists are aware that the choice of units of many universal constants, such as 0 and µ0, is completely arbitrary in current Physics. For example, Prof. Littlejohn of University of California at Berkeley expresses the following in his lecture notes on Quantum Mechanics: “In Gaussian units, the unit of charge is defined to make Coulomb’s law look simple, that is, with a force constant equal to 1 (instead of the 1/4π0 that appears everywhere in SI units). This leads to a simple rule for translating formulas of electrostatics (without D) from SI to Gaussian units: just replace 1/4π0 by 1. Thus, there are no 0’s in Gaussian units. There are no µ0’s either, since these can be 2 expressed in terms of the speed of light by the relation 0µ0 = 1/c . Instead of 0 and µ0’s, one sees only factors of c in Gaussian units.” [33]. The definition of 0, as the inverse of the speed of light, is strictly necessary for 0, since the electromagnetic waves displace at the speed of light, but µ0 does not appear to have such requirement. Therefore, assuming that 0 = 1/c, let us replace it in the known rela- 2 tion: q = 2 α h 0 c, [34]. This imposes the requirement that the elementary charge, q, be redefined as equal to the square root of Planck’s constant, h, multiplied, for macroscopic applications, by double the fine structure constant α: √ q = 2 α h (64) 1 − 1 Therefore, the dimensions of charge become [M 2 LT 2 ]. 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 conversion constant is used, the electrical units, such as the coulomb and ampere, are not anymore indispensable, except for compatibility with previous knowledge: √ 2 α h 1 − 1 −1 C = [M 2 LT 2 coul ] (65) q The square of C divided by 2 α is the von Klitzing constant (about 25812.808 ohm). CODATA 2002 defines this constant as independent and has about seven digits precision (in 1990 the CIPM adopted exact values for the von Klitzing constant). With the above mentioned proposal the von Klitzing constant is a function of Planck constant and of the value of the elementary charge. The number of correct digits can be duplicated. Several other constants such as the elementary charge, Planck’s constant, fine structure constant and electron mass, can also be obtained with several additional digits of 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 profitably to improve the precision of several constants. Algebra for Vectors and an Application 21

Using equations 50-53 let us prove that the electric and magnetic ﬁelds satisfy the d’Alembert wave equation. In both cases it is necessary to use the following vector identity, for any vector X:

∇ × (∇ × X) = ∇(∇ · X) − ∇2X (66)

First, let us take the time derivative of the proposed Faraday equation 52:

2 ∂ B µ0 ∂E 2 = − ∇ × (67) ∂t 0 ∂t Replacing here the definition of ∂E/∂t obtained from the proposed Ampere equation 53: 2 ∂ B µ0 1 1 2 = − ∇ × ( ∇ × B + ∇s) (68) ∂t 0 0 µ0 0 By vector calculus, the curl of any gradient is always zero, so the second term at the right-hand side vanishes. Applying vector equivalence, 66, and using 2 Gauss’ equation for the magnetic field, 51, the result B = 0 is immediate: ∂2B −2 + ∇2B = 0 (69) 0 ∂t2 In a similar form, for the electric field, let us take the curl of the proposed Faraday’s equation, 52, and use the vector equivalence 66:

∂B ∇(∇ · E) − ∇2E = − 0 ∇ × (70) µ0 ∂t Replacing the divergence of E by its equivalent from the proposed Gauss’ equation for the electric ﬁeld, 50, we ﬁnd the equation:

∂s 2 0 ∂B ∇ 0 − ∇ E = − ∇ × (71) ∂t µ0 ∂t Equating this to the time derivative of the proposed Ampere equation, 53, 2 multiplied by 0, the result E = 0 is immediate: ∂2E −2 + ∇2E = 0 (72) 0 ∂t2 After some standard and very well known operations within the study of electrodynamics, we have arrived to the corresponding d’Alembert equation (or “four-dimensional Laplace equation”) for the spatial components, E and B, of the electromagnetic ﬁeld.

In the following Subsections, similar wave equations are inferred for other physical variables. 22 Diego Sa´a 1

5.2. Electromagnetic four-vector from potential four-vector Let us deﬁne the potential four-vector as: Aµ = −(i φ, A) (73) To obtain the rank 2 electromagnetic tensor (Faraday’s tensor) (here multiplied by the speed of light), in current Physics it is necessary to carry out the following tensor operations [32]:

F µν = ∂µAν − ∂ν Aµ (74) All the signs of the elements of this rank 2 tensor appear in the same order as the one provided in the product for four-vectors, deﬁned in section 3.2:

 0 Ex Ey Ez  x z y µν  −E 0 c B −c B  F =   (75)  −Ey −c Bz 0 c Bx  −Ez c By −c Bx 0 With four-vectors we deﬁned the simple electromagnetic four-vector 42, which contains more information than this rank 2 tensor (the four-vector includes information about the scalar component, which is new to Physics, and the four-vector product 18 generates the correct signs in identical positions as in Faraday’s tensor 75).

Therefore, in covariant notation, Faraday’s tensor should be written as: F µν = ∂µAν − ∂ν Aµ + gµν s (76) The scalar field s and the electromagnetic fields E and B can be defined by performing the following four-vector product: M = Aµ ∂ (77) Replacing the four-gradient ∂ by its definition 38, where the inverse of the speed of light is assumed from now on for both 0 and µ0, by relation 63, and expanding this product with 19 we get 1 ∂φ 1 ∂A M = (s, i E + cB) = − − ∇ · A, −i − i ∇φ + ∇ × A (78) c ∂t c ∂t Equating the corresponding real and imaginary components of this identity, we find the definition of the electromagnetic scalar in terms of potentials as: 1 ∂φ s = − − ∇ · A (79) c ∂t Its form is identical to the Lorenz gauge. This “gauge” is normally assumed equal to zero in current Physics. However, if this were the case, the electromagnetic scalar, s, would be zero and, according to the present theory, we would get the homogeneous Maxwell’s equations. This means that the Lorenz gauge amounts to assuming that the charge and current densities are zero. Precisely some studies conclude that the Lorenz gauge is unphysical. For example, in [35], the authors say “we see that the Lorenz gauge is in conflict Algebra for Vectors and an Application 23 with the physical phenomena. This is a crash of the Lorenz gauge. Now it is essential to find a new transformation of the equations.” The spatial components contain the well known definitions for the electric field in terms of potentials: 1 ∂A E = − − ∇φ (80) c ∂t 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 φ and A. Thus, replacing first in Gauss’ electric field law, 50, we derive the wave equation for the scalar potential: 1 ∂s ∇ · E = (82) c ∂t 1 ∂A 1 ∂ 1 ∂φ ∇ · − − ∇φ = − − ∇ · A (83) c ∂t c ∂t c ∂t Both terms containing A are identical since the time and space derivatives can be interchanged. Then 1 ∂2φ − + ∇2φ = 0 (84) c2 ∂t2 2 Which is the wave equation for the scalar potential, φ = 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 Faraday’s law, 52, ∂B ∇ × E = − (86) ∂t and replace in it the definitions of the electric and magnetic fields in terms of potentials: 1 ∂A 1 ∂ ∇ × − − ∇φ = − ∇ × A (87) c ∂t c ∂t The time and space derivatives of the A potential from the first term can be interchanged and in this way becomes identical to the first term in the right-hand side, being simplified. The curl of a gradient is always zero, so we obtain an identity. Finally, from Ampere’s law, 53, 1 ∂E c ∇ × B = − ∇s (88) c ∂t by replacing the definitions of the fields in terms of potentials we derive the wave equation for the vector potential: 1 ∂ 1 ∂A 1 ∂φ ∇ × ∇ × A = − − ∇φ − ∇ − − ∇ · A (89) c ∂t c ∂t c ∂t 24 Diego Saá 1

Apply the vector identity 66 to the left-hand side term and simplify the two terms containing φ: 1 ∂2A ∇∇ · A − ∇2A = − + ∇∇ · A (90) c2 ∂t2 2 Then simplify the terms at the margins, obtaining A = 0: 1 ∂2A − + ∇2A = 0 (91) c2 ∂t2 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-hand side becomes zero because the curl of any gradient is zero. This proves that the (local) circulation of current is zero. This is new to Physics. Next we can derive the equation of conservation of charge by obtaining the divergence of the classical Ampere’s law, 57, and replacing the divergence of E by its definition in 54. But, instead of that, let us take the divergence of our new equation 53. The left-hand side becomes zero because the divergence of any curl is zero. We obtain: 1 ∂ 0 = (∇ · E) + ∇ · (−∇s) (92) c ∂t Using the Gauss’ electric field law, 50: 1 ∂ 1 ∂s 0 = ( ) + ∇ · (−∇s) (93) c ∂t c ∂t From here and using equations 61 and 58 it is easy to recover the conservation of charge equation: ∂ρ ∇ · J = − (94) ∂t However, equation (93) is equivalent to this one and constitutes the wave 2 equation for the electromagnetic scalar, s = 0: 1 ∂2s − ∇2s = 0 (95) c2 ∂t2 There is no need to continue with these operations. It is more profitable to question whether four-vectors can generate this kind of equations. The answer is, 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 the current four-vector: 1 ∂s J µ = ∂(−s) = i , −∇s (96) c ∂t In other words, again by equations 61 and 58, we arrive at the classic charge- current four-vector J µ = i ρ c, J (97) Algebra for Vectors and an Application 25

5.4. Generalized charge-current continuity equations Next, apply (of course, this means “apply the four-vector product”) the gradient four-vector to the current four-vector just defined: 1 ∂ ∂J µ = − i , ∇i ρ c, J (98) c ∂t The resulting four-vector, equated to zero, produces three equations, which constitute the generalized charge-current continuity equations. In particular, the first one is the well known equation for conservation of charge: ∂ρ + ∇ · J = 0 (99) ∂t 1 ∂J + ∇ρ c = 0 (100) c ∂t ∇ × J = 0 (101) The second equation can also be derived by taking the gradient of Gauss’ electric field law, equating with the time derivative of Ampere’s law, applying the vector equivalence 66 and simplifying the emerging wave equation for the electric field, which is known to be zero. Similarly, the last expression is a completely reasonable vector equation within our theory, since in equations 96 and 97 (or in 58) we had defined the current (3D) vector as the gradient of the electromagnetic scalar. Re- membering a theorem (see Feynman [21]) of differential calculus of vectors that says that the curl of a gradient is zero, the identity is proved.

Finally, taking the time derivative of 99 and subtracting the divergence 2 of 100 we find that the charge density satisfies the wave equation ρ = 0: 1 ∂2ρ − + ∇2ρ = 0 (102) c2 ∂t2 Also, take the time derivative of 100, subtract the gradient of 99, apply the vector equivalence 66 and simplify with 101. With which we conclude 2 that the current density satisfies the wave equation J = 0: 1 ∂2J − + ∇2J = 0 (103) c2 ∂t2 5.5. Some electromagnetism in classical Physics Classical electromagnetic theory finds inhomogeneous wave equations for both potentials and electromagnetic fields, whereas we found homogeneous equations for all physical variables. Jackson, [38] p.246, shows the equation for the B field: 1 ∂2B ∇2B − = −µ ∇ × J (104) c2 ∂t2 0 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, it would be equal to the gradient of some scalar potential. With this, the curl of a gradient is automatically zero. 26 Diego Saá 1

Jackson, [38] p.246, also shows the equation for the E field: 2 2 1 ∂ E 1 1 ∂J ∇ E − 2 2 = − − ∇ρ − 2 (105) c ∂t 0 c ∂t Whose right-hand side is, again, considered as different from zero in classical Physics, whereas in our theory it is immediately zero after replacing the definitions of charge and current densities in terms of our proposed electromagnetic 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 equates to zero the definition of the electromagnetic scalar of our theory, effectively forcing to zero the charge and current densities, which conform the right-hand sides 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 partial 2 differential equation u = 0: 1 ∂2u ∂2u ∂2u ∂2u − + + + = 0 (106) c2 ∂t2 ∂x2 ∂y2 ∂z2 where c is a constant with the dimensions of speed. The classical solution of this equation is a periodic function. On the other hand, Coulomb’s law and the potentials, in the solutions obtained by Liénardand Wiechert, are not periodic. One might wonder how are we going to reproduce such results. As long as the present writer knows, Coulomb’s law has never been derived from first principles, and even Newton’s law of gravitation has the same form and, therefore, it is very probable, and rather obvious, that its origin is also in the wave equation. The answer is that we have to pick the new form as the ansatz of the solution: a u(r, t) = (107) ω t ± k · r This form (for electromagnetic and gravitational potentials), or some small integer power of it (square for gravitational, electric and magnetic forces), 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 function u(r, t) = f(ω ·t±k·r), such as the suggested above, or any linear combination of such solutions, satisfies the wave equation, where ω is angular velocity, k is a wave vector pointing in any direction, and r is a position vector from an arbitrary 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 dispersion relation is obtained: ω = ± c · k (108) Algebra for Vectors and an Application 27

5.7. Covariance of physical laws In section 4.2 it was shown the form of the diﬀerential of interval. Now we have to question whether such form is preserved when a relativistic boost is applied. Let us assume the usual Lorentz transformations, which deﬁne a boost in the x direction: 0 vdx q v2 dt = (dt − c2 )/ 1 − c2 , (109)

0 q v2 dx = (dx −vdt)/ 1 − c2 , (110) dy0 = dy, (111) dz0 = dz. (112) Then, let us replace these values into the space-time four-vector

dS = (i c dt0, dx0, dy0, dz0) (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 square of the interval:

(−c2 dt2 + dx2 + dy2 + dz2, 0, 0, 0) (114) This is important for guaranteeing that the four-vector product preserves the covariance of physical laws. As Feynman explains, “The fact that the Maxwell equations are simple in this particular [four-vector] notation is not a miracle, because the notation was invented with them in mind. But the interesting physical thing is that every law of physics[...] must have this same invariance under the same transformation. Then when you are moving at a uniform velocity in a spaceship, all of the laws of nature transform together in such a way that no new phenomenon will show up. It is because the principle of relativity is a fact of nature that in the notation of four-dimensional vectors the equations of the world will look simple.” [21] This is the reason why it is of paramount importance to represent all physical variables with four-vectors and, besides, that the operations over them should preserve their form. As we have seen, current Physics does not integrate the electromagnetic fields E and B into a four-vector, as was explained before, because it lacks the equivalent of the time component, which is represented by our scalar field, s. In the following section we will reveal other 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 in its rest-frame. It is, of course, well known that the Lorentz force must be modified anyway to include the effect of radiation damping, when a charge produces electromagnetic waves due to strong acceleration. Whether a mod- ification of the Lorentz force alone leaves [Maxwell] equations intact, is an 28 Diego Saá 1 open question. In 1890 Hertz was aware of the fact that the final forms of the forces are not yet found” (emphasis in the original). In the following, a new formula is suggested for completing the computation of 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 or the 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 addition of the scalar electromagnetic field and the additional term for force that is proposed below, as a function of the current (flow of electrons in the A-B effect).

The forces over a test charge are computed in classical Physics by means of the classical “Lorentz force” equation, by multiplying the charge, q, by an expression reminiscent of the electromagnetic four-vector:

F = qE + v B (115)

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

We propose the following formula to compute the forces associated with currents in an electromagnetic ﬁeld, as an extrapolation and correction of the previous one. It is reached by multiplying the inverse of the speed of light by the current four-vector and by the electromagnetic four-vector (see Jackson [38], p. 611): 1 F = J µM = i ρ c, Js , i E + c B (116) c

By expanding the indicated product we get:

1 1 s F = i(ρ s + J · E) + J · B, i ( J × E − ρ cB) + ( J + ρ E + J × B) (117) c c c

Whittaker [39], proposed a scalar force. Jackson [38], in page 611, except for the minor appearance of the speed of light in the last term, shows 1 our second scalar term, that is c J · E, together with the two classical vector terms, which appear at the end of (117). Prykarpatsky and Bogolubov [40] and Martins and Pinheiro [41] have obtained part of our ﬁrst real vector term, (s J/c), since our deﬁnition of the electromagnetic scalar in terms of potentials includes the divergence of the vector potential (refer to expression (79) ), which those authors multiply by the charge and velocity. This is simply recognized as current, J. Algebra for Vectors and an Application 29

6. Discussion Four-vectors, in the form proposed by the present author, emerge in this paper as a possibly more appropriate mathematical tool for the handling of vectors and the study of fundamental physical variables and their describing equations.

This new mathematical structure seems to be a formalization of the classical vectors. Its simplicity contributes to the possibility of more extended and fruitful uses in all branches of science.

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

All the classical physical variables mentioned in the present paper, such as charge and current densities, scalar and vector potentials, electric, magnetic and the new electromagnetic scalar fields have been proved here to satisfy the homogeneous wave equation, which gives a strong argument to conclude that our universe is of a single wave-like constitution. In this paper, the author proposed several wave equations, where some of them 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 well tested. However, a new scientific theory, with the aim of being worthy and successful, should really understand and describe how nature works. More- over, it should be falsifiable, and make testable predictions of experimental outcomes not yet put to test. In the reader should remain the questioning whether the theory that has been exposed represents reality or not. To answer this, numerous differences have been proposed in this paper with respect to the current accepted models, which should make it easy for the physicists to locate the discrepancies, and reject 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 physicists and mathematicians, to locate the failures or problems, if any. The reproduction of many known equations, some of them rather complex, such as all the Maxwell’s equations, originated on simple four-vector products, provides a very strong reassurance in favor of the proposed vector algebra as one of the most correct and powerful mathematical tools to apply in Physics.

The periodic solutions of the wave equation are well known, but the non-periodic solutions are not known, or have been ignored, despite their importance to Physics. The classical Coulomb and Newtonian equations for the electrostatic and gravitational forces and potentials seem to be traceable 30 Diego Sa´a 1 to the wave equation, which is rather natural after acceptance of equation 64 for the conversion between mass and charge. The present author has pos- tulated the non-periodic solutions of the d’Alembert equation as the correct expressions 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 the wave equations, were equated to zero in order to generate the Maxwell’s equations and others. Why should it be so? The present author does not have the answer. The question should be posited to Nature and is left to the reader to discover. This problem has the signature of the situations mentioned in Subsection 2.2, according to which some complex four-vectors may have zero magnitude despite the fact that they are non-zero. Feynman [11] in chapter 25 concludes that “All of the laws of physics can be contained in one equation. That equation is U = 0” [his emphasis]. Our statement is stricter, in the sense that the relations satisﬁed by the physical variables are not arbitrary but all of them are just standard linear wave equations (d’Alembert equation).

The proposal of changing the existing deﬁnitions of the dielectric permittivity and magnetic permeability of vacuum appears as too radical. However, with the proviso that the theory proposed in this paper is correct and even without it, the new deﬁnitions allow to simplify and simultaneously preserve the coherence of all Physics.

A proposal was given to dismiss the classical electromagnetic units called “coulomb” and “ampere”. Therefore, the mechanical units, kilogram, meter and second, are the only ones that remain as required to study Physics in general and electromagnetism in particular.

Acknowledgment The author wishes to express his gratitude to Dr. Delbert Larson for performing a comprehensive and exhaustive revision of the paper and for providing multiple suggestions for improvement. Also, thanks to Dr. César Costa of Escuela PolitécnicaNacional, who provided some important observations. Of course, any remaining errors are full responsibility of the present author. Finally my deep thanks to Dr. Bertfried Fauser, who kindly provided the respective endorsement to a slightly different version of the present paper, in order that it be published in ArXiv, no matter if it was later suppressed pub- lication by anonymous managers of such Cornell web site, without providing any explanation. Algebra for Vectors and an Application 31

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Diego Saá 1 1 Emeritus. Departamento de Ciencias de Informacióny Computación,Escuela Politécnica Nacional, Ladrónde Guevara E11-253, Quito – Ecuador. Tel. (593-2) 2567-849. e-mail: [email protected]