Dimensional scaffolding of electromagnetism using geometric algebra
11 Dimensional scaffolding of electromagnetism using geometric algebra
Xabier Prado Orbán and Jorge Mira Departamento de Didácticas Aplicadas, Universidade de Santiago de Compostela, E-15782 Santiago de Compostela, Spain Departamento de Física Aplicada – área de electromagnetismo, Universidade de Santiago de Compostela, E-15782 Santiago de Compostela, Spain E-mail: [email protected]
Abstract
Using geometric algebra and calculus to express the laws of electromagnetism we are able to present magnitudes and relations in a gradual way, escalating the number of dimensions. In the one-dimensional case, charge and current densities, the electric field E and the scalar and vector potentials get a geometric interpretation in spacetime diagrams. The geometric vector derivative applied to these magnitudes yields simple expressions leading to concepts like displacement current, continuity and gauge or retarded time, with a clear geometric meaning. As the geometric vector derivative is invertible, we introduce simple Green´s functions and, with this, it is possible to obtain retarded Liénard-Wiechert potentials propagating naturally at the speed of light. In two dimensions, these magnitudes become more complex, and a magnetic field B appears as a pseudoscalar which was absent in the one-dimensional world. The laws of induction reflect the relations between E and B, and it is possible to arrive to the concepts of capacitor, electric circuit and Poynting vector, explaining the flow of energy. The solutions to the wave equations in this two-dimensional scenario uncover now the propagation of physical effects at the speed of light. This anticipates the same results in the real three-dimensional world, but endowed in this case with a nature which is totally absent in one or three dimensions. Electromagnetic waves propagating entirely at the speed of light can thus be viewed as a consequence of living in a world with an odd number of spatial dimensions. Finally, in the real three-dimensional world the same set of simple multivector differential expressions encode the fundamental laws and concepts of electromagnetism. Keywords: geometric algebra, geometric calculus, electromagnetism, spacetime dimensions, algebraic dimensions
1. Introduction
The concept of a geometric algebra was first established by Grassmann [1, 2] to encode the main elements of a geometric space. Clifford [3, 4] later added the quaternions developed by Hamilton [5-7], which encoded the operations on these same spaces, together with the main concept of a geometric product between these elements. Greatly overshadowed by the vector algebra developed at the time by Gibbs [8] and Heaviside [9], which was thoroughly employed to express and to manipulate Maxwell equations for electromagnetism, geometric algebra has nonetheless survived and it has been revived by authors like Hestenes [10, 11] -which has shown that the geometric product and its algebraic properties can be efficiently applied to differential operators - as a new language for mechanics, electromagnetism and modern physics. For readers who are not familiar with geometric algebra and calculus, Appendix 1 presents its main concepts and relations: multivectors, geometric product, spacetime directions, indexes and their rules, vector derivative, spacetime split, relative vectors, spatial directions, and cross product. In this framework, the set of Maxwell equations can be unified in the simple expression ∇ F = J (see section A1.3 of Appendix 1 for the definition of F, J and ∇ ) , which is the electromagnetic field equation (or simply, field equation) in geometric algebra. With the help of a spacetime split, this formula can be translated to relative vectors, getting in this way the four Maxwell equations in the usual vector formulation. Natural units (like the Heaviside-Lorentz ones) where μ = 1 and ε = 1 are being used consistently and, consequently, the speed of light has the value c=1. Applying a spacetime split to the field bivector F yields γ Fγ = γ (F·γ ) + γ (F ∧γ ) = E + i B where E is a relative vector (the electric field), corresponding to the temporal component of the bivector F, and B is the magnetic field vector. The product i B corresponds to the purely spatial part of the spacetime bivector F. The symbol i identifies the euclidean pseudoscalar (i n in our notation), in order to keep the expression valid for dimensions other than 3. The bivector F can be derived from a vector-valued potential function A by means of the formula: ∇A = F together with a gauge to eliminate ambiguities. We will follow the Lorenz gauge: ∇ ·A = F The form of the field equation in geometric algebra using this vector potential is ∇F = ∇(∇A) = ∇ A = J The rest of this paper is devoted to the application of this geometric algebraic strategy to spaces of increasing dimensions, where we will use the convention ND to refer to a space of N dimensions and (1, N) to its corresponding spacetime. The formulation for electromagnetism as provided by the unified language for Physics [12] developed by Hestenes with the names of geometric algebra [13] and geometric calculus [14] provides us thus with a mathematical tool capable of working in different dimensions, and we will use this capability in order to present the main electromagnetic concepts in a sequential way, beginning with the lowest number of dimensions (1,1) and reaching finally its highest order (1,3). This process was named by Wheeler [15] as dimensional reduction, although using a different mathematical approach. In every step of this dimensional escalating sequence we will begin with the separation of the field equation in its multivector components, identifying the resulting expressions as the equivalents of Maxwell´s laws for this dimensional world. After applying the corresponding spacetime splits, we get these laws expressed in the more conventional vector algebra formulation, allowing us to identify the relevant electromagnetic magnitudes and their properties. In every step we will rely –if not otherwise stated- in the results obtained in the previous –lower- dimension, which explains the name of dimensional scaffolding applied to the resulting dimensional increasing sequence. We will try to illustrate the resulting concepts and relations in a visual way using geometric figures in purely spatial or spacetime diagrams. The sequence of diagrams, beginning with a one-dimensional (linear) 1D space and its (1,1) spacetime, followed by a two-dimensional (plane) 2D space and the corresponding (1,2) spacetime, will provide a gradual visualization of the main electromagnetic concepts. Otherwise elusive concepts like the displacement current, the retarded time and its corresponding Liénard-Wiechert potentials and fields, the null cones and the propagation of signals, will get a clear geometric meaning which will evolve gradually as we increase the dimensionality of the physical space. In a few cases – like the Green´s functions and their effects on the propagation of potentials and fields - a striking difference appears between dimensions, showing that the real physical world as we know it would be quiet different in other dimensions.
2. Dimensional scaffolding
In order to present the electromagnetic magnitudes in a sequential way, we will use a strategy of dimensional scaffolding, beginning with a one-dimensional world and adding successive spatial dimensions to get finally the whole set of magnitudes. This process can be understood in two complementary ways: Either as a series of toy universes where the laws of electromagnetism keep the same overall form, or else as a series of special situations, where the symmetry of the arrangement allows us to keep some of the spatial dimensions out of the equations. The first step in this scaffolding process will be a one-dimensional space. In this toy universe, objects can move only in one line, with two possible directions, given by σ and its opposite (- σ ), as in figure 1. Figure 1: one-dimensional space To produce an arrangement with this desired symmetry, we could think about a situation where a great cloud of uniformly distributed charges move only in one direction. This can happen, for instance, in the interior of a plasma cloud, or in a waveguide. Adding the time dimension, this produces a (1,1) spacetime. Figure 2 shows a diagram of this one-dimensional spacetime, which is divided vertically between past and future, being the present the real world, represented by the horizontal x-axis, with a spatial direction γ . The vertical direction γ represents the flow of time. A dashed line stands for lightlike (or null) directions. In natural units, the speed of light c =1, corresponds to those diagonal lines with unitary slope. Two curved arrows try to show the effect of a new bivector magnitude γ The fact that γ γ = γ and γ γ = γ shows that the effect of γ as an operator is equivalent to a stretching towards one of the diagonals: (γ + γ ) γ = (γ + γ ) and a squeezing towards the other diagonal line. (γ - γ ) γ = - (γ - γ ) Figure 2: (1,1) spacetime The second step in the dimensional scaffolding process is a two-dimensional space. In this toy universe, objects can move on a plane. All the possible directions are given by the linear combinations of two basis elements: ( σ , σ ). Together with the time dimension, this produces a (1,2) spacetime. Geometric Algebra introduces a new kind of direction, a bivector one, with two possible orientations, given by the pseudoscalar i = σ and its opposite -i = σ . The operation of this pseudoscalar on any direction is given by σ i = σ σ = σ = σ and σ i = σ σ = σ = - σ = - σ Pseudoscalar magnitudes act thus as rotating operators, either clockwise ( σ ) or anti-clockwise ( σ ), like those shown in figure 3. Figure 3: two-dimensional space This two-dimensional toy universe is equivalent to an arrangement of indefinitely long charged rods placed perpendicularly to a given plane, which would be our two-dimensional space. This can happen, for instance, in the interior of a large coil, or in the vicinity of a planar current. Together with the time dimension, this produces a (1,2) spacetime (figure 4). Figure 4: (1,2) spacetime Time is the vertical axis, space is a horizontal plane with coordinates (x,y) and light cones separate two interior parts (future and past) from the exterior part with all the possible “presents” for a given spacetime point (event) P. Two timelike bivectors ( γ , γ ) with the same meaning as in figure whose effect is opposite to that of σ due to the fact that σ = γ γ = γ = - γ = - γ Finally, the third step in our dimensional scaffolding corresponds to the real three-dimensional space and the (1,3) Minkowski spacetime. In any of those steps we will work initially with spacetime multivectors on the field equation, separating them in different multivector directions. Using the correspondences obtained at the spacetime split, we will express later the same equations using their spatial electric and/or magnetic components, which yields one of the Maxwell equations. In this way we assure both formal coherence due to the fact that the field equation does not change at all in this process (except for the reduction in dimensions), and at the same time we keep the physical flavour of the well-known Maxwell expressions with electric and magnetic vectors. The following expressions will try to exploit the fact that Geometric Algebra allows to write down most of its equations in a synthetic, coordinate-free flavour. A great advantage of this approach is that it enables us to get rid of what Hestenes [16] called the “coordinate mathematical virus”, meaning the widespread misbelief that vectorial calculations need to be made using coordinates all the way. In Appendix 2 the same expressions are developed using coordinate expansions, in order to get into the subtleties of these equations.
3. One-dimensional electromagnetism
The electromagnetic multivectors taking part at the field equation in this one-dimensional case are: ∇ = ∂ γ + ∂ γ F = F γ J = J γ + J γ The spacetime split delivers the following correspondences: Jγ = J + J σ = ⍴ + (- j ) The minus sign for the current vector j is needed to assure a perfect accordance between the spacetime current and its spatial counterpart, as shown in Appendix 2. It means that the spacetime vector J is spatially reflected with respect to the current vector j , and this does not imply any change in the latter. ∇ γ = ∂/∂t + σ ∂/∂x = ∂/∂t + ∇ γ (Fγ ) =γ (F γ γ ) = F σ = E σ = E and there is no equivalent to a magnetic field in one dimension. The field equation in this one-dimensional world would be ∇ F = J = J γ + J γ which can be separated in two multivector directions timelike vectors: ( ∇ ·F) t = ∂ γ F 𝛾𝛾 = J γ spacelike vectors: ( ∇ ·F) s = ∂ γ F γ = J γ The first part of the one-dimensional field equation corresponds to the timelike component of the current vector (J ): ∂ γ F γ = ∂ F γ = J γ implying ∂ F = J Translating to spatial vectors: ∂E/∂x = ρ or, equivalently: ∇ ·E = ρ This equation is known as the differential form of Gauss´s law. It states the strength of the electric field as due to the amount of charges, which act as the sources of the electric field.
Visualization of Gauss´s law in one dimension
Figure 5 shows a set of 8 electric charges (for the sake of simplicity, we suppose that all charges are positive and equal in value) at rest in a given reference frame. The figure shows that the charge density ρ (which is inversely proportional to the separation between charges) is greater to the right than to the left. We can see also the electric field E as a vector pointing to the direction σ at the right side, and to the opposite direction - σ at the left side. These vectors show the direction of the force which would feel a test particle with positive charge placed in these positions. Figure 5: charges in one dimension The corresponding spacetime diagram can be seen in figure 6, where the set of charges at rest is represented by parallel vertical lines. Figure 6: Charges at rest in spacetime. The vertical lines represent the position of the charges. The value of the field in each segment is indicated by the numbers at the bottom. The field is constant (4 or -4) beyond the position of the set of charges. The current vector J is represented as an arrow in two cases (both vertically, because the current component is zero), and the field bivector F γ has the positive direction γ at the right side, and the negative direction -F
01 01 at the left side. The curved arrows representing these bivectors show how the movement of a test particle with positive charge placed in these points would be. We can recognize that the value of the electric field increases horizontally as if it were a counter of charges. The grey bars stand for the value of the charge density, which is inversely proportional to the spacing between adjacent charges. The Gauss equati on ∂E/∂x = ρ states that the charge density is equal to the slope of the function E(x). It is interesting to note that the Gauss equation can be interpreted also as the way in which the field E derives from the electric charges which are its sources. In this case, where all the charges have the same sign, the electric field is proportional to the amount of charges included in the interior of a Gauss surface (in one dimension, the surface is formed by a pair of equidistant points from the center). We can even analyze how this applies to a situation with the needed symmetry, as it would be the case of having an arrangement of several charged plates. Figure 7 shows two different cases: the left case corresponds to a pair of infinite plates charged with the same sign. As in the previous figure, the field E diverges outside the plates, and in this case it is equally null between the plates. The right case can be used to present the concept of a capacitor, made by two plates (which in this one-dimensional world would be reduced to two points) charged with opposite signs. Gauss law can be applied to see that the field is only nonzero between the plates, where it points from the positive to the negative charges. Figure 7: Charged plates Ampère´s law in one dimension: charge and displacement currents
The second part of the one-dimensional field equation corresponds to the spacelike component of the current vector (J ): ∂ γ F γ = ∂ F γ = J γ , implying ∂ F = J Translating to spatial vectors: ∂E/∂t = - j or, equivalently: ∂ E /∂t = - j We can use this result to introduce the concept of displacement current, j D = ∂ E /∂t which was proposed ad-hoc by Maxwell in order to establish a coherent set of equations. As we have seen, this term arises naturally using Geometric Algebra without further justification, and in the one-dimensional world it reflects the effect of a time-varying electric field on the electric current: j D = J = - j We could view this equation also as the form of the Ampère Law in a one-dimensional world: j D + j = 0 This equation shows us that a “charge current” j produced by the movement of charges (or, alternatively, by the movement of the observer´s reference system) is somehow compensated by a displacement current j D which is due to the time evolution of the electric field. Visualization of Ampère´s law in one dimension
In order to visualize Ampère´s law in this one-dimensional world, we can begin with an arrangement of charges at rest but viewed from a reference system which is in movement with respect to the charges. They will no longer appear at rest but they will have a “drift” velocity due to the relative movement of our reference system. Figure 8: drift current Figure 8 represents the same situation as in figure 6, but viewed from 1234 a reference system moving to the left. The charges have now a “drift” velocity to the right. We have thus created a charge current j (to the right) which is compensated by a displacement current j D = ∂E/∂t The figure shows that the electric field E is decreasing (from 3 to 1 at the right side) as time passes. The time derivative of E(t) will be negative and as a consequence the displacement current j D will point to the left. A surprising feature in figure 8 is the fact that the charge-current vector J does not point in the direction of the particles worldlines but it is reflected horizontally, following the direction of the displacement current j D instead of the charge current j. This reflection of the spacetime current vector with respect to the current lines is a geometric feature of spacetime that does not depend on the choice of spacetime split, as we will see in Appendix 2. Vector potential in one dimension: null directions and retarded potentials
The potential vector function in this one-dimensional world has the form A = A γ + A γ The field equation with this potential becomes ∇ A = J ( ∂ - ∂ )(A γ + A γ ) = J γ + J γ Separating the temporal and spatial components we get a pair of very similar differential equations: ( ∂ - ∂ ) A = J and ( ∂ - ∂ ) A = J which are the Poisson equations. The most interesting case appears when we consider a chargeless space or vacuum: J = 0 In this case, both differential equations reduce to the same form: (∂ - ∂ )A= 0 or, equivalently, ∂ A /∂x - ∂ A /∂x = 0 ∂ A/∂t = ∂ A/∂x This is the Laplace equation, which is also the wave equation in one dimension, and its solutions have the form A R (x-t) or A L (x+t) corresponding to waves propagating in vacuum with speed c = 1. A R (x-t) propagates to the right, and A L (x+t) propagates to the left. A general solution for this one-dimensional wave equation has the form A = A γ +A γ , with A = a (x-t) + a (x+t) A = a (x-t) + a (x+t) where (a , a , a , a ) are differentiable functions in both variables (x,t). Defining a phase function φ either as φ R = x- t or φ L = x+t we have two kinds of functions with the general form a(φ), with derivatives a´(φ) = da/dφ. Separating them in R and L functions a R ( φ R ) = a R (x-t) , a L ( φ L ) = a L (x+t) and applying the chain rule, we get ∂ a R = - a´ R , ∂ a R = a´ R ∂ a R = a´ L , ∂ a L = a´ L The Lorenz gauge ∇ ·A = 0 implies ∂ A = ∂ A and, since ∂ A = -a´ (φ R ) + a´ (φ L ) ∂ A = -a´ (φ R )+ a´ (φ L ) we arrive to a´ = a´ and a´ = a´ In other words, A and A differ only by a constant: A = A + k and the general expression for the one-dimensional wave potential obeying the Lorentz gauge will have the form A = A γ + (A +k)γ = A (γ +γ )+kγ Where A is a linear combination of A R (x-t) and A L (x+t) The electromagnetic field derived from this potential has the form F = ∇ A = ∇∧ A = (∂ A - ∂ A )γ After calculating ∂ A = ∂ [a (x-t) + a (x+t)] = -a´ +a´ ∂ A = ∂ [a (x-t) + a (x+t)] = -a´ +a´ we arrive finally to F = (∂ A - ∂ A )γ = 0 This means that every possible solution for a one-dimensional wave equation in the form of potentials will yield a null electric field. In other words, all the possible solutions do not produce any physically observable effect. Nevertheless, it is interesting to discuss here the relation of the potentials and the null cones. The potential function A has the same form in any pair of spacetime points (events) which can be linked by a diagonal line. In other words, if the potential function has the value A in some point, it will keep this value in all the points of its null-cone. It is useful to use the concept of retarded potential as the one that depends on the retarded time, which can be defined at this early level as t r = t - |x|, which is a measure of the time lapse of the potential change from the point of view of an observer receiving its influence. When dealing with higher dimensions, the existence of this retarded potential will be fundamental to explain the existence and properties of electromagnetic waves. Wheeler [15, 17], using the language of p-forms, has found interesting additional properties in (1,1) dimensions which enhance what he calls the “tutorial potential” of EM as presented in this dimensionally reduced spacetime. He assumes the absence of electric charges as an initial condition for a one-dimensional EM with electric (E) and magnetic fields (B) which generate one another under the rule of differential equations. This produces a consistent mathematical structure but at the cost of losing contact with the consideration of electromagnetism as a theory for the behaviour of electric charges. The presentation of EM at this extremely simplified (1,1) level should therefore be taken with care in order to avoid unnecessary assumptions, and this is also valid for our formulation of a wave equation for the potentials. This mathematical result can be simply a consequence of our consistent choice of the Lorenz gauge from the beginning. It is only interesting from a conceptual point of view, as it allows for the presentation of the key concepts of the speed of light and retarded time at a very early stage, but, as we have shown, they do not produce any observable effect since the resulting electric field is constant. This should be stressed when presenting EM with this dimensional upgrading approach. The concept of gauge as a way to regulate redundant degrees of freedom in the Lagrangian is key for the understanding of gauge theories, which are possibly the best way to describe the interactions between charges and fields, and it could be introduced at this level to explain the difference between mathematical results and their physical meaning. The general expression for the one-dimensional wave potential obeying the Lorentz gauge, as we have seen, has the form A = A γ + (A +k)γ = A (γ +γ )+kγ Where A is a linear combination of A R (φ R ) and A L (φ L ), being φ R and φ L are the phase functions for a right-propagating wave and a left-propagating wave respectively. Figure 9 is a representation of the main magnitudes taking part in such a solution for the wave equation. It is possible to recognise that the lines of constant phase (dashed lines) correspond to worldlines with c = 1, and this is the speed of light. We can also observe the directions (γ +γ ) and (γ - γ ) for any one-dimensional vector potential in spacetime which is a solution of the wave equation with the Lorentz gauge. These are null vectors, and we have also seen that such null vector potentials do not produce any field F, which implies that there are no physical effects associated with such potential waves in a one-dimensional spacetime. Figure 9: Potential wave directions and retarded time in (1,1) spacetime. The retarded time between the emitter P and the observer P´, t R , is also shown. It reflects the time lapse (as measured by P´) from any influence coming from P. The retarded time from every point in the future light cone of P is zero. Figure 10 shows how this can be applied to explain the redshift due to the movement of the source P from P to P (separated by a time delay T, which we can consider equivalent to a period for harmonic waves). Figure 10: Retarded time and redshift. For observer A, lying in the direction of the advance of the source, the retarded time shows a delay T A while observer B, in the opposite side, measures a delay T B . The redshift is clear from the figure, as T B > T A Wheeler [15] has shown that even in this one-dimensional case it is possible to present in a visual way Green´s functions, which are the key elements to construct retarded potentials and fields from given sources.
4. Two-dimensional electromagnetism
In analogy with the alternative definition of a cross-product as explained in Appendix 1: v ✕ w = - v ·(i w) we can define a two-dimensional vector derivative on a scalar-valued function f(x,y) as ∇ ✕ f = - ∇ ( if) = ∂f/∂ y σ - ∂f/∂ x σ which is coincident with the definition of a two-dimensional vector derivative -based on the notion of a perpendicular vector in two dimensions- given by McDonald [18]: ∇ ⊥ = (∂/∂y , - ∂/∂x ) This is an operator with a clear geometric interpretation and a deep explanatory potential as we will see later. The electromagnetic magnitudes taking part at the field equation in this case are: J = J γ + J γ + J γ F = F γ + F γ + F γ ∇ = ∂ γ + ∂ γ + ∂ γ where ∂ = ∂/∂t , ∂ = ∂/∂x , ∂ = ∂/∂y Figure 11 is a visualization of the vector derivative as defined for the two-dimensional space: ∇ ✕ f = - ∇ ( if) = ∂f/∂ y σ - ∂f/∂ x σ Figure 11: Vector derivative in two spatial dimensions The two-dimensional vector derivative of a scalar field f produces a vector which lies on the lines of constant f and points towards the direction of the combined flux of (if) on both sides of this same line.
Spacetime split in two dimensions
The two-dimensional spacetime split yields the following correspondences (see Appendix 2 for more detailed calculations): Jγ = ⍴ + (- j ) with j = j x σ + j y σ γ (Fγ ) = E + i B where E = E x σ + E y σ and B is a scalar magnitude γ ∇ = ∂/∂ t + ∇ where ∇ = σ ∂/∂x + σ ∂/∂y is the usual vector derivative in two dimensions. Spacetime and vectorial components are thus related by F = E x , F = E y , F = B J = 𝜌𝜌 , J = -j x , J = -j y The field equation in this two-dimensional world would be ∇ F = J = J γ + J γ + J γ which can be separated in three multivector directions timelike vectors: ( ∇ ·F) t = J γ spacelike vectors: ( ∇ ·F) s = J γ + J γ trivectors: ∇∧ F = 0
Gauss´s law in two dimensions: field divergence
The timelike vector component of the two-dimensional field equation can be written as: ( ∇ ·F) t = J γ implying ∂ F +∂ F = J Translating to spatial vectors: ∂E x /∂x +∂E y /∂y = ρ or, equivalently: ∇ ·E = div (E) = ρ which has the same form as the usual expression for Gauss´s law. The physical meaning of this expression is, as we have seen previously for the one-dimensional case, that the sources for the electric field are the electric charges. The two-dimensional approach allows us to introduce the concept of divergence. Visualization of Gauss´s law in two dimensions
Figure 12 represents an escalation from the (1,1)-dimensional case represented in figure 6 to a (1,2)-dimensional analogous situation. In both cases, a set of positive charges at rest are represented as vertical lines pointing upwards in spacetime. Applying Gauss´s law results in divergent bivectors on opposite sides of the charges. On the upper side of the figure the spacetime split is represented as a horizontal plane with two spatial directions ( σ , σ ) and the divergence of the electric field E is clearly evident as outwards-pointing vectors. Figure 12: Field divergence in (1,2) spacetime Taking in consideration the whole figure, it is possible to recognize that the positive charges are the source of the electric field lines, while negative charges would be its sinks. A closed line around the group of positive charges would be crossed outwards by a number of field lines which is proportional to the net amount of positive charges in its interior. Ampère´s law in two dimensions: magnetic field
The spacelike component of the two-dimensional field equation corresponds to the spacelike components (J ,J ) of the current vector J: ( ∇ ·F) s = J γ + J γ , implying ∂ F - ∂ F = J ∂ F + ∂ F = J Translating to spatial vectors and rearranging terms we get: ∂E x /∂t + j x = ∂B/∂y ∂E y /∂t + j y = - ∂B/∂x Recalling that i B =B σ - ∇ (i B) = - ∂B/∂x σ + ∂B/∂y σ we can write ∇ ✕ B = - ∇ (i B) = ∂ E /∂t + j which can be interpreted as Ampère´s equation for a two-dimensional world. Introducing again the displacement current j D = ∂ E /∂t Ampère´s equation can be written as j D + j = - ∇ (i B) The displacement current j D appeared in the one-dimensional world (where no equivalent to a magnetic field does exist) as a compensating effect (produced by the rate of temporal change in the electric field) for a charge current. In a two-dimensional world a new component of the field (iB) must be taken into consideration. It is a pseudoscalar, and this means that we can now interpret the magnetic field as the pseudoscalar magnitude resulting from the combined effects of both currents j D and j . This leads to an understanding of the electric field E as the timelike component of the electromagnetic field bivector F, whilst the magnetic field iB corresponds to its spacelike component. Visualization of Ampère´s law in two dimensions
Ampère´s equation for a two-dimensional world ∂ b F = ∂ F - J a (a = 1, 2) is also ∇ ✕ B = ∂ E /∂t + j Introducing the displacement current j D = ∂ E /∂t - ∇ (i B) = j D + j E is the timelike component of the electromagnetic field bivector F, and we can interpret the magnetic field iB as the pseudoscalar magnitude resulting from the combined effects of both currents j D and j . We will consider first a special electrostatic situation where the electric field does not change in time: ∂ E /∂t = 0 , and Ampère´s law can be written simply as ∇ ✕ B = j In this case, the current vector j can be regarded as the source for the magnetic field. Figure 13 shows an example of the magnetic field generated by a linear current in a two-dimensional space, where we can see that the bivector field iB has opposite curling directions on both sides of the current. Figure 13: Current as the source for the magnetic field Figure 13 can be also interpreted as the way a spatially changing magnetic field B generates a current. If this process happens in vacuum, it produces again a pure displacement current, shown in figure 16. This adds an interesting feature to the understanding of electromagnetic waves, as we will see at the end of section 4.5.1. Figure 14 shows a circular wire stretched by the passage of time as a cylinder in (1,2) spacetime with a steady current flowing through it -shown by several inclined particle worldliness- and producing a uniform field B in its interior. A corresponding two-dimensional spacetime split is shown at the right. The same figure could be obtained by cutting a horizontal section through an indefinite vertical solenoid, which is an arrangement with the needed symmetry to be represented by a two-dimensional section showing a circular wire with a uniform current. Figure 14: Two-dimensional model of a solenoid The curling direction of the field iB follows the direction of the circular current j. It is possible to go a step further ahead in order to present the concept of an electric circuit in an alternative way, as shown in figure 15, where a battery to the left and a resistance to the right are joined by a conducting wire. Figure 15: Electric and magnetic fields in a simple electric circuit. We have now an additional electric field E. We represent with the letter S the energy flow from the battery to the resistance. It is the Poynting vector. If the electric field is not constant, the displacement current has to be considered also as a source for the magnetic field. This is evident in the model of a charging capacitor, as presented in figure 16. Two parallel conducting plates (placed vertically in the figure) receive a current j , whose effect is to increase the charge density in the left plate and to decrease it (or to increase the negative charge density, which is equivalent) on the right plate. This produces a time-increasing electric field whose partial time de rivative ∂ E /∂t points to the same direction as the current j . The resulting displacement current j D acts therefore again as a compensation effect for the absence of a material charge current in the interior of the capacitor. Figure 16: Two-dimensional model of a charging capacitor Faraday´s law in two dimensions: electromagnetic induction
The charge-current J is by definition a vector quantity, and this implies that any trivector part of the field equation must vanish identically. In other words, ∇∧ F = 0 or, equivalently, ∂ F = - ∂ F + ∂ F Translated to the corresponding vector magnitudes: ∂ t B = - ∂ E y /∂x + ∂E x /∂y This equation can be further modified taking into account that i B = B σ ∇ = σ ∂/∂ x+ σ ∂/∂y E = E x σ + E y σ Using again the two-dimensional vector derivative ∇ ✕ E = - ∇ (iE ) = σ ∂E/∂ y - σ ∂E/∂x we get ∂ t B = - ∂E y /∂x + ∂E x /∂y = - ∇ ✕ E which is formally equivalent to Faraday´s law of induction. Visualization of Faraday´s law in two dimensions
Figure 17 shows a spacetime diagram for a set of two circular wires, the exterior one having initially a clockwise current (producing a negative field B) which changes to a counterclockwise current which produces a positive field. The time derivative of B will be thus positive. Faraday´s equation ∂ t B = - ∂E y /∂x + ∂E x /∂y = - ∇ ✕ E tells us that the interior circular wire (shown in grey) will receive an induced electromotive force which produces a clockwise current on it. This is in accordance with Lenz´s law which states that the induced current tries to compensate the cause of the induction (in this case, producing a clockwise current). Figure 17: Induced electromotive force and current Vector potentials in two dimensions: electromagnetic waves
The potential vector function in a two-dimensional world has the form A = A γ + A γ + A γ The field equation using this potential becomes ∇ A = J, or ( ∂ - ∂ - ∂ )(A γ +A γ +A γ ) = J γ + J γ + J γ Separating the different components we get three similar Poisson´s differential equations: ( ∂ - ∂ - ∂ )A k = J k , with k having values (0, 1, 2) We consider again first a chargeless space (vacuum): J = 0 In this case, all three differential equations reduce to the same form: (∂ - ∂ - ∂ )A= 0 or, equivalently, ∂ A/∂t = ∂ A/∂x + ∂ A/∂y These are two-dimensional Laplace´s equations, known also as wave equations, whose solutions can be expressed as linear combinations of plane waves travelling at the speed of light. Let us take one of these solutions (a general plane wave travelling to the right on the x-axis) to explore its physical meaning: A = A γ + A γ + A γ A = a (x- t)γ +a (x- t)γ + a (x- t)γ Applying the Lorenz gauge condition: ∇ ·A = 0 we get -a´ - a´ - 0 = 0 , so that a´ = - a´ and, except for a constant factor, we can write the wave potential as A = a (x- t)(γ - γ )+ a (x- t)γ and the corresponding electromagnetic field will be F = ∇∧ A = (∂ γ +∂ γ +∂ γ ) A = (∂ γ +∂ γ ) [a (x- t)(γ - γ )+ +a (x- t)γ ] = -a´ γ - a´ γ - a´ γ +a´ γ = a´ (- γ + γ ) This expression can be written in the multivector spacetime flavour as the product of a null direction (- γ +γ ) with a vector direction γ : F = a´ (- γ + γ ) γ Alternatively, in the vector spatial formulation we can express F as the sum of an electric field with a direction (- γ = σ ) and a magnetic field with a directio n (γ = σ ), both having the same numerical value a´ (in natural units). We can consider a pure plane wave solution as an approximation which is valid when we are very far away from the localized source charge-current J. When this approximation is not valid, the general solution will be a superposition or linear combination of such plane waves. A general approach to radiation should begin with the Poisson's equations which we have already seen having the form (∂ - ∂ - ∂ )A k = J k McDonald [18], citing Hadamard [19, 20] and Ehrenfest [21, 22], stated that in this 2-dimensional case the general solution has contributions from times earlier than the purely retarded time. This behaviour is clearly different from that of the radiation in 3 spatial dimensions, where only retarded times contribute to the radiation in potentials. This difference, nevertheless, vanishes when the distance from the source gets high enough, as we have seen. The same effect applies to the case of a vanishing loop current, which in the 3D case produces a vanishing magnetic field, but in the 2D situation the magnetic field has always a non-vanishing value.
Visualization of vector potentials in two dimensions
After presenting the concept of retarded time in the one-dimensional world, where it was useful to explain the concept of redshift, it can be used in the two-dimensional case to explain further concepts like the production and characteristics of plane electromagnetic waves. Figure 18: Retarded time for a moving charge Figure 18 is the equivalent of figure 10 for the two-dimensional spacetime. We can see the light cones produced by a uniformly moving charge in two instants P and P and how they are perceived by two observers A and B. Figure 19 tries to present this same situation in the two-dimensional space, where the light cones are drawn as circles and the radial lines correspond to the electric field of the charged particle. We can appreciate that the distortion of the field lines is due to the displacement of the circles, which is a consequence of the movement of the source. Figure 19: Electric field of a moving charge Electromagnetic waves are produced when the source undergoes any change from the uniform movement. These Liénard-Wiechert potentials and fields are described for the two-dimensional case by McDonald [18]. For the three-dimensional space, they were derived via Lorentz transformations [23] and discussed thoroughly in [24]. Figure 20 depicts one of the simplest changes, consisting in a brief displacement into a space direction and an equally brief returning displacement. This is called a burst. The displacement of the light cones following this burst produces a displacement of the field lines which is maximal in the transversal direction and diminishes in intensity until it apparently disappears in the direction of the burst. This explains the absence of electromagnetic waves in the one-dimensional case. Figure 20: Spacetime propagation of a burst The same situation is presented in the two-dimensional space in figure 21, where we can see the displacement of the light cones (represented by circles, as in figure 19), and the distortions in the field lines caused by the burst. In this figure, the burst happens in the horizontal (x) direction) direction, while the distortion is greater in the vertical direction (y) transversal to the burst, decreasing as we shift the direction to the horizontal axis, where there is no distortion at all. Figure 21: Electric field after a burst Figure 23 depicts the result of an oscillating charged particle, which can be seen as a series of consecutive bursts. The light cones from different positions of the source, when seen in a horizontal plane, produce an oscillating pattern.
Figure 22: Propagation of a dipole oscillation When a charged particle is displaced by a forcing field, the position formerly occupied by the particle remains with an opposite charge, like a small instantaneous dipole. This is the reason for calling it a dipole oscillation. Figure 23 shows this more clearly, where the displacement of the charge from the equilibrium position produces a dipolar field. The field points in the same direction as the displacement because the oscillating particle, in this case, has a negative charge.
Figure 23: Lateral displacement of the light cone surfaces and electric fields We have seen that in the vector spatial formulation the wave field F is the sum of an electric field E with a direction (- γ = σ ) and a magnetic field iB with a direction (γ = σ ), both having the same numerical value a ´. This is shown in figure 24, where a plane wave travelling in the x-axis presents a transversal oscillating electric field (straight arrows) and a corresponding magnetic field (curved arrows). Figure 24: Electromagnetic plane wave. Comparing with figure 15, it is possible to identify visually the propagating direction of the wave with the direction of energy transfer in an electric circuit. This is shown in the figure by the Poynting vector S . A further comparison with figure 13 leads us to understand that the points where electric and magnetic fields vanish (called nodes) are occupied by a displacement current j D as shown in figure 25. Figure 25: Displacement currents in an electromagnetic plane wave.
5. Three-dimensional electromagnetism
For the sake of simplicity, we introduce in the three-dimensional case the convention of representing the spatial indexes (1,2,3) of the vectorial components by the letter k, and the components of a given multivector direction by the ordered letters (a,b,c), grouping the various directions in a common symbol, as in γ a γ b = γ ab γ a γ b γ c = γ abc Appendix 2 shows a more detailed calculation using these notations; we have also completely separated the indexes in order to check easily the resulting expressions. The electromagnetic magnitudes that take part of the field equation in the general three-dimensional case are ∇ = ∂ γ + ∂ k γ k F = F γ + F ba γ ba J = J γ + J k γ k where ∂ = ∂/∂ t , ∂ k = ∂/∂x k The spacetime split gives the following correspondences: Jγ = (J γ +J k γ k )γ = ⍴ + (- j ) where j = j k σ k γ (Fγ ) = F σ k + F ba σ ab = E + i B where E = E k σ k and B = B k σ k γ ∇ = ∂/∂t + ∇ Spacetime and vectorial components are related by F = E k , F ba = B c , J = ρ , J k = -j k The field equation in this three-dimensional world would be ∇ F = J = J γ + J k γ k which can be separated in four multivector directions timelike vectors: ( ∇ ·F) t = J γ spacelike vectors: ( ∇ ·F) s = J k γ k timelike trivectors: ( ∇∧ F) t = 0 spacelike trivectors: ( ∇∧ F) s = 0 The timelike vector component of the three-dimensional field equation can be written as: ( ∇ ·F) t =∂ k F γ = J γ implying ∂ k F = J Translating to spatial vectors: ∂E k /∂x k = ρ or, equivalently: ∇ ·E = div (E) = ρ which is Gauss law. The physical meaning of this expression is the same as in the two-dimensional case, with the difference that the divergence operator extends now to the whole space instead of a single plane. The spacelike component of the three-dimensional field equation corresponds to the spacelike components J k of the current vector J: ( ∇ ·F) s = J k γ k implying ∂ F - ∂ b F ba + ∂ c F ac = J a Translating to spatial vectors ∂E a /∂ t - ∂B c /∂x b + ∂B b /∂x c = - j a Rearranging terms and recalling that ∇ = σ k ∂/∂x k and B = B k σ k ∇∧ B = (- ∂B b /∂x a + ∂B a ∂x b ) σ ba -i( ∇∧ B) = (∂B b /∂x a - ∂B a ∂x b ) σ c ∂ E /∂t = σ k ∂E k /∂t j = j k σ k we can rewrite the spatial vector term of the field equation as -i( ∇∧ B) = ∂ E /∂t + j Remembering the definition of the cross-product as v ✕ w = -i( v ∧ w) we can write ∇ ✕ B = -i( ∇∧ B) to get ∇ ✕ B = ∂ E /∂t + j which is Ampère´s equation. We introduced an alternative definition for the cross-product (to allow a smooth transition between three and two dimensions) which allowed us to write ∇ ✕ B = -i( ∇∧ B) = - ∇ ·(iB) It is interesting to compare both definitions at the same example we have already seen: - ∇ ·(iB) = - ∇ ·(σ abc B) = - ∇ ·(σ abc B k σ k ) = = - (σ k ∂/∂x k )·( B a σ bc ) =(∂B b ∂x a - ∂B a ∂x b ) σ c which coincides with -i( ∇∧ B) Any trivector part of the field equation must vanish identically, and this is true both for the timelike and the pure spatial trivectors. The timelike trivector expression at the three-dimensional field equation is: ( ∇∧ F) t = 0 or, equivalently, ∂ F ba = - ∂ a F + ∂ b F Translated to the corresponding vector magnitudes: ∂B c /∂t = - ∂E b /∂x a + ∂E a /∂x b The right sides are the components of the cross-product with the vector derivative ∇ ✕ E = - i ( ∇∧ E ) = (∂E b /∂x a - ∂E a /∂x b ) σ c we can thus write ∂ t B = - ∇ ✕ E which is Faraday´s law of induction. The spacelike trivector expression at the three-dimensional field equation must also be equally null, so: ( ∇∧ F) s = 0 or, equivalently, ∂ a F cb = 0 Translated to the corresponding vector magnitudes: ∂ B k /∂x k = 0 which can be expressed also as ∇ · B = 0 Which is the Gauss law for the magnetic field (implying the non-existence of magnetic sources like magnetic monopoles). The potential vector function in a three-dimensional world has the form A = A γ + A k γ k The field equation using this potential becomes ∇ A = J Or (∂ - ∂ k2 )(A γ +A k γ k ) = J γ + J k γ k Separating the different components we get four Poisson´s differential equations: (∂ - ∂ k2 ) A = J ( ∂ - ∂ k2 ) A k = J k The most interesting case appears again when we consider a chargeless space (vacuum): J = 0 In this case, all four differential equations reduce to the same form: (∂ - ∂ k2 )A = 0 or, equivalently, ∂ A/∂x - ∂ A/∂x k2 = 0 ∂ A/∂t = ∂ A/∂ x + ∂ A/∂y + ∂ ª/ ∂z This is the three-dimensional Laplace´s equation, whose solutions can be expressed as linear combinations of plane waves travelling at the speed of light. The form of any of those plane waves is similar to the one we have already seen for the two-dimensional case.
6. Conclusions
We have tried to show how Geometric Algebra and Geometric Calculus can be used to present a smooth sequence for the understanding of the basic concepts of electromagnetism, by doing what we call a dimensional scaffolding. This was possible with the application of the synthetic formulation for the electromagnetic field equation provided by Geometric Algebra and Geometric Calculus in different dimensions. In any dimension, we begin with the spacetime multivector formulation of the field equation, separating it into homogeneous parts and translating the results to the usual language of vector calculus. In this process, the main electromagnetic magnitudes and concepts have been appearing gradually, showing their main characteristics and allowing for their visualization. The two-dimensional vector derivative as presented by McDonald [18] has been deduced using Geometric Algebra and it has proven to have deep explanatory and visual capabilities throughout the corresponding two-dimensional toy universe. 6.1
Gauss´s law: charge, electric field and divergence
In one dimension, the electric field appears as a temporal bivector in spacetime with a boost-like effect, together with the concepts of spatial derivative and charge density in figure 6. Charges are clearly seen as sources for the electric field, and a toy model for a capacitor has been also presented in figure 7. In two dimensions, the concept of divergence gets a clear visualization by comparison between the spacetime diagram and its planar spacetime split (figure 12). 6.2
Ampère´s law: magnetic field, displacement current 16
The concepts of temporal derivative and current density appear already in one dimension and could be explained in figure 8 as the application of a simple boost to the diagram for Gauss´s law. The same figure allows us to visualize a particular and intriguing feature of the charge-current vector in spacetime, which is reflected with respect to the worldliness of the charged particles. This is an unavoidable characteristic of spacetime, as we show in Appendixes 1 and 2, but it has no counterpart in the physical description of electromagnetism. Although there is no equivalent to a magnetic field in one dimension, the elusive concept of displacement current could be introduced, showing a characteristic compensating attitude. In two dimensions, the magnetic field makes its appearance as a purely spatial bivector with a rotating attitude in figures 13 and 14, allowing for the introduction and basic understanding of new concepts like solenoids, capacitors and electric circuits (figures 14 to 16). The flow of energy represented by the Poynting vector, as well as the displacement current in a charging capacitor could even be addressed at this early level, showing how the magnetic field results from the combined effect of both charge and displacement currents. 6.3
Faraday´s law: electromotive force, induction
The magnetic field is absent in one dimension, and so does Faraday´s law, but in two dimensions it has been possible to explain -with help of a spacetime diagram as in figure 14- the induction of an electromotive force as a consequence of a time-varying magnetic field. The resulting induced current stands against the change that produces it, and this is in accordance with Lenz´s law too. 6.4
Poisson´s equation: potentials, waves, retarded time, Doppler effect.
The potential formulation for the field equation in one dimension allowed us to arrive in this extremely simple situation to the concept of wave equation, whose solutions are plane waves with the speed of light as shown in figure 9. Retarded time could also be presented and visualized in figure 10, allowing for the understanding of the redshift and retarded potentials. No physical meaning, nevertheless, could be assigned to those potential waves in a one-dimensional world because they render identically null electric fields. In two dimensions, the potentials for moving charges could be visualized as effects propagating with the speed of light in figures 18 to 22. The analysis of a plane wave solution together with its physical meaning as a combination of equal-valued electric and magnetic fields normal to the propagating direction were shown in figures 23 to 25. 6.5
Final remarks
We have shown how the gradual presentation of the fundamental electromagnetic concepts by a scaffolding process over increasing dimensions has been possible in a unified way using the extremely simplicity of the geometric algebra formulations for the field and potential equations. Nevertheless, the whole algebraic three-dimensional structure should be needed to introduce and explain some remaining concepts like the complementarity between electric and magnetic fields –as well as the absence of magnetic monopoles-, electromagnets and electric motors, rotating wires and generators, and wave polarization, for example.
7. Acknowledgements
We thank Kirk Mc Donald, from the Department of Physics of Princeton University, for fruitful discussions and the incorporation of one of our views as additional remarks in [18], and Rosana Rodríguez López, from the Departamento de Estatística, Análise Matemática e Optimización of the Universidade de Santiago de Compostela, for her help.
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The Relation Between Expressions for Time-Dependent Electromagnetic Fields Given by Jefimenko and by Panofsky and Phillips http://physics.princeton.edu/~mcdonald/examples/jefimenko.pdf Appendix 1 A1 Main features of Geometric Algebra and notations used in this work
Geometric algebra is based mainly on the concept of multivector as well as the existence of the geometric product and its strong property of being associative. The geometric product of two vectors can be decomposed as ab = a·b + a ∧ b where a·b = b·a is the inner product, whose result is a scalar. and a ∧ b = - b ∧ a is the outer product, whose result, as we will see, is called a bivector. The elements of an n-dimensional Geometric Algebra are called multivectors, and they can be expressed in a basis of 2 n elements, which can be arranged in blades of the same dimension such as scalars (dim = 0), vectors (dim =1), bivectors (dim = 2), trivectors (dim =3) or, alternatively, by decreasing dimensionalities such as pseudoscalars (dim = n), pseudovectors (dim = n-1), and so on. This is because the dimension of every blade is the number of vector directions that must be combined by the geometric product to produce the respective directions. For example, if the space has dimension N=3, the possible blades are scalars (dim=0), vectors (dim =1), bivectors. (dim=2) and trivectors (dim =3). The respective combinations of indexes are: none for the unique scalar direction, (1,2,3) for the three possible vector directions, (12, 23, 31) for the three bivector directions, and (123) for the unique trivector direction (which is also a pseudoscalar in this 3-dimensional space). The sum of all these numbers of multivector directions gives 1+3+3+1 = 8 = 2³. The number of directions of a D-dimensional blade D in a space of dimension N is the number of possible combinations of N elements taken in groups of D. The ordered arrangement of these dimensions gives the Fibonacci series, where every horizontal line corresponds to a certain space dimension N, and the elements are its blades in increasing order of their dimension D: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 The sum of all the elements in a horizontal line gives the dimension of the multivector algebra in the corresponding N-dimensional space: The elements of any of these blades can be expressed in an orthonormal basis, beginning with scalars (whose basis consists only of the number 1) and vectors (whose basis is a set of n ortho normal vectors {γ , γ , γ , ...γ n }) These basis elements are called directions, and they obey the following rules: γ i γ i = γ i ·γ i = ±1 (+1 for i = 0 and -1 for i = 1,2,3, …) This fundamental distinction between a timelike direction γ and a spacelike dire ction γ i≠0 is a characteristic feature of spacetime. An important consequence of it is the existence of null-directions like, for example, γ +γ . They form the so-called “null-cone” or “light cone”. Elements of higher dimensionality are consistently created by means of the outer product: γ k γ l = γ k ∧ γ l = - γ l γ k In this process, when two equal indexes are adjacent, they are removed and the direction is multiplied by the sign of their product. Two different adjacent indexes can exchange positions, changing the sign of the direction. Bivectors have basis elements like {γ γ , γ γ , γ γ , γ γ , γ γ , γ γ }. The inner product, in general, is the grade-lowering part of the geometric product, and the outer product is its grade-rising part, as we have seen for vectors. This is even true for a vector (v) and a bivector (W), whose geometric product can be decomposed as vW = v·W + v ∧ W where v·W = -W·v is the inner product and v ∧ W = W ∧ v is the outer product. The commutation properties, therefore, are not the same as in the case of vectors. This is due to the properties of the direction products. In the case of vectors, γ a γ b = γ ab = - γ ba = - γ b γ a for different indexes, while γ a γ a = γ a γ a for equal indexes is evident. In the case of a vector with a bivector, γ a γ bc = γ abc = γ bca = γ bc γ a for different indexes, whilst γ a γ ab = γ aab = - γ aba = γ ab γ a and γ a γ ba = γ aba = - γ baa = γ ba γ a if one of the indexes is shared. A1.1 Geometric Calculus
Geometric calculus is founded on the concept of a vector derivative ∇ , which can be expressed as ∇ = ∂ γ + ∂ γ + ∂ γ … where ∂ i means the partial derivative with respect to the coordinate x i ∂ i = ∂/∂x i The operator ∇ can be treated as a vector: ∇ = ∇ · ∇ = ∂ - ∂ - ∂ - ∂ = = ∂ /∂x - ∂ /∂x - ∂ /∂x - ∂ /∂x where the signs arise from the products γ i γ i . The Fundamental Theorem of Geometric Calculus, which embodies in a common way all sort of Stoke´s-type formulas in different dimensions and signatures, opens the possibility of inverting the vectorial derivative. The resulting antiderivative depends mainly on the use of the appropriate Green´s functions. A1.2 Spacetime Split
In order to connect spacetime with the usual framework of an euclidean space together with a separate time coordinate, we will use the so-called spacetime split. This begins with the selection of a certain timelike direction in spacetime which we will call for convenience γ . It can be identified with the observer's reference system and it is sometimes called simply “the observer”. The split is made by the geometric product of the observer with the physical magnitudes in spacetime. To see how this works we will present a pair of examples, the first for a spacetime vector V and the second for a spacetime bivector B. The split of a vector V is produced by Vγ = V· γ + V ∧ γ = v + v k where v is a scalar and v k is a vector in the euclidean space. These vectors are written in boldface type, and they are often called “relative vectors” to distinguish them from the more general spacetime vectors. The reason is that they depend on the - relative - choice of the observer γ . The split of a bivector W is produced by (Wγ ) γ =(W·γ ) γ +(W ∧ γ ) γ = w +i w k where both w and w k are relative vectors and i stands for the pseudoscalar. We introduce the notation i n to identify specifically the pseudoscalar in the n-dimensional euclidean space and i for the (1,n)-dimensional spacetime pseudoscalar, the former being the even subalgebra of the latter. If {γ , γ , ... , γ n } form an orthonormal basis for the (1,n)-dimensional spacetime, then its pseudoscalar will be given by the expression i = γ γ ... γ n = γ ∧ γ ∧ ... ∧ γ n We can build a basis for the corresponding n-dimensional euclidean space defining its elements as σ k = γ k0 = γ k γ = γ k ∧ γ In the three-dimensional space, they are equivalent to the directions σ = i , σ = j , σ = k The orthogonality is assured because of σ k2 = σ k σ k = γ k γ γ k γ = - γ k γ k γ γ = 1 (using γ γ = 1 and γ k γ k = -1) σ k σ l =γ k γ γ l γ = - γ k γ l γ γ = - γ k γ l = γ l γ k σ l σ k =γ γ l γ γ k = - γ γ γ l γ k = - γ l γ k = - σ k σ l We can now define the pseudoscalar i n = σ ... n = σ ... σ n = σ ∧ ... ∧ σ n Both types of pseudoscalar are related: i = σ = γ l0 = - γ = - i i = σ = γ l020 = - γ = -i i = σ = γ l02030 = i The euclidean space is 3-dimensional, therefore we can use i or i indistinctly, but for lower dimensionalities this is not the case, and we must specify which type of pseudoscalar is being used. The spacetime split of the vector derivative ∇ yields: ∇ γ = ∇ ·γ + ∇∧ γ = ∂ o + ∂ k γ k0 = = ∂/∂t + σ ∂/∂x + σ ∂/∂y + σ ∂/∂z = ∂/∂ t + ∇ The cross-product of two vectors can be defined as v ✕ w = -i( v ∧ w) The following identity: v ∧ w = - (i v ) · (i w ) allows us to use an alternative but equivalent definition for the cross product: v ✕ w = - v ·(i w) This alternative definition will be useful for the building of an analogue to the cross-product in the two-dimensional case.
A1.3 Electromagnetism in geometric algebra
The set of Maxwell´s Equations can be unified in the following expression: ∇ F = J This is the electromagnetic field equation in Geometric Algebra, and in this context we can call it simply the field equation. The symbol ∇ stands for the spacetime vector derivative as presented into the framework of Geometric Calculus. F is a bivector field: F = F γ + F ba γ ba which should be interpreted like a sum of terms where k can have the values (1,2,3) and we define the pairs of subindexes ba as having the values (21, 32, 13) With help of a spacetime split, this formula can be translated to relative vectors, getting in this way the four Maxwell equations in the usual vector formulation. Natural units (like Heaviside- Lorentz) where μ = 1 and ε = 1 are being used consistently and, consequently, the speed of light c = 1. Applying a spacetime split to the field bivector F yields γ Fγ = γ (F·γ ) + γ (F ∧ γ ) = E + i B where E is a relative vector (the electric field), corresponding to the temporal component of the bivector F, and B is the magnetic field vector. The product i B corresponds to the purely spatial part of the spacetime bivector F. The symbol i corresponds to the euclidean pseudoscalar (i n in our notation), in order to keep the expression valid for dimensions other than 3. F is thus splitted in two parts, the first part corresponding to the electric field E : F = E k and the second part to the magnetic field B : F ba = B c or, equivalently, F = B , F = B , F = B J is a vector field: J = J γ + J i γ i Jγ = J·γ + J ∧ γ = ⍴ + (- j ) where the scalar part corresponds to the charge density (J = ρ), and the vector part to the vector current dens ity j : J i = - j i The negative sign appearing at this last identity is needed to assure a perfect correspondence between the spacetime formulation and the usual vectorial set of Maxwell´s formulas, as explained in Appendix 1. The bivector F can be derived from a vector-valued potential function A by means of the formula: ∇ A = F together with a gauge to eliminate ambiguities. We will follow the Lorenz gauge: ∇ ·A = F The form of the field equation in GA using this vector potential is ∇ F = ∇ ( ∇ A) = ∇ A = J
A1.4 Correspondence between spacetime and purely spatial magnitudes
We begin with the electromagnetic equations on both formulations: ∇ F = J where ∇ = ∂ γ + ∂ γ + ∂ γ + ∂ γ = ∂ γ + ∂ k γ k (k = 1,2,3) F = F γ +F γ + +F γ + F γ + F γ + F γ = F γ + F ba γ ba (k =1,2,3) and (ab =21, 32, 13) J = J γ + J γ + J γ + J γ = J γ + J k γ k (k = 1,2,3) for the spacetime multivector formulation, and the set of four Maxwell´s laws for the purely spatial vector formulation: Gauss: ∇ · E = ⍴ Faraday: ∇⨉ E = - ∂ B /∂t Ampère: ∇⨉ B = j + ∂ E /∂ t Gauss for B: ∇ · B = 0 where ∇ = σ ∂/∂x + σ ∂/∂y + σ ∂/∂z = σ k ∂ k is the spatial vector derivative E = E σ + E σ + E σ = E k σ k B = B σ + B σ + B σ = B k σ k are the electric and magnetic fields, and ⨉ is the usual cross- product, which can be expressed also as v ⨉ w = -i v ∧ w , being i the pseudoscalar i = σ σ σ Beginning with Gauss´s law for the electric field: ∇ · E = ⍴ σ k ∂ k E k σ k = ∂ k E k = ⍴ the corresponding term in spacetime is the timelike vector relation: ∂ k γ k F γ = J γ ∂ k F γ k γ = - ∂ k F γ kk0 = ∂ k F γ and ∂ k F = J identifying J with the charge density ⍴ , it follows that F = E k (F = E , F = E , F = E ) Faraday´s equation ∇⨉ E = - ∂ B / ∂ t can be written as ∂ a E b - ∂ b E a = ∂ t B c where the indexes (a, b, c) stand for (1, 2, 3), (2, 3, 1) or (3, 1, 2) in order to write down three different terms. In spacetime, the corresponding equation has to be obtained from the timelike trivector part: ∂ F ba γ + ∂ a F γ a0b + ∂ b F γ b0a = 0 (because J is only a vector-valued function in spacetime, without trivector parts) Rearranging terms to get the same direction: ∂ F ba γ ba0 + ∂ a F γ ba0 - ∂ b F γ ba0 = 0 we get ∂ F ba + ∂ a F - ∂ b F = 0 where we can substitute F with E k to obtain ∂ F ba + ∂ a E b - ∂ b E a = 0 an d ∂ a E b - ∂ b E a = - ∂ F ba A direct comparison with the vectorial expression obtained earlier ∂ a E b - ∂ b E a = ∂ t B c allows us to write F ba = B c (F = B , F = B , F = B ) Recalling finally Ampère´s law ∇ ⨉ B = j + ∂ E / ∂ t which can be written as ∂ a B b - ∂ b B a = j c + ∂ t E c using the same convention for the combinations of subindexes, we can compare it with the the purely spatial vector expression in spacetime ∂ F γ + ∂ b F cb γ bcb + ∂ a F ac γ aac = (∂ F + ∂ b F cb - ∂ a F ac )γ c = J c γ c ∂ F + ∂ b F cb - ∂ a F ac = J c ∂ a F ac - ∂ b F cb = - J c + ∂ F we can substitute again the spacetime components with their space equivalents as obtained previously to get ∂ a B b - ∂ b B a = - J c + ∂ E c and, recalling our previous expression ∂ a B b - ∂ b B a = j c + ∂ t E c we arrive to the equivalence J c = - j c (J = - j , J = - j , J = - j ) We have already introduced this minus sign in section 3, where we explained that it does not imply any change in the physical magnitude j . Its meaning can be interpreted as the spacetime vector J being spatially reflected with respect to the current vector, and this is a striking geometric feature with no physical counterpart. Appendix 2 A2. Detailed calculations using components
As we have seen, Geometric Algebra offers the possibility of expressing the electromagnetic formulas in very synthetic coordinate-free expressions which hold formally for several dimensions. Nevertheless, more detailed calculations using coordinates are also possible, and we include here some calculations of the results we have seen previously as an exercise for readers who are not yet familiar with the subtleties of Geometric Algebra and want to check where they arise from and how they function. The following tables may be helpful to understand in a straightforward way how the directions combine in geometric algebra, applied to the field equation in the four-dimensional spacetime and its corresponding three-dimensional space. The field equation ∇ F = J bears a geometric product between the vector derivative ∇ ∇ = ∂ γ + ∂ γ + ∂ γ + ∂ γ and the field bivector F F = F γ +F γ +F γ +F γ +F γ +F γ The result of a geometric product between a vector and a bivector consists, in general, in the sum of a vector v and a trivector T: ∇ F = v+T Where v = ∇ ·F, T = ∇∧ F In the special case of the electromagnetic field equation, the vector part is the charge-current vector J, and the trivector part is equally null: ∇ F = ∇ ·F = J A series of tables will show us how the four vector directions (γ , γ , γ , γ ) combine with the six bivector directions (γ , γ , γ , γ , γ , γ ). The first step consists in creating all the possible ordered arrays of the indexes, as shown in table 1: γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ Table 1: geometric product of vector and bivector directions. We apply now the rules for the combination and exchange of indexes in order to write them in a coherent way in table 2: γ = 1, γ kk = -1 (for k = 1, 2, 3), γ rs = - γ sr (for r ≠ s) γ γ γ γ γ γ γ γ γ γ γ γ γ γ - γ γ γ γ - γ - γ γ γ γ γ γ γ γ - γ - γ γ γ γ - γ γ Table 2: combination and reordering of indexes We can simplify this into table 3 by taking into account the properties of the pseudoscala r i = γ iγ = γ = - γ , iγ = γ = - γ , iγ = γ = - γ , iγ = γ = - γ γ γ γ γ γ γ γ γ γ γ γ γ γ γ - iγ iγ iγ γ - iγ - iγ iγ γ iγ iγ iγ - iγ γ - γ - γ - iγ γ γ - γ - iγ Table 3: introducing the pseudoscalar i We can recognize in table 3 the following groups of directions: Timelike vectors < ∇ ·F> t : direction γ Spacelike vectors < ∇ ·F> s : directions (γ , γ , γ ) Spacelike trivectors < ∇∧ F> s : direction iγ = - γ Mixed trivectors < ∇∧ F> t : directions (iγ , iγ , iγ ) Now we write the partial derivatives of the bivector components in table 4 following the same order. F F F F F F ∂ ∂ ∂ ∂ ∂ F ∂ F ∂ F ∂ F ∂ F ∂ F ∂ F ∂ F ∂ F ∂ F ∂ F ∂ F ∂ F ∂ F ∂ F ∂ F ∂ F ∂ F ∂ F ∂ F ∂ F ∂ F ∂ F ∂ F Table 4: partial derivatives of the field bivector F The final expressions will be obtained by the combination of the terms in table 4 that share the same group of indexes in table 3. The resulting vector J = ∇ ·F has two parts: -A timelike component < ∇ ·F> t wit h direction γ , (∂ F +∂ F +∂ F )γ =J γ which can be written as ∂ k F = J (F1) -A spacelike component < ∇ ·F> s whose direction is a combination of (γ , γ , γ ) (∂ F + ∂ F - ∂ F ) γ = J γ (∂ F + ∂ F - ∂ F ) γ = J γ (∂ F + ∂ F - ∂ F ) γ = J γ Which can be written as ∂ F + ∂ b F ab - ∂ c F ca = J a (F2) Where the indexes (a,b,c) can have the following values: (1,2,3), (2,3,1) or (3,1,2) The directions showing the pseudoscalar i correspond to the components of the trivector parts, whose combinations are identically null for the electromagnetic field equation. The corresponding equation ∇∧ F = 0 can be separated also in two parts: -A purely spacelike component < ∇∧ F> s with direction iγ = - γ (∂ F +∂ F +∂ F )iγ =0 which can be written as ∂ a F ab = 0 (F3) -A partially timelike component < ∇∧ F> t whose direction is a combination of (iγ , iγ , iγ ) (∂ F + ∂ F - ∂ F ) iγ = 0 (∂ F + ∂ F - ∂ F ) iγ = 0 (∂ F + ∂ F - ∂ F ) iγ = 0 Which can be written as ∂ F bc + ∂ c F - ∂ b F = 0 (F4) The resulting field equations (F) F1: ∂ k F = J F2: ∂ F + ∂ b F ab - ∂ c F ca = J a F3: ∂ a F ab = 0 F4: ∂ F bc + ∂ c F - ∂ b F = 0 can be compared with the corresponding Maxwell´s equations (M): -Gauss: ∇ ·E = ρ or, equivalently, ∂ k E k = ρ (M1) -Ampère: ∇ xB = ∂ E /∂t + j implying ∂ b B c - ∂ c B b = ∂E a /∂t + j a (M2) -Gauss for B: ∇· B = 0, or ∂ a B a = 0 (M3) -Faraday: ∇ x E = - ∂ B /∂t ∂ b E c - ∂ c E b = - ∂B a /∂t (M4) We can now make a stepwise comparison between the expressions resulting from the field equation (F) and those obtained from Maxwell´s laws (M) in order to establish direct coherent correspondences between the spacetime magnitudes and their related spatial equivalents. Beginning with F1 and M1 (Gauss): ∂ k F = J , ∂ k E k = ρ It is evident that the natural choice of J as equivalent to the charge density (J = ρ ) leads to the identification between the temporal components of the field bivector and those of the electric field (F = E k ). We can now resort to F4 ∂ F bc + ∂ c F - ∂ b F = 0, rewrite it as ∂ F bc + ∂ c E b - ∂ b E c = 0 or ∂ b E c - ∂ c E b = ∂ F bc and make a comparison with M4 (Faraday) ∂ b E c - ∂ c E b = - ∂B a /∂t The partial spacetime derivative ∂ is equivalent to the time derivative ∂/∂t, and as a result the components of the magnetic field are related to the purely spatial components of the field bivector by the expression F bc = -B a which embodies three identities: F = -B , F = -B , F = -B Finally, F2: ∂ F + ∂ b F ab - ∂ c F ca = J a rewritten as ∂ E a - ∂ b B c + ∂ c B b = J a or ∂ b B c - ∂ c B b = ∂ E a - J a can be compared with M2 (Ampère) ∂ b B c - ∂ c B b = ∂E a /∂t + j a arriving finally to the expression J a = - j a A final remark: we could have arrived to a different set of equivalences changing the signs in all of them, beginning with J = - ρ which applying Gauss´s law would lead to F = -E k Faraday´s law would require in this case that F bc = -B a And finally applying Ampère´s law we would arrive to J a = j a Although this can seem more natural, changing the sign in J0