# Electrodynamics in geometric algebra

Sylvain D. Brechet11footnotemark: 1 Institute of Physics, Station 3, Ecole
Polytechnique Fédérale de Lausanne - EPFL, CH-1015 Lausanne, Switzerland
<EMAIL_ADDRESS>
###### Abstract

We consider the electrodynamics of electric charges and currents in vacuum and
then generalise our results to the description of a dielectric and magnetic
material medium : first in spatial algebra (SA) and then in space-time algebra
(STA). Introducing a polarisation multivector
$\tilde{P}=\boldsymbol{\tilde{p}}-\,\frac{1}{c}\,\boldsymbol{\tilde{M}}$ and
an auxiliary electromagnetic field multivector
$G=\varepsilon_{0}\,F+\tilde{P}$, we express the Maxwell equation in the
material medium in SA. Introducing a bound current vector
$\tilde{J}=J-\,c\,\nabla\cdot\tilde{P}$ in space-time, the Maxwell equation is
then expressed in STA. The wave equation in the material medium is obtained by
taking the gradient of the Maxwell equation. For a uniform electromagnetic
medium consisting of induced electric and magnetic dipoles, the stress-energy
momentum vector is written as
$\dot{T}\left(\dot{\nabla}\right)=\frac{1}{c}\,J\cdot F=f$ where $f$ is the
electromagnetic force density vector in space-time. Finally, the Maxwell
equation in the material medium can be written in STA as a wave equation for
the potential vector $A$.

††journal: Journal of LaTeX Templates

###### Contents

  1. 1 Introduction
  2. 2 Maxwell equations in vector space (VS)
  3. 3 Maxwell equations in spatial algebra (SA)
  4. 4 Maxwell equation in vacuum (SA)
  5. 5 Electromagnetic waves in vacuum (SA)
  6. 6 Electromagnetic energy and momentum in vacuum (SA)
  7. 7 Poynting theorem in vacuum (SA)
  8. 8 Electric and magnetic potentials (SA)
  9. 9 Maxwell equation in matter (SA)
  10. 10 Electromagnetic waves in matter (SA)
  11. 11 Electromagnetic energy and momentum in matter (SA)
  12. 12 Poynting theorem in matter (SA)
  13. 13 Maxwell equation in vacuum (STA)
  14. 14 Maxwell equation in matter (STA)
  15. 15 Electromagnetic waves in vacuum (STA)
  16. 16 Electromagnetic waves in matter (STA)
  17. 17 Electromagnetic fields (STA)
  18. 18 Stress energy momentum in vacuum (STA)
  19. 19 Stress energy momentum in matter (STA)
  20. 20 Electromagnetic potential (STA)
  21. 21 Conclusion
  22. A Spatial algebra (SA)
  23. B Duality in spatial algebra (SA)
  24. C Differential duality in spatial algebra (SA)
  25. D Algebraic identities in spatial algebra (SA)
  26. E Differential algebraic identities in spatial algebra (SA)
  27. F Space-time algebra (STA)

##  1 Introduction

In his seminal paper of $1865$ entitled “A dynamical theory of the
electromagnetic field” [1], James Clerk Maxwell obtained a fully self
consistent field description of electromagnetic phenomena. The electromagnetic
fields he used were introduced by Michael Faraday in order to accurately
present the results of his experiments. In his earlier attempts to establish a
theory of electromagnetic phenomena, Maxwell first used mechanical analogies
to explain these phenomena in mechanical terms [2]. With great insight, he
understood that since the mechanical scaffolding of his theory was neither
relevant nor needed, he could simply get rid of it. By doing so, Maxwell
revealed a beautiful field theory that served as a template for numerous other
field theories. It was a paramount paradigm shift in the history of physics
that paved the way for the discovery of special relativity presented by Albert
Einstein as “the electrodynamics of moving bodies” [3]. The prediction of
electromagnetic waves played historically a key role for the foundations of
quantum mechanics that resulted from the synthesis of the wave mechanics of
Erwin Schrödinger [4] and the matrix mechanics of Werner Heisenberg, Max Born
and Pascual Jordan [5, 6]. The importance of Maxwell’s work was clearly stated
by Einstein on the centenary of Maxwell’s birth : “We may say that, before
Maxwell, physical reality, in so far as it was to represent the process of
nature, was thought of as consisting in material particles, whose variations
consist only in movements governed by partial differential equations. Since
Maxwell’s time, physical reality has been thought of as represented by
continuous fields, governed by partial differential equations, and not capable
of any mechanical interpretation. This change in the conception of Reality is
the most profound and the most fruitful that physics has experienced since the
time of Newton.” [7] Maxwell understood the importance of his theory of
electromagnetism and shared his enthusiasm with his cousin Charles Cay : “I
have also a paper afloat, containing an electromagnetic theory of light,
which, till I am convinced to the contrary, I hold to be great guns”. [8]

Maxwell’s equations are usually presented in textbooks as a set of four
vectorial equations [9, 10]. This was not the direct result of Maxwell’s work.
It was in fact Oliver Heaviside [11] who gathered Maxwell’s equations into a
set of four vectorial equations in $1884$ using the vector calculus he
coformulated with Josiah Willard Gibbs [12]. In his initial paper published in
$1865$, Maxwell wrote $20$ equations in components. Subsequently, in his
“Treatise on Electricity and Magnetism” [13], Maxwell recast his set of $20$
equations in terms of quaternions discovered by William Rowan Hamilton in
$1843$. Since, the vector space of Heaviside and Gibbs is ideally suited to
describe translations, it was adopted historically as the main geometric
framework of physics. However, the quaternion algebra $\mathbb{H}$ is far more
suited for the description of rotations than the vector space $\mathbb{R}^{3}$
where pseudovectors need to be introduced in order to treat rotations. This
naturally prompts the question : “Do we really need to choose or is it
possible to take advantage of both framework at the same time ?” In fact, the
good news is that both frameworks belong to a broader mathematical framework
called either geometric algebra (GA) $\mathbb{G}^{3}$ or Clifford algebra
$C\ell_{3}\left(\mathbb{R}\right)$ that was discovered by William Kingdom
Clifford in $1878$ in an article entitled “Applications of Grassmann’s
extensive algebra” [14].

In geometric algebra (GA) $\mathbb{G}^{3}$, there are four types of natural
geometric entities. First, there are geometric entities of dimension $0$
called scalars, which are oriented points defined by their magnitude and
orientation (e.g. positive or negative). Second, there are geometric entities
of dimension $1$ called vectors, which are oriented lines defined by their
magnitude and orientation. Third, there are geometric entities of dimension
$2$ called bivectors, which are oriented surfaces defined by their magnitude
and orientation. Fourth, there are geometric entities of dimension $3$ called
trivectors or pseudoscalars, which are oriented volume defined by their
magnitude and orientation (e.g. positive or negative). The geometric algebra
(GA) $\mathbb{G}^{3}$ is a vector space consisting of multivectors, which are
linear combinations of scalars, vectors, bivectors and pseudoscalars. The
quaternion algebra $\mathbb{H}$, which consists of linear combination of
scalars and bivectors is the even subalgebra of the geometric algebra (GA)
$\mathbb{G}^{3}$. The vector space $\mathbb{R}^{3}$, which consists of linear
combination of vectors, is a subspace of the geometric algebra (GA)
$\mathbb{G}^{3}$. The geometric algebra (GA) $\mathbb{G}^{3}$ is a vector
space endowed with two composition laws : the inner and outer product of two
multivectors. The algebraic sum of the inner and outer product of two vectors
is called the geometric product, as explained in Appendix A.

The geometric algebra (GA) $\mathbb{G}^{3}$ is also called the spatial algebra
(SA). Since special relativity was based on electromagnetism, it is relevant
to generalise the spatial algebra (SA) to include an additional temporal
dimension. The generalisation of the spatial algebra (SA) $\mathbb{G}^{3}$ to
space-time is the geometric algebra (GA) $\mathbb{G}^{1,3}$ or the Clifford
algebra $C\ell_{1,3}\left(\mathbb{R}\right)$ called the space-time algebra
(STA) developed by Hestenes [15]. The spatial algebra (SA) $\mathbb{G}^{3}$ is
isomorphic, or equivalent in structure, to the even subalgebra of the space-
time algebra (STA) $\mathbb{G}^{1,3}$. Formally, bivectors consisting of the
outer product of a space vector and a time vector in $\mathbb{G}^{1,3}$ are
isomorphic to spatial vectors in $\mathbb{G}^{3}$, the outer product of these
bivectors in $\mathbb{G}^{1,3}$ are isomorphic to spatial bivectors in
$\mathbb{G}^{3}$ and the pseudoscalar $I$ in $\mathbb{G}^{3}$ is isomorphic to
the pseudoscalar in $\mathbb{G}^{1,3}$.

Geometric algebra in its various forms, namely spatial algebra (SA)
$\mathbb{G}^{3}$ or space-time algebra (STA) $\mathbb{G}^{1,3}$, appears to be
the most appropriate language to describe physical phenomena since its
structure is based on the geometry underlying the natural world. In his memoir
entitled “Reapings and Sowings”, Alexander Grothendieck, one of the greatest
mathematician of the $20^{\text{th}}$ century states [16] : “If there is one
thing in mathematics that fascinates me more than anything else (and doubtless
always has), it is neither ‘number’ nor ‘size,’ but always form. Among the
thousand-and-one faces whereby form chooses to reveal itself to us, the one
that fascinates me more than any other and continues to fascinate me, is the
structure hidden in mathematical things.”

This article is structured as follows. In the first part, we begin by stating
Maxwell’s equations in vector space (VS) in Sec. 2. Then, we recast them in
spatial algebra (SA) in Sec. 3. In order to do so, we introduce a magnetic
induction field bivector $\boldsymbol{B}$ and a auxiliary magnetic field
bivector $\boldsymbol{H}$ by duality with the corresponding vectors
$\boldsymbol{b}$ and $\boldsymbol{h}$. The duality operation in spatial
algebra (SA) is defined in Appendix B.

In the second part, presented in Sec. 4-7, we examine the foundations of
electromagnetism in spatial algebra (SA) in the absence of a material medium
which we refer to as “vacuum”. The term “vacuum” does not mean here that there
are no electric charges or no electric current but simply that the medium is
vacuum. By introducing an electromagnetic multivector
$F=\boldsymbol{e}+c\,\boldsymbol{B}$ in spatial algebra (SA) that is
algebraically isomorphic to the complex Riemann-Silberstein vector
$\boldsymbol{f}=\boldsymbol{e}+i\,c\,\boldsymbol{b}$ in complex vector space
$\mathbb{C}^{3}$, we then show in Sec 4. that the four Maxwell equations are
written as a single equation in spatial algebra (SA). In Sec. 5, we show that
electromagnetic waves are described quite naturally in spatial algebra (SA).
In Sec 6, we obtain an expression of the energy density and momentum density
in terms of the electromagnetic field multivector
$F=\boldsymbol{e}+c\,\boldsymbol{B}$, its reverse
$F^{{\dagger}}=\boldsymbol{e}-\,c\,\boldsymbol{B}$, the auxiliary
electromagnetic field multivector
$G=\boldsymbol{d}+\frac{1}{c}\,\boldsymbol{H}$ and its reverse
$G^{{\dagger}}=\boldsymbol{d}-\,\frac{1}{c}\,\boldsymbol{H}$, which allows us
to establish Poynting’s theorem in spatial algebra (SA) in Sec (7). In Sec. 8,
the electromagnetic field multivector $F$ is expressed in terms of the
electric scalar potential $\phi$ and the magnetic vector potential
$\boldsymbol{a}$.

In the third part, shown in Sec. 9-12, we examine the foundations of
electromagnetism in spatial algebra (SA) for a material medium with electric
and magnetic dipoles, which we refer to as “matter”, where we perform a
similar analysis as in “vacuum”.

In the fourth part, presented in Sec. 13-19, we recast the results obtained
thus far within the space-time algebra (STA). In Sec. 13 and 14, Maxwell’s
equation in vacuum and matter is written very elegantly in space-time algebra
(STA). In Sec. 17, the electromagnetic field multivectors $F$ and $G$ and the
electric polarisation multivector $\tilde{P}$ are expressed as bivectors in
space-time algebra (STA). In Sec. 18 and 19, the stress energy momentum vector
is established in vacuum and matter in space-time algebra (STA). In Sec 19, we
show that the electromagnetic field bivector in space-time $F$ is the curl of
the potential vector $A$ in space-time.

Appendix A sets the foundations of spatial algebra (SA) and provides
identities for the products of vectors, bivectors and the pseudoscalar. The
duality between geometric entities in spatial algebra is presented in Appendix
B. The duality of the gradient, the divergence or the curl is establish in
Appendix C. Dual identities involving geometric entities in spatial algebra is
derived in Appendix D. Dual identities involving the gradient, the divergence
or the curl are shown in Appendix E. Finally, Appendix F sets the foundations
of space-time algebra (STA) and provides identities for the products of
vectors, bivectors and multivectors.

The notation convention adopted in this paper is the following. To distinguish
typographically different types of entities in the spatial algebra (SA), we
will use in general lower case letters for scalars like the charge density
$q$, lower case bold letters for vectors like the electric field
$\boldsymbol{e}$, uppercase bold letters for bivectors like the magnetic
induction field $\boldsymbol{B}$ and uppercase letters for multivectors like
the electromagnetic field $F$. In the space-time algebra (STA), we will in
general use uppercase letters for vectors like the electric current density
$J$ and for bivectors like the auxiliary electromagnetic field $G$.

##  2 Maxwell equations in vector space (VS)

In vector space (VS), electromagnetism is described by a set of four vector
fields : the electric field vector $\boldsymbol{e}$, the magnetic field
pseudovector $\boldsymbol{b}$, the electric displacement field vector
$\boldsymbol{d}$, the auxiliary magnetic field pseudovector $\boldsymbol{h}$.
The Maxwell equations relate these fields to the electric charge distribution
described by the scalar density field $q$ and the electric current
distribution described by the vector density field $\boldsymbol{j}$, [9, 10]

$\displaystyle\boldsymbol{\nabla}\cdot\boldsymbol{d}=q$ (1)
$\displaystyle\boldsymbol{\nabla}\times\boldsymbol{h}=\partial_{t}\,\boldsymbol{d}+\boldsymbol{j}$
(2)
$\displaystyle\boldsymbol{\nabla}\times\boldsymbol{e}=-\,\partial_{t}\,\boldsymbol{b}$
(3) $\displaystyle\boldsymbol{\nabla}\cdot\boldsymbol{b}=0$ (4)

The electric Gauss equation (1) is a scalar equation representing the local
electric flux and the magnetic Gauss equation (4) is a pseudoscalar equation
representing the local magnetic flux. The Maxwell-Ampère equation (2) is a
vectorial equation representing the local electric circulation and the Faraday
equation (3) is pseudovectorial equation representing the local magnetic
circulation.

The divergence of the Maxwell-Ampère relation (2) is written as,

$\boldsymbol{\nabla}\cdot\left(\boldsymbol{\nabla}\times\boldsymbol{h}\right)=\partial_{t}\left(\boldsymbol{\nabla}\cdot\boldsymbol{d}\right)+\boldsymbol{\nabla}\cdot\boldsymbol{j}=0$
(5)

Substituting the electric Gauss equation (1) into relation (5), we obtain the
electric continuity equation,

$\partial_{t}\,q+\boldsymbol{\nabla}\cdot\boldsymbol{j}=0$ (6)

The Lorentz force density vector is given by, [9, 10]

$\boldsymbol{f}=q\left(\boldsymbol{e}+\boldsymbol{v}\times\boldsymbol{b}\right)$
(7)

##  3 Maxwell equations in spatial algebra (SA)

In spatial algebra (SA), pseudovectors are replaced by bivectors [17].
Pseudovectors are the dual of bivectors. Bivectors respresent oriented plane
elements that are orthogonal to the dual pseudovectors. The area or
“magnitude” of the bivector corresponds to the length or “norm” of the dual
pseudovector. The orientation of this duality is given by the right hand rule.
If the right hand rotates along the oriented plane element defined by the
bivector then the dual pseudovector is oriented along the direction of the
thumb of the right hand. The converse duality is given by the left hand rule.
If the thumb of the left hand is oriented along the pseudovector then the left
hand rotates along the oriented plane element defined the dual bivector. Note
that this orientation is the opposite of the one that would be obtained with
the right hand rule. Thus, the dual of the dual of a bivector yields a
bivector with the opposite orientation, which is the opposite of the initial
bivector. Similarly, the dual of the dual of a vector yields a vector with the
opposite orientation, which is the opposite of the initial vector. [18] In
view of this duality, we will now recast the Maxwell equations (1)-(4) in
terms of vectors and bivectors.

The auxiliary magnetic field is generated by an electric current circulating
in a loop, which is an oriented plane element. The auxiliary magnetic field
pseudovector in vector space $\boldsymbol{h}$ is orthogonal to the plane of
the loop. It is the dual (401) of the auxiliary magnetic field bivector
$\boldsymbol{H}$ oriented along the circulation of the current density
$\boldsymbol{j}$ [18],

$\boldsymbol{H}^{\ast}=\boldsymbol{h}\qquad\text{where}\qquad|\boldsymbol{H}|=|\boldsymbol{h}|$
(8)

where the duality is denoted with a $\vphantom{a}\ast$. The dual of this
duality (394) is,

$\boldsymbol{h}^{\ast}=\left(\boldsymbol{H}^{\ast}\right)^{\ast}=-\,\boldsymbol{H}$
(9)

According to the vectorial duality (435), the curl of the auxiliary magnetic
field pseudovector $\boldsymbol{h}$ is the opposite of the divergence of the
auxiliary magnetic field bivector $\boldsymbol{H}$,

$\boldsymbol{\nabla}\times\boldsymbol{h}=-\,\boldsymbol{\nabla}\cdot\boldsymbol{H}$
(10)

Substituting relation (10) into the Maxwell-Ampère equation (2), it becomes,

$\boldsymbol{\nabla}\cdot\boldsymbol{H}=-\,\partial_{t}\,\boldsymbol{d}-\,\boldsymbol{j}$
(11)

In vacuum, the magnetic induction field is generated by an electric current
circulating in a loop, which is an oriented plane element. The magnetic
induction field pseudovector in vector space $\boldsymbol{b}$ is orthogonal to
the plane of the loop. It is the dual (401) of the magnetic induction field
bivector $\boldsymbol{B}$ oriented along the circulation of the current
density $\boldsymbol{j}$,

$\boldsymbol{B}^{\ast}=\boldsymbol{b}\qquad\text{where}\qquad|\boldsymbol{B}|=|\boldsymbol{b}|$
(12)

The dual of this duality (394) is,

$\boldsymbol{b}^{\ast}=\left(\boldsymbol{B}^{\ast}\right)^{\ast}=-\,\boldsymbol{B}$
(13)

According to the vectorial duality (435), the curl pseudovector of the
electric field vector $\boldsymbol{e}$ is the dual of the curl bivector of the
electric field vector $\boldsymbol{e}$,

$\boldsymbol{\nabla}\times\boldsymbol{e}=\left(\boldsymbol{\nabla}\wedge\boldsymbol{e}\right)^{\ast}$
(14)

In view of relations (12) and (14), the Faraday equation (3) is recast as,

$\left(\boldsymbol{\nabla}\wedge\boldsymbol{e}\right)^{\ast}=-\,\partial_{t}\,\boldsymbol{B}^{\ast}$
(15)

The opposite of the dual of the Faraday equation (15) is given by,

$\boldsymbol{\nabla}\wedge\boldsymbol{e}=-\,\partial_{t}\,\boldsymbol{B}$ (16)

Using the vectorial duality (436) and (437), the dual of the divergence of the
magnetic induction field pseudovector $\boldsymbol{b}$ is the opposite of the
curl of the magnetic induction field bivector $\boldsymbol{B}$,

$\left(\boldsymbol{\nabla}\cdot\boldsymbol{b}\right)^{\ast}=\boldsymbol{\nabla}\wedge\boldsymbol{b}^{\ast}=-\,\boldsymbol{\nabla}\wedge\boldsymbol{B}$
(17)

In view of relation (17), the dual the magnetic Gauss equation (4) is given
by,

$\boldsymbol{\nabla}\wedge\boldsymbol{B}=0$ (18)

Thus, in spatial algebra (SA), the Maxwell equations are written as,

$\displaystyle\boldsymbol{\nabla}\cdot\boldsymbol{d}=q$ (19)
$\displaystyle\boldsymbol{\nabla}\cdot\boldsymbol{H}=-\,\partial_{t}\,\boldsymbol{d}-\,\boldsymbol{j}$
(20)
$\displaystyle\boldsymbol{\nabla}\wedge\boldsymbol{e}=-\,\partial_{t}\,\boldsymbol{B}$
(21) $\displaystyle\boldsymbol{\nabla}\wedge\boldsymbol{B}=0$ (22)

The electric Gauss equation (19) is a scalar equation, the Maxwell-Ampère
equation (20) is a vectorial equation, the Faraday equation (21) is a
bivectorial equation or pseudovectorial equation, and the magnetic Gauss
equation (22) is a trivectorial equation, or pseudoscalar equation.

The divergence of the Maxwell-Ampère relation (20) is written as,

$\boldsymbol{\nabla}\cdot\left(\boldsymbol{\nabla}\cdot\boldsymbol{H}\right)=-\,\partial_{t}\left(\boldsymbol{\nabla}\cdot\boldsymbol{d}\right)-\,\boldsymbol{\nabla}\cdot\boldsymbol{j}=0$
(23)

Substituting the electric Gauss equation (19) into relation (23), we obtain
the electric continuity equation,

$\partial_{t}\,q+\boldsymbol{\nabla}\cdot\boldsymbol{j}=0$ (24)

According to the vectorial duality (412), (413) and (13), and to the
antisymmetry of the inner product with a bivector (360), the cross product of
the velocity vector $\boldsymbol{v}$ and the magnetic induction field
pseudovector $\boldsymbol{b}$ is the inner product of the magnetic induction
field bivector $\boldsymbol{B}$ and the velocity vector $\boldsymbol{v}$,

$\boldsymbol{v}\times\boldsymbol{b}=\left(\boldsymbol{v}\wedge\boldsymbol{b}\right)^{\ast}=\boldsymbol{v}\cdot\boldsymbol{b}^{\ast}=-\,\boldsymbol{v}\cdot\boldsymbol{B}=\boldsymbol{B}\cdot\boldsymbol{v}$
(25)

In view of the vectorial duality (25), the Lorentz force density vector (7) is
recast as,

$\boldsymbol{f}=q\left(\boldsymbol{e}+\boldsymbol{B}\cdot\boldsymbol{v}\right)$
(26)

##  4 Maxwell equation in vacuum (SA)

We now show that in vacuum, the Maxwell equations (19)-(22) reduce to a single
equation. The term “vacuum” does not mean here that there are no electric
charges or no electric current but simply that the medium is vacuum. In
vacuum, the vectorial linear electric constitutive relation is given by, [9,
10]

$\boldsymbol{d}=\varepsilon_{0}\,\boldsymbol{e}$ (27)

and the pseudovectorial linear magnetic constitutive relation is given by, [9,
10]

$\boldsymbol{b}=\mu_{0}\,\boldsymbol{h}$ (28)

The dual of the constitutive relation (28) is written as,

$\boldsymbol{b}^{\ast}=\mu_{0}\,\boldsymbol{h}^{\ast}$ (29)

Using the duality relations (9) and (13), relation (29) yields the bivectorial
linear magnetic constitutive relation,

$\boldsymbol{B}=\mu_{0}\,\boldsymbol{H}$ (30)

where the vacuum electric permittivity $\varepsilon_{0}$ and the vacuum
magnetic permeability $\mu_{0}$ are related to the speed of light $c$ by, [9,
10]

$\varepsilon_{0}\,\mu_{0}=\frac{1}{c^{2}}$ (31)

Substituting the linear constitutive relations (27) and (30) into the electric
Gauss equation (19) and the Maxwell-Ampère equation (20) and taking into
account the definition (31) of the speed of light, we obtain,

$\displaystyle\boldsymbol{\nabla}\cdot\boldsymbol{e}=\frac{q}{\varepsilon_{0}}$
(32)
$\displaystyle\boldsymbol{\nabla}\cdot\boldsymbol{B}=-\,\frac{1}{c^{2}}\,\partial_{t}\,\boldsymbol{e}-\,\mu_{0}\,\boldsymbol{j}$
(33)

In vector space, the Riemann-Silberstein vector $\boldsymbol{f}$ is defined
as, [19]

$\boldsymbol{f}=\boldsymbol{e}+i\,c\,\boldsymbol{b}$ (34)

where $i$ is the unit imaginary number. The Riemann-Silberstein vector
$\boldsymbol{f}\in\mathbb{C}^{3}$ is a complex vector field. In spatial
algebra (SA), the unit imaginary number $i$ is replaced by the unit
pseudoscalar $I$ (347) that has the same essential property, namely
$i^{2}=I^{2}=-\,1$. Thus, the Riemann-Silberstein vector $\boldsymbol{f}$
becomes a multivector called the electromagnetic multivector field,

$F=\boldsymbol{e}+c\,I\,\boldsymbol{b}$ (35)

In three dimensions, the pseudoscalar $I$ commutes with the vector field
$\boldsymbol{b}$. Thus, according to the duality (398) and (13),

$I\,\boldsymbol{b}=\boldsymbol{b}\,I=-\,\boldsymbol{b}\,I^{-1}=-\,\boldsymbol{b}^{\ast}=\boldsymbol{B}$
(36)

This duality operation is defined in terms of the pseudoscalar in Appendix
(B). Substituting identity (36) into relation (35), the electromagnetic
multivector field (35) becomes,

$F=\boldsymbol{e}+c\,\boldsymbol{B}$ (37)

The very elegant expression (37) of the electromagnetic multivector $F$ was
used only by few authors like Arthur [20] and Macdonald [18] whereas most
authors like Doran and Lasenby [21] or Hestenes [15] write the electromagnetic
field multivector (35) in terms of the magnetic induction field pseudovector
$\boldsymbol{b}$. The choice made here seem much more natural in a sense since
it reflects the underlying geometry of space. In view of the Faraday equation
(21), the Maxwell-Ampère relation (20) and the speed of light (31), the
partial time derivative of the electromagnetic multivector field (37) yields,

$\partial_{t}\,F=\partial_{t}\,\boldsymbol{e}+c\,\partial_{t}\,\boldsymbol{B}$
(38)

The gradient of the electromagnetic multivector field (37) is given by,

$\boldsymbol{\nabla}\,F=\boldsymbol{\nabla}\cdot F+\boldsymbol{\nabla}\wedge
F=\boldsymbol{\nabla}\cdot\boldsymbol{e}+\boldsymbol{\nabla}\wedge\boldsymbol{e}+c\,\boldsymbol{\nabla}\cdot\boldsymbol{B}+c\,\boldsymbol{\nabla}\wedge\boldsymbol{B}$
(39)

In view of the electric Gauss equation (32), the Faraday equation (21), the
Maxwell-Ampère relation (33), the magnetic Gauss equation (22), the linear
electric constitutive relation (27), the linear magnetic constitutive relation
(30) and the speed of light (31), the gradient of the electromagnetic
multivector field (39) becomes,

$\boldsymbol{\nabla}\,F=\frac{q}{\varepsilon_{0}}-\,\partial_{t}\,\boldsymbol{B}-\,\frac{1}{c}\,\partial_{t}\,\boldsymbol{e}-\,\frac{1}{\varepsilon_{0}\,c}\,\boldsymbol{j}$
(40)

Taking into account the partial time derivative (38) and the gradient (40) of
the electromagnetic multivector field and the definition (31) of the
propagation velocity, we obtain, [21]

$\left(\frac{1}{c}\,\partial_{t}+\boldsymbol{\nabla}\right)\varepsilon_{0}\,F=q-\,\frac{1}{c}\,\boldsymbol{j}$
(41)

which is the Maxwell equation in spatial algebra. In vacuum, the auxiliary
electromagnetic multivector field is defined as,

$G=\varepsilon_{0}\,F$ (42)

which is the linear electromagnetic constitutive relation. Using the
definition (42) of the auxiliary electromagnetic multivector field $G$ in
vacuum, the Maxwell equation (41) is recast as,

$\left(\frac{1}{c}\,\partial_{t}+\boldsymbol{\nabla}\right)G=q-\,\frac{1}{c}\,\boldsymbol{j}$
(43)

Using the constitutive relations (27) and (30) and the speed of light (31) the
auxiliary electromagnetic multivector field $G$ is recast as, [21]

$G=\boldsymbol{d}+\frac{1}{c}\,\boldsymbol{H}$ (44)

The electric Gauss equation (19) and the Maxwell-Ampère equation (20)
describing the electrodynamic phenomena driven by electric charges and
currents are expressed as the divergence of the auxiliary electromagnetic
multivector field $G$ in vacuum,

$\boldsymbol{\nabla}\cdot
G=\boldsymbol{\nabla}\cdot\boldsymbol{d}+\frac{1}{c}\,\boldsymbol{\nabla}\cdot\boldsymbol{H}=q-\,\frac{1}{c}\left(\boldsymbol{j}+\partial_{t}\,\boldsymbol{d}\right)$
(45)

Similarly, the Faraday equation (21) and the Magnetic Gauss equation (22)
describing the electrodynamic phenomena that do not involve directly electric
charges and currents are expressed as the curl of the electromagnetic
multivector field $F$,

$\boldsymbol{\nabla}\wedge
F=\boldsymbol{\nabla}\wedge\boldsymbol{e}+c\,\boldsymbol{\nabla}\wedge\boldsymbol{B}=-\,\partial_{t}\,\boldsymbol{B}$
(46)

##  5 Electromagnetic waves in vacuum (SA)

In the spatial algebra (SA), electromagnetic waves in vacuum are a direct and
straightforward consequence of the Maxwell equation (43), as expected, but it
is nonetheless quite beautiful. Indeed, multiplying the Maxwell equation (43)
by $c^{-1}\partial_{t}-\,\boldsymbol{\nabla}$ and using the relations,

$\left(\frac{1}{c}\,\partial_{t}-\,\boldsymbol{\nabla}\right)\left(\frac{1}{c}\,\partial_{t}+\boldsymbol{\nabla}\right)G=\left(\frac{1}{c^{2}}\,\partial_{t}^{2}-\,\boldsymbol{\nabla}^{2}\right)G$
(47)

and

$\left(\frac{1}{c}\,\partial_{t}-\,\boldsymbol{\nabla}\right)\left(q-\,\frac{1}{c}\,\boldsymbol{j}\right)=\frac{1}{c}\,\partial_{t}\,q-\,\boldsymbol{\nabla}\,q-\,\frac{1}{c^{2}}\,\partial_{t}\,\boldsymbol{j}+\frac{1}{c}\,\boldsymbol{\nabla}\cdot\boldsymbol{j}+\frac{1}{c}\,\boldsymbol{\nabla}\wedge\boldsymbol{j}$
(48)

together with the electric continuity equation (24), we obtain,

$\left(\frac{1}{c^{2}}\,\partial_{t}^{2}-\,\boldsymbol{\nabla}^{2}\right)G=-\,\boldsymbol{\nabla}\,q-\,\frac{1}{c^{2}}\,\partial_{t}\,\boldsymbol{j}+\frac{1}{c}\,\boldsymbol{\nabla}\wedge\boldsymbol{j}$
(49)

which is the wave equation for the auxiliary electromagnetic multivector field
$G$ in vacuum. Substituting the auxiliary electromagnetic multivector field
(44) in vacuum into the wave equation (49) yields,

$\left(\frac{1}{c^{2}}\,\partial_{t}^{2}-\,\boldsymbol{\nabla}^{2}\right)\left(\boldsymbol{d}+\frac{1}{c}\,\boldsymbol{H}\right)=-\,\boldsymbol{\nabla}\,q-\,\frac{1}{c^{2}}\,\partial_{t}\,\boldsymbol{j}+\frac{1}{c}\,\boldsymbol{\nabla}\wedge\boldsymbol{j}$
(50)

Identifying the vectorial terms in relation (50), we obtain the wave equation
for the electric displacement field $\boldsymbol{d}$,

$\left(\frac{1}{c^{2}}\,\partial_{t}^{2}-\,\boldsymbol{\nabla}^{2}\right)\boldsymbol{d}=-\,\boldsymbol{\nabla}\,q-\,\frac{1}{c^{2}}\,\partial_{t}\,\boldsymbol{j}$
(51)

Identifying the bivectorial terms in relation (50), we obtain the wave
equation for the auxiliary magnetic field $\boldsymbol{H}$,

$\left(\frac{1}{c^{2}}\,\partial_{t}^{2}-\,\boldsymbol{\nabla}^{2}\right)\boldsymbol{H}=\boldsymbol{\nabla}\wedge\boldsymbol{j}$
(52)

Dividing the wave equation (51) for the electric displacement field
$\boldsymbol{d}$ by $\varepsilon_{0}$ yields the wave equation for the
electric field $\boldsymbol{e}$,

$\left(\frac{1}{c^{2}}\,\partial_{t}^{2}-\,\boldsymbol{\nabla}^{2}\right)\boldsymbol{e}=-\,\frac{1}{\varepsilon_{0}}\,\boldsymbol{\nabla}\,q-\,\frac{1}{\varepsilon_{0}\,c^{2}}\,\partial_{t}\,\boldsymbol{j}$
(53)

Multiplying the wave equation (52) for the auxiliary magnetic field
$\boldsymbol{H}$ by $\mu_{0}$ yields the wave equation for the magnetic field
$\boldsymbol{B}$,

$\left(\frac{1}{c^{2}}\,\partial_{t}^{2}-\,\boldsymbol{\nabla}^{2}\right)\boldsymbol{B}=\mu_{0}\,\boldsymbol{\nabla}\wedge\boldsymbol{j}$
(54)

##  6 Electromagnetic energy and momentum in vacuum (SA)

The electromagnetic field multivector $F$ is a linear combination of the
electric field vector $\boldsymbol{e}$ and the magnetic induction field
bivector $\boldsymbol{B}$ according to relation (37). Similarly, the auxiliary
electromagnetic field multivector $G$ is a linear combination of the electric
displacement field vector $\boldsymbol{d}$ and the auxiliary magnetic field
bivector $\boldsymbol{H}$ according to relation (44). Since the
electromagnetic energy density $e$ and the electromagnetic momentum density
$\boldsymbol{p}$ in vacuum are determined by these fields, they can be recast
in terms of the multivectors $F$ and $G$. The electromagnetic energy density
in vacuum is written in vector space (VS) as, [9, 10]

$e=\frac{1}{2}\,\boldsymbol{d}\cdot\boldsymbol{e}+\frac{1}{2}\,\boldsymbol{h}\cdot\boldsymbol{b}$
(55)

Using the linear electric constitutive relation (27) and the linear magnetic
constitutive relation (28), relation (55) is recast as,

$e=\frac{1}{2}\,\varepsilon_{0}\,\boldsymbol{e}^{2}+\frac{1}{2\,\mu_{0}}\,\boldsymbol{b}^{2}$
(56)

According to relations (12), (381), (382), the square of the magnetic
induction field bivector $\boldsymbol{B}$ is the opposite of the square of the
magnetic induction field pseudovector $\boldsymbol{b}$,

$\boldsymbol{b}^{2}=|\boldsymbol{b}|^{2}=|\boldsymbol{B}|^{2}=\boldsymbol{B}\,\boldsymbol{B}^{{\dagger}}=-\,\boldsymbol{B}^{2}$
(57)

where $\boldsymbol{B}^{{\dagger}}=-\,\boldsymbol{B}$ is the reverse (378) of
the magnetic field bivector $\boldsymbol{B}$. In view of relation (57), the
electromagnetic energy density (58) is recast in spatial algebra (SA) as,

$e=\frac{1}{2}\,\varepsilon_{0}\,\boldsymbol{e}^{2}-\,\frac{1}{2\,\mu_{0}}\,\boldsymbol{B}^{2}$
(58)

Taking into account the linear electric constitutive relation (27) and the
linear magnetic constitutive relation (28), the electromagnetic energy density
(58) is recast as,

$e=\frac{1}{2}\,\boldsymbol{d}\cdot\boldsymbol{e}-\,\frac{1}{2}\,\boldsymbol{H}\cdot\boldsymbol{B}$
(59)

The electromagnetic momentum density in vacuum is written in vector space (VS)
as, [9, 10]

$\boldsymbol{p}=\frac{1}{c^{2}}\,\boldsymbol{e}\times\boldsymbol{h}$ (60)

where $\boldsymbol{e}\times\boldsymbol{h}$ is the Poynting vector. By duality
using the identities (407), (410), (9) and (360), the electromagnetic momentum
density (60) in recast in spatial algebra (SA) as,

$\boldsymbol{p}=\frac{1}{c^{2}}\left(\boldsymbol{e}\wedge\boldsymbol{h}\right)^{\ast}=\frac{1}{c^{2}}\,\boldsymbol{e}\cdot\boldsymbol{h}^{\ast}=-\,\frac{1}{c^{2}}\,\boldsymbol{e}\cdot\boldsymbol{H}=\frac{1}{c^{2}}\,\boldsymbol{H}\cdot\boldsymbol{e}$
(61)

Thus, in spatial algebra, in view of the magnetic constitutive relation (30),
the electromagnetic momentum density in vacuum (61) is written as,

$\boldsymbol{p}=\frac{1}{c^{2}}\,\boldsymbol{H}\cdot\boldsymbol{e}=\frac{1}{c^{2}}\,\frac{\boldsymbol{B}}{\mu_{0}}\cdot\boldsymbol{e}$
(62)

where $\boldsymbol{H}\cdot\boldsymbol{e}$ is the Poynting vector. According to
the definition (37), the reverse of the electromagnetic multivector field $F$
is written as,

$F^{{\dagger}}=\boldsymbol{e}^{{\dagger}}+c\,\boldsymbol{B}^{{\dagger}}=\boldsymbol{e}-\,c\,\boldsymbol{B}$
(63)

The reversion operation, defined in Appendix (B), reverses the order of the
basis vectors in the different geometric entities of a multivector in spatial
algebra. In view of the multivectors (37) and (63), the electric vector field
is recast as,

$\boldsymbol{e}=\frac{1}{2}\left(F+F^{{\dagger}}\right)$ (64)

and the magnetic induction bivector field is recast as,

$\boldsymbol{B}=\frac{1}{2\,c}\left(F-\,F^{{\dagger}}\right)$ (65)

According to the definition (44), the reverse of the auxiliary electromagnetic
multivector field $G$ is written as,

$G^{{\dagger}}=\boldsymbol{d}^{{\dagger}}+\frac{1}{c}\,\boldsymbol{H}^{{\dagger}}=\boldsymbol{d}-\,\frac{1}{c}\,\boldsymbol{H}$
(66)

The reverse of the linear electromagnetic constitutive relation (42) is
written as,

$G^{{\dagger}}=\varepsilon_{0}\,F^{{\dagger}}$ (67)

In view of the multivectors (37) and (63), the electric displacement vector
field is recast as,

$\boldsymbol{d}=\frac{1}{2}\left(G+G^{{\dagger}}\right)$ (68)

and the auxiliary magnetic bivector field is recast as,

$\boldsymbol{H}=\frac{c}{2}\left(G-\,G^{{\dagger}}\right)$ (69)

Using the symmetry of the inner product of two vectors (349) and two bivectors
(368), the electromagnetic energy density (59) is recast as,

$e=\frac{1}{2}\left(\boldsymbol{d}\,\boldsymbol{e}+\boldsymbol{e}\,\boldsymbol{d}\right)-\,\frac{1}{2}\left(\boldsymbol{H}\,\boldsymbol{B}+\boldsymbol{B}\,\boldsymbol{H}\right)$
(70)

Using relations (64), (64), (68) and (69), the electromagnetic energy density
(70) is expressed in terms of the geometric product of the electromagnetic
multivectors as,

$\begin{split}&e=\frac{1}{16}\Big{(}\left(G+G^{{\dagger}}\right)\left(F+F^{{\dagger}}\right)+\left(F+F^{{\dagger}}\right)\left(G+G^{{\dagger}}\right)\Big{)}\\\
&\phantom{e=}-\,\frac{1}{16}\Big{(}\left(G-\,G^{{\dagger}}\right)\left(F-\,F^{{\dagger}}\right)+\left(F-\,F^{{\dagger}}\right)\left(G-\,G^{{\dagger}}\right)\Big{)}\end{split}$
(71)

which reduces to,

$e=\frac{1}{8}\left(G\,F^{{\dagger}}+F\,G^{{\dagger}}+G^{{\dagger}}F+F^{{\dagger}}G\right)$
(72)

Using the antisymmetry of the inner product of a vector and a bivector (360)
the electromagnetic momentum density (60) is recast as,

$\boldsymbol{p}=\frac{1}{2\,c^{2}}\left(\boldsymbol{H}\,\boldsymbol{e}-\,\boldsymbol{e}\,\boldsymbol{H}\right)$
(73)

Using relations (64), (64), (68) and (69), the electromagnetic momentum
density (73) is expressed in terms of the geometric product of the
electromagnetic multivectors as,

$\boldsymbol{p}=\frac{1}{8\,c}\Big{(}\left(G-\,G^{{\dagger}}\right)\left(F+F^{{\dagger}}\right)-\,\left(F+F^{{\dagger}}\right)\left(G-\,G^{{\dagger}}\right)\Big{)}$
(74)

which is expressed as,

$\begin{split}&\boldsymbol{p}=\frac{1}{8\,c}\left(G\,F^{{\dagger}}+F\,G^{{\dagger}}-\,G^{{\dagger}}F-\,F^{{\dagger}}G\right)\\\
&\phantom{\boldsymbol{p}=}+\frac{1}{8\,c}\left(G\,F+F^{{\dagger}}G^{{\dagger}}-\,F\,G-\,G^{{\dagger}}F^{{\dagger}}\right)\end{split}$
(75)

In vacuum, in view of the linear electromagnetic constitutive relations (42),
(67), the electromagnetic energy density (72) reduces,

$e=\frac{1}{8}\,\left(G\,F^{{\dagger}}+F\,G^{{\dagger}}+G^{{\dagger}}F+F^{{\dagger}}G\right)=\frac{1}{4}\,\varepsilon_{0}\left(F\,F^{{\dagger}}+F^{{\dagger}}F\right)$
(76)

and the electromagnetic momentum density (62) reduces to, [20]

$\boldsymbol{p}=\frac{1}{8\,c}\left(G\,F^{{\dagger}}+F\,G^{{\dagger}}-\,G^{{\dagger}}F-\,F^{{\dagger}}G\right)=\frac{1}{4}\,\frac{\varepsilon_{0}}{c}\left(F\,F^{{\dagger}}-\,F^{{\dagger}}F\right)$
(77)

Thus, according to relations (76) and (77) in vacuum, [20]

$e+\boldsymbol{p}\,c=\frac{1}{4}\left(G\,F^{{\dagger}}+F\,G^{{\dagger}}\right)=\frac{1}{2}\,\varepsilon_{0}\,F\,F^{{\dagger}}$
(78)

##  7 Poynting theorem in vacuum (SA)

The continuity equation for the electromagnetic energy, called the Poynting
theorem, is written in vacuum in terms of the time derivative of the energy
density and the energy flux which is a multiple of the electromagnetic
momentum density. The time derivative of the electromagnetic energy density
(58) in vacuum is given by,

$\partial_{t}\,e=\varepsilon_{0}\,\boldsymbol{e}\cdot\partial_{t}\,\boldsymbol{e}-\,\frac{1}{\mu_{0}}\,\boldsymbol{B}\cdot\partial_{t}\,\boldsymbol{B}$
(79)

Using the electric constitutive equation (27) and the magnetic constitutive
equation (30), relation (79) is recast as,

$\partial_{t}\,e=\boldsymbol{e}\cdot\partial_{t}\,\boldsymbol{d}-\,\boldsymbol{H}\cdot\partial_{t}\,\boldsymbol{B}$
(80)

Using the Mawell-Ampère equation (20) and the Faraday equation (21), the time
derivative of the electromagnetic energy density (80) is recast as,

$\partial_{t}\,e=-\,\boldsymbol{e}\cdot\left(\boldsymbol{\nabla}\cdot\boldsymbol{H}+\boldsymbol{j}\right)+\boldsymbol{H}\cdot\left(\boldsymbol{\nabla}\wedge\boldsymbol{e}\right)$
(81)

Thus,

$\partial_{t}\,e+\left(\boldsymbol{\nabla}\cdot\boldsymbol{H}\right)\cdot\boldsymbol{e}-\,\boldsymbol{H}\cdot\left(\boldsymbol{\nabla}\wedge\boldsymbol{e}\right)=-\,\boldsymbol{j}\cdot\boldsymbol{e}$
(82)

Using the algebraic identity (478),

$\boldsymbol{\nabla}\cdot\left(\boldsymbol{H}\cdot\boldsymbol{e}\right)=\left(\boldsymbol{\nabla}\cdot\boldsymbol{H}\right)\cdot\boldsymbol{e}-\,\boldsymbol{H}\cdot\left(\boldsymbol{\nabla}\wedge\boldsymbol{e}\right)$
(83)

relation yields (82) the Poynting’s theorem,

$\partial_{t}\,e+\boldsymbol{\nabla}\cdot\left(\boldsymbol{H}\cdot\boldsymbol{e}\right)=-\,\boldsymbol{j}\cdot\boldsymbol{e}$
(84)

where $\boldsymbol{H}\cdot\boldsymbol{e}$ is the Poynting vector, which is the
electromagnetic current density. In view of the electromagnetic momentum
density (62), Poynting’s theorem (84) in vacuum is recast as, [9, 10]

$\frac{1}{c}\,\partial_{t}\,e+\boldsymbol{\nabla}\cdot\left(\boldsymbol{p}\,c\right)=-\,\frac{1}{c}\,\boldsymbol{j}\cdot\boldsymbol{e}$
(85)

##  8 Electric and magnetic potentials (SA)

In vector space (VS), the electric vector field $\boldsymbol{e}$ and the
magnetic induction pseudovector field $\boldsymbol{b}$ can be expressed
entirely in terms of the electric scalar potential $\phi$ and magnetic vector
potential $\boldsymbol{a}$. This implies that the electromagnetic multivector
field $F$ can be entirely expressed in terms of these potentials. The electric
vector field $\boldsymbol{e}$ is expressed in terms of the electric scalar
potential $\phi$ and the magnetic vector potential $\boldsymbol{a}$ as, [9,
10]

$\boldsymbol{e}=-\,\boldsymbol{\nabla}\,\phi-\,\partial_{t}\,\boldsymbol{a}$
(86)

and the magnetic induction vector field $\boldsymbol{b}$ is expressed in terms
of the vector potential $\boldsymbol{a}$ as, [9, 10]

$\boldsymbol{b}=\boldsymbol{\nabla}\times\boldsymbol{a}$ (87)

According to relation (13) and identity (432), the magnetic induction bivector
field $\boldsymbol{B}$ is the dual of the magnetic induction vector field
$\boldsymbol{b}$,

$\boldsymbol{B}=-\,\boldsymbol{b}^{\ast}=-\,\left(\boldsymbol{\nabla}\times\boldsymbol{a}\right)^{\ast}=\boldsymbol{\nabla}\wedge\boldsymbol{a}$
(88)

In view of relations (37) and (88), the electromagnetic multivector field $F$
is written in terms of the potentials as,

$F=\boldsymbol{e}+c\,\boldsymbol{B}=-\,\boldsymbol{\nabla}\,\phi-\,\partial_{t}\,\boldsymbol{a}+c\,\boldsymbol{\nabla}\wedge\boldsymbol{a}$
(89)

The Lorentz gauge in vector space (VS) is defined as, [9, 10]

$\frac{1}{c^{2}}\,\partial_{t}\,\phi+\boldsymbol{\nabla}\cdot\boldsymbol{a}=0$
(90)

##  9 Maxwell equation in matter (SA)

We now show that in matter, the Maxwell equations (19)-(22) reduce to a single
equation. The term “matter” refers here to a material medium consisting of
continuum of electric and magnetic dipoles. In a dielectric and magnetic
medium, the vectorial linear electric constitutive relation is given by, [9,
10]

$\boldsymbol{d}=\varepsilon_{0}\,\boldsymbol{e}+\boldsymbol{\tilde{p}}$ (91)

where $\boldsymbol{\tilde{p}}$ is the matter electric polarisation vector
field, and the pseudovectorial linear magnetic constitutive relation is given
by, [9, 10]

$\boldsymbol{b}=\mu_{0}\left(\boldsymbol{h}+\boldsymbol{\tilde{m}}\right)$
(92)

where $\boldsymbol{\tilde{m}}$ is the matter magnetisation pseudovector field.
The dual of the constitutive relation (92) is written as,

$\boldsymbol{b}^{\ast}=\mu_{0}\left(\boldsymbol{h}^{\ast}+\boldsymbol{\tilde{m}}^{\ast}\right)$
(93)

The magnetisation field pseudovector $\boldsymbol{\tilde{m}}$ in vector space
is the dual of the auxiliary magnetisation field bivector
$\boldsymbol{\tilde{M}}$ in spatial algebra (398),

$\boldsymbol{\tilde{M}}^{\ast}=\boldsymbol{\tilde{m}}\qquad\text{where}\qquad|\boldsymbol{\tilde{M}}|=|\boldsymbol{\tilde{m}}|$
(94)

The dual of this duality is,

$\boldsymbol{\tilde{m}}^{\ast}=\left(\boldsymbol{\tilde{M}}^{\ast}\right)^{\ast}=-\,\boldsymbol{\tilde{M}}$
(95)

Using the dualities (13),(9) and (95), relation (93) becomes the bivectorial
linear magnetic constitutive relation,

$\boldsymbol{B}=\mu_{0}\left(\boldsymbol{H}+\boldsymbol{\tilde{M}}\right)$
(96)

Substituting the constitutive relations (91) and (96) into the auxiliary
electromagnetic multivector field (44) in vacuum and taking into account the
speed of light (31), we obtain,

$G=\left(\varepsilon_{0}\,\boldsymbol{e}+\boldsymbol{\tilde{p}}\right)+\frac{1}{c}\left(\frac{\boldsymbol{B}}{\mu_{0}}-\,\boldsymbol{\tilde{M}}\right)=\varepsilon_{0}\left(\boldsymbol{e}+c\,\boldsymbol{B}\right)+\left(\boldsymbol{\tilde{p}}-\,\frac{1}{c}\,\boldsymbol{\tilde{M}}\right)$
(97)

Defining the electromagnetic polarisation multivector as,

$\tilde{P}=\boldsymbol{\tilde{p}}-\,\frac{1}{c}\,\boldsymbol{\tilde{M}}$ (98)

and substituting the multivectors (42) and (98) into relation (97) we obtain
the multivectorial linear electromagnetic constitutive relation,

$G=\varepsilon_{0}\,F+\tilde{P}$ (99)

Substituting the multivectorial linear electromagnetic constitutive relation
(99) into the Maxwell equation (43), we obtain the Maxwell equation for a
dielectric and magnetic medium,

$\left(\frac{1}{c}\,\partial_{t}+\boldsymbol{\nabla}\right)\left(\varepsilon_{0}\,F+\tilde{P}\right)=q-\,\frac{1}{c}\,\boldsymbol{j}$
(100)

which is in agreement with the result derived by Arthur [20]. The dynamics of
the electromagnetic polarisation of the medium is described by the equation,

$\left(\frac{1}{c}\,\partial_{t}+\boldsymbol{\nabla}\right)\tilde{P}=\frac{1}{c}\,\partial_{t}\,\tilde{P}+\boldsymbol{\nabla}\cdot\tilde{P}+\boldsymbol{\nabla}\wedge\tilde{P}$
(101)

Using the definition (98) of the electromagnetic polarisation multivector, we
obtain the following relation,

$\frac{1}{c}\,\partial_{t}\,\tilde{P}+\boldsymbol{\nabla}\cdot\tilde{P}=\frac{1}{c}\,\partial_{t}\,\boldsymbol{\tilde{p}}+\boldsymbol{\nabla}\cdot\boldsymbol{\tilde{p}}-\,\frac{1}{c^{2}}\,\partial_{t}\,\boldsymbol{\tilde{M}}-\,\frac{1}{c}\,\boldsymbol{\nabla}\cdot\boldsymbol{\tilde{M}}$
(102)

Using relations (98), (101) and (102), the Maxwell equation (100) is recast
as,

$\begin{split}&\left(\frac{1}{c}\,\partial_{t}+\boldsymbol{\nabla}\right)\varepsilon_{0}\,F=\left(q-\,\boldsymbol{\nabla}\cdot\boldsymbol{\tilde{p}}\right)-\,\frac{1}{c}\left(\boldsymbol{j}+\partial_{t}\,\boldsymbol{\tilde{p}}-\,\boldsymbol{\nabla}\cdot\boldsymbol{\tilde{M}}\right)\\\
&\phantom{\left(\frac{1}{c}\,\partial_{t}+\boldsymbol{\nabla}\right)\varepsilon_{0}\,F=}-\,\boldsymbol{\nabla}\wedge\boldsymbol{\tilde{p}}+\frac{1}{c}\,\boldsymbol{\nabla}\wedge\boldsymbol{\tilde{M}}\end{split}$
(103)

The total electric charge density $\tilde{q}$ in matter is the sum of the free
electric charge density $q$ and the bound electric charge density
$-\,\boldsymbol{\nabla}\cdot\boldsymbol{\tilde{p}}$,

$\tilde{q}=q-\,\boldsymbol{\nabla}\cdot\boldsymbol{\tilde{p}}$ (104)

and total electric current density $\boldsymbol{\tilde{j}}$ in matter is the
sum of the free electric currant density $\boldsymbol{j}$ and the bound
electric current density
$\partial_{t}\,\boldsymbol{\tilde{p}}-\,\boldsymbol{\nabla}\cdot\boldsymbol{\tilde{M}}$,

$\boldsymbol{\tilde{j}}=\boldsymbol{j}+\partial_{t}\,\boldsymbol{\tilde{p}}-\,\boldsymbol{\nabla}\cdot\boldsymbol{\tilde{M}}$
(105)

Using the definitions (98), (104) and (105), the Maxwell equation (103)
reduces to,

$\left(\frac{1}{c}\,\partial_{t}+\boldsymbol{\nabla}\right)\varepsilon_{0}\,F=\tilde{q}-\,\frac{1}{c}\,\boldsymbol{\tilde{j}}-\,\boldsymbol{\nabla}\wedge\boldsymbol{\tilde{p}}+\frac{1}{c}\,\boldsymbol{\nabla}\wedge\boldsymbol{\tilde{M}}$
(106)

Taking into account the definition (37) of the electromagnetic multivector
field $F$, the trivectorial part of the Maxwell equation (106) reduces to,

$c\,\varepsilon_{0}\,\boldsymbol{\nabla}\wedge\boldsymbol{B}=\frac{1}{c}\,\boldsymbol{\nabla}\wedge\boldsymbol{\tilde{M}}$
(107)

In view of the magnetic Gauss equation (22), relation (107) yields the
condition,

$\boldsymbol{\nabla}\wedge\boldsymbol{\tilde{M}}=0$ (108)

Taking into account the definition (37) of the electromagnetic multivector
field $F$, the bivectorial part of the Maxwell equation (106) reduces to,

$\varepsilon_{0}\left(\partial_{t}\,\boldsymbol{B}+\boldsymbol{\nabla}\wedge\boldsymbol{e}\right)=-\,\boldsymbol{\nabla}\wedge\boldsymbol{\tilde{p}}$
(109)

Using the Faraday equation (21), relation (109) yields the condition,

$\boldsymbol{\nabla}\wedge\boldsymbol{\tilde{p}}=0$ (110)

In view of the conditions(108) and(110), the Maxwell equation (106) reduces
to,

$\left(\frac{1}{c}\,\partial_{t}+\boldsymbol{\nabla}\right)\varepsilon_{0}\,F=\tilde{q}-\,\frac{1}{c}\,\boldsymbol{\tilde{j}}$
(111)

which differs from the result derived by Arthur [20] since the conditions
(108) and (110) were not properly identified. The Maxwell equation (111) in a
dielectric and magnetic medium can be obtained by replacing the electric
charge density $q$ by the total electric charge density $\tilde{q}$ and the
electric current density $\boldsymbol{j}$ by the total electric current
density $\boldsymbol{\tilde{j}}$ in the Maxwell equation (41) in vacuum. The
difference between the Maxwell equations (100) and (111) yields the relation,

$\left(\frac{1}{c}\,\partial_{t}+\boldsymbol{\nabla}\right)\tilde{P}=\left(q-\,\tilde{q}\right)-\,\frac{1}{c}\left(\boldsymbol{j}-\,\boldsymbol{\tilde{j}}\right)$
(112)

In view of relation of the electric charge density in matter (104) and the
electric current density in matter (105), relation (112) is recast as,

$\left(\frac{1}{c}\,\partial_{t}+\boldsymbol{\nabla}\right)\tilde{P}=\boldsymbol{\nabla}\cdot\boldsymbol{\tilde{p}}+\frac{1}{c}\left(\partial_{t}\,\boldsymbol{\tilde{p}}-\,\boldsymbol{\nabla}\cdot\boldsymbol{\tilde{M}}\right)$
(113)

In view of relations (108) and (110), the curl of the electromagnetic
polarisation multivector (98) vanishes,

$\boldsymbol{\nabla}\wedge\tilde{P}=0$ (114)

According to relation (114), the geometric product on the left-hand side of
relation (113) becomes a inner product,

$\left(\frac{1}{c}\,\partial_{t}+\boldsymbol{\nabla}\right)\cdot\tilde{P}=\boldsymbol{\nabla}\cdot\boldsymbol{\tilde{p}}+\frac{1}{c}\left(\partial_{t}\,\boldsymbol{\tilde{p}}-\,\boldsymbol{\nabla}\cdot\boldsymbol{\tilde{M}}\right)$
(115)

##  10 Electromagnetic waves in matter (SA)

In the spatial algebra (SA), electromagnetic waves in vacuum are a direct and
straightforward consequence of the Maxwell equation (43), as expected, but it
is nonetheless quite beautiful. Indeed, multiplying the Maxwell equation (111)
by $c^{-1}\partial_{t}-\,\boldsymbol{\nabla}$ yields,

$\left(\frac{1}{c}\,\partial_{t}-\,\boldsymbol{\nabla}\right)\left(\frac{1}{c}\,\partial_{t}+\boldsymbol{\nabla}\right)\varepsilon_{0}\,F=\left(\frac{1}{c}\,\partial_{t}-\,\boldsymbol{\nabla}\right)\left(\tilde{q}-\,\frac{1}{c}\,\boldsymbol{\tilde{j}}\right)$
(116)

which is written explicitly as,

$\left(\frac{1}{c^{2}}\,\partial_{t}^{2}-\,\boldsymbol{\nabla}^{2}\right)\varepsilon_{0}\,F=\frac{1}{c}\,\partial_{t}\,\tilde{q}-\,\boldsymbol{\nabla}\,\tilde{q}-\,\frac{1}{c^{2}}\,\partial_{t}\,\boldsymbol{\tilde{j}}+\frac{1}{c}\,\boldsymbol{\nabla}\cdot\boldsymbol{\tilde{j}}+\frac{1}{c}\,\boldsymbol{\nabla}\wedge\boldsymbol{\tilde{j}}$
(117)

In view of the definitions of the total electric charge density (104) and the
total electric current density (105), we obtain,

$\partial_{t}\,\tilde{q}+\boldsymbol{\nabla}\cdot\boldsymbol{\tilde{j}}=\partial_{t}\,q-\,\partial_{t}\left(\boldsymbol{\nabla}\cdot\boldsymbol{\tilde{p}}\right)+\boldsymbol{\nabla}\cdot\boldsymbol{j}+\partial_{t}\left(\boldsymbol{\nabla}\cdot\boldsymbol{\tilde{p}}\right)-\,\boldsymbol{\nabla}\cdot\left(\boldsymbol{\nabla}\cdot\boldsymbol{\tilde{M}}\right)$
(118)

Taking into account the electric continuity equation (24) for the free
electric charge density $q$, relation (118) yields the electric continuity
equation for the total electric charge density $\tilde{q}$,

$\partial_{t}\,\tilde{q}+\boldsymbol{\nabla}\cdot\boldsymbol{\tilde{j}}=0$
(119)

Using the continuity equation (119), relation (117) yields the electromagnetic
wave equation for the electromagnetic multivector field $F$ in a dielectric
and magnetic medium,

$\left(\frac{1}{c^{2}}\,\partial_{t}^{2}-\,\boldsymbol{\nabla}^{2}\right)\varepsilon_{0}\,F=-\,\boldsymbol{\nabla}\,\tilde{q}-\,\frac{1}{c^{2}}\,\partial_{t}\,\boldsymbol{\tilde{j}}+\frac{1}{c}\,\boldsymbol{\nabla}\wedge\boldsymbol{\tilde{j}}$
(120)

Taking into account the definition (37) of the electromagnetic multivector
field $F$, the vectorial part of the wave equation (120) yields the wave
equation for the electric field $\boldsymbol{e}$,

$\left(\frac{1}{c^{2}}\,\partial_{t}^{2}-\,\boldsymbol{\nabla}^{2}\right)\boldsymbol{e}=-\,\frac{1}{\varepsilon_{0}}\,\boldsymbol{\nabla}\,\tilde{q}-\,\frac{1}{\varepsilon_{0}\,c^{2}}\,\partial_{t}\,\boldsymbol{\tilde{j}}$
(121)

Taking into account the definition (37) of the electromagnetic multivector
field $F$ and using the speed of light (31), the bivectorial part of the wave
equation (120) yields the wave equation for the magnetic field
$\boldsymbol{B}$,

$\left(\frac{1}{c^{2}}\,\partial_{t}^{2}-\,\boldsymbol{\nabla}^{2}\right)\boldsymbol{B}=\mu_{0}\,\boldsymbol{\nabla}\wedge\boldsymbol{\tilde{j}}$
(122)

##  11 Electromagnetic energy and momentum in matter (SA)

The electromagnetic field multivector $F$ is a linear combination of the
electric field vector $\boldsymbol{e}$ and the magnetic induction field
bivector $\boldsymbol{B}$ according to relation (37). Similarly, the auxiliary
electromagnetic field multivector $G$ is a linear combination of the electric
displacement field vector $\boldsymbol{d}$ and the auxiliary magnetic field
bivector $\boldsymbol{H}$ according to relation (44). Moreover, the electric
polarisation multivector $\tilde{P}$ is a linear combination of the electric
polarisation vector $\boldsymbol{\tilde{p}}$ and the magnetisation bivector
$\boldsymbol{\tilde{M}}$ according to relation (98). Since the electromagnetic
energy density $e$ and the electromagnetic momentum density $\boldsymbol{p}$
in matter are determined by these fields, they can be recast in terms of the
multivectors $F$, $G$ and $\tilde{P}$. In view of the linear electric
constitutive relation in matter (91) and the linear magnetic constitutive
relation (96), relation (59) yields the electromagnetic energy density in a
dielectric and magnetic medium, [9, 10]

$e=\frac{1}{2}\,\boldsymbol{d}\cdot\boldsymbol{e}-\,\frac{1}{2}\,\boldsymbol{H}\cdot\boldsymbol{B}=\frac{1}{2}\,\varepsilon_{0}\,\boldsymbol{e}^{2}-\,\frac{1}{2\,\mu_{0}}\,\boldsymbol{B}^{2}+\frac{1}{2}\,\boldsymbol{\tilde{p}}\cdot\boldsymbol{e}+\frac{1}{2}\,\boldsymbol{\tilde{M}}\cdot\boldsymbol{B}$
(123)

In view of the linear magnetic constitutive relation in matter (96), relation
(62) yields the electromagnetic momentum density in matter,

$\boldsymbol{p}=\frac{1}{c^{2}}\,\boldsymbol{H}\cdot\boldsymbol{e}=\frac{1}{c^{2}}\left(\frac{\boldsymbol{B}}{\mu_{0}}-\,\boldsymbol{\tilde{M}}\right)\cdot\boldsymbol{e}$
(124)

The reverse of the linear electromagnetic constitutive relation in matter (99)
is written as,

$G^{{\dagger}}=\varepsilon_{0}\,F^{{\dagger}}+\tilde{P}^{{\dagger}}$ (125)

According the definition (98), the reverse of the electromagnetic polarisation
tensor is given by,

$\tilde{P}^{{\dagger}}=\boldsymbol{\tilde{p}}^{{\dagger}}-\,\frac{1}{c}\,\boldsymbol{\tilde{M}}^{{\dagger}}=\boldsymbol{\tilde{p}}+\frac{1}{c}\,\boldsymbol{\tilde{M}}$
(126)

In view of the multivectors (98) and (126), the electric polarisation vector
field is recast as,

$\boldsymbol{\tilde{p}}=\frac{1}{2}\left(\tilde{P}+\tilde{P}^{{\dagger}}\right)$
(127)

and the magnetisation bivector field is recast as,

$\boldsymbol{\tilde{M}}=-\,\frac{c}{2}\left(\tilde{P}-\,\tilde{P}^{{\dagger}}\right)$
(128)

Comparing the expressions (59) and (72) for the electromagnetic energy density
in vacuum to the expression (123) of the electromagnetic energy density in
matter, we conclude that the electromagnetic energy density in matter (123) is
recast as,

$e=\frac{1}{8}\left(G\,F^{{\dagger}}+F\,G^{{\dagger}}+G^{{\dagger}}F+F^{{\dagger}}G\right)$
(129)

In view of the electromagnetic constitutive equations (99) and (125), the
electromagnetic energy density in matter (129) is recast as,

$e=\frac{1}{4}\,\varepsilon_{0}\left(F\,F^{{\dagger}}+F^{{\dagger}}F\right)+\frac{1}{8}\left(P\,F^{{\dagger}}+F\,P^{{\dagger}}+P^{{\dagger}}F+F^{{\dagger}}P\right)$
(130)

Comparing the expressions (62) and (75) for the electromagnetic momentum
density in vacuum to the expression (124) of the electromagnetic momentum
density in matter, we conclude that the electromagnetic momentum density in
matter (124) is recast as,

$\begin{split}&\boldsymbol{p}=\frac{1}{8\,c}\left(G\,F^{{\dagger}}+F\,G^{{\dagger}}-\,G^{{\dagger}}F-\,F^{{\dagger}}G\right)\\\
&\phantom{\boldsymbol{p}=}+\frac{1}{8\,c}\left(G\,F-\,F\,G+F^{{\dagger}}G^{{\dagger}}-\,G^{{\dagger}}F^{{\dagger}}\right)\end{split}$
(131)

In view of the electromagnetic constitutive equations (99) and (125), the
electromagnetic momentum density in matter (129) is recast as,

$\boldsymbol{p}=\frac{1}{4\,c}\,\varepsilon_{0}\left(F\,F^{{\dagger}}-\,F^{{\dagger}}F\right)+\frac{1}{8\,c}\left(P\,F-\,F\,P+F^{{\dagger}}P^{{\dagger}}-\,P^{{\dagger}}F^{{\dagger}}\right)$
(132)

According to relations (129) and (131),

$e+\boldsymbol{p}\,c=\frac{1}{4}\left(G\,F^{{\dagger}}+F\,G^{{\dagger}}\right)+\frac{1}{8}\left(G\,F-\,F\,G+F^{{\dagger}}G^{{\dagger}}-\,G^{{\dagger}}F^{{\dagger}}\right)$
(133)

In view of the electromagnetic constitutive equations (99) and (125), relation
(133) in matter is recast as,

$e+\boldsymbol{p}\,c=\frac{1}{2}\,\varepsilon_{0}\,F\,F^{{\dagger}}+\frac{1}{8}\left(P\,F-\,F\,P+F^{{\dagger}}P^{{\dagger}}-\,P^{{\dagger}}F^{{\dagger}}\right)$
(134)

##  12 Poynting theorem in matter (SA)

The continuity equation for the electromagnetic energy, called the Poynting
theorem, is written in matter in terms of the time derivative of the energy
density and the energy flux which is a multiple of the electromagnetic
momentum density. In linear electromagnetism, the matter electric polarisation
vector field $\boldsymbol{\tilde{p}}\left(\boldsymbol{e}\right)$ induced by
the electric field $\boldsymbol{e}$ is a linear map of the electric vector
field $\boldsymbol{e}$,

$\boldsymbol{\tilde{p}}\left(\boldsymbol{e}\right)=\varepsilon_{0}\,\boldsymbol{\chi}_{\boldsymbol{e}}\left(\boldsymbol{e}\right)$
(135)

where $\boldsymbol{\chi}_{\boldsymbol{e}}\left(\boldsymbol{e}\right)$ is the
electric susceptibility vector field, and the matter magnetisation bivector
field $\boldsymbol{\tilde{M}}\left(\boldsymbol{B}\right)$ induced by the
magnetic induction field $\boldsymbol{B}$ is a linear map of the magnetic
induction bivector field $\boldsymbol{B}$,

$\boldsymbol{\tilde{M}}\left(\boldsymbol{B}\right)=\frac{1}{\mu_{0}}\,\boldsymbol{\chi}_{\boldsymbol{B}}\left(\boldsymbol{B}\right)$
(136)

where $\boldsymbol{\chi}_{\boldsymbol{B}}\left(\boldsymbol{B}\right)$ is the
electric susceptibility bivector field. In view of the linear maps (135) and
(136) the electric displacement vector field (91) is recast as,

$\boldsymbol{d}\left(\boldsymbol{e}\right)=\varepsilon_{0}\,\Big{(}\boldsymbol{e}+\boldsymbol{\chi}_{\boldsymbol{e}}\left(\boldsymbol{e}\right)\Big{)}$
(137)

and the magnetic auxiliary bivector field (96) is recast as,

$\boldsymbol{H}\left(\boldsymbol{B}\right)=\frac{1}{\mu_{0}}\,\Big{(}\boldsymbol{B}-\,\boldsymbol{\chi}_{\boldsymbol{B}}\left(\boldsymbol{B}\right)\Big{)}$
(138)

In view of linear relations (135) and (136), the electromagnetic energy
density (123) is recast as,

$e\left(\boldsymbol{e},\boldsymbol{B}\right)=\frac{1}{2}\,\varepsilon_{0}\,\Big{(}\boldsymbol{e}+\boldsymbol{\chi}_{\boldsymbol{e}}\left(\boldsymbol{e}\right)\Big{)}\cdot\boldsymbol{e}-\,\frac{1}{2\,\mu_{0}}\,\Big{(}\boldsymbol{B}-\,\boldsymbol{\chi}_{\boldsymbol{B}}\left(\boldsymbol{B}\right)\Big{)}\cdot\boldsymbol{B}$
(139)

The time derivative of the linear electromagnetic energy density in matter
(139) reads,

$\displaystyle\partial_{t}\,e\left(\boldsymbol{e},\boldsymbol{B}\right)=\varepsilon_{0}\,\boldsymbol{e}\cdot\partial_{t}\,\boldsymbol{e}+\frac{1}{2}\,\varepsilon_{0}\,\boldsymbol{\chi}_{\boldsymbol{e}}\left(\boldsymbol{e}\right)\cdot\partial_{t}\,\boldsymbol{e}+\frac{1}{2}\,\varepsilon_{0}\,\partial_{t}\Big{(}\boldsymbol{\chi}_{\boldsymbol{e}}\left(\boldsymbol{e}\right)\Big{)}\cdot\boldsymbol{e}$
(140)
$\displaystyle\phantom{\partial_{t}\,e\left(\boldsymbol{e},\boldsymbol{B}\right)=}-\,\frac{1}{\mu_{0}}\,\boldsymbol{B}\cdot\partial_{t}\,\boldsymbol{B}+\frac{1}{2\,\mu_{0}}\,\boldsymbol{\chi}_{\boldsymbol{B}}\left(\boldsymbol{B}\right)\cdot\partial_{t}\,\boldsymbol{B}+\frac{1}{2\,\mu_{0}}\,\partial_{t}\Big{(}\boldsymbol{\chi}_{\boldsymbol{B}}\left(\boldsymbol{B}\right)\Big{)}\cdot\boldsymbol{B}$

The electric susceptibility vector
$\boldsymbol{\chi}_{\boldsymbol{e}}\left(\boldsymbol{e}\right)$ is a linear
map of the electric field vector $\boldsymbol{e}$ and the magnetic
susceptibility bivector
$\boldsymbol{\chi}_{\boldsymbol{B}}\left(\boldsymbol{B}\right)$ is a linear
map of the magnetic induction field bivector $\boldsymbol{B}$. Furthermore, we
assume that the electric and magnetic properties of the material medium are
constant, which implies that,

$\begin{split}&\partial_{t}\Big{(}\boldsymbol{\chi}_{\boldsymbol{e}}\left(\boldsymbol{e}\right)\Big{)}\cdot\boldsymbol{e}=\boldsymbol{\chi}_{\boldsymbol{e}}\left(\boldsymbol{e}\right)\cdot\partial_{t}\,\boldsymbol{e}\\\
&\boldsymbol{\chi}_{\boldsymbol{B}}\left(\boldsymbol{B}\right)\cdot\partial_{t}\,\boldsymbol{B}=\partial_{t}\Big{(}\boldsymbol{\chi}_{\boldsymbol{B}}\left(\boldsymbol{B}\right)\Big{)}\cdot\boldsymbol{B}\end{split}$
(141)

In view of the conditions (141), relation (140) reduces to,

$\partial_{t}\,e\left(\boldsymbol{e},\boldsymbol{B}\right)=\varepsilon_{0}\,\boldsymbol{e}\cdot\partial_{t}\,\Big{(}\boldsymbol{e}+\boldsymbol{\chi}_{\boldsymbol{e}}\left(\boldsymbol{e}\right)\Big{)}-\,\frac{1}{\mu_{0}}\,\Big{(}\boldsymbol{B}-\,\boldsymbol{\chi}_{\boldsymbol{B}}\left(\boldsymbol{B}\right)\Big{)}\cdot\partial_{t}\,\boldsymbol{B}$
(142)

Using the electric constitutive equation (91) and the magnetic constitutive
equation (96), relation (142) is recast as,

$\partial_{t}\,e=\boldsymbol{e}\cdot\partial_{t}\,\boldsymbol{d}-\,\boldsymbol{H}\cdot\partial_{t}\,\boldsymbol{B}$
(143)

Using the Mawell-Ampère equation (20) and the Faraday equation (21), the time
derivative of the electromagnetic energy density (143) is recast as,

$\partial_{t}\,e=-\,\boldsymbol{e}\cdot\left(\boldsymbol{\nabla}\cdot\boldsymbol{H}+\boldsymbol{j}\right)+\boldsymbol{H}\cdot\left(\boldsymbol{\nabla}\wedge\boldsymbol{e}\right)$
(144)

Thus,

$\partial_{t}\,e+\left(\boldsymbol{\nabla}\cdot\boldsymbol{H}\right)\cdot\boldsymbol{e}-\,\boldsymbol{H}\cdot\left(\boldsymbol{\nabla}\wedge\boldsymbol{e}\right)=-\,\boldsymbol{j}\cdot\boldsymbol{e}$
(145)

Using the algebraic identity (478),

$\boldsymbol{\nabla}\cdot\left(\boldsymbol{H}\cdot\boldsymbol{e}\right)=\left(\boldsymbol{\nabla}\cdot\boldsymbol{H}\right)\cdot\boldsymbol{e}-\,\boldsymbol{H}\cdot\left(\boldsymbol{\nabla}\wedge\boldsymbol{e}\right)$
(146)

relation yields (145) the Poynting’s theorem,

$\partial_{t}\,e+\boldsymbol{\nabla}\cdot\left(\boldsymbol{H}\cdot\boldsymbol{e}\right)=-\,\boldsymbol{j}\cdot\boldsymbol{e}$
(147)

where $\boldsymbol{H}\cdot\boldsymbol{e}$ is the Poynting vector, which is the
electromagnetic current density. In view of the electromagnetic momentum
density (124), Poynting’s theorem (147) in vacuum is recast as,

$\frac{1}{c}\,\partial_{t}\,e+\boldsymbol{\nabla}\cdot\left(\boldsymbol{p}\,c\right)=-\,\frac{1}{c}\,\boldsymbol{j}\cdot\boldsymbol{e}$
(148)

Poynting’s theorem (148) in matter is valid under the assumption that the
electric and magnetic properties of the material medium are constant. In
linear electromagnetism, according to relations (137) and (138), the auxiliary
electromagnetic field multivector (44) is written as,

$G=\boldsymbol{d}\left(\boldsymbol{e}\right)+\frac{1}{c}\,\boldsymbol{H}\left(\boldsymbol{B}\right)=\varepsilon_{0}\,\Big{(}\boldsymbol{e}+\boldsymbol{\chi}_{\boldsymbol{e}}\left(\boldsymbol{e}\right)\Big{)}+\frac{1}{\mu_{0}\,c}\,\Big{(}\boldsymbol{B}-\,\boldsymbol{\chi}_{\boldsymbol{B}}\left(\boldsymbol{B}\right)\Big{)}$
(149)

In view of identity (31), the auxiliary electromagnetic field multivector
(149) is recast as,

$G=\varepsilon_{0}\left(\boldsymbol{e}+c\,\boldsymbol{B}\right)+\varepsilon_{0}\,\Big{(}\boldsymbol{\chi}_{\boldsymbol{e}}\left(\boldsymbol{e}\right)-\,c\,\boldsymbol{\chi}_{\boldsymbol{B}}\left(\boldsymbol{B}\right)\Big{)}$
(150)

The susceptibilities are linear maps of the electromagnetic fields
$\boldsymbol{e}$ and $\boldsymbol{B}$ that are expressed in relations (64) and
(65) as linear combinations of the electromagnetic multivector $F$ and its
reverse $F^{{\dagger}}$. Thus, we define an electromagnetic susceptibility
multivector in space-time as a linear map of the electromagnetic field
bivector $F$,

$\chi_{F}\left(F\right)=\boldsymbol{\chi}_{\boldsymbol{e}}\left(\boldsymbol{e}\right)-\,c\,\boldsymbol{\chi}_{\boldsymbol{B}}\left(\boldsymbol{B}\right)$
(151)

In view of relations (37) and (151), the auxiliary electromagnetic field
multivector (150) is written as a linear map of the electromagnetic field
multivector $F$,

$G\left(F\right)=\varepsilon_{0}\,\Big{(}F+\chi_{F}\left(F\right)\Big{)}$
(152)

In view of expressions (99) and (152) for the electromagnetic field
multivector $G\left(F\right)$, the electromagnetic polarisation multivector
$\tilde{P}\left(F\right)$ is a linear map of the electromagnetic field
multivector $F$,

$\tilde{P}\left(F\right)=\varepsilon_{0}\,\chi_{F}\left(F\right)$ (153)

##  13 Maxwell equation in vacuum (STA)

Special relativity, that is rooted in electromagnetism, is described in space-
time. To describe relativistic theories like electromagnetism in vector space
(VS), an additional time dimension is added such that the generalisation of
the spatial vector space $\mathbb{R}^{3}$ becomes the space-time vector space
$\mathbb{R}^{1,3}$. Similarly, the spatial algebra (SA) $\mathbb{G}^{3}$ is
generalised by including an additional time dimension and called the space-
time algebra (STA) $\mathbb{G}^{1,3}$. [15] As we will show, the fundamental
equations of electromagnetism reach the highest degree of simplicity and
beauty because it turns out that the natural language of relativistic physical
phenomena is precisely space-time algebra (STA). We consider an orthonormal
vector frame $\\{e_{0},e_{1},e_{2},e_{3}\\}$ in space-time in order to recast
Maxwell’s equation in vacuum. The gradient operator $\nabla$ is a covariant
vector in space-time is written in coordinates as,

$\nabla=e^{\mu}\,\partial_{\mu}=e^{0}\,\partial_{0}+e^{i}\,\partial_{i}=e^{0}\,\frac{1}{c}\,\partial_{t}+e^{i}\,\partial_{i}$
(154)

In view of relation (520), the inner product of the gradient $\nabla$ with the
time vector $e_{0}$ yields,

$\nabla\cdot e_{0}=e_{0}\cdot\nabla=\partial_{0}=\frac{1}{c}\,\partial_{t}$
(155)

According to relation (521), the outer product of the gradient $\nabla$ with
the time vector $e_{0}$ yields,

$\nabla\wedge
e_{0}=-\,e_{0}\wedge\nabla=-\,\boldsymbol{e}_{i}\,\partial^{i}=-\,\boldsymbol{\nabla}$
(156)

In view of the inner product (520) and the outer product (521), the geometric
product of the gradient $\nabla$ and the time vector $e_{0}$ yields,

$\nabla\,e_{0}=\nabla\cdot e_{0}+\nabla\wedge
e_{0}=\frac{1}{c}\,\partial_{t}-\,\boldsymbol{\nabla}$ (157)

Taking into account the properties (494) and (496), the geometric product in
reverse order is given by,

$e_{0}\,\nabla=e_{0}\cdot\nabla+e_{0}\wedge\nabla=\nabla\cdot
e_{0}-\,\nabla\wedge e_{0}=\frac{1}{c}\,\partial_{t}+\boldsymbol{\nabla}$
(158)

The electric current density $J$ is a contravariant vector in space-time
defined as,

$J=J^{\mu}\,e_{\mu}=J^{0}\,e_{0}+J^{i}\,e_{i}=qc\,e_{0}+j^{i}\,e_{i}$ (159)

In view of relation (509), the inner product of the electric current density
$J$ with the time vector $e_{0}$ yields,

$J\cdot e_{0}=e_{0}\cdot J=J^{0}=qc$ (160)

According to relation (510), the outer product of the electric current density
$J$ with the time vector $e_{0}$ yields,

$J\wedge e_{0}=-\,e_{0}\wedge J=J^{i}\left(e_{i}\wedge
e_{0}\right)=j^{i}\,\boldsymbol{e}_{i}=\boldsymbol{j}$ (161)

In view of the inner product (509) and the outer product (510), the geometric
product of the electric current density $J$ and the time vector $e_{0}$
yields,

$J\,e_{0}=J\cdot e_{0}+J\wedge e_{0}=qc+\boldsymbol{j}$ (162)

Taking into account the properties (494) and (496), the geometric product in
reverse order is given by,

$e_{0}\,J=e_{0}\cdot J+e_{0}\wedge J=J\cdot e_{0}-\,J\wedge
e_{0}=qc-\,\boldsymbol{j}$ (163)

The Maxwell equation (43) in vacuum is written in spatial algebra as,

$\left(\frac{1}{c}\,\partial_{t}+\boldsymbol{\nabla}\right)G=\frac{1}{c}\left(qc-\,\boldsymbol{j}\right)$
(164)

In view of the geometric product (158) between the time vector $e_{0}$ and the
gradient operator $\nabla$ and the geometric product (163) of the time vector
$e_{0}$ and the electric current density $J$, the Maxwell equation (164) in
vacuum is recast in space-time algebra as,

$e_{0}\,\nabla\,G=\frac{1}{c}\,e_{0}\,J$ (165)

The time vector $e_{0}$ defines a relative spatial frame
$\\{\boldsymbol{e}_{1},\boldsymbol{e}_{2},\boldsymbol{e}_{3}\\}$ that is
orthogonal to it. Taking the geometric product of the time vector $e_{0}$ and
the Maxwell equation (165) in spatial algebra, the latter is projected onto an
orthonormal frame $\\{e_{0},e_{1},e_{2},e_{3}\\}$ with a specific time vector
$e_{0}$. Thus, in space-time algebra the frame independent Maxwell equation in
vacuum reads,

$\nabla\,G=\frac{1}{c}\,J$ (166)

The gradient of the auxiliary electromagnetic field on the left-hand side of
the Maxwell equation (166) in vacuum can be written as the sum of the
divergence and the curl,

$\nabla\cdot G+\nabla\wedge G=\frac{1}{c}\,J$ (167)

Since the electric current density $J$ is a vector in space-time and the the
auxiliary electromagnetic multivector field $G$ is the sum of a linear
combination (44) of the vector field $\boldsymbol{d}$ and the auxiliary
magnetic bivector field $\boldsymbol{H}$, the curl of the auxiliary
electromagnetic multivector field, which is a bivector, has to vanish,

$\nabla\wedge G=0$ (168)

which is the homogeneous Maxwell equation in vacuum that is independent of the
electric current density $J$. In view of the homogeneous Maxwell equation
(168) in vacuum, the Maxwell equation (167) in vacuum yields the inhomogeneous
Maxwell equation in vacuum,

$\nabla\cdot G=\frac{1}{c}\,J$ (169)

In view of the electromagnetic constitutive relation in vacuum (42), the
homogeneous Maxwell equation (168) in vacuum is recast as,

$\nabla\wedge F=0$ (170)

and the inhomogeneous Maxwell equation (169) in vacuum is recast as,

$\nabla\cdot F=\frac{1}{\varepsilon_{0}\,c}\,J$ (171)

The Maxwell equation in vacuum is the sum of the homogeneous and inhomogeneous
Maxwell equations (168) and (169) in vacuum,

$\nabla\,F=\frac{1}{\varepsilon_{0}\,c}\,J$ (172)

##  14 Maxwell equation in matter (STA)

In order to recast Maxwell’s equation in matter, we follow a similar approach
as in vacuum except that the electromagnetic constitutive equation (99) is
different due to the electric polarisation multivector $\tilde{P}$. In view of
the gradient (158) and the electric current density (163), the Maxwell
equation (100) is recast as,

$e_{0}\,\nabla\left(\varepsilon_{0}\,F+\tilde{P}\right)=\frac{1}{c}\,e_{0}\,J$
(173)

Since $e_{0}^{2}=1$, multiplying relation (173) by $e_{0}$ yields,

$\nabla\left(\varepsilon_{0}\,F+\tilde{P}\right)=\frac{1}{c}\,J$ (174)

In view of the electromagnetic constitutive equation (99), the Maxwell
equation in matter (174) reduces to,

$\nabla\,G=\frac{1}{c}\,J$ (175)

We now recast the electromagnetic constitutive equation in terms of the
electric current density in matter $\tilde{J}$. The electric current density
in a dielectric and magnetic medium $\tilde{J}$ is a contravariant vector in
space-time defined as,

$\tilde{J}=\tilde{J}^{\mu}\,e_{\mu}=\tilde{J}^{0}\,e_{0}+\tilde{J}^{i}\,e_{i}=\tilde{q}c\,e_{0}+\tilde{j}^{i}\,e_{i}$
(176)

In view of relation (509), the inner product of the electric current density
in matter $\tilde{J}$ with the time vector $e_{0}$ yields,

$\tilde{J}\cdot e_{0}=e_{0}\cdot\tilde{J}=\tilde{J}^{0}=\tilde{q}c$ (177)

According to relation (510), the outer product of the electric current density
in matter $\tilde{J}$ with the time vector $e_{0}$ yields,

$\tilde{J}\wedge e_{0}=-\,e_{0}\wedge\tilde{J}=\tilde{J}^{i}\left(e_{i}\wedge
e_{0}\right)=\tilde{j}^{i}\,\boldsymbol{e}_{i}=\boldsymbol{\tilde{j}}$ (178)

In view of the inner product (509) and the outer product (510), the geometric
product of the electric current density in matter $\tilde{J}$ and the time
vector $e_{0}$ yields,

$\tilde{J}\,e_{0}=\tilde{J}\cdot e_{0}+\tilde{J}\wedge
e_{0}=\tilde{q}c+\boldsymbol{\tilde{j}}$ (179)

Taking into account the properties (494) and (496), the geometric product in
reverse order is given by,

$e_{0}\,\tilde{J}=e_{0}\cdot\tilde{J}+e_{0}\wedge\tilde{J}=\tilde{J}\cdot
e_{0}-\,\tilde{J}\wedge e_{0}=\tilde{q}c-\,\boldsymbol{\tilde{j}}$ (180)

In view of relations (104) and (105), relation (179) is recast as,

$e_{0}\,\tilde{J}=qc-\,\boldsymbol{j}-\,c\,\boldsymbol{\nabla}\cdot\boldsymbol{\tilde{p}}-\,\left(\partial_{t}\,\boldsymbol{\tilde{p}}-\,\boldsymbol{\nabla}\cdot\boldsymbol{\tilde{M}}\right)$
(181)

Using the gradient (158), relation (115) is recast as,

$e_{0}\,\nabla\cdot\tilde{P}=\boldsymbol{\nabla}\cdot\boldsymbol{\tilde{p}}+\frac{1}{c}\left(\partial_{t}\,\boldsymbol{\tilde{p}}-\,\boldsymbol{\nabla}\cdot\boldsymbol{\tilde{M}}\right)$
(182)

In view of the electric current density (163) and the divergence of the
electromagnetic polarisation multivector (182), the electric current density
in matter (181) reduces to,

$e_{0}\,\tilde{J}=e_{0}\,J-\,e_{0}\,c\ \nabla\cdot\tilde{P}$ (183)

Since $e_{0}^{2}=1$, taking the geometric of the vector $e_{0}$ and equation
(183) yields,

$\tilde{J}=J-\,c\ \nabla\cdot\tilde{P}$ (184)

which is quite a beautiful result. The Maxwell equation (185) in a dielectric
and magnetic medium is written in spatial algebra $\mathbb{G}^{3}$ as,

$\left(\frac{1}{c}\,\partial_{t}+\boldsymbol{\nabla}\right)\varepsilon_{0}\,F=\tilde{q}-\,\frac{1}{c}\,\boldsymbol{\tilde{j}}$
(185)

In view of the geometric product (158) between the time vector $e_{0}$ and the
gradient operator $\nabla$ and the geometric product (180) of the time vector
$e_{0}$ and the electric current density in matter $\tilde{J}$, the Maxwell
equation (164) in matter is recast in space-time algebra $\mathbb{G}^{3}$ as,

$e_{0}\,\nabla\,F=\frac{1}{\varepsilon_{0}\,c}\,e_{0}\,\tilde{J}$ (186)

Taking the geometric product of the time vector $e_{0}$ and the Maxwell
equation (186) in spatial algebra $\mathbb{G}^{3}$, the latter is projected
onto an orthonormal frame $\\{e_{0},e_{1},e_{2},e_{3}\\}$ with a specific time
vector $e_{0}$. Thus, in space-time algebra $\mathbb{G}^{3}$, the frame
independent Maxwell equation in matter reads,

$\nabla\,F=\frac{1}{\varepsilon_{0}\,c}\,\tilde{J}$ (187)

The gradient of the auxiliary electromagnetic field on the left-hand side of
the Maxwell equation (187) in matter can be written as the sum of the
divergence and the curl,

$\nabla\cdot F+\nabla\wedge F=\frac{1}{\varepsilon_{0}\,c}\,\tilde{J}$ (188)

Since the electric current density $J$ is a vector in space-time and the the
electromagnetic multivector field $F$ is the sum of a linear combination (37)
of the vector field $\boldsymbol{e}$ and the magnetic bivector field
$\boldsymbol{B}$, the curl of the electromagnetic multivector field, which is
a bivector, has to vanish,

$\nabla\wedge F=0$ (189)

which is the homogeneous Maxwell equation in matter that is independent of the
electric current density in matter $\tilde{J}$. In view of the homogeneous
Maxwell equation (189) in matter, the Maxwell equation (188) in matter yields
the inhomogeneous Maxwell equation in matter,

$\nabla\cdot F=\frac{1}{\varepsilon_{0}\,c}\,\tilde{J}$ (190)

##  15 Electromagnetic waves in vacuum (STA)

In the space-time algebra (STA), electromagnetic waves in vacuum are a direct
and straightforward consequence of the Maxwell equation (166) as in space-
algebra but it is a way that is even simpler. The gradient of the Maxwell
equation (166) in space-time is in fact the electromagnetic wave equation. The
gradient of the Maxwell equation (166) in vacuum reads,

$\nabla^{2}\,G=\frac{1}{c}\,\nabla\,J$ (191)

The gradient of the electric current density on the right-hand side of
relation (191) can be written as the sum of the divergence and the curl,

$\nabla^{2}\,G=\frac{1}{c}\,\nabla\cdot J+\frac{1}{c}\,\nabla\wedge J$ (192)

Since the Laplacian $\nabla^{2}$ is a scalar operator and the auxiliary
electromagnetic multivector field $G$ is the sum of a linear combination (44)
of the electric displacement vector field $\boldsymbol{d}$ and the auxiliary
magnetic bivector field $\boldsymbol{H}$, the divergence of the current
density $\nabla\cdot J$ on the right-hand side of relation (192), which is a
scalar, has to vanish,

$\nabla\cdot J=0$ (193)

The electric charge conservation law (193) is recast in spatial algebra as the
electric continuity equation (6) using to the identity (525) for the scalar
product of two vectors in space-time and the definition of the current density
vector (165),

$\nabla\cdot J=\partial_{t}\,q+\boldsymbol{\nabla}\cdot\boldsymbol{j}=0$ (194)

In view of the electric charge conservation law (193), relation (192) reduces
to,

$\nabla^{2}\,G=\frac{1}{c}\,\nabla\wedge J$ (195)

which is the electromagnetic wave equation in vacuum in space-time algebra. To
show this, we recast now this equation in spatial algebra. In view of the
geometric products (157), (158) and of the identity $e_{0}\,e_{0}=1$, the
Laplacian operator called the d’Alembertian operator is written in a specific
space-time frame defined by a time vector $e_{0}$ as,

$\nabla^{2}=\nabla\,\nabla=\left(\nabla\,e_{0}\right)\left(e_{0}\,\nabla\right)=\left(\frac{1}{c}\,\partial_{t}-\,\boldsymbol{\nabla}\right)\left(\frac{1}{c}\,\partial_{t}+\boldsymbol{\nabla}\right)=\frac{1}{c^{2}}\,\partial^{2}_{t}-\,\boldsymbol{\nabla}^{2}$
(196)

In view of identity (196), the left-hand side of relation (195) is recast in
spatial algebra as,

$\nabla^{2}\,G=\left(\frac{1}{c^{2}}\,\partial^{2}_{t}-\,\boldsymbol{\nabla}^{2}\right)G$
(197)

In view of identity (528), the right-hand side of relation (195) is recast in
spatial algebra as,

$\frac{1}{c}\,\nabla\wedge
J=-\,\frac{1}{c^{2}}\,\partial_{t}\boldsymbol{j}-\,\boldsymbol{\nabla}\,q+\frac{1}{c}\,\boldsymbol{\nabla}\wedge\boldsymbol{j}$
(198)

According to relations (197) and (198), the gradient of the Maxwell equation
(195) in vacuum is recast in spatial algebra as the electromagnetic wave
equation (49),

$\left(\frac{1}{c^{2}}\,\partial^{2}_{t}-\,\boldsymbol{\nabla}^{2}\right)G=-\,\boldsymbol{\nabla}\,q-\,\frac{1}{c^{2}}\,\partial_{t}\boldsymbol{j}+\frac{1}{c}\,\boldsymbol{\nabla}\wedge\boldsymbol{j}$
(199)

##  16 Electromagnetic waves in matter (STA)

In the space-time algebra (STA), electromagnetic waves in matter are a direct
and straightforward consequence of the Maxwell equation (175) or (188) as in
space-algebra but it is a way that is even simpler. The gradient of the
Maxwell equation (175) or (188) in space-time is in fact the electromagnetic
wave equation. The gradient of the Maxwell equation (175) in matter reads,

$\nabla^{2}\,G=\frac{1}{c}\,\nabla\,J$ (200)

In view of equation (193), the electromagnetic wave equation in matter (200)
becomes,

$\nabla^{2}\,G=\frac{1}{c}\,\nabla\wedge J$ (201)

The gradient of the inhomogeneous Maxwell equation (188) in matter is given
by,

$\nabla^{2}\,F=\frac{1}{\varepsilon_{0}\,c}\,\nabla\,\tilde{J}$ (202)

Since the Laplacian $\nabla^{2}$ is a scalar operator and the electromagnetic
multivector field $F$ is the sum of a linear combination (37) of the vector
field $\boldsymbol{e}$ and the magnetic bivector field $\boldsymbol{B}$, the
divergence of the current density in matter $\nabla\cdot\tilde{J}$ on the
right-hand side of relation (202), which is a scalar, has to vanish,

$\nabla\cdot\tilde{J}=0$ (203)

In view of the electric charge conservation law (203), relation (202) reduces
to,

$\nabla^{2}\,F=\frac{1}{\varepsilon_{0}\,c}\,\nabla\wedge\tilde{J}$ (204)

which is the electromagnetic wave equation in matter in space-time algebra. To
show this, we recast now this equation in spatial algebra. In view of identity
(196), the left-hand side of relation (204) is recast in spatial algebra as,

$\nabla^{2}\,F=\left(\frac{1}{c^{2}}\,\partial^{2}_{t}-\,\boldsymbol{\nabla}^{2}\right)F$
(205)

In view of identity (528), the right-hand side of relation (204) is recast in
spatial algebra as,

$\frac{1}{\varepsilon_{0}\,c}\,\nabla\wedge\tilde{J}=-\,\frac{1}{\varepsilon_{0}\,c^{2}}\,\partial_{t}\boldsymbol{\tilde{j}}-\,\frac{1}{\varepsilon_{0}}\,\boldsymbol{\nabla}\,\tilde{q}+\frac{1}{\varepsilon_{0}\,c}\,\boldsymbol{\nabla}\wedge\boldsymbol{\tilde{j}}$
(206)

According to relations (205) and (206), the gradient of the Maxwell equation
(204) in matter is recast in spatial algebra as the electromagnetic wave
equation (49),

$\left(\frac{1}{c^{2}}\,\partial^{2}_{t}-\,\boldsymbol{\nabla}^{2}\right)F=-\,\frac{1}{\varepsilon_{0}}\,\boldsymbol{\nabla}\,\tilde{q}-\,\frac{1}{\varepsilon_{0}\,c^{2}}\,\partial_{t}\boldsymbol{\tilde{j}}+\frac{1}{\varepsilon_{0}\,c}\,\boldsymbol{\nabla}\wedge\boldsymbol{\tilde{j}}$
(207)

##  17 Electromagnetic fields (STA)

The electromagnetic multivectors $F$ and $G$ and the electromagnetic
polarisation multivector $\tilde{P}$ in spatial algebra (SA) are in fact
bivectors in space-time algebra (STA), as we will show in this section. We
will also determine the components of these bivectors with respect to an
orthonormal space-time frame. The electromagnetic multivector (37) is written
in terms of the relative spatial vectors as,

$F=\boldsymbol{e}+c\,\boldsymbol{B}=e^{i}\,\boldsymbol{e}_{i}+\frac{1}{2}\,c\,B^{\,ij}\,\boldsymbol{e}_{i}\wedge\boldsymbol{e}_{j}$
(208)

Using the definition (498) of the relative spatial vectors, the
electromagnetic multivector (208) is recast as,

$F=e^{i}\left(e_{i}\,e_{0}\right)+\frac{1}{2}\,c\,B^{\,ij}\left(e_{i}\,e_{0}\right)\wedge\left(e_{j}\,e_{0}\right)$
(209)

The electric field vector $\boldsymbol{e}$ and magnetic induction field
bivector $\boldsymbol{B}$ are frame dependent since they are defined in a
specific spatial algebra $\mathbb{G}^{3}$. In contrast, the electromagnetic
field multivector $F$ is frame independent since it is defined in the space-
time algebra $\mathbb{G}^{1,3}$. Using identities (495) and (496), we obtain,

$\begin{split}&e_{i}\,e_{0}=e_{i}\wedge e_{0}\\\
&\left(e_{i}\,e_{0}\right)\wedge\left(e_{j}\,e_{0}\right)=e_{i}\,e_{0}\,e_{j}\,e_{0}=-\,e_{i}\,e_{j}=-\,e_{i}\wedge
e_{j}\end{split}$ (210)

In view of identities (210), the electromagnetic field multivector (209) is
recast as,

$F=e^{i}\,e_{i}\wedge e_{0}-\,\frac{1}{2}\,c\,B^{\,ij}\,e_{i}\wedge e_{j}$
(211)

In the space-time algebra, the electromagnetic field multivector (211) is in
fact a bivector written as,

$F=\frac{1}{2}\,F^{\mu\nu}\,e_{\mu}\wedge e_{\nu}$ (212)

According to relations (212), (561) and (562), the components of the
electromagnetic field bivector $F$ are given by,

$F^{\mu\nu}=\left(e^{\nu}\wedge e^{\mu}\right)\cdot
F=F\cdot\left(e^{\nu}\wedge e^{\mu}\right)$ (213)

According to relations (515) and (545), the components of the electromagnetic
field bivector $F$ are written as,

$F^{\mu\nu}=e^{\nu}\cdot\left(e^{\mu}\cdot F\right)=\left(F\cdot
e^{\nu}\right)\cdot e^{\mu}$ (214)

According to identity (496), the components of the electromagnetic field
bivector $F$ are antisymmetric,

$F^{\mu\nu}=-\,F^{\nu\mu}$ (215)

In view of identities (496) and (215), the electromagnetic field bivector
(212) is split as,

$F=\frac{1}{2}\left(F^{i0}\,e_{i}\wedge e_{0}+F^{0i}\,e_{0}\wedge
e_{i}+F^{ij}\,e_{i}\wedge e_{j}\right)=F^{i0}\,e_{i}\wedge
e_{0}+\frac{1}{2}\,F^{ij}\,e_{i}\wedge e_{j}$ (216)

In view of relations (211), (216) and (215), the components of the
electromagnetic field bivector $F$ are given by,

$\displaystyle F^{\mu\mu}=-\,F^{\mu\mu}=0$ $\displaystyle
F^{i0}=-\,F^{0i}=e^{i}$ (217) $\displaystyle F^{ij}=-\,F^{ji}=-\,c\,B^{ij}$

Thus, the components of the electromagnetic bivector field $F$ are written as
an antisymmetric matrix,

$F^{\mu\nu}=\begin{pmatrix}0&-\,e^{1}&-\,e^{2}&-\,e^{3}\\\
e^{1}&0&-\,c\,B^{12}&-\,c\,B^{13}\\\ e^{2}&c\,B^{12}&0&-\,c\,B^{23}\\\
e^{3}&c\,B^{13}&c\,B^{23}&0\\\ \end{pmatrix}$ (218)

The components of the electromagnetic field bivector $F^{\mu\nu}$ can be
recast in terms of the components of the dual magnetic induction field vector
$\boldsymbol{b}$ where $\boldsymbol{b}=\boldsymbol{B}^{\ast}$,

$b^{1}\,\boldsymbol{e}_{1}+b^{2}\,\boldsymbol{e}_{2}+b^{3}\,\boldsymbol{e}_{3}=B^{12}\left(\boldsymbol{e}_{1}\wedge\boldsymbol{e}_{2}\right)^{\ast}+B^{23}\left(\boldsymbol{e}_{2}\wedge\boldsymbol{e}_{3}\right)^{\ast}+B^{31}\left(\boldsymbol{e}_{3}\wedge\boldsymbol{e}_{1}\right)^{\ast}$
(219)

According to the duality (410),

$\displaystyle\left(\boldsymbol{e}_{1}\wedge\boldsymbol{e}_{2}\right)^{\ast}=\boldsymbol{e}_{1}\times\boldsymbol{e}_{2}=\boldsymbol{e}_{3}$
$\displaystyle\left(\boldsymbol{e}_{2}\wedge\boldsymbol{e}_{3}\right)^{\ast}=\boldsymbol{e}_{2}\times\boldsymbol{e}_{3}=\boldsymbol{e}_{1}$
(220)
$\displaystyle\left(\boldsymbol{e}_{3}\wedge\boldsymbol{e}_{1}\right)^{\ast}=\boldsymbol{e}_{3}\times\boldsymbol{e}_{1}=\boldsymbol{e}_{2}$

which implies that the identity (219) is recast as,

$b^{1}\,\boldsymbol{e}_{1}+b^{2}\,\boldsymbol{e}_{2}+b^{3}\,\boldsymbol{e}_{3}=B^{23}\,\boldsymbol{e}_{1}+B^{31}\,\boldsymbol{e}_{2}+B^{12}\,\boldsymbol{e}_{3}$
(221)

Thus,

$B^{12}=b_{3}\qquad\text{and}\qquad
B^{13}=-\,B^{31}=-\,b_{2}\qquad\text{and}\qquad B^{23}=b_{1}$ (222)

Using relations (222), the components of the electromagnetic field bivector
(218) are recast in vector space as,

$F^{\mu\nu}=\begin{pmatrix}0&-\,e^{1}&-\,e^{2}&-\,e^{3}\\\
e^{1}&0&-\,c\,b^{3}&c\,b^{2}\\\ e^{2}&c\,b^{3}&0&-\,c\,b^{1}\\\
e^{3}&-\,c\,b^{2}&c\,b^{1}&0\\\ \end{pmatrix}$ (223)

The frame dependence of the electric field vector $\boldsymbol{e}$ and
magnetic induction field bivector $\boldsymbol{B}$ is made explicit using a
geometric product with the time vector $e_{0}$. In view of relation (211), the
double geometric product of the electromagnetic multivector $F$ and the time
vector $e_{0}$ yields,

$\begin{split}&e_{0}\,F\,e_{0}=e^{i}\,e_{0}\,e_{i}\,e_{0}+\frac{1}{2}\,c\,B^{\,ij}\,e_{0}\left(e_{i}\wedge
e_{j}\right)e_{0}\\\
&\phantom{e_{0}\,F\,e_{0}}=-\,e^{i}\,e_{i}+\frac{1}{2}\,c\,B^{\,ij}\left(e_{i}\wedge
e_{j}\right)\end{split}$ (224)

In view of relations (208), (209) and (224), we obtain,

$e_{0}\,F\,e_{0}=-\,\boldsymbol{e}+c\,\boldsymbol{B}=-\,F^{{\dagger}}$ (225)

Using relations (225), (64) and (65), the electric field vector
$\boldsymbol{e}$ is written entirely in terms of the electromagnetic field
multivector $F$ and the time vector $e_{0}$ as,

$\boldsymbol{e}=\frac{1}{2}\left(F-\,e_{0}\,F\,e_{0}\right)$ (226)

and the magnetic field bivector $\boldsymbol{B}$ is written as,

$\boldsymbol{B}=\frac{1}{2\,c}\left(F+e_{0}\,F\,e_{0}\right)$ (227)

The auxiliary electromagnetic multivector (44) is written in terms of the
relative spatial vectors as,

$G=\boldsymbol{d}+\frac{1}{c}\,\boldsymbol{H}=d^{\,i}\,\boldsymbol{e}_{i}+\frac{1}{2\,c}\,H^{\,ij}\,\boldsymbol{e}_{i}\wedge\boldsymbol{e}_{j}$
(228)

Using the definition (498) of the relative spatial vectors, the auxiliary
electromagnetic multivector (228) is recast as,

$G=d^{\,i}\left(e_{i}\,e_{0}\right)+\frac{1}{2\,c}\,H^{\,ij}\left(e_{i}\,e_{0}\right)\wedge\left(e_{j}\,e_{0}\right)$
(229)

The electric displacement field vector $\boldsymbol{d}$ and auxiliary magnetic
field bivector $\boldsymbol{H}$ are frame dependent since they are defined in
a specific spatial algebra $\mathbb{G}^{3}$. In contrast, the electromagnetic
field multivector $G$ is frame independent since it is defined in the space-
time algebra $\mathbb{G}^{1,3}$. In view of identities (210), the auxiliary
electromagnetic field multivector (229) is recast as,

$G=d^{\,i}\,e_{i}\wedge e_{0}-\,\frac{1}{2\,c}\,H^{\,ij}\,e_{i}\wedge e_{j}$
(230)

In the space-time algebra, the electromagnetic field multivector (230) is in
fact a bivector written as,

$G=\frac{1}{2}\,G^{\mu\nu}\,e_{\mu}\wedge e_{\nu}$ (231)

According to relations (231), (561) and (562), the components of the
electromagnetic field bivector $G$ are given by,

$G^{\mu\nu}=\left(e^{\nu}\wedge e^{\mu}\right)\cdot
G=G\cdot\left(e^{\nu}\wedge e^{\mu}\right)$ (232)

According to relations (515) and (545), the components of the electromagnetic
field bivector $G$ are written as,

$G^{\mu\nu}=e^{\nu}\cdot\left(e^{\mu}\cdot G\right)=\left(G\cdot
e^{\nu}\right)\cdot e^{\mu}$ (233)

According to identity (496), the components of the electromagnetic field
bivector $G$ are antisymmetric,

$G^{\mu\nu}=-\,G^{\nu\mu}$ (234)

In view of identities (496) and (496), the auxiliary electromagnetic field
bivector (231) is split as,

$G=\frac{1}{2}\left(G^{i0}\,e_{i}\wedge e_{0}+G^{0i}\,e_{0}\wedge
e_{i}+G^{ij}\,e_{i}\wedge e_{j}\right)=G^{i0}\,e_{i}\wedge
e_{0}+\frac{1}{2}\,G^{ij}\,e_{i}\wedge e_{j}$ (235)

In view of relations (230), (231) and (234), the components of the
electromagnetic field bivector $G$ are given by,

$\displaystyle G^{\mu\mu}=-\,G^{\mu\mu}=0$ $\displaystyle
G^{i0}=-\,G^{0i}=d^{i}$ (236) $\displaystyle
G^{ij}=-\,G^{ji}=-\,c^{-1}\,H^{ij}$

Thus, the components of the electromagnetic bivector field $G$ are written as
an antisymmetric matrix,

$G^{\mu\nu}=\begin{pmatrix}0&-\,d^{1}&-\,d^{2}&-\,d^{3}\\\
d^{1}&0&-\,c^{-1}\,H^{12}&-\,c^{-1}\,H^{13}\\\
d^{2}&c^{-1}\,H^{12}&0&-\,c^{-1}\,H^{23}\\\
d^{3}&c^{-1}\,H^{13}&c^{-1}\,H^{23}&0\\\ \end{pmatrix}$ (237)

The components of the electromagnetic field bivector $G^{\mu\nu}$ can be
recast in terms of the components of the dual auxiliary magnetic field vector
$\boldsymbol{h}$ where $\boldsymbol{h}=\boldsymbol{H}^{\ast}$,

$h^{1}\,\boldsymbol{e}_{1}+h^{2}\,\boldsymbol{e}_{2}+h^{3}\,\boldsymbol{e}_{3}=H^{12}\left(\boldsymbol{e}_{1}\wedge\boldsymbol{e}_{2}\right)^{\ast}+H^{23}\left(\boldsymbol{e}_{2}\wedge\boldsymbol{e}_{3}\right)^{\ast}+H^{31}\left(\boldsymbol{e}_{3}\wedge\boldsymbol{e}_{1}\right)^{\ast}$
(238)

According to the duality (17), the identity (238) is recast as,

$h^{1}\,\boldsymbol{e}_{1}+h^{2}\,\boldsymbol{e}_{2}+h^{3}\,\boldsymbol{e}_{3}=H^{23}\,\boldsymbol{e}_{1}+H^{31}\,\boldsymbol{e}_{2}+H^{12}\,\boldsymbol{e}_{3}$
(239)

Thus,

$H^{12}=h_{3}\qquad\text{and}\qquad
H^{13}=-\,H^{31}=-\,h_{2}\qquad\text{and}\qquad H^{23}=h_{1}$ (240)

Using relations (240), the components of the electromagnetic field bivector
(237) are recast in vector space as,

$G^{\mu\nu}=\begin{pmatrix}0&-\,d^{1}&-\,d^{2}&-\,d^{3}\\\
d^{1}&0&-\,c^{-1}\,h^{3}&c^{-1}\,h^{2}\\\
d^{2}&c^{-1}\,h^{3}&0&-\,c^{-1}\,h^{1}\\\
d^{3}&-\,c^{-1}\,h^{2}&c^{-1}\,h^{1}&0\\\ \end{pmatrix}$ (241)

The frame dependence of the electric displacement field vector
$\boldsymbol{d}$ and auxiliary magnetic field bivector $\boldsymbol{H}$ is
made explicit using a geometric product with the time vector $e_{0}$. In view
of relation (230), the double geometric product of the auxiliary
electromagnetic multivector $G$ and the time vector $e_{0}$ yields,

$\begin{split}&e_{0}\,G\,e_{0}=d^{i}\,e_{0}\,e_{i}\,e_{0}+\frac{1}{2\,c}\,H^{\,ij}\,e_{0}\left(e_{i}\wedge
e_{j}\right)e_{0}\\\
&\phantom{e_{0}\,G\,e_{0}}=-\,e^{i}\,e_{i}+\frac{1}{2\,c}\,H^{\,ij}\left(e_{i}\wedge
e_{j}\right)\end{split}$ (242)

In view of relations (228), (229) and (242), we obtain,

$e_{0}\,G\,e_{0}=-\,\boldsymbol{e}+\frac{1}{c}\,\boldsymbol{H}=-\,G^{{\dagger}}$
(243)

Using relations (243), (68) and (69), the electric displacement field vector
$\boldsymbol{d}$ is written entirely in terms of the auxiliary electromagnetic
field multivector $G$ and the time vector $e_{0}$ as,

$\boldsymbol{d}=\frac{1}{2}\left(G-\,e_{0}\,G\,e_{0}\right)$ (244)

and the auxiliary magnetic field bivector $\boldsymbol{H}$ is written as,

$\boldsymbol{H}=\frac{c}{2}\left(G+e_{0}\,G\,e_{0}\right)$ (245)

The electromagnetic polarisation multivector (98) is written in terms of the
relative spatial vectors as,

$\tilde{P}=\boldsymbol{\tilde{p}}-\,\frac{1}{c}\,\boldsymbol{\tilde{M}}=p^{\,i}\,\boldsymbol{e}_{i}-\,\frac{1}{2\,c}\,M^{\,ij}\,\boldsymbol{e}_{i}\wedge\boldsymbol{e}_{j}$
(246)

Using the definition (498) of the relative spatial vectors, the
electromagnetic polarisation multivector (246) is recast as,

$\tilde{P}=p^{\,i}\left(e_{i}\,e_{0}\right)-\,\frac{1}{2\,c}\,M^{\,ij}\left(e_{i}\,e_{0}\right)\wedge\left(e_{j}\,e_{0}\right)$
(247)

The electric polarisation vector $\boldsymbol{\tilde{p}}$ and magnetisation
bivector $\boldsymbol{\tilde{M}}$ are frame dependent since they are defined
in a specific spatial algebra $\mathbb{G}^{3}$. In contrast, the
electromagnetic polarisation multivector $\tilde{P}$ is frame independent
since it is defined in the space-time algebra $\mathbb{G}^{1,3}$. In view of
identities (210), the electric polarisation multivector (229) is recast as,

$\tilde{P}=\tilde{p}^{\,i}\,e_{i}\wedge
e_{0}+\frac{1}{2\,c}\,\tilde{M}^{\,ij}\,e_{i}\wedge e_{j}$ (248)

In the space-time algebra, the electromagnetic polarisation multivector (248)
is in fact a bivector written as,

$\tilde{P}=\frac{1}{2}\,\tilde{P}^{\mu\nu}\,e_{\mu}\wedge e_{\nu}$ (249)

According to relations (249), (561) and (562), the components of the
electromagnetic polarisation bivector $\tilde{P}$ are given by,

$\tilde{P}^{\mu\nu}=\left(e^{\nu}\wedge
e^{\mu}\right)\cdot\tilde{P}=\tilde{P}\cdot\left(e^{\nu}\wedge e^{\mu}\right)$
(250)

According to relations (515) and (545), the components of the electromagnetic
polarisation bivector $\tilde{P}$ are written as,

$\tilde{P}^{\mu\nu}=e^{\nu}\cdot\left(e^{\mu}\cdot\tilde{P}\right)=\left(\tilde{P}\cdot
e^{\nu}\right)\cdot e^{\mu}$ (251)

According to identity (496), the components of the electromagnetic
polarisation bivector $\tilde{P}$ are antisymmetric,

$\tilde{P}^{\mu\nu}=-\,\tilde{P}^{\nu\mu}$ (252)

In view of identities (496) and (496), the electromagnetic polarisation
bivector (249) is split as,

$\tilde{P}=\frac{1}{2}\left(\tilde{P}^{i0}\,e_{i}\wedge
e_{0}+\tilde{P}^{0i}\,e_{0}\wedge e_{i}+\tilde{P}^{ij}\,e_{i}\wedge
e_{j}\right)=\tilde{P}^{i0}\,e_{i}\wedge
e_{0}+\frac{1}{2}\,\tilde{P}^{ij}\,e_{i}\wedge e_{j}$ (253)

In view of relations (248), (249) and (252), the components of the
electromagnetic polarisation bivector $\tilde{P}$ are given by,

$\displaystyle\tilde{P}^{\mu\mu}=-\,\tilde{P}^{\mu\mu}=0$
$\displaystyle\tilde{P}^{i0}=-\,\tilde{P}^{0i}=\tilde{p}^{i}$ (254)
$\displaystyle\tilde{P}^{ij}=-\,\tilde{P}^{ji}=c^{-1}\,\tilde{M}^{ij}$

Thus, the components of the electromagnetic polarisation bivector $\tilde{P}$
are written as an antisymmetric matrix,

$\tilde{P}^{\mu\nu}=\begin{pmatrix}0&-\,\tilde{p}^{1}&-\,\tilde{p}^{2}&-\,\tilde{p}^{3}\\\
\tilde{p}^{1}&0&c^{-1}\,\tilde{M}^{12}&c^{-1}\,\tilde{M}^{13}\\\
\tilde{p}^{2}&-\,c^{-1}\,\tilde{M}^{12}&0&c^{-1}\,\tilde{M}^{23}\\\
\tilde{p}^{3}&-\,c^{-1}\,\tilde{M}^{13}&-\,c^{-1}\,\tilde{M}^{23}&0\\\
\end{pmatrix}$ (255)

The components of the electromagnetic polarisation bivector
$\tilde{P}^{\mu\nu}$ can be recast in terms of the components of the dual
auxiliary magnetisation vector $\boldsymbol{\tilde{m}}$ where
$\boldsymbol{\tilde{m}}=\boldsymbol{\tilde{M}}^{\ast}$,

$\tilde{m}^{1}\,\boldsymbol{e}_{1}+\tilde{m}^{2}\,\boldsymbol{e}_{2}+\tilde{m}^{3}\,\boldsymbol{e}_{3}=\tilde{M}^{12}\left(\boldsymbol{e}_{1}\wedge\boldsymbol{e}_{2}\right)^{\ast}+\tilde{M}^{23}\left(\boldsymbol{e}_{2}\wedge\boldsymbol{e}_{3}\right)^{\ast}+\tilde{M}^{31}\left(\boldsymbol{e}_{3}\wedge\boldsymbol{e}_{1}\right)^{\ast}$
(256)

According to the duality (17), the identity (256) is recast as,

$\tilde{m}^{1}\,\boldsymbol{e}_{1}+\tilde{m}^{2}\,\boldsymbol{e}_{2}+\tilde{m}^{3}\,\boldsymbol{e}_{3}=\tilde{M}^{23}\,\boldsymbol{e}_{1}+\tilde{M}^{31}\,\boldsymbol{e}_{2}+\tilde{M}^{12}\,\boldsymbol{e}_{3}$
(257)

Thus,

$\tilde{M}^{12}=\tilde{m}_{3}\qquad\text{and}\qquad\tilde{M}^{13}=-\,\tilde{M}^{31}=-\,\tilde{m}_{2}\qquad\text{and}\qquad\tilde{M}^{23}=\tilde{m}_{1}$
(258)

Using relations (258), the components of the electromagnetic polarisation
bivector (255) are recast in vector space as,

$\tilde{P}^{\mu\nu}=\begin{pmatrix}0&-\,\tilde{p}^{1}&-\,\tilde{p}^{2}&-\,\tilde{p}^{3}\\\
\tilde{p}^{1}&0&c^{-1}\,\tilde{m}^{3}&-\,c^{-1}\,\tilde{m}^{2}\\\
\tilde{p}^{2}&-\,c^{-1}\,\tilde{m}^{3}&0&c^{-1}\,\tilde{m}^{1}\\\
\tilde{p}^{3}&c^{-1}\,\tilde{m}^{2}&-\,c^{-1}\,\tilde{m}^{1}&0\\\
\end{pmatrix}$ (259)

The frame dependence of the electric polarisation vector
$\boldsymbol{\tilde{m}}$ and magnetisation bivector $\boldsymbol{\tilde{M}}$
is made explicit using a geometric product with the time vector $e_{0}$. In
view of relation (230), the double geometric product of the electromagnetic
polarisation multivector $P$ and the time vector $e_{0}$ yields,

$\begin{split}&e_{0}\,\tilde{P}\,e_{0}=\tilde{p}^{i}\,e_{0}\,e_{i}\,e_{0}-\,\frac{1}{2\,c}\,\tilde{M}^{\,ij}\,e_{0}\left(e_{i}\wedge
e_{j}\right)e_{0}\\\
&\phantom{e_{0}\,P\,e_{0}}=-\,\tilde{p}^{i}\,e_{i}-\,\frac{1}{2\,c}\,\tilde{M}^{\,ij}\left(e_{i}\wedge
e_{j}\right)\end{split}$ (260)

In view of relations (246), (247) and (242), we obtain,

$e_{0}\,\tilde{P}\,e_{0}=-\,\boldsymbol{\tilde{p}}-\,\frac{1}{c}\,\boldsymbol{\tilde{M}}=-\,\tilde{P}^{{\dagger}}$
(261)

Using relations (261), (127) and (128), the electric polarisation field vector
$\boldsymbol{\tilde{p}}$ is written entirely in terms of the electromagnetic
polarisation multivector $P$ and the time vector $e_{0}$ as,

$\boldsymbol{\tilde{p}}=\frac{1}{2}\left(\tilde{P}-\,e_{0}\,\tilde{P}\,e_{0}\right)$
(262)

and the magnetisation field bivector $\boldsymbol{\tilde{M}}$ is written as,

$\boldsymbol{\tilde{M}}=-\,\frac{c}{2}\left(\tilde{P}+e_{0}\,\tilde{P}\,e_{0}\right)$
(263)

##  18 Stress energy momentum in vacuum (STA)

In the space-time algebra (STA) as opposed to to tensor calculus, the stress-
energy momentum is not a tensor but a vector. In fact it is a self-adjoint
linear application of a space-time vector $V$ in vacuum. If this vector $V$ is
the space-time gradient $\nabla$, the stress-energy momentum vector
$\dot{T}\left(\dot{\nabla}\right)$ reduces to the electromagnetic force
density vector $f$ in space-time as we show below. In order to show this, we
begin by writing the electromagnetic momentum density $P$ in space-time. The
electromagnetic momentum density $P$ is a contravariant vector in space-time
defined as,

$P=P^{\mu}\,e_{\mu}=P^{0}\,e_{0}+P^{i}\,e_{i}=\frac{e}{c}\,e_{0}+\tilde{p}^{i}\,e_{i}$
(264)

According to the gradient (154) and the identity (525) for the scalar product
of two vectors in space-time, the divergence of the electromagnetic momentum
density (264) is given by,

$\nabla\cdot
P=\frac{1}{c^{2}}\,\partial_{t}\,e+\boldsymbol{\nabla}\cdot\boldsymbol{p}$
(265)

In view of relations (37), (162), and $e_{0}^{2}=1$, the inner product of the
electric current density vector $J$ and the electromagnetic multivector $F$ is
written as,

$J\cdot F=\left(J\,e_{0}\right)e_{0}\cdot
F=\left(qc+\boldsymbol{j}\right)e_{0}\cdot\left(\boldsymbol{e}+c\,\boldsymbol{B}\right)$
(266)

Since the time vector $e_{0}$ anticommutes with the vector $\boldsymbol{e}$
and commutes with the bivector $\boldsymbol{B}$, using the identity (360),
relation (266) is recast as,

$J\cdot
F=-\,\left(\boldsymbol{j}\cdot\boldsymbol{e}\right)e_{0}-\,c\left(q\,\boldsymbol{e}+\boldsymbol{B}\cdot\boldsymbol{j}\right)e_{0}$
(267)

where $\boldsymbol{j}\cdot\boldsymbol{e}$ is the electromagnetic power density
and $q\,\boldsymbol{e}+\,\boldsymbol{B}\cdot\boldsymbol{j}$ is the Lorentz
force density in spatial algebra $\mathbb{G}^{3}$. This means that the vector
$\frac{1}{c}\,\,J\cdot F$ in the space-time algebra $\mathbb{G}^{1,3}$
represents the electromagnetic force density,

$f=\frac{1}{c}\,J\cdot F$ (268)

In view of relations (159) and (212) and identities (215) and (546), the
electromagnetic force density (268) in space-time is given by,

$f_{\mu}\,e^{\mu}=\frac{1}{2\,c}\,J^{\rho}\,F_{\nu\mu}\,e_{\rho}\cdot\left(e^{\nu}\wedge
e^{\mu}\right)=\frac{1}{2\,c}\left(J^{\nu}\,F_{\nu\mu}-\,J^{\nu}\,F_{\mu\nu}\right)e^{\mu}=\frac{1}{c}\,J^{\nu}\,F_{\nu\mu}\,e^{\mu}$
(269)

Using relations (208) and (211), the electromagnetic force density (268) in
space-time is written in the orthonormal basis
$\\{\,e_{0},\,e_{1},\,e_{2},\,e_{3}\,\\}$ as,

$f=\frac{1}{c}\,J\cdot
F=\frac{1}{c}\,\left(J^{i}\,F_{i0}\right)e_{0}+\frac{1}{c}\,\left(J^{i}\,F_{ij}\right)e_{j}$
(270)

In view of relations (17), (267) and (270) and identity (495),

$f\cdot e_{0}=\frac{1}{c}\,\left(J\cdot F\right)\cdot
e_{0}=\frac{1}{c}\,\left(J^{i}\,F_{i0}\right)=\frac{1}{c}\,\left(j^{i}\,e_{i}\right)=-\,\frac{1}{c}\,\boldsymbol{j}\cdot\boldsymbol{e}$
(271)

In view of relations (265) and (271), Poynting’s theorem in vacuum (85) is
recast as,

$\nabla\cdot P=\frac{1}{c^{2}}\,\left(J\cdot F\right)\cdot
e_{0}=\frac{1}{c}\,f\cdot e_{0}$ (272)

The geometric product of the electromagnetic momentum density (264) and the
time vector $e_{0}$ yields,

$P\,e_{0}=\frac{e}{c}\left(e_{0}\,e_{0}\right)+\tilde{p}^{i}\left(e_{i}\,e_{0}\right)=\frac{1}{c}\left(e+\boldsymbol{p}\,c\right)$
(273)

Since $e_{0}^{2}=1$, multiplying relation (273) by $e_{0}$ we obtain,

$P=\frac{1}{c}\left(e+\boldsymbol{p}\,c\right)e_{0}$ (274)

In view of relations (78) and (225),

$e+\boldsymbol{p}\,c=-\,\frac{1}{4}\left(G\,e_{0}\,F\,e_{0}+F\,e_{0}\,G\,e_{0}\right)=-\,\frac{1}{2}\,\varepsilon_{0}\,F\,e_{0}\,F\,e_{0}$
(275)

and using the identity $e_{0}^{2}=1$, the momentum vector (274) is recast as,

$P=-\,\frac{1}{4\,c}\left(G\,e_{0}\,F+F\,e_{0}\,G\right)=-\,\frac{1}{2\,c}\,\varepsilon_{0}\,F\,e_{0}\,F$
(276)

The stress energy momentum vector $T\left(e_{0}\right)$ is defined as a linear
mapping of the time vector $e_{0}$,

$T\left(e_{0}\right)=P\,c=-\,\frac{1}{4}\left(G\,e_{0}\,F+F\,e_{0}\,G\right)=-\,\frac{1}{2}\,\varepsilon_{0}\,F\,e_{0}\,F$
(277)

Since the stress energy momentum vector $T\left(e_{0}\right)$ depends on the
time vector $e_{0}$, it is frame dependent. In view of relation (277),
Poynting’s theorem (272) is recast as,

$\nabla\cdot T\left(e_{0}\right)=\frac{1}{c}\,\left(J\cdot F\right)\cdot
e_{0}=f\cdot e_{0}$ (278)

To generalise the specific stress energy momentum vector (277), a stress
energy momentum vector $T\left(V\right)$ can be defined as a linear mapping of
a vector $V$,

$T\left(V\right)=-\,\frac{1}{4}\left(G\,V\,F+F\,V\,G\right)=-\,\frac{1}{2}\,\varepsilon_{0}\,F\,V\,F$
(279)

The inner product of a vector $U$ and the stress energy momentum vector
$T\left(V\right)$ yields a scalar,

$U\cdot
T\left(V\right)=-\,\frac{1}{4}\,U\cdot\left(G\,V\,F+F\,V\,G\right)=-\,\frac{1}{4}\,\Big{(}\langle\,U\,G\,V\,F\,\rangle+\langle\,U\,F\,V\,G\,\rangle\Big{)}$
(280)

where the angle brackets denote the scalar part of the multivector. Similarly,
the inner product of the vector $V$ and the stress energy momentum vector
$T\left(U\right)$ also yields a scalar,

$V\cdot
T\left(U\right)=-\,\frac{1}{4}\,V\cdot\left(G\,U\,F+F\,U\,G\right)=-\,\frac{1}{4}\,\Big{(}\langle\,V\,G\,U\,F\,\rangle+\langle\,V\,F\,U\,G\,\rangle\Big{)}$
(281)

Using the invariance under cyclic permutation of the scalar part of the
geometric product of four multivectors (543), we obtain,

$\langle\,U\,G\,V\,F\,\rangle=\langle\,V\,F\,U\,G\,\rangle\qquad\text{and}\qquad\langle\,U\,F\,V\,G\,\rangle=\langle\,V\,G\,U\,F\,\rangle$
(282)

Thus, according to relations (280), (281) and (282) in vacuum, the stress
energy momentum vector satisfies the symmetry, [21]

$U\cdot T\left(V\right)=V\cdot T\left(U\right)$ (283)

which means that $T\left(V\right)$ is a self-adjoint linear application of
$V$. In particular, for $U=\nabla$ and $V=e_{0}$, relation (283) becomes,

$\nabla\cdot
T\left(e_{0}\right)=e_{0}\cdot\dot{T}\,(\dot{\nabla})=\dot{T}\,(\dot{\nabla})\cdot
e_{0}$ (284)

In view of relation (284), Poynting’s theorem (278) is recast as,

$\dot{T}\,(\dot{\nabla})\cdot e_{0}=\frac{1}{c}\,\left(J\cdot F\right)\cdot
e_{0}=f\cdot e_{0}$ (285)

which means that Poynting’s theorem is an explicit expression of the stress
energy momentum vector,

$\dot{T}\,(\dot{\nabla})=\frac{1}{c}\,J\cdot F=f$ (286)

According to relation (279), the stress energy momentum of the gradient is
given by,

$\dot{T}\,(\dot{\nabla})=-\,\frac{1}{4}\,\dot{G}\,\dot{\nabla}\,F-\,\frac{1}{4}\,G\,\dot{\nabla}\,\dot{F}-\,\frac{1}{4}\,\dot{F}\,\dot{\nabla}\,G-\,\frac{1}{4}\,F\,\dot{\nabla}\,\dot{G}$
(287)

where the overdot denotes on which bivector the gradient operates to the left
or the right. The stress energy momentum vector (287) is the electromagnetic
force density vector in space-time. It is explicitly written as,

$\dot{T}\,(\dot{\nabla})=\frac{1}{4}\left(\nabla\,G\right)F-\,\frac{1}{4}\,G\left(\nabla\,F\right)+\frac{1}{4}\left(\nabla\,F\right)G-\,\frac{1}{4}\,F\left(\nabla\,G\right)$
(288)

where the positive signs are due to the anticommutation between the vector
$\nabla$ and the bivectors $G$ and $F$. Equivalently, in vacuum, using the
electromagnetic constitutive equation (42), relations (287) and (288) for the
stress energy momentum of the gradient are recast as, [21]

$\dot{T}\,(\dot{\nabla})=-\,\frac{1}{2}\,\varepsilon_{0}\,\dot{F}\,\dot{\nabla}\,F-\,\frac{1}{2}\,\varepsilon_{0}\,F\,\dot{\nabla}\,\dot{F}=\frac{1}{2}\,\varepsilon_{0}\left(\nabla\,F\right)F-\,\frac{1}{2}\,\varepsilon_{0}\,F\left(\nabla\,F\right)$
(289)

To show the consistency of our approach, we use the Maxwell equation (166) and
(171), the electromagnetic constitutive equation (42) and the identity (555)
in order to recast the stress energy momentum vector (288) as,

$\displaystyle\dot{T}\,(\dot{\nabla})=\frac{1}{4}\left(\frac{1}{c}\,J\right)F-\,\frac{1}{4}\,\varepsilon_{0}\,F\left(\frac{1}{c\,\varepsilon_{0}}\,J\right)+\frac{1}{4}\,\left(\frac{1}{c\,\varepsilon_{0}}\,J\right)\varepsilon_{0}\,F-\,\frac{1}{4}\,F\left(\frac{1}{c}\,J\right)$
$\displaystyle\phantom{\dot{T}\,(\dot{\nabla})}=\frac{1}{2\,c}\left(J\,F-\,F\,J\right)=\frac{1}{c}\,J\cdot
F=f$ (290)

We now show that we recover the expression of the components of the stress
energy momentum tensor in vacuum usually obtained in the framework of tensor
calculus. In view of identity (283), the symmetric components of the stress
energy momentum tensor are written as,

$T^{\mu\nu}=e^{\mu}\cdot
T\left(e^{\nu}\right)=\langle\,e^{\mu}\,T\left(e^{\nu}\right)\,\rangle=\langle\,e^{\nu}\,T\left(e^{\mu}\right)\,\rangle=e^{\nu}\cdot
T\left(e^{\mu}\right)=T^{\nu\mu}$ (291)

In view of relation (279), relation (291) becomes,

$\begin{split}&T^{\mu\nu}=-\,\frac{1}{4}\,\Big{(}\,\langle\,e^{\mu}\,G\,e^{\nu}\,F\,\rangle+\langle\,e^{\mu}\,F\,e^{\nu}\,G\,\rangle\,\Big{)}\\\
&\phantom{T^{\mu\nu}}=-\,\frac{1}{4}\,\Big{(}\,\langle\,e^{\nu}\,G\,e^{\mu}\,F\,\rangle+\langle\,e^{\nu}\,F\,e^{\mu}\,G\,\rangle\,\Big{)}=T^{\nu\mu}\end{split}$
(292)

Using the invariance under cyclic permutation of the scalar part of the
geometric product of four multivectors (543), we obtain,

$\langle\,e^{\mu}\,G\,e^{\nu}\,F\,\rangle=\langle\,e^{\nu}\,F\,e^{\mu}\,G\,\rangle\qquad\text{and}\qquad\langle\,e^{\mu}\,F\,e^{\nu}\,G\,\rangle=\langle\,e^{\nu}\,G\,e^{\mu}\,F\,\rangle$
(293)

In view of identity (293), relation (292) is recast as,

$T^{\mu\nu}=-\,\frac{1}{4}\,\Big{(}\,\langle\,e^{\mu}\,G\,e^{\nu}\,F\,\rangle+\langle\,e^{\nu}\,G\,e^{\mu}\,F\,\rangle\,\Big{)}$
(294)

According to the symmetry (293), the scalar components (292) of the symmetric
stress energy momentum tensor are the result of the inner product of vectors
and the double inner product of bivectors,

$T^{\mu\nu}=-\,\frac{1}{2}\,\left(e^{\mu}\cdot G\right)\cdot\left(e^{\nu}\cdot
F\right)-\,\frac{1}{2}\,\left(e^{\nu}\cdot G\right)\cdot\left(e^{\mu}\cdot
F\right)+\frac{1}{2}\,\left(e^{\mu}\cdot e^{\nu}\right)\,G\colon\\!F$ (295)

where the positive sign in front of the last term is due to the
anticommutation of the vectors $e^{\mu}$ and $e^{\nu}$ with the bivector $G$.
In view of relations (215), (214), (234) and (233),

$\begin{split}&e^{\nu}\cdot\left(e^{\mu}\cdot
F\right)=F^{\mu\nu}=-\,F^{\nu\mu}=-\,\left(F\cdot e^{\mu}\right)\cdot
e^{\nu}\\\ &e^{\nu}\cdot\left(e^{\mu}\cdot
G\right)=G^{\mu\nu}=-\,G^{\nu\mu}=-\,\left(G\cdot e^{\mu}\right)\cdot
e^{\nu}\\\ \end{split}$ (296)

which implies that,

$e^{\mu}\cdot F=-\,F\cdot e^{\mu}\qquad\text{and}\qquad e^{\mu}\cdot
G=-\,G\cdot e^{\mu}$ (297)

In view of the symmetry (297) and identity (494), relation (295) reduces to,

$T^{\mu\nu}=\frac{1}{2}\,\left(e^{\mu}\cdot G\right)\cdot\left(F\cdot
e^{\nu}\right)+\frac{1}{2}\,\left(e^{\nu}\cdot G\right)\cdot\left(F\cdot
e^{\mu}\right)+\frac{1}{2}\,\left(e^{\mu}\cdot e^{\nu}\right)\,G\colon\\!F$
(298)

According to relations (212), (215), (231), (234), (515), (545) and (548),

$\begin{split}&e^{\mu}\cdot
G=\frac{1}{2}\,G^{\rho\sigma}\,e^{\mu}\cdot\left(e_{\rho}\wedge
e_{\sigma}\right)=\frac{1}{2}\,G^{\rho\sigma}\,\Big{(}\left(e^{\mu}\cdot
e_{\rho}\right)e_{\sigma}-\,\left(e^{\mu}\cdot
e_{\sigma}\right)e_{\rho}\Big{)}\\\ &\phantom{e^{\mu}\cdot
G}=\frac{1}{2}\,G^{\mu\sigma}\,e_{\sigma}-\,\frac{1}{2}\,G^{\rho\mu}\,e_{\rho}=G^{\mu\sigma}\,e_{\sigma}\\\
&F\cdot e^{\nu}=\frac{1}{2}\,F^{\rho\sigma}\,\left(e_{\rho}\wedge
e_{\sigma}\right)\cdot
e^{\nu}=\frac{1}{2}\,F^{\rho\sigma}\,\Big{(}\left(e^{\nu}\cdot
e_{\sigma}\right)e_{\rho}-\,\left(e^{\nu}\cdot
e_{\rho}\right)e_{\sigma}\Big{)}\\\ &\phantom{e^{\nu}\cdot
F}=\frac{1}{2}\,F^{\rho\nu}\,e_{\rho}-\,\frac{1}{2}\,F^{\nu\sigma}\,e_{\sigma}=F^{\rho\nu}\,e_{\rho}\end{split}$
(299)

In view of relations (215), (297), (299) and (494),

$\begin{split}&\left(e^{\mu}\cdot F\right)\cdot\left(G\cdot
e^{\nu}\right)=\left(G^{\mu\sigma}\,e_{\sigma}\right)\cdot\left(F^{\rho\nu}\,e_{\rho}\right)=G^{\mu\sigma}\,F^{\rho\nu}\left(e_{\sigma}\cdot
e_{\rho}\right)\\\ &\phantom{\left(e^{\mu}\cdot F\right)\cdot\left(G\cdot
e^{\nu}\right)}=G^{\mu\sigma}\,F^{\rho\nu}\,\eta_{\sigma\rho}=G^{\mu\rho}\,F_{\rho}^{\phantom{\rho}\nu}\end{split}$
(300)

According to relations (212), (215), (231), (515) and (562),

$\begin{split}&G\colon\\!F=\frac{1}{4}\,G_{\rho\sigma}\,F^{\mu\nu}\,\left(e_{\rho}\wedge
e_{\sigma}\right)\cdot\left(e^{\mu}\wedge e^{\nu}\right)\\\ &\phantom{G\cdot
F=}=\frac{1}{4}\,G_{\rho\sigma}\,F^{\mu\nu}\,\Big{(}\left(e_{\sigma}\cdot
e^{\mu}\right)\left(e_{\rho}\cdot e^{\nu}\right)-\,\left(e_{\sigma}\cdot
e^{\nu}\right)\left(e_{\rho}\cdot e^{\mu}\right)\Big{)}\\\ &\phantom{G\cdot
F=}=\frac{1}{4}\,\Big{(}G_{\rho\sigma}\,F^{\sigma\rho}-\,G_{\rho\sigma}\,F^{\rho\sigma}\,\Big{)}=-\,\frac{1}{2}\,G_{\rho\sigma}\,F^{\rho\sigma}\end{split}$
(301)

In view of relations (300) and (301), the components of the stress energy
momentum tensor (298) are explicitly given by,

$T^{\mu\nu}=G^{\mu\rho}\,F_{\rho}^{\phantom{\rho}\nu}-\,\frac{1}{4}\,\eta^{\mu\nu}\,G_{\rho\sigma}\,F^{\rho\sigma}$
(302)

or equivalently by lowering the second index of the components
$T^{\mu\lambda}$ with the Minkowski metric $\eta_{\lambda\nu}$,

$T^{\mu}_{\phantom{\mu}\nu}=G^{\mu\rho}\,F_{\rho\nu}-\,\frac{1}{4}\,\delta^{\mu}_{\nu}\,G^{\rho\sigma}\,F_{\rho\sigma}$
(303)

In vacuum, the components of the electromagnetic constitutive relation (42)
are given by,

$e_{\nu}\cdot\left(e^{\mu}\cdot
G\right)=\varepsilon_{0}\,e_{\nu}\cdot\left(e^{\mu}\cdot F\right)$ (304)

In view of identities (214) and (233), it is recast as,

$G^{\mu}_{\phantom{\mu}\nu}=\varepsilon_{0}\,F^{\mu}_{\phantom{\mu}\nu}$ (305)

In vacuum, using the constitutive relation (305), the components of the stress
energy momentum tensor (303) are recast as,

$T^{\mu}_{\phantom{\mu}\nu}=\varepsilon_{0}\left(F^{\mu\rho}\,F_{\rho\nu}-\,\frac{1}{4}\,\delta^{\mu}_{\nu}\,F^{\rho\sigma}\,F_{\rho\sigma}\right)$
(306)

##  19 Stress energy momentum in matter (STA)

In matter the stress-energy momentum is also a vector. As we show in this
section, it is a self-adjoint linear application of a space-time vector $V$ in
matter as in vacuum. If this vector $V$ is the space-time gradient $\nabla$,
the stress-energy momentum vector $\dot{T}\left(\dot{\nabla}\right)$ reduces
to the electromagnetic force density vector $f$ in space-time as we show
below. In spatial algebra (SA), the Poynting theorem in matter (148) has the
same algebraic structure as the Poynting theorem in vacuum (85). Thus, in
space-time algebra, the Poynting theorem in matter has the same structure the
Poynting theorem in vacuum (278),

$\nabla\cdot T\left(e_{0}\right)=\frac{1}{c}\,\left(J\cdot F\right)\cdot
e_{0}=f\cdot e_{0}$ (307)

where according to equations (274) and (277), the stress energy momentum
tensor in matter $T\left(e_{0}\right)$ is given by,

$T\left(e_{0}\right)=P\,c=\left(e+\boldsymbol{p}\,c\right)e_{0}$ (308)

In view of the identities (133) and (225), we obtain,

$e+\boldsymbol{p}\,c=-\,\frac{1}{4}\left(G\,e_{0}\,F\,e_{0}+F\,e_{0}\,G\,e_{0}\right)-\,\frac{1}{8}\left(F\,G-\,G\,F+e_{0}\,F\,G\,e_{0}-\,e_{0}\,G\,F\,e_{0}\right)$
(309)

Using relation (309), and the identity $e_{0}^{2}=1$, the stress energy
momentum tensor in matter (308) is recast as,

$T\left(e_{0}\right)=-\,\frac{1}{4}\left(G\,e_{0}\,F+F\,e_{0}\,G\right)-\,\frac{1}{8}\Big{(}\left(F\,G-\,G\,F\right)e_{0}+e_{0}\left(F\,G-\,G\,F\right)\Big{)}$
(310)

In view of the commutator (565) of the bivectors $F$ and $G$, the stress
energy momentum vector (310) is recast as,

$T\left(e_{0}\right)=-\,\frac{1}{4}\left(G\,e_{0}\,F+F\,e_{0}\,G\right)-\,\frac{1}{4}\Big{(}\left(F\times
G\right)\cdot e_{0}+e_{0}\cdot\left(F\times G\right)\Big{)}$ (311)

Since $F\times G$ is a bivector and $e_{0}$ is a vector, according to identity
(554),

$\left(F\times G\right)\cdot e_{0}=-\,e_{0}\cdot\left(F\times G\right)$ (312)

which implies that the stress energy momentum vector (311) reduces to,

$T\left(e_{0}\right)=-\,\frac{1}{4}\left(G\,e_{0}\,F+F\,e_{0}\,G\right)$ (313)

To generalise the specific stress energy momentum vector (310), a stress
energy momentum vector $T\left(V\right)$ can be defined as a linear mapping of
a vector $V$,

$T\left(V\right)=-\,\frac{1}{4}\left(G\,V\,F+F\,V\,G\right)$ (314)

According to relations (280), (281) and (282) in matter, the stress energy
momentum vector satisfies the symmetry,

$U\cdot T\left(V\right)=V\cdot T\left(U\right)$ (315)

which means that $T\left(V\right)$ is a self-adjoint linear application of
$V$. In particular, for $U=\nabla$ and $V=e_{0}$, relation (315) becomes,

$\nabla\cdot
T\left(e_{0}\right)=e_{0}\cdot\dot{T}\,(\dot{\nabla})=\dot{T}\,(\dot{\nabla})\cdot
e_{0}$ (316)

In view of relation (316), Poynting’s theorem (307) is recast as,

$\dot{T}\,(\dot{\nabla})\cdot e_{0}=\frac{1}{c}\,\left(J\cdot F\right)\cdot
e_{0}=f\cdot e_{0}$ (317)

which means that Poynting’s theorem is an explicit expression of the stress
energy momentum vector,

$\dot{T}\,(\dot{\nabla})=\frac{1}{c}\,J\cdot F=f$ (318)

According to relation (279), the stress energy momentum of the gradient is
given by,

$\dot{T}\,(\dot{\nabla})=-\,\frac{1}{4}\,\dot{G}\,\dot{\nabla}\,F-\,\frac{1}{4}\,G\,\dot{\nabla}\,\dot{F}-\,\frac{1}{4}\,\dot{F}\,\dot{\nabla}\,G-\,\frac{1}{4}\,F\,\dot{\nabla}\,\dot{G}$
(319)

where the overdot denotes on which bivector the gradient operates to the left
or the right. The stress energy momentum vector (319) is the electromagnetic
force density vector in space-time. It is explicitly written as,

$\dot{T}\,(\dot{\nabla})=\frac{1}{4}\left(\nabla\,G\right)F-\,\frac{1}{4}\,G\left(\nabla\,F\right)+\frac{1}{4}\left(\nabla\,F\right)G-\,\frac{1}{4}\,F\left(\nabla\,G\right)$
(320)

where the positive signs are due to the anticommutation between the vector
$\nabla$ and the bivectors $G$ and $F$. The electromagnetic susceptibility
bivector in space-time $\chi_{F}\left(F\right)$ is a linear map of the
electromagnetic field bivector $F$. Furthermore, we assume that the
electromagnetic properties of the material medium are constant and uniform,
which implies that,

$F\,\left(\nabla\chi_{F}\left(F\right)\right)=\chi_{F}\left(F\right)\left(\nabla\,F\right)$
(321)

According to relations (152) and (321),

$\begin{split}&F\,\Big{(}\nabla\,G\left(F\right)\Big{)}=F\,\Big{(}\nabla\left(F+\chi_{F}\left(F\right)\right)\Big{)}=F\left(\nabla\,F\right)+F\,\left(\nabla\chi_{F}\left(F\right)\right)\\\
&G\left(F\right)\left(\nabla\,F\right)=\Big{(}F+\chi_{F}\left(F\right)\Big{)}\,\left(\nabla\,F\right)=F\left(\nabla\,F\right)+\chi_{F}\left(F\right)\left(\nabla\,F\right)\end{split}$
(322)

In view of relations (321) and (322), we deduce the following identity,

$G\left(F\right)\left(\nabla\,F\right)=F\,\Big{(}\nabla\,G\left(F\right)\Big{)}$
(323)

Thus, the stress energy momentum vector (319) in matter reduces to,

$\dot{T}\,(\dot{\nabla})=\frac{1}{2}\left(\nabla\,G\right)F-\,\frac{1}{2}\,F\left(\nabla\,G\right)$
(324)

To show the consistency of our approach, we use the Maxwell equation (175),
relation (318) and identity (555) to recast the stress energy momentum vector
(324) as,

$\displaystyle\dot{T}\,(\dot{\nabla})=\frac{1}{2}\left(\frac{1}{c}\,J\right)F-\,\frac{1}{2}\,F\left(\frac{1}{c}\,J\right)=\frac{1}{2\,c}\left(J\,F-\,F\,J\right)=\frac{1}{c}\left(J\cdot
F\right)=f$

For an non-uniform material material medium, an additional ponderomotive force
density vector in space-time has to be added to the electromagnetic force
density vector $f$ in space-time. Since the stress energy momentum vector in
matter (320) has the same structure as the stress energy momentum vector in
vacuum (288), the components of the stress energy momentum tensor in matter
have the same structure as the components of the stress energy momentum vector
in vacuum,

$T^{\mu}_{\phantom{\mu}\nu}=G^{\mu\rho}\,F_{\rho\nu}-\,\frac{1}{4}\,\delta^{\mu}_{\nu}\,G^{\rho\sigma}\,F_{\rho\sigma}$
(325)

In matter, the components of the electromagnetic constitutive relation (99)
are given by,

$e_{\nu}\cdot\left(e^{\mu}\cdot
G\right)=\varepsilon_{0}\,e_{\nu}\cdot\left(e^{\mu}\cdot
F\right)+\varepsilon_{0}\,e_{\nu}\cdot\left(e^{\mu}\cdot\tilde{P}\right)$
(326)

In view of identities (214), (233) and (251), it is recast as,

$G^{\mu}_{\phantom{\mu}\nu}=\varepsilon_{0}\,F^{\mu}_{\phantom{\mu}\nu}+\tilde{P}^{\mu}_{\phantom{\mu}\nu}$
(327)

In matter, using the constitutive relation (327), the components of the stress
energy momentum tensor (325) are recast as,

$\begin{split}&T^{\mu}_{\phantom{\mu}\nu}=\varepsilon_{0}\left(F^{\mu\rho}\,F_{\rho\nu}-\,\frac{1}{4}\,\delta^{\mu}_{\nu}\,F^{\rho\sigma}\,F_{\rho\sigma}\right)\\\
&\phantom{T^{\mu}_{\phantom{\mu}\nu}=}+\tilde{P}^{\mu\rho}\,F_{\rho\nu}-\,\frac{1}{4}\,\delta^{\mu}_{\nu}\,\tilde{P}^{\rho\sigma}\,F_{\rho\sigma}\end{split}$
(328)

##  20 Electromagnetic potential (STA)

In vector space (VS), the electric scalar potential $\phi$ and the magnetic
vector potential $\boldsymbol{a}$ are the temporal and spatial parts of a
quadrivector $A$. Similarly, in the space-time algebra (STA), we define an
electromagnetic vector potential $A$ in space-time and express the
electromagnetic field bivector $F$ in terms of $A$. The electromagnetic vector
potential $A$ is a contravariant vector in space-time defined as,

$A=A^{\mu}\,e_{\mu}=A^{0}\,e_{0}+A^{i}\,e_{i}=\frac{\phi}{c}\,e_{0}+a^{i}\,e_{i}$
(329)

In view of relation (509), the inner product of the electromagnetic vector
potential $A$ with the time vector $e_{0}$ yields,

$A\cdot e_{0}=e_{0}\cdot A=A^{0}=\frac{\phi}{c}$ (330)

According to relation (510), the outer product of the electromagnetic vector
potential $A$ with the time vector $e_{0}$ yields,

$A\wedge e_{0}=-\,e_{0}\wedge A=A^{i}\left(e_{i}\wedge
e_{0}\right)=a^{i}\,\boldsymbol{e}_{i}=\boldsymbol{a}$ (331)

In view of the inner product (509) and the outer product (510), the geometric
product of the electromagnetic vector potential $A$ and the time vector
$e_{0}$ yields,

$A\,e_{0}=A\cdot e_{0}+A\wedge e_{0}=\frac{\phi}{c}+\boldsymbol{a}$ (332)

Taking into account the properties (494) and (496), the geometric product in
reverse order is given by,

$e_{0}\,A=e_{0}\cdot A+e_{0}\wedge A=A\cdot e_{0}-\,A\wedge
e_{0}=\frac{\phi}{c}-\,\boldsymbol{a}$ (333)

In view of identity (528), the curl of the electromagnetic vector potential
$A$ is written as,

$\nabla\wedge
A=-\,\frac{1}{c}\,\partial_{t}\,\boldsymbol{a}-\,\boldsymbol{\nabla}\left(\frac{\phi}{c}\right)+\boldsymbol{\nabla}\cdot\boldsymbol{a}$
(334)

According to relation (334), the electromagnetic field bivector (89) is the
curl of the electromagnetic potential up to a factor $c$,

$F=c\,\nabla\wedge A$ (335)

In view of identity (525), the divergence of the vector potential $A$ is
written as,

$\nabla\cdot
A=\frac{1}{c}\,\partial_{t}\left(\frac{\phi}{c}\right)+\boldsymbol{\nabla}\cdot\boldsymbol{a}$
(336)

In view of relations (336) and (90), the Lorentz gauge in space-time algebra
is written as,

$\nabla\cdot A=0$ (337)

In the Lorentz gauge (337), the electromagnetic field bivector (335) is recast
as,

$F=c\,\nabla\wedge A=F=c\,\nabla\,A-\,c\,\nabla\cdot A=c\,\nabla\,A$ (338)

Thus, in the Lorentz gauge (337), the gradient of the electromagnetic field
bivector (338) is given by,

$\nabla\,F=c\,\nabla^{2}A$ (339)

In view of relations (339) and (31), the Maxwell equation in vacuum (172) is
recast in terms of the electromagnetic potential vector $A$ as,

$\nabla^{2}\,A=\mu_{0}\,J$ (340)

In view of relations (339) and (31), the Maxwell equation in matter (187) is
recast in terms of the electromagnetic potential vector $A$ as,

$\nabla^{2}\,A=\mu_{0}\,\tilde{J}$ (341)

##  21 Conclusion

Geometric algebra (GA) has been the subject of intense study for the past half
century [22, 15, 21, 18, 23]. Electrodynamics has been examined in spatial
algebra (SA) and space-time algebra (STA) by numerous authors [15, 21, 17, 24,
20, 25]. This prompts the following question : “Is there any need to discuss
the matter further ?” The answer is “yes” for two main reasons. First, the
magnetic induction field $\boldsymbol{b}$ and the auxiliary magnetic field
$\boldsymbol{h}$ are treated in general as pseudovectorial fields by the vast
majority of authors with the notable exception of MacDonald [18] who
introduced magnetic induction field bivector $\boldsymbol{B}$ and the
auxiliary magnetic field bivector $\boldsymbol{H}$ but he did not investigate
the foundations of electromagnetism in spatial algebra (SA) or space-time
(STA) further. Since the magnetic fields clearly geometrically as bivectors,
it is necessary to treat them accordingly and this has been done in this
publication. Second, the electric polarisation and magnetisation in matter was
not studied in depth so far in spatial algebra (SA) and space-time (STA).
Although, there was an attempt by Arthur [20] to describe electromagnetism in
matter in spatial algebra (SA) but these results remain inconclusive.

In this publication, we considered an electric polarisation vector
$\boldsymbol{\tilde{p}}$ induced linearly by the electric field vector
$\boldsymbol{e}$ and a magnetisation bivector $\boldsymbol{\tilde{M}}$ induced
linearly by the magnetic induction field bivector $\boldsymbol{B}$.

First, in the spatial algebra (SA), we introduced a polarisation multivector
$\tilde{P}=\boldsymbol{\tilde{p}}-\,\frac{1}{c}\,\boldsymbol{\tilde{M}}$ and
an auxiliary electromagnetic field multivector
$G=\varepsilon_{0}\,F+\tilde{P}$ we were able to express the Maxwell equation
in matter (175) very elegantly. Using the bound electric charge density
$\tilde{q}$ and the bound electric current density $\boldsymbol{\tilde{j}}$,
the Maxwell equation (187) could then be written entirely in terms of the
electromagnetic field multivector $F$. This allowed us to derive the
electromagnetic wave equation in matter (120) straightforwardly in the spatial
algebra (SA). Writing the electromagnetic energy density $e$ and momentum
density $\boldsymbol{p}$ in terms of the electromagnetic multivectors $F$, $G$
and $\tilde{P}$, we show that the electromagnetic polarisation multivector
$\tilde{P}$ is a linear map of the electromagnetic multivector $F$ written as
$\tilde{P}\left(F\right)=\varepsilon_{0}\,\chi\left(F\right)$ where
$\chi\left(F\right)$ is the electromagnetic susceptibility multivector.

Second, in the space-time algebra (STA), the electromagnetic multivectors $F$,
$G$ and $\tilde{P}$ defined in the spatial algebra (SA) could be interpreted
as bivectors in the space-time algebra (STA). Writing the Maxwell equation
(175) in terms of the auxiliary electromagnetic field multivector $G$, the
stunningly elegant equation $\nabla\,G=\frac{1}{c}\,J$ had the same analytical
expression as in vacuum (166). By introducing a bound current vector
$\tilde{J}=J-\,c\,\nabla\cdot\tilde{P}$ in space-time, the Maxwell equation in
matter (187) could be written also in terms of the electromagnetic field
bivector $F$ as $\nabla\,F=\frac{1}{\varepsilon_{0}\,c}\,\tilde{J}$. In the
space-time algebra (STA), the wave equation in matter (201) is obtained by
simply taking the gradient of the Maxwell equation in matter (175), namely
$\nabla^{2}\,G=\frac{1}{c}\,\nabla\wedge J$ since $\nabla\cdot J=0$, or
equivalently
$\nabla^{2}\,F=\frac{1}{\varepsilon_{0}\,c}\,\nabla\wedge\tilde{J}$ since
$\nabla\cdot\tilde{J}=0$. The stress energy momentum vector $T\left(V\right)$
in space-time (STA) describes the energy flux across the hypersurface
orthogonal to $V$. For a uniform electromagnetic medium consisting of induced
electric and magnetic dipoles, the stress-energy momentum vector is a linear
mapping (314) of the vector $V$ given by
$T\left(V\right)=-\,\frac{1}{4}\left(G\,V\,F+F\,V\,G\right)$. It turns out
that the stress energy momentum vector has the same analytical structure in
matter as in vacuum (279). Moreover, for the gradient $V=\nabla$, the stress
energy momentum vector (318) is written as
$\dot{T}\left(\dot{\nabla}\right)=\frac{1}{c}\,J\cdot F=f$ where $f$ is the
electromagnetic force density vector in space-time. Finally, in the space-time
algebra (STA), the Maxwell equation in matter (341) can be written as a wave
equation for the potential vector $A$, namely $\nabla^{2}A=\mu_{0}\,\tilde{J}$
since $F=\nabla\wedge A$.

## Acknowledgements

The author would like to thank Jean-Philippe Ansermet and François A. Reuse
for insightful discussions as well as his PhD thesis adviser Anthony Lasenby
for introducing him to geometric algebra.

##  Appendix A Spatial algebra (SA)

We consider an orthonormal vector spatial frame
$\\{\boldsymbol{e}_{1},\boldsymbol{e}_{2},\boldsymbol{e}_{3}\\}$. The
geometric product of two basis vectors reads, [21]

$\boldsymbol{e}_{i}\,\boldsymbol{e}_{j}=\boldsymbol{e}_{i}\cdot\boldsymbol{e}_{j}+\boldsymbol{e}_{i}\wedge\boldsymbol{e}_{j}\qquad\text{with}\qquad
i,j=1,2,3$ (342)

where,

$\boldsymbol{e}_{1}^{2}=\boldsymbol{e}_{2}^{2}=\boldsymbol{e}_{3}^{2}=1\qquad\text{and}\qquad\boldsymbol{e}_{i}\wedge\boldsymbol{e}_{i}=\boldsymbol{0}$
(343)

The inner product of two basis vectors is symmetric and defined as,

$\boldsymbol{e}_{i}\cdot\boldsymbol{e}_{j}=\boldsymbol{e}_{j}\cdot\boldsymbol{e}_{i}=\frac{1}{2}\left(\boldsymbol{e}_{i}\,\boldsymbol{e}_{j}+\boldsymbol{e}_{j}\,\boldsymbol{e}_{i}\right)$
(344)

and the outer product is antisymmetric,

$\boldsymbol{e}_{i}\wedge\boldsymbol{e}_{j}=-\,\boldsymbol{e}_{j}\wedge\boldsymbol{e}_{i}=\frac{1}{2}\left(\boldsymbol{e}_{i}\,\boldsymbol{e}_{j}-\,\boldsymbol{e}_{j}\,\boldsymbol{e}_{i}\right)$
(345)

The $8$ basis elements of the geometric algebra (GA) $\mathbb{G}^{3}$ called
the spatial algebra are :

  * 1.

one scalar : $\ 1$

  * 2.

three vectors : $\ \boldsymbol{e}_{1}$, $\ \boldsymbol{e}_{2}$, $\
\boldsymbol{e}_{3}$

  * 3.

three bivectors : $\ \boldsymbol{e}_{1}\wedge\boldsymbol{e}_{2}$, $\
\boldsymbol{e}_{2}\wedge\boldsymbol{e}_{3}$, $\
\boldsymbol{e}_{3}\wedge\boldsymbol{e}_{1}$

  * 4.

one pseudoscalar : $\
\boldsymbol{e}_{1}\wedge\boldsymbol{e}_{2}\wedge\boldsymbol{e}_{3}$

where the number of elements of each type $1$, $3$, $3$, $1$ are the binomial
coefficients in three dimensions. In $\mathbb{G}^{3}$, there are $2^{3}$ basis
elements. The geometric interpretation of the basis elements is clear. A
scalar $s$ is oriented point, a vector is an oriented line element, a bivector
is an oriented plane element and a pseudoscalar is an oriented volume element.
The notation used is the following : scalars $s$ are written in lower case
letters, vectors $\boldsymbol{v}$ are written in lower case bold letters,
bivectors $\boldsymbol{B}$ are written in upper case bold letters and
pseudoscalars are written as $s^{\prime}I$. A multivector $M$, which is linear
combination of elements of $\mathbb{G}^{3}$, is written in upper case letters,

$M=s+\boldsymbol{v}+\boldsymbol{B}+s^{\prime}I$ (346)

Taking into account the definition (342), the bivectors and the pseudoscalar