relativistic electrodynamics with minkowski spacetime algebra · relativistic electrodynamics with...

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Relativistic Electrodynamics with Minkowski Spacetime

Algebra

Walid Karam Azzubaidi

Dissertation submitted for obtaining the degree of

Master in Electrical and Computer Engineering

Jury

Supervisor: Carlos Manuel dos Reis Paiva, PhD

Co-Supervisor: António Luís Campos da Silva Topa, PhD

President: José Manuel Bioucas Dias, PhD

Member: José Júlio Alves Paisana, PhD

December 2009

iii

Acknowledgements

Acknowledgements First, I would like to thank both supervisors Carlos Manuel dos Reis Paiva and António Luís Campos

da Silva Topa for giving me the opportunity to perform this thesis. Along these several months, both

Professors have shown much availability and patience towards all of the students, which helped

greatly to accomplish this work. Professor Carlos Manuel dos Reis Paiva helped, not only with this

thesis, but also giving advices which supported my intellectual growth as a person, giving different

points of view on various subjects and also serving as an inspiration, for the comprehensive

knowledge he possesses and having the ability to discuss all kinds of subjects with certainty.

I would also like to thank my parents and grandmother, which are no longer among us, for educating

me the proper way, in order to become a complete person. Muna for being always there for me and for

being such a good sister; the rest of my family, in particular aunties Leila and Muna, uncles Eduardo

and José, who have also helped a great deal, not only on my education, but also giving good advises

and helping me when dealing with big challenges. I only got this far in terms of academic and human

level thanks to their support and belief in me even on harder moments, so they are especially

important. My cousins Ana, Maria, Marta and Pedro, for providing good human values, a lot of cheerful

moments and always being ready to help when needed. Rui, for helping me move from the old house

to the new one, when most needed and for being a good influence. A big appreciation to Inês, for

putting up with me, helping on difficult decisions and supporting me on hard times, always giving love.

Her presence in my life has helped me a lot, providing tranquility and patience, in order to clear

various obstacles.

Finally, I would like to thank my friends, in particular during these years I have spent on this university.

iv

Abstract

Abstract The aim of this work is to study several electrodynamic effects using spacetime algebra – the

geometric algebra of spacetime, which is supported on Minkowski spacetime. The motivation for

submitting onto this investigation relies on the need to explore new formalisms which allow attaining

simpler derivations with rational results. The practical applications for the examined themes are

uncountable and diverse, such as GPS devices, Doppler ultra-sound, color space conversion,

aerospace industry etc. The effects which will be analyzed include time dilation, space contraction,

relativistic velocity addition for collinear vectors, Doppler shift, moving media and vacuum form

reduction, Lorentz force, energy-momentum operator and finally the twins paradox with Doppler shift.

The difference between active and passive Lorentz transformation is also established. The first

regards a vector transformation from one frame to another, while the passive transformation is related

to user interpretation, therefore considered passive interpretation, since two observers watch the

same event with a different point of view.

Regarding time dilation and length contraction in the theory of special relativity, one concludes that

those parameters (time and length) are relativistic dimensions, since their interpretation depends on

the frame where an observer is located. The concept of simultaneity being relative is also introduced –

the same event is interpreted on different ways, depending on the frame of reference.

The relativistic velocity addition for collinear vectors is analyzed without resorting to Einstein’s second

postulate, which states that the velocity of transmission of light in vacuum has to be considered equal

to � for all inertial frames (non-accelerated), therefore the framework of special relativity does not

depend on electromagnetism.

Moving media is a subject which is addressed on this work as well, using the vacuum form reduction.

It is concluded that applying this method to plane wave propagation in moving isotropic media (on its

own frame) is considered a bianisotropic media on the rest frame.

The famous twins paradox is solved using the Doppler shift – the observers, opposed to expected,

actually agree with the time divergence between them. This supposed deviation in calculations is

explained by time dilation, therefore does not sustain as a mystery or inconsistency.

Keywords

Geometric algebra, Minkowski spacetime, Maxwell equations and Lorentz force, Vacuum Form

Reduction, Doppler shift, Energy-momentum operator

Resumo

Resumo O objectivo deste trabalho é estudar diversos efeitos electrodinâmicos, usando a álgebra de espaço-

tempo, sendo suportada pelo espaço-tempo de Minkowski. A motivação para realizar esta

investigação reside na necessidade de estabelecer uma álgebra que seja intuitiva e versátil. Os

efeitos que serão analisados incluem a dilatação do tempo, contração do espaço, adição relativista de

velocidades para vectores colineares, efeito de Doppler, meios em movimento e redução à forma de

vácuo, força de Lorentz, operador momento-energia e o famoso problema do paradoxo dos gémeos

recorrendo ao efeito de Doppler.

A diferença entre as transformações activa e passiva de Lorentz também será estabelecida. A

primeira situação trata da transformação de um vector de um certo referncial para outro, enquanto

que a transfomação passiva está relacionada com a forma como os observadores de refernciais

diferentes observam um mesmo evento – cada um deles tem o seu próprio ponto de vista.

Em relação à dilatação do tempo e contração do espaço, pode concluir-se que esses parâmetros

(tempo e espaço) são dimensões relativas, dado que as suas interpretações dependem do referencial

em que um observador esteja localizado. O conceito de simultaneidade será também introduzido – o

mesmo evento é interpretado de forma diferente, dependendo do referncial de referência.

A adição relativista de velocidades para vectores colineares é analizada, sem recorrer ao segundo

postulado de Einstein, que afirma que a velocidade de propagação da luz no vácuo é igual a �, para

todos os referenciais de inércia (não acelerados), por conseguinte o referencial da relatividade restrita

não depende do electromagnetismo.

Os meios em movimento constituem um tema clássico da teoria electromagnética, e também será

tratado neste trabalho. Ao invés de utilizar cálculos tensoriais, que são bastante complexos,

recorrendo à redução à forma de vácuo poder-se-à obter o mesmo resultado mas de foma muito mais

simplificada. Aplicando este método a uma opnda plana propagando num meio isotrópico (no seu

referncial), pode ser considerado como um meio bianisotropico no referncial de repouso.

O paradoxo dos gémeos é abordado utilizando o efeito Doppler – os observadores, ao contrário do

expectável, concordam com a divergência de tempo medida entre ambos. Este suposto erro de

cálculos é explicado pela dilatação do tempo, portanto não constitui qualquer inconsistência.

Palavras-chave

Álgebra geométrica, Espaço-tempo de Minkowski, Equações de Maxwell e força de Lorentz, Redução

à Forma de Vácuo, Efeito Doppler, Operador momento-energia

vi

Table of Contents

Table of Contents

Acknowledgements ................................................................................. iii

Abstract ................................................................................................... iv

Resumo .................................................................................................... v

Table of Contents .................................................................................... vi

List of Figures .......................................................................................... ix

List of Tables ........................................................................................... xi

List of Acronyms ..................................................................................... xii

List of Symbols ....................................................................................... xiii

List of Software ...................................................................................... xvii

1 Introduction .................................................................................... 1

1.1 State of the Art and Motivation .................................................................. 2

1.1.1 The Beginning of Algebra (Euclides) ..................................................................... 2

1.1.2 Quaternions (Hamilton) ......................................................................................... 3

1.1.3 Electromagnetism (Maxwell) ................................................................................. 3

1.1.4 Extensive Algebra (Grassmann) ............................................................................ 3

1.1.5 Vector Calculus (Gibbs) ......................................................................................... 4

1.1.6 Clifford Algebra (Clifford) ....................................................................................... 4

1.1.7 Newton vs. Maxwell Electromagnetic Theories ..................................................... 4

1.1.8 Spinors (Cartan, Dirac and Pauli) .......................................................................... 5

1.1.9 Modern GA (David Hestenes) ............................................................................... 5

1.2 Objectives ................................................................................................. 5

1.3 Structure and Organization ....................................................................... 6

1.4 Original Contributions ............................................................................... 7

1.5 Bibliography and References .................................................................... 8

2 Geometric Algebra ........................................................................ 9

2.1 Introduction ............................................................................................. 10

2.2 Geometric Algebra of the Plane .............................................................. 10

2.2.1 Bivectors .............................................................................................................. 14

2.2.2 Rotations .............................................................................................................. 16

2.3 Geometric Algebra of the Space ............................................................. 19

2.3.1 Properties ............................................................................................................ 19

2.3.2 Trivectors ............................................................................................................. 21

2.3.3 Contractions and Rotations ................................................................................. 26

2.4 Conclusions ............................................................................................ 29

2.5 Bibliography and References .................................................................. 30

3 Spacetime Algebra ...................................................................... 31

3.1 Introduction ............................................................................................. 32

3.2 Properties ................................................................................................ 32

3.3 �ℓ�,� and Boosts ..................................................................................... 35

3.4 Minkowski Diagram ................................................................................. 43

3.4.1 Time Dilation ........................................................................................................ 45

3.4.2 Length Contraction .............................................................................................. 47

3.5 Relativistic Velocity Addition for Collinear Vectors .................................. 50

3.6 Doppler Shift ........................................................................................... 53

3.7 Conclusions ............................................................................................ 57


4 Electrodynamic and Relativistic Effects ....................................... 59

4.1 Introduction ............................................................................................. 60

4.2 Maxwell Equations in �ℓ� algebra ........................................................... 60

4.2.1 Maxwell-Boffi Equations ...................................................................................... 62

4.3 Maxwell equations in �ℓ�,� algebra ......................................................... 63

4.4 Moving Media and Vacuum Form Reduction .......................................... 66

4.5 Relativistic Dynamics for a Particle ......................................................... 69

4.5.1 Lorentz Force ....................................................................................................... 71

4.5.2 Energy-Momentum Operator ............................................................................... 74

4.6 Twins Paradox with Doppler Shift ........................................................... 75

4.7 Conclusions ............................................................................................ 78


5 Conclusions ................................................................................. 81

5.1 Conclusions and Discussion ................................................................... 82

5.2 Future Work and Developments ............................................................. 83

viii

A Proper and Relative Velocities .................................................... 85

B Boost Application ......................................................................... 89

C Bondi k-Calculus ......................................................................... 95

D Quaternions ............................................................................... 101

E STA Pins and Spins .................................................................. 107

F Maxwell Equations Auxiliary Calculations ................................. 109

G Doppler Effect Application ......................................................... 115

Bibliography and References ............................................................... 118

ix

List of Figures

List of Figures Figure 1.1 – Evolution of algebras to the present day .............................................................................. 2

Figure 1.2 – Flowchart with investigated concepts on this thesis ............................................................ 7

Figure 2.1 – Pascal triangle to demonstrate �ℓ� algebra dimension .....................................................11

Figure 2.2 –Bivector described by a frame .............................................................................................12

Figure 2.3 – Geometric interpretation of a bivector ................................................................................14

Figure 2.4 – Anti-symmetry of a bivector ................................................................................................15

Figure 2.5 – Anti-Symmetry of a bivector ...............................................................................................15

Figure 2.6 – Area of a bivector ...............................................................................................................16

Figure 2.7 – Clock-wise rotation .............................................................................................................18

Figure 2.8 – Counter clock-wise rotation ................................................................................................19

Figure 2.9 – Oriented planes ..................................................................................................................21

Figure 2.10 – Associativity of the outer product: example 1 ..................................................................22

Figure 2.11 - Associativity of the outer product: example 2 ...................................................................23

Figure 2.12 - Pascal triangle to demonstrate �ℓ� algebras dimension ..................................................23

Figure 2.13 – Geometric interpretation of duality ...................................................................................24

Figure 2.14 – Geometrical interpretation of Gibbs cross product ...........................................................25

Figure 2.15 – Geometrical interpretation of a left contraction ................................................................27

Figure 2.16 – Rotation from vector � to �’ ..............................................................................................28

Figure 3.1 – Spacetime trajectories ........................................................................................................33

Figure 3.2 – Light cone for demonstration of causality property. ...........................................................33

Figure 3.3 - Pascal triangle to demonstrate �ℓ�,� algebras dimension ..................................................34

Figure 3.4 - �ℓ�,� frame ..........................................................................................................................37

Figure 3.5 – Intermediate scenario to determine parameter ζ ................................................................38

Figure 3.6 – Representation of a boost ..................................................................................................38

Figure 3.7 – � vector equal to 1 .............................................................................................................41

Figure 3.8 – � vector equal to -1 ............................................................................................................41

Figure 3.9 – Minkowski diagram with � and � frames ............................................................................43

Figure 3.10 - Line of simultaneity in � frame ..........................................................................................44

Figure 3.11 - Line of simultaneity in � frame ..........................................................................................44

Figure 3.12 – Time dilation given an observer on the � frame ...............................................................45

Figure 3.13 – Time dilation given an observer on the � frame ...............................................................46

Figure 3.14 – Length contraction given an observer on the � frame .....................................................47

Figure 3.15 – Length contraction given an observer on the � frame .....................................................49

Figure 3.16 – Collinear velocity vectors .................................................................................................50

Figure 3.17 - �ℓ� velocity addition ..........................................................................................................52

Figure 3.18 - �ℓ�,� velocity addition ........................................................................................................53

Figure 3.19 – Addressed problem on Doppler shift from a certain frame ..............................................55

x

Figure 4.1 – Geometric representation of particle’s relativistic dynamics ..............................................70

Figure 4.2 – Geometric interpretation of the problem ............................................................................76

Figure B.1 - Boost application for � = 90° and � = 0 .............................................................................90

Figure B.2 - Boost application for � = 90° and � = 0.5 ..........................................................................90

Figure B.3 - Boost application for � = 180° and � = 0.5 ........................................................................91

Figure B.4 - Boost application for � = 270° and � = 0.5 ........................................................................91

Figure B.5 - Boost application for � = 0° and � = 0.5 ............................................................................92



Figure C.1 – Spacetime diagram for representation of the radar method presentation .........................96

Figure C.2 – Geometrical representation of the problem .......................................................................97

Figure C.3 – Geometrical representation of the problem .......................................................................98

Figure D.1 – Representation of rings, division rings, bodies and integral domains ............................ 103

Figure G.1 – Doppler effect application for � = 0.1 and � ∈ [0, 2 ] .................................................... 116

Figure G.2 - Doppler effect application for � = 0.1 and � ∈ [0, 2 ] ..................................................... 116

Figure G.3 - Doppler effect application for � = 0.75 and � ∈ [0, 2 ] ................................................... 117

Figure G.4 - Doppler effect application for � = 0.95 and � ∈ [0, 2 ] ................................................... 117

xi

List of Tables

List of Tables Table 2.1 - �ℓ� subspaces ......................................................................................................................11

Table 2.2 - �ℓ� multiplicative table .........................................................................................................13

Table 2.3 - �ℓ� subspaces ......................................................................................................................23

Table 2.4 - �ℓ� multiplicative table .........................................................................................................26

Table 3.1 - �ℓ�,� subspaces ....................................................................................................................34

Table D.1 – Isomorphism between ℍ and �ℓ�' .................................................................................... 104

Table D.2 - Isomorphism between ℂ2 and �ℓ� .................................................................................... 105

Table D.3 – Quaternion multiplication table ........................................................................................ 105

xii

List of Acronyms

List of Acronyms AC After Christ

BC Before Christ

EMF Electromotive force

GA Geometric algebra

Iff If and only if

STA Spacetime algebra

VFR Vacuum Form Reduction

xiii

List of Symbols

List of Symbols �┴ Perpendicular part of vector � �∥ Parallel part of vector � + Magnetic field , Basis set for a proper space on an own frame ,- Basis set for a space on a relativistic frame � Normalized relative velocity � Speed of light �. Transformation from time to space on the own frame �. Transformation from time to space on another observer’s frame �ℓ� Euclidean algebra with 2 space axis �ℓ� Euclidean algebra with 3 space axis �ℓ�' Even part of �ℓ� algebra �ℓ�/ Odd part of �ℓ� algebra �ℓ�,� Euclidean algebra with 1 space axis and 1 time axis �ℓ�,� Euclidean algebra with 3 space axis and 1 time axis ℂ Complex space ℂ021 Complex space representation on a 2 by 2 matrix Δ�.34 Time difference between events �.5 and �.6 7 Electric displacement vector ∂ Dirac operator ∂ ⋅ : Divergence of : ∂ ∧ : Curl of : <=> Kronecker delta ? Electric field vector @A Time vector of the Minkowski spacetime frame @� First space vector of a Minkowski spacetime frame @� Second space vector of a Minkowski spacetime frame @� Third space vector of a Minkowski spacetime frame BA Electric permittivity in free space B Electric permittivity C Total energy for a particle CA Proper energy for a particle CD Energy’s electromagnetic density

xiv

ηA Vacuum impedance F Faraday bivector FG Faraday bivector on a relativistic frame H Lorentz force IJ Relative Lorentz force HA Time vector of a relativistic Minkowski spacetime frame H� First space vector of a relativistic Minkowski spacetime frame H� Second space vector of a relativistic Minkowski spacetime frame H� Third space vector of a relativistic Minkowski spacetime frame I Linear frequency K Maxwell bivector LJ Linear momentum’s volume density M Length and time correction factor; ratio between the derivative of proper times . and . M Ratio between the derivative of proper times . and . N Magnetic field strength ℍ Hypercomplex space O Relative velocity of a time event of rest point P ℎ Proper velocity of a time event of rest point P OR Unit vector for the relative velocity of a time event of rest point P S Pseudoscalar T Identity matrix T0U1 Particle’s inertia coefficient V Quaternions imaginary unit element W Quaternions imaginary unit element X Current density XY Total current density XZ Magnetized current density X[ Polarized current density \ Kinetic energy for a particle ] Quaternions imaginary unit element ^ Wave vector and Bondi k-factor _ Velocity of an event inferior to � L Boost a Lorentz transformation matrix and length on a frame aA Length on a relativistic frame b� Vector’s length c Magnetization vector d Mass of a particle de Derivative of the mass of a particle

fA Refractive index ℕ ⊂ Natural numbers space i0j1 Number of received signals i0k1 Number of sent signals iA0j1 Total number of received signals l Relative velocity of a time event of rest point m n Proper velocity of a time event of rest point m lo Unit vector for the relative velocity of a time event of rest point m P Rest point on an own frame p Polarization vector q Proper linear momentum for a particle rJ Relative linear momentum for a particle ℘0t1 Real part of a quaternion t u Angle between two vectors on a relativistic frame t Electric charge and quaternion m Rest point on a relativistic frame ℚ Rational numbers space ℜ0t1 Real part of a quaternion t x Rotor y Spacetime coordinates for an event on an own Minkowski spacetime frame zJ Spatial coordinates for an event on an own Minkowski spacetime frame y{

Spacetime coordinates for an event on a relativistic Minkowski spacetime frame zJ{ Spatial coordinates for an event on a relativistic Minkowski spacetime frame y| Constant spatial vector z|0.1 Time representation for an event on an own event z|0. 1 Time representation for an event on a relativistic event z}0.1 Time representation for an event on an own event

z}0. 1 Time representation for an event on a relativistic event

y}0.1 Time-dependant spatial vector

ℝ Real numbers space ℝ�,� Minkowski spacetime with 1 time and space axis ℝ�,� Minkowski spacetime with 1 time axis and 3 space axis ℝ� Bi-dimensional space ℝ� Tri-dimensional space ℝ� Quadri-dimensional space � Electric charge density �[ Polarized electric charge density

�Y Total electric charge density �J Unit vector on the space axis

xvi

� Proper velocity of a rest point P

� Own frame and Poynting vector � Other relativistic frame . Time on own frame . Time on relativistic frame � Space on own frame � Space on relativistic frame ζ Rapidity of a boost, which gives its intensity � Energy-momentum operator � Transposed matrix and time on a frame �A Time on a relativistic frame θ Angle between two vectors on the own frame ⋃ Rotation U Multivector and particle’s velocity on a rest point m � Proper velocity of an event �e Proper acceleration of an event �U�� Projection of U regarding blade ] μA Magnetic permeability in free space � Magnetic permeability G Proper velocity of an event on a relativistic frame �j Angular frequency on the receiver side �k Angular frequency on the emitter side

z Doppler magnitude of redshift ℤ Whole numbers space Z0�ℓ�1 Center of an f dimensional algebra

� ℝ�=

Subspace V for a given space f

⨁ Direct sum � � Geometric product of � with � � ∧ � Exterior, outer or wedge product of � with � � ∙ � Inner or dot product of � with � � × � Cross product of � with � � � Left contraction of � with � � � Right contraction of � with � @= ↦ @> Application from @� to @� ∇A��J Gradient of A��J ∇ × A��J Curl of A��J ∇ ∙ A��J Divergence of A��J ∘ Binary operation

xvii

List of Software

List of Software Adobe Acrobat 8.1 Professional

Animation Shop 2

Matlab R2007b

Mathtype 6.5

Microsoft Excel 2007

Microsoft Powerpoint 2007

Microsoft Visio 2007

Microsoft Word 2007

Paint Shop Pro 6

Photofiltre 6.2.3

Windjview 0.5

1

Chapter 1

Introduction 1 Introduction

This chapter will be used to introduce the concepts used on the current thesis, focusing on the

objectives and motivation, as well as a detailed synopsis of its structure. The evolution of the several

algebras that led to STA will be presented in order to give the reader a proper idea of the

fundamentals that underlie this algebra, and how it has developed during several ages.

2

Relativistic Electrodynamics with Minkowski Spacetime Algebra

(...) “that, in a few years, all great physical constants will have been approximately estimated, and that the

only occupation which will be left to men of science will be to carry these measurements to another place of decimals.”

J. C. Maxwell

1.1 State of the Art and Motivation

Since it is not possible to cite all physicists, mathematicians and scientists who supported the

evolution of algebra from its primordial form to the current days, only the crucial ones will be featured.

As figure 1.1 shows, GA results from a blending of several algebras.

Figure 1.1 – Evolution of algebras to the present day [1]

1.1.1 The Beginning of Algebra (Euclides)

Algebra had its dawn alongside with the birth of one of the most important mathematics of all times.

Referring to Euclid (330 BC – 260 BC), or Euclid of Alexandria, as he was know. He was born in Syria,

studied in Athens, and later on moved to Alexandria, where he gave math lectures on a school known

as “museum” created by Ptolomeu Soter (306 BC – 283BC). Thus, he achieved great prestige due to

the brilliant way he taught geometry and algebra [2]. In his book Elements, he presented a solid and

3

Introduction

consistent framework, which still nowadays remains the basis of mathematics. That algebra is

commonly known as Euclidean geometry to differentiate from non-Euclidean geometries, which were

discovered in the 19th century, as STA, for instance.

1.1.2 Quaternions (Hamilton)

Sir William Rowan Hamilton (1805-1865) was the precursor of quaternions – his intention was to

expand complex numbers, which could be viewed on a plane. He tried to apply on the three-

dimensional space, however it was proven impossible. Therefore he had to use his theory with four

dimensions. November 13th 1843, he presented a paper [3] which contained his theory of

quaternions. Its properties may be read on Appendix D – Quaternions. Multiplication rules were

created, which could be directly compared, hence it was considered an extension of complex

numbers. He represented a vector by a pure quaternion and wrote the outcome of the product of two

pure quaternions as the sum of a scalar part and a vector part, which are equal to the negative of the

dot product and to the cross product, respectively. [4]

1.1.3 Electromagnetism (Maxwell)

James Clerk Maxwell (13 June 1831 – 5 November 1879) is responsible for developing the classical

electromagnetic theory, uniting separate theories for magnetism, electricity and optics. He

demonstrated that electrical fields travel through space in the form of waves, and at the constant

speed of light. He suggested as well that light consisted in undulations in the same media that is the

result of electric and magnetic effects The initial eight Maxwell equations were presented as well [5]

Electromagnetism is the result of a unified theory of several centuries of experiments with this class of

phenomena. On mid 19th century, thanks to the investigations of Faraday and Ampère, the relation

between electricity and magnetism was already known. However, a theory which could join these

theories lacked – there were several proposals, among those one may pinpoint some of them, as

Maxwell’s [6] and Webber’s [7]. The first one prevailed of those theories, thanks to several factors,

such as the recognition that optical effects are of electrical and magnetic nature. The applications

which may relate to this topic are uncountable, as GPS devices, TV, mobile phones, etc… On

Maxwell’s’ book, his own equations were presented on the scalar form – which was clearly influenced

by Hamilton’s quaternions theory [8]. However, it was thanks to Heaviside and Gibbs the more modern

notation which allows writing the same equations with scalars.

1.1.4 Extensive Algebra (Grassmann)

Hermann Grassmann (1809-1877) taught mathematics at Stettin Gymnasium, Germany. In 1844 he

presented his famous treatise Die Lineale Ausdehnungslehre, where innovative theories (general

theory of analysis of manifolds of arbitrary dimensions) were exposed. Although being farfetched and

pioneering, his publication did not get much attention due to the way its content was exposed – a very

philosophical language. In 1862 he released a second edition, which received better reviews and led

to the progress of Clifford Algebra, among other different forms, as vector analysis and linear algebra.

4


The algebraic entities present on the theory of Extension are denominated extensive quantities.

Among the several elements introduced on this algebra, Grassmann created the inner product for

vectors to define a quadratic form on a vector space over the field of real numbers, as well as the

exterior product to form the diverse kinds of multivectors, which will be studied later on. [4]. So, it may

be stated that Grassmann was the initial discoverer of GA. [9]

1.1.5 Vector Calculus (Gibbs)

Josiah Willard Gibbs (1839-1903) - an American theoretical physicist, chemist and mathematician,

who invented vector analysis (a remake of quaternions) and Oliver Heaviside (1850-1925), took

advantage on the fact that the fundamentals of GA were established in the 19th century as well as the

early death of Clifford, to create the geometric language of the 20th century. This language was used

to re-write Maxwell’s equations, which added some beauty. A necessity of geometric objects that

would be better applied to physics than quaternions, motivated Gibbs to build a three-dimensional

vector analysis based on Hamilton’s works. For instance, the cross product was replaced by the

exterior product and axial vectors by bivectors. Regarding axial vectors, it was possible to make this

substitution since both elements are equivalent, on a geometrical point of view – magnitude and

direction represent the oriented area formed by a bivector. Gibbs also created dyadics, being inspired

on the Grassmann’s tensor product, earlier called indeterminate product. [2b] [2c]

1.1.6 Clifford Algebra (Clifford)

William Kingdon Clifford (1845-1879) was a philosopher and geometer who also taught mathematics

at a London university. In 1876 he wrote the abstract for a paper [10], which remained unfinished due

to his health problems. He was reformulating the products which had been created previously by

Grassmann. Besides this accomplishment, he was also the first person to suggest that gravitation

could be a demonstration of an underlying geometry.

1.1.7 Newton vs. Maxwell Electromagnetic Theories

Late XIX century there were problems related to physics and as a result Albert Einstein published in

1905 the Special Theory of Relativity, which answered to several questions, such as the contradiction

between Newtonian and Maxwell’s Electromagnetic Theory. The first approach made physics

universal, since it explained planetary motion as well as events occurred on earth, such as the

movement of a vehicle, for instance. On the other hand, Maxwell’s Electromagnetic Theory explained

the propagation of light as well as the behavior of magnetism and electricity. According to Newton, all

inertial frames are the same, opposed to Maxwell, who stated all inertial observers measure the same

value for speed of light. An inertial frame is a frame which has no forces applied to it, which implies it

has no acceleration, based on one of Newton’s three laws of motion - more specifically, the first law,

also known as the law of inertia, which states that an object at rest will remain at rest unless acted by

an unbalanced force. An object in motion continues in motion with constant velocity and in the same

direction unless acted upon by an unbalanced force. So, according to Newton, the addition of

5

Introduction

velocities is additive, given by � + � = 2�, where � represents the speed of light. On the other hand,

James Clerk Maxwell derived the electromagnetic wave equation in 1864, which sustained the idea

that “light and magnetism are affections of the same substance, and that light is an electromagnetic

disturbance propagated through the field according to electromagnetic laws”. In addition, this theory

stated that light travelled at � velocity with respect to a chosen frame. Both theories contradict among

each other, so Einstein’s special theory of relativity demystified this subject, which affirmed that the

relative velocity between two objects never exceeds the velocity of light, �, giving reason to Maxwell’s

theory, since it is always valid, while Newton’s is only valid until certain speeds. Using GA of the

spacetime, it is possible to deduce the relativistic addition of velocities expression [11].

All these referred applications are essential on diverse industries, such as aerospace [12], Doppler

ultra-sounds [13], radars, GPS devices, color space conversion 14]. GA has proven to be a powerful

analytical tool for symbolic calculations, and has been implemented on a FPGA, showing a 4x

speedup compared to general algebra. [15] [16].

1.1.8 Spinors (Cartan, Dirac and Pauli)

Spinors were invented by Élie Cartan in 1913. [17] They are very important in mathematics and

physics concerning the theory of orthogonal groups, as rotations and Lorentz transformations. They

are so essential because the structure of rotations in a certain dimension requires more dimensions to

fully characterize it. Using Clifford or other algebra, it is possible to obtain spinors from an Euclidean or

Minkowski space. The spinors Cartan created were adopted for particular application and is still used

nowadays. Dirac spinors are required to describe the state of relativistic particles. Appendix E contains

a thorough explanation of spinors. In 1926, Pauli described the interaction of electron spin with an

external magnetic field.

1.1.9 Modern GA (David Hestenes)

David Hestenes is a physicist, who is currently the most important worldwide researcher and

responsible for the evolution of GA in the 20th century, as well as other algebras, in order to formalize

theoretical physics.

1.2 Objectives

The main objective of this work is to study several electrodynamic and relativistic effects, using the

most appropriate algebra for that meaning – STA. [18] In order to do so, one must first understand the

basic concepts of Euclidean algebra, which is the foundation of STA. The difference between both

relies on the fact that Euclidean algebra only uses vectors in ordinary three-dimensional space, while

STA unites spatial and temporal vectors, resulting in a non-Euclidean algebra. STA is supported on

Minkowski spacetime, which symmetry group is the Poincaré group. To understand the applications

6


which will be presented, it is also necessary to introduce Maxwell’s equations, including Lorentz force,

since they are the basis of electromagnetism. Other applications which will take part on this work

relate to kinematics and Doppler effect, which will be subsequently used to solve the famous problem

of the twins’ paradox. The moving media problem is also addressed, and instead of grasping with

tensors, which has proven to be a rather complex task, the vacuum form reduction method will be

applied.

1.3 Structure and Organization

This master’s dissertation is divided in five chapters: the present chapter – Introduction gives the

reader an overall idea of how STA was invented, its successive evolutions and revolutions, always

depending on the intensive work of physicists and mathematics over the centuries. In order to reach a

higher and further level in science, it is common to have divergences among scientists, since some

defend a certain idea, while others support another theory – this gives place to critical thinking, and is

the path to new and better thoughts.

In chapter two – GA, the reader will have the opportunity to grasp the fundamentals of this algebra,

such as bivectors and trivectors – and its geometric interpretation in the corresponding algebra space, �ℓ� and �ℓ�. It is essential to understand these algebras, as they are the foundation of STA, since

they led to the latter and more perfect form. The essential rules will be presented, as well as the

application of rotations and contractions, which will be useful on subsequent chapters. The relation of

GA with dot product and wedge product will be studied as well. Gibbs’ and Grassmann’s algebras are

related as well, their relation and review will be given on this chapter.

The third chapter – Spacetime Algebra, will introduce the framework which will support the

applications. The major strength of STA is the ability to treat boosts, since it is the basis of all

presented applications. The difference of STA’s indefinite metric to the former algebras will be

presented. That is the big advantage of STA, when compared to the Euclidean algebras – space and

time are connected. The presented concepts on this chapter will allow the reader to understand and

interpret the applications, such as time dilation and space contraction, as well as relativistic velocity

addition for collinear vectors and Doppler shift as well.

The fourth and main chapter of this thesis - Electrodynamic and Relativistic Effects, will present the

interaction of this new algebra with electromagnetism. Maxwell equations will be the kernel of this

chapter, since those equations give the best characterization of the electric and magnetic fields and

relate them to their sources, charge density and current density. These equations are used to show

that light is an electromagnetic wave. The four equations, together with the Lorentz force law are the

complete set of laws of classical electromagnetism. The Lorentz force law itself was actually derived

by Maxwell under the name of "Equation for Electromotive Force" and was one of an earlier set of

eight equations by Maxwell.

7

Introduction

The fifth and last chapter of this thesis will give the reader the attained conclusions regarding all

chapters and some suggestions for future work, since the evolution of science requires a continuous

effort, and not a temporary one. Each chapter, except for the first and latter ones will provide an

intermediate introduction and conclusion, in order to give the reader a greater monitoring, which will

hopefully provide a greater awareness of the presented concepts.

Figure 1.2 – Flowchart with investigated concepts on this thesis

1.4 Original Contributions

Most concepts which appear on this work are already known, the difference is the used formalism – in

this case STA is the chosen foundation. This master’s dissertation gives a global, yet, in-depth study

of diverse applications that STA may help grasp. STA shows its enormous ability to solve problems,

which formerly were very complicated to solve, since required much more extended calculations and

major complexity level. On this thesis, VFR will be used to solve and analyze the STA constitutive

relation on an isotropic media; this has been subject of PhD students and therefore presents a novelty

on the master’s degree. Another innovation on this thesis is the resolution of the relativistic velocity

addition for collinear vectors, without Einstein’s second postulate, which states that the velocity of

transmission of light in vacuum has to be considered equal to �.

8


1.5 Bibliography and References

[1] http://modelingnts.la.asu.edu/gif/FamilyTree.jpeg

[2] http://www.educ.fc.ul.pt/docentes/opombo/seminario/euclides/euclides.htm

[3] W. R. Hamilton, On a New Species of Imaginary Quantities Connected with the Theory of

Quaternions; 1843, edited by David R. Wilkins, 1999

[4] A. Diek and R. Kantowski, “Some Clifford Algebra History”; Dept. of Physics and Astronomy Univ.

of Oklahoma Norman

[5] J. C. Maxwell, A Dynamical Theory of the Electromagnetic Field, Philosophical Transactions of the

Royal Society of London, 1865.

[6] J. C. Maxwell, A Treatise on Electricity & Magnetism; Dover Publications, Inc, New York, 1954.

[7] S. Webber and W. H Keesom, Communs. Icamerlingh Onnes Lab. Univ. Leiden, NO. 223, b.1932

[8] W. R. Hamilton, Lectures on Quaternions; Hodges and Smith, Dublin, 1853

[9] R. G. Calvet, Treatise of Plane Geometry through Geometrical Algebra, electronic edition, June

2000

[10] W. K. Clifford, “Mathematical Papers”, MacMillan Co. London, 1882, edited by R. Tucker

[11] http://video.sc.edu/astronomy/units/unit56.mov

[12] D. Hestenes, “Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics”,

American Journal of Physics, Vol. 71, Issue 2, pp. 104-121, February 2003

[13] http://www.ob-ultrasound.net/doppler_a.html

[14] L. Dorst, S. Mann and D. Fontijne, Geometric Algebra for Computer Science: An Object-Oriented

Approach to Geometry; Elsevier Science & Technology Books, April 2007

[15] A. Gentile et al, “CliffoSor: a Parallel Embedded Architecture for Geometric Algebra and

Computer Graphics”; Palermo, Italy, July 2006

[16] Silvia Franchini et al, “An embedded, FPGA-based computer graphics coprocessor with native

geometric algebra support Source Integration”; the VLSI Journal archive, Vol. 42, Issue 3, 2009, pp.

346-355

[17] E. Cartan, "Les groupes projectifs qui ne Laissent Invariante Aucune Multiplicité Plane"; Bul. Soc.

Math. France, 1913, 41, pp 53–96

[18] D. Hestenes and G. Bobczyk, Clifford Algebra to Geometric Calculus: A Unified Language for

Mathematics and Physics; Kluwer Academic Publishers, Dordrecht, 1984

9

Chapter 2

Geometric Algebra 2 Geometric Algebra

This chapter provides an in depth study of the basic aspects of GA in �ℓ� and �ℓ� spaces, which will

serve as the groundwork for STA. In a geometric point of view, a relation between Grassmann’s and

Gibbs’ algebras will be attained, as well as an interpretation of bivectors and trivectors in �ℓ� and �ℓ�,

respectively.

10


“(…) plus une déduction est rigoureause, plus elle est facile à entendre: car la rigueur consiste

à reduire tout aux principes les plus simples (…)” J. R. D’Alembert

2.1 Introduction

This chapter aims to develop a GA as a symbolic system, which will represent the fundamental

geometrical concepts of dimension, orientation, direction, velocity, acceleration, force, etc, which will

then change into a fully characterized mathematical language. All these concepts serve as the

foundation to spacetime algebra, which will be studied later on and allow the reader to understand the

applications regarding electromagnetism and mechanics. The basics of GA in two and three

dimensions will be presented with a formal approach.

GA is a multilinear algebra, which may be conidered as a Clifford algebra which includes a geometric

product. Given this fact, it allows the theory to be built in a logic way, which eases its understanding.

This algebra has diverse applications, which include computer vision, biomechanics and robotics,

spaceflight dynamics among others. GA is a powerful computational system due to its compactness.

One of the most powerful applications of GA is the ability to handle rotations, which will be extensively

dissected on this chapter. Rotations on a plane are well grasped with complex numbers, therefore this

idea will pass on to three dimensional space as well. A relation between complex calculus and GA

may be established, which was the insightful idea Clifford had while formulating this new algebra

2.2 Geometric Algebra of the Plane

Considering a two-dimensional space, spanned by two orthonormal vectors @� and @� which satisfy

the conditions

@�� = @�� = 1 (2.1)

@� ∙ @� = 0 (2.2)

A convention to use bases in which all vectors are orthogonal among each other is chosen

@= ∙ @> = <=> (2.3)

According to the Kronecker delta function, defined by

<=> = �0, if i ≠ j1, if i = j¢ (2.4)

A vector y ∈ ℝ� will also be considered. Its length is given by the modulus |y|�, which can be called

the signature of vector y. Using Euclidean metric associated to the inner product |y|� = y ∙ y. The goal

11

Geometric Algebra

is to introduce a new product between vectors, called geometric product, which will allow to recover |y|�. �ℓ� is a notation given to GA of the plane. These are the axioms which define the algebra in

analysis:

Associativity � 0� ¤1 = 0� �1¤ (2.5)

Distributivity � 0� + ¤1 = �� + � ¤ (2.6)

Anti-symmetry � � = −� � (2.7)

Contraction rule � � = �� = |�|� (2.8)

Positive �� = |�|� > 0 (2.9)

Scalar multiplication § � = � § (2.10)

The connection between Clifford algebras and quadratic forms comes from the contraction rule. �ℓ�

algebra is spanned by the basis set , = {1, @�, @�, @��}, so the subspaces of �ℓ� are

Table 2.1 - �ℓ� subspaces

1 scalar 1

2 vectors {@�, @�}

1 bivector @� @� = @��

So it can be written that �ℓ� = ℝ ⊕ ℝ�⨁ « ℝ��, which means the vector

y = rA + r� @� + r� @� + r�� @�� ∈ �ℓ� (2.11)

The corresponding dimension is given by 2�, where n, in this case, is 2, therefore, this algebra’s

dimension is equal to 4, as demonstrated next.

Figure 2.1 – Pascal triangle to demonstrate �ℓ� algebra dimension

Bivector @�� is the highest grade of �ℓ� algebra, and pseudoscalar is the highest grade of a given

algebra. So it can be asserted that @�� is the pseudoscalar of �ℓ� algebra. Now, the pseudoscalar’s

square will be determined

12


@�� = 0@� @�1� = 0@� @�10@� @�1 (2.12)

Employing the associative rule

0@� @�10@� @�1 = @� 0@� @�1 @� (2.13)

And now using the anti-commutative rule, the equality is obtained

@�� = @�0@� @�1 @� = −@�0@� @�1 @� − @�� @�� = −1 (2.14)

The conclusion is that when the square of @�� is negative, it is considered a bivector, since it’s neither

a scalar nor a vector. Figure 2.2 shows that the bivector is defined by an orientated frame

Figure 2.2 –Bivector described by a frame

Considering the vector y ∈ ℝ�, where y = x @� + y @�, y� will be deduced

y� = 0x @� + y @�10x @� + y @�1 = (2.15)

x� @� @� + y� @� @� + x y 0@� @� + @� @�1 (2.16)

Given that y� = |y|�,

@� @� + @� @� = 0 (2.17)

Then

@� @� = −@� @� (2.18)

This proves equation (2.7) is correct. Back to equation (2.16) we have

y� = x� @� @� + y� @� @� (2.19)

From equation (2.1) it can be written

y� = x� + y� (2.20)

�ℓ� algebra follows rules according to the table presented next, three different types of grades:

scalars, vectors and bivectors.

13

Geometric Algebra

Table 2.2 - �ℓ� multiplicative table

↗ @� @� @��

@� 1 @�� @�

@� @�� = −@�� 1 −@�

@�� −@� @� -1

Now, the existing relation between inner product and geometric product is going to be determined. For

that purpose, considering vector y = � + � with �, � ∈ ℝ� and Not imposing any restriction regarding

the new product’s commutative rule

y� = y y = 0� + �10� + �1 = �� + �� + � � + � � (2.21)

y� = y ∙ y = 0� + �1 ∙ 0� + �1 = |�|� + |�|� + 20� ∙ �1 (2.22)

Given that �� = |�|� and �� = |�|�, it can be deduced

� � + � � = 20� ∙ �1 (2.23)

Then

� ∙ � = �� 0� � + � �1 = α ∈ ℝ (2.24)

Where α is a scalar. Now considering vectors � = a� @� + a� @� and � = b� @� + b� @�, the result of

geometric product will be observed, for further analysis.

� � = 0a� @� + a� @�10b� @� + b� @�1 (2.25)

= 0a� b� @� @� + a� b� @� @� + a� b� @� @� + a� b� @� @�1 (2.26)

Knowing @�� = 1, @�� = 1 and @� @� = −@� @�, it is obtained

� � = 0a� b� + b� a�1 + 0 a� b� − b� a�1 @� @� (2.27)

The reversion of bivector @�� is represented by @²�� = @�� = −@��. Comparing equation (2.27) with

equation (2.24), it can be concluded that the first parcel is a scalar and the operation is an inner

product. Regarding the second parcel, it is a bivector, and the operation involved is called outer

product (Grassmann’s), as follows

� ∧ � = 0a�b� − b�a�1@�� = + @�� (2.28)

Knowing that the bivector @�� is anti-commutative, @�� = −@��, it can be concluded � ∧ � = −� ∧ �.

It is possible to observe that this new operation is not commutative like the inner product – it is anti-

commutative. Now, a relation between geometric product and wedge product is the purpose, not as

written on equation (2.27), since it includes the inner product. � � = � ∙ � + � ∧ � (2.29) � � = � ∙ � + � ∧ � = � ∙ � − � ∧ � (2.30)

14


Summing equations (2.29) and (2.30), equation (2.24) is obtained. Subtracting (2.30) from (2.29), the

equation we were looking for is deduced

� ∧ � = �� 0� � − � �1 (2.31)

2.2.1 Bivectors

The nature of the bivector is going to be studied next

+� = 0� ∧ �1� = 0� ∧ �10� ∧ �1 (2.32)

Substituting from equations (2.27) and (2.29) in order to � ∧ �, on equation (2.31)

= 0� � − � ∙ �10� ∙ � − � �1 (2.33)

= 0� ∙ �10� � + � �1 − 0� ∙ �1� − � � � � (2.34)

= 0� ∙ �102� ∙ �1 − 0� ∙ �1� − 0� �1� (2.35)

= 20� ∙ �1� − 0� ∙ �1� − 0� �1� (2.36)

= 0� ∙ �1 − �� (2.37)

= −|�|�|�|� sin� θ (2.38)

In ℝ�, a geometric interpretation for the inner product regarding angle θ = ∢0�, �1 exists

µ� ∙ � = |�||�| cos θ +� = −|�|�|�|� sin� θ ≤ 0 ¢ (2.39)

When θ ≠ 0, yields +� < 0, so as analyzed before, bivector) + is not just a scalar nor a vector. It is a

new type of object, which results from the wedge (exterior) product of two vectors. An important

conclusion in ℝ�, is that the bivector’s square is always a negative number. θ = 0 is a minor case

which its square is null.

|+| = β = |�||�| sin θ (2.40)

β represents the area of a parallelogram, defined by the bivector +, which forms an oriented frame.

This parallelogram is shown next on the next figure

Figure 2.3 – Geometric interpretation of a bivector

15

Geometric Algebra

Nevertheless having considered that bivector + corresponds to the oriented parallelogram on the

previous figure, the exact form is irrelevant. Only its frame, orientation and area matter. For example,

a circle could also be used to show the same as a parallelogram, although the latter is more

suggestive. The next two figures show that bivector + is anti-symmetric.

Figure 2.4 – Anti-symmetry of a bivector

Figure 2.5 – Anti-Symmetry of a bivector

Now, it will be proven that the area formed by a bivector is given by equation (2.28). The rhomb

presented on figure (2.6) is formed by the outer product between vectors � and �, so in order to

discover the desired area, the best way is to calculate the area of the big square, then subtract the

area of the two smaller (black and white) squares, and finally subtract the area related to the four

rectangle triangles, as written on equation (2.41).

16


Figure 2.6 – Area of a bivector

A»¼½¾¿ = A¿=À ÁÂÃÄ»Å − 2 AÁ¾ÄÆÆ ÁÂÃÄ»Å − 4 AÈ»=Ä�ÀÆÅÁ (2.41)

A¿=À ÁÂÃÄ»Å = 0a� + b�10a� + b�1 (2.42)

AÁ¾ÄÆÆ ÁÂÃÄ»Å = a� b� (2.43)

AÈ»=Ä�ÀÆÅ = �� a� a� = �� b� b� (2.44)

Replacing equations (2.42), (2.43) and (2.44) on (2.45) yields

A»¼½¾¿ = 0a� + b�10a� + b�1 − [2a� b� + 2 �� a� a� + 2b� b�] (2.45)

= a� a� + a� b� + b� a� + b� b� − 2a� b� − a� a� − b� b� (2.46)

As only scalars are being dealt with, the commutative property applies in this situation, so equation

(2.28) is correct, as represented on equation (2.47) Since negative areas don’t exist, the modulus of

the result is introduced

A»¼½¾¿ = |a�b� − a�b�| (2.47)

2.2.2 Rotations

Now the aim is to study rotations in �ℓ�, algebra, which result from applying a rotor to a certain vector.

This will originate another vector rotated form the initial one, both forming a certain angle, which is

given by the one defined on the rotor’s expression. Let’s consider + = β @��.

exp0+1 = exp 0β @��1. (2.48)

The starting point will be equation (2.14). From that equation it can be written

@�� = 0−11� (2.49)

17

Geometric Algebra

So another deduction can be taken

@��'� = @�� @�� = 0−11� @�� (2.50)

Taylor series expansion will have to be used in order to solve the problem, Therefore, next are

presented equations which will be useful

exp0x1 = Ë ÌÍ�!Ï�ÐA , ∀x (2.51)

Equation (2.52) defines even numbers, while equation (2.53) defines the odd ones, so equation (2.51)

is the direct sum of both equations, since it encloses both

cos0x1 = Ë 0−11� ÌÒÍ0��1!Ï�ÐA (2.52)

sin0x1 = Ë 0−11� ÌÒÍÓÔ0��'�1!Ï�ÐA (2.53)

Applying equation (2.51) on (2.48), the following equation is attained

x = exp0β @��1 = Ë 0Õ @ÔÒ1Í�!Ï�ÐA (2.54)

Given the relation between the exponential and both trigonometric functions presented above, the

following is achieved

= Ö ÕÒÍ @ÔÒÒÍ0��1!

Ï�ÐA + Ö ÕÒÍÓÔ @ÔÒÒÍÓÔ

0��'�1!Ï�ÐA (2.55)

Comparing the first parcel to equation (2.49) and the second to (2.50)

x = exp0β @��1 = Ö β�� 0−11�02k1!Ï

�ÐA+ Ö β��'� 0−11� @��02k + 11!

Ï�ÐA

= cos0β1 + sin 0β1@�� (2.56)

The right-hand side of the previous equation represents the Euler formula. Now applying this result to

vectors @� and @�, exchanging β for θ, as convention, it is obtained

exp0θ @��1 @� = [cos0θ1 + sin0θ1@��] @� = cos0θ1 @� + sin0θ1 @�� (2.57)

Using the anti-symmetric property

= cos0θ1 @� − sin0θ1@� (2.58)

exp0θ@��1@� = [cos0θ1 + sin0θ1@��] @� = cos0θ1 @� + sin0θ1@�� (2.59)

Once more, using the anti-symmetric property

= sin0θ1@� + cos0θ1 @� (2.60)

Another way of writing down equations (2.58) and (2.60) is using the following notation

@� ↦ H� = cos0θ1 @� − sin0θ1@� (2.61)

18


@� ↦ H� = sin0θ1@� + cos0θ1 @� (2.62)

This means that vector @� generates function H� and vector @� generates function H�. Imposing a

variation on angle θ, it is possible to interpret geometrically the function

θ = 0 ⇒ �H� = @�H� = @� ¢ (2.63)

θ = Ù� ⇒ �H� = −@�H� = @� ¢ (2.64)

The result is presented on the next figure

Figure 2.7 – Clock-wise rotation

The observed transformation is a rotor which originates a 90º clock-wise rotation. In order to induce a

90º counter clock-wise rotation, equation (2.54) suffers a modification

xÚ = exp0θ @��1 = exp 0−θ @��1 = cos0θ1 − sin 0θ1 @�� (2.65)

In an analogous line of thought, the generated functions in this case are

@� ↦ H� = cos0θ1 @� + sin0θ1@� (2.66)

@� ↦ H� = −sin0θ1@� + cos0θ1 @� (2.67)

Again, in order to interpret the geometrical meaning, values to angle θ will be imposed

θ = 0 ⇒ �H� = @�H� = @� ¢ (2.68)

θ = Ù� ⇒ �H� = @� H� = −@� ¢ (2.69)

Next figure shows the result of the application. As stated, it’s a 90º CCW rotation.

19

Geometric Algebra

Figure 2.8 – Counter clock-wise rotation

So, given a certain pair of vectors �, G ∈ �ℓ�, where � is the initial vector and G the final, a clock-wise

rotation is given by the following equation

� = exp0θ@��1 G (2.70)

While a counter-clockwise is given by

� = exp0−θ @��1 G = exp0θ@��1 G (2.71)

Analyzing these results, it can be observed an analogy between multivectors in GA and complex

numbers. GA can be seen the same way as an addition of a real and an imaginary number. Therefore,

the result is not purely real nor purely imaginary, but a blend of both. GA is isomorphic to complex

numbers, which can be supported by @�� = Û� = −1.

2.3 Geometric Algebra of the Space

2.3.1 Properties

On this algebra, the existence of a linear (or vectorial) space ℝ� on space ℝ (of the real numbers) is

admitted. On the previous chapter �ℓ� 0f = 21 was studied, now f = 3 will be considered, also called

GA of the space. GA also allows studying other dimensions and metrics. On the particular case of

Minkowski spacetime, the metric is not Euclidean. That will be dealt with later on, now �ℓ� will be

studied. The essential definition of GA is geometric product (or Clifford’s) between vectors. So, let’s

consider the linear space ℝ� with the usual Euclidean metric, as considered previously on �ℓ�. Defining an orthonormal basis (all basis vectors are orthogonal among each other and their length is

normalized to the unit) , = {@�, @�, @�} of this algebra

W, ] ∈ {1,2,3} → @Ý ∙ @Þ = ßÝÞ = �1, W = ]0, W ≠ ]¢ (2.72)

in a way that

|@�| = |@�| = |@�| = 1. (2.73)

20


Given a vector y = x @� + y @� + z @� ∈ ℝ�, its length is

|y| = √y ∙ y = âx� + y� + z� ≥ 0 (2.74)

Now, let’s define a product between vectors, or geometric product (or yet Clifford’s product), so that

with the square of vector y, y y = y� the square of its length is attained. �ℓ� fundamental axiom is

Contraction y� = |y|� (2.75)

In this particular case, @�� = |@�|� = 1, @�� = |@�|� = 1, @�� = |@�|� = 1

If it is not considered that the product is necessarily commutative

|y|� = x� + y� + z� (2.76)

But it is also known

y� = 0x @� + y @� + z @�10x @� + y @� + z @�1 = |y|� + x y0@� @� + @� @�1 +

+x z0@� @� + @� @�1 + y z0@� @� + @� @�1 (2.77)

So, equaling equations (2.74) and (2.75) y� = |y|� The following equation system is obtained

ä@� @� = −@� @�@� @� = −@� @�@� @� = −@� @�¢ (2.78)

Therefore, another axiom of geometric product is that it is anti-commutative, according to equations

(2.76). However, it is associative. Considering �, �, ¤ ∈ ℝ�, the following relation can be written

0� �1 ¤ = � 0� ¤1 (2.79)

As deduced on equation (2.14) regarding �ℓ�, on �ℓ� algebra there are analogous equations

�@�� = −1@�� = −1¢ (2.80)

Geometric product of two generic vectors �, � ∈ ℝ�, with � = a� @� + a� @� + a� @� and � = b� @� +b� @� + b� @� is the one defined on equation (2.29), while the inner (or scalar) product is given by

� ∙ � = a� b� + a� b� + a� b� (2.81)

The outer product (Grassmann’s) is more complex when compared to �ℓ� alegbra, as presented

below

� ∧ � = ä@�� @�� @��a� a� a�b� b� b� ä = (2.82)

= 0a� b� − a� b�1 @�� + 0a� b� − a� b�1 @�� + 0a� b� − a� b�1 @�� ∈ å ℝ�� (2.83)

U = � � is the graded sum of a scalar α with the bivector F

� � = � ∙ � + � ∧ � ∈ ℝ⨁ å ℝ�� (2.84)

Two properties of this operation are symmetry of the inner product and anti-symmetry of the outer

product (Grassmann’s), as shown next

21

Geometric Algebra

� ∙ � = � ∙ � (2.85)

� ∧ � = −� ∧ � (2.86)

When both vectors are parallel amongst each other, this means the outer product between them is

null, therefore, the geometric product is equal to the inner product. On the other hand, when they are

perpendicular, the inner product is null, therefore the geometric product is equal to the outer product of

the vectors. Subspace å ℝ�� has a vector basis , = {@��, @��, @��}, so its dimension is equal to 3.

Figure 2.9 – Oriented planes

Concerning the outer product, the geometric product has the major advantage of being invertible.

However, not like Grassmann’s outer product, the geometric product depends only on one metric

�� = � � = � ∙ � = |�|�, � ∈ ℝ. (2.87)

Next is shown the ability to invert of the geometric product. Still considering vectors � and �,

ä�/� = ��Ò�/� = ��Ò¢ (2.88)

and considering U = ��, it is obtained

ä� = U�/ç � = �/�U U/� = �/��/� ¢ (2.89)

2.3.2 Trivectors

Just like the outer product of two vectors originates a bivector (or oriented frame), if three vectors are

considered, their outer product results in a trivector, or an orientated volume. Trivector is also

considered the pseudoscalar of �ℓ� algebraic structure. The trivector is represented by

22


S = @�� = @� @� @� = @� ∧ @� ∧ @� (2.90)

It is possible to define the trivector’s reversion by @²�� = @�� = −@��. As verified with the bivector, the

trivector’s square is a negative number

S� = @�� = @�� @�� = @� @� @� @� @� @� = @� @�@� @� @� @� =

−@� @� @� @� @� @� = (2.91)

= −@�� @�� @�� = −1 (2.92)

The trivector’s outer product is represented by a similar but more complex notation as the bivector’s

� ∧ � ∧ ¤ = äa� a� a�b� b� b�c� c� c� ä @�� = β @�� (2.93)

= [a�0b� c� − b� c�1 + a�0b� c� − b� c�1 + a�0b� c� − b� c�1] @�� ∈ å ℝ�� (2.94)

It is verified that

è� = 0� ∧ � ∧ ¤1� = 0β S�1� = −β� (2.95)

The outer product has the property of being associative, consequently it can be written

0� ∧ �1 ∧ ¤ = � ∧ 0� ∧ ¤1 = � ∧ � ∧ ¤ (2.96)

This may be interpreted from the following duo of figures

Figure 2.10 – Associativity of the outer product: example 1

23

Geometric Algebra

Figure 2.11 - Associativity of the outer product: example 2

After the brief introduction to the trivectors and corresponding operations on �ℓ� algebra, it is now

possible to characterize properly this algebraic structure, defined on three dimension Euclidean space ℝ�. �ℓ� algebra consists on the graded sum of subspaces, as shown next �ℓ� = ℝ ⊕ ℝ�⨁ « ℝ�� ⨁ « ℝ��.

Table 2.3 - �ℓ� subspaces

1 scalar 1

3 vectors {@�, @�, @�}

3 bivectors {@��, @��, @��} 1 trivector @��

This algebra is spanned by the basis set , = {1, @�, @�, @�, @��, @��, @��, @��}, so its dimension is equal

to 8, which obviously follows the dim 0�ℓ�1 = 2�.

Figure 2.12 - Pascal triangle to demonstrate �ℓ� algebras dimension

�ℓ� multivector is

U = �U�A + �U�� + �U�� + �U�� = α + � + F + è ∈ �ℓ� (2.97)

24


Where

êë ∈ å ℝ� = ℝA ì ∈ å ℝ� = ℝ�� í ∈ å ℝ� = ℝ�åℝ� �î ∈ å ℝ� = ℝ�åℝ�åℝ��¢ (2.98)

�U�� is the projection of the multivector U regarding degree ], which is the dimension of subspace å ℝ�Þ . A k-blade of �ℓ algebra is an element UÞ such that UÞ = �u��, where �u�� is a homogeneous

element of a certain grade k, and results from the outer product of one or more vectors. It is possible

to define an operation called duality, given a multivector U = α + � + �@�� + β@�� ∈ �ℓ�, its Clifford’s

dual is defined as being the multivector ð = U @�� = −β − � + �@�� + α@�� ∈ �ℓ�. An example is

given next concerning the referred operation

êê1 @�� = @�� @� @�� = @� @� @� @� = @� @� @�� = @� @� @�� @�� = @� @� @� @� @� = −@�� @�� @� = −@�@�� @�� = −1 ¢ (2.99)

With this transformation, it can be observed that a scalar turns into a trivector (and the opposite as

well) and a vector is transformed into a bivector (and vice-versa). Next figure shows geometrically this

operation

Figure 2.13 – Geometric interpretation of duality

Given these concepts on the subject of duality, it is possible to establish the connection between

Gibbs’ cross product with �ℓ� GA. It is only possible to define the cross product in ℝ� space, opposed

to Grassmann’s outer product. Next is defined Gibb’s cross product

¤ = � × � = ä@� @� @�ñ� ñ� ñ�ò� ò� ò�ä = ä@�� @�� @��ñ� ñ� ñ�ò� ò� ò� ä @�� = −0� ∧ �1 @�� (2.100)

The following relation is concluded

0� ∧ �1 = 0� × �1 @�� (2.101)

This results in

� × � = −0� ∧ �1 @�� (2.102)

Next figure explains the equation to give a better idea of the equation

25

Geometric Algebra

Figure 2.14 – Geometrical interpretation of Gibbs cross product

While Gibbs’ cross product depends on the metric and is neither associative nor invertible,

Grassmann’s doesn’t depend on its metric, and besides that, it’s associative and invertible. �ℓ�

algebra apprehends several algebraic structures within, it is possible to define the even part of �ℓ�,

denoted as �ℓ�' and the odd part (�ℓ�/) as those structures, which result from the geometric product of

an even and odd number of vectors, respectively of ℝ� space. Only the even part incorporates sub-

algebra, since the odd part is not closed regarding the geometric product.

ä�ℓ�' = ℝ ⨁ « ℝ� ��ℓ�/ = ℝ�⨁ « ℝ�� ¢ (2.103)

One may define the center of algebra as the aggregate of the several elements which commute with

all the algebra’s elements. Therefore, �ℓ� has only two subalgebras: its center, defined by Cen0�ℓ�1 or Z0�ℓ�1 and its even part, �ℓ�'. Cen0�ℓ�1 = ℝ ⨁ « ℝ��. The even part �ℓ�' is isomorphic to Hamilton’s

division ring (Appendix D), while its center, Cen0�ℓ�1 is isomorphic to the complex body,

�Cen0�ℓ�1 = Z0�ℓ�1 ≃ ℂ�ℓ�' ≃ ℍ ¢ (2.104)

Next is presented the multiplicative table related to �ℓ� algebra

26


Table 2.4 - �ℓ� multiplicative table

↗ @� @� @� @�� @�� @�� @��

@� 1 @�� −@�� @� @�� −@� @��

@� −@�� 1 @�� -@� @� @�� @��

@� @�� -@�� 1 @�� -@� @� @��

@�� -@� @� @�� -1 -@�� @�� -@�

@�� @�� -@� @� @�� -1 -@�� -@�

@�� @� @�� -@� -@�� @�� -1 -@�

@�� @�� @�� @�� -@� -@� -@� -1

Rotor is an operator which generates rotations, previously studied on �ℓ� algebra. Defining two unitary

vectors �, G ∈ ℝ�, the rotor is given by the geometric product R = � G, so Rö = G �. The following

property is attained

Rö R = R Rö = 0� G10G �1 = �� G� = 1 (2.105)

It is possible to obtain �ℓ� Euler’s formula, which is similar to the one presented on equation (2.55), R = cos0θ1 + +Dsin0θ1 = exp 0θ+D1, where +D represents the unitary bivector. Separating the rotor on 0-

blade and 2- blade, it can be written

��x�A = � ∙ G = cos0θ1 �x�� = � ∧ G = +Dsin0θ1 ¢ (2.106)

2.3.3 Contractions and Rotations

In GA, it’s possible to define an operation called contraction, existing both left and right contractions. A

vector � ∈ ℝ÷ and a bivector + ∈ å ℝ÷ø will form a trivector defined by the geometric product U = � + =� 0� ∧ ¤1 ∈ �ℓ�. Expanding the geometric product, the following is obtained:

� 0� ∧ ¤1 = �� 0� ¤ − ¤ �1 = �� 0� �1 ¤ − �� 0� ¤1� (2.107)

=�� [20� ∙ �1 − � �] ¤ − �� [20�. ¤1 − ¤ �] � (2.108)

= 0� ∙ �1 ¤ − 0�. ¤1 � − �� 0� � ¤ − ¤ � �1 (2.109)

Where it’s possible to expand the third and fourth parcels of the previous equation

�� 0� � ¤ − ¤ � �1 = �� 0� ¤1 − �� ¤ 0� �1 (2.110)

27

Geometric Algebra

= �� [20� ∙ ¤1 − ¤ �] − �� ¤ [20� ∙ �1 − � �] (2.111)

= 0� ∙ ¤1 � − 0� ∙ �1 ¤ − �� 0� ¤ � − ¤ � �1 (2.112)

= 0� ∙ ¤1 � − 0� ∙ �1 ¤ − �� 0� ¤ − ¤ �1� (2.113)

= 0� ∙ ¤1 � − 0� ∙ �1 ¤ − 0� ∧ ¤1� (2.114)

Replacing equation (2.112) on (2.107), it’s possible to conclude the following equation:

� + − + � = 20� ∙ �1 ¤ − 20� ∙ ¤1 � ∈ ℝ� (2.115)

So, the left contraction is defined on the next equation. This operation’s result is a decrease of the 2-

grade bivector + to the 1-grade vector �.

� + = �� 0� + − + �1 (2.116)

On the other hand, there is also the right contraction, where the result is anti-symmetric to the one

observed for the left contraction

+ � = �� 0+ � − � +1 (2.117)

These operations are anti-symmetric among them, as observed on the next equation

� + = −+ � (2.118)

Figure 2.15 – Geometrical interpretation of a left contraction

One may also write the fundamental rule of left contraction, defined by

� 0� ∧ ¤1 = 0� ∙ �1¤ − 0� ∙ ¤1� (2.119)

This is the dual rule to the known bac-cab rule from Gibbs’ vectorial algebra,

� × 0� × ¤1 = 0� ∙ �1¤ − 0¤ ∙ �1� (2.120)

The operation � + serves as a reduction - from a certain vector + with a degree equal to two, it

reduces to a degree equal to one, present on a vector �. Vector � has a perpendicular �┴ and parallel

28


�∥ components, both enclosed on bivector +. Using Figure 2.15 as a generic example, the result of a

left contraction between � and + is given by the geometric product of vector a perpendicular

component and bivector +.

� = � + = �∥ + (2.121)

When vector � only possesses a perpendicular element, it can be written that � + = 0 and when � is

equal to its parallel component, � ∧ + = 0. Analogously to �ℓ� algebra, it is possible to define a

rotation on �ℓ� algebra. A certain vector � on a plane defined by bivector + = ù ú is rotated to another

vector �′ through a transformation defined by the rotors x and xÚ. Since x = exp 0− ü� +1 and xÚ =exp ýü� +þ, both rotors, when multiplied, equal to 1, as shown previously. The equation which describes

the rotation is presented next

�{ = x � xÚ (2.122)

It is equally possible to attain the inverse transformation, given by � = xÚ �{ x (2.123)

Vectors � and �′ can be decomposed into perpendicular and parallel components. Next figure shows

both vectors (including their components) and bivector as well.

Figure 2.16 – Rotation from vector � to �’ When the transformation from vector � to �’ occurs, one concludes that the perpendicular component

of vector � remains unchanged, which can be proven analytically. The purpose is to prove the

following equation.

�┴ xÚ = xÚ �┴{ (2.124)

Where

�x = cos ýü�þ − + �Vf 0ü�1xÚ = cos ýü�þ + + �Vf 0ü�1¢ , + = ù ú (2.125)

29

Geometric Algebra

Therefore

�┴ + = + �┴ (2.126)

Will be deduced, which proves the perpendicular component remains unchanged

�┴ + = �┴ ù ú = −ù �┴ ú = ù ú �┴ = + �┴ Q.E.D (2.127)

The previous equation supports the following one

�┴ = �┴{ (2.128)

Concerning the parallel component, it changes with the referred transformation. So, the following

equation describes the modification on that given component of vector �.

+ �∥ = −�∥ + ⟹ µxÚ �∥ = �∥ xx �∥ = �∥ xÚ ¢ (2.129)

So, it can be concluded that

x �∥ xÚ = x x �∥ = x� �∥ = exp 0− ü� +1� �∥ = exp 0−�+1 �∥ (2.130)

Analyzing the former equation, it can be deduced that a ü� degree rotor induces a � on vector �, which

is contained on the frame defined by bivector + = � @çø÷. This can be understood with the following

equation.

�∥{ = exp 0−� +1 �∥ (2.131)

2.4 Conclusions

From chapter 2 one may conclude that there is a relation between the geometric, inner and outer

products, which is given by � � = � ∙ � + � ∧ �. The outer (or wedge) product is anti-commutative,

therefore the geometric product may be written as � � = � ∙ � − � ∧ � and is also anti-commutative.

A bivector is a new kind of operator, which is not considered a scalar neither a vector, it results from

the wedge product of two vectors, so it represents an area with an oriented plane. A particularity of the

outer product is relies on the fact it possesses a geometric meaning, which constitutes an advantage

when compared to Gibbs cross product. Its square is always negative in �ℓ� algebra.

A rotation is an easy object to handle, whether on the plane or on space geometries. On the plane it is

represented by � = exp0θ@��1 G, when a clockwise rotation is required and � = exp0−θ @��1 G = exp0θ@��1 G on the anti-clockwise situation.

When both vectors are parallel amongst each other, this means the outer product between them is

null, therefore, the geometric product is equal to the inner product. On the other hand, when they are

perpendicular, the inner product is null, therefore the geometric product is equal to the outer product of

the vectors.

30


Just as bivectors characterize an oriented plane when considering �ℓ� algebra, the trivectors have the

same function regarding �ℓ� algebra, where a volume is represented. This new operator is defined by

the outer product of three vectors, or a vector with a bivector. While Gibbs’ cross product depends on

the metric and is neither invertible nor associative, Grassmann’s outer product doesn’t, and besides

that, it’s associative and invertible, so it holds more advantages, i.e. being more flexible to handle.

One may define the center of algebra as the aggregate of the several elements which commute with

all the algebra’s elements. The center of �ℓ� algebra is given by Cen0�ℓ�1 = Z0�ℓ�1 ≃ ℂ. This means

that it is an isomorphism to the complex structure.

A rotor x = exp 0− ü� +1 is an operator which constitutes rotations. In �ℓ� algebra, a rotation is

represented by �{ = x � xÚ. Whether working on a two or three-dimensional space, the corresponding

algebra is very useful on handling rotations, allowing computational geometry without matrices or

tensors, reducing complexity and reducing rotations to algebraic multiplication.


C. R. Paiva, "Lição de Síntese", Departamento de Engenharia Electrotécnica e de Computadores,

Instituto Superior Técnico

C. R. Paiva, "Álgebra Geométrica do Espaço", Departamento de Engenharia Electrotécnica e de

Computadores, Instituto Superior Técnico

C. Doran and A. Lasenby, Geometric Algebra for Physicists, Cambridge University Press, 2003

G. L. Naber, The Geometry of Minkowski Spacetime An Introduction to the Mathematics of the Special

Theory of Relativity; Springer, 1992

D. Hestenes, Spacetime Physics with Geometric Algebra; Department of Physics and Astronomy,

Arizona State University, Tempe, Arizona, in Am. J. Phys. 71, 691, 2003

31

Chapter 3

Spacetime Algebra 3 Spacetime Algebra

This chapter will consist on interpreting the different kinds of vectors and their behavior concerning

trajectories. The concept of a boost will be introduced as well as Minkowski diagram and its inherent

ability to analyze several applications, such as time dilation and space contraction, among other

relativistic effects.

32


“Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union

of the two will preserve an independent reality” H. Minkowski

3.1 Introduction

Opposite to tridimensional vectorial algebra, which is based on Gibbs’ cross product, GA is not

confined to ℝ� space nor Euclidean metric. Minkowski spacetime frame implies a connection between

space and time, differing from the restrict relativity, where space and time are distinguished apart.

There is a contradiction between Newton and electrodynamic mechanics, and the only way to surpass

this issue is to change the linear space from ℝ� to ℝ�. Besides that, Euclidean metric (�ℓ�) which is

associated to the vectorial basis , = {@�, @ç, @ø, @÷}, where @A� = @�ø = @�� = @�� = 1 is not satisfactory for

spacetime, therefore the desired metric will be �ℓ�,�. This is defined as a negative indefinite metric,

which consists on one time (@�) and three space (@�, @� and @�) dimensions with @A� = 1, @�� = @�� = @�� =−1. There is the option to define a positive indefinite metric (�ℓ�,�), with @A� = −1, @�� = @�� = @�� = 1

which has no difference to the negative one, it is just a matter of convention.

3.2 Properties

The first form 0�ℓ�,�1 will be used, since it’s the most used on literature. It is possible to define the

trajectory equation, also known as event

y0.1 = 0�.1@A + �0.1@� + �0.1@� + �0.1@� ∈ ℝ�,� (3.1)

With

y� = 0�.1� − 0�� + �� + ��1 = 0�.1� − �� ∈ ℝ (3.2)

An event is described by three kinds of spacetime trajectories (defined by a vector) in Minkowski

spacetime:

� y� = 0 lightlike y� > 0 .VdbV] y� < 0 �rñ�bV] ¢ (3.3)

That event is defined by a vector, which may be considered lightlike when it describes the movement

of light at speed equal to �, timelike when time prevails over space and spacelike when space

dominates over time.

33

Spacetime Algebra

Figure 3.1 – Spacetime trajectories

On the previous figure, the region known as elsewhere is not accessible since it is necessary to travel

at a certain speed greater than �, which is not possible, as will be shown later on this chapter. A

property which must be satisfied on STA is that of causality, which will be explained next.

Figure 3.2 – Light cone for demonstration of causality property

Regarding the figure above, A, B and C are events which occur ate certain time and space. Interval A-

B is time-like, since there is a frame of reference in which both events A and B occur at different times

but at the same location in space. Considering that event A happens before B in that frame, than A

happens always before event B in all frames, therefore it is possible to travel from A to B, hence there

is a causal relationship (being A the cause and B the effect).

34


On the other hand, interval A-C is space-like, which means events A and C occur at the same time but

separated in space. In this case there are also frames in which A (cause) happens before C (effect)

and other frames where the opposite happens (C precedes A). For example, if A was the cause, and

C the effect, then there would be frames of reference in which the effect preceded the cause. Although

this itself won't give rise to a paradox, faster than light signals can be sent back into one's own past.

[1][2][3] So, a consequence of special relativity is that nothing can travel at a velocity greater than �,

assuming that causality is preserved. �ℓ�,� consists on a graded sum of the several subspaces

�ℓ�,� = ℝ ⊕ ℝ�,�⨁ � ℝ�,�� ⨁ � ℝ�,�⨁ � ℝ�,��

Table 3.1 - �ℓ�,� subspaces

1 scalar 1

4 vectors {@A, @�, @�, @�}

6 bivectors {@A�, @A�, @A�, @��, @��, @��} 4 trivector {@A��, @A��, @A��, @��}

1 quadrivector S = {@A��}

This algebra is spanned by the following basis set

, = {1, @A, @�, @�, @�, @A�, @A�, @A�, @��, @��, @��, @A��, @A��, @A��, @��, @A��}, so its dimension is equal to 16, represented on the correct notation, dim(�ℓ�,�)=16.

Figure 3.3 - Pascal triangle to demonstrate �ℓ�,� algebras dimension

The multivector is represented by the k-blades, from 0 to 4, as shown next

U = �U�A + �U�� + �U�� + �U�� + �U�� = α + � + F + � + è ∈ �ℓ�,� (3.4)

35

Spacetime Algebra

Where

êêë ∈ å ℝ�,� = ℝA ì ∈ å ℝ�,� = ℝ�,��í ∈ å ℝ�,� = ℝ�,�� ∈ å ℝ�,�� î ∈ å ℝ�,��

¢ (3.5)

An important rule in �ℓ�,� algebra is that of commutation: while vectors and trivectors anti-commute

with the pseudoscalar S, bivectors commute with S. Just like the studied algebra, it’s possible to define

Clifford’s dual for a generic multivector U = α + � + F + � S + β S, which results in vector ð = U S =U @�çø÷. ð = −β − � + F S + � S + α S (3.6)

Representing both vectors’ blades:

êê �U�A = α �U�� = � �U�� = F �U�� = � S�U�� = β S

¢ (3.7)

êê �ð�A = −β �ð�� = −� �ð�� = F S �ð�� = � S �ð�� = α S

¢ (3.8)

3.3 �,� and Boosts

Bivectors can be classified regarding their blades; while on �ℓ� algebra all bivectors are simple, which

means they possess a blade with a negative square, on �ℓ�,� algebra not all bivectors are simple. As

for instance, bivector F = @� ∧ @ç + @ø ∧ @÷ is not simple since its square is different from zero. Simple

bivectors can be classified just like the vectors, as parabolic, hyperbolic and elliptical. For example,

bivector @�ç = @� ∧ @ç is hyperbolic (timelike vector), @ø÷ = @ø ∧ @÷ is elliptical (spacelike vector) and F = @ç ∧ @� + @ç ∧ @÷ is parabolic (timelike vector). The major difference from elliptical and hyperbolic

bivectors is the fact that the first set generates rotations (approached earlier on equation (2.65)), while

the second generates a boost, also known as Lorentz active transformation. Now, the geometric result

of the boost will be studied. Considering the rotor

exp 0ζ @�A1. (3.9)

Exposing the fundamental “tools” to be used subsequently

@�A� = @� 0@A @�1 @A = −@� 0@� @A1 @A = −@�� @A� = 1 (3.10)

It is precisely where this demonstration differs from the one relative to �ℓ� or �ℓ� algebra, on that

previous case, @�� = −1 (for �ℓ�) or @�� = −1 (for �ℓ�). Now it can be written

36


@�A�� = 1 (3.11)

So another deduction can be taken

@�A��'� = @�A (3.12)

Taylor series expansion will have to be used in order to solve the problem, Therefore, equation (2.51)

and the following ones

cosh0x1 = Ë ÌÒÍ0��1!Ï�ÐA (3.13)

sinh 0x1 = Ë ÌÒÍÓÔ0��'�1!Ï�ÐA (3.14)

will come in handy to find the solution, so applying equation (2.51) on (3.9), the following is attained

+ = exp0ζ @�A1 = Ë 0� @Ô 1Í�!Ï�ÐA (3.15)

Given the relation between the exponential and both trigonometric functions presented above, the

following is achieved

= Ë 0� @Ô 1ÒÍ0��1!Ï�ÐA + Ë 0� @Ô 1ÒÍÓÔ0��'�1!

Ï�ÐA (3.16)

Comparing the first parcel to equation (3.13) and the second to (3.14)

+ = exp0ζ@ �A1 = Ë �ÒÍ0��1!Ï�ÐA + Ë �ÒÍÓÔ @Ô 0��'�1!

Ï�ÐA = cos h ζ + @�A sinh ζ (3.17)

Using the correct notation to represent the boost

@A ↦ HA = [cos h ζ + @�Asinh ζ] @A (3.18)

= @A cos h ζ + sinh0ζ1 @�A @A (3.19)

= @A cos h ζ + sinh0ζ1 @� @A @A (3.20)

= @A cos h ζ + @� sinh ζ (3.21)

@� ↦ H� = [cos h ζ + sinh0ζ1 @�A] @� (3.22)

= @�cos h ζ + sinh0ζ1 @�A @� (3.23)

= @�cos h ζ + sinh0ζ1 @� @A @� (3.24)

= @A cos h ζ − sinh0ζ1 @� @� @A (3.25)

= @A cos h ζ + @A sinh ζ (3.26)

Considering

ä M = cosh ζ � = tanh ζ M � = � M = sinh ζ ¢ (3.27)

Parameter M may be obtained from the Pythagorean identity for the hyperbolic frame, as follows

37

Spacetime Algebra

cos� ζ − sin� ζ = 1 (3.28)

Replacing the set of equations (3.27) on (3.28) it is obtained

�M� = cosh� ζ M� �� = sinh� ζ¢ (3.29)

Hence

M� − M� �� = 1 (3.30)

M� − M� �� = 1 (3.31)

M� 01 − ��1 = 1 (3.32)

Finally the desired parameter is attained

M = �â�/�Ò (3.33)

Equations (3.21) and (3.26) are re-written

�HA = γ 0@A + β@�1H� = γ 0@� + β@A1¢ (3.34)

Or representing using a matrix

�H HÔ� = γ �� @ @Ô� (3.35)

Figure 3.4 - �ℓ�,� frame

The goal is to determine how the function reacts to a variation of the rapidity parameter ζ, so assigning

values, the result of the functions HA and H� will be observed.

ζ = 0 ⇒ � β = 0γ = 1¢ ⇒ � HA = @AH� = @� ¢ (3.36)

ζ = ∞ ⇒ �β = 1 γ = ∞ ¢ ⇒ � HA =? ?H� =? ? ¢ (3.37)

38


It’s not possible to analyze conveniently the results, therefore an intermediate value between zero and

infinite will be considered. Using the following equation which relates � to �

� = �� ln 0�'��/�1 (3.38)

Assigning a number that interests to �

ζ = √�� ⇒ � β = �� γ = �√� ≈ 1.2¢ ⇒ � HA = 1.2@A + 0.6@�H� = 0.6@A + 1.2@� ¢ (3.39)

Representing these results on the appropriate frame

Figure 3.5 – Intermediate scenario to determine parameter ζ Analyzing the previous figure, one can deduce the higher the value assigned to �, both functions HA

and H� will tend to the bisection of the angle, as presented next:

ζ → ∞ ⇒ β → tan ýÙ�þ = 1 ⇒ � HA → @A + @�H� → @A + @� ¢ (3.40)

So, it can be concluded that HA → H�. As stated earlier, this is the result of a boost.

Figure 3.6 – Representation of a boost

39

Spacetime Algebra

An event is described by a person standing on a frame, which can be represented on the Minkowski

diagram according to the expression which links both time and space

y = 0�.1 @A + � @�. (3.41)

The same event can be expressed by a second person who is located on a different axis, according to

the next equation

y = 0�. 1 HA + � H� (3.42)

This can be expanded, through the substitution of both equations (3.34) on (3.42), so

y = 0�. 1 γ 0@A + β@�1 + � γ 0@� + β@A1 (3.43)

= � . γ @A + � . γ β @� + � γ @� + � γ β @A (3.44)

= [� . γ + � γ β] @A + [� . γ β + � γ]@� (3.45)

Comparing the previous equation to (3.41), it can be deduced

�� . = � . γ + � γ β� = � . γ β + � γ ¢ (3.46)

Representing on a matrix

�� Y� � = M �� Y � � (3.47)

This matrix can also be called Lorentz transformation matrix since it allows obtaining parameters ct and � from ct and x�.

a = � �� (3.48)

It’s possible to obtain the inverse matrix, which results on a passive Lorentz transformation

a/� = ��ÅÈ0�1 × 0−11�'Ýa�: V, W ∈ {1,2} (3.49)

Replacing equation (3.48) on (3.49) yields

a/� = ��ÅÈ0�1 × � �/�� /�� (3.50)

Where det0a1 = M� − M�β�, which is equal to 1, according to equation (3.32). Consequently,

a/� = M � �/� /�� (3.51)

Now it is possible to obtain parameters ct and x� from the other ones which are associated to the

observer on the first frame.

��Y � � = M � �/� /�� Y� � (3.52)

Hence

40


�@�@ç� = � ½Á¼ �/ Á=�¼ � / Á=�¼ � ½Á¼ � � �H�Hç� (3.53)

To build the diagram, on must bear in mind that axis �. corresponds forcing � = 0, which corresponds

to � = ð. = �0�.1.

�. axis⇒ tan0�1 = ��Y = � ⇐ "� = γ[x − β0ct1]� = 0 ¢ (3.54)

On a similar analysis, one can affirm that axis � corresponds forcing �. = 0, which corresponds to �. = ��

� axis⇒ tan0�1 = �Y� = � ⇐ ��. = γ[0ct1 − βx]�. = 0 ¢ (3.55)

An in-depth study on the action of a boost on a certain vector enclosed on the hyperbolic frame

(�ℓ�,�algebra) will be made next. Considering a negative indefinite metric, @�� = −@ç� = 1, so the active

transformation will be analyzed. Considering a boost

� ↦ G = exp 0�@ç�1� (3.56)

Where vectors � and G possess time (UA = �.) and space (U� = �) components

"� = UA @� + U� @çG = ðA @� + ð� @ç ¢ (3.57)

Considering vector � is a unit vector

�UA = Á=� üâ|Á=�Òü/ ½ÁÒü|U� = ½Á üâ|Á=�Òü/ ½ÁÒü|¢ (3.58)

Which results on the angle between both components

tan θ = # #Ô (3.59)

since the metric of ℝ�,� space is indefinite. The following equation gives the square of the vector’s

length

b�� = |UA� − U��| (3.60)

If the set of equations on (3.58) weren’t normalized, vector b� would not be always unitary because on

that case the length would be given by

b�� = | sin �� − cos �� | (3.61)

Only for a given angle � = 0. b�� would be equal to 1 for particular cases, for instance, � = $� , , �$� .

Vector � can be described as a passive transformation using the following parametric representations

when % ∈] − ∞,∞[ �UA = �. = cosh %U� = � = sinh % ⟶ �ø = c�t� − x� = 1¢ (3.62)

41

Spacetime Algebra

�UA = �. = −cosh %U� = � = sinh % ⟶ �ø = c�t� − x� = 1¢ (3.63)

�UA = �. = sinh %U� = � = cosh % ⟶ �ø = c�t� − x� = −1¢ (3.64)

�UA = �. = sinh % U� = � = −cosh % ⟶ �ø = c�t� − x� = −1¢ (3.65)

Figure 3.7 – � vector equal to 1

Figure 3.8 – � vector equal to -1

From equation (3.17) and knowing that

"@�A @A = @�@�A @� = @A ¢ (3.66)

One can obtain the matrix with vector G in order to vector �, as follows

42


�' 'Ô� = � ½Á¼ (Á=�¼ ( Á=�¼ ( ½Á¼ (� �# #Ô� (3.67)

So, writing the previous matrix as a set of equations,

�ðA = UAcosh � + U� sinh �ð� = UAsinh � + U� cosh �¢ (3.68)

Therefore the angle which describes the slope of vector � is achieved

tan u = ' 'Ô = ÈÄ� ü'ÈÄ�¼ (�'ÈÄ�¼ ( ÈÄ� ü (3.69)

It must be pointed that the length of vector � remains constant on this transformation

b# = b' (3.70)

So the following relation is reached

UA� − U�� = ðA� − ð�� (3.71)

Thus, vector G will be as well a unitary vector. However, when angle � tends to $�, the length of vector � tends to zero and � tends to ∞. Consequently, one will not be able to define the direction of a null

vector. Given a certain angle �, vector � with a unitary length is defined. Subsequently for a given

intensity �, vector G has a slope specified by angle u. On the particular situation of � equal to zero, this

means vectors G and � will be the same, therefore it will be called a identity transformation. Other

cases will be shown next

� = µ0 0� = @ç1 $� 0� = @�1 ¢ → �tan u = tanh � 0G = Hç = @�sinh � + @çcosh �1cotu = tan � 0G = H� = @�cosh � + @çsinh �1 ¢ (3.72)

The next step consists on obtaining the expression which defines vector G.Knowing that vector G is

given by

G = [ðA ð�] �@�@ç� (3.73)

Replacing equation (3.53) on the previous one, it is obtained

G = [ðA ð�] � cosh ζ− sinh ζ − sinh ζ cosh ζ� �H�Hç� (3.74)

Hence

G = [ðA ð�] �H�Hç� (3.75)

This may be written on an equation form as well

G = UAH� + U�Hç (3.76)

Some analytical examples are shown next, in order to help understand the action of a boost

� = µ 0� = −@ç1 − $� 0� = −@�1¢ → �G = −HçG = −H� ¢ (3.77)

43

Spacetime Algebra

The geometric results may be found on Appendix B - Boost Application

3.4 Minkowski Diagram

All necessary elements to build Minkowski diagram are exposed, so next figure presents the referred

diagram. It encloses two grids, each corresponding to a certain person who is observing the event

occurring. Frame � is formed by axis �. and �, while � is formed by axis �. and � . �.5 corresponds to

the time associated to a certain event ), while �5 refers to its position, both on � frame. So, �. 5 and � 5

correspond to time and event on � frame axis, respectively.

Figure 3.9 – Minkowski diagram with � and � frames

Any two events ) and * which occur on a line parallel (also known as line of simultaneity in � ) to axis � , (defined as �. A), happen simultaneously regarding an observer on the � frame. This is not valid for

another observer who is placed on the � frame, which is shown on next figure. The difference in time

events regarding � frame is given by Δ�.56 = �.6 − �.5, which is different from zero.

44


Figure 3.10 - Line of simultaneity in � frame

Following the same thread of thought, Any two events + and , which occur on a line parallel to axis �,

defined as �.A, also known as line of simultaneity in � happen simultaneously regarding an observer

on the � frame. This is not valid for another observer who is placed on the � frame, which is shown on

next figure. The difference in time events regarding � frame is given by Δ�. 34 = �. 3 − �. 4, which is

different from zero.

Figure 3.11 - Line of simultaneity in � frame

So, one can conclude that simultaneity is a relative fact, since if a certain event is simultaneous on �

frame, it is not simultaneous on � frame and vice-versa. Minkowski diagram allows an easy

explanation of both time dilation and length contraction situations, as it is going to be demonstrated

next.

45

Spacetime Algebra

3.4.1 Time Dilation

3.4.1.1 Given an observer located on the - frame

Figure 3.12 – Time dilation given an observer on the � frame

From the previous figure it can be inferred that vector )*�� is equal to the sum of other two vectors, as

follows:

)*�� = )�� + �*�� (3.78)

Where each of the vectors can be described by their respective components, regarding the existing

axis on the Minkowski diagram:

)*�� = ��AHA (3.79)

)�� = ��@A (3.80)

�*�� = ð�@� (3.81)

Replacing equations (3.79) to (3.81) on (3.78) yields

��A HA = �� @A + ð� @� (3.82)

Multiplying the previous equation by @A gives

��A HA @A = �� @A @A + ð� @� @A (3.83)

@A @A is equal to 1, since the axis are collinear and @� @A is equal to 0 because they are perpendicular

among each other. The other geometric product will have to be determined. Using the first equation on

(3.34), the following relation is obtained

HA @A = γ @A @A + γβ @� @A (3.84)

Using the multiplication property present on equation (2.10), it can be deduced:

HA @A = γ @ A @A + γβ @A @� (3.85)

= γ × 1 + γβ × 0 = γ (3.86)

46


So, replacing the result on the previous equation on (3.83), it can be obtained

�Aγ = � (3.87)

This yields on the final equation

γ�A = � (3.88)

Replacing equation (3.33) on (3.88), the following equation is obtained

� â�/�Ò = � (3.89)

3.4.1.2 Given an observer located on the -- frame

Figure 3.13 – Time dilation given an observer on the �� frame


follows:

)*�� = )�� + �*�� (3.90)



)*�� = ��HA (3.91)

)�� = ��A@A (3.92)

�*�� = ð�H� (3.93)


�� HA = ��A @A + ð� H� (3.94)

Multiplying the previous equation by HA gives

�� HA HA = ��A @A HA + ð� H� HA (3.95)

HAHA is equal to 1, since the axis are collinear and H�HA is equal to 0 because they are perpendicular

47

Spacetime Algebra

among each other. The other geometric product will have to be determined. Once again the first

equation on (3.34) will be used:

@A HA = @A γ @A + @A γβ @� (3.96)


@A HA = γ @A @A + γβ @A@ � (3.97)

= γ × 1 + γβ × 0 = γ (3.98)

So, replacing the result on the previous equation on (3.95), it can be obtained

�� = ��Aγ + 0 (3.99)

Which yields on the final equation

γ�A = � (3.100)


� â�/�Ò = � (3.101)

As expected, deriving the previous equation gives the same as equation (3.89).These equations mean

that time is not absolute for observers on different frames. Given an observer located on the � frame,

time urges faster on that frame comparing to another observer who is standing on frame � , with a

certain speed. � is the time associated to the moving frame, while �A indicates the time running on a

frame which has no speed.

3.4.2 Length Contraction

3.4.2.1 Given an observer located on the -- frame

Figure 3.14 – Length contraction given an observer on the � frame

48



follows:

)*�� = )�� + �*�� (3.102)



)*�� = aH� (3.103)

)�� = aA@� (3.104)

�*�� = �a@A (3.105)


a H� = aA @� + �a @A (3.106)

Multiplying the previous equation by @� results

a H� @� = aA @� @� + �a @A @� (3.107)

@� @� is equal to -1, since the axis are collinear, but is a negative indefinite metric, and @A @� is equal to

0 because they are perpendicular among each other. The other geometric product will have to be

determined. Once again the second equation on (3.34) will be used:

H� @� = γβ @A @� + γ@ � @� (3.108)

Using the multiplication property present on equation (2.10), it can be deduced that

= γβ × 0 + γ × 0−11 = −γ (3.109)

So, replacing the result of the previous equation on (3.107), it can be obtained

−γa = −aA + 0 (3.110)


� . = a (3.111)


aAâ1 − �� = a (3.112)

49

Spacetime Algebra

3.4.2.2 Given an observer located on the - frame

Figure 3.15 – Length contraction given an observer on the � frame


follows:

)*�� = )�� + �*�� (3.113)



)*�� = aAH� (3.114)

)�� = a@� (3.115)

�*�� = �aHA (3.116)


aAH� = a@� + �aHA (3.117)

Multiplying the previous equation by H�, which can be described by the second equality present on the

set of equations (3.34):

aA H� H� = a @� H� + �a HA H� (3.118)

H� H� is equal to -1, since the axis are collinear, but is a negative indefinite metric, and HA H� is equal to 0

because they are perpendicular among each other. The other geometric product will have to be

determined. Once again the second equation on (3.34) will be used:

@� H� = @� γβ @A + @� γ @� (3.119)


@� H� = γβ @� @A + γ @� @� (3.120)

50


= γβ × 0 + γ × 0−11 = −γ (3.121)

So, replacing the result of the previous equation on (3.96), it can be obtained

−aA = −γa + 0 (3.122)


� . = a (3.123)


aAâ1 − �� = a (3.124)

The previous equation means that length is not absolute for observers on different frames. For

example, a certain observer standing on � frame is watching a train move. If the train has null speed, it

will be observed as having its real size. On the other hand, if the train is moving, the observer will

perceive it’s length as being narrower than the initial situation. As the train’s speed increases, the

observer will see it as getting smaller. The spacetime model does not accept the fact that time and

space are absolute. Instead of that, this model has made the light cone absolute, which means the

only absolute parameter if the speed of light, since it is more fundamental than time and space.

3.5 Relativistic Velocity Addition for Collinear Vectors

According to Newton, the addition of velocities is additive

� + � = 2� (3.125)

Where � represents the speed of light. This was contradicted by James Clerk Maxwell, according to

equation (3.126). Considering two vectors, ð� and ð�which are collinear, as represented on the

following figure,

Figure 3.16 – Collinear velocity vectors

it is possible to attain the relativistic addition of velocities of two particles, given by the next equation.

ð = 'Ô''Ò�'/ÔÓ/Ò0Ò (3.126)

If both particles’ velocities are equal to the speed of light, then the addition of both equals to � as well,

contrarily to Newton’s theory. This demonstration requires the use of a previously studied operation:

the boost, represented on equation (3.56): The special feature concerning the derivation is the fact

51

Spacetime Algebra

that it will not be using Einstein’s second postulate, which states that the velocity of transmission of

light in vacuum has to be considered equal to � for all inertial frames (non-accelerated) This means

that the framework of special relativity does not depend on electromagnetism. [4]

� ↦ G = exp0� @ç�1 � (3.127)

Where G is a vector defining a certain event enclosed on an inertial frame �, which possesses a proper

velocity given by G = � @�. Parameters M, � and � are directly related to each other

M� = cosh �� = �1�/�2Ò , V = 1,2 (3.128)

Particle ) has a speed relative to G given by

ð� = M�0� @� + ð1 (3.129)

While particle * has a speed relative to ð� given by

ð� = M�0� @� + ð�1 (3.130)

The two previous equations can be written on a form of a boost

�ð� = exp0�� @ç�1 ð ð� = exp0�� @ç�1 ð� ¢ (3.131)

Therefore one can relate particle *’s speed to the inertial frame, which is equal to

ð� = exp0�� @ç�1 exp0�� @ç�1 ð = exp0� @ç�1 ð (3.132)

Given that ð, ð�and ð� are all in the same direction, it can be written

exp{0�� + ��1 @ç�} ð = exp0� @ç�1 ð (3.133)

So, one can deduce the following relation

� = �� + �� (3.134)

Applying the hyperbolic cosine to the equation

cosh � = cosh 0�� + ��1 (3.135)

And the hyperbolic tangent as well, which gives the relative velocity each observer measures

tanh � = tanh 0�� + ��1 = ÈÄ�¼ (Ô'ÈÄ�¼ (Ò �'ÈÄ�¼ (Ô ÈÄ�¼ (Ò (3.136)

According to the center equation on (3.27), it is possible to re-write the previous equation

� = �Ô'�Ò�'�Ô�Ò (3.137)

Finally, applying the equality which relates the velocity of a certain particle to the speed of light

� = '� , �� = '2� , V = 1, 2 (3.138)

It is obtained

52


'�/Ô0 '/Ò0�'/Ô0 /Ò0 (3.139)

One may conclude that the relativistic velocity addition concerning exclusively collinear vectors is

given by equation (3.126), as desired. It can be concluded that addition of velocities in spacetime

consists on a generalized rotation enclosed in a hyperbolic space, which differs from Newton’s theory,

who stated it consisted on a mere vector addition of velocities. It is also going to be proven that the

best framework for velocity addition is �ℓ�,� and not �ℓ� algebra. In order to do so, velocity addition for

two particles in �ℓ� algebra is given by [5]

� = �Ô'�Ò�/�Ô�Ò (3.140)

On the next figure, the behavior of � in order to �� and �� is given, where �� is equal to ��. According to

this addition, there doesn’t exist a maximum boundary for a certain particle’s velocity, since � tends to

infinite when �� tends to the inverse of ��, in this case, 2. On the other hand, the addition of two

positive velocities �� > 0, �� > 0 results on a negative velocity � < 0 when �� > ��Ò, which is totally

unacceptable on a physical point of view.

Figure 3.17 - �ℓ� velocity addition

The only way to surpass this contradiction is to question Euclidean relativity and to grasp the idea that

there in fact a maximum limit for a particle’s velocity. In order for that limit to exist, one must adopt a

non-euclidean metric, instead of the former one, presented on equation (3.140). Equation (3.137)

gives the addition of velocities for two certain particles, where the same consideration regarding �� is

taken, as it is equal to ��. Next figure presents the variation of �.

53

Spacetime Algebra

Figure 3.18 - �ℓ�,� velocity addition

It may be concluded that there is indeed a maximum velocity for particles, which is equal to �, as

referred previously. Having derived the Lorentz transformation without Einstein’s second postulate,

which states that the speed of light is the same for all inertial observers, one may affirm that special

relativity is independent from electromagnetism, rendering that postulate redundant. Another

particularity regarding the boosts may be concluded – they do not form a group. This is supported on

the idea that if two successive boosts are non-collinear, then the combined transformation is the

composite of a boost and a rotation.[6][7]

3.6 Doppler Shift

STA is very effective since it can analyze applications with ease, compared to other algebras, such as

the three-dimensional space. There must be a link between time and space in order to approach

certain problems, as studied previously. General Doppler shift is another classic example – for one to

treat this matter correctly, it must be done using STA (it properly connects space to a certain observer

with a specific proper time), since transverse Doppler shift is without a doubt, an exclusive relativistic

effect. Other approaches could be taken, as for instance, the dyadic notation or Gibbs’ vector analysis

in ordinary three-dimensional space. These two alternatives lack the flexibility of STA, since the cross

product of vectors, which is the foundation of those notations, can only be defined in three or seven

dimensions, which is not useful at all. Besides that fact, the cross product is not associative. The goal

is to calculate the Doppler shift defined by

1 + � = 3435 (3.141)

Everyone is familiar with the Doppler shift of sound waves. If we stand by the tracks and listen to a

54


train pass, the sound of its horn will be shifted to higher pitch as the train approaches, then abruptly to

lower pitch as it passes by us and recedes. Doppler shift can also be used to make a spectral analysis

of the sun’s light, for instance. To say that an image is blueshifted (where a blueshift occurs) means

that an observer is looking at that part of the sun that is moving towards him, or the light is

compressed to shorter wavelengths, so the frequency is increasing. Likewise, the opposite can be

said about a red image, where the opposite takes place, also called as a redshift. In a more general

point of view, the Doppler shift is defined by a source which emits photons with a certain wave vector,

while a moving observer watches the photon with the same wave vector. An event in �ℓ�,� algebra is

characterized by y = 0�.1 @A + zJ ∈ ℝ�,�, with zJ = � @� + � @� + � @� and the wave vector as

^ = ý3�þ @A + ]�J ∈ ℝ�,� (3.142)

Wave vector represents the tangent vector to the trajectory taken by the photon, where ]�J = ]� @� +]� @� + ]� @�. So, for a plane wave propagation of the given by exp[−V0^ ∙ y1] one may calculate the

inner product operation, represented next

^ ∙ y = �]�0@� ∙ @�1 + �]�0@� ∙ @�1 + �]�0@� ∙ @�1 = ]�J ∙ zJ (3.143)

Given an electromagnetic wave propagating in a stationary medium (which can result from the

interference of two waves travelling in opposite directions) described by refractive index fA, one has

]�J = fA ý3�þ �JA (3.144)

So, it is possible to re-write equation (3.142)

^ = ý3�þ @A + fA ý3�þ �JA = 3� 0@A + fA�JA1 (3.145)

�JA is a unit vector defined on each of the space axis. It is also known that 0�JA1� = −1 since a negative

indefinite metric is adopted. One may conclude that the wave vector is a lightlike vector, since ^ø = 0

From the former statement and considering the standard orthonormal basis set ,6 = {@A, @�, @�, @�}, one obtains @�ø = 1 and @çø = @øø = @÷ø = −1. Considering that the emitting photon is in vacuum, the

refractive index is equal to 1, therefore equation (3.145) outcomes on the following one

^ = 34� 0@A + fA�Jk1 (3.146)

Where the frequency of the emitter is given by parameter

�k = ^ ∙ G (3.147)

G = � @A is the emitter’s velocity. The same photon is viewed from another frame defined by the

(orthonormal) basis set ,′6 = {HA, H�, H�, H�} with H�ø = 1 and Hçø = Høø = H÷ø = −1. This photon, although

being the same as the one considered on equation (3.147), is received with a different frequency, from

the one it possessed when it was emitted, which yields on an equation for the wave vector on the

receiver’s viewpoint.

^ = 35� 0HA + fA�Jj1 (3.148)

55

Spacetime Algebra

Where the frequency of the receiver is given by parameter

�j = ^ ∙ � (3.149)

� = � HA is the receiver’s velocity. Reminding that the relation between two frames (and its basis sets)

is given by a boost, it is possible to establish HA in order to @A. To do so, one will have to define a unit

bivector +Dj = �J@A, which square is equal to 1, since it’s hyperbolic

HA = exp0� +D j1@A (3.150)

Replacing the bivector on the previous equation one obtains

HA = cosh �@A + sinh ��J (3.151)

Usually, due to the rotation, unit vector �Jj might not be aligned with the relative velocity U�J. In that case,

there is another equation which describes the relative velocity, represented next

U�J = ��J (3.152)

With

�J ∙ �Jk = − cos u (3.153)

So, in order to obtain the receiving photon from the emitting one, a rotation and a boost will occur. As

explained earlier, the rotation will change vector �J, while the same will not happen with the boost,

since the unit vector at hand will modify. The addressed transformation is shown on the following

figure

Figure 3.19 – Addressed problem on Doppler shift from a certain frame

Now, rotation ⋃ is going to be applied on unit vector �Jk in order to obtain

�J = 7�Jk7ö (3.154)

Considering 7 = exp0u+D 2⁄ 1 and 7ö = exp0− u+D 2⁄ 1 then

7� = exp0u+D1 = cos u + +D sin u. (3.155)

From equation (3.154) one concludes that

�J = 7��Jk = −�J�Jk�Jk (3.156)

This results on

7� = −�J�Jk (3.157)

Comparing equations (3.156) and (3.157), it can be deduced that �J ∧ �Jk = sin u+D and +D ø = −1.

56


Consequently

+D = 7+D@7ö = �J@� (3.158)

Concerning the boost L0@�1, it allows to obtain frame H� from @�, according to the next equation

H� = a@�LÚ = L�@� (3.159)

Where L = exp0(+Dy� 1, L = exp0− (+Dy� 1 and

L� = exp0� +D y1 = cosh � + +Dy sinh � (3.160)

From equation (3.159) one concludes that

H� @� = L� @� @� (3.161)

Knowing that @� @� = @�ø = 1

L� = H� @� (3.162)

Comparing equations (3.160) and (3.162), it can be deduced that H� ∙ @� = cosh �, H� ∧ @� = +Dy sinh �

and finally � = tanh �. With the boost operation, bivector +Dy remains unchanged, as proven next

µL0�J1 = L �J LÚ = �Jj L9+Dy: = L +Dy LÚ = +Dy ¢ (3.163)

On the other hand, relative velocity G = � @A suffers a change concerning its frame, while its speed is

the same and equal to �. So

� = � HA (3.164)

Replacing equation (3.151) on the previous one, the following is obtained

� = � HA = c0cosh �@A + sinh ��J1 (3.165)

Bearing in mind that cosh � = M and sinh � = M�

� = � HA = c0@AM + M��J1 = �M0@A + ��J1 (3.166)

Now it is possible to obtain the ratio which defines the Doppler shift using equations (3.146), (3.149)

and (3.166)

�j = ^ ∙ � = � ∙ ^ = ��k �; 0@A + �Jk1�[�M0@A + ��J1] (3.167)

Which results on �j �k; = �� M + �� M��Jk�J = M + M��Jk�J (3.168)

Using the property on equation (3.153) �j �k; = M − M� cos u = M01 − � cos u1 (3.169)

Replacing equation (3.33) on the previous one, the general formula for the Doppler shift is obtained �j �k; = �/� ½Á<â�/�Ò (3.170)

57

Spacetime Algebra

Two Doppler effects may be differentiated: the longitudinal effect, which occurs when angle u = 0 and

the transverse one, which happens when u = 2; . When considering u = 0, equation (3.170) yields

�j �k; = �/�â�/�Ò = �/�â0�/�10�'�1 = 1 0�/�1Ò0�/�10�'�1 = 10�/�10�'�1 (3.171)

When �j < �k, a redshift is obtained, the frequency is reducing, which happens for example, when the

train has passed by an observer, so it is receding. On the other hand, the blueshift happens when �j > �k, so the frequency is increasing, thus the train is approaching the observer. Another example

may be given. For example, a moving observer increases his speed tending to the speed of light.

While he’s gaining speed and looks behind, it will seem as the image is turning red, while if he looks

ahead it will have a blueish color, hence calling redshift and blueshift respectively.[7]

3.7 Conclusions

Spacetime algebra is non-Euclidean, while possessing one time and three spatial vectors. It is known

as �ℓ�,� algebra. An event in this algebra is characterized by vector y and may be lightlike, spacelike

or timelike, whether y� = 0, y� < 0 or y� > 0 correspondingly. As spacetime algebra is supported on

Minkowski spacetime, which is closely related to special relativity, then it is imposed that no particle

may travel at a velocity greater than �, assuming that causality is preserved. This means that if a

certain even occurs before another event, then it will always take place on that order, no matter what

frame is considered.

Several possible applications were studied, such as Doppler shift, relativistic velocity addition, time

dilation and length contraction. Using these effects, involving boosts or active Lorentz transformation,

one may differentiate that action from a passive Lorentz transformation. The boost is a transformation

which takes a certain vector {@�, @ç} and transforms it to {H�, Hç}. It is a mere transformation, thus it

does not associate to a particular frame and hence an observer is not involved in the process. On the

other hand, passive Lorentz transformation is related to a user interpretation – passive interpretation.

A certain event is processing, and two observers who are on different frames see the same event but

in a different way, in respect to time and space. The boost is an analogous operation to the rotation

although its transformation is not linear. Appendix B shows in detail the result of the referred

operation. The application is expressed by � ↦ G = exp 0�@ç�1�. It is the basis of spacetime algebra,

since all applications involve the boost. It represents a great advantage comparing to Euclidean

algebra, since it transforms a non-linear application to a single multiplication, taking particular

importance in electromagnetism. It allows to put aside matrices and tensors, when performing

computational geometry.

Concerning time dilation and space contraction, the correction factor M = �â�/�Ò allows the observers

(on certain frames) to agree on the apparent time and space divergence. This application is especially

important on GPS devices; if this correction factor would not exist, it would render the system useless,

58


since its calculations would be totally inaccurate.

One may conclude that simultaneity is a relative fact, since if a certain event is simultaneous on a

certain � frame, then it is not simultaneous on another relativistic � frame and vice-versa.

When performing a relativistic velocity addition for collinear vectors, one may conclude the appropriate

equations that represents is given by '� = /Ô0 '/Ò0�'/Ô0 /Ò0 , opposite to the Euclidean algebra, which states that

it is equal to '� = /Ô0 '/Ò0�//Ô0 /Ò0 . The presented graphics show and support this theory, since there is a

velocity limit given by �. This derivation was complete without employing Einstein’s second postulate,

which states that the velocity on any inertial frame (non-accelerated) is limited to �.

Regarding Doppler shift, one may conclude its general expression is given by �j �k; = �/� ½Á<â�/�Ò . Two

distinct cases derive from this – the longitudinal 0u = 01 and transverse 9u = 2; : effects. Graphics in

Appendix G show the function of the equation on the longitudinal effect for different speeds, where � = ð�.


[1] http://prola.aps.org/pdf/PRD/v2/i2/p263_1

[2] http://www.scar.utoronto.ca/~pat/fun/RELATVTY/PDF/RLTVTY1.pdf

[3] G. L. Naber, The Geometry of Minkowski Spacetime An Introduction to the Mathematics of the

Special Theory of Relativity; Springer, 1992

[4] C. R. Paiva, “Passive Lorentz Transformations with Spacetime Algebra”; Departamento de

Engenharia Electrotécnica e de Computadores, Instituto Superior Técnico

[5] C. R. Paiva, "Lição de Síntese", Departamento de Engenharia Electrotécnica e de Computadores,


[6] C. R. Paiva and M. A. Ribeiro, “Generalized relativistic velocity addition with spacetime algebra”;

Departamento de Engenharia Electrotécnica e de Computadores, Instituto Superior Técnico

[7] C. R. Paiva and M. A. Ribeiro, “Doppler Shift from a Composition of Boosts with Thomas Rotation:

A Spacetime Algebra Approach”; Departamento de Engenharia Electrotécnica e de Computadores,


59

Chapter 4

Electrodynamic and Relativistic

Effects 4 Electrodynamic and Relativistic Effects

This chapter’s purpose is to address several applications regarding electromagnetism and relativistic

effects using the studied concepts on the former chapters. An explanation of Maxwell’s equations shall

be given, in order to fully understand the applications at hand.

60


“The possibility that mathematical tools used today were invented

to solve problems in the past and might not be well

suited for current problems is never considered.”

D. Hestenes

4.1 Introduction

As proven, GA possesses innumerous techniques for studying problems in electrodynamics and

electromagnetism. This chapter proposes several applications regarding the matters referred above,

which will emphasize the power of GA compared to traditional algebras. On a first approach, Maxwell

equations shall be presented whether on regular three-dimensional space or spacetime. As one will

see later on, this is a great quality of STA, since it will shed light on a somewhat dark matter. It will

simplify the study of how electromagnetic fields appear to observers on different frames. Another big

improvement of this treatment, is a more recent and compact formulation on Maxwell’s equations.

Geometric product and derivative vector will permit to transform Maxwell’s four equations into an

astonishing individual one. Relative and absolute vectors shall be distinguished, which will simplify the

formulation of the equations.

4.2 Maxwell Equations in � algebra

Maxwell's equations represent one of the most elegant and concise ways to state the fundamentals of

electricity and magnetism. From them one can develop most of the working relationships in the field.

Because of their concise statement, they embody a high level of mathematical sophistication and are

therefore not generally introduced in an introductory treatment of the subject, except perhaps as

summary relationships. In order to introduce the Maxwell equations, working on either �ℓ� or �ℓ�,�

algebra, one must first define ∇, also known as nabla or yet del operator. It is present on all Maxwell

equations, although it possesses a particular action on each one of them. The following equation

characterizes the referred operator

∇ = @� ==�Ô + @� ==�Ò + @� ==�> ∈ ℝ0,3 (4.1)

Del is a vector differential operator used for mathematical notation, and makes equation easier to

interpret. It may represent a gradient, which describes the slope, divergence, illustrating the degree to

which something converges or diverges, and finally the curl, which expresses the rotational motion at

points in a fluid. Del may be seen as the derivative in a multi-dimensional space. If that space

61

Electrodynamic and Relativistic Effects

possesses solely one axis, then the del operator will act as a standard derivative tool. There are four

Maxwell equations, which may be represented on the differential or integral form, and each one of

those forms may be formulated in terms of free charge and current or total charge and current. The

equations may be catalogued in pairs – the first is the Faraday group, which includes Maxwell-

Faraday’s equation, also known as the Faraday’s law of induction and Gauss’s law for magnetism,

present on the following couple of equations, respectively. On this thesis the differential form with

respect to free charge and current shall be taken

µ ∇ × ? = − =+=Y ∇ ∙ + = 0 ¢ (4.2)

? ∈ ℝ÷ represents the intensity of the electric field- it is defined as the electric force per unit charge.

The direction of the field is taken to be the direction of the force it would exert on a positive test

charge. The electric field is radially outward from a positive charge and radially in toward a negative

point charge. + ∈ ℝ÷ represents the intensity of the magnetic field. These two fields are denominated

as strength values, since characterize how strong the fields are. The second pair of equations is

identified as the Maxwell group – both Ampère’s circuital law with Maxwell’s correction and Gauss’s

law are present, as the following set of equations show

µ ∇ × N = X + =7=Y∇ ∙ 7 = −� ¢ (4.3)

7 ∈ ℝ÷ represents the electric displacement, while N ∈ ℝ÷ represents the magnetic field strength.

These two vectors reflect a certain material’s excitation amount, consequently are known as excitation

values. It is possible to describe 7 and N in order to ? and +, although using another two vectors as

well: polarization p ∈ ℝ÷ and magnetization c ∈ ℝ÷. The indicated equations are represented next

µ7 = BA ? + p N = �μ + − c¢ (4.4)

The following relations are valid as well

ê�[ = −∇ ∙ pX[ = =p=È XZ = ∇ × c¢ (4.5)

Where the total electric charge density and current are given

µ�Y = � + �[ ∈ ℝ XY = X + X[ + XZ ∈ ℝ� ¢ (4.6)

Finally one may attain the light speed on vacuum

� = 1 âBAμA? (4.7)

62


4.2.1 Maxwell-Boffi Equations

This formulation states that a media is a mere collection of currents and charges on vacuum. Opposite

to Maxwell’s equations, the auxiliary fields 7 and N are not present. Therefore it is considered a

fundamental and reductionist viewpoint, since one intends to explain the most using the fewest

principles possible [1]

This new formulation is very attractive since both curl and divergence of field quantities ? and +,

respectively, are specified. According to the Helmholtz theorem, a field vector requires its curl and

divergence to be given, in order to be fully specified.[2] One may separate the four equations

regarding magnetic flux conservation and charge-current conservation, according to equations (4.8)

and (4.9), respectively

µ ∇ × ? = − =+=Y ∇ ∙ + = 0 ¢ (4.8)

�∇ × + = μA XY + � Ò =?=Y∇ ∙ ? = @AB ¢ (4.9)

The next goal is to write these equations as a single one. In order to do so, one must replace the next

couple of equations on the previous ones

�∇ ∧ ? = 0∇ × ?1 @�� ∇ 0+ @��1 = −∇ × +¢ (4.10)

So, the following set of equations is obtained. Each equation has its own grade, represented on the

left side

êê 0 ↦ ∇ ∙ ý� ?þ = ηA�Y 1 ↦ � ==È ý� ?þ + ∇ 0+ 1 = −μA XY 2 ↦ � ==È 0+ @��1 + ∇ ∧ ý� ?þ = 0 3 ↦ ∇ ∧ 0+ @��1 = 0

¢ (4.11)

Where the vacuum impedance is equal to

ηA = 1μ B (4.12)

Therefore the following equations are obtained

�ηAc = �B C = μA ¢ (4.13)

Replacing the following equation set (4.11) and performing a graded sum, one obtains equation (4.15)

� ∇ ? = ∇ ∙ ? + ∇ ∧ ? ∇ 0+ @��1 = ∇ 0+ @��1 + ∇ ∧ 0+ @��1 ∇ ý� ? + + @��þ = � ∇ ? + ∇ 0+ @��1 ¢ (4.14)

63


ý� ==Y + ∇þ ý� ? + + @��þ = ηA ý�Y − � XYþ (4.15)

This equation is the most compressed form one may write using �ℓ� algebra, which is not the case in

STA. With this new notation, it is possible to re-write Maxwell equations, as follows

µ ==Y 0+ @��1 + ∇ ∧ ? = 0∇ ∙ + = 0 ¢ (4.16)

µ=7=È + ∇ 0N @��1 = −X∇ ∙ 7 = � ¢ (4.17)

4.3 Maxwell equations in �,� algebra

In order to write Maxwell equations in �ℓ�,� algebra, one must start from the equations in Gibbs’

algebra. To integrate the equations into STA, the usual ℝ� vectors, which are the same as ℝ�,A, now

must be considered on ℝA,� space. Considering the usual negative indefinite metric on �ℓ�,� algebra, @�ø = 1 and @çø = @øø = @÷ø = −1. Given a certain event, described by y = 0�.1 @A + zJ ∈ ℝ�,� and zJ =ñ�@� + ñ�@� + ñ�@� ∈ ℝA,� which meets the contraction rule 0ñJ1� = −|ñJ|� = −0a�� + a�� + a��1, as

analyzed previously. Hence, one may define the ∇ operator as equation (4.1), though the

D’Alembertian is ∇� = − D =Ò=EÔÒ + ==EÒÒ + ==E>Ò F ∈ ℝ [3]

Maxwell’s four equations are the same as presented on equations (4.2) and (4.3) and the field vectors

remain the same as well

êê G�J = G�@� + G�@� + G�@� ∈ ℝA,� *�J = *�@� + *�@� + *�@� ∈ ℝA,� H��J = H�@� + H�@� + H�@� ∈ ℝA,� I��J = I�@� + I�@� + I�@� ∈ ℝA,� � ∈ ℝ JJ = J�@� + J�@� + J�@� ∈ ℝA,�

¢ (4.18)

In order to characterize Maxwell equations in �ℓ�,� algebra, one must introduce their relative vectors

(hyperbolic bivectors)

êê? = G�J@A ∈ « ℝ�,��+ = *�J@A ∈ « ℝ�,��7 = H��J@A ∈ « ℝ�,��N = I��J@A ∈ « ℝ�,��

¢ (4.19)

Besides that, the Dirac operator and electric charge – current densities are also necessary

�X = �@A + �� JJ = �@A + �� {J�@� + J�@� + J�@�} ∈ ℝ�,� ∂ = �� @A ==Y + @� ==�Ô + @� ==�Ò + @� ==�> ∈ ℝ�,� ¢ (4.20)

64


Now it is possible to define the essential bivectors of the electromagnetic field, which are absolute

vectors – Faraday and Maxwell bivectors

�F = �� ? + S + K = 7 + �� S N¢ (4.21)

Considering the following definition for each relative vector

�? = G�J @A = G� @�A + G� @�A + G� @�A S ? = ? S = −G� @�� − G� @�� − G� @�� ¢ (4.22)

�+ = *�J @A = *� @�A + *� @�A + *� @�A S + = + S = −*� @�� − *� @�� − *� @�� ¢ (4.23)

�7 = H��J @A = H� @�A + H� @�A + H� @�A S 7 = 7 S = −H� @�� − H� @�� − H� @�� ¢ (4.24)

�N = I��J @A = I� @�A + I�@�A + I� @�A S N = N S = −I� @�� − I� @�� − I� @�� ¢ (4.25)

Replacing each relative vector on the absolute vectors, on equation set (4.21), the following is

obtained

�F = �� 0G� @�A + G� @�A + G� @�A1 − 0*� @�� + *� @�� + *� @��1 K = 0H� @�A + H� @�A + H� @�A 1 − �� 0I� @�� + I� @�� + I� @��1¢ (4.26)

Having introduced all parameters to define the Maxwell equations, it is now possible to write them for

STA. There are two equations, the homogeneous and non-homogeneous, corresponding to the

Faraday and Maxwell’s groups, respectively, as presented on equations (4.27) and (4.28)

∂ ∧ F = 0 → ä∇ × G�J = − =6�J=Y ∇ ∙ *�J = 0 ¢ (4.27)

An explanation of the equations regarding each group will be given next – starting with Faraday’s law

of induction. In a general way, this equation states that the induced EMF in any closed circuit is equal

to the time rate of change of the magnetic flux through the circuit. The line integral of the electric field

around a closed loop is equal to the negative of the rate of change of the magnetic flux through the

area enclosed by the loop. This line integral is equal to the generated voltage or EMF in the loop, so

Faraday's law is the basis for electric generators. It also forms the basis for inductors and

transformers. Gauss’s Law for magnetism states that the magnetic field + has divergence equal to

zero, in other words, that it is a solenoid vector field. It is equivalent to the statement that magnetic

monopoles do not exist. Rather than "magnetic charges", the basic entity for magnetism is the

magnetic dipole. The net magnetic flux out of any closed surface is zero. This amounts to a statement

about the sources of magnetic field. For a magnetic dipole, any closed surface the magnetic flux

directed inward toward the South Pole will equal the flux outward from the North Pole. The net flux will

always be zero for dipole sources. If there were a magnetic monopole source, this would give a non-

zero area integral. The divergence of a vector field is proportional to the point source density, so the

65


form of Gauss' law for magnetic fields is then a statement that there are no magnetic monopoles.

Addressing Maxwell’s group equations

K L = M → äN × I��J = JJ + O6�JOY N ∙ *�J = −� ¢ (4.28)

Ampere’s Law states that in the case of static electric field, the line integral of the magnetic field

around a closed loop is proportional to the electric current flowing through the loop. This is useful for

the calculation of magnetic field for simple geometries, while Gauss’s law for electricity affirms that the

electric flux out of any closed surface is proportional to the total charge enclosed within the surface.

The integral form of Gauss' Law finds application in calculating electric fields around charged objects.

In applying Gauss' law to the electric field of a point charge, one can show that it is consistent with

Coulomb's law. While the area integral of the electric field gives a measure of the net charge enclosed,

the divergence of the electric field gives a measure of the density of sources. It also has implications

for the conservation of charge. Having made an introduction to Maxwell’s equations, one may

differentiate absolute and relative vectors. Another goal which may be achieved is to reduce Maxwell’s

equations to a single one, which is only possible using STA. In vacuum, the constitutive relation is

given by

K = çC F (4.29)

One can write the geometric product as

∂ F = ∂ F − ∂ ∧ F (4.30)

As ∂ ∧ F = 0, one obtains

∂ F = ∂ F (4.31)

Knowing that ∂ K = X and using the constitutive relation in vacuum, one obtains

∂ çη0 F = ∂ çη0 F = X (4.32)

So, in �ℓ�,� algebra, both Maxwell equations may be written as a single one, presented next

∂ F = ηA X = ηA P�@A + �� 0J�@� + J�@� + J�@�1Q (4.33)

If the considered media does not have sources, then the right-side of the previous equation is equal to

0. One can distinguish relative and absolute vectors on Maxwell equations. Vectors ? and + are both

relative vectors, while F is not, even being composed by the two previous vectors. The same can be

verified for 7 and N, which are relative but K is an absolute vector. Consequently, vectors F and K

don´t depend on any observer, so they represent the electromagnetic field and can be designated as

covariant forms of the electromagnetic field. It is possible to reduce Maxwell’s equations to a sole one,

since in vacuum the existence of vector K is not required.

66


4.4 Moving Media and Vacuum Form Reduction

�ℓ�,� algebra is useful to study a great variety of problems. One of the classic electrodynamics

problems is moving media. Considering a simple isotropic medium, in �ℓ� algebra is defined by the

following constitutive relations.

µH��J = BABG�J*�J = �A�I��J¢ (4.34)

Those vectors can be written as relative vectors, concerning STA.

�7 = BAB ? + = �A� N¢ (4.35)

In order of the magnetic excitation N yields

µ7 = BAB ? N = �R R +¢ (4.36)

K = 7 + � S N = BAB ? + � �R R S + (4.37)

Considering

�ë� = BC ë� = �C R¢ (4.38)

K = 7 + 1 c; S N = ë� �; ? + ë�S + (4.39)

As G = � @�, represents the frame where the media is at rest, one obtains

FG = G/� F G = @A F @A (4.40)

= @A ý�� ? + S +þ @A (4.41)

= @A ý�� ? + S +þ @A (4.42)

This can be defined by the electric and magnetic field intensity relative vectors

= @A ý�� G�J @A + S *�J @Aþ @A (4.43)

= @A ý�� G�J @A + S *�J @Aþ @A (4.44)

= −@A @A ý�� G�J @A + S *�J @Aþ (4.45)

= @A �� G�J @A @A + @A S *�J @A @A (4.46)

= @A �� G�J + @A S *�J (4.47)

= − �� G�J @A − S @A *�J (4.48)

67


= − �� G�J @A + S *�J @A (4.49)

Concluding,

FG = − �� ? + S + (4.50)

This represents vector F seen by an observer on frame @A. For a certain linear combination of F with FG, comes

K = �� F + �ø FG (4.51)

So

K = �� ý�� ? + S +þ + �� ý− �� ? + S +þ (4.52)

Using the distributive property

�� 0�� − ��1 ? + 0�� + ��1 S + (4.53)

Comparing the previous equation with (4.39), the following is deduced

�ë� = �� − ��ë� = �� + �� ¢ (4.54)

�� = ë� + �� = ë� − �� ¢ (4.55)

Replacing the second equation on the first one

�� = ë� + ë� − �� (4.56)

This results on

2�� = ë� + ë� (4.57)

So

�� = �� 0ë� + ë�1 (4.58)

From equation (4.55), by replacing the first equation on the second one, the following is obtained

�� = ë� − ë� − �� (4.59)

2�� = ë� − ë� (4.60)

�� = �� 0ë� − ë�1 = − �� 0ë� − ë�1 = (4.61)

Replacing the set of equations (4.38) on (4.58) and (4.61) results

�� = �� ý BC + �C Rþ = ��C ýB + �Rþ �� = − �� ý BC − �C Rþ = − ��C ýB − �Rþ¢ (4.62)

Replacing the set of equations (4.62) on (4.51) yields

68


K = ��C ýB + �Rþ F − ��C ýB − �Rþ FG (4.63)

The previous equation represents the local form of the universal constitutive relation. It is possible to

simplify the previous equation, in terms of the hyperbolic functions and media’s refractive index fA as

well

êêcosh � = �� 1RB ýB + �Rþ = � Ò'�� sinh � = �� 1RB ýB − �Rþ = � Ò/�� ¢ (4.64)

These equations meet the following rules

�cosh� � − sinh� � = 1 cosh � + sinh � = fA = √B�¢ (4.65)

Considering

�S = 1R B 1RB = 1RB SA yT0F1 = FG = G/� F G¢ (4.66)

Equation (4.63) is expressed on a simpler way

K = �U 0cosh % F − sinh % FG1 (4.67)

= �U 0cosh % − yT sinh %1 F (4.68)

= �U exp0−% yT1 F (4.69)

This expression is very useful due to its simplicity – it gives the ability to transform immediately a

media which is located on its proper frame to the laboratory’s frame, where all particles displace with a

proper velocity equal to � with

� = exp0ζ VA1 G = W0G + U�J1 (4.70)

So, equation (4.69) may be re-written as follows

K = �U exp0−% yÃ1 F (4.71)

Where

yÃ0F1 = F� = �/� F � (4.72)

Combining with equation (4.70), one obtains

�/� = G/� exp0−ζ VA1 = ��Ò � = ��Ò exp0ζ VA1 G (4.73)

Hence

yÃ0F1 = F� = �/� F � = ��Ò � F � (4.74)

69


The local form of the constitutive relation represents a projection of the constitutive relation on a

particular observer, who is at rest regarding the media. In other words, it could be the media’s own

observer. It is possible to split the constitutive relation K = K0F1 into two constitutive relations H��J =H��J0G�J,*�J1 and I��J = H��J0G�J,*�J1, as presented on the equation set below

�H��J = �Ò�ÒR �0fA� − ��1 G�J + 0fA� − 119�J ∙ G�J: �J + �0fA� − 119�J × �:�

I��J = �Ò�R �0fA� − 119�J × G�J: + �01 − fA��1 *�J − �0fA� − 119�J ∙ �: �J� ¢ (4.75)

These constitutive relations show that the media at hand is bianisotropic, which means that both H��J and I��J depend, not only on the electric field intensity G�J, but also on magnetic field intensity *�J. According to these results it can be concluded that an isotropic media on its own frame is considered

as a bianisotropic medium on the rest frame. [4]

The following transformation helps grasp equation (4.71), which is the isotropic constitutive relation

�F = �U exp ýX� yÃþ F{ K = �U exp ý− X� yÃþ K{ ¢ → �F{ = �U exp ý− X� yÃþ F K{ = �U exp ýX� yÃþ K ¢ (4.76)

This results on

�U exp ý− X� yÃþ K{ = �U exp0−% yÃ1 �U exp ýX� yÃþ F′ (4.77)

And finally

K{ = �U F′ (4.78)

It may be concluded that with the presented transformation, a constitutive relation is obtained, which

has the same build as the constitutive relation concerning vacuum, therefore it is called VFR. With this

transformation, the Minkowski spacetime structure has been changed, since it now corresponds to a

fictitious spacetime, different from the original one. [5]

4.5 Relativistic Dynamics for a Particle

Another field where STA excels is the one of electrodynamics and kinematics, where it is used to

correct Einstein’s very well acknowledged mass-energy equivalence. But first one must describe the

present characters on this application. Defining the proper linear momentum of a certain particle with

mass m will be given by

q = d � (4.79)

Which is related to the particle’s total energy C. The mass-energy equivalence states the following

rA = Md� = C� (4.80)

70


Hence, combining both equations presented above, one may deduce

q = C� @A + rJ = 0Md�1 @A + MdU�J (4.81)

Therefore

CA = d�� → YC = MCArJ = MdU�J¢ (4.82)

Forcing the following equality

r = |rJ| → q� = 0d�1� = ýC�þ� − r� (4.83)

This yields

C = âCA� + 0�r1� (4.84)

CA is the particle’s proper energy, and one may attain the kinetic energy, represented by

\ = C − CA = 0M − 11CA → C = CA + \ = MCA (4.85)

The following figure represents the application’s geometry

Figure 4.1 – Geometric representation of particle’s relativistic dynamics

The equations that reproduce the problem’s geometry are presented next

êêsin u = � = #� sec u = � ½Á< = M = 1 â1 − ��? = CC ¢ (4.86)

It must be stressed that Newton’s mechanics limit - which happens when speed � reaches infinite, may

be attained from this definition of kinetic energy, as demonstrated next. From equation (4.85) one

obtains the expression for kinetic energy

71


\ = 0M − 110d��1 (4.87)

In order to achieve the proposed goal, one must start by determining 0M − 11

0M − 11 = 1 â1 − ��? − 1 (4.88)

This is described by

�√$∑ ��! Γ ýf + ��þ �� = �� + �\ �� + �]�\ �^ + ⋯Ï�Ð� (4.89)

Hence

\0U1 = �� dU� + �\ d #`�Ò + �]�\ d #a�` (4.90)

Newton’s mechanics is expressed by having � → ∞. Applying that condition on the former equation

one obtains

\ = �� dU� (4.91)

It is wrong to state that the inertia of energy is given by CA = d��, what it describes is the following

C = T0U1�� = M0U1d�� =

= d�� + [M0U1 − 1]0d��1 = CA + \ (4.92)

T0U1 = M0U1d is the inertia coefficient and the proper energy at rest is a kind of potential energy where CA = C0U = 01. Einstein’s equation is given by C = T0U1�� and not as formerly known. Einstein and

other physicists used an equation which regarded variable mass, a concept which is nowadays not

acceptable. Saying that a mass is moving doesn’t make sense, there is only an invariant mass, that

doesn’t depend on the particle’s rest state nor its movement. The mass is always measured on the

particle’s own frame, where it is at rest and that frame is constantly changing according to its position.

4.5.1 Lorentz Force

Another concept which was wrongly introduced it that of transverse and longitudinal mass.[6] STA is

useful to define Lorentz force, which is formed by space and time components, as represented next

H = d�e = t0F �1 (4.93)

A particle with a given mass d is being considered, as well as a certain electric charge t, on an

electromagnetic field described by Faraday’s bivector F. The particle’s proper velocity is

� = c HA = γ0G + U�J1 (4.94)

where G = c @A is an observer’s proper speed, which is located on the rest frame. Recalling Faraday’s

bivector, which is the first equation on the equation set (4.21), where ? is a hyperbolic bivector and S +

is an elliptical bivector. As usual, the pseudoscalar’s square - which on �ℓ�,� algebra is the

quadrivector - is equal to -1, and besides that, the trivector’s square is equal to 1, opposite to �ℓ�

72


algebra, where it’s symmetrical. Given these facts, the right contraction present on equation (4.93) will

be investigated. Replacing equations (4.21) and (4.94) on (4.93)

ý� G�J@A + *�J@��þ [γ0c @A + U�J1] (4.95)

Using the distributive property on the former equation, results

� G�J@A γc @A + � G�J@A γU�J + *�J@�� γc @A + *�J@�� γU�J (4.96)

Analyzing each parcel separately, yields

� G�J@A γc @A = γG�J@A @A (4.97)

Since it is a right contraction among two vectors, this operation is equivalent to a geometric product,

hence

γG�J@A@A = γG�J (4.98)

Now, analyzing the second parcel one must compare the previous equation to (2.118), which yields

ý� G�J@Aþ γU�J (4.99)

= − �γU�J ý� G�J ∧ @Aþ � (4.100)

Using the fundamental rule of left contraction, present on equation (2.119)

− �ýγU�J ⋅ � G�Jþ @A − 0γU�J ⋅ @A1 � G�J � = − ýγU�J ⋅ � G�Jþ @A = − . 9U�J ⋅ G�J:@A (4.101)

Studying the third parcel, one concludes it is equal to zero, therefore

*�J@�� γc @A = 0 (4.102)

Regarding the last equation, one may write

*�J@�� γU�J = −γ�U�J 9*�J@��: � = γU�J × *�J (4.103)

Based on the fact that ℝA,� is different than ℝ� = ℝ�,A ⊂ �ℓ�, one may still consider the sum

orthogonal, so ℝ�,� = ℝ�,A ⊥ ℝA,� ⊂ �ℓ�,�. That is the reason why equation (4102) is valid. Now,

expanding the wedge product, bearing in mind that U�J = U�@� and *�J = *�@�, yields [6]

γU�J × *�J = γ ä @�U�*� @� U�*�

@� U�*� ä (4.104)

Hence

= γ[0 U�*� − U�*�1@� + 0 U�*� − U�*�1@� + 0 U�*� − U�*�1@�] (4.105)

Replacing equations (4.98) and (4.101) to (4.103) on (4.96) yields

F � = γG�J − . 9U�J ⋅ G�J:@A + γU�J × *�J = γ9U�J × *�J + G�J: − . 9U�J ⋅ G�J:@A (4.106)

The particle’s relative velocity U is given by

73


U = |U�J| = cCc[ (4.107)

Then, from energy’s inertia theorem, on equation (4.84), one may obtain its derivative, given by

2C cCc[ = 2��r (4.108)

Therefore

U = cCc[ = �Ò[C (4.109)

The proper force which acts on the particle is equal to

H = cqcY = cqcY cYcd = M cqcY = M ý�� cCcY @A + IJ þ (4.110)

Where IJ = erJ e.; is the relative force. As q = d�e , one may obtain equation (4.93) with �e = �Ã�f. Having � = M 0� @A + U�J1, one obtains

H ∙ � = γ� ýcCcY + IJ ∙ U�Jþ = d0�e ∙ �1 (4.111)

Regarding that

�� = � ∙ � = �� (4.112)

Then

� ∙ �e + �e ∙ � = 0 → �e ∙ � = 0 (4.113)

Equation (4.111) is equal to zero, hence

− cCcY = IJ ∙ U�J (4.114)

Going back to equation (4.93)

H = M ý�� cCcY @A + IJ þ = t0F �1 = γ �q9U�J × *�J + G�J: − h 9U�J ∙ G�J:@A� (4.115)

This may be divided on two expressions – the first one doesn’t have any important meaning, while the

second one reflects Lorentz force on three dimensions.

äcCcY = − h 9U�J ∙ G�J:@AIJ = t9U�J × *�J + G�J: ¢ (4.116)

From the most important of the previous equations, one may write its inner product with the particle’s

relative velocity

IJ ∙ U�J = t9G�J ∙ U�J: = − cCcY (4.117)

This equation is valid for a pure force, where the particle’s proper mass doesn’t alter along the time, as

for example on an explosion. For a non-pure force, one would write

H = c0Z�1cY = d �e + d e �, de = cZecd (4.118)

74


H ∙ � = m0� ∙ �e 1 + de 0� ∙ �1 = �� cZcd (4.119)

However, given a sole electromagnetic force, one has cZcd = 0, thus H ∙ � = 0, then IJ ∙ U�J = − cCcY .

Bivector F expresses the intensity of the electromagnetic field, which impacts on the Lorentz force, as

the studied case – electromagnetic field on a charged particle. An electric charge t has and electric

field applied on it even when it is not in motion (electrostatic). When it is moving, the magnetic field

acts on the particle, joining the electric field, using its relative velocity. One may conclude that the

magnetic field only makes itself visible when there are electric charges moving. Therefore the

magnetic field is an electric field in and the relative motion. This implies that on magnetostatic, where

there is a magnetic field formed by a static magnet, the magnetic field must be somehow a

combination of the electric charges’ inner motion that constitute the magnet itself (on this case). That

movement results on Foucault (or Eddy) currents.

4.5.2 Energy-Momentum Operator

One may use STA to characterize and analyze the energy-momentum operator, defined by �:ℝ�,� →ℝ�,�: � ↦ � = �0�1 with

�0�1 = − ��µ 0F � F1 = ��µ 9F � FÚ: (4.120)

Regarding that Faraday’s bivector F is the first equation on set (4.21). Expanding that equation yields

F = �� G�J@A + *�J@�� (4.121)

�0@A1 = − ��μ 0F @A F1 = − ��μ ý�� G�J@A + *�J@��þ @A ý�� G�J@A + *�J@��þ (4.122)

= − ��μ ý�� G�J + *�J@��Aþ ý�� G�J@A + *�J@��þ (4.123)

= − ��μ ý�� G�J + *�J@��Aþ ý�� G�J@A + *�J@��þ (4.124)

= − ��μ ý�� G�J �� G�J@A + �� G�J*�J@�� + *�J@��A �� G�J@A + *�J@��A*�J@��þ (4.125)

= − ��μ ý ��Ò 9G�J:�@A + �� G�J*�J@�� − �� *�JG�J@��AA − *�J*�J@��A@��þ (4.126)

= − ��μ ý ��Ò 9G�J:�@A + �� G�J*�J@�� − �� *�JG�J@�� + *�J*�J@�� @Aþ (4.127)

Given that on �ℓ�,�, @�� = 1 results

− ��μ ý ��Ò 9G�J:�@A + �� G�J*�J@�� − �� *�JG�J@�� + 9*�J:�@Aþ (4.128)

Grouping the parcels according to their grade, one obtains

− ��μ P� ��Ò 9G�J:� + 9*�J:�� @A + �� 9G�J*�J − *�JG�J:@��Q (4.129)

From equation (2.31), one may collapse the second parcel of the previous equation to the following

75


ýB � iG�Ji� + ��μ i*�Ji�þ @A − � μ 9G�J ∧ *�J:@�� (4.130)

Concluding

�0@A1 = CD@A + �� J (4.131)

Where the energy’s electromagnetic density is given by

CD = B � iG�Ji� + ��μ i*�Ji� (4.132)

The Poynting vector is defined as follows

�J = − �μ 9G�J ∧ *�J: = �μ 9G�J × *�J: (4.133)

The linear momentum’s volume density and flux density are given by

�LJ = ��Ò �J�LJ = �� J¢ (4.134)

4.6 Twins Paradox with Doppler Shift

Considering two observers, ) and *. The first one leaves the location next to observer * with a

constant velocity, given by ð = ��. After a while, he inverts the way he was moving and encounters

observer * once again, with ð = ��. Presuming that either observer ) and * send electromagnetic

signals to each other, on a uniform way regarding their respective frame. Let I be that emission’s

frequency, on this case it is equal to the number of signals each one sends. According to observer *,

observer ) takes a total time of � = 2a/ð to make the trip, where a is the maximum distance where

observer ) and * part from each other. Using the Doppler effect, it will be shown that from the point of

view of observer ), his journey lasts �A = 2a/0Mð1. So, �A = a/M and M is given on equation (3.33).

Opposite to expected, both observers will concur with this difference in measured times. Observer * is

located on frame S, hence his universe line corresponds to a segment defined by kℬm��, as

represented on figure 4.2. Regarding observer ), his universe line corresponds to the segment kHm��.

76


Figure 4.2 – Geometric interpretation of the problem

Considering the length and time represented on the former figure and yet given the segment kH�� =��A/2, using the Pythagorian theorem, one attains

�� ý� � þ� = �� ý��þ� − ð� ý��þ� (4.135)

Taking the point of view of observer *, observer )’s trip until a certain distance a took a total time of � = 2a/ð, then one may conclude the number of signals observer * sent to the other one, as follows

i0k1 = I� = �n�' (4.136)

Observer * is able to detect when observer ) inverts his direction, which happens on time instant .�.

Expressing that time in order of distances and velocities yields

.� = �' + �� = �' 01 + �1 (4.137)

Therefore, the number of signals received by * on this one way journey is given by .�I�, where the

relativistic Doppler effect is given by equation (3.171). So, replacing that equation on .�I� yields

.�I� = n�' 01 + �1101−�101+�1 = n�' â1 − �� (4.138)

The return journey, which can be comprehended as the point where both observers meet once again,

occurs on a certain time equal to .�

77


.� = �' − �� = �' 01 − �1 (4.139)

So, the total number of received signals by observer * while observer A returns is given by .�I�, where

I� = I10�'�10�/�1 (4.140)

Hence

.�I� = n�' 01 − �1101+�101−�1 = n�' â1 − �� (4.141)

The total number of signals received by * is equal to

i0j1 = .�I� + .�I� = �n�' â1 − �� = �n��' (4.142)

Observer * concludes that the in the other observer’s point of view, his journey took

�A = i0z1I = .1I1+.2I2I = 2aMð (4.143)

Changing to the perspective of observer ), the journey took a certain time equal to

�A = ��' (4.144)

So he sent a number of signals

iA0k1 = I�A = �n��' = i0j1 (4.145)

This concurs with the number of signals received by observer *. According to observer ), the time

instant when he inverted his direction is

. � = ��' (4.146)

During the period of time when left observer *, he received a number of signals given by

. �I� = n��'101−�101+�1 = n�' 01 − �1 (4.147)

The return journey took a total time of .�- and the total amount of signals received on this time period is

. � = ��' → .�- I� = n��'101+�101−�1 = n�' 01 + �1 (4.148)

On the whole journey, that number of received signals is

iA0j1 = . �I� + .�- I� = �n�' = i0k1 (4.149)

This result corresponds to the number of signals observer * sent. Observer ) will then conclude that

the total time that passed for observer * while he was on the journey is equal to

� = o 051n = . 1nÔ'. 2nÒn = ��' (4.150)

78


When the observers meet once again on point p, they agree that the total elapsed time is equal to �

for observer * and �A = �/M for observer ). Therefore this is a simple matter of time dilation, which

was studied on chapter 3, thus constitutes no inconsistency. Time dilation is a reciprocal effect,

although observer ) crosses two different frames while traveling, which results on the concluded time

difference for both observers. This problem is not feasible in real life since the acceleration suffered by

the travelling twin is infinite. This occurs because his velocity changes from � to −� in an infinitesimal

time fraction. Considering that observer * travels at a speed given by ð and changes to −ð on the

point where it loses reciprocity, the associated acceleration is given by lim∆Y→A '/0/'1∆Y = �'∆Y → ∞.

4.7 Conclusions

It may be asserted that Maxwell’s equation in STA ∂ F = ηA X constitutes the most innovative way to

be expressed. Not only for the simplicity, but also for the amount of information they apprehend. It is

possible to distinguish absolute 0F, K1 and relative vectors 0?, +, 7, N1: the absolute vectors are don’t

depend on the frame the observer is located on; on the other hand relative vectors depend on the

work frame.

As emphasized by David Hestenes, STA is the best framework to tackle electromagnetism in the

context of Special Relativity, [7] due to the inherent reduced complexity, as well as the coherent

results obtained. In fact, the two Maxwell equations in STA are reduced to a single one, as the

vacuum situation. This was able by applying VFR to a plane wave propagation in moving isotropic

media. From the point of view of the laboratory, it may be concluded that it is actually a nonreciprocal

bianisotropic media, which means its constitutive expressions H��J and I��J depend on G�J and *�J. Applying STA to a particle’s relativistic dynamics, allowed to correct some concepts, as the inertia of

energy – Einstein’s expression CA = d�� is incoherent, as well as an idea of a variable mass some

physicists had.

From the Lorentz force application, one may conclude that the expression on three dimensions IJ = t9U�J × *�J + G�J: is equivalent to the one on spacetime, given by H = d�e = t0F �1. As it is proven,

the inherent complexity on the spacetime situation is inferior, and besides that, it involves commutative

operations. On the other hand, the three dimensional equation - Gibbs cross product appears, which is

a major flaw, since it is not associative and depends on its metric, so turns out more complicated to

compute with.

A relation between the energy-momentum operator, Poynting vector and energy’s electromagnetic

density has been established as well, using STA. and one of the most famous paradoxes - Twins

paradox has been disclosed, using the Doppler shift. It was concluded that the apparent inconsistency

of results is acceptable, being explained by time dilation.

79



[1] C. R. Paiva, "Lição de Síntese", Departamento de Engenharia Electrotécnica e de Computadores,


[2] E. J. Rothwell and M. J. Cloud, Electromagnetics; CRC Press, New York, 2001

[3] D. Hestenes, Spacetime Physics with Geometric Algebra; Department of Physics and Astronomy,

Arizona State University, Tempe, Arizona

[4] C. R. Paiva and M. A. Ribeiro, “Relativistic optics in moving media with spacetime algebra”,

Departamento de Engenharia Electrotécnica e de Computadores, Instituto Superior Técnico

[5] M. A. Ribeiro and C. R. Paiva, “An equivalence principle for electromagnetics through Clifford’s

geometric algebras”; Departamento de Engenharia Electrotécnica e de Computadores, Instituto

Superior Técnico, 2008

[6] C. R. Paiva, “Óptica Relativista"; Departamento de Engenharia Electrotécnica e de Computadores,

Instituto Superior Técnico, March 2004



80


81

Chapter 5

Conclusions 5 Conclusions

The current chapter finalizes this thesis, summarizing conclusions regarding the analyzed applications

and pointing out aspects to be developed in future work.

82


"We may always depend on it that algebra which cannot be translated into good English and sound common sense, is bad algebra.”

W. K. Clifford

5.1 Conclusions and Discussion

Regarding GA, some main features may be concluded from the performed investigation: the ease one

has when applying rotations in spatial dimensions and boosts (or Lorentz transformations); the

invertibility of all Clifford objects, which is a great advantage when applying to computational systems

or simply performing calculations. Another important characteristic is the inherent geometric

interpretation one has, which makes this algebra so intuitive to understand and use, hence attractive

for beginners and motivating new users. These factors have been contributing for the success of GA

and its continuous evolution – as Maxwell stated concerning physical constants, they are constantly

evolving, are not simply discovered and some details are added. The same applies to GA and other

algebras. Initially, Grassmann’s algebra was exposed and did not get good reviews, only years later,

when he reformulated, it proved a success.

This work’s foundations rely on not only in GA, but on special relativity and Lorentz transformations as

well, so several conclusions shall be presented. Simultaneity is a crucial concept when dealing with

these matters: observers on different frames interpret events, each on a particular way. The correction

factor M = �â�/�Ò allows a temporal or spatial transformation from one frame to another. This explains

the different measurements acquired by observers on certain frames – time dilation and space

contraction are the effects which occur.

When performing a relativistic velocity addition for collinear vectors, one may conclude the appropriate

equations that represents is given by '� = /Ô0 '/Ò0�'/Ô0 /Ò0 . Graphics prove this theory, since there is a velocity

limit given by �. This derivation was complete without employing Einstein’s second postulate, which

states that the velocity on any inertial frame (non-accelerated) is limited to �.

Regarding Doppler shift, one may conclude its general expression is given by �j �k; = �/� ½Á<â�/�Ò . Two

distinct cases derive from this – the longitudinal 0u = 01 and transverse 9u = 2; : effects. Graphics in

Appendix G show the function.

∂ F = ηA X is the most condensed way to write Maxwell equations in STA. It is possible to distinguish

absolute 0F, K1 and relative vectors 0?, +, 7, N1: the absolute vectors are don’t depend on the frame the

observer is located on; on the other hand relative vectors depend on the work frame.

As emphasized by David Hestenes, STA is the best framework to tackle electromagnetism in the

context of Special Relativity [1] due to the inherent reduced complexity, as well as the coherent results

obtained. In fact, the two Maxwell equations in STA are reduced to a single one, as the vacuum

situation. This was able by applying VFR to a plane wave propagation in moving isotropic media. From

83

Conclusions

the point of view of the laboratory, it may be concluded that it is actually a nonreciprocal bianisotropic

media, which means its constitutive expressions H��J and I��J depend on G�J and *�J. Applying STA to a particle’s relativistic dynamics, allowed to correct some concepts, as the inertia of

energy – Einstein’s expression CA = d�� is incoherent, as well as an idea of a variable mass some

physicists had.

From the Lorentz force application, one may conclude that the expression on three dimensions IJ = t9U�J × *�J + G�J: is equivalent to the one on spacetime, given by H = d�e = t0F �1. As it is proven,

the inherent complexity on the spacetime situation is inferior, and besides that, it involves commutative

operations. On the other hand, the three dimensional equation - Gibbs cross product appears, which is

a major flaw, since it is not associative and depends on its metric, so turns out more complicated to

compute with.

A relation between the energy-momentum operator, Poynting vector and energy’s electromagnetic

density has been established as well, using STA. and one of the most famous paradoxes - Twins

paradox has been disclosed, using the Doppler shift. It was concluded that the apparent inconsistency

of results is acceptable, being explained by time dilation.

One must stress the importance of quaternions – due to its properties, it has allowed aerospace

scientists to save up to 25% in calculations for rocket computers, comparing to other methods, which

meant a great increase in terms of productivity and energy, given the obvious complexity of the

technology at hand.

5.2 Future Work and Developments

Since this work has approached several subjects, it has a great range of future master dissertations.

The topic of VFR allows future students to solve the given problem using Doppler shift instead of the

way used on this thesis. Regarding the same topic, the climax conclusion on this work is that the

Minkowski spacetime structure is altered and has the same form as vacuum, although the spacetime

is not the same as the former one. This may lead to future study, as Maxwell equations are also

modified, and it would be interesting to discovery their nature.

On chapter three, this work also analyzed the relativistic addition for collinear vectors. Therefore it

would be a very good suggestion to investigate further ahead on the non-collinear vectors, and obtain

a generalized equation for both collinear and non-collinear cases. Given the infinite applications the

presented themes may be applied to, it is certain that future work is guaranteed. Doppler shift from a

composition of boosts would also be a challenging topic to study in future master’s thesis.

An integration of GA with computer processing to implement specific applications is another possible

suggestion.

The twins paradox problem could be studied in a way that the traveling twin could bear the

84


accelerations on his journey. This could be done using circular or hyperbolic movement.

85

Appendix A

Proper and Relative Velocities 6 Proper and Relative Velocities

86


It is necessary to define the correspondence between an event represented by a vector y, opposed to

the four-dimensional Minkowski spacetime r. Considering an orthogonal basis , = {@A, @�, @�, @� }, where @A is the unit vector for the time axis, with

@A� = 1, @�� = @�� = @�� = −1 (A.1)

Such that

y = 0_.1 @A + zJ, zJ = � @� + � @� + � @� (A.2)

_ is a constant with dimension of velocity, which transforms unit of time into unit of length. It is possible

to define the same event on another frame ,- = {HA, Hç, H�, H� } with

HA� = 1, H�� = H�� = H�� = −1 (A.3)

Such that

y{ = 0_. 1 HA + zJ{, zJ{ = � H� + �� H� + � H� (A.4)

A rest point P in reference frame � is represented by the world line z|0.1 = 0_.1 @A + y|, where r| is a

constant vector and does not depend on proper time t. In frame � the referred rest point will be defined

by the world line z|0. 1 = 0_. 1HA + y |0. 1, where vector y | depends on proper time . . The proper velocity

of rest point P is defined by

ℎ = cjscY = cjscY cY cY (A.5)

This results on the following

ℎ = _ @A = M0_ HA + O1, O = cj scY (A.6)

Where . is the proper time of � frame and M = cY cY. Analogously, considering a certain rest point defined

by point m in frame � , the world line is z}0. 1 = 0_. 1 HA + y } for an observer located on that frame.

Therefore, for an observer on frame �, its world line is z}0.1 = 0_.1 @A + y}0.1 and the proper velocity

of rest point m will be given by

n = cjtcY = cjscY cYcY (A.7)

And hence

n = _ HA = M 0_ @A + l1, l = cytcY (A.8)

Where M = cYcY .. According to .the principle of relativity, the proper time on frames � and � should be

valid, which means M = γ. Under those circumstances equations (A.6) and (A.8) result on the following

pair of equations

�ℎ = _ @A = M0U + O1n = _ HA = M0� + l1 ¢ (A.9)

On the previous couple of equations, one can differentiate proper velocities, represented by ℎ and n,

87

Proper and Relative Velocities

while O and l stand for relative velocities. O is the velocity that � is moving away from an observer

located on frame � and l is the velocity that � is moving away from an observer on frame � . The

magnitudes of either O and l should be equal, according to the equations presented next

�O = �_ OR l = −�_ lo ¢ (A.10)

Where OR and lo are unit vectors and comply to OR ø = loø = 1, in accordance to the adopted convention

on equations (A.1) and (A.3). If one would have adopted for the other convention, where OR ø = loø =−1, strange effects would occur, as for instance, by adding two velocities in the same direction, the

result would be a velocity on the opposite direction, which is totally inadequate. This is the reason why

the metric of Minkowski spacetime is not Euclidean but Lorentz metric. From equations on (A.9) and

(A.10), the following ones are obtained

µ@A = M0HA − � lo1 HA = M9@A + � OR : ¢ (A.11)

Consequently the unit vectors can be attained

�OR = M0lo − �Ò/��Ò� H�1lo = M0OR + �Ò/��Ò� @�1¢ (A.12)

88


89

Appendix B

Boost Application 7 Boost Application

90


Figure B.1 - Boost application for � = 90° and � = 0

Figure B.2 - Boost application for � = 90° and � = 0.5

91

Boost Application



92




93

Boost Application


94


95

Appendix C

Bondi k-Calculus 8 Bondi k-Calculus

96


Bondi k-Calculus is an alternative and more intuitive method of obtaining the same results as using

special relativity, although only requiring basic mathematics, in opposition to the latter method, which

has many implicit concepts. It is taught to undergraduate students due to its simplicity, which may help

them to shed some light on physical aspects and derivations, which would remain a mystery. This

method was created by Professor Sir Hermann Bondi and presented in his book Relativity and

Common Sense, first published in 1962 and then in 1980. In his work, he begins with the Doppler

factor, represented by the letter k, and then explains several questions concerning special relativity,

like the twins paradox, relativity of simultaneity, time dilation and length contraction. Other aspects are

also dealt with, as for instance the Lorentz transformation and the relationship between velocity and

the factor k.

Radar Method Presentation

Considering a �ℓ�,� space with the typical metric characterized by @A� = −@�� = 1, one obtains the

already known Minkowski diagram, with a certain event described on it, according to the following

figure

Figure C.1 – Spacetime diagram for representation of the radar method presentation

First one must bear in mind the following equalities

�t� = 2ðtA + �t� (C.1)

�tA = �ÈÒ'�ÈÔ� (C.2)

Line A represents an event which is not moving, but time is urging in his frame, while line B represents

an event that changes position while time is passing by, with a constant velocity. Each of those points

represent an observer which meet at point 0 and reset their clocks to t=0. This synchronization will

give the possibility to conclude how much time has passed by on each of the frames. On this particular

time instant, observer A sends a light signal during ct� seconds, while observer B receives the same

97

Bondi k-Calculus

light signal during �t Ì = k0�t�1. At this moment, neither observer A or B know which one of them is

moving. Besides that, as the light beam reaches observer B, it is reflected again to observer A for a

certain time equal to �t� = k0k�t�1 = k��t� seconds. So

t� = k�t� (C.3)

The purpose now is to determine �t� in order to �t�, or putting on another way, to determine the value

of k or � = ð/�. As Figure C.1 may not give a good representation of the geometrical problem, so the

next one is more insightful.

Figure C.2 – Geometrical representation of the problem

Since light beams are being sent, the represented angle is 45º, corresponding to a light speed equal to �. Dividing equation (C.1) by c one obtains

t� − t� = 2ð/�tA = 2�tA (C.4)

Now applying equation (C.2) on the former one, the following is attained

t� − t� = β0t� + t�1 (C.5)

So, grouping according to the respective times

t�01 − β1 = t�01 + β1 (C.6)

Hence

ÈÒÈÔ = �'Õ�/Õ (C.7)

Substituting equation (C.7) on (C.3) one obtains

k = 1�'Õ�/Õ (C.8)

It is then possible to obtain β in order to k

k�01 − β1 = 1 + β (C.9)

k� − k�β = 1 + β (C.10)

98


k� − 1 = k�β + β (C.11)

k� − 1 = β0k� + 11 (C.12)

β = '� = �Ò/��Ò'� (C.13)

This is the expected equation, since it expresses the velocity of B relative to A.

Relativistic Velocity Addition

Considering an observer O, located on earth (null speed) and a spaceship, represented by letter A

and a certain velocity β�. This last element will fire a missile, represented by B and velocity β�. The

goal is to determine the velocity of B relative to O.

Figure C.3 – Geometrical representation of the problem

Supposing that observer O sends a light beam to observer during T seconds. Observer A will not

interpret that time interval as T but as

Τ� = k�T (C.14)

Then, observer A will send a light beam to the missile (observer B) for Τ� seconds, so the latter

observer will perceive as being

Τ� = k�T� (C.15)

Now one may obtain parameters k� and k�, as follows

99

Bondi k-Calculus

êêk� = 1�'ÕÔ�/ÕÔk� = 1�'ÕÒ�/ÕÒ¢ (C.16)

It is possible to deduce a relation between times T� and T�,

T� = ëT� = k�k�T (C.17)

So substituting equations belonging to set (C.16) on equation (C.17) yields

k = 11 + β 1 − β; = �'ÕÔ�/ÕÔ �'ÕÒ�/ÕÒ (C.18)

Hence

� = ÕÔ'ÕÒ�'ÕÔÕÒ (C.19)

The attained equation is the desired, since it characterizes the addition of relativistic velocities, as

studied on the third chapter. One may encounter on equation (3.137) the same equation as (C.19).

The difference is that the one which is located on chapter 3 uses GA to grasp the final result, which is

not the case of the one using the k calculus.

100


101

Appendix D

Quaternions 9 Quaternions

102


Quaternions were invented by Sir William Hamilton in 1843. t ∈ ℍ are hypercomplex numbers defined

by the Hamilton quaternion division ring ℍ. Having t = w + V� + W� + �] ∈ ℍ, one obtains its

dimension, which is equal to 4. Its basis set is defined by {1, V, W, ]}.w, �, �, � ∈ ℝ and the generalized

imaginary units V, W, ] comply with the following equations

�V� = W� = ]� = −1 VW = −WV = ], W] = −]W = V, ]V = −V] = W ¢ (D.1)

Concluding

V� = W� = ]� = VW] = −1 (D.2)

The multiplication operation is non-commutative, although associative, therefore ℍ is considered a

division ring but not a body, since the latter one is a commutative division ring. The groups ℕ, ℤ, ℚ, ℝ,ℂ and ℍ represent respectively the following numbers: natural (except to zero), whole, rational, real,

complex and Hamilton’s quaternions, where ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ ⊂ ℍ.

A binary operation ∘: x� → x, defined on the x set, has a neutral element ∈ x iff � ∘ = ∘ � = for

any � ∈ x. An element is told to be invertible iff there exists a � ∈ y such that � ∘ � = � ∘ � = .

Element � is the inverse of �. The algebraic structure 0x,∘1 is called a monoid if is possesses an

identity on x and if the binary operation is associative, 0� ∘ �1 ∘ � = � ∘ 0� ∘ �1 for any �, �, � ∈ x.

A monoid 0z,∘1 is a group iff all elements of z are invertible. For instance, the monoid 0ℝ,×1 is not a

group since the number 0, which is an absorbing element 0� × 0 = 0 × �1 for any � ∈ ℝ, does not have

an inverse (the neutral element is = 1). The group is abelian if the operation ∘ is commutative.

A ring is defined by 0), +,×1, where 0), +1 is an abelian group, the product × is associative, the sum

and subtraction are both distributive. If ring ) has identity for the product, then 0),×1 is a monoid and

the ring is unitary. It is common to define the sum’s neutral element as zero and the product’s neutral

element as the identity. A ring is abelian if its multiplication is commutative. )' is the aggregate of all

the monoid 0),×1 invertible element. For instance, ℤ∗ = {−1,1}, if ) is a unitary ring then 0)∗,×1 is a

group. There are also rings which comply to ñ × ò = 0 where ñ and ò are different from zero (ñ and ò

are known as zero dividers, in this case). Next an example of this situation will be given. Considering a

zero divider on the unitary ring ℝ021 = Mat02, ℝ1 of a 2 by 2 matrix defined on the real space ℝ.

"� ≠ 0� ≠ 0¢ → ñ = ��A AA� ≠ 0, ò = �AA A}� ≠ 0 → ñ × ò = �AA AA� = 0 (D.3)

A division ring is a unitary ring ) such that )∗ = ) − {0}, which means that all non-null elements are

invertible. The ring which defines the whole space ℤ is not a division ring, while the ones which define

the rational, real, complex and quaternion spaces are all division rings. A body is a division rings that

is abelian at the same time. For example, the quaternion space is not a body. A certain ring

represented by ) complies with the commutative ring with no zero divisors rule. if for any ñ, ò, � ∈ )

one has [� ≠ 0 and 0a × c = b × c or c × a = c × b1], which means that ñ = �. Recalling the former

example represented on equation (D.1), one can state that the commutative ring with no zero divisors

rule cannot be complied, as demonstrated next

103

Quaternions

ñ = �AA A��, ò = �AA AA�, � = ��A AA� ≠ 0 → �~×�Ð�×�ÐA�×~Ð�×�ÐA� ← ñ ≠ ò (D.4)

An integral domain is an abelian unitary ring ) ≠ {0} on which it has a commutative ring with no zero

divisors. The following figure represents all these definitions, as rings, division rings, bodies and

integral domains.

Figure D.1 – Representation of rings, division rings, bodies and integral domains

The quaternion multiplication rules can be shortened as follows V� = W� = ]� = VW] = −1 It is possible

to define a conjugate quaternion, just as the complex case. The representation is the same as the

latter one

t = w + V� + W� + �] ∈ ℍ (D.5)

Its conjugate is given by

t� = w − V� − W� − �] ∈ ℍ (D.6)

The square of a quaternion is

t� = 0w� − �� − �� − ��1 + 2w0V� + W� + �]1 (D.7)

And its modulus

|t|� = tt� = w� + �� + �� + �� (D.8)

The inverse of a quaternion may be written as well

t/� = h�|h|Ò = ��'��'Ý}'�Þ = �/��/Ý}/�Þ�Ò'�Ò'}Ò'�Ò (D.9.9)

Hence

t� − 2wt + |t|� = 0 (D.10)

Analogously to the complex numbers, quaternions can be decomposed on the real part ℜ0t1 = w ∈ ℝ

as well as the pure part ℘0t1 = V� + W� + �] ∈ ℝ�, which can be better understood on the following

equation.

104


t = tA + � ∈ ℍ → �tA = ℜ0t1� = ℘0t1 ¢ → ℍ = ℝ ⊕ ℝ� (D.11)

Considering two quaternions ñ and ò, with a real and a pure part

�ñ = ñA + � ∈ ℍò = òA + � ∈ ℍ¢ (D.12)

Their multiplication is given by

ñ ò = ñA òA − � ∙ � + ñA� + òA� + � × � (D.13)

Equaling the real part to zero, one obtains the quaternion’s pure part

ñA = òA = 0 → �� = ñ�� + ñ�� + ñ�^� = ò�� + ò�� + ò�^¢ (D.14)

�� ∙ � = ñ�ò� + ñ�ò� + ñ�ò� � × � = 0ñ�ò� − ñ�ò�1� + 0ñ�ò� − ñ�ò�1� + 0ñ�ò� − ñ�ò�1^¢ (D.15)

And considering that the geometric product is defined by

� � = −� ∙ � + � × � (D.16)

One may define Gibbs cross product for the quaternions

� × � = ℘0� �1 (D.17)

There is an isomorphism between the division ring ℍ and the even subalgebra of �ℓ�: ℍ ≃ �ℓ�', where �ℓ�' = ℝ ⊕ « ℝ��, which can be supported by the following correspondence:

Table D.1 – Isomorphism between ℍ and �ℓ�'

ℍ �ℓ÷'

i −@��

j −@��

k −@��

There is also another isomorphism, between �ℓ� algebra and ℂ021 = Mat02, ℂ1, as follows

105

Quaternions

Table D.2 - Isomorphism between ℂ021 and �ℓ�

ℂ0ø1 = c��0ø, ℂ1 �ℓ÷

�=identity matrix 1

��,��,�� @�, @�, @�

��,��,�� @��, @��, @��

�� @��

��,��,�� represent Pauli’s matrices and are given by

�� = �0 11 0� ,�� = �0 −VV 0 � ,�� = �1 00 −1� (D.18)

Knowing that

ê�� = �� = �� = T �� = −�� = V�� = −�� = V�� = −�� = V�� ¢ (D.19)

Therefore quaternion t ∈ ℍ is isomorph to the matrix � ∈ ℂ021, which yields

t = w + V� + W� + �] ≃ � = �w − V� −� − V�� − V� w + V� � (D.20)

Table D.3 – Quaternion multiplication table

1 -1 i -i j -j k -k

1 1 -1 i -i j -j k -k

-1 -1 1 -i i -j j -k k

i i -i -1 1 k -k -j j

-i -i i 1 -1 -k k j -j

j j -j -k k -1 1 i -i

-j -j j k -k 1 -1 -i i

k k -k j -j -i i -1 1

-k -k k -j j i -i 1 -1

106


107

Appendix E

STA Pins and Spins 10 STA Pins and Spins

108


Considering �, � ∈ ℝ and U, ð,w ∈ �ℓ�,� one can define a left contraction in STA

�� 0U ∧ ð1 = 0� U1 ∧ ð + u� ∧ 0� ð10U ∧ ð1 w = U 0ð w1 ¢ (E.1)

Now considering two vectors ð,w ∈ �ℓ�,�, one obtains the relation

|U|� = |�UU²�A| (E.2)

Where 0Uð1~ = ð²U² is an anti-automorphism, and ð = U � is the Clifford’s dual.

A multivector in STA is composed by a scalar, vector, bivector, trivector and quadrivector (or

pseudoscalar) U = ë + � + F + � S + � S ∈ �ℓ�,�, that complies with the following relation

S U S = −ë + � − F + � S − � S, so U± = �� 0U ∓ S U S1 ∈ �ℓ�,�± (E.3)

There are three Lorentz groups, Pin01,31, Spin01,31, and Spin'01,31.

êPin01,31 = �U ∈ �ℓ �,�' ∪ �ℓ�,�/ |UU² = ±1�Spin01,31 = �U' ∈ �ℓ�,�' |UU² = ±1� Spin'01,31 = �U' ∈ �ℓ�,�' |UU² = 1� ¢ (E.4)

Each group has a correspondence to a certain Lorentz transformation, represented next

�� ↦ U � U�/�, U ∈ Pin01,31 � ↦ U � U�/�, U ∈ Spin01,31 � ↦ U � U�/�, U ∈ Spin'01,31¢ (E.5)

The latter Lorentz group presented above is known as the special ortochronous Lorentz group, since it

preserves the spacetime orientation. Those which preserve orientation are called proper and have

determinant equal to 1. The improper Lorentz transformations have determinant symmetric to the

proper one.

109

Appendix F

Maxwell Equations Auxiliary

Calculations 11 Maxwell Equations Auxiliary

Calculations

110


This appendix will aim on proving Maxwell’s homogeneous and non-homogeneous equations. The first

derivation will regard the homogeneous equation.

∂ ∧ F = D1� @A ∂∂. + @� ∂∂�� + @� ∂∂�� + @� ∂∂��F ∧

∧ P�� 0G� @�A + G� @�A + G� @�A1 − 0*� @�� + *� @�� + *� @��1Q (F.1)

As the involved operations are distributive, one obtains

1� @A ∂∂. ∧ 1� 0G� @�A1 + 1� @A ∂∂. ∧ 1� 0G� @�A1 + 1� @A ∂∂. ∧ 1� 0G� @�A1 −

− 1� @A ∂∂. ∧ *� @�� 1� @A ∂∂. ∧ *�@�� − 1� @A ∂∂. ∧ *�@�� +

+@� ∂∂�� ∧ 1� 0G�@�A1 + @� ∂∂�� ∧ 1� 0G�@�A1 + @� ∂∂�� ∧ 1� 0G�@�A1 −

−@� ∂∂�� ∧ *�@�� − @� ∂∂�� ∧ *�@�� − @� ∂∂�� ∧ *�@�� +

+@� ∂∂�� ∧ 1� 0G�@�A1 + @� ∂∂�� ∧ 1� 0G�@�A1 + @� ∂∂�� ∧ 1� 0G�@�A1 −

−@� ∂∂�� ∧ *�@�� − @� ∂∂�� ∧ *�@�� − @� ∂∂�� ∧ *�@�� +

+@� ∂∂�� ∧ 1� 0G�@�A1 + @� ∂∂�� ∧ 1� 0G�@�A1 + @� ∂∂�� ∧ 1� 0G�@�A1 −

−@� ==�> ∧ *�@�� − @� ==�> ∧ *�@�� − @� ==�> ∧ *�@�� (F.2)

Knowing that @�AA = @�AA = @�AA = @��A = @�A� = @�� = @�� = @�A� = @��A = @�� = @�� = @�A� =@��A = @�� = @�� = 0 yields

∂ ∧ F = − 1� ∂*�∂. @A�� − 1� ∂*�∂. @A�� − 1� ∂*�∂. @A�� −

+ 1� ∂G�∂�� @A�� − 1� ∂G�∂�� @A�� − ∂*�∂�� @�� −

− 1� ∂G�∂�� @A�� + 1� ∂G�∂�� @A�� − ∂*�∂�� @�� +

+ �� =�Ô=�> @A�� − �� =�Ò=�> @A�� − =6>=�> @�� = 0 (F.3)

The parcels’ sum regarding the 4 previous equations gives the desired formula.

When forcing ∂ ∧ F = − ý∇ ∙ *��Jþ @�� − �� =*��J=Y S − �� ý∇ × G��Jþ S = 0, one obtains

− 1� D∂*�∂. @A�� + ∂*�∂. @A�� + ∂*�∂. @A��F +

111

Maxwell Equations Auxiliary Calculations

+ 1� ∂G�∂�� @A�� − 1� ∂G�∂�� @A�� − ∂*�∂�� @�� −

− 1� ∂G�∂�� @A�� + 1� ∂G�∂�� @A�� − ∂*�∂�� @�� +

+ �� =�Ô=�> @A�� − �� =�Ò=�> @A�� − =6>=�> @�� = 0 (F.4)

Grouping the equation in order to the respective trivectors

− Y∂*�∂�� + ∂*�∂�� + ∂*�∂�� @�� − 1� Y∂*�∂. @A�� + ∂*�∂. @A�� + ∂*�∂. @A�� −

− �� P=�Ô=�Ò − =�Ò=�ÔQ @A�� − �� P=�>=�Ô − =�Ô=�>Q @A�� − �� P=�Ò=�> − =�>=�ÒQ @A�� = 0 (F.5)

The purpose is to classify physically the previous equation.

�� 9∇ × G�J: S = �� @�==�Ô G�

@� ==�Ò G� @� ==�> G�

� @A�� (F.6)

= �� ý∂G3∂�2 − ∂G2∂�3þ @� + �� ý=�Ô=�> − =�>=�Ôþ @� + �� ý=�Ò=�Ô − =�Ô=�Òþ @�� @A�� (F.7)

Considering the following equalities

ä@� @A�� = −@A�� = @A�� @� @A�� = @A�� = −@A�� =@� @A�� = −@A�� = @A�� ¢ @A�� (F.8)

One obtains

1� �D∂G�∂�� − ∂G�∂��F @A�� + D∂G�∂�� − ∂G�∂��F @A�� + D∂G�∂�� − ∂G�∂��F @A�� =

= − �� ý=�Ò=�> − =�>=�Òþ @A�� − ý=�>=�Ô − =�Ô=�>þ @A�� − ý=�Ô=�Ò − =�Ò=�Ôþ @A�� (F.9)

− 1� ∂*�J∂. S = − 1� @A ∂∂. 0*�@� + *�@� + *�@�1S =

− �� P=6Ô=Y @A�� + =6Ò=Y @A�� + =6>=Y @A��Q (F.10)

9∇ ∙ *�J:@�� = Y∂*�∂�� 0@� ∙ @�1 + ∂*�∂�� 0@� ∙ @�1 + ∂*�∂�� 0@� ∙ @�1� @��

= − P=6Ô=�Ô + =6Ò=�Ò + =6>=�>Q @�� (F.11)

So, if

9∇ ∙ *�J: @�� = 0 (F.12)

Then the following is valid as well

112


−9∇ ∙ *�J: @�� = 0 (F.13)

Concerning the non-homogeneous equation, its expansion is going to be studied next

∂ K = D1� @A ∂∂. + @� ∂∂�� + @� ∂∂�� + @� ∂∂��F

P0H�@�A + H�@�A + H�@�A1 − �� 0I�@�� + I�@�� + I�@��1Q (F.14)

First of all, one must bear in mind that

∂ K = ∂ K + ∂ ∧ K (F.15)

Then, both geometric and wedge products must be calculated separately, in order to obtain the left

contraction

∂ K = D1� @A ∂∂. + @� ∂∂�� + @� ∂∂�� + @� ∂∂��F

P0H�@�A + H�@�A + H�@�A 1 − �� 0I�@�� + I�@�� + I�@��1Q (F.16)

= 1� @A ∂∂. 0H�@�A1 + 1� @A ∂∂. 0H�@�A1 + 1� @A ∂∂. 0H�@�A1 −

− 1� @A ∂∂. 1� 0I�@��1 − 1� @A ∂∂. 1� 0I�@��1 − 1� @A ∂∂. 1� 0I�@��1 −

+@� ∂∂�� 0H�@�A1 + @� ∂∂�� 0H�@�A1 + @� ∂∂�� 0H�@�A1 −

−@� ∂∂��1� 0I�@��1 − @� ∂∂��

1� 0I�@��1 − @� ∂∂��1� 0I�@��1 +

+@� ∂∂�� 0H�@�A1 + @� ∂∂�� 0H�@�A1 + @� ∂∂�� 0H�@�A1 −

−@� ∂∂��1� 0I�@��1 − @� ∂∂��

1� 0I�@��1 − @� ∂∂��1� 0I�@��1 +

+@� ∂∂�� 0H�@�A1 + @� ∂∂�� 0H�@�A1 + @� ∂∂�� 0H�@�A1 −

−@� ==�> �� 0I�@��1 − @� ==�> �� 0I�@��1 − @3 ∂∂�3 1� 0I3@121 (F.17)

Using the operations regarding geometric product, the following equality is obtained

∂ K = − 1� ∂H�∂. @� − 1� ∂H�∂. @� − 1� ∂H�∂. @� − 1�� ∂I�∂. @A�� − 1�� ∂I�∂. @A�� − 1�� ∂I�∂. @A�� +

+ ∂H�∂�� @A − ∂H�∂�� @A�� + ∂H�∂�� @A�� + 1� ∂I�∂�� @�� + 1� ∂I�∂�� @� − 1� ∂I�∂�� @� +

+ ∂H�∂�� @A�� + ∂H�∂�� @A − ∂H�∂�� @A�� − 1� ∂I�∂�� @� + 1� ∂I�∂�� @�� + 1� ∂I�∂�� @� −

113

Maxwell Equations Auxiliary Calculations

− =�Ô=�> @A�� + =�Ò=�> @A�� + =�>=�> @A + �� =�Ô=�> @� − �� =�Ò=�> @� + �� =�>=�> @�� (F.18)

∂ ∧ K = D1� @A ∂∂. + @� ∂∂�� + @� ∂∂�� + @� ∂∂��F

∧ P0H�@�A + H�@�A + H�@�A 1 − �� 0I�@�� + I�@�� + I�@��1Q (F.19)

Having considered that @�AA = @�AA = @�AA = @��A = @�A� = @�� = @�� = @�A� = @��A = @�� = @�� =@�A� = @��A = @�� = @�� = 0 yields

= − 1�� ∂I�∂. @A�� − 1�� @A ∂I�∂. @A�� − 1�� ∂I�∂. @A�� −

− ∂H�∂�� @A�� + ∂H�∂�� @A�� + 1� ∂I�∂�� @�� −

+ ∂H�∂�� @A�� − ∂H�∂�� @A�� + 1� ∂I�∂�� @�� −

− =�Ô=�> @A�� + =�Ò=�> @A�� + �� =�>=�> @�� (F.20)

Replacing the two previous equalities on the former one, yields

∂ K = ∂ K − ∂ ∧ K =

− 1� ∂H�∂. @� − 1� ∂H�∂. @� − 1� ∂H�∂. @� +

+ ∂H�∂�� @A + 1� ∂I�∂�� @� − 1� ∂I�∂�� @� +

+ ∂H�∂�� @A − 1� ∂I�∂�� @� + 1� ∂I�∂�� @� +

+ =�>=�> @A + �� =�Ô=�> @� − �� =�Ò=�> @� (F.21)

Equaling Maxwell’s bivector to the electric current density absolute vector comes

∂ K = − �� =H��J=Y − ý∇ ∙ H��Jþ @A + �� ∇ × I��J = �@A + �� JJ = X (F.22)

− 1� ∂H�∂. @� − 1� ∂H�∂. @� − 1� ∂H�∂. @� +

+ ∂H�∂�� @A + 1� ∂I�∂�� @� − 1� ∂I�∂�� @� +

+ ∂H�∂�� @A − 1� ∂I�∂�� @� + 1� ∂I�∂�� @� +

+ =�>=�> @A + �� =�Ô=�> @� − �� =�Ò=�> @� = �@A + �� {J�@� + J�@� + J�@�} (F.23)

∇ ∙ H��J = P=�Ô=�Ô 0@� ∙ @�1 + =�Ò=�Ò 0@� ∙ @�1 + =�>=�> 0@� ∙ @�1Q = − ý=�Ô=�Ô + =�Ò=�Ò + =�>=�>þ (F.24)

114


So

− ý∇ ∙ H��Jþ @A = − P=�Ô=�Ô + =�Ò=�Ò + =�>=�>Q @A = �@A (F.25)

And finally

∇ ∙ H��J = −� (F.26)

From the non-homogeneous equation, one obtains

− �� =H��J=Y + �� ∇ × I��J = �� JJ (F.27)

This will be demonstrated next

1� ∇ × I��J = êê@1∂∂�1I1

@2 ∂∂�2I2@3 ∂∂�3I3

êê =

= �� P=�>=�Ò − =�Ò=�>Q @� + P=�Ô=�> − =�>=�ÔQ @� + P=�Ò=�Ô − =�Ô=�ÒQ @�� (F.28)

− 1� ∂H��J∂. = − 1� ∂∂. 0H�@� + H�@� + H�@�1 =

= − �� ý=�Ô=Y @� − =�Ò=Y @� − =�>=Y @�þ (F.29)

Concluding 1� �Y∂I3∂�2 − ∂I2∂�3 � @1 + Y∂I1∂�3 − ∂I3∂�1 � @2 + Y∂I2∂�1 − ∂I1∂�2 � @3� =

�� ýJ1 + ∂H1∂. þ @1 + ýJ2 + ∂H1∂. þ @2 + ýJ3 + ∂H1∂. þ @3� (F.30)

115

Appendix G

Doppler Effect Application 12 Doppler Effect Application

116


Figure G.1 – Doppler effect application for � = 0.1 and � ∈ [0, 2 ]

Figure G.2 - Doppler effect application for � = 0.1 and � ∈ [0, 2 ]

117

Doppler Effect Application



118

Bibliography and References





G. F. R. Ellis and R. M. Williams, Flat and Curved Space-Times; Oxford University Press, Oxford,

2000, 2nd ed., pp. 49–121

www.arxiv.org/PS_cache/physics/pdf/0108/0108012v2.pdf

http://www.geocities.com/ResearchTriangle/System/8956/Bondi/intro

C. R. Paiva, “Óptica Relativista"; Departamento de Engenharia Electrotécnica e de Computadores,

Instituto Superior Técnico, March 2004

C. R. Paiva, "Lição de Síntese", Departamento de Engenharia Electrotécnica e de Computadores,


relativistic electrodynamics with minkowski spacetime algebra · relativistic electrodynamics with...

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