Relativistic Electrodynamics with Minkowski Spacetime Algebra

Walid Karam Azzubaidi nº 52174 Mestrado Integrado em Eng. Electrotécnica e de Computadores, Instituto Superior Técnico, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal. e-mail: [email protected]

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 approached 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 sustains as a mystery or inconsistency.

Keywords Geometric algebra, Minkowski spacetime, Maxwell equations and Lorentz force, Vacuum Form Reduction (VFR), Doppler shift, Energy-momentum operator

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1. Introduction relativistic velocity addition for collinear vectors and Doppler shift as well. The interaction of spacetime algebra with electromagnetism will be presented as well. Maxwell The main objective of this work is to study several equations will be the kernel of this chapter, since those electrodynamic and relativistic effects, using the most equations give the best characterization of the electric and appropriate algebra for that meaning – STA (spacetime magnetic fields and relate them to their sources, charge algebra). In order to do so, one must first understand the density and current density. These equations are used to basic concepts of Euclidean algebra, which is the foundation show that light is an electromagnetic wave. The four of STA. The difference between both relies on the fact that equations, together with the Lorentz force law are the Euclidean algebra only uses vectors in ordinary three- complete set of laws of classical electromagnetism. The dimensional space, while STA unites spatial and temporal Lorentz force law itself was actually derived by Maxwell vectors, resulting in a non-Euclidean algebra. STA is under the name of "Equation for Electromotive Force" and supported on Minkowski spacetime, which symmetry group was one of an earlier set of eight equations by Maxwell. is the Poincaré group. To understand the applications which will be presented, it is also necessary to introduce Maxwell’s equations, including Lorentz force, since they 2. Geometric Algebra are the basis of electromagnetism. Other applications which
will take part on this work relate to kinematics and Doppler Considering a C n algebra with n = 2 (n = 3) , which effect, which will be subsequently used to solve the famous means it is a two (three)-dimensional space, all its vectors problem of the twins’ paradox. The moving media problem follow the rule is also addressed, and instead of grasping with tensors, δ ei⋅ e j = ij (1) which has proven to be a rather complex task, the vacuum form reduction method will be applied. According to the Kronecker delta function, defined by During the investigation which lead to this dissertation, several concepts had not yet been written, whether as a 0, if i≠ j δ master’s dissertation or as literature. Most concepts which i j = (2) appear on this work are already known, the difference is the 1, if i=j used formalism – in this case STA is the chosen foundation. This is the first master’s dissertation which gives a global, Those algebras are spanned by the basis sets yet, in-depth study of diverse applications that STA may B = B = {1,e1 , e 2 , e 3 } and {1,,,,eeee1 2 3 12 , e 23 , e 31 , e 123 } . These help grasp. STA shows its enormous ability to solve algebras are Euclidean, since the vectors’ square is always problems, which formerly were very complicated to solve, 2 2 2 since required much more extended calculations and major equal to 1. As follows e1= e 2 = e 3 = 1. The axioms which complexity level. On this thesis, VFR will be used to solve define the algebras at hand are the following and analyze the STA constitutive relation on an isotropic media and presents a novelty on a master’s thesis. Another Associativity a(bc)=(ab)c innovation on this thesis is the resolution of the relativistic velocity addition for collinear vectors, without Einstein’s Distributivity a(b+c) = ab +ac second postulate, which states that the velocity of transmission of light in vacuum has to be considered equal Anti-symmetry ba = -ab to . The reader will have the opportunity to grasp the fundamentals of this algebra, such as bivectors and 2 2 trivectors – and its geometric interpretation in the Contraction rule aa=a= a

corresponding algebra space, C 2 and C 3 . It is essential 2 to understand these algebras, as they are the foundation of Positive a2 = a > 0 STA, since they led to the latter and more perfect form. The essential rules will be presented, as well as the application Scalar multiplication λa = a λ of rotations and contractions, which will be useful on subsequent chapters. The relation of GA (geometric It is possible to obtain a certain algebra’s dimension, which algebra) with dot product and wedge product will be studied is given by dimC
= 2 n , therefore the dimensions are 4 as well. Gibbs’ and Grassmann’s algebras are related as ( n ) well, their relation and review will be given on this chapter. and 8, which may be corroborated by the number of objects
Some applications shall be addressed, therefore an for a given algebra. C 2 algebra is composed by one scalar, introduction to the framework will be made previously. The two vectors and one bivector, while C
algebra has one major strength of STA is the ability to treat boosts, since it 3 is the basis of all presented applications. The difference of scalar, three vectors, three bivectors and one trivectors. For

STA’s indefinite metric to the former algebras will be their corresponding algebras (C2 and C 3 ) , bivectors and presented. That is the big advantage of STA, when trivectors are also known as pseudoscalars, being the compared to the Euclidean algebras – space and time are element with the highest grade. 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

- 2 -  vu= exp (β ev12 ) (4)

Figure 1 – Bivector described by a frame

Figure 3 – Clock-wise rotation

It is also possible to attain a counter clock-wise rotation, given by the following rotor and its application to a vector

 R =exp ()()βe21 =cos β − sin( β) e 12 (6)

 vu=exp ( −β ev12) = exp (β ev 21 ) (7)

Figure 2 - Trivector described by oriented frames

Bivectors and trivectors may be considered as oriented planes, formed by the outer product of vectors. As for Figure 4 – Counter clock-wise rotation instance, a bivector is formed by the outer product of two These rotations are similar to the complex numbers. vectors e=ee12 1∧ 2=− e 12 = e 21 = e 2 ∧ e 1 and a trivector by Performing a rotation with rotors gives a different result three vectors, e123 =e 1∧ e 2 ∧ e 3 . Being an oriented plane when that rotor is on the left or right side of the vector to be means that the figure representing the object doesn’t matter, just its orientation and area. The orientation is given by the rotated. On the other hand, with complex numbers it is the
vectors and area by a certain scalar. The outer product is an same to apply on the right or left side. Regarding C 3 anti-commutative operation, opposite to the inner product. algebra, one may highlight the property of duality, where The geometric product will be introduced as the sum of a F e inner and outer product of two vectors, as shown = - 123 (8) ab = ab ⋅ + a ∧ b . This product is anti-commutative due to And the inner product, so, ba =⋅+ ba b ∧=⋅−∧ a ab a b . F = ae 123 (9) 1 Summing both equations one obtains a∧ b =() abba − . It is possible to establish the connection between Gibbs’ 2 cross product with C
GA. It is only possible to define the Now the aim is to study rotations in C
, algebra, which 3 2 »3 result from applying a rotor to a certain vector. This will cross product in space, opposed to Grassmann’s outer originate another vector rotated form the initial one, both product, as follows forming a certain angle, which is given by the one defined cab= × =−( abe ∧ ) 123 (10) on the rotor’s expression. Considering B= β e 12 , one obtains While Gibbs’ cross product depends on the metric and is the rotor neither associative nor invertible, Grassmann’s doesn’t depend on its metric, and besides that, it’s associative and R =exp βe =cos β + sin( β) e (3) ( 12) ( ) 12
invertible. C 3 algebra apprehends several algebraic Applying the rotor to a vector, it is possible to obtain the structures within, it is possible to define the even part of final and rotated vector, which will be a clock-wise rotation.

+
− C 3 , denoted as C 3 and the odd part C 3 as those structures, which result from the geometric product of an

- 3 - even and odd number of vectors, respectively of »3 space. quadrivector, hence the dimension is equal to 16. An event Only the even part incorporates sub-algebra, since the odd may be described by the equation 1,3 part is not closed regarding the geometric product. On r(t) =( ct) e0 + xt( ) e 1 + yt( ) ee 2 + zt( ) 3 ∈ (14)
It is possible to distinguish three kinds of vectors, directly C 3 algebra it is also possible to apply rotations, which is related to the spacetime trajectory – lightlike, when the given by movement is made at light speed c and r2 = 0 ; spacelike (space predominates over time and r2 < 0 ) and timelike a a' = R a R (11) (time predominates over space and r2 > 0 ) 2 2222 22 r =()ct −++=( x1 x 2 x 3 ) () ct −∈ x (15) It is possible to explain trajectories using the lightcone

Figure 5 – Rotation on space algebra

It may be concluded that when performing a rotation, the parallel component of the vector is changes, while the perpendicular component remains unchanged. Another important operation which will be introduced is the Figure 6 - Spacetime trajectories represented on a lightcone contraction – it may be done to the left or to the right. The consequence of the left operation is to reduce the degree of On the previous figure, the region known as elsewhere is a bivector, assigned by letter B to a unitary degree, not accessible since it is necessary to travel at a certain represented by a speed greater than c , which is not possible. An important
rule in C 1,3 algebra is that of commutation: while vectors  1 and trivectors anti-commute with the pseudoscalar , aB=( aB − Ba )  2  (12) bivectors commute with . Just like the studied algebra, it’s 1 Ba= ( Ba − aB )  2 possible to define Clifford’s dual for a generic multivector u =+++α a F bI + β I , which results in vector

It is an anti-symmetric operation, hence v= uI = u e 0123 . Then, the resulting vector is
v =−−+β b FI + aI + α I . In C 1,3 algebra bivectors may aB= − Ba = aB (13) || be classified as simple (hyperbolic), when F=e∧ e → F 2 = 1 , simple (elliptical) 0 1 3. Spacetime Algebra F=e∧ e → F 2 =− 1 , simple (parabolic) 2 3 2 F=e1∧ e 0 +∧ e 1 e 3 → F = 0 or non-simple, when STA is useful to handle certain kinds of problems which 2 F=e0∧+ ee 1 2 ∧ e 3 → F = 2 I . The most important cannot be addressed by Euclidean algebras. So, this algebra highlight is that elliptical bivectors originate rotations differs from the latter ones on a basic principle: the square ( exp(θe23 )= cos θ + e 23 sin θ ), while hyperbolic bivectors of the vectors is not always equal to 1, as follows originate boosts ( coshζ+ e10 sinh ζ ). A boost is a similar e2=1, ee 2 = 2 = e 2 =− 1 . The algebra is spanned by the 0 1 2 3 transformation to rotation, although not as simple. ζ is the B rapidity parameter, which controls the intensity of the boost, basis set = {1,,,,,eeeeeeeeeee0 1 2 3 01 , 02 , 03 , 12 , 13 , 23 , 012 , e 013 , so the goal is to expose that tool. A boost is also known as active Lorentz transformation and performs a e, e , e } . It is constituted by one scalar, four vectors, 023 123 0123 transformation of a certain frame defined by (e, e ) to six bivectors, four trivectors and the pseudoscalar – 0 1

- 4 - 1 of vectors, the passive merely shows the diverse points of another defined by (f0, f 1 ) . γ =cosh ζ = is the 1− β 2 view for s particular event. Any two events A and B correction factor, which depends directly on the velocity of which occur on a line parallel (also known as line of a certain particle. β =v = tanh ζ is the normalized simultaneity in S ) to axis x , (defined as ct 0 ), happen c velocity hence γ β = sinh ζ . When the rapidity tends to simultaneously regarding an observer on the S frame. This π  is not valid for another observer who is placed on the S infinite, the normalized velocity tends to tan  = 1 , which 4  frame, which is shown on next figure. The difference in time events regarding S frame is given by means that f0→+ ee 01 and f 1 →+ ee 01 , therefore tending to the bisection, as represented next ∆ctAB= c t A − c t B , which is different from zero. Following the same thread of thought, Any two events A and B

which occur on a line parallel to axis x , defined as ct 0 , also known as line of simultaneity in S happen simultaneously regarding an observer on the S frame. This is not valid for another observer who is placed on the S frame, which is shown on next figure. The difference in time events regarding S frame is given by

∆ctAB= c t A − c t B , which is different from zero. Figure 7 - Representation of a boost

Boosts result Minkowski diagrams, which will be the used Several application shall be addressed on this paper, starting tool to study relativistic effects. For a purpose of simplicity, with time dilation and space contraction. Either effect may

be performed considering the rest frame as S or S , as the one will use a C 1,1 diagram instead of C 1,3 result is the same. For a matter of convention, the first approach shall be taken.

Figure 8 – Minkowski diagram

An event is described by a person standing on a frame, which can be represented on the Minkowski diagram Figure 9 – Time dilation according to the expression which links both time and space. An observer located on the rest frame S shall From the previous figure it can be inferred that vector AB interpret an event r=(ct) e0 + x e 1 , whicle another observer is equal to the sum of other two vectors, as follows: located on the S will have another interpretation AB= AC + CB (16) r=(ct) f0 + x f 1 , which means that the same event has diverse point of view, therefore it is not possible to define This results on the following equation an absolute time or space. Simultanity is a relative concept, and the passive Lorentz transformation explains that. While cT00f= cT e 0 + vT e 1 (17) the active transformation performs an actual transformation

- 5 - And therefore velocity greater than c . This was contradicted by Maxwell, who stated there is the need to perform a correction to

fe00=γ ee 00 + γβ ee 10 (18)
Newton’s equation. Using C 4 to determine the velocity Given that f0 e 0 = 0 , e0 e 0 =1and e1 e 0 = 0 , yields composition, one obtains equation

T v+ v Tγ =0 = T (19) v = 1 2 (24) 0 2 v v 1− β 1− 1 2 c2 Now, analyzing the space contraction, the geometric equation which translates the problem is given by Simulations were made, and it was concluded that it is incorrect – adding two particles’ velocities greater when the AB= AC + CB (20) 1 condition β1 > is complied, results on a velocity inferior β2 to 0, which is physically unacceptable. Besides that fact, the velocity addition tends to infinite when both particles’ velocities tend to c , which is also incorrect.

Figure 10 – Length contraction

This is equal to Figure 11 - C 4 velocity addition

Lf1= L 01 e + β L e 0 (21) Striving for the same objective, C 1,3 algebra, with its union between time and space, accomplished a clear result as Thus crystal – not only the equation corroborates the physical effects, as well as the velocity addition tends to c , as fe=γβ ee + γ ee (22) 11 01 11 supposed to. The correct formula is represented next, as well as its simulation Given that f1 e 1 = 0 , e1 e 1 =1and e0 e 1 = 0 , yields v+ v v = 1 2 (25) L v v 0 =L1 −β 2 = L (23) 1+ 1 2 γ 0 c2

It may be concluded that the correction factor is the parameter that allows to transform the time or length observed on a sprecific frame to another which may not be at rest.

The importance of using C 1,3 algebra instead of C 4 may be demonstrated when determining the relativistic velocity addition, in this case, for collinear vectors. According to Newton, the velocity addition for two particles traveling at a speed equal to c is equal to 2c , constituted by a simple
addition. This is not possible, since no particles travel at a Figure 12 - C 1,3 velocity addition

- 6 - Doppler shift is another application which deserves special 4. Electrodynamic and Relativistic Effects ω attention. Its equation is given by 1+z = e , where z is ω As shown, GA possesses innumerous techniques for r studying problems in electrodynamics and the Doppler magnitude of redshift, ωr (ωe ) it the frequency electromagnetism. This chapter proposes several on the receiver (emmiter) side. The goal is to attain the applications regarding the matters referred above, which relation among both latter parameters. In order to reach the will emphasize the power of GA compared to traditional algebras. On a first approach, Maxwell equations shall be relation, on other words, to determine the receiving photon presented whether on regular three-dimensional space or from the emmiter, a rotation and a boost shall occur, which spacetime. As one will see later on, this is a great quality of is represented on the next flowchart 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. In order to study electromagnetism and electrodynamics, one must introduce Figure 13 – Addressed problem on Doppler shift a fundamental opearator – del. ∂ ∂ ∂  ∇=e + e + e ∈ 0,3 (27) Bivector B defines the relation between two frames and its 1 2 3 ∂x1 ∂ x 2 ∂ x 3 basis sets – given by a boost, thus it is possible to establish
It is possible to represent the Maxwell equations in C 3 on a f in order to e . When a rotation occurs, its frame 0 0 most known structure, where one may distinguish two maintains, while the obtained angle suffers an alteration as groups of equations - Faraday group, which includes represented on Figure 13. The referred angle is given by Maxwell-Faraday’s equation, also known as the Faraday’s   law of induction and Gauss’s law for magnetism s⋅ s = − cos φ . When applying a rotation, just the vector’s e ∂B ∇×E =− angle changes, while the modulus remains unaltered. On the ∂t (28) other hand, when a boost is applied, the photon’s frame is ∇ ⋅B = 0 altered. On both applications, the photon’s velocity does not E∈ 3 represents the intensity of the electric field- it is change, its is always equal to the speed of light. The defined as the electric force per unit charge. The direction of expression which gives the desired relation is presented next the field is taken to be the direction of the force it would exert on a positive test charge. The electric field is radially ω 1− β cos φ outward from a positive charge and radially in toward a r = (26) ω 2 3 e 1− β negative point charge. B∈ represents the intensity of the magnetic field. These two fields are denominated as It is the general formula for Doppler shift – when angle strength values, since characterize how strong the fields are The second pair of equations is identified as the Maxwell π  φ = 0 φ =  results on the longitudinal (transverse) group – both Ampère’s circuital law with Maxwell’s 2  correction and Gauss’s law are present, as the following set Doppler effect. The first situation degenerates on of equations show (1− β ) ∂D ωr ∇×H = J + = thus on a redshift, as ωr< ω e ; the second ωe ()1+ β ∂t (29) ∇⋅D =− ω 1 situation results on r = or a blueshift, which 3 3 ω 2 D∈ represents the electric displacement, while H ∈ e 1− β represents the magnetic field strength. These two vectors happens when ω> ω . Doppler shift may be used to make r e reflect a certain material’s excitation amount, consequently a spectral analysis of the sun’s light, for instance. To say are known as excitation values. It is possible to describe D that an image is blueshifted (where a blueshift occurs) and H in order to E and B although using another two means an observer is looking at that part of the sun that is moving towards him, or the light is compressed to shorter vectors as well: polarization P ∈ 3 and magnetization wavelengths, so the frequency is increasing. Likewise, the M ∈ 3 . The indicated equations are represented next opposite can be said about a red image, where the opposite takes place, also called as a redshift.

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D=ε 0 E + P In order to characterize Maxwell equations in C 1,3 algebra, 1 (30) H= B − M one must introduce their relative vectors (hyperbolic µ0 bivectors)

 2 The following relations are valid as well 1,3 E=E e 0 ∈Λ  2 1,3 P B=B e 0 ∈Λ p = −∇⋅ (38)  2 ∂P 1,3 D=D e 0 ∈Λ J p = (31) ∂t  2 1,3 H=H e 0 ∈Λ Jm = ∇ × M Besides that, the Dirac operator and electric charge – Where the total electric charge density and current are given current densities are also necessary

1 1 t= + p ∈ 1,3 (32) Je=+=+0J e 0{ JJJ 112233 eee ++ } ∈ 3 c c Jt=+ JJ p + J m ∈ (39) 1 ∂ ∂ ∂ ∂ 1,3 ∂=e0 + e 1 + e 2 + e 3 ∈ Finally one may attain the light speed on vacuum ct∂ ∂ x1 ∂ x 2 ∂ x 3 Now it is possible to define the essential bivectors of the c = 1 (33) electromagnetic field, which are absolute vectors – Faraday ε µ 0 0 and Maxwell bivectors

Opposite to Maxwell’s equations, the auxiliary fields D 1 and H are not present. Therefore it is considered a F= E + IB c fundamental and reductionist viewpoint, since one intends (40) 1 to explain the most using the fewest principles possible. G= D + IH c This new formulation is very attractive since both curl and Now, STA Maxwell shall be presented divergence of field quantities E and B , respectively, are   ∂B specified. According to the Helmholtz theorem, a field ∇×E =− ∂∧F =0 → (41) vector requires its curl and divergence to be given, in order  ∂t to be fully specified. One may separate the four equations ∇ ⋅B = 0 regarding magnetic flux conservation and charge-current    ∂ B conservation ∇×H = J + ∂B ∂G = J → ∂t (42) ∇×E =−  ∂t (34) ∇⋅B =− ∇ ⋅B = 0 Having made an introduction to Maxwell’s equations, one 1 ∂E may differentiate absolute and relative vectors. Another ∇×B =µ J + 0 t c2 ∂t goal which may be achieved is to reduce Maxwell’s (35) equations to a single one, which is only possible using STA. ∇ ⋅E = t ε 0 In vacuum, the constitutive relation is given by Using some mathematic principles, it is possible to reduce 1 those equations to a more compressed set, as follows G= F (43) η ∂ 0 ()Be+∇∧ E = 0 ∂t 123 (36) Knowing that∂ G = J and using the constitutive relation ∇ ⋅B = 0 in vacuum, one obtains 1  ∂D ∂== FJη0 η 0 e 0 +()J 11 eee ++ J 22 J 33  (44) +∇()He123 =− J c  ∂t (37) ∇ ⋅D = If the considered media does not have sources, then the right-side of the previous equation is equal to 0. One can

- 8 -  distinguish relative and absolute vectors on Maxwell on magnetic field intensity B . According to these results it equations. Vectors E and B are both relative vectors, can be concluded that an isotropic media on its own frame is while F is not, even being composed by the two previous considered as a bianisotropic medium on the rest frame. vectors. The same can be verified for D and H , which are From equation (49), one may obtain the following relation relative but G is an absolute vector. Consequently, vectors F and G don’t depend on any observer, so they represent 1 G' = F ' (51) the electromagnetic field and can be designated as covariant η forms of the electromagnetic field. It is possible to reduce From equation It may be concluded that with the presented Maxwell’s equations to a sole one, since in vacuum the transformation, a constitutive relation is obtained, which has existence of vector G is not required. the same build as the constitutive relation concerning vacuum, therefore it is called VFR. With this
C 1,3 algebra is useful to study a great variety of problems. transformation, the Minkowski spacetime structure has been One of the classic electrodynamics problems is moving changed, since it now corresponds to a fictitious spacetime,
different from the original one. media. Considering a simple isotropic medium, in C 3 Another field where STA excels is the one of algebra is defined by the following constitutive relations electrodynamics and kinematics, where it is used to correct (using STA) Einstein’s very well acknowledged mass-energy equivalence. But first one must describe the present D= ε0 ε E characters on this application. Defining the proper linear 1 (45) H= B momentum of a certain particle with mass m will be given µ µ 0 by Then, one may determine vector G α p= m u (52) GD=+1 IH =1 E + α IB (46) c c 2 Which is related to the particle’s total energy . The mass-

As v= c e 0 represents the frame where the media is at rest, energy equivalence states the following one obtains 1 F= − EIB + (47) p0 =γ mc = (53) v c c This represents vector F seen by an observer on frame e . And 0   For a certain linear combination of F with F , comes p= e +=p()γmc e + γ m u (54) v c 0 0 Therefore 1 1  1  1  G=ε +−  F  ε −  F (48) 2η µ 2η  µ  v 0 0  = γ 2  0 Then 0 =mc →  (55)  p = γ m u 1 G=exp() − ξ ru F (49) This yields η 2 2 With =0 + ()cp (56)

−1 1 0 is the particle’s proper energy, and one may attain the rFu () = Fu = uFu = 2 uFu (50) c kinetic energy, represented by The local form of the constitutive relation represents a projection of the constitutive relation on a particular K = γ 0 (57) observer, who is at rest regarding the media. In other words, The following figure represents the application’s geometry it could be the media’s own observer. It is possible to split the constitutive relation G= G( F ) into two constitutive         relations D= DEB( , ) and H= HEB( , ) .This means that   the media at hand is bianisotropic, and both D and H  depend, not only on the electric field intensity E , but also

- 9 - on the point where it loses reciprocity, the associated v−( − v ) 2 v acceleration is given by lim = → ∞ . ∆t → 0 ∆t ∆ t

5. Conclusion

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, Figure 14 - Geometric representation of particle’s which is a great advantage when applying to computational relativistic dynamics systems or simply performing calculations. Another important characteristic is the inherent geometric interpretation one has, which makes this algebra so intuitive It must be stressed that Newton’s mechanics limit - which to understand and use, hence attractive for beginners and happens when speed c reaches infinite, may be attained motivating new users. from this definition of kinetic energy This work’s foundations rely on not only in GA, but on special relativity and Lorentz transformations as well, so K=()γ − 1 ( mc 2 ) (59) several conclusions shall be presented. Simultaneity is a crucial concept when dealing with these matters: observers One obtains on different frames interpret events, each on a particular 1 3u4 15 u 6 way. The correction factor allows a temporal or spatial K() u= mu2 + m + m (60) 2 8c2 48 c 4 transformation from one frame to another When performing a relativistic velocity addition for c Newton’s mechanics is expressed by having → ∞ . collinear vectors, one may conclude there is a velocity limit Applying that condition on the former equation one obtains given by c This derivation was complete without employing Einstein’s second postulate. 1 2 Regarding Doppler shift, one concluded its general K= mu (61) 2 expression, where it is possible to distinguish the It is wrong to state that the inertia of energy is given by longitudinal and transverse situations. ∂F = η J is the most condensed way to write Maxwell = mc 2 , what it describes is the following 0 0 equations in STA. It is possible to distinguish absolute (F,G ) and relative vectors (E,B,D,H ) : the absolute =Iuc2 =γ umc 2 + K (62) ( ) ( ) 0 vectors are don’t depend on the frame the observer is located; on the other hand relative vectors depend on the Iu( ) = γ ( um ) is the inertia coefficient and the proper work frame. As emphasized by David Hestenes, STA is the best energy at rest is a kind of potential energy where framework to tackle electromagnetism in the context of 2 0 =(u = 0) . Einstein’s equation is given by = I( u) c Special Relativity, due to the inherent reduced complexity, and not as formerly known. 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 Twins paradox may be solved using Doppler shift. plane wave propagation in moving isotropic media. From Conclusion is that observers measure different times. This is the point of view of the laboratory, it may be concluded that T  it is actually a nonreciprocal bianisotropic media, which a simple matter of time dilation T = 0  , thus constitutes   γ  means its constitutive expressions D and H depend on no inconsistency. Time dilation is a reciprocal effect, E and B . although one observer crosses two different frames while Applying STA to a particle’s relativistic dynamics, allowed travelling, which results on the concluded time difference to correct some concepts, as the inertia of energy – 2 for both observers. This problem is not feasible in real life Einstein’s expression 0 =mc is incoherent, as well as an since the acceleration suffered by the travelling twin is idea of a variable mass some physicists had. infinite. This occurs because his velocity changes from c to One of the most famous paradoxes – Twins paradox has been disclosed, using the Doppler shift. It was concluded −c in an infinitesimal time fraction. Considering that one that the apparent inconsistency of results is acceptable, observer travels at a speed given by v and changes to −v being explained by time dilation.

- 10 -