An Introduction to Relativistic Electrodynamics Part I: Calculus with 4-Vectors and 4-Dyadics

An introduction to relativistic electrodynamics Part I: Calculus with 4-vectors and 4-dyadics

Jonas Larsson¦ Department of Physics, Ume˚aUniversity, SE-901 87 Ume˚a,Sweden

Karl Larsson: Department of Mathematics and Mathematical Statistics, Ume˚aUniversity, SE-901 87 Ume˚a,Sweden (Dated: October 22, 2018) The conventional way of introducing relativity when teaching electrodynamics is to leave Gibbs’ vector calculus for a more general tensor calculus. This sudden change of formalism can be quite problematic for the students and we therefore in this two-part paper consider alternate approaches. In this Part I we use a simplified tensor formalism with 4-vectors and 4-dyadics (i.e., second order tensors built by 4-vectors) but with no tensors of higher order than two. This allows for notations in good contact with the coordinate-free Gibbs’ vector calculus that the students already master. Thus we use boldface notations for 4-vectors and 4-dyadics without coordinates and index algebra to formulate Lorentz transformations, Maxwell’s equations, the equation of the motion of charged particles and the stress-energy conservation law. By first working with this simplified tensor formalism the students will get better prepared to learn the standard tensor calculus needed in more advanced courses.

I. INTRODUCTION electromagnetic field tensor happens to be just of order two is it possible to postpone the use of index algebra to In a first course in electrodynamics, on the level of say more advanced courses. Griffith’s textbook1, the mathematical formalism is es- Instead, since the students are confident with vector sentially Gibbs’ vector calculus with one exception; the calculus, it is beneficial to keep in contact with that for- chapter on relativistic electrodynamics. There new math- malism. The students also have some basic knowledge ematics seems to be needed and instead of the coordinate- of abstract linear algebra with its axiomatic structure as free language with bold face letters for vector fields one well as matrix algebra. This knowledge may be used in now uses coordinates, matrices and tensor calculus with a more efficient way than is usually the case. index algebra. This change of formalism causes problems The Minkowski vector space M is easily defined ax- 3 for many students. As Hestenes states:2 iomatically in the language of abstract linear algebra. The space M is then a 4D real vector space with some Einstein’s special theory of relativity has been additional structure. From a mathematical point of view incorporated into the foundations of theoreti- this is a simple construction but from a physical point cal physics for the better part of a century, yet of view it implies the loss of the concepts of absolute it is still treated as an add-on in the physics time and simultaneity. This causes considerable prob- curriculum. Even today, a student can get a lems since our intuitive understanding of time turns out Ph.D. in physics with only superficial knowl- to be wrong. In this case simple mathematics may help edge of relativity theory and its import. I us to deal correctly with a strange reality. Basic linear submit that this sorry state of affairs is due, algebra is certainly within the curriculum of most stu- to a large part, to serious language barriers. dents studying relativistic electrodynamics so it is a bit The standard tensor algebra of relativity the- surprising that this rigorous but elementary formulation ory so differs from ordinary vector algebra of special relativity is not explicitly included in most in- that it amounts to a new language for stu- troductory textbooks.4,5 dents to learn. The appearance of 4-by-4 matrices in relativity sig- The solution suggested by Hestenes is to use a differ- nals the use of linear transformations in terms of coordi- ent mathematical language built on the Clifford algebra nates. In fact, the wish to represent linear transformations without using coordinates motivated Gibbs to in- STA (Spacetime Algebra) and thus avoiding tensor calcu- 6 lus. This (very radical) suggestion is outside the scope of troduce dyadics in his vector algebra. Dyadics are math- the present paper and instead we consider the option to ematically speaking second order tensors in a notation make the transition to tensor calculus in smaller steps. that fits in with usual vector algebra. However, dyadics The tensor calculus used in relativity generalizes stan- have largely disappeared from modern textbooks. The dard vector calculus in many different ways and we must reason for this is perhaps that they are considered obso- avoid introducing too many new ideas at once. It is for lete, being just a special case of general tensors. This is example not necessary to start with tensor index alge- unfortunate because it promotes the use of unnecessar- bra. Actually, the index calculus is needed for dealing ily complicated notations when introducing relativistic with tensors of higher order than two and because the electrodynamics. 2

The problem with dyadics (like the problem with ma- of all 4-vectors that are orthogonal to the observer. The K trices) is that they may not be used when tensors of order space e0 is 3D and spacelike. It inherits from the space- three or higher are required. Attempts to generalize the time geometry exactly the structure of an Euclidean ori- dyadic formalism to such situations have not been very ented space, i.e., all structure we need to construct the K successful. However, in situations where we can manage usual Gibbs’ vector calculus. No coordinates on e0 are in without higher order tensors, the dyadic notation has ad- principle needed in this process. The linear algebra lan- vantages. It is easy to grasp, it is free from indices and guage make this observer-dependent split of spacetime more obviously geometric and actually quite perfect for into space and time mathematically clear in an elemen- introducing special relativity and relativistic electrody- tary way and should increase the students’ understanding namics. of this fundamental structure of special relativity. We start by defining the mathematical structure of the In summary, the relativistic structure of electrodynam- usual 3D space in terms of linear algebra.7 For students ics may be described as follows. First we may deal with with a first standard course in linear algebra this should spacetime as a geometric unity without the need for any cause no problems since only very basic and elementary observer or coordinates. The equations for the world line concepts from linear algebra are used. Furthermore we of a charged particle and the Maxwell’s equations may be presume just about everyone have our usual 3D space as given in such a spacetime geometric form. However, our geometric model when concepts like vector spaces, lin- understanding of spacetime much depend on a split into ear independence of vectors, linear transformations and separate time and space. This split is achieved by the in- so on are considered. Some simple dyadic notations are troduction of some observer e0. By introducing e0 into introduced and exemplified with rotations. our spacetime geometric equations we find the dynami- Next we generalize from 3D space to 4D spacetime. An cal equation for a charged particle as well as Maxwell’s advantage with the linear algebra approach is that we equations in their standard space geometric form on the K naturally get similar mathematical formulations for both 3D space e0 of the observer. space and spacetime. Our good understanding of space This paper is organized as follows. In Section II the now makes it easier to get confidence in the spacetime linear algebra formulation of usual 3D space is presented, theory. Also of importance is the fact that the linear al- some dyadic notation is introduced and the rotations of gebra formulation of special relativity is quite elementary space are considered. Section III deals with the corre- and conceptually simple. This is a most welcome feature sponding linear algebra formulation for 4D spacetime. when we try to understand the so unintuitive spacetime The observer e0 and the associated spacetime split is where the concepts of simultaneity and universal time considered. The world-line concept is defined and we have been lost. check that our observer concept corresponds to iner- In usual 3D vector calculus we write formulas in tial observers by demonstrating that different observers coordinate-free geometric form using boldface notations moves with constant velocity with respect to each other. for vector fields. Many times we can both in the deriva- Dyadics for spacetime are introduced and (active) proper tions and in the formulation of results gain clarity by orthochronous Lorentz transformations of spacetime are avoiding the use of coordinates. 3D space is treated as defined in close analogy with rotations in ordinary space. a geometric unity with spatial derivatives in the form of It is shown that an observer may express any proper grad, div and curl operators. Time appears as a real pa- orthochronous Lorentz transformation as a rotation fol- rameter. In the calculus of spacetime we like to treat lowed by a boost. spacetime as a geometric unity. The equation for the In Section IV coordinates and matrices are introduced. world line of a charged particle as well as Maxwell’s Up to this section the presentation is coordinate free, equations may be written in such a geometric spacetime this in order to make the geometric nature of the the- form. No observer or coordinates are necessary in these ory evident. However the theory needs numbers since at equations.8 some point experimental numbers should be compared However, while we all have a good understanding of with theoretic predictions. In our linear algebra setting space geometry our intuitive understanding of spacetime we get coordinates by introducing bases for the vector geometry is, at least from daily life experiences, rather spaces. The dyadic formalism is essentially a kind of non-existent. Certainly we would prefer to split space- coordinate-free matrix theory and the step to standard time into the separate parts of time and space. It is numerical matrices turns out to be an easy one. In rela- however a most essential feature of special relativity that tivity we compare measurements between inertial frames such a split is observer dependent so that there is no by means of passive Lorentz transformations. The re- unique split but rather many different ones. By an ob- lations between active and passive transformations are server in special relativity one usually means a frame of considered and explicit matrix expressions for boosts and orthonormal appropriately oriented coordinates for all of rotations are derived. spacetime. However, following Hestenes, we will define Sections V and VI include some mathematical theory an observer by just specifying a future oriented unit vec- that is needed in the formulation of relativistic electrody- 9 K tor e0. The 3D space for this observer is denoted e0 and namics. Section V concerns the anti-symmetric 4-dyadics defined to be the linear subspace of spacetime consisting and what they look like in terms of an observer. Also the 3 star operator is defind as a linear transformation of anti- coordinates. However, many times it is convenient to symmetric 4-dyadics. Section VI is about the natural use coordinates in parts of the algebra but still formulate derivatives in spacetime corresponding to the grad, div the final results without coordinates and thereby giving and curl derivatives in ordinary space. them increased clarity. Also, the geometric formulation Relativistic electrodynamics is presented in Sec- of Maxwell’s equations makes it easy to rewrite them in tion VII. The electromagnetic dyadic field is introduced. terms of any special coordinate system that simplifies The spacetime geometric equations for the world line of a some particular problem we study. charged particle and for Maxwell’s equations are given. A spacetime split is obtained by introducing an observer e0 and now the electromagnetic dyadic is related to the elec- B. Dyadics in space tric and magnetic fields of this observer. The equation of the motion of a charged particle and Maxwell’s equations In order to deal with linear transformations of E in a appear in their standard form. The transformation of the geometric coordinate-free way Gibbs introduced dyadics b electric and magnetic fields between two observers is ob- (or in modern terminology second order tensors E 2 tained. The electromagnetic potential and corresponding EbE). Standard dyadic notion by Gibbs omits the tensor formulation of Maxwell’s equations are included. We also symbol b but we keep it in this paper. The dyadic K consider the electromagnetic stress energy tensor. The fi- is a sum of terms of the form A b B where A and B nal Section VIII is a discussion. are vectors in E (each such term is called a dyad), i.e., K A b B C b D ¤ ¤ ¤ , and we define for dyads

II. THE LINEAR ALGEBRA FORMULATION pA b Bq C ApB Cq (1) OF SPACE C pA b Bq pC AqB (2)

A. The axioms of space From these relationships we define the dot product between dyadics and vectors in the obvious way. For each b The symbol E will be used to denote ordinary space dyadic K P E 2 we have the corresponding linear trans- p p and space vectors (E as in Euclidean). Let us define E formation K : E Ñ E defined by KC K C. We may axiomatically in terms of abstract linear algebra by the also take the dot product between two dyadics so that a b b b following three axioms: mapping E 2 ¢E 2 Ñ E 2 is obtained. This dot product is for dyads Axiom 1E. E is a 3-dimensional real vector space. Axiom 2E. An inner product (that we denote the dot pA b Bq pC b Dq pB CqA b D (3) product A B) is defined on E with signature p q, i.e., there exist some ordered basis and the general case with dyadics follows from this. The r s dot product of dyadics corresponds to the usual compo- e1 e2 e3 of E such that ej ek δjk. sition between linear operators. Thus for dyadics K and Axiom 3E. E is oriented, i.e., for any ordered basis like L we have the one in Axiom 2E we can tell if it is right x p p or left handed. M KL ô M K L (4) An orthonormal right handed basis of is in this paper E The transpose of an dyad is defined by called a normal basis. In the next section we will consider a normal basis for spacetime (but then there are T pA b Bq B b A (5) two orientations resulting in time orientation and space orientation). and from this we get the transpose of a dyadic in the ¢ The cross product A B of two vectors have a well obvious way. The relation defined geometric meaning and may now be constructed from the axioms using the dot product and the orienta- T T T pK Lq L K (6) tion. The grad, div and curl operators are geometrical operators associated with the three products (the prod- follows easily for general dyadics by first checking it in uct of vector with a real number, the dot product and the special case of dyads. the cross product). By “geometrical” we mean that these We define the antisymmetric tensor product of vectors operators may be defined without reference to any coor- by dinate system. The standard form of Maxwell’s equations is geometrical in space (while time is treated just A ^ B A b B ¡ B b A (7) as a parameter). The popularity of Gibbs’ vector calculus in electrodynamics is much due to this feature of and find Maxwell’s equation as well as the possibility to manipu- late equations without needing to introduce distracting pA ^ Bq C ¡ pA ¢ Bq ¢ C (8) 4

For a normal basis e re1 e2 e3s we use the notations It is easy to conﬁrm that (16) satisfy (19) by using the identities e e b e , e^ e ^ e (9) ij i j ij i j rn¢sT ¡ rn¢s (20) b2 r ¢s r ¢s ¡p ¡ b q A (non-ordered) basis for E is the nine dyads eij. The n n I n n (21) P unit dyadic, deﬁned by I A A for any vector A E, The second identity is proved by “dotting” it from the may be written right with an arbitrary vector A

I e11 e22 e33 (10) n ¢ pn ¢ Aq ¡A npn Aq (22) The space of antisymmetric dyadics is defined The exponential of a dyadic is defined by ( K K K K K K ^2 P b2 | T ¡ e.K exp .K I ... (23) E K E K K (11) 1! 2! 3! ^2 r ^ ^ ^ s By use of (6) in (23) we get and an ordered basis for E is e23 e31 e12 . We define the dyadic rA¢s associated with the cross product by pexp .KqT exp .pKT q (24)

rA¢s B A ¢ B for all B P E (12) The addition formula for the exponential function becomes It follows from (8) that e.K e.M e.K M if K M M K (25) A ^ B ¡ rpA ¢ Bq ¢s (13) To prove (25) we like to verify that ¢ K K2 K3 In terms of a normal basis we have the expansion I ... 1! 2! 3! r ¢s ¡ 1 ^ ¡ 2 ^ ¡ 3 ^ ¢ A A e23 A e31 A e12 (14) M M2 M3 I ... 1 2 3 where A A e1 A e2 A e3. 1! 2! 3! K M pK Mq2 pK Mq3 I ... (26) 1! 2! 3! C. Rotations in space if K M M K. In (26) we use notations K2 K K, K3 K K K, etc. One may now expand both sides and p Ñ Consider now a rotation R : E E with axis n, where show that the coeﬃcient of Kl Mm is the same on both n n 1, and angle φ in accordance with the right hand sides. The calculation becomes the same as if K and M rule. We get in terms of standard vector algebra would be real numbers and the dot usual multiplication. But in that case we know the result from elementary p ¢ RA cos φ AK sin φ n AK Ak (15) calculus and thus (26) is true.

where Ak pA nqn and AK A ¡ Ak. In dyadic Theorem II.1. Let the dyadic K be antisymmetric. p K p notation we write RA R A with the dyadic Then the dyadic R e. is associated with a rotation R p deﬁned by RA R A. Furthermore, all rotations can R cos φ pI ¡ n b nq sin φ rn¢s n b n (16) be obtained in this way. p The expression (15) is obtained directly from our geo- Proof. To show that the linear transformation R corre- K metric understanding of rotations. However, when we sponding to the dyadic R e. indeed is a rotation we in the next section deal with 4D spacetime the natural must verify the algebraic conditions (17)–(18). Condi- generalization of rotations will be proper orthochronous tion (17) is equivalent to (19) and that (19) is true fol- Lorentz transformations. Then this geometric “method” lows directly by use of (24) and (25). From (19) we also p is not available to us. Instead we will use an algebraic get det R ¨1 and we need to show that the correct procedure and it is instructive to do so also in ordinary sign here is plus to verify ¨ (18). Consider the continuous 3D space. The algebraic conditions for Rˆ to be a rotation function ψpsq det e.sK for s ¥ 0. This function can are take values -1 or +1 but, due to continuity, it can just take one of these values. From ψp0q 1 it follows that p p pRAq pRBq A B for all A, B P E (17) det Rp ψp1q 1. det Rp ¡ 0 (18) Next we prove that all rotations can be obtained using the exponential representation in this theorem. Just where the second condition is necessary to keep the ori- choose the dyadic K φ rn¢s and ﬁnd that 10 entation unchanged. In dyadic notation we may write R exp . pφ rn¢sq (27) equation (17) as11 is the same dyadic as in (16). The simple derivation is RT R I (19) an application of equations (21) and (23). 5

III. THE LINEAR ALGEBRA FORMULATION An event x P M may accordingly be written OF SPACETIME x cte0 ~x (30) A. The axioms of spacetime P K where ~x e0 and c is the speed of light. By dotting with e we get the relations Let us now consider special relativity from a linear al- 0 gebra perspective. The reader is assumed to be familiar ¡1 t ¡c e x , ~x x pe xqe (31) with some introductory special relativity.3 We use the 0 0 0 symbol M to denote 4D spacetime and the space of 4- The time of the event x is t and its spatial position is vectors ( as in Minkowski). The linear algebra for- K K M ~x P e , both according to observer e0. The space e mulation of special relativity starts with the following 0 0 represents simultaneous events for e0, i.e., two events x axiomatic definition of M: and y are simultaneous if they happen at the same time t, i.e., if x ¡ y P eK. The concept of simultaneous events Axiom 1M. is a 4-dimensional real vector space. 0 M is thus observer dependent and not geometric. The interpretation of eK as the usual 3D space ac- Axiom 2M. An inner product is defined on M with sig- 0 p¡ q cording to observer e0 is more formally motivated by the nature , i.e., there exist some K r s following fact: The subspace e0 of M is 3D and spacelike ordered basis e e0 e1 e2 e3 of M such K ¡ (i.e., all nonzero vectors in e0 are spacelike). Note that that e0 e0 1, e1 e1 e2 e2 e3 e3 K 12 re e e s is a normal basis of e if e re e e e s 1 and eµ eν 0 if µ ν. 1 2 3 0 0 1 2 3 is a normal basis of M. It is easy to see that if we take K Axiom 3M. M is oriented with respect to space and E e0 and use the orientation and dot product induced time, i.e., for any ordered basis e like the from the M-axioms we get all the E-axioms satisfied. On K one in Axiom 2M we can tell if re1 e2 e3s e0 we now have the whole machinery of Gibbs’ vector is right or left handed and if e0 is future or calculus available. The dot product is directly inherited 13 past oriented. from M but the cross product depend on e0. The cross K product is extended from e0 to all of M by The formal similarity with the axioms defining E is obvious. However, the indefinite structure of the relativistic A ¢ B A~ ¢ B~ (32) dot product results in three different kinds of 4-vectors. A 4-vector A is timelike if A A 0, spacelike if A A ¡ 0 where we recall that the meaning of the arrows from (31). and null or lightlike if A A 0. Furthermore, if A is Note that both the arrow and the cross product are timelike or null then it is either future or past oriented. observer dependent. If there are two observers e0 and r s We define a normal basis e0 e1 e2 e3 of M if it sat- e01 present in the problem we use notations Ñ and ¢ isfy the properties in Axiom 2M and, in accordance with respectively Ñ1 and ¢1. Axiom 3M, re1 e2 e3s is right handed and e0 future It will be shown in Section VII D that we from oriented.14 the geometric (observer and coordinate-free) version of Maxwell’s equations may derive Maxwell’s equations in their standard form where the electric and magnetic fields p q p q K B. The split of spacetime into time-plus-space E t,~x and B t,~x take values in e0 (or we may say that they are 3-vectors understanding now that the term 3- The p¡ q signature of the dot product on M in- vector is observer dependent in contrast to the term 4- dicates one time dimension and three space dimensions. vector that is spacetime geometric). However, there exist no geometric (observer and/or coordinate independent) separation of spacetime into a space part plus a time part. In particular there is no universal C. The world line of a point object time so the statement that two events are simultaneous is not defined from the axioms of M. In order to obtain a The world line of a point object A is the set A M split of M into time-plus-space the additional structure of of all events it experiences (we here use the same name an observer is needed. Following Hestenes we associate for the object and for the set defining its world line). We an (inertial) observer with each future oriented vector assume the world line may be parameterized such that P ¡ e0 M of unit length e0 e0 1. We then get a space- t p q | P u time split of M into the following time and space parts A x s s R (33) K t u t P | u The time parameter s may be chosen quite arbitrarily as span e0 , e0 x M x e0 0 (28) long as the parameterization is sufficiently smooth and dx so that increases towards the future, i.e., we assume ds is timelike and future oriented. A choice of particular interest is the t u K M span e0 e0 (29) time shown by a clock traveling with the particle. We 6 denote this parameter s τ and it is defined by the where by choice t1 t 0 at some event P on the world- condition line. Clearly the velocity of the object A is v according dx dx to e0. ¡c2 (34) We may of course directly write the corresponding re- dτ dτ lations from the perspective of observer e01 and find the 1 ¡ K 1 ¡ 1 P Note that the world line is a geometric concept and no co- velocity v cγ e0 ce0 e01 of any point object ordinates of spacetime are needed for its definition. How- that is not moving according? to e0. The speeds are the 1 1 1 1 1 ever, as already mentioned, spacetime M has no intrinsic same, i.e., v v where v v v , but v ¡v while separation between space and time. A split of spacetime equality would have been expected from a nonrelativistic into time and space is obtained by an observer e0 and point of view. the worldline of an point object A may be written We may notice again and again that in terms of linear algebra special relativity is quite simple but in terms p q p q A : x t cte0 ~x t (35) of intuitive understanding it is difficult because it is This should be interpreted so that t is time according to counter-intuitive to our normal perceptions of space and time. We need both aspects and the linear algebra may the observer e0 and ~xptq is the path in the corresponding K help us to get confidence in calculations and improve our 3D space e0 and the set defining the worldline is A t p q | P u geometric feeling for the relativistic spacetime. cte0 ~x t t R . With two observers e0 and e01 it may of course be of 1 interest to compare e0-time t with e01 -time t . From the E. Dyadics in spacetime first equation in (31) we define t : M Ñ R and in the 1 same way we have t : M Ñ R. Since M is 4D and R is 1 1D it follows that in general we do not get t as a function In order to deal with linear transformations of M in of t. However, replacing with the 1D set A of a world a geometric coordinate-free way we may use 4-dyadics, M b line we obtain such a relation i.e., second order tensors in M 2 M b M. The procedure is the same as for and all relations in Section II B 1 1 p q E t tA t (36) generalize straightforwardly except for equation (8) that involves the cross product. We use in this subsection For the worldline A in terms of both observers we may A, B, C, D for 4-vectors and K, M, N for 4-dyadics. write A 4-dyadic K is a sum of 4-dyads A b B, i.e., K 1 1 1 b b ¤ ¤ ¤ A : cte0 ~xptq ct e01 xpt q (37) A B C D . We define for 4-dyads

1 1 K p q P p b q p q Here we use the notation x t e01 . The equality (37) A B C A B C (42) deﬁnes (36). A pB b Cq pA BqC (43)

From these we define the dot product of a 4-dyadic with D. Inertial observers a 4-vector in the obvious way. For each 4-dyadic K we associate a linear transformation Kˆ : M Ñ M defined by The observers we have defined are to be interpreted as inertial observers. Given two observers e0 and e01 we KAˆ K A (44) split e01 into time plus space according to e0 and define P P K γ R and v e0 so that The product of two 4-dyads is defined

1 p { q e0 γ e0 v c (38) pA b Bq pC b Dq pB CqA b D (45)

This implies that b from which we obtain the product K M P M 2 of two ¡1 γ ¡e0 e01 and v cγ e01 ¡ ce0 (39) 4-dyadics. This product corresponds to the composition of linear transformations From (38) and (39) we easily ﬁnd that? this is the familiar γ in relativity so with notation v v v Nˆ Kˆ Mˆ ô N K M (46) 1 γ a (40) The transpose of a 4-dyadic follow from its deﬁnition on ¡ 2{ 2 1 v c an 4-dyad

1 The observer e0 moves according to e0 with velocity v T as we can see by considering the world line of any point pA b Bq B b A object A that does not move according to e01 . From the The relation perspectives of e01 and e0 we ﬁnd the worldline

1 T T T A : ct e01 P cte0 tv P (41) pK Mq M K (47) 7 is valid for 4-dyadics K and M. We also deﬁne the “dou- In dyadic notation we may write (58) as (cf. Ref. 11) ble dot” product between 4-dyadics K : M P R, which T follows from the deﬁnition for dyads L L I (59)

pA b Bq : pC b Dq pA CqpB Dq (48) Equation (58) implies § § § § We define the anti-symmetric tensor product of 4-vectors det ˆL ¨1 , e0 ˆLe0 ¥ 1 (60) by where e0 is any observer. The first equation in (60) may A ^ B A b B ¡ B b A (49) be proved by first introducing a basis in (59) and then use matrix algebra.15 To derive the second equation in and the symmetric tensor product by (60) we write ¨ K _ b b ˆ ¡ ˆ P A B A B B A (50) Le0 e0 Le0 e0 a where a e0 (61) The space of anti-symmetric 4-dyadics is defined and obtain by use of (58) ( ¨ ^2 b2 T 2 K P | K ¡K (51) M M ¡ ¡e0 ˆLe0 a a ¡1 (62)

The unit 4-dyadic I is defined so that Î is the identity whereby the equation follows. transformation on M. With each observer e0 we associate In Section II we considered rotations in ordinary 3D K the 3D space e0 on which all the structure of the usual space E, which are precisely the transformations that 3D vector calculus is inherited from the axioms of M. We preserves all structure of the axioms for . In complete ^ E define a 4-dyadic rA¢s P M 2 such that analogy we now define a proper orthochronous Lorentz transformation so that all the structure of the axioms r ¢s ¢ A B A B (52) for M is preserved. The conditions for ˆL to be a proper orthochronous Lorentz transformation may be expressed for any 4-vector B. Consider a normal basis e as equation (59) together with the conditions re0 e1 e2 e3s and use the notations det ˆL 1 , e ˆLe ¤ ¡1 for any observer e (63) b ^ b ¡ b 0 0 0 eµν eµ eν , eµν eµ eν eν eµ (53) The conditions (63) may be interpreted as the conserva- Then tion of spacetime orientation and time orientation of the transform ˆL.16 I ¡e00 e11 e22 e33 (54) Example 1. A boost (also called pure Lorentz transfor- b2 A basis for M is the sixteen 4-dyads eµν mation) for the observer e0 is a proper orthochronous ^ and the space M 2 has a basis of six 4-dyadics Lorentz transformation ˆL Pˆ (we use P as in “pure”) t ^ ^ ^ ^ ^ ^ u e01, e02, e03, e12, e13, e23 . We get where the associated 4-dyadic may be written ¡ ^ ^ ^ p q 1 ^ r ¢s ~ ¢ ¡ 1 ¡ 2 ¡ 3 P e0, v γIk IK γc e0 v (64) A A A e23 A e31 A e12 (55) with The cross product is P K ¡ ¡2 b ¡ ¨ v e0 , Ik e00 v v v, IK I Ik (65) A ¢ B A~ ¢ B~ A2B3 ¡ A3B2 e (56) ¨1 1 2 2 1 Note that the boost (64) satisfies the relation e01 A B ¡ A B e3 ¨ Ppe0, vq e0 where e01 is the observer that has velocity v A3B1 ¡ A1B3 e 2 according to observer e0, see Section III D. By construction, the following identities hold A relation similar to (13) is

Ik Ik Ik , IK IK IK , Ik IK IK Ik 0 (66) A~ ^ B~ ¡ rpA ¢ Bq ¢s (57)

Ik pe0 ^ vq pe0 ^ vq Ik e0 ^ v (67)

IK pe0 ^ vq pe0 ^ vq IK 0 (68)

F. Active Lorentz transformations 2 pe0 ^ vq pe0 ^ vq v Ik (69)

An (active) Lorentz transformation is a linear trans- It is now easy to check by straightforward algebra that formation ˆL : M Ñ M that satisfies (64) satisfies (59). The conditions in (63) follow because P pe0, vq is continuously connected to the unity 4-dyadic pˆ q pˆ q P p q Ñ Ñ 16 LA LB A B for all A, B M (58) I (just note that P e0, v I when v 0). A boost 8 is a particularly simple Lorentz transformation because Solving these equations gives it is trivial on a 2D subspace a4 ¨γ, a2 ¨βγ (81) K Ppe0, vq A A for A P span te0, vu (70) and we thus obtain The boost is thus nontrivial only on spante , vu so it is 0 ¡ ¨ ¨ ¡ essentially a transformation just of this 2D space. Phys- P γe00 γe11 βγe01 βγe10 IK (82) ically we interpret (70) as the fact that the transforma- With the plus sign this is exactly (64). But why neglect tion Ppe , vq involves no spatial rotation according to 0 the minus sign solution above? One way to see this is to observer e . 0 consider the limit of small β and require that P I in By use of (38) and the first equation in (39) we may this limit.16 rewrite the boost expression (64) in terms of e0, e01 and Example 2. A rotation for the observer e0 is a Lorentz γ ¡e e 1 . We get 0 0 transformation ˆL Rˆ that may be written

1 _ I pe00 e0101 ¡ γe 1 1 q (71) Rpe , φnq I cos φ IK sin φ rn¢s (83) k γ2 ¡ 1 0 0 0 k ¡1 γc e0 ^ v e001 ¡ e010 (72) where n n 1 and P K ¡ b ¡ where n e0 , Ik e00 n n , IK I Ik (84)

p q e001 e0 b e01 , e010 e01 b e0 (73) Note that e0 R e0, φn e0 and this property exactly _ 1 1 1 b 1 1 b b 1 defines those proper orthochronous Lorentz transforma- e0 0 e0 e0 , e010 e0 e0 e0 e0 (74) tions that are rotations according to the observer e0. p q and find Also, n R e0, φn n so the rotation is trivial on a 2D space Ppe0, vq Ppe0 Ñ e01 q ˆ p q P t u 1 _ (75) R e0, φn A A for A span e0, n (85) p 1 1 q ¡ 1 I e00 e0 0 e001 2e0 0 γ 1 If we consider the restriction of the transformation ˆ p q K Can we be sure that equation (75) defines the only boost R e0, φn to e0 we get a map (according to e ) such that e Ñ e 1 ? Yes, this is implied 0 0 0 Rˆ pe , φnq : eK Ñ eK (86) by the following derivation. 0 0 0 Derivation of the boost formula (64): Given the observers which is an ordinary 3D rotation with axis n as discussed e0 and e01 we like to find a proper orthochronous Lorentz in Section II. Ñ 1 transformation that maps e0 e0 and which also is free Let us now consider the generalization of Theorem II.1 from spatial rotation according to observer e0. Such a to spacetime. The exponential of a 4-dyadic is analo- transformation is called a boost and the associated 4- gously to (23) defined dyadic P must satisfy

K K K K K K K 1 e. exp .K I ... (87) P e0 e0 (76) 1! 2! 3! ^ and Theorem III.1. Any K P M 2 determines a proper or- ˆ K thochronous Lorentz transformation LA L A where L P A A for A P span te0, e01 u (77) is the 4-dyadic

K An orthonormal basis of the subspace spante0, e01 u is L e. (88) te0, e1u where e1 v{v. We make an ansatz All proper orthochronous Lorentz transformations are ob- P a1e00 a2e01 a3e10 a4e11 IK (78) tained in this way. For boosts and rotations we have so that (77) is satisﬁed for all real a1, a2, a3 and a4. From Ppe0, vq exp . rpw{vqe0 ^ vs (89) (38) we have e 1 γe γβe where β v{c so by use 0 0 1 Rpe0, φnq exp . pφ rn¢sq (90) of (76) and (78) we get where the rapidity w wpvq may be obtained from ¡ ¡ a1 γ, a3 γβ (79) cosh w γ.

T Using P P I we find Proof. The equations (24) and (25) are valid for 4-dyadics K, M and, in complete analogy with the case of rotations 2 ¡ 2 a1a2 a3a4, a4 a2 1 (80) in E, it follows that L defined in the theorem satisfies 9

(59). It remains to be shown that also (63) is satisﬁed. IV. COORDINATES AND MATRICES IN We deﬁne the 4-dyadic RELATIVITY

sK Ls e. (91) Linear transformations on M are called active transformations because they move one 4-vector to a diﬀerent and consider the functions of a real parameter s 4-vector. No coordinates are in principle needed for active transformations but it is anyhow often useful to use

φpsq e0 ˆLspe0q and ψpsq det ˆLs (92) coordinates in their study. This may be achieved by using a single ordered basis for M. Passive transformations refer to coordinate changes so for these two ordered bases Since Ls satisfies (60) we know that must be involved. If these are normal bases then the pas- |φ psq| ¥ 1 and |ψ psq| 1 (93) sive transformation is interpreted in terms of change of inertial observer and the coordinate transformation represents comparison between measurements of the two ob- Continuity of these functions and φp0q ¡1 and ψp0q servers. Thus, passive transformations are of particular 1 now implies that φpsq ¤ ¡1 and ψpsq 1 for all s and significance in special relativity. However, choosing the ˆ in particular for s 1. We conclude that L is a proper linear algebra perspective of the present paper, it is nat- orthochronous Lorentz transformation. ural to also consider active transformation. Active and We omit proving that every proper orthochronous passive transformations are related in an essentially triv- Lorentz transform have an exponential representation ial manner but in practice anything concerned with vari- (88) but rather consider the important cases of boosts able transformations easily causes confusion. To avoid and rotations. To verify that all boosts can be ob- such problems we introduce simple and practical rules of tained using the exponential representation (89) we take ^ ^ ^ manipulation that builds on formal matrix algebra. ¡ K we01. Using the identities e01 e01 e00 e11 Ik ^ ^ and Ik e01 e01 in the power series defining the exponential function we find from familiar potential series17 A. 4-Vectors, 4-dyadics and matrices

p ^ q ^ L exp we01 cosh wIk sinh we01 IK (94) In this subsection we consider a general ordered basis r s f f 0 f 1 f 2 f 3 for the vector space M. We use the By choosing w such that letter f to remind the reader that we allow for general bases while in the particular case of a normal basis we tanh w β v{c (95) use the letter e like in e re0 e1 e2 e3s. The reason to consider general ordered bases in this paper is partly we get cosh w γ and sinh w βγ and the boost 4- pedagogical, as it is no more difficult than to consider dyadic (64) follows. In a similar way we verify that the just normal bases and we avoid introducing additional exponential representation for rotations (90) is equivalent structure at the start. General ordered bases are also to (83). common in applications even though we give no such examples here. For a 4-vector A we define a 4-by-1 (column) matrix Any proper orthochronous Lorentz transformation is, r µs P 4¢1 18 Af A R so that for an observer e0, a unique product of a rotation followed by a boost. A0 A fA f ¤ ¤ ¤ f . (97) Theorem III.2. Let ˆL be a proper orthochronous f 0 3 . A3 Lorentz transformation and e0 an observer. Define the p Ñ ˆ q T 0 3 4-dyadics P P e0 Le0 and R P L. Then A f 0 ¤ ¤ ¤ A f 3 (98) ˆL Pˆ Rˆ where Pˆ is a boost and Rˆ is a rotation according With a 4-dyadic K we associate a 4-by-4 matrix K f to the observer. ¢ rKµν s P R4 4 so that T ¸ Proof. We have P R P P L and it follows, by use K Kµν f (99) of equation (59), that P R L. Thus ˆL Pˆ Rˆ . By µν ˆ definition we know P is a boost so is only remains to be summing over all µ, ν 0, 1, 2, 3 (16 terms). The matrix shown that Rˆ is a rotation according to the observer. We ηf is defined by have ¡ © ηf rf µ f ν s (100) ¡1 ˆ ˆ ˆ ˆ Ñ ˆ Re0 P Le0 P Le0 e0 Le0 e0 (96) We may use the following formulas to translate expressions in 4-vector and 4-dyadic formalism to matrix alge- and the theorem follows. bra. The superscript T is already used for transposing 10 dyadics and it will now also be used on matrices. Theorem IV.1. (a) The change of basis for a vector A and a 4-dyadic K are given by the matrix formulas p b q T A B f Af Bf (101) T A 1 f 1 A , K 1 f 1 K pf 1 q (112) p q T f f f f f f f A B f Af ηf Bf (102) p q 1 K M f K f ηf Mf (103) (b) The transformation Sˆ Sˆ pf Ñ f q satisfies the matrix formulas pK Aq K f ηf Af (104) f ¡ © p q T 1 1 A K Af ηf K f (105) ˆ ˆ ˆ f SA f Af , SA Af , Sf ff (113) f 1 We also, by use of (103) with K M I, find that 1 1 (c) The matrices ff and ff satisfy ¡1 If ηf (106) 1 1 ff ff 14¢4 (114) For the associated linear transformation Kˆ : Ñ we M M where 1 ¢ denotes the unit 4-by-4 matrix. define a 4-by-4 matrix Kˆ in the standard way so that 4 4 f (d) The change of basis for a linear transformation Kˆ is given by KAˆ B ô Kˆ f Af Bf (107) 1 ˆ 1 1 ˆ K f ff K f ff (115) By definition KAˆ K A so by use of (104) we get Proof. The proofs consist of simple observations guided Kˆ f K f ηf (108) in particular by the formal matrix notations. (a). The calculation B. Active and passive linear transformations in A0 matrix language . 1 r 1 ¤ ¤ ¤ 1 s ff Af f 0f f 3f . (116) A3 A linear transformation Kˆ : M Ñ M is always an ac- 0 3 tive transformation because it actually changes a phys- A f 1 ¤ ¤ ¤ A f 1 (117) 0f 3¨f ical event or 4-vector. No coordinates are essential to 0 3 A f ¤ ¤ ¤ A f A 1 (118) define an active transformation but it is of course pos- 0 3 f 1 f sible to express it in terms of coordinates like in (108) gives the first equation in (112). For the corresponding ˆ 4¢1 Ñ 4¢1 where K f : R R .A passive linear transfor- result for the 4-dyadic K it is sufficient to consider the mation concerns changes of coordinates so the use of two 1 special case of a 4-dyad K A b B. Using (101), the r s r 1 1 1 1 s ordered bases f f 0 f 1 f 2 f 3 and f f 0 f 1 f 2 f 3 first equation in (112) and matrix algebra we now get is needed. We then would like to know how to get Af 1 , 1 ˆ 1 ˆ T K f and K f from Af , K f and K f . p b q 1 A B 1 Af Bf 1 (119) In order to obtain simple expressions and transparent f T derivations of the basically trivial, but yet in practice pff 1 Af q pff 1 Bf q (120) often confusing, questions concerning active and passive T T ff 1 Af B pff 1 q (121) transformations the following notation is useful f T 1 p b q p 1 q ff A B f ff (122) r s r s ABCD f Af Bf C f Df (109) and the second equation in (112) follows. A0 B0 C0 D0 (b). To obtain these results we start with . . . . . . . . (110) 3 3 3 3 0 3 A B C D SAˆ A f 01 ¤ ¤ ¤ A f 31 (123)

Thus a lower index f on a 1-by-4 matrix with entries whereby the two ﬁrst results in (113) directly follow. This from M results in a 4-by-4 real matrix. Our applications is just about true also for the third result but we include of this notation involve the 4-by-4 real matrices the following short derivation ¡ © 1 f rf 1 f 1 f 1 f 1 s rf 1 f 1 f 1 f 1 s (111) ˆ 0 1 3 1 f 0 1 2 3 f 0 f 1 f 2 f 3 f SA A f 0 f ¤ ¤ ¤ A f 3 f (124) f 0 and the corresponding for ff 1 . A 1 . 1 With the two bases f and f we also associate the linear 1 ¤ ¤ ¤ 1 1 f 0 f f 3 f . ff Af (125) transformation Sˆ Sˆ pf Ñ f q deﬁned by Sfˆ f 1 for . µ µ 3 µ 0, 1, 2, 3. A 11

(c). This result is obtained by use of the first equation appear for example in the real matrices xe and Je where in (112) and the corresponding formula where f and f 1 x is the 4-vector of an event and J is the 4-current den- change place: sity. Another example is F e where F is the electromagnetic field dyadic (i.e., a second order tensor). We may in 1 1 1 1 ff ff Af ff Af Af (126) these matrices identify numbers for position, time, charge and current densities, electric and magnetic fields as mea- (d). This result is obtained as sured in the inertial frame defined by the basis e. 1 ¡ © ¡ © Given a second normal basis e re01 e11 e21 e31 s we Kˆ f 1 Af 1 KAˆ ff 1 KAˆ (127) would like to be able to compare the measurements in e f 1 f and e1. From Theorem IV.1 we get the matrix relations 1 1 ˆ 1 ˆ 1 ff K f Af ff K f ff Af (128) T xe1 ee1 xe , Je1 ee1 Je , F e1 ee1 F epee1 q (132) and this concludes the proof. The matrix e 1 in (132) determines the passive Lorentz The dot product was not used at all in the above e transformations x Ñ x 1 and J Ñ J 1 as well as the derivations but, by definition (44), it appears in the rela- e e e e corresponding transformation F Ñ F 1 . Also an active tion between K and Kˆ . As a check let us now show that e e Lorentz transformation ˆL : M Ñ M is associated with the second equation in (112) involving K and the equa- the two given normal bases defined by tion (115) involving Kˆ are consistent with this relation. By use of (108) in (115) we get ˆLeµ eµ1 for µ 0, 1, 2, 3 (133) 1 ¡1 1 1 K f ff K f ηf ff ηf 1 (129) The corresponding 4-dyadic is then For (129) to be the same as the second equation in (112) L ¡e010 e111 e212 e313 (134) the following must hold

1 1 b 1 ¡1 T where eµ µ eµ eµ. An active Lorentz transformation η f η 1 f 1 (130) f f f f may in principle be deﬁned without refering to any basis. However, we are free use a normal basis e and discuss By use of (114) and (106) this condition may be written active Lorentz transformations using real matrices ˆLe : 4¢1 4¢1 T Ñ 1 1 R R . From (113) we have If ff If ff 1 (131) ˆ 1 which is a valid relation because it is just a special case Le ee (135) of the second equation in result (112). and we note that the matrix of the active Lorentz transformation (135) is the inverse of the matrix for the pas- C. Passive Lorentz transformations and sive one in (132). The inverse of a Lorentz transformation measurements (135) is most easily obtained by the use of the relation (59) from which we get the matrix formula In the two previous subsections A and B we allowed ¡ for general ordered bases. We will now consider nor- 1 1 T 1 mal bases. Physically a choice of normal basis e e 1 η pe q η where η (136) e e e e e 1 re0 e1 e2 e3s is interpreted as an observer e0 equipped 1 with a right handed orthonormal basis re1 e2 e3s for the K 3D space e0 . Measurements in terms of real numbers

Let us now give explicit expressions for the matrices in the case of a boost. The boost dyadic is given by (64) and (65). We get from these

e01 Ppe0, vq e0 γpe0 v{cq (137) 2 ek1 Ppe0, vq ek ek pγ ¡ 1qvkv{v γvke0{c for k 1, 2, 3 (138) where vk v ek. We get γ γv1{c γv2{c γv3{c 2 2 2 2 1 γv1{c 1 pγ ¡ 1q v {v pγ ¡ 1q v1v2{v pγ ¡ 1q v1v3{v 1 1 1 1 1 ee e0 e e1 e e2 e e3 e { p ¡ q { 2 p ¡ q 2{ 2 p ¡ q { 2 (139) γv2 c γ 1 v1v2 v 1 γ 1 v2 v γ 1 v2v3 v { p ¡ q { 2 p ¡ q { 2 p ¡ q 2{ 2 γv3 c γ 1 v1v3 v γ 1 v2v3 v 1 γ 1 v3 v 12

The matrix appearing in the passive boost is the inverse of this matrix which may easily be found by use of equation (136) as γ ¡γv1{c ¡γv2{c ¡γv3{c 2 2 2 2 ¡γv1{c 1 pγ ¡ 1q v {v pγ ¡ 1q v1v2{v pγ ¡ 1q v1v3{v 1 1 ee ¡ { p ¡ q { 2 p ¡ q 2{ 2 p ¡ q { 2 (140) γv2 c γ 1 v1v2 v 1 γ 1 v2 v γ 1 v2v3 v ¡ { p ¡ q { 2 p ¡ q { 2 p ¡ q 2{ 2 γv3 c γ 1 v1v3 v γ 1 v2v3 v 1 γ 1 v3 v

We may write the passive boost The Lorentz transformation most textbooks start with is for reference frames in so called standard conﬁguration. 1 xe1 ee1 xe (141) In terms of two normal bases e and e this corresponds to the conditions that e2 e21 and e3 e31 . The matrix in a compact and familiar “vector” form by using nota- 1 1 1 1 for the corresponding passive Lorentz transformation is tions v pv1, v2, v3q, r px, y, zq and r px , y , z q 1 1 obtained from (140) with v2 v3 0 and may be written where x cte0 xe1 ye2 ze3 ct e01 x e11 1 1 y e21 z e31 . We use the “dot” as the standard inner product on R3 and may then we may write 1 ¡ t γpt ¡ v r{c2q (142) γ γβ 0 0 ¡ 1 γβ γ 0 0 p ¡ q e 1 where β v{c (146) r rK γ rk vt (143) e 0 0 1 0 0 0 0 1 2 where rk pr vqv{v and rK r ¡ rk. The inverse 1 1 transformation xe eexe may be obtained by changing v to ¡v and we get

1 1 t γpt v r {c2q (144) We note that the matrices appearing are symmetric but this is a feature of boosts and not of general Lorentz 1 p 1 1q r rK γ rk vt (145) transformations.

Let us for example consider rotations. The associated 4-dyadic is given by (83) and (84). We get from these

e01 Rpe0, φnq e0 e0 (147)

ek1 Rpe0, φnq ek cos φek p1 ¡ cos φqnkn sin φn ¢ ek for k 1, 2, 3 (148)

We then get the non-symmetric matrix 1 0 0 0 p ¡ q 2 p ¡ q ¡ p ¡ q 1 0 cos φ 1 cos φ n1 1 cos φ n1n2 sin φ n3 1 cos φ n1n3 sin φ n2 e e p ¡ q p ¡ q 2 p ¡ q ¡ (149) 0 1 cos φ n1n2 sin φ n3 cos φ 1 cos φ n2 1 cos φ n2n3 sin φ n1 p ¡ q ¡ p ¡ q p ¡ q 2 0 1 cos φ n1n3 sin φ n2 1 cos φ n2n3 sin φ n1 cos φ 1 cos φ n3

D. The determinant and trace of a linear Kˆ : M Ñ M we then define the determinant and trace by transformation det Kˆ det Kˆ f , tr Kˆ tr Kˆ f (150) The determinant and trace of a linear transformation where the right hand sides are known from matrix theory may be defined without reference to any basis. This may and f is a general ordered basis. In order for equation be unfamiliar to some students who have only consid- (150) to make sense we must check that this definition ered the determinant and trace of a square matrix. How- does not depend on our choice of basis f. This follows ever, the most satisfactory way of introducing these con- from the fact that the matrices Kˆ f and Kˆ f 1 are similar cepts in linear algebra is for a linear transformation using (see equations (115) and (114)). coordinate-free definitions (see Halmos in Ref. 7). In this There is a relation between the double dot product section we will however refer to square matrices assuming (48) and the trace the reader already knows some basic properties of deter- ¨ minants and traces for these. For a linear transformation {T G : H tr G H (151) 13

To prove this we first need the trace of a linear transfor- Then the coefficients in the expansions mation associated with a dyad ¸3 ¸3 ¨ 1 { a aiei , a ai1 ei1 (159) tr A b B A B (152) i1 i11 Using a normal basis e we easily see that (152) follows and in the analogous expansions for b and b1 satisfy from the definition of the trace (150) and the relation ¨ a11 a1 , b11 b1 {b T A B e AeBe ηe (153) a21 γ pa2 βb3q , b21 γ pb2 ¡ βa3q (160) a 1 γ pa ¡ βb q , b 1 γ pb βa q It is sufficient to prove equation (151) when G A b B 3 3 2 3 3 2 and H is a general dyadic. Using equation (152) we get Proof. From (158) we get the matrix relation ¨ {T tr G H tr B b pA Hq (154) 0 a1 a2 a3 b ¡ ¡ a1 0 b3 b2 p q : Ge M A, B (161) A H B G H (155) ¡a2 b3 0 ¡b1 ¡a3 ¡b2 b1 0 and the result (151) follows. where A pa1, a2, a3q and B pb1, b2, b3q. We also have the corresponding equation in the primed system. We V. ANTI-SYMMETRIC 4-DYADICS get the result (160) by use of (146) in

T We have seen that the anti-symmetric 4-dyadics play a Ge1 ee1 Gepee1 q (162) role in Lorentz transformations and also, the electromagnetic field dyadic is anti-symmetric (see the next section). cf. (112). The importance of anti-symmetric 4-dyadics in relativ- ^2 Ñ The observer e0 defines the star operator : M ity is rather obvious from a mathematical point of view ^2 M by because they represent elements of the Lie algebra associated with the Lorentz group. We now consider a cou- G pe0 ^ a rb¢sq ¡ ra¢s e0 ^ b (163) ple of simple but important properties of anti-symmetric P K 4-dyadics needed in the formulation of basic electromag- where a, b e0 . netic theory. Theorem V.3. The star operator is actually geometric P ^2 in the sense that it does not depend on the choice of ob- Theorem V.1. Let G M and e0 be an observer. P K server e0. Then there the exist a, b e0 such that 1 ^ Proof. Let us define a second star operator : M 2 Ñ G e0 ^ a rb¢s (156) ^2 M by referring to a second observer e01 so that Furthermore, these a and b are uniquely determined. 1 1 1 1 G ¡ a ¢ e01 ^ b (164) Proof. First we prove the existence of a and b. It is We must prove that 1G G. Choosing normal bases sufficient to consider the particular case when G C ^D 1 e re e e e s and e re 1 e 1 e 1 e 1 s in standard P 0 1 2 3 0 1 2 3 where C, D M because a general anti-symmetric 4- configuration we have dyadic may be written as a sum of such terms. We easily ¨ ¨ find p q p ¡ q 1 1 ¡ 1 G e M B , A , G 1 M B , A (165) e C ^ D e ^ pC0D~ ¡ D0C~ q pD~ ¢ C~ q¢ (157) p1 q p q 0 It is sufficient to prove that G e1 G e1 . We have T p q 1 p q p 1 q from (132) that G e1 ee G e ee so we just need where we used an equality like (57). Now (157) is of the the result same form as (156) and the existence part of the theorem ¨ 1 1 T follows. For the uniqueness part it is sufficient to show M B , ¡A ee1 M pB , ¡Aq pee1 q (166) that if G 0 then a b 0. This follows by dotting (156) with a general 4-vector. The proof of (166) is just a straightforward but slightly tedious calculation involving products of 4-by-4 matrices. r s 1 Theorem V.2. Let e e0 e1 e2 e3 and e It is also possible to find (166) without calculations by r 1 1 1 1 s e0 e1 e2 e3 be two normal bases in standard configu- use of equation (162), i.e., P ^2 P K 1 1 P K ration and let G M . Take a, b e0 and a , b e01 ¨ 1 1 T such that (these are by Theorem V.1 uniquely defined) M A , B ee1 M pA , Bq pee1 q (167) 1 1 1 G e0 ^ a rb¢s e01 ^ a b ¢ (158) and inspection of (160). 14

VI. THE NATURAL DERIVATIVES IN The vector and scalar d’Alembertians satisfy the relation SPACETIME ¸3 ∆ A p∆ Aµq e (177) In ordinary vector calculus the grad, div and curl M M µ µ0 derivatives appear naturally as geometric derivatives, i.e., they are independent of coordinates. The gradient has We would like to have coordinate free definitions in a simple geometric interpretation. The divergence and spacetime of the derivatives that correspond to the grad, Stoke’s theorems may be used to show the geometric div and curl derivatives in ordinary space. Let us first meaning of the div and curl derivatives. The formal nabla define the spacetime gradient ∇∇∇∇∇∇∇φ : M Ñ M of a scalar B B B operator ∇ ex x ey y ez z is expressed in terms function φ : M Ñ R by of Cartesian coordinates but it is essentially a coordinate independent “vector operator” and turns out to be an φpx ∆xq ¡ φpxq ∆x ∇∇∇∇∇∇∇φpxq O p|∆x ∆x|q (178) extremely convenient notational tool while dealing with p q the div, grad and curl derivatives. Examples of much Then we define the divergence of a 4-vector field φ x A p q used identities in electromagnetic theory are the scalar and a 4-dyadic field φ x K where A is a constant 4- Laplace operator vector and K a constant 4-dyadic by

∇∇∇∇∇∇∇ pφAq p∇∇∇∇∇∇∇φq A and ∇∇∇∇∇∇∇ pφKq p∇∇∇∇∇∇∇φq K (179) ∆φ divpgradpφqq ∇ p∇φq (168) The general case is obtained from the fact that any 4- and the vector Laplace operator vector or 4-dyadic field may be written as a sum of such terms. A “curl-like” derivative ∇∇∇∇∇∇∇^ may be defined in a ∆A gradpdiv Aq ¡ curlpcurl Aq (169) similar way ∇p∇ Aq ¡ ∇ ¢ p∇ ¢ Aq ∇∇∇∇∇∇∇ ^ pφAq p∇∇∇∇∇∇∇φq ^ A (180) as well as a couple of identities involving second order derivatives Three identities involving second order derivatives in spacetime are given in the following theorem. curlpgrad φq ∇ ¢ p∇φq 0 (170) Theorem VI.1. (a) Any field φ : M Ñ R satisfies divpcurl Aq ∇ p∇ ¢ Aq 0 (171) ∇∇∇∇∇∇∇ ^ p∇∇∇∇∇∇∇φq 0 (181) There are similar coordinate-free spacetime relations ^ that may be used in the 4D geometric formulation of (b) Any field G : M Ñ M 2 satisfies Maxwell’s equations. The nabla operator in Minkowski space becomes ∇∇∇∇∇∇∇ p∇∇∇∇∇∇∇ Gq 0 (182) Ñ ¡ B (c) Any field A : M M satisfies ∇∇∇∇∇∇∇ ¡e c 1 ∇∇∇∇∇∇∇~ (172) 0 Bt pp ^ qq B B B B ∇∇∇∇∇∇∇ ∇∇∇∇∇∇∇ A 0 (183) ¡1 ¡e0c e1 e2 e3 (173) Bt Bx1 Bx2 Bx3 Proof. (a) Straightforward. (b) It is sufficient to prove the identity (182) in the particular case G pxq φ pxq C ^ r s where e e0 e1 e2 e3 is a normal basis. In spite D where C and D are constant 4-vectors. In this case of the use of a basis in the expression (172) it is in fact the derivation of the result is straightforward. (c) It is a coordinate-free object in spacetime when it appears in sufficient to prove the identity (183) in the particular case the expressions below. This is suggested by the defini- A pxq φ pxq C. Then tions of the 4D gradient, the 4D divergence and a 4D curl-like operator given below in this section. ∇∇∇∇∇∇∇ ^ A p∇∇∇∇∇∇∇φq ^ C (184) The scalar Laplace operator in space corresponds to the scalar d’Alembertian in spacetime We write this in space-plus-time form ¡ © B2 B2 B2 B2 p q ^ ^ ¡ ¡1B ~ ¡ 0 ~ ¡ ¡2 ∇∇∇∇∇∇∇φ C e0 c tφ C C ∇∇∇∇∇∇∇φ (185) ∆M c 2 2 2 2 (174) ¡¡ © © Bt Bx1 Bx2 Bx3 ¡ ∇∇∇∇∇∇∇~ φ ¢ C~ ¢ For scalar fields we get and use the definition (163) of the star operator p q ∆M φ ∇∇∇∇∇∇∇ ∇∇∇∇∇∇∇φ (175) ¡ © ¡1 ~ 0 p∇∇∇∇∇∇∇φq ^ C c Btφ C C ∇∇∇∇∇∇∇~ φ ¢ (186) For vector d’Alembertian in spacetime ¡¡ © © ~ ¡ e0 ^ ∇∇∇∇∇∇∇~ φ ¢ C ∆M A ∇∇∇∇∇∇∇p∇∇∇∇∇∇∇ Aq ∇∇∇∇∇∇∇ p∇∇∇∇∇∇∇ ^ Aq (176) 15

To compute the divergence of this term the following B. The electromagnetic ﬁeld dyadic and the Ñ K two identities are useful. By viewing a ﬁeld a : M e0 equation of the motion for a charged particle as a function of time and space, i.e., apxq apt,~xq, we deduce the following relationships We consider equation (190) in terms of an observer K e0. The space e0 with the dot product and orientation 1 Ba p ^ q ¡ p ~ q inherited from M satisfy all the three E-axioms. Any ∇∇∇∇∇∇∇ e0 a ∇∇∇∇∇∇∇ a e0 (187) ^ c Bt F P M 2 may, in one and only one way, be written (see ∇∇∇∇∇∇∇ ra¢s ∇∇∇∇∇∇∇~ ¢ a (188) Section V) ¡1 ^ ¡ r ¢s where 4D Nabla appears on the left hand sides and 3D F c e0 E B (194) Nabla on the right. Using these identities we readily get P K with E and B e0 . Consider now the worldline of a charged particle A. ∇∇∇∇∇∇∇ rp∇∇∇∇∇∇∇φq ^ Cs 0 (189) We use proper time τ or e0-time t as parameter for the worldline. The condition that they correspond to the and the result follows. same event of the worldline give relations t tpτq or equivalently τ τptq. On the worldline of A VII. ELECTRODYNAMICS x cte0 ~x (195) A. Geometric equations for electrodynamics take the τ derivative

In this subsection the geometric nature of the ba- dx dt pce0 vq (196) sic equations in electrodynamics is emphasized. The dτ dτ equations are therefore given without any choice of ob- d~x server or coordinates. The electromagnetic ﬁeld dyadic where v dt is the ordinary velocity according to ob- ^2 server e . Using (196) in (34) we get F : M Ñ M determines the worldline xpτq of a charged 0 particle by the equation dt γ (197) dτ d2x dx m qFpxq (190) dτ 2 dτ The equation (190) may by use of (194), (196) and (197) be written The condition (34) for τ to be proper time is consistent d with (190) thanks to the anti-symmetry of F . m pγce0 γvq (198) The Maxwell’s equations in geometric form may now dτ ¨ ¡1 be written q c e0 ^ E ¡ rB¢s pγce0 γvq which we rewrite as the two equations ∇∇∇∇∇∇∇ F ¡µ0J , ∇∇∇∇∇∇∇ pFq 0 (191)

dγ ¡ 1 where the field J : M Ñ M is a current density spacetime mc γqc v E dτ (199) vector field. The divergence of a vector field J is defined d in a similar way as the divergence on a 4-dyadic field. m pγvq γqpE v ¢ Bq The equation for charge conservation is dτ d ¡1 d From (197) we get dt γ dτ which we use to rewrite ∇∇∇∇∇∇∇ J 0 (192) (199) as

This equation is built into the mathematical structure of dγ mc2 qv E Maxwell’s equations since dt (200) d m pγvq qpE v ¢ Bq ∇∇∇∇∇∇∇ p∇∇∇∇∇∇∇ Fq 0 (193) dt and thus as follows from Theorem VI.1. The equations above are coordinate-free in spacetime dE and no observer has been introduced. How are they re- qv E dt (201) lated to the standard Maxwell equations and the stan- dp qpE v ¢ Bq dard equation of motion for a charged particle acted on dt by the Lorentz force? This will be explained in the following subsections. where E γmc2 and p γmv. 16

C. Transformation of the electromagnetic field from Theorem VI.1 if we define a vectorfield A as the electromagnetic potential by Consider the electromagnetic tensor F P ^2 in terms M F ∇∇∇∇∇∇∇ ^ A (210) of a normal basis e. From (194) Then what remains of Maxwell’s equations become 0 E1{c E2{c E3{c ¡E1{c 0 B3 ¡B2 ∇∇∇∇∇∇∇ p∇∇∇∇∇∇∇ ^ Aq ¡µ0J (211) F e (202) ¡E2{c ¡B3 0 B1 However, this equation does not determine A because to ¡E3{c B2 ¡B1 0 any solution you may add a an arbitrary 4D gradient, i.e., Consider now a new normal basis e1. We then get from A Ñ A ∇∇∇∇∇∇∇φ, to get a new solution. We need a gauge (132) condition to fix this. One possibility is the Lorentz gauge

T ∇∇∇∇∇∇∇ A 0 (212) F e1 ee1 F epee1 q (203) We then may rewrite (211) as Let us take the new basis e1 is in the standard conﬁgura- tion with respect to e so that we may use equation (146) ∆M A ¡µ0J (213) and get where ∆M is the d’Alembert vector operator (176). E 1 E ,B 1 B 1 1 1 1 ¨ 2 E 1 γ pE ¡ vB q ,B 1 γ B vE {c (204) 2 2 3 2 2 3 ¨ F. The stress-energy dyadic 2 E31 γ pE3 vB2q ,B31 γ B3 ¡ vE2{c The stress-energy dyadic is deﬁned in geometric space- These are the standard relations appearing in many text- time notation by books. ¢ 1 1 T ¡ F F pF : FqI (214) µ0 4 D. Maxwell’s equations in standard form It is symmetric and the trace of the associated linear transformation is zero, i.e. Let us write Maxwell’s equations in terms of the observer e0. By use of (187) and (188) we obtain T T T , tr Tˆ 0 (215)

¡ BE ¡ It also satisfy the conservation relation ∇∇∇∇∇∇∇ F c 2 ¡ c 1p∇∇∇∇∇∇∇~ Eqe ¡ ∇∇∇∇∇∇∇~ ¢ B (205) Bt 0 ∇∇∇∇∇∇∇ T F J 0 (216) ¡ BB ¡ p q 1 ¡ p ~ q 1 ~ ¢ ∇∇∇∇∇∇∇ F c B ∇∇∇∇∇∇∇ B e0 c ∇∇∇∇∇∇∇ E (206) t The equations (215) follows from (214), the anti- With symmetry of F and the equations (47) and (151). The conservation law (216) may be obtained from the ~ Maxwell’s equations (191) and some algebra. The nota- J cρe0 J (207) tion we rewrite ∇∇∇∇∇∇∇ F ¡µ J as 0 ∇∇∇∇∇∇∇ eµB (217) ¢ µ BE ρ ~ is convenient to get the derivatives right. Here the sum- ∇∇∇∇∇∇∇~ ¢ B µ0 J ε0 , ∇∇∇∇∇∇∇~ E (208) B 0 t ε0 mation over µ 0, 1, 2, 3 is understood. Also e ¡e0 j B B and e ej for j 1, 2, 3 and µ Bxµ . Applying the p q From ∇∇∇∇∇∇∇ F 0 we get the remaining two Maxwell’s 4-divergence to equation (214) gives equations ¢ 1 BB ∇∇∇∇∇∇∇ T ¡ p∇∇∇∇∇∇∇ Fq F (218) ~ ¢ ¡ ~ µ0 ∇∇∇∇∇∇∇ E B , ∇∇∇∇∇∇∇ B 0 (209) t 1 eµ F B F ∇∇∇∇∇∇∇pF : Fq µ 4 E. Maxwell’s equations in terms of potentials which by use of Maxwell’s equations and the anti- symmetry of F may be written With relativistic potentials we like to represent the ¢ electromagnetic dyadic ﬁeld F so that the equations (209) 1 µ 1 ∇∇∇∇∇∇∇ T F J ¡ e F BµF ∇∇∇∇∇∇∇pF : F (219) are automatically taken care of. This happens, as follows µ0 4 17

We must now show that the right hand side of (219) is VIII. DISCUSSION zero. Let us introduce an observer e0 and write F ^ { ¡ r ¢s e0 E c B . We get after a little algebra This is the first part of a two-part paper with the Minkowski vectorspace as a common starting-point. In F : F ¡2E E{c2 2B B (220) this Part I we use a simplified 4-tensor calculus involving only 4-vectors and 4-dyadics but no higher order tensors. and This makes it possible to use coordinate-free spacetime geometric notations, more closely related to Gibbs’ vector ¢ 1 calculus than the standard tensor formalism. The grad, eµ F B F ¡e B E E (221) div and curl derivatives generalize to space-time and we µ 0 0 2 may write Maxwell’s equations in a space-time geometric 2 ¡ E ¢ B0B{c E ∇∇∇∇∇∇∇~ E{c form. We also write such equations for Lorentz transformations and for the motion of charged particles. By ¡ B p∇∇∇∇∇∇∇~ ¢ Eqe0{c p∇∇∇∇∇∇∇~ BqB ¢ introducing an observer e0 we then split space-time into 1 separate time and space as span te u eK. From the ¡ ∇∇∇∇∇∇∇~ B B M 0 0 2 space-time geometric equations we get the usual vector K equations on the 3D space e0 . For example, the geo- The two homogeneous Maxwell’s equations (209) are metric form of Maxwell’s equations (191) becomes the then used and we obtain after some further algebra familiar 3D equations (208)–(209). In Part II of this paper we use complex 4-vectors.19 It ¢ ¢ is well known that complex vectors, complex quaternions, µ 1 2 1 e F BµF ∇∇∇∇∇∇∇ E E {c ¡ ∇∇∇∇∇∇∇ B B (222) Clifford algebras and spinors may be used for alternative 2 2 compact formulations of Maxwell’s equations as well as of other equations in special relativity.20 In Part II we where we have used the identity consider such equations. The Lorentz transformations ¢ and the Maxwell’s equations then involves complex 3- K 1 vectors, i.e., elements in the 3D complex vectorspace e0 pE ∇∇∇∇∇∇∇~ qE E ¢ p∇∇∇∇∇∇∇~ ¢ Eq ∇∇∇∇∇∇∇~ E E (223) K 2 ie0 , rather than anti-symmetric 4-dyadics. For example, the electromagnetic dyadic field is transformed as

The conservation law (216) now follows. ¡1 ¡1 c e0 ^ E ¡ rB¢s Ñ c E iB (226) After a little algebra we ﬁnd that the observer e0 may write the stress-energy dyadic as The complex vectors are somewhat easier to deal with than the anti-symmetric 4-dyadics. The generalization T U e pS{cq _ e (224) of Gibbs’ vector calculus to the complex case is trivial ¢00 0 1 while dealing with anti-symmetric 4-dyadics require some ¡ ε0E b E B b B ¡ UIK results presented in Sections V and VI above. On the µ 0 other hand, in the complex case we need to introduce a certain algebra product on the complex Minkowski space where (essentially a Cliﬀord algebra product). Furthermore, the equations are not spacetime geometric because they de- ε0 1 1 U E E B B , S E ¢ B (225) pend, as is obvious in equation (226), explicitly on some 2 2µ0 µ0 observer. The equations in Part II have a “spinor” form. In terms of coordinates there appear 2-by-2 complex ma- and IK I e00. The term in the last bracket in (224) trices in Part II rather than the 4-by-4 real matrices in is the Maxwell stress tensor. Part I.

¦ [email protected] damental motivations for special relativity and the reader : [email protected] is assumed to already be familiar with some basic the- 1 D. J. Griffiths, Introduction to electrodynamics; 4th ed. ory like essentially the content of sections 12.1 and 12.2 (Pearson, Boston, MA, 2013). in Griffith’s textbook (Ref. 1). These sections include time 2 D. Hestenes, “Spacetime physics with geometric algebra,” dilatation, Lorentz contraction, the velocity addition for- Am. J. Ph. 71, 691–714 (2003). mula, the world line concept, spacetime diagrams, the in- 3 The use of the Minkowski space as a starting point for spe- variant spacetime interval and the Lorentz transformation cial relativity must of course be motivated. The present for observers in standard configuration. paper does not include any discussion explaining the fun- 4 G. L. Naber, The geometry of Minkowski spacetime, Ap- 18

plied Mathematical Sciences, Vol. 92 (Springer, New York, 11 This follows from 2012) This textbook has a linear algebra perspective on p p special relativity. It is however coordinate oriented from pRAq pRBq pR Aq pR Bq T the start and the contact with usual vector analysis is not pA R q pR Bq stressed at all. It is also more advanced than the present T paper and not intended to serve as an introduction to rel- A pR Rq B ativistic electrodynamics. 12 5 R. Penrose and W. Rindler, Spinors and space-time. The dot-products appearing in this paper are defined for Vol. 1 , Cambridge Monographs on Mathematical Physics 3D space E and 4D spacetime M. For E it is the standard p q (Cambridge University Press, Cambridge, 1984). Most dot product in vector algebra which has signature , , of this book requires a background in general relativity as defined in Axiom 2M. The dot product on M is the p¡ q and/or differential geometry so the level is far more ad- Lorentz inner product with signature , , , as de- vanced than the present paper. However, parts of the first fined in Axiom 2M. This choice of signature, rather than p ¡ ¡ ¡q three chapters have been useful both for this paper and for the alternative , , , , makes it possible to (with a forthcoming paper on 2-component relativistic spinors. In some abuse of language) extend the 3D spatial dot to all the first chapter of the book the Minkowski vector space of spacetime. 13 is defined axiomatically in terms of linear algebra. The For the mathematics of spacetime, space and time orien- importance of coordinate free notations is much stressed. tations of M we refer to pages 1–4 in Chapter 1 of Ref. 5. 14 This may seem to contradict the appearance of numerous In this paper the concept of a normal basis is defined for E indices in most formulas. However, these abstract indices and M. For E it is just an orthonormal right handed basis r s r s are not associated with coordinates but there is still an in- e1 e2 e3 . For M it is an orthonormal basis e0 e1 e2 e3 r s dex algebra that needs to be studied. For the present paper such that e0 is future oriented while e1 e2 e3 is right p (Part I and Part II) we manage without any index algebra handed. A rotation R : E Ñ E maps a normal ba- because only 4-vectors and 4-dyadics will be used, in the sis re1 e2 e3s to a new normal basis re11 e21 e31 s where p forthcoming paper about spinors we also manage without ek1 Rek. In spacetime we have a corresponding sit- index algebra because only spin-vectors and spin-dyadics uation. A proper orthochronous Lorentz transformation ˆ will be needed. This allows for a notation in close contact L : M Ñ M maps a normal basis re0 e1 e2 e3s to a new with ordinary vector calculus in the present paper. normal basis re01 e11 e21 e31 s where eµ1 ˆLeµ. The con- 6 E. B. Wilson, Vector analysis, A textbook for the use of cept of normal basis will also be used for spinvectors S students of mathematics and physics, founded upon the (where S is a 2D complex vector space) in a forthcoming lectures of J. Willard Gibbs (Yale University Press, 1901). paper about 2-spinors for spacetime. A normal basis for M 7 P. R. Halmos, Finite-dimensional vector spaces, The Uni- is the same as an admissible basis in Ref. 4 or an restricted versity Series in Undergraduate Mathematics (D. Van Minkowski basis in Ref. 5. Nostrand Co., Inc., Princeton-Toronto-New York-London, 15 Let f be an ordered basis. Then from (59) and equations 1958) 2nd ed. The effectiveness of usual vector calculus is in Section IV A we get the matrix equations associated with its coordinate-free formulation. It is impor- T ¡1 ñ r p qs2 ñ pˆ q2 tant that also the linear algebra theory the students have Lf ηf Lf ηf det Lf ηf 1 det Lf 1 been studied has been done in the coordinate-free way, like in Halmos’ textbook. This is probably usually the case but and thus we have det ˆL ¨1. there are exceptions. 16 In the terminology of continuous groups (i.e., Lie-groups) 8 The principle of covariance in special relativity states that the proper orthochronous Lorentz transforms constitute the laws of physics are the same in every Lorentz reference the subgroup of Lorentz transformations that is contin- system. By writing the equations in a spacetime geomet- uously connected to the identity, cf. the proof of Theo- ric form we do not refer to any reference system and the rem III.1 on how this relates to the conditions (63). 2 4 3 principle is obviously satisfied. 17 cosh w 1 w w ¤ ¤ ¤ , sinh w w w ¤ ¤ ¤ 9 2! 4! 1! 3! The concept of observer we use is the same as in Ref. 2 by 18 The lowering and raising of tensor indices are important Hestenes. The approach of Hestenes is however completely operations in tensor index algebra. Most of this is outside different. The term spacetime split is used by Hestenes but the scope of the present paper but we use upper indices in he defines it in a specific way related to the Clifford algebra Section IV A and B for consistency with that formalism. formalism he uses. In the present paper we just find the However, we like to note that for a 4-vector A and a normal term useful for expressing the fact that the choice of an k basis e re0 ¤ ¤ ¤ e3s we have A Ak A ek for k observer allow us to deal separately with space and time. 0 1, 2, 3 and ¡A A0 A e0. This is mathematically expressed by equation (29). 19 J. Larsson and K. Larsson, “An introduction to relativis- 10 The determinant of a linear transformation may be defined tic electrodynamics, Part II: Calculus with complex 4- without the need to use coordinates at all (See Halmos in vectors,” Report, Ume˚aUniversity (2018). Ref. 7). Its value is the same as the usual determinant of the 20 P. Lounesto, Clifford algebras and spinors, 2nd ed., London matrix for the transformation obtained by the use of any Mathematical Society Lecture Note Series, Vol. 286 (Cam- ordered basis. A direct simple proof that the determinant bridge University Press, Cambridge, 2001). Some historical of the matrix associated with a linear transformation is the references p.115 and p.326. same for any choice of basis is given in Subsection IV D for linear transformations on spacetime.