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 for- malism 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 transforma- tions 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.
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