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Bernard Jancewicz Directed Quantities in Electrodynamics Bernard Jancewicz Directed Quantities in Electrodynamics BernardJancewicz InstituteofTheoreticalPhysics UniversityofWrocław Wrocław,Poland ISBN978-3-030-90470-8 ISBN978-3-030-90471-5 (eBook) https://doi.org/10.1007/978-3-030-90471-5 MathematicsSubjectClassification:15A75,53Z05,78A25,78A40,83A05 ©SpringerNatureSwitzerlandAG2021 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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ThisbookispublishedundertheimprintBirkhäuser,www.birkhauser-science.com,bytheregistered companySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface The first extensive description of electromagnetism using the notion of a field— introduced by Michael Faraday—was in A Treatise on Electricity and Magnetism byJamesClerkMaxwell,publishedin1873.Sincethen,Maxwell’sequationshave constitutedthecoreofelectrodynamics—theclassicaltheoryoftheelectromagnetic field. The traditional mathematical tools for the presentation of Maxwell’s theory were created by Josiah Willard Gibbs in his vector calculus, a blend of algebraic and geometric notions. A typical example is the noncommutative vector product. SuchaproductwasintroducedbyHermannGuenterGrassmanninabout1840and called the exterior product. However, the result of this product was not a vector, but a bivector—an element of an exterior algebra of multivectors, later called a Grassmann algebra. It consists of scalars, vectors, bivectors, trivectors (generally k-vectors)andtheirlinearcombinations. Grassmanndidnotdevelopfullyalgebraicstructurescorrespondingtogeometric notions, because he lacked the idea of duality, that is, a notion of linear form over given linear space. Duality allows one to replace the scalar product with one factor fixed by a linear form. (The reader will see this in the example of the wave vectorreplacedbyalinearformofwavefronts.)Moreover,thescalarproductwith two arbitrary factors can be replaced by a symmetric bilinear form. An object dual to a kvector is kform, which is an exterior form of kth order. A set of such forms constitutes an exterior algebra, also called a Grassmann algebra. Exterior forms, depending on points in space, called differential forms, are appropriate to integrateovercurves,surfacesorthree-dimensionalregions,withoutreferringtoa scalar product. They were introduced by Gregorio Ricci-Cubastro in about 1880 and developed by Elie Joseph Cartan at the turn of the nineteenth and twentieth centuries. The presentation of electrodynamics with a wide use of differential forms has become quite popular; see Refs. [1–9, 11–14, 52]. Some authors address electromagnetism through differential forms in relation to electrical engineering [15–21, 23, 25]. Such a formulation ensures a deep synthesis of formulae and simplifiesmanydeductions.Usually,geometricimagingdoesnotfollowtheformal v vi Preface mathematical definitions of the notions, which could be done by analogy with vectorsandcanbeeasilydepicted. Other mathematical structures have been applied to physics, namely Clifford algebras, see [22, 24, 26, 27], where basic ingredients are multivectors as a generalization of vectors. The present book is intended to keep a balance between multivectorsandexteriorformsastwomathematicaltools.Asbothphysicistsand engineersknow,avectorcanbeshownasadirectedsegmentwhichhasadirection (a straight line with an orientation) and the magnitude (length). The linear form, as the dual object, also has a direction and magnitude. In addition to vectors and linearforms,wemayintroducealso multivectorsandtheirduals,namelyexterior forms of higher grade (known also as multilinear forms when acting on vectors). Here, I call these objects directed quantities because each of them has a direction and magnitude. Schouten [46] was probably the first author who considered the directions of multivectors and exterior forms. They were attractively illustrated in the book of Misner, Thorne, and Wheeler [40]. Similar pictures can be seen in publications of Burke [6–8] and Schutz [47], in an article by Warnick et al. [54]. Hehl and Obukhov illustrate them richly in their book [20]. However, the most careful presentation of the directed quantities in many domains of physics is contained in the book by Tonti [51] under the name “physical variables”. The discussion of the directed nature of physical quantities is not covered in such depth here as in Tonti’s book. Nevertheless, the present book is focused on the elaboration of the graphical presentation of directed quantities in mathematics, thereby deepening the understanding of their properties and the role played in physics. I have used as many diagrams as possible with the hope that this helps withvisualization. Not all publications using differential forms mention odd forms. These were introduced by Weyl [55] and developed by Schouten [46], who called them W-p- vectors (for Weyl). De Rham [10] called them odd forms, as did Ingarden with Jamiołkowski[28]afterhim.ThetermtwistedformswasusedbyFrankel[17]and Burke[7,8].Edelen[15],andlaterFrankel[18],usedthetermpseudo-forms. AsitisshowninChap.1,sixteendirectedquantitiesareconnectedwiththethree- dimensional vector space. We demonstrate in Chap.2, that when a scalar product is present, the whole variety of directed quantities becomes unnecessary. There thenappearsanaturalwaytoreplacethemwithvectors,pseudovectors, scalarsor pseudoscalars—inthiswayonecanreturntothedescriptionknownfromthetime ofGibbsandemployedintraditionaltextbooks. Anaturalquestionarises:whydoweintroducesixteendirectedquantitieswhen four of them can also do the job in three dimensions with the scalar product? In addition to the above mentioned simplicity of deductions, there are two extra reasons for this, which are discussed in Chap.4. First, the authors of textbooks on electrodynamics, using exterior forms, have not stressed sufficiently strongly that to some extent one can proceed without the scalar product, i.e. without the metric. Van Dantzig [52] and Post [44] could be mentioned among authors treating a metric-free approach to electromagnetism, and more recent is the book by Hehl and Obukhov [20]. One may pose the question: how far can one go in Preface vii electrodynamicswithoutthemetric?Myansweristhatwecanonlydefinephysical quantities and formulate principal equations (Maxwell’s equations, Lorentz force, continuity equation, gauge transformation, continuity conditions on interfaces and electromagnetic stress tensor). The part of it that can be formulated in a scalar product-independent way has been called premetric electrodynamics. The best advocates of this attitude are Hehl and Obukhov [20, 23], who also build up an axiomaticstructureofelectromagnetism. Itshouldbementionedthatthepresentbookisanexpandedversionofmybook [30],thePolishtitleofwhichshouldbetranslatedasDirectedQuantitiesinElectro- dynamics.Thisisnotatextbookonclassicalelectrodynamics,andaccordingly,its materialisneitherstandardnorcomplete.Itisintendedforreaderswhohavegained atleastabasicknowledgeofelectrodynamicsandspecialrelativity.Italsodemands competence in the advanced calculus, linear algebra and differential equations. At advanced undergraduate and graduate levels in departments of mathematics and physics, the book will be useful for those lecturers and students who want to deepen their understanding of fundamental aspects of classical field theory and electrodynamics. The inclusion of exercises of different levels of complexity also contributes to the possibility of using the book as supplementary reading in combinationwiththemorestandardandcompletetextbooks. I want to express my gratitude to Zbigniew Oziewicz who, when living in Wrocław,gavemanylecturesandseminarsaboutdifferentialgeometry.Theyhave revealedthebeautyofthisbranchofmathematicsanditsusefulnesswhenapplied to physics. I am also grateful to Friedrich Hehl for plenty of valuable discussions on the subjects present in this book. However, not all his suggestions have been fulfilled.ThanksareduetoArkadiuszJadczykforcooperationwhenpreparingthe Appendix. Wrocław,Poland BernardJancewicz April2021 Introduction BydirectedquantitiesImeanmultivectorsand,dualtothem,exteriorforms.They maybedividedintotwogroups.Thequantitiesofthefirstgroup,namelyordinary (even) multivectors, have internal orientations, and the ordinary (even) forms have external orientations, whereas the quantities of the second group are defined conversely: odd multivectors have external orientations odd forms have internal orientations. A detailed discussion of the adjectives “internal” and “external” is containedinthebookbyTonti[51].Icanillustratethisdistinctionwiththeexample of vectors: an even (ordinary) vector has an orientation chosen on its straight segment;anodd(pseudo-)vectorhasitsorientationasadirectedcirclearoundthe straightsegment.Thenames“even”and“odd”arenotgoodbecausetheyhavetoo many meanings and connotations. For instance, the expression “even form” may equallywelldenoteaformofevengrade.Ifyoucalltheoddobjectspseudo-forms or pseudo-multivectors, there is no natural word to describe objects that are not “pseudo”.Thename“twistedquantities”isnotappropriateforallquantitiesofthe second group: for instance, an odd vector has its orientation marked as a directed ringsurroundingastraightsegment,andhencethiscanbeseenastwisteddirection, but the odd two-form has its orientation placed on a straight line, so this does not fit the intuitive meaning of the word “twist”. Nevertheless, I accept the opinion of Burkein[8,p.183],that“thelanguageisforcedonusbyhistory”,andIshalluse theadjective“nontwisted”forthefirstgroup,and“twisted”forthesecondgroupof quantities. Sixteendirectedquantitiesofthethree-dimensionalvectorspaceareintroduced in Chap.1 with careful definitions of their directions and magnitudes. In this mathematical chapter, Sect.1.4 is an exception because it gives examples of physicalquantities,showingthatallthesixteenquantitieshaveatleastonephysical counterpart.Whatisstrikingisthateachofthefourphysicalquantitiesdescribing the electromagnetic field, E—electric field strength, D—electric induction, H— magneticfieldstrengthandB—magneticinduction,isanexteriorformofadifferent directed nature. Some examples of physical quantities are also placed in other sections of the two mathematical chapters when the mathematical tools necessary fortheirintroductionbecomeavailable. ix x Introduction Chapter 2 describes linear spaces of directed quantities, their transformation propertiesandthestatusofthescalarproduct.Franklyspeaking,thewholevarietyof directedquantitiesbecomesunnecessarywhenascalarproductisgiveninthelinear space of vectors and, therefore, also a metric defined by it. There then appears a naturalwaytoreplacethemwithvectors,pseudovectors,scalarsorpseudoscalars— in this way one can return to the description known from the times of Gibbs and employedintraditionaltextbooks. Chapter 3 is devoted to algebra and analysis of the directed quantities: exterior multiplication,contractions,linearoperatorsastensors,differentialforms,exterior derivativeandintegration.AmoststrikingfeatureisvisibleinSect.3.5,namelythat thedifferentialequationscanbewritteninthesameshapeinanarbitrarysystemof curvilinear coordinates. This property goes under the name of general covariance. A well-recognized fact is that the calculus of differential forms was developed specificallytounifyandsimplifytheintegrationtheory.Wepresentitratherbriefly, onlytoprovidetoolsforapplicationsinelectromagnetism. Relevantphysicalquantities,theprincipalrelationsbetweenthem,anddifferen- tialequationsarepresentedinChap.4,inmetric-freemanner.However,inorderto writedownthesolutionsofpreviouslyintroducedequations,ametricisnecessary, which enters the constitutive relations D = ε(E), B = μ(H). Then the Coulomb field, the Biot-Savart law, the field of rectilinear current, and plane waves can be obtained. The Laplace and d’Alembert equations also belong to this part of electrodynamics.Itisusefultomentionalessondrawnfromtheseconsiderations: the constitutive relations are needed to get solutions to Maxwell’s equations. The authors who have stressed this role are, among others, van Dantzig [52], Post [44] and Edelen [15], who wrote on page 372: “...wave properties of solutions arise from the electromagnetic constitutive relations and the resulting differential relationsbetweenquantities...”. The second reason is that in anisotropic media, it is worth introducing another metric, dependent on the electric permittivity or magnetic permeability. Then the prescription for reducing forms to vectors or pseudovectors can be arranged so that the medium looks like an isotropic one. Thus, the typical solutions of the electrostaticandmagnetostaticproblemscanbeeasilytransposedfromthoseknown inisotropicmedia.Chapter4isdevotedtothis,sotheCoulombfieldandBiot-Savart lawarediscussedthere. Electromagnetic waves are treated in Chap.5, and not only plane ones. An attempt is made to find waves with planar wave fronts, but not constant fields on them. For this purpose, cylindrical and spherical coordinates are used and so- calledsemiplanewavesareintroduced.Planewavesareconsideredinananisotropic medium.Inthiscase,thetraditionalwavevectordoesnotdescribethepropagation directionofthewave:thisroleistakenbyaPoyntingtwo-form,whereasthewave vectorisreplacedbyaone-formofphasedensity,wellsuitedtosurfacesofconstant phase.So-calledeigenwavesareusefulinsuchamedium,asthephasevelocityhas adefinitevalueforthem. Considerations of interfaces between two anisotropic media are placed in Chap.6. This starts with the mathematical Sect.6.1, which was not included in Introduction xi earlier chapters because it is not necessary for understanding Chaps.4 and 5. Directed quantities of two-dimensional space (i.e. a plane) are introduced—only twelve of them are possible—and a transition from two- to three-dimensional directed quantities is discussed. The continuity conditions for the electromagnetic field on interfaces are derived. They follow from (integral) Maxwell’s equations only,andthereforetheybelongtothepremetricelectrodynamics.Theseconditions help to obtain electromagnetic fields generated by infinite plates with uniform charges or currents, and fields of ideal electric and magnetic capacitors. The electromagneticstresstensorisintroduced.Reflectionandrefractionofplanewaves areconclusionsfromtheinterfacialconditions. The described transition from three-dimensional space to a plane helps us to move one step higher and predict what kinds of directed quantities can be present in four dimensions. There are twenty of them—they are introduced in Chap.7. In the existing literature, the prolongation of the exterior forms from three dimensions to four was considered to be automatic, whereas—as I show— a distinguished time-like vector is necessary for this purpose, and without it, the directions of the prolongated forms cannot be defined. This knowledge has been used in Sects.7.2 and 7.3 to present physical quantities appropriate in the spacetime and the four-dimensional formulation of electrodynamics. After introducing a Faraday two-form F=B+E∧dt and a Gauss two-form G=D−H∧dt, it is sufficient to write down the two Maxwell equations in very compact shape. Theuseofdifferentialformsandexteriorderivativeensuresthattheyareinvariant underarbitrarycoordinateandframetransformationsandindependentofthemetric of spacetime see [20]. Since the metric of spacetime serves as the gravitational potential in general relativity, it is useful to know that there is a gravity-free way of formulating Maxwell’s equations. Accordingly, the equations are valid in a flat space-timeandcanbetransferredtoacurvedone. The plane electromagnetic wave in vacuum is discussed for four-dimensional space-time. The distinguished role of its phase velocity is used to find the Minkowskiscalarproduct.Areasonisshownwhyametrictensorwithoneminus and three pluses is better than the alternative with opposite signs on the diagonal. TheconstitutiverelationbetweenFandGisfound.ItcontainsaHodgemapwith themetricincluded.ThisrelationallowsonetoexpresstwoMaxwellequationsby the coordinates of the Faraday field only. The homogeneous equation is generally covariant,whereastheinhomogeneousone(withthefour-currentontheright-hand side)isonlyLorentzcovariant.