WAVE MOTION AND OPTICS
WAVE MOTION AND OPTICS PHY 315
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INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS I Basic Principles 1 II Three Dimensional Spaces 4 III Physical Vectors 8 IV Examples Cylindrical and Spherical Coordinates 9 V Application Special Relativity including Electromagnetism 10 VI Covariant Di erentiation 17 VII Geodesics and Lagrangians 21 I Basic Princi les We shall treat only the basic ideas which will suffice for much of physics The objective is to analyze problems in any coordinate system the variables of which are expressed as qjxi or q jqi where xi Cartesian coordinates i 123 N for any dimension N Often N3 but in special relativity N4 and the results apply in any dimension Any wellde ned set of q will do Some explicit requirements will be speci ed later An invariant is the same in any system of coordinates A vector however has components which depend upon the system chosen To determine how the components change transform with system we choose a prototypical vector a small displacement dx1 Of course a vector is a geometrical object which is in some sense independent of coordinate system but since it can be prescribed or quantified only as components in each particular coordinate system the approach here is the most straightforward By the chain rule dq1 Bql Bx de where we use the famous summation convention of tensor calculus each repeated index in an expression here j is to be summed from 1 to N The relation above gives a prescription for transforming the contravariant vector dx1 to another system This establishes the rule for transforming any contravariant vector from one system to another 7 A1q7axAJx E i E E 39i in mhmaafk Akltx aqJ Aiq39lAiq A qx E Contravariant vector transform The contravariant vector is a mathematical object whose representation in terms of components transforms according to this rule The conventional notation represents only the object Ak without indicating the coordinate system To clarify this discussion of transformations the coordinate system will be indicated by Akx but this should not be misunderstood as implying that the components in the quotxquot system are actually expressed as functions of the XI The choice of variables to be used to express the results is totally independent of the choice coordinate system in which to express the components Ak The Akq might still be expressed in terms of the xi or Akx might be more conveniently expressed in terms of some ql Distance is the prototypical invariant In Cartesian coordinates ds2 Sij dxi dxj where EU is the Kroneker delta unity if ij 0 otherwise Using the chain rule INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS dxi lt dqi Bi Bxl ds2 ailaf xgpqudql gk1qqudq1 Bxi Bxl gkl q aqik aiql 51 definition of the metric tensor One is thus led to a new object the metric tensor a covariant tensor and by analogy the covariant transform coefficients AjiqX E 3 2 Covariant vector transform More generally one can introduce an arbitrary measure a generalized notion of 39distance in a chosen reference coordinate system by ds2 gkl 0 qu dq1 and that measure will be invariant if gkl transforms as a covariant tensor A space having a measure is a metric space Unfortunately the preservation of an invariant has required two different transformation rules and thus two types of vectors covariant and contravariant which transform by definition according to the rules above The root of the problem is that our naive notion of 39vector is simple and well defined only in simple coordinate systems The appropriate generalizations will all be developed in due course here Further we define tensors as objects with arbitrary covariant and contravariant indices which transform in the manner of vectors with each index For example quot i 39 1 mn like 2 Am qx A11ltqx Akqx T1 00 The metric tensor is a special tensor First note that distance is indeed invariant d82q gk1 q dq k dq 1 E E Elk s all t Mk aq1g1q aqs dq aqt dq a i a 39k 3139 a 391 gij q x 3115 x 311 qucht u U W L a qs 51s SJI gij q dqi dqj E dszq There is also a consistent and unique relation between the covariant and contravariant components of a vector There is indeed a single object39 with two representations in each coordinate system dqj E gji dqi INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS a k a 1 3 vi dq j E gji q39dqi1 7gk1 q q pdqp 51p qu 1 qu 7 aqj gk1qdq 7 aqj qu Thus it transforms properly as a covariant vector These results are quite general summing on an index contraction produces a new object which is a tensor of lower rank fewer indices 39139 k U TiltG1 R1 The use of the metric tensor to convert contravariant to covariant indices can be generalized to raise and 39lower indices in all cases Since gij Sij in Cartesian coordinates dXi dxi there is no difference between co and contravariant Hence gij Sij too and one can thus define gij in other coordinates More generally if an arbitrary measure and metric have been defined the components of the contravariant metric tensor may be found by inverting the NNl 2 equations symmetric g of g1 0 gik0 gnj0 gkn0 The matrices are inverses Nws waw l 39k it 335 E 3 K l L 5m EEi ig xhp q Thus g is a unique tensor which is the same in all coordinates and the Kroneker delta is sometimes written as 5 to indicate that it can indeed be regarded as a tensor itself Contraction of a pair of vectors leaves a tensor of rank 0 an invariant Such a scalar invariant is indeed the same in all coordinates y I y 7 aqvi aqk 7 I A1qB1q7 HMJGD aq iBkq 7 51k AJQBkQ 7quw It is therefore a suitable definition and generalization of the dot or scalar product of vectors Unfortunately many of the other operations of vector calculus are not so easily generalized The usual definitions and implementations have been developed for much less arbitrary coordinate systems than the general ones allowed here For example consider the gradient of a scalar One can define the covariant derivative of a scalar as INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS ao 30730 3x1 0X1 0Q1 a qia qi The covariant derivative thus de ned does indeed transform as a covariant vector The comma notation is a conventional shorthand However it does not provide a direct generalization of the gradient operator The gradient has special properties as a directional derivative which presuppose orthogonal coordinates and use a measure of physical length along each perpendicular direction We shall return later to treat the restricted case of orthogonal coordinates and provide specialized results for such systems All the usual formulas for generalized curvilinear coordinates are easily recovered in this limit A covariant derivative may be defined more generally in tensor calculus the comma notation is employed to indicate such an operator which adds an index to the object operated upon but the operation is more complicated than simple differentiation if the object is not a scalar We shall not treat the more general object in this section but we shall examine a few special cases below 11 Three Dimensional Spaces For many physical applications measures of area and volume are required not only the basic measure of distance or length introduced above Much of conventional vector calculus is concerned with such matters Although it is quite possible to develop these notions generally for an Ndimensional space it is much easier and quite sufficient to restrict ourselves to three dimensions The appropriate generalizations are straightforward fairly easy to perceive and readily found in mathematics texts but rather cumbersome to treat or writing compact expressions for determinants and various other quantities we introduce the permutation symbol which in three dimensions is 11k 1 for ijk123 or an even permutation thereof ie 231 or 312 1 for ijk an odd permutation ie 132 or 213 or 3 2 1 0 otherwise ie there is a repeated index 113 etc The determinant of a 3x3 matrix can be written as lal silk an azj a3k Another useful relation for permutation symbols is gijk gum 5 5km Sjm 51d Furthermore gijk glmn and 3 where Sign is a multidimensional form of the Kroneker delta which is 0 except when ijk and lmn are each distinct triplets Then it is 1 if lmn is an even permutation of ijk 1 if it is an odd permutation These symbols and conventions may seem awkward at first but after some practice they become extremely useful tools for manipulations Fairly complicated vector identities and rearrangements as one often encounters in electromagnetism texts are made comparatively simple Although the permutation symbol is not a tensor two related objects are Sijk xfg gijk and Silk VLLg gijk where g E gijl with absolute value understood if the determinant in negative This surprising result may be confirmed by noting that the expression for the determinant given above may also be written as 4 INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS 21m lal eijk an amj ank which is certainly true for lmnl23 and a little thought will show it to be true in all cases The transformation law for g may then be obtained as 21m lgq l eijk gq 1i gq mj gq nk 7 BqP qu qu qu igllka qraq laqmw aqu aqv aqvn Mm gpqq grsq guV q tensor transform of metric tensors E E P E r E u 3 a jrrxac llmxaj mgpqgrsguv considering the terms with indices ijk eqsv 2P1 gq I 33 aqP aq 1 qu aq m aq aq considering the terms with indices qsv in constituting a determinant as above a 2 glmn gq l aiq l forming another determinant as above thus establishing that g transforms with the square of the Jacobian determinant For the putatively covariant form of the permutation tensor t gijk I if I v gq gijk gq the form desired Raising indices in the usual way will produce the contravariant form by arguments similar to those applied above The permutation tensors enable one to construct true vectors analogous to the familiar ones The vector or cross product becomes Ai Sijk Ck although again we have both co and contravariant forms INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS The invariant measure of volume is easily constructed as d id 39dk AV Silk 16 q which is explicitly an invariant by construction and can be identified as volume in Cartesian coordinates This is a general method of argument in tensor calculus If a result is stated as an equation between tensors or vectors or scalars if it can be proven or interpreted in any coordinate system it is true for all That is the power of tensor calculus and its general properties of transformation between coordinates Note that the application of this relation for AV in terms of dq1 and transforming directly from Cartesian dx1 gives immediately the familiar relation AV J dqldq2dq3 J 2 the Jacobian For the volume integrals of interest note that I I Sijk dqi dqj qu for I invariant is invariant but TV Sijk dq1 qu qu is not a vector because the transformation law for TV in general changes over the volume e operators of divergence and curl require more care Just as the gradient has a direct physical significance these operators are constructed to satisfy certain Green s theorems Gauss and Stokes law These must be preserved if their utility is to continue One can prove a beautiful general theorem in spaces of arbitrary dimension from which all common vector theorems are simple corollaries but the proof requires extensive formal preparation Instead we shall provide straightforward if lengthy proofs of the two specific results desired or Gauss39 law we require a relation which is a proper equation between invariants and further reduces to the usual result in Cartesian coordinates i 39 k IdivTm Sijkw ITl dSi the choice dSi Sijk dqj qu is explicitly a covariant vector making the right integral invariant and it gives the correct result in Cartesians On the left we require a suitable operator We shall next prove that 1 a g T g 9 is such an invariant It certainly gives the usual Cartesian divergence but the inspiration for this guess must remain obscure for it is deep in the development of general covariant differentiation and Christofel symbols Fortunately that need not concern us Proof that this expression is indeed invariant requires proving that the form is the same in any two systems ai JV T i alwg k v qu v Bql 2 Ti 7 where g k E JVE V g 3x1 8 1 INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS as shown above introducing J for the Jacobian determinant The expression in the new coordinates can then be written 1 9 IE TH aqi Tkl3Jl E qu XEHTl 3 where the first term is simply the desired expression in xi by the chain rule and we must show that the second term the portion in brackets H is then zero That term may be written Bx a i 32 i a 1 i A X E q1 Bxk Jaxk 3x1 aql vl and the first term converted using J 1 to gik a 3 q J2 BX J2 2C1 Bxk 3X Bxk 3x1 Bql thereby canceling the second term and proving the assertion The last step requires some algebra to con rm but it is straightforward using the methods used above for writing a determinant considering all the terms present and inserting a i 331 ax 5139 BxsBqJ with appropriate choice of indices the inverse of the usual procedure The tensorial39 form of the divergence theorem is therefore an equality of invariants I 1 BHVg ImI dgidgjdgk 7 i k 7 8 k i T 81k qu dq Vg qu J 3 J i Furthermore the familiar result div0A 0divA VQA remains as div0A 0divA 3 Ai 1 Fortunately Stokes theorem is somewhat easier there is only one subtlety The naive generalization is l 811k a qj gist dqs dqt l Ti dq1 which again obviously reduces to the usual result in Cartesian coordinates and would be explicitly a good 39tensor equation between invariants if BTkqu were indeed a covariant tensor of rank two It is not but the portion used in the equation above is In general 7 INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS Rij Ril392R39i Ril39 2Rl39i the sum of a symmetric and antisymmetric part For contractions with the antisymmetric permutation symbol as used above only the antisymmetric part contributes replacing an 61 aqj 39 9 qu 7 2 is equivalent and gives the identical Cartesian reduction The antisymmetric expression is easily shown to be a tensor as follows BTi BT39 3Tquot 3Tquot 1 1 1 R11 ax Bxi and R11 qu qi but by the laws of tensor transformation this should also be k k B T1434 a de4 R i A ka ai 1 qu 3 k1 3 qu 7 BTk axk BTk axk 32Xk azxk 7 qu qu 39 qu qu lTk quaqi 39Tk quaqj where the last two terms cancel and the first two using the chain rule 23quBBxk3xkqu give the required tensor transform of Rij We therefore have the desired tensor form of the divergence and curl operators and the corresponding integral theorems Note also that the important results curl grad 0 0 and div curl Ai 0 both follow easily from these forms by symmetry 32 11k 7 0 g a liBQj 111 Physical Vectors The distinction between covariant and contravariant vectors is essential to tensor analysis but it is a complication which is unnecessary for elementary vector calculus In fact the usual formulation of vector calculus can be obtained from tensor calculus as a special case that being one in which the coordinate system is orthogonal Most practical coordinate systems are of this type for which tensor analysis is not really necessary but a few are not For example in plasma physics the natural coordinates may be ones determined by the magnetic geometry and not be orthogonal In orthogonal systems with positive metric one can define 39physical39 vectors which are neither covariant nor contravariant Nevertheless they have welldefined transformation properties among orthogonal systems and they have simple physical significance For example all components of a displacement vector have the dimensions of length They are the vectors of traditional vector calculus For orthogonal systems of this type gij h2i Sij hi is not a vector no summation INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS A AG 5 hi A1 no summation 1 for the components of the physical39 vector The usual dot or scalar product is simply AiAi and produces the same result as given above In this special case the metric tensor can be 39put into the vector in a natural manner All the usual vector formulas can be obtained from the preceding tensor expressions by consistently converting to physical vectors Note that g h1h2h32 and Sijk h gllk using h h1h2h3 Ci A0 X Bk giik A0 Bk grad 0x0 1hi 8 eaqi div A 1hBhAihiqu curl Ai hih giik ahkAkqu Volume d3v h gijk dqidqjqu d31 ijk dlidljdlk Integrations are over physical volumes areas and lengths If the integrals are set up in coordinates like dq the necessary factors must be inserted to give the physical units as illustrated here for volume IV Examples Cylindrical coordinates simple example to illustrate the ideas is provided by cylindrical coordinates Xrcos9 rlX2y2 y r sin 9 9 tan 1 yX 22 ij l 2 3 Ai a qi c39oseey smeey d j 39 s1n r cos r BX 0 0 l ij J BXI cos 9 sin 9 0 Ai E if rsin 9 rcos 9 0 0 0 l l 0 0 l 0 0 gij 0 r2 0 g1 0 r392 0 0 0 l 0 0 l gr2 hilrl INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS Spherical Coordinates A second example of broad utility is spherical coordinates Xrsin9cosQ r39lX2y222 VXZ y2 z yrsin9sinQ 9tan 1 z r cos 9 Q tan 391yX sin 9 cos Q sin 9 sin Q cos 9 Ai 7 cos 9 cos Qr cos 9 sin Qr sin 9r J 3 sin Q cos Q 0 r sin 9 r sin 9 ij BXJ sin 9 cos Q sin 9 sin Q cos 9 Ali E aqi r cos 9 cos Q r cos 9 sin Q r sin 9 r sin 9 sin Q r sin 9 cos Q 0 l 0 0 l 0 0 gij 0 r2 0 gij 0 r2 0 0 0 r2sin29 0 0 r392sin3929 g r4 sin2 9 hi lrr sin 9 h r2sin 9 V Application Special Relativity Special relativity is generally introduced without tensor calculus but the results often seem rather ad hoc Einstein used the ideas of tensor calculus to develop the theory and it certainly assumes its most natural and elegant formulation using tensors The arguments are easily stated The use of tensors is natural for it guarantees that if the laws of physics are properly formulated as equations between scalars vectors or tensors a result or equality in one coordinate system will be true in any Special relativity is based on only two postulates The first is that all coordinate systems moving uniformly with respect to one another are equivalent ie indistinguishable from one another The second is that the speed of light is constant in all such systems The first was a longstanding principle The second was the implication of the MichelsonMorley experiment These are easily phrased in tensor calculus The first implies that the metric tensor must be the same in all equivalent systems otherwise the differences would provide a basis for distinguishing among them The second is achieved by introducing a space of four dimensions with Cartesian coordinates Xyzct and choosing the metric tensor to be gHV 0 0 1 0 0 0 0 1 This is one of many equivalent choices none of which has become standard Sometimes the time is placed first the indices may run from 03 instead of 14 and the factors of c can be put into g instead of into the coordinates 10 INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS The resulting invariant measure quotlengthquot is d2039 gm Xm l39dXV d2s c2d2t introducing the usual convention that Greek indices range 14 whereas Latin indices range only over 13 the spatial dimensions d2s XmXm Xl l39 Xyzct X1 ct It is this measure of quotlengthquot sometimes called proper distance39 no better a choice of words which makes c a unique constant You may be more familiar with this invariant called 39proper time39 d l do c Speci cally a disturbance propagating at c in one system dsdtc in that system will produce events in that system for which d 039 0 Since this quotlengthquot is invariant it will be the same in all systems d2039 0 for the events transformed to any other system and they will thus also appear to move at ds dt c For all equivalent uniformly moving systems which have the metric above a speed of c will be invariant This argument is carefully phrased to avoid quotthe speed of lightquot although quotthe speed of light in vacuumquot would suffice If light is observed in a medium which is difficult to avoid the medium introduces a preferred reference frame and the speed is no longer strictly invariant It remains only to obtain the transformation law between uniformly moving coordinate systems which will preserve the metric Let the origins coincide at t0 and the origin of one system 0ct move with velocity v in the other along X If one looks for the simplest covariant transform which could accomplish this AOOB B cc 0100 cc 0010 gHVAHAVg0 B COOD BZ A2 0 0 BDAC 0 1 0 0 WV 0 0 0 BDAC 0 0 D202 where one must be careful if one does the tensor contraction as matrix multiplication transposes must sometimes be used to obtain the proper indeX matching The requirements are thus ACBD BZ A2 1 D2 021 0ct gt Bct00Dct gt BD vc EB where the signs come from using covariant displacements to employ the transform law above but one is not concerned about the sign of v Note that co and contravariant vectors differ but only in sign of the spatial part The unique solution to these four equations in four unknowns is Y 0 0 BY 0 1 0 0 0 0 1 0 AOLH BY 0 0 Y Y 0 0 BY 0 1 0 0 0 0 1 0 AOLH BY 0 0 v INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS 1 52 which give the rules for transforming tensors between uniformly moving systems Note that the metric is not positive de nite here The notion of physical vectors introduced in Section III cannot be employed to disguise a difference between co and contravariant An attempt to do so introduces Wl the origin of the ubiquitous is which permeate nontensor treatments of special relativity It is ironic that the attempt to quothidequot the metric by introducing quotphysicalquot vectors should result in the rather unphysical appearance of imaginary dimensions Because the metric does not depend upon position we have the useful generalization already YE employed above that not only is the displacement dxH a contravariant vector as it always must be but the coordinates 0r vector position of a point x is also a vector which is not true in general and constitutes a major conceptual subtlety in tensor calculus This is a great simplification for special relativity and it means that the law above for transformation of contravariant vectors is also the law for coordinate transformations Finally note that gHV guy which can be con rmed by direct calculation As noted earlier the two must be matrix inverses of one another All the usual relativistic effects follow in a straightforward manner from these equations An event at x0 cto occurs at Yx0 Bcto Yct0 on in the moving system The origin of the initial coordinates appears to be moving at v in the new system whereas the origin in the new system appears to be moving at v in the initial system Events at the point x0 but separated by Ato occur at different points and different times the time difference being 7 Ato the wellknown time dilation A stationary bar with ends 0ct0 and Lct1 appears at Bth0 Y cto and YLBct1 Yct1 BL Expressed in terms ofa new t39 t39 Yto and t39 Yt1 BLc Bct ct39 and UV Bct ct which implies that the ends appear separated by a distance LY the contraction of length if they are observed measured simultaneously in the new system The velocity addition formula follows simply by applying two successive transformations 1BB39W o o BB39W39 vquot 0 0 mu 0 1 0 0 7 0 1 0 0 0 0 1 0 0 0 1 0 BB W 0 0 1BB YY 45m 0 0 yquot BB39 Bquot a BB rlt1BBW NBquot but note that the addition of two velocities in different directions gives much more complicated results the transformations do not even commute If the physical laws are expressed in terms of relativistic vectors and tensors they will transform properly with coordinate system and have the same form in any system as desired The analog of velocity is v E dxHd c do cd l INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS v Y viYc pH mvlL pi Ec These relations for the fourvelocity follow directly if xi vit d2xi v2d2t d2 d2t d2xi c2 1B2d2t d2t W2 This is a wellformed vector which reduces to the usual velocity for V ltlt c it is the only useful relativistic expression for velocity and thus momentum The fourth component of the momentum vector is identi ed as E because it becomes mc2 l2mv2 KE constant in the usual limit Because of the tensor transformation law if plll pzll in one system p39llL p zlL in any other and only momentum defined in this way will be conserved in all systems if it is conserved any system Because the conserved momentum is that given by these expression the relativistic equations are often described as giving a mass increase Ym because pi Ymvi The generalization of energy is E Ymcz Since only the rest mass ever appears we shall omit m0 and keep all factors on explicit The equations of mechanics are d H d H fll E L Yg YF1Pc F1dp1dt Newtonian force F d E d1 t dt de de a E W Example Uniform Acceleration39 To illustrate the use of these equations consider a particle subjected to a constant force e g an electron in a constant electric field starting from rest The spatial part of Newton39s law canceling Ys i d i is simply ddii F1 For motion in one dimension X05 and Ell t 1662 CLO ocot 1 10th at small t and B N l at large t and Yt j l XOth This is a solution for the motion in a fixed reference frame in which the particle was originally at rest From the view of the particle things are more complicated for the particle does not define an inertial frame At best one can consider a succession of inertial frames in which the particle is instantaneously at rest From the solution d v Yv00c and all Yc0co00 Yc0co00 500 the contravariant vector transform to This may be integrated directly to give Bt which has the necessary vat behavior the particle 39rest39 frame at v gives v H 000c as it should and a H c0LO000 The constant force in the laboratory frame implies a constant acceleration in the instantaneous rest frame the power 3 is always zero in that frame because F39V 0 there INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS Another very useful fourvector is the wave vector k ki Dc such that kHXl l39 is an invariant krr ot the phase of a wave The formal argument is just the reverse The phase of a wave must be an invariantall observers can identify a peak Since kHXl l39 is the phase kHXl l39 must be an invariant and hence k must transform as a covariant vector Transforming this as a fourvector easily gives the Doppler shift of frequency and the change in wavelength in a new system accurate for all values of v Maxwell39s equations and the equations of electromagnetism are comparatively straightforward in fourvector form The current vector is 39u jHEGch BJ divj BpBt0 BXH the natural form of a conservation law Compare discussion above for case here where Wlgll and therefore there are no contributions from g to the derivatives For a constant metric covariant differentiation reduces to partial differentiation in the sense that BBxl simply adds a wellformed covariant indeX This the unique wellformed tensor equation which guarantees that if charge is conserved in one reference frame it is conserved in all Charge conservation means that charge is an invariant e g all observers agree on e for the electron but note that j transforms as a vector and that dilTerent observers measure dilTerent currents and charge densities The potentials also make a natural fourvector AM a Ai0c AM a Ai0c The argument is straightforward A tensorial differential operator an invariant is easily formed as gHV i i which is familiar as the operator of the wave equation V2 axu axv 0239 equations for the potentials in the Lorentz gauge can therefore be expressed as gMV i AJAMHOJ39M E0 The usual Bx BXV Bx with the choice of A above and these are proper tensor equations if A is a fourvector Furthermore 7 BAH BAV T L V BXV BX is by the arguments above also a good tensor whose components are in fact INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS rm 0 BZ By EXc B 0 B c z X By THV By BX 0 EZc EX c Ey c EZ c 0 rm 0 BZ By EX c B 0 B z X By c Tm By BX 0 EZ c EX c Ey c EZ c 0 new expresses the two Maxwell39s equations with sources Gauss and Ampere s Laws directly Since the elds are constructed from the potentials using the usual equations the other two Maxwell39s equations are automatically satis ed but they can also be expressed as gums BTW 0 3x5 noting that the simple permutation symbols are tensors when HgHl absolute value of the determinant of the metric tensor a simple generalization of the arguments of Section 11 One can construct two interesting invariants from the elds as TMV TV 13P lEclz and 206W T045 Tye 2 EB These have important physical consequences implying that if the eld is purely electric in one frame there will be a dominant electric eld in all frames and vise versa Conversely if there are both electric and magnetic elds in some frame it is possible to nd a frame in which one vanishes An important consequence of the second invariant is that if the fields are transverse in one frame perpendicular to one another they will be so in all frames The Lorentz force expressions may also be constructed fv THV TVHjlquot39 and fv qTHV Vl l39 qTVH V The covariant force density fv appropriate to a continuous system with a currentdensity charge density fourvector j is to be distinguished from the fourvector force fV which acts on a particle of char e q These expressions are explicitly formed as invariant tensor expressions and may be directly computed to verify that they give the familiar results of electromagnetism for the contravariant form f pEi j x B jEc fH qYEi v x B vEc The fourvector force fli which appears here has the same factor of Y multiplying the familiar terms as did the corresponding fourvector in the tensor form of Newton39s law 15 INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS These constructions of the tensor equivalents of mechanics and electromagnetism may appear to lack rigor but that is not the case If an equation is written as a proper equation among tensors tensor calculus guarantees that it will remain true in all coordinate systems Therefore an equation of the proper form which is correct in one coordinate system will be universal You may nd more detailed arguments helpful in understanding relativistic effects but they are not necessary For example to prove that j is a fourvector it is not necessary to examine current densities and charge densities in one coordinate system and determine their complex transformations as velocities and volumes transform between systems It suffices to declare that charge conservation is a physical law BjH Only 7 0 with j Oipc being a genuine fourvector is a proper tensor equation which BxlL provides the usual form of the charge conservation equation in a reference system Therefore j must be a fourvector It is a symptom of the Lorentz invariance of electromagnetism that the equation of charge conservation indeed has the familiar form in all inertial coordinate systems However the tensor equations for mechanics involving fll etc include factors of Y and reduce to the familiar forms only for low velocity T N l The most important application of this argument is to the electromagnetic field the tensor and transformation character of which would otherwise require considerable tedious argument The argument above shows that the fields are thoroughly linked being components of a single tensor Since E and B are conventionally vectors one might have expected analogous fourvectors but that would create a conceptual difficulty in expressing a fourvector force coupling fourvector fields and the fourvector velocity a difficulty which is obviated by the tensor force expressions above The field tensor transforms normally for reference the result is shown here rm 0 YBZ BEyc way BEZc EXc 4032 BEyc 0 BX YBBZ Eyc T YBy BEZc 13X 0 YBBy EZc V EXc YBBZ Eyc YBBy EZc 0 The familiar VXB contribution to the new E is present but there are factors of Y and contributions to B as well The tensor form of energy conservation may be obtained by similar arguments or it can be obtained as follows using methods analogous to those of the classical argument Since momentum energy conservation already involves tensors the fourvector analog is not particularly easy to construct BTHOL onc l fv TquM Tuv Ho 0L 1 T BTHOL 1 Tl l39 THOL BTHV aXOL 2H0 i V BXOL BXOL INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS 3T 3T 3T Since i amp 0 if the indices are distinct this is one of Maxwell39s XOL BXV BX BTBY X5 equations SOLBYS 0 otherwise is it true by the antisymmetry of T E X 0 3T 3T 1 ET 13TuT THOL 0111 VOLJ T 2M0 V BXOL 2M0 BXOL BXV BX fv 1101 E T THOL E THOL T 8T L THV 8T L uv 15 0110me VOL 2M0 BXOL 2M0 BXOL BXV BX 1 BTW 1 3Tuv TW 1 3TW Tocu 3TW Tvoc BTW Tm E TVOL 2M0 BXOL 2M0 BXOL BXV BX BX By changing dummy indices and using the antisymmetry of T the rst and last terms cancel and the second and fourth terms are identical leaving 3G L EXTHV THOL 1 3THOL THOL V Ho BXOL 4 aXv BXH V for the relativistic stress tensor It can be converted to other forms for example 1 1 GMV H gag TBV TOW Z TOLB Tug gHV 0 u 1 1 u G 7 H0 TOW Tom 4 NB Tag 5V which is clearly symmetric but the elements remain complicated functions of the fields This completes the fundamental formulation of 39 39 and 39 J 39 in 39 quot 39 quot form VI Covariant Differentiation Differentiation of tensors is not simple The partial derivatives of an invariant form a good covariant vector and certain antisymmetric forms have been shown above to be tensors but generally speaking the partial derivatives of vectors and perforce tensors introduce derivatives of the transform law and metric Only for constant gij eg Cartesian coordinates and special relativity but not even cylindrical or spherical coordinates do partial derivatives produce tensors The formulation of derivatives ie finding definitions for derivatives of a tensor absolute and covariant differentiation which do behave properly is subtle Several approaches are possible the one here is geometric rather than formal and strives to provide a basis for and understanding of the complications which arise Nevertheless not all steps can be well motivated and certain choices will become clear only in retrospect Since only derivatives of invariants have tensor character we begin by considering a simple fundamental object the tangent to a curve q1u l7 INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS 1 131 1 This is a wellformed contravariant vector from which an invariant w gijpipj may be constructed Its derivative must likewise be an invariant 39 E ETW 2 gijpl 73 1311ka 2 which can be written in this simple symmetric form because of the de nition of pi above A fundamental and rather obvious theorem of tensor calculus sometimes called the quotient rule implies that if AiB1 0 an invariant and B1 is an arbitrary contravariant vector then Ai must be a covariant vector One can thus factor out a term pi from this expression and conclude that the remainder is a good covariant vector However i is a dummy index any of the three p factors in the nal product could be extracted In fact a particular combination is particularly useful the sum of the two symmetric forms in ij minus the form using k quotdipl 39k l agjk agik 3amp1 f1 egg du uk1p1p 11k 2 aqi aqj aqk 3 where the bracket de nes a famous object the Christoffel symbol of the rst kind It is clear that this fi is not the only covariant vector involving 1111 but the special symmetry of the Christoffel symbol makes it an advantageous choice There is an obvious corresponding contravariant vector d 1 i i f t jkPJPk jk5g 1 k 4 which employs the Christoffel symbol of the second kind These objects are not tensors their transformation law remaining to be inferred from the known transformation character of the other terms in the equation but raised and lowered indices are used to indicate the indices with which they are to be summed in the usual convention This leads one to de ne a derivative the absolute derivative of p1 as the contravariant vector 5pi 7 dpi i 39 k 5 g a t jk P1P The signi cance of the Christoffel symbols may be understood as follows If u is chosen to be length 5 1 s then a 39strarght lrne39 would have a constant tangent 8i 0 as an invariant property In Cartesran s d i coord1nates that is equivalent to 1 1 0 but this de nition 1mp11es otherwrse 1f the Christoffel symbols are nonzero In fact in 39curved coordinates even ones as simple as cylindrical or spherical d 1 systems d ps 0 for a straight line and a constant tangent vector has varying r9 components along the line in general The Christoffel symbols embody this curvature and introduce it into the equations i 5 i guaranteeing that only the proper 11 will produce 3 0 Very similar equations and calculations s to those here appear in the rigorous generalization of a straight line which is a geodesic a curve of variationally stationary length or simply the shortest distance if the metric is positive de nite As mentioned above the transformation laws for Christoffel symbols may be adduced from the tensorial form of the terms in 3 and 4 Speci cally from 4 INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS 7 dlqi i 39 dqj qu iaqi dlxj j dxl dxm 1 d2u jk EH T d2u 1m 5 du 6 Bqi A111 Mew Lqiaxiiqlmi 822639 any an all ax duaI du MEI d2u ax aqaqp du du i 82 d2u ext aqjaqk du du The second derivatives of qi match leaving i39cLIid35 qi 82 iA39djEii t BX BxmgAdi Jk du du aXt aqjaqk du du kg 1111 aqj aqk du du i 397 qu 32X n 3X1 me qu jk aqjaqk 1ma qjaq k w 7 This implies that the Christoffel symbols transform like tensors but with an additional term which involves the second derivatives of the coordinate transformations They therefore remain zero for all linear transformations like rotation and the Lorentz group They are nonzero in cylindrical and spherical coordinates and the transformation law 7 from Cartesian can be as convenient for calculation as the definition 4 The same procedure may be applied to 3 to give i39k39 a Xlr BZXm 1mn13 Xi 3 K axn 8 1 glmaql anaqF aql an ER i An absolute derivative was defined for the contravariant vector p1 E 11 but the calculation depended on the special properties of p However a straightforward generalization is possible based on the invariant 0 gijplTJ for any vector eld TJ de ned along the curve qJu d0 di de 3gquot m ij gijPlE xii P TJPk using 3 dTJ 3gquot fj 11911131135TJ gij 1315 434 p Tka d0 dTJ a ijJ gij p1 du Dkdl plTka 9 Since the quantity on the left in an invariant so is the right and factoring out pi implies that de gij a mm Tka fi a covariant vector The corresponding contravariant vector is the appropriate absolute derivative g tjk m H INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS The vector character of 10 may also be con rmed by direct transformation using 7 and the procedure used to obtain 7 A similar procedure gives the form for the absolute derivative of a covariant vector Ri An invariant may be formed with any T1 and an additional derivative invariant likewise as dRiTi7dRi dTi7 dRi STi i qu T ETHRIE mTltRlg39jk TJW 11 5T1 Choosing the arbitrary T1 such that 8 0 means that the coef cient of T1 is a vector u g RJW By forming an invariant with a collection of arbitrary vectors each of which has zero absolute derivative the absolute derivative of any tensor de ned by analogy with the form below is easily shown to have the same tensor character as the tensor itself 5Tij dTij 7k7k111111111111 13 511 du 111 k du 111 k du kn l du This one could be proved using 0 TE AiBj Ck Mathematicians typically strive for the greatest generality meaning minimal assumptions In this case the vectors and tensors need only be de ned along the curve eg Riu However we are generally concerned with vector and tensor elds meaning objects which are de ned at all points in k k space In this case one can use i di 1 and since di is now an arbitrary contravariant du du aqk du vector factor in the absolute derivative its coef cient must be a tensor and covariant in that index One thereby de nes the covariant derivative 7 dTi i T1 Tk 14 dTi k Tia qu 39ij Tk 15 with the obvious generalization to tensors of higher rank Other common notations are T1 j and T1 j These results are susceptible to some helpful and intuitive interpretation In general if the derivative of a function is zero the function is constant in some sense This idea may be pursued by noting that gi39k 0 Veri cation is straightforward using the extension of 15 for two covariant indices with tlie definitions 3 and 4 of the Christoffel symbols it is a further illustration of the advantage of choosing the particular symmetry for fi in 3 The sense in which gij having zero covariant derivative is constant is both special and signi cant Since gij is a rather arbitrary symmetric tensor it certainly varies with position in general and none of its partial derivatives with respect to q1 need be zero In fact its covariant derivative through the Christoffel symbols has been implicitly constructed to be zero The metric tensor de nes the space 39changes39 in the metric tensor are changes in the space itself The tensor derivatives show changes with respect to the space Almost by de nition the space does not change with respect to itself and gij should be a constant with respect to the space de ned by gij 20 INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS The concept of constancy may be developed by noting that l 1 may be written as d RTi STi 8R Ri 1Tl 16 u 5u 5u STi Applying this to a Single vector if 8 0 then the length of T1 remains constant along u u Furthermore the angle between two vectors may be de ned in the usual sense as RiTi lRiHTilcos 9 lTil l TiT1 If both vectors have zero absolute derivative along u then their lengths and the angle 5T1 between them remain constant For this reason if 8 0 the vector T1 is cons1dered to be u propagated parallel to itself along u Parallelism is easily de ned at a point in the usual sense that RiTi i lRiHTil but vectors at different points cannot generally be compared This offers a generalization which preserves most of the usual properties Unfortunately uniqueness is not one of them different curves u between a pair of points A B may lead to different T1 at B starting from a given T1 at A 5T1 An interpretation of Christoffel symbols can again be given from noting their role in a 8 0 u 1 11 causing T1 to change to compensate for the 39curvature39 of the space condition as that of driving VII Geodesics and Lagrangians As noted above the concepts of parallelism straight line and really all nonlocal global comparisons require some specialization in general metric space They cannot be carried over with all 8 1 i their familiar properties A primitive if correct notion of straight line as si 0 p1 lg was s introduced in the previous section in interpreting the meaning of absolute differentiation but a more general formulation is useful The fundamental formulation is based on a variational principle and such principles are also important for mechanics To review if a definite integral 1 whose value is expressed as a functional of functions of a parameter u between fixed end points is to have an extremum maximum minimum or possibly an in ection point u2 dq1 51 7 5 L duq1uu du 7 0 1 111 By the usual argument in calculus of variations if the set of functions qu is a solution then for a small variation about that qi q Sqi 51 must be second order in Sqi and the first order variation is zero This is simply a generalization of the fact that the first derivative of a function is zero at extrema If L is then regarded as a function of the functions listed above regarded as independent u2 39 7 a d5q1 BL 7 qu 51 7 ujldu du a qi 5q1 7 0 where q1 7E 2 and a sum over the index i is understood The first term may be integrated by parts and if the end points are prescribed so that 5q1u1 5q1u2 0 the condition may be written as 21 INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS 112 d BL BL d 7 510 3 ujl u du aqq aql q since the Sqi are arbitrary the integral will be zero only if all its coef cients zero which are the well known EulerLagrange equations for the variational problem dBL BL 0 4 E aqvi 39 aqi The application to straight lines arises because a straight line is among other things the shortest distance between two points and this criterion can be formulated in any metric space A geodesic is de ned as a curve for whic u2 didj 515 gij ldu 0 5 ul and in cases like special relativity for which the metric is not positive de nite and there are curves of zero length the integral may be a maximum In any case the solutions q1u are geodesics and the best generalization of a quotstraight linequot in a general metric space The Euler equations are thus 1M1 M01L LE f dqidqi d Bq l Bql du N Bq 1W3q1 or W g1 du du i and if u is chosen to be the measure of distance ds2 gi39 du2 w l andd W 0 leaving Jdu du ds 16w m012cni E djk 6 dsaq39i aqi ds g1 dSJ39qu ds ds as the equation of the geodesic Computing the derivative through the qi dependence and rearranging dummy indices produces 2 A9gimqidi gikdj igj o 7 g1 dSZ aqk ds ds aqj ds ds aql ds ds which through no accident can be written d2qJ dqiqu dzqi i dqiqu gu wkvll o r md 8 which are the standard forms for these equations Variational principles are also used to form the Lagrangian and related equations of motion The familiar results may be extended to construct relativistically proper forms but somewhat indirectly The normal construction of L TV with jdt has no clear tensor equivalent Instead we must try to nd an invariant L such that deu generates the correct equations of motion For example 22 INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS dXOl dXB L 10 goals Tu W 9 is manifestly invariant and also independent of position only derivatives enter and thus a possible starting point as the Lagrangian for a free particle The Euler equations are g dXB a 7 macTl A 0 10 CECE go B du du If u is now chosen to be the invariant parameter 5 the radical becomes the invariant constant c and the equations reduce to the standard equation of motion for a free particle dZXOL d 0 L 0 P m d EZ d1 11 With this start the Lagrangian for a particle in an electromagnetic eld could be 1 dXOl dXB dXOl 7 7 7 12 L mc gOLB du du q gOLB du AB I which is again an invariant and linear in q v and AB one can argue the second term as the only plausible one The equations of motion thereby implied are dXB d 10 gocB d dXH aAB 11 q gocB AB qguBWB OL 13 X du dXOl dXB gals W W and with the same choice of u as 17 and extraction of the 17 dependence of AB through the x dZXB dXH 3A6 3A6 J go Bm dr2 q gHB ax oc39gw Bx H dr axoc axu Xm l39 q Tuocfoc 14 the same equation as obtained previously thus con rming the choice of L above The procedures of classical mechanics may be continued to construct a Hamiltonian from the conjugate momenta 23