Maxwell equations in Lorentz covariant integral form

Transcription

1 ENEÑANZA REVITA MEXIANA DE FÍIA E 52 ( JUNIO 2006 Maxwell equations in Lorentz covariant integral form E. Ley Koo Instituto e Física, Universia Nacional Autónoma e México, Apartao Postal , México D.F., México. Recibio el 1 e septiembre e 2005; aceptao el 21 e febrero e 2006 Most textbooks of electromagnetism give comparable weights to the presentation of Maxwell equations in their integral an ifferential forms. The same books, when ealing with the Lorentz covariance of the Maxwell equations, limit themselves to the iscussion of their ifferential forms, an make no reference to their integral forms. uch a gap in the iactic literature is brige in this paper by explicitly constructing the latter via the integration of the former, for the source-epenent an source-inepenent cases, over a four-vector an a rank-3 tensor hypersurfaces, respectively. Keywors: Electromagnetismo; Maxwell equations; integral an ifferential forms; stanar an Lorentz-covariant forms. La mayoría e los textos e electromagnetismo an pesos comparables a la presentación e las ecuaciones e Maxwell en sus formas integrales y iferenciales. Los mismos libros, al tratar la covariancia e Lorentz e las ecuaciones e Maxwell, se limitan a la iscusión e sus formas iferenciales, y no hacen referencia a sus formas integrales. Tal laguna en la literatura iáctica se elimina en este artículo, construyeno explícitamente las últimas por meio e la integración e las primeras, para los casos epenientes e inepenientes e las fuentes, sobre hipersuperficies cuarivectorial y tensorial e rango 3, respectivamente. Descriptores: Electromgnetismo; ecuaciones e Maxwell; formas integrales y iferenciales; formas estánar y covariantes e Lorentz. PA: 41.20; p 1. Introuction Introuctory [1 3], intermeiate [4, 5], avance [6 8] an grauate [9 12] books on electromagnetism usually present Maxwell equations in their integral forms first, an then make use of them an of the Gauss an toke theorems to obtain their ifferential forms. The balance presentation of both forms at this initial stage can be contraste with the one sie iscussions of the Lorentz covariant ifferential forms of Maxwell equations in the same texts, in which the integral forms are practically absent. The purpose of this work is to provie a balance in the stuy of the Lorentz covariance of Maxwell equations in both forms, filling in an obvious gap in the books. ection 2 contains a brief review of Maxwell equations, the connections between their integral an ifferential stanar forms, an their Lorentz covariant ifferential forms, with emphasis on their physical contents an the mathematical arguments to go between their various forms. ection 3 presents the Lorentz covariant integral forms of Maxwell equations, constructe by integrating the corresponing ifferential forms over a four-vector an a rank-3 tensor hypersurfaces for the source-epenent an source-inepenent cases, respectively. ection 4 contains a iscussion of the aitional physical insights, incluing important relativistic effects, that follow from the connections between the Lorentz covariant an stanar integral forms of Maxwell equations. 2. Maxwell equations in stanar an Lorentz covariant ifferential forms Maxwell equations are the mathematical expressions of the laws of electromagnetism. These laws are escribe in wors first, an their successive mathematical forms are presente next. Gauss electric law: Electric charges are sources of electric flux. Ampére-Maxwell law: Electric currents an isplacement currents are sources of magnetic circulation. Gauss magnetic law: There are no magnetic monopoles as sources of magnetic flux. Faraay electromagnetic inuction law: The time rate of change of magnetic flux is a source of electric circulation. The first two laws are ientifie as the source-epenent laws an the last two as the source-inepenent laws. The former are consistent with the law of conservation of electric charge, while the latter make the escription of the electromangetic phenomena possible in terms of potentials, as shown later on. Now we procee to express the same laws in the form of Maxwell equations in the stanar integral forms: E( r, t a = 4πQ(t (1 B( r, t l = 4π c I(t + 1 c E( r, t a (2 B( r, t a = 0 (3 E( r, t l = 1 c B( r, t a (4 Here the surface integrals over close surfaces of Eqs. (1 an (3, of the respective electric intensity fiel E

2 MAXWELL EQUATION IN LORENTZ OVARIANT INTEGRAL FORM 85 an magnetic inuction fiel B, correspon to the mathematical flux integrals. In Eq. (1, Q represents the electric charge within the volume limite by the close surface of integration, while in the magnetic case, Eq. (3, the corresponing term is zero ue to the non existence of magnetic monopoles. The line integrals over close curves of Eqs. (2 an (4, correspon to circulation integrals for the respective fiels. In Eq. (2, I represents the intensity of the electric current crossing the open surface limite by the curve, while the isplacement current is associate with the time rate of change of the electric flux across the same open surface. imilarly, in Eq. (4, the right han sie epenens on the time rate of change of the magnetic flux across the open surface limite by the curve. Also, the positive signs on the r.h.s. of Eq. (2 are consistent with Ampere s right han law etermining the relative irection of the currents an the magnetic inuction fiel, while the negative sign on the r.h.s. of Eq. (4 is require by Lenz s left han law, in orer to guarantee energy conservation In orer to go from the integral forms of Maxwell s equations (1- (4 to the respective ifferential forms, use is mae of the Gauss an tokes theorems for any vector fiel V ( r, t: V ( r, t a = V ( r, tτ (5 V V ( r, t l = V ( r, t a, (6 also known as the flux (or ivergence an circulation (or curl theorems. The first one gives the flux integral as a volume integral of the ivergence erivative of the fiel, V, while the secon expresses the circulation integral as the surface integral of the curl erivative of the fiel, V. The point limits of Eqs. (5 an (6 lea to the geometrical interpretation of such erivatives: V V = lim ( r, t a (7 τ 0 τ ( V V ˆn = lim ( r, t l (8 a 0 a as the flux per unit volume an the circulation per unit area, respectively. Therefore, the stanar ifferential forms of Maxwell s equations follow immeiately from Eqs. (1- (4 an (7- (8: E( r, t = 4πρ( r, t (9 B( r, t = 4π c J( r, t + 1 c E( r, t t (10 B( r, t = 0 (11 E( r, t = 1 c B( r, t, (12 t where ρ( r, t = Q/ τ is the electric charge volume ensity an J ˆn = I/ a is the electric current ensity. When the ivergences of both sies of Eq. (10 are evaluate, the l.h.s. vanishes, an the ivergence of the electric intensity fiel on the r.h.s. can be substitute by its value from Eq. (9 with the final result J + ρ = 0, (13 t which is the continuity equation expressing the conservation of electric charge. The solenoial character of the magnetic inuction fiel as expresse by Gauss law, Eq. (11, is immeiately satisfie when written as B = A (14 in terms of the vector potential A. ubstitution of Eq. (14 in (12 allows us, in turn, to write E = φ 1 A (15 c t in terms of the scalar potential φ. Equations (14 an (15 are solutions of the ifferential Eqs. (11 an (12, but the potentials A an φ are not uniquely efine. We are free to change them via the so-calle gauge transformations: an A A + χ (16 φ φ χ ct, (17 leaving the force fiels B an E with the same values, or invariant. The ifferential equations satisfie by the potentials are obtaine by substituting Eqs. (14 an (15 in the sourceepenent Maxwell s equations (9 an (10. The resulting equations, 2 φ 1 c t ( A = 4πρ (18 ( A 2 A 4π = J c 1 ( φ c t 1 c 2 2 A t 2, (19 are couple equations in both potentials. They can be uncouple by choosing the Lorentz gauge A + φ = 0, (20 ct with the result that both potentials obey the inhomogeneous wave equations 2 φ 1 2 φ c 2 = 4πρ (21 t2 2 A 1 c 2 2 A t 2 = 4π c J, (22

3 86 E. LEY KOO with the electric charge ensity ρ an the electric current ensity J as their respective sources. Notice that for the time inepenent situations, Eqs. (21 an (22 become the Poisson equations of electrostatics an magnetostatics, respectively; while the Lorentz gauge of Eq. (20 reuces to the transverse gauge. Now we are in a position to ientify the tensorial characteristics of the sources, potentials an force fiels uner Lorentz transformations, by writing their respective equations in obviously covariant forms. The starting point is to introuce the four-vector x µ (x, y, z, ict efining the spacetime position of each event, where µ = 1, 2, 3, 4. Then the space-time isplacement between two neighboring events is given by x µ ( x, y, z, ic t. orresponingly, the space-time rate of change involves the four-vector irectional erivative / (/x, /y, /z, /ict. Fourscalar quantities can be constructe by contracting fourvectors: 4 x µ x µ = x 2 + y 2 + z 2 c 2 t 2 (23 µ=1 is the square of space-time interval, an 4 µ=1 = 2 x y z 2 1 c 2 2 t 2, (24 is the wave operator or D Alambertian involve in Eqs. (21 an (22. Notice the importance of the presence of the imaginary unit i in the fourth or time component in the fourvectors, which translates into the negative sign in Eqs. (23 an (24, as require by the Minkowski metric. Einstein s summation convention consists in ropping the summation sign in Eqs. (23 an (24 an simply summing the successive terms over the repeate inex as µ = 1, 2, 3, 4; this convention an some symbols are use in the following: x µ x µ = ( σ 2 = c 2 ( τ 2, (25 where σ is the norm of the space-time interval an τ is the proper time, an = 2 = c 2 t 2 (26 is the D Alambertian, the natural space-time extension of the Laplacian operator. Einstein s special relativity principle states that all laws of physics are vali in any inertial frame of reference. Lorentz covariance implements this principle by writing the laws of physics in terms of tensor equations, thus guaranteeing that they keep the same form when changing from one frame to another. The implementation for the laws of electromagnetism is carrie out next. We start out with the conservation of charge, Eq. (13, in which the presence of the components of / is recognize. If the zero on the r.h.s. is to be the same in all inertial frames, the reasonable an simplest assumption is that it is a scalar, which suggests that the current an charge ensities must be the components of a four-vector J µ ( J, icρ. Then the four-vector form of Eq. (13 becomes J µ = 0. (27 We continue with the Lorentz gauge conition of Eq. (20, which by the same reasoning of the previous paragraph takes the form A µ = 0, (28 an allows the ientification of the four-vector potential A µ ( A, iφ. The presence of the D Alambertian in Eqs. (21 an (22 ha alreay been pointe out, an they become a single fourvector equation connecting the four-vector potentials an sources: 2 A µ = 4π c J µ. (29 Equations (14 an (15 are combine in a single antisymmetric secon rank tensor equation, f µν = A ν A µ x ν, (30 expressing the connection between the force fiel an the erivatives of the potentials. omparison of the components of Eq. (30, for µ=1, 2, 3, 4 an ν=1, 2, 3, 4, with those of Eqs. (14 an (15 leas to the ientification of the electromagnetic fiel antisymmetric tensor 0 B z B y ie x f µν = B z 0 B x ie y B y B x 0 ie z, (31 ie x ie y ie z 0 where the space-space components correspon to the magnetic fiel an the space-time an time-space components correspon to the electric fiel. In turn, the source epenent Maxwell s equations (9 an (10 become a single four-vector equation f µν = 4π c J ν (32 involving the four-ivergence of the fiel tensor. imilarly, the source-inepenent Maxwell s equations (11 an (12 are combine in a rank-3 tensor equation f µν x λ + f νλ + f λµ x ν = 0 (33 involving the simmetrize combination of four-graients of the antisymmetric fiel tensor.

4 MAXWELL EQUATION IN LORENTZ OVARIANT INTEGRAL FORM 87 The reaer can verify that Eq. (32 correspons to Eq. (9 for ν = 4, an to Eq.(10 for ν = 1, 2, 3; an Eq. (33 to Eq. (11 for λ, µ, ν = P (1, 2, 3, to Eq. (12 for λ, µ, ν = P (1, 2, 4, P (3, 1, 4 an P (2, 3, 4, where P symbolizes the permutations of the inices, an to the ientity 0 = 0 for all other combination of the tensor inices. Thus the presentation of Maxwell equations in their stanar integral forms, Eqs. (1-(4, stanar ifferential forms, Eqs. (9-(12, an Lorentz covariant ifferential forms, Eqs. (32-(33, is complete. This section can be conclue by pointing out how we can go back an forth between the successive forms. While the steps from the stanar integral forms to the stanar ifferential forms have alreay been escribe by using the briges of Eqs. (5-(8, the return trip from Eqs. (9-(12 to Eqs. (1 an (4 uses the same briges after integrating the ivergence Gauss laws over a finite volume, an the curl Ampere-Maxwell an Faraay s laws over an open surface. The steps back an forth between the sets of Eqs. (9-(12 an (32-(33 have alreay been escribe in the previous paragraph. 3. Maxwell equations in Lorentz covariant integral form The absence of the Lorentz invariant integral form of Maxwell equations in the textbooks is intentionally mirrore in the previous section. Here we simply continue with the obvious task of constructing such integral forms by integrating the ifferential forms of Eqs. (32 an (33 over the appropriate omains. In the process the circle is complete by showing the connections between the Lorentz covariant an stanar integral forms. The key elements for the construction are the tensorial natures of the equations to be integrate an of the elements of integration, which must be properly matche in orer to guarantee the return to Eqs. (1 an (4. It was alreay recognize that Eq. (32 is a four-vector equation an Eq. (33 is a rank- 3 tensor equation. Each of them inclues one stanar ivergence an one stanar curl equation, which upon integration over a three-imensional volume an a two-imensional surface, respectively, become the stanar flux an circulation integral forms of the corresponing Maxwell equations. The elements of integration of the Lorentz covariant ifferential Eqs. (32 an (33 must inclue the volume an surface elements in the respective stanar integral forms. On the other han, the choices of elements of integration in the four-imensional space time inclue a four-vector line element x µ, a rank-2 tensor surface element x µ x ν, a rank-3 tensor hypersurface element x λ x µ x ν, a four vector hypersurface element 3 x µ (yzc, xzc, xyc, ixyz, an a scalar hypervolume element 4 x = x 1 x 2 x 3 x 4. The nee for integration over both volume an surface elements in the stanar forms automatically exclues the line an surface elements in the covariant case. From the remaining elements, the three-imensional hypersurfaces satisfy the conitions of incluing the 3-D stanar volume an the 2-D stanar surface elements involve in the stanar Maxwell equations (1-(4. Thus, they are the natural caniates to be chosen as the omains of integration of Eqs. (32 an (33. The integration of Eq. (32 over the four-vector hypersurface leas to the rank-2 tensor equation fµν 3 x λ = 4π J ν 3 x λ. (34 c Let us analyze some of its components in orer to ientify those that are connecte with Eqs. (1-(2. We start with the time-time component ν = 4, λ = 4, ( f14 + f 24 + f 34 + f 44 x 1 x 2 x 3 x 4 = 4π c 3 x 4 J 4 3 x 4. (35 When the space-time components of the fiel-tensor, Eq. (28, an the explicit forms of the integration element an the value J 4 = icρ are use, the result is ( Ex x + E y y + E z z = 4π c xyz ρxyz, (36 which is ientifie as the electric Gauss law of Eq. (1. Next, we write one of the iagonal space-space components of Eq. (31, ν = λ = 1, with the result ( B Ex x yz + ct yz = 4π J x yz, (37 c an similarly for ν = λ = 2 an ν = λ = 3. When the three iagonal space-space component equations are ae, the integrans of the time integration are connecte by the Ampére-Maxwell law, Eq. (2. The conclusion is that Eq. (34 correspons to the Lorentz covariant integral form of the source-epenent Maxwell equations, with the ientifications of its time-time component as the Gauss law, Eq. (1, an of the trace of its space part as the Ampére-Maxwell law, Eq. (2, before the common time integration in Eq. (37. imilarly, the integration of Eq. (33 over the rank-3 tensor hypersurface allows us to write the rank-6 tensor equation ( fµν x λ + f νλ + f λµ x ν x α x β x γ = 0. (38 The only combinations of the tensor inices in the integran have alreay been ientifie in the paragraph after Eq. (33;

5 88 E. LEY KOO an the tensor inices of the integration element must match them α, β, γ = P (λ, µ, ν in orer to connect with the stanar form of the source-inepenent Maxwell equations (3 an (4. In fact, the space-space-space component becomes ( f23 x 1 + f 31 x 2 + f 12 x 3 x 1 x 2 x 3 = 0, (39 an, in terms of the magnetic fiel components, ( Bx x + B y y + B z xyz = 0, (40 z which is recognize as the magnetic Gauss law of Eq. (3. In the same way, the space-space-time components, P (i, j, 4P (i, j, 4 of Eq. (38 with i j, lea to the form ( fj4 x i + f 4i x j + f ij x 4 x i x j x 4 = 0. (41 In turn, by substituting the electric an magnetic fiel component, an ientifying the integration element of area a k = x i x j for (i, j, k = P (1, 2, 3, as well as x 4 = ic, Eqs. (41 become [ Ej E i + B ] k x i x j x i x j ct [ ( E + 1 ] B a k = 0 (42 s c t k The integrans of the time integral in this equation, summe over the inex k = 1, 2, 3 lea to Faraay s law of Eq, (4. Therefore, the Lorentz covariant integral form of the source-inepenent Maxwell equations is given by Eq. (38, in which the space-space-space components correspon to the magnetic Gauss law, an the space-space-time components in the time integrans of Eq. (42 a up to Faraay s law, Eq. (4. 4. Discussion The absence of the Maxwell equations in Lorentz covariant integral form in textbooks of electromagnetism has been pointe out in ec. 1, recognizing the nee to fill in such a gap. ection 2 presente the laws of electromagnetism in the forms of Maxwell equations in their stanar integral [Eqs. (1-(4] an ifferential [Eqs. (9-(12] versions, an also their Lorentz covariant Eqs. (32 an (33, versions, mirroring the presentation in the text books. In ec. 3, the Lorentz covariant integral forms of Maxwell equations (34 an (38 were constructe by integrating Eqs. (32 an (33 over the four-vector an rank-3 tensor hypersurface elements, respectively, completing an balancing the stuy of the four ifferent forms. The remaining iscussion complements the connections between Eqs. (34-(38 an Eqs. (1-(4, explaining some of the mathematics an physics behin them. We call the reaer s attention to the facts that the steps from Eq. (34 to Eq. (1 an from Eq. (38 to Eq. (3, relate to the electric an magnetic Gauss laws, are irect, while the steps from Eq. (34 to Eq. (2 an from Eq. (38 to Eq. (4, relate to the circulation laws of Ampére-Maxwell an Faraay, involve the summations of the integrans in the respective time integrations. We start with the source-epenent Maxwell equation (34, involving rank-2 tensors. Its iagonal timetime [Eq. (35] an space-space [Eq. (37] components lea to Eq. (1 an (2 by integrating over the time an space components of the four-vector hypersurface elements 3 x µ ( ac, iv, respectively, as etaile in ec. 3. The 3D volume integration in Eq.(36 gives irectly the connection with Eq. (1. In contrast, Eq. (37 involves both the time integration an one of the components of the 2D surface integration, thus requiring the comparison of integrans an the summation over components in orer to connect with Eq. (2. On the other han, since Eq. (34 is an equality between rank-2 tensors, the traces of such tensors are scalars an equal to each other. The traces are obtaine by contracting the inices ν=λ in Eq. (34, which is equivalent to summing Eqs. (37 with ν=1, 2, 3 an (36 with ν=4. Notice that, in fact, Eq. (1 involves the electric flux an the electric charge that are recognize to be Lorentz scalars. In contrast, the magnetic circulation an its source currents must be integrate over time, as shown by Eq. (37, written as B l= [ 4π c J a+ 1 c ] E a, (43 in orer to show the associate Lorentz scalars, an the connection with Eq. (2. Also, in other wors, the time integrate magnetomotive force aroun curve is the linear combination of the electric charge an the electric flux crossing surface. In the case of the source-inepenent Maxwell equations (38, the integran is a symmetric-antisymmetric rank-3 tensor an the integration element is a symmetric rank-3 tensor. Let us recall that the ifferential form, Eq. (33, correspons to Bianchi s ientity involving the antisymmetric fiel tensor, Eq. (31, an the symmetric combination of its graients. The selection of matching tensor inices in Eq. (38, to be of the space-space-space an space-space-time for both sets λ, µ, ν an α, β, γ involves again V = xyz an x i x j c, Eqs. (39 an (41, respectively. orresponingly, Eq. (39 connects immeiately with (3, but (42 requires the comparison of time integrans an the summation in orer to connect with Eq. (4. The contraction of inices λ=α, µ=β, ν=γ in Eq. (38 also leas to our equality of Lorentz scalars, constructe as the sum of Eqs. (40 an (42 for k = 1, 2, 3. In fact,the magnetic flux of Eq. (3 is a scalar too, an the connection between Eqs. (42 an (4 requires the time integration in-

6 MAXWELL EQUATION IN LORENTZ OVARIANT INTEGRAL FORM 89 volving the scalars E l = [ 1 c ] B a (44 which are ientifie as the time integrate electromotive force aroun an the magnetic flux crossing. In conclusion, the scalar invariants associate with the rank-2 tensor equation (34 an the rank-6 tensor equation (38, contain the stanar integral forms of the sourceepenent an source-inepenent Maxwell equations, respectively; irectly for the Gauss laws, an as integrans in the circulation laws. Upon recognition of the scalar nature of Eqs. (1 an (43, an Eqs. (3 an (44, their valiity for all inertial frames is establishe beyon any oubt. Obviously, Eqs. (34 an (38 have much more mathematical an physical content than Eqs. (1 - (4. This contribution covers only the connections between the respective integral forms of Maxwell equations, involving only the iagonal components of the tensor equations, with the istinction between the time-time an space-space contributions. The tensors in Eqs. (34 an (38 also have off-iagonal components, an their analysis may lea to future work an results. 1. D. Halliay an R. Resnick, Physics for tuents of cience an Engineering (Wiley, New York E.M. Purcell, Electricity an Magnetism (Berkeley Physics ourse, McGraw-Hill, New York, 1965 Vol P. Lorrain an D.R. orson, Electromagnetism Principles an Applications, econ Eition (Freeman, New York, R.P. Feynman, R. Leighton, M. ans, The Feynman Lectures on Physics (Aison-Wesley, Reaing. Mass O.D. Jeffimenko, Electricity an Magnetism: An introuction to the theory of electric an magnetic fiels (Apple-entury- rofts, New York, J. R. Reitz an F.J. Milfor, Founations of Electromagnetic Theory, econ Eition (Aison-Wesley, Reaing, Mass D.J. Griffiths, Introuction to Electroynamics, Thir Eition (Prentice Hall, Upper ale, N.J L. Eyges, The lassical Electromagnetic Fiel (Dover, New York, W.K.H. Panofsky an M. Phillips, lassical Electricity an Magnetism (Aison-Wesley, Reaing, Mass J.D. Jackson, lassical Electroynamics, econ Eition (Wiley, New York M. Bréov, V. Rumiántsev, an I. Toptiguin, Electroinámica lásica (Mir, Moscú W. Greiner, Electroynamics pringer (Heielberg, 1996.