Reply to "Comment on `Can Maxwell's equations be obtained from the continuity equation?'" by E. Kapuscik [Am. J. Phys. 77, 754 (2009)]
aa r X i v : . [ phy s i c s . c l a ss - ph ] D ec Reply to “Comment on ‘Can Maxwell’s equations be obtainedfrom the continuity equation?”,’ by E. Kapu`scik [Am. J. Phys.77, 754 (2009)]
Jos´e A. Heras
Departamento de Ciencias B´asicas,Universidad Aut´onoma Metropolitana, Unidad Azcapotzalco,Av. San Pablo No. 180, Col. Reynosa, 02200,M´exico D. F. M´exico and Departamento de F´ısica y Matem´aticas,Universidad Iberoamericana, Prolongaci´on Paseode la Reforma 880, M´exico D. F. 01210, M´exico on my derivation of Maxwell’s equations from the continuityequation are based on the idea that I neither used relativistic notation nor consideredmaterial media. My derivation has also been commented on by Jefimenko. Kapu`scik’sobjections provide the opportunity to clarify certain important points not emphasized inmy original paper. Kapu`scik invokes the well-known result that given the continuity equation in spacetime ∂ ν J ν = 0 , (1)we can write the solution of this equation as J ν = ∂ µ H µν , (2)where H µν is an antisymmetric tensor. Kapu`scik points out that for fixed J ν = [ cρ, ( J ) j ],Eq. (2) becomes a differential equation for H µν that is equivalent to the inhomogeneousMaxwell equations. He concludes that the correct statement should be that the inhomo-geneous Maxwell equations follow from continuity equation. However, on the basis only ofEq. (2) with J ν = [ cρ, ( J ) j ], the tensor H µν is not necessarily the electromagnetic tensorfield, that is, Eq. (2) does not necessarily represent the inhomogeneous Maxwell equations.In (1+3) notation, Eq. (2) implies that[ cρ, ( J ) j ] = ( ∂ i H i , ∂ H j + ∂ i H ij ) . (3)But we are free to specify the components H i and H ij . For sources in vacuum, only thespecific choice (Gaussian units): H i = c π ( E ) i and H ij = − c π ε ijk ( B ) k , (4)allows us to identify Eq. (3) with the equations: ∇ · E = 4 πρ and ∇ × B − c ∂ E ∂t = 4 πc J . (5)For different choices of the components H i and H ij , Eq. (3) cannot be identified withEq. (5). Furthermore, we cannot yet claim that Eq. (5) represents the inhomogeneousMaxwell’s equations because E and B in Eq. (5) are not necessarily electric and magneticfields. To be sure that E and B in Eq. (5) are electric and magnetic fields the quantities ∇ · B and ∇ × E + (1 /c ) ∂ B /∂t need to be specified as ∇ · B = 0 and ∇ × E + 1 c ∂ B ∂t = 0 , (6)2r equivalently as ∂ µ ∗ H µν = 0, but this last relation is not mentioned in Ref. 1 (the dual of H µν is defined as ∗ H µν = (1 / ε µνκσ H κσ ). In other words Eq. (2) cannot yet be identifiedwith the inhomogeneous Maxwell equations because H µν is not completely determined byEq. (2).From the Helmholtz theorem for antisymmetric tensors a tensor H µν that vanishes atinfinity is completely determined by specifying its divergence ∂ µ H µν and the divergence of itsdual ∂ µ ∗ H µν . Equation (2) of Ref. 1 specifies ∂ µ H µν , but ∂ µ ∗ H µν has not been specified. If wemaliciously specify ∂ µ ∗ H µν by the relation ∂ µ ∗ H µν = S ν , where S ν = ( ∇ · s , ∇ × s − (1 /c ) ∂ s /∂t )with s being an arbitrary vector (note that ∂ ν S ν = 0), then the fields E and B in Eq. (5)are not Maxwell’s fields and therefore Eq. (2) cannot be identified with the inhomogeneousMaxwell equations.Kapu`scik’s argument has been explored on the basis of the de Rham theorem in differen-tial forms, which implies that given J ν satisfying ∂ ν J ν = 0, there exists an antisymmetrictensor F µν such that J ν = ∂ µ F µν . This F µν is not uniquely determined. We can addto it the antisymmetric tensor ε µναβ ∂ α χ β , where χ β is an arbitrary four-vector, and therelation J ν = ∂ µ F µν does not change. This ambiguity in the definition of F µν preventsus from giving a physical meaning to this field. From the invariance of ∂ µ F µν = J ν un-der the field transformation F µν → e F µν = F µν + ε µναβ ∂ α χ β , that is, ∂ µ e F µν = J ν , wemay explore the idea of fixing the function χ β so that ∂ µ ∗ e F µν = 0. Suppose that F µν satisfies ∂ µ F µν = J ν , but not ∂ µ ∗ F µν = 0. This means that ∂ µ ∗ F µν = g ν = 0. Thenwe make a field transformation to F µν and require that e F µν satisfies ∂ µ ∗ e F µν = 0, that is, ∂ µ ∗ e F µν = 0 = g ν + ∂ µ ∂ µ χ ν − ∂ ν ∂ µ χ µ . The result ∂ µ ∗ e F µν = 0 would be possible only if wecould find the solution of ∂ µ ∂ µ χ ν − ∂ ν ∂ µ χ µ = − g ν . Such a solution is not known and so wecannot guarantee the existence of the function χ µ , and thus the relation ∂ µ ∗ e F µν = 0 cannotgenerally be established. The derivation of Maxwell’s equations based only on the de Rahmtheorem turns out to be unsatisfactory and further additional assumptions are required.Kapu`scik also claims that my derivation that the homogeneous Maxwell equations followfrom the continuity equation must be considered to be incorrect because the homogeneousMaxwell equations are based on a tensor field F µν that is conceptually independent of thesources of the electromagnetic field present in the inhomogeneous Maxwell equations. Inmy derivation I considered only sources in vacuum satisfying the continuity equation andtherefore E and B obtained in this way (or equivalently F µν ) are causally generated by the3ources ρ and J (or equivalently J ν ) as is seen in Eq. (28) of Ref. 2. That is, in the contextof my derivation the fields E and B (or equivalently F µν ) in the homogeneous Maxwellequations depend on the sources present in the inhomogeneous Maxwell equations.Kapu`scik claims that my derivation of Maxwell’s equations applies only in vacuum andthat Maxwell equations apply to arbitrary media. Equation (6) holds in material media,but Eq. (5) must be replaced by ∇ · D = 4 πρ and ∇ × H − c ∂ D ∂t = 4 πc J , (7)where D and H are related to E and B through the polarization P and the magnetization M of the material media by D = E + 4 π P and H = B − π M . (8)I agree with Griffiths who has written: “Some people regard . . . [Eqs. (6) and (7)] as the“true” Maxwell’s equations, but please understand that they are in no way more “general”than . . . [Eqs. (5) and (6)]; they simply reflect a convenient division of charge and currentinto free and nonfree parts.” From Eqs. (7) and (8) we obtain ∇ · E = 4 π ( ρ − ∇ · P ) , (9) ∇ × B − c ∂ E ∂t = 4 πc (cid:18) J + c ∇ × M + ∂ P ∂t (cid:19) . (10)The right-hand sides of Eqs. (9) and (10) display the division of charge and current intofree and nonfree parts. The derivation of Maxwell’s equation given for free sources in Ref. 2can be generalized to free and nonfree sources by making the substitutions ρ → ρ T and J → J T , where ρ T and J T are the total sources defined by ρ T = ρ f − ∇ · P and J T = J f + c ∇ × M + ∂ P ∂t , (11)where ρ f and J f are free parts and − ∇ · P and c ∇ × M + ∂ P /∂t are nonfree parts. Thereforethe derivation of Maxwell’s equations from the continuity equation applies not only to thevacuum but also to material media.Kapu`scik claims that the homogeneous Maxwell’s equations ∂ µ F νλ + ∂ λ F µν + ∂ ν F λµ = 0 , (12)can be obtained from the equation ∂ µ F νλ + ∂ λ F µν + ∂ ν F λµ = J µνλ , because the Faradayinduction law leads to J µνλ ( x ) = 0. Unfortunately, his argument is circular. It considers4he Faraday induction law, ∇ × E + (1 /c ) ∂ B /∂t = 0, as one of Maxwell’s equations toobtain Eq. (12). But it does not make sense to assume one of Maxwell’s equations to deriveMaxwell’s equations themselves.Kapu`scik claims that my result should be treated as finding a particular solution ofthe vacuum Maxwell equations rather than proving that Maxwell equations for arbitrarymedia follow from the continuity equation. As noted, Maxwell’s equations in matter canalso be obtained from the continuity equation. I agree with Kapu`scik’s comment that myderivation involves only retarded fields and that Maxwell’s equations admit also advancedfields. However, we can derive Maxwell’s equations from the continuity equation usingthe free-space Green function of the wave equation, which admits retarded and advancedforms. The fields E and B obtained in such a derivation would be defined in terms of thisGreen function and would represent either retarded or advanced fields when the retarded oradvanced form of the Green function is made explicit.I emphasize that the derivation of Maxwell’s equations from the continuity equationformally arises from applying an existence theorem. If the proof of the theorem is incor-rect or the mentioned application is inconsistent, then the derivation could be questioned.Kapu`scik has not questioned the validity of the theorem. Acknowledgments
The author thanks Edward Kapu`scik for useful comments. The support of the FondoUIA-FICSAC is gratefully acknowledged. E. Kapu`scik, “Comment on the paper ‘Can Maxwell’s equations be obtained from the continuityequation? by J. A. Heras [Am. J. Phys. , 652–657 (2007)]”’ Am. J. Phys. , – (2009). J. A. Heras, “Can Maxwell’s equations be obtained from the continuity equation?,” Am. J.Phys. , 652–657 (2007). O. D. Jefimenko, “Causal equations for electric and magnetic fields and Maxwell’s equations:Comment on a paper by Heras,” Am. J. Phys. , 101 (2008). J. A. Heras, “Author’s response,” Am. J. Phys. , 101–102 (2008). Greek indices µ, ν, κ . . . run from 0 to 3; Latin indices i, j, k, . . . run from 1 to 3; x = x µ = ( ct, x )is the field point and x ′ = x ′ µ = ( ct ′ , x ′ ) the source point; the signature of the metric of spacetimeis (+ , − , − , − ); ε µναβ is the totally antisymmetric four-dimensional tensor with ε = 1 and ε ijk is the totally antisymmetric three-dimensional tensor with ε = 1. Summation convention onrepeated indices is adopted. A four-vector is represented in the (1+3) notation as F ν = ( f , F ). We could make the choice H i = − [ c/ (4 π )]( E ) i and H ij = − [ c/ (4 π )] ε ijk ( B ) k , for which Eq. (3)does not yield Eq. (5). Equation (5) also represents the inhomogeneous field equations of a Galilean electromagnetictheory, which also involves the homogeneous field equations ∇ · B = 0 and ∇ × E = 0. See M.Jammer and J. Stachel, “If Maxwell had worked between Amp´ere and Faraday: An historicalfable with a pedagogical moral,” Am. J. Phys. , 5–7 (1980); J. A. Heras, “Instantaneous fieldsin classical electrodynamics,” Europhys. Lett. , 1–7 (2005). D. H. Kobe, “Helmholtz theorem for antisymmetric second-rank tensor fields and electromag-netism with magnetic monopoles,” Am. J. Phys. , 354–358 (1984); J. A. Heras, “A shortproof of the generalized Helmholtz theorem,” Am. J. Phys. , 154–155 (1990). See also E.Kapu`scik, “Generalized Helmholtz theorem and gauge invariance of classical field theories,”Nuovo Cimento , 263-266 (1985). F. W. Hehl, Y. Itin, and Y. N. Obukhov, “Recent developments in premetric classical electro-dynamics,” arXiv:physics/0610221. D. J. Griffiths,
Introduction to Electrodynamics (Prentice Hall, Upper Saddle River, NJ, 1999),3rd ed., p. 330.(Prentice Hall, Upper Saddle River, NJ, 1999),3rd ed., p. 330.