As Regards the Speed in a Medium of the Electromagnetic Radiation Field

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1 Journal of Modern Physis, 6, 7, 3-33 Published Online July 6 in SiRes. As Regards the Speed in a Medium of the letromagneti Radiation Field Robert M. Yamalee, A. R. Rodríguez-Domínguez Joint Institute for Nulear Researh, Dubna, Russia Instituto de Físia, Uniersidad Autónoma de San Luis Potosí, San Luis Potosí, Méxio Reeied June 6; aepted July 6; published 5 July 6 Copyright 6 by authors and Sientifi Researh Publishing In. This work is liensed under the Creatie Commons Attribution International Liense (CC BY). Abstrat The eloity of the eletromagneti radiation in a perfet dieletri, ontaining no harges and no ondution urrents, is explored and determined on making use of the Lorentz transformations. Besides the idealised blakbody radiation, whose auum propagation eloity is the uniersal onstant, being this alue independent of the obserer, there is another behaiour of eletromagneti radiation, we all inertial radiation, whih is haraterized by an eletromagneti iner- x, and therefore, it happens to be desribed by a time-like Poynting four-etor tial density ( ) field ( x ν ) whih propagates with eloity < β. ( x ) is found to be a relatiisti inariant expressible in terms of the relatiisti inariants of the eletromagneti field. It is shown that there is a rest frame, where the Poynting etor is equal to zero. Both phase and group eloities of the eletromagneti radiation are ealuated. The wae and eikonal equations for the dynamis of the radiation field are formulated. Keywords Inertial Radiation Field, Mass Field Density, Rest State, Poynting Vetor, Wae and ikonal quations of the Radiation Field Dynamis. Introdution In his famous leture deliered for almost 93 years now to the Nordi Assembly of Naturalists at Gothenburg [], A. instein expressed his liely interest onerning the problem of identifying graitational and eletromagneti fields not as quite independent manifestations of nature, but just as two manifestations of same nature, or of a same entity. Although some preious efforts hae been already onduted looking for generalized and dual How to ite this paper: Yamalee, R.M. and Rodríguez-Domínguez, A.R. (6) As Regards the Speed in a Medium of the letromagneti Radiation Field. Journal of Modern Physis, 7,

2 R. M. Yamalee, A. R. Rodríguez-Domínguez expressions of the eletromagneti field, where the graitational field should already appear lodging there []-[6], the idea of present work is to deelop a ontribution to see through as diretly as possible, not where the onept of unifiation is now playing a deep role; but where the onept of inertiality appears learly as an intrinsi hidden property of the eletromagneti field. So, the problem is to derie a formula for eloity of the eletromagneti radiation as a ertain funtion of the field strengths [7]. In this ontext, let us reall that in the ase of the dynamis of massie point partiles, it an onentionally be presented by two expressions, where the first one ontains eolution equations for the energy and momentum, and the seond part ontains a onnetion among energy, momentum and the eloity of the partile. The relationship of the energy with the eloity is the main part of the partile dynamis whih is indispensable in order to define a trajetory of motion of the point partile. The aim of the present paper is to elaborate a method for finding the eloity of the e-m radiation, orresponding to the transport of the energy, momentum and angular momentum, as a funtion of the field strengths. Z. Oziewiz [8] (998) proposed that the transportation of the energy by e-m radiation field is possible if only if the density of the momentum does not anish with respet to all inertial obserers. He has argued that this may happen only in the system of referene moing with light-eloity equal to, beause for other systems of referene moing with eloities less than light-eloity, one may find suh a system where the Poynting etor anishes. Furthermore, he elaborated a method of alulation of the eloity of the system of referene where the Poynting etor anished [9]. In this paper, we explore the dynamis of energy and momentum of the eletromagneti field by introduing the onepts of eloity of the radiation and the field mass density. Our method is based on the same idea of Z. Oziewiz proposed to identify the eloity of the radiation with the eloity of the system of referene, where the Poynting etor anished. Howeer, oppositely to our result, he onluded that there was no physial rest frame, with <, where the Poynting etor might anish, but the frame of light. In order to define a eloity for the eletromagneti radiation, a simple logi sheme has to be used: there exists a ertain inertial referene system where the density of the momentum of the inertial e-m radiation field is equal to zero. The eloity of this inertial system is identified with the eloity of the radiation. So, aording to this onept, in order to define the eloity of the field, it is suffiient to know the transformation laws of the field under the Lorentz-group. The Conept of the Inertial letromagneti Field In the solution of any eletromagneti problem the fundamental relations that must be satisfied are the four field equations Maxwell equations []. Consider the partiular ase of eletromagneti phenomena in a perfet dieletri ontaining no harges and no ondution urrents. For this ase the Maxwell equations beome [ B] =, (.a) t ( ) =, (.b) B (.a) t [ ] + =, ( B ) =. (.b) Aording to Maxwell theory the eloity of the eletromagneti (e-m) waes is defined ia permittiity and permeability of the medium. In the same way, in the auum the eloity is defined by the uniersal onstant =. (.3) The physial sense of the speed of the eletromagneti radiation is attahed to the eloity of the flux of radiation transporting energy, momentum and angular momentum. Sine the speed of the light in the medium is defined by formula =, the eloity of eletromagneti radiation depends of the index of refration n, n =, i.e, the eloity of the e-m radiation in the medium depends on the refratie index n of the medium. 3

3 R. M. Yamalee, A. R. Rodríguez-Domínguez The refratie index n is related with the phase eloity and the waelength aording to formulae where n = =, (.4) ph λ λ π λ = (.5) ω is the auum waelength. We are addressing both ases ouring n > and n <. The transersal fields in the medium are related as follows B = n[ k ], (.6) and [ ] B = n k, (.7) ( n ) B =. (.8) It is seen, that in the medium the alue of the e-m field harateristis hange in suh a way that the property of transersality keeps onsered, but the relationship B =, (.9) whih is alid for the radiation field in the auum, in the medium holds no more true. This argument is taken as a pioting idea to introduing the onept of inertial e-m radiation field, as a transersal e-m field, whih is prinipally haraterized by the main ondition B. (.) One the onept of inerial eletromagneti field is defined, the rest of the paper is organized as follows. First we present an alternatie oariant double-salar potential formalism for the representation of transersal eletromagneti fields (Setion ). Thereafter we derie the formula for the phase eloity of the eletromagneti radiation field (Setion 3). As an appliation of this formula, we write the eikonal equation of the geometrial optis, and the wae equation for eletromagneti radiation field for a definite energy density, Poynting etor and mass field density (Setion 4). Finally, in Setion 5 the main onlusions of the work are thrown.. Double Potential Representation of the Transersal letromagneti Field In Ref. [] we hae suggested another look on the nature of the transersal e-m fields. Aording to this iewpoint, for the transerse e-m fields, instead of using four potential funtions, it is better to use a pair of Lorentz salar funtions. This representation automatially proides transersality of the strength etors. Furthermore, the pair of salar potentials are solutions of the Klein-Gordon equations, whereas the fourpotentials play the role of a urrent density, and the Lorentz gauge ondition for the four potentials takes the form of a ontinuity equation for the urrent density. The Maxwell s equations in the auum are equations for the eletri field strength and the magneti flux density B. The four-potentials Φ, Ax, Ay, Az were introdued into the Maxwell s eletrodynamis in order to simplify the system of wae equations. Howeer, thanks prinipally to the quantum mehanis, the essential oneptual and theoretial role of the potentials has been manifested. Indeed, in the Shrödinger, Dira and Klein-Gordon equations the field strengths do not any longer appear, but the four potentials atually do instead. xternal e-m fields were introdued into quantum mehanis ia four-potentials already at the leel of Hamilton-Jaobi equations. It is to be aknowleged, that the priniple of gauge inariane was born thanks to the four-potential representation of the e-m fields. Howeer, the form of the potential representation has a great signifiane in the quantum field theory, in the theory of superondutiity, in the theory of the magneti harge. The puzzling role playing by the etor potential representation in the quantum mehanial motion of partiles was strikingly illustrated by the Aharono-Bohm effet []. As we ery well know, the equation 3

4 R. M. Yamalee, A. R. Rodríguez-Domínguez ( B ) =. (.) is automatially satisfied, if the magneti-flux density is represented as B = [ A ], (.) beause of the identity ( [ ]) =. The expression for the eletri field strength is in turn expressed by A (.3) A = Φ. (.4) t These formulae are suh that the seond group of Maxwell equations representing Faraday s law and the absene of magneti harges are automatially satisfied. The first group of Maxwell equations are redued to the following two equations for the potentials A A+ ( A ) + Φ + =, (.5a) t t Φ + =. t A (.6b) quations (.6a,.6b) an be separated by hoosing the so alled Lorentz gauge ondition ( A ) + Φ=. (.7) t Substitution of quation (.7) into (.6a,.6b) yields wae equations for the four-potentials A =, Φ Φ =. t A t (.8) Now, let us rewrite these formulae in their tensorial form. From the potentials Φ and A we pass into the four-etor representation by Further, let us introdue four-oordinates ( A) ( A ) A : = Φ,, A : = Φ, ( r) ( r ) x : = t,, x : = t,. In this notation, the etors of eletri field strength and magneti flux density may be ast into the form of a srew-symmetri tensor ( ) Then, formulae (.4) and (.5) are joined into one expression F : = B ν,. (.9) Fν = Aν ν A, with : =. (.) x The first group of Maxwell equations takes the form =, (.a) F ν whereas the seond part of Maxwell equations admits the following form ρ Fν + Fνρ + ν Fρ =. (.b) In order to obtain the wae equation for the four-potential etor, usually, the Lorentz-gauge ondition is used =. (.) A 33

5 R. M. Yamalee, A. R. Rodríguez-Domínguez With this ondition, the Maxwell equations are redued to the wae equation for the four-potential etor A =. (.3) ν With respet to Lorentz transformations the eletromagneti fields are haraterized to possess two inariants I = ( B ) and I = B. Among them, the field defined as a pure radiation field, or null field, for whih the two inariants are identially zero, has a peuliar status. This field is a propagating field with the wellknown properties B k, where k is the diretion of propagation. Under these onditions, we refer to the transersal harater of the e.m. radiation field []. Howeer, the potential representation of this field meets diffiulties onerning with its Lorentz oariant desription. The radiation fields are desribed by two degrees of freedom, whereas the theory desribes them by a four potential etor. Therefore, it is neessary for the theory to introdue subsidiary onditions, like the radiation gauge, or the Coulomb gauge. These Lorentz nonoariant onditions are used besides the Lorentz-oariant Lorentz gauge equation. These are the main diffiulties whih appear in the ommon aepted formulation. Alternatiely we an handle the formalism as follows First of all let us notie that the equation ( B) =, (.4) is satisfied by defining the magneti-flux density as B = [ φ ψ]. (.5) For the eletri field strength one obtains φ ψ = ψ φ. t t (.6) In a tensorial form these formulas are gien by = φ ψ φ ψ. (.7) F ν ν ν Obiously, the funtions φψ, (two-potentials) are inariant with respet to Lorentz-transformations. Within these formulae, the field Lorentz-transformations diretly follow from the oordinate Lorentz-transformations. The four-potential etor is expressed from two-potentials as follows ( ), ψ φ A = φ ψ ψ φ Φ= φ ψ, (.8) t t or in tensorial notation A = ( φ ψ ψ φ). (.9) The adantage of using this two-potential representation resides in the fat that we automatially introdue the two desired degrees of freedom. The first main onsequene of this approah is the property of transersality of the eletromagneti field. In fat, in this representation, the field strength etors satisfy the equation ( ) I : = B =. (.) Notie, howeer, that the dealing of the seond inariant I is not triial. It is to be notied, that by reduing the degrees of freedom from four to two, additional algebrai relations between the fields and the four potential etor arise, they are namely [ ] ( ) A Φ B =, A B =. (.) What kind of interpretation an be gien to these equations? Looking for an answer, we refer the reader to Ref. [3] and referenes therein. Stating it briefly, the authors hae found the following topologial inariant S = ( A B )d V. (.) Furthermore, it is shown that for the stati magneti field B, S haraterizes to what extent the magneti lines are oupled to eah other. For a single magneti line this alue estimates the srewness of the line. The 34

6 R. M. Yamalee, A. R. Rodríguez-Domínguez relatiisti generalization of this inariant was done in Ref. [4], as Or in the omponents The four-etor satisfies the equation From this last equation it follows that ν ν ναβ = F Aν, F = Fαβ, (.3) = Φ, =. [ A ] B ( A B) ( B) Lorentz Gauge quation as a Continuity quation =. (.4) is onsered only for transersal fields. The Lorentz gauge quation (.) appears written in our two-potential representation as whih an be ealuated to ( ), φ ψ ψ φ = This equation separates into two Klein-Gordon type equations φ ψ ψ φ =, with =. (.5) t ψ λ = ψ, φ = λ φ, (.6) where the parameter λ is the Compton s length of the wae. Furthermore, two real Klein-Gordon equations (.6) may be united into one Klein-Gordon type equation for the omplex-alued funtion Ψ= φ + iψ. We may also define in our formalism the urrent density as = ( φ ψ ψ φ). (.7) j Furthermore, this urrent density satisfies the ontinuity equation =, (.8) j whih arose aboe as the Lorentz-gauge ondition (.) for the potential A. 3. The Conept of Veloity for the Inertial Radiation Field Consider two obserers and with relatie eloity. Then the e-m fields are transformed aording to the Lorentz formulae where B B B B ( ) = ( ( ) ) + γ ( ) + ( ) ( ) = ( ( ) ) + γ ( ) ( ) γ = Suppose that the eloity of -obserer is perpendiular to eletri and magneti fields deteted by - obserer: ( ) B( ).,, (3.) (3.) = =. (3.3) 35

7 R. M. Yamalee, A. R. Rodríguez-Domínguez Then formulae (3.) are redued to Heaiside formulae ( ) = γ ( ) + B( ), B( ) = γ B( ) ( ). (3.4) These transformation formulae keep unhanged both inariants: The Poynting s etor is transformed as follows ( ) I I = B, = B. (3.5) { } ( ) ( ) = γ [ ] + [ ] [ ] [ ] = [ B] [ ] = ( [ B] ) ( [ B] ) = ( [ B] ) B B B B B (Hereafter we omit the -obserer designation.) Now suppose that in the -system of referenes moing with eloity with respet to the -system the Poynting etor is equal to zero Then, ( ) B( ) =. [ ] ( ) ( [ ]). (3.6) B + + B + B =. (3.7) From this equation it follows that the eloity etor is parallel (or, anti-parallel) to the Poynting etor B. This proposition is ompatible with the ondition (3.3). Thus, we may take [ ] = λ [ B ]. (3.8) In order to onstrut an equation for the unknown onstant λ, from (3.8) is substituted into (3.7). In this way we obtain and ( ) [ ] λ[ ]( ) λ [ ] [ ] B + B + B + B B =. (3.9) Suppose that [ B ]. Then quation (3.9) is redued into the following quadrati equation for λ : Let us introdue the following quantities idently, is a funtion of the two inariants I, I : ( ) ( ) λ [ ] + λ + B + B =. (3.) P = ( + B ), P = [ B ] (3.) = P P. (3.) ( ) ( ) = B + 4 B = I + 4 I. (3.3) Consequently, is itself also an inariant. This alue we interpret as the inertial field density. In this notation, the quadrati equation for λ appears as whih has two different roots λ P + λp + =, (3.4) P λ = P +, P λ = P. By using this quantities in (3.8) we obtain two kinds of eloities [ ] ( P ) = λ B =, (3.5a) P + 36

8 R. M. Yamalee, A. R. Rodríguez-Domínguez and [ ] The first eloity is less than the speed of light. Notie also that ( P + ) = λ B =. (3.5b) P., and other one is greater than the speed light These eloities are expliitly expressed ia field strengths as follows = = = (3.6) + B, + B + + B +. + B (3.7a) (3.7b) Thus, if, then there always exists a system of referene where the Poynting s etor is equal to zero. In other words, the rest state is ahieable. The situation is quite analogous to the ase of massie partile where the state with P = an be ahieed. Now, we are in position to speify the onept of inertial field density. Define this quantity as follows In this notation, the formula for the eloity is written as =. (3.8) P =. P + From this formula it follows the expression for the density of the energy and for the Poynting s etor Let ψ be the rapidity, so that, Then, =, P (3.9) + (3.) P =. (3.) = tanh ψ. (3.) ( ψ) P ( ψ) P = osh, = sinh. (3.3) Comparing these formulae with the analogous formulae for the relatiisti point-partile, it is seen, that in the ase of e-m field, for theenergy and the momentum, the rapidity appears multiplied by the fator. We hae deried two kinds of eloities < and >. In order to interpret these eloities we hae to refer to formulae (.6)-(.8) for transerse fields in the medium with refratie index n. Within our notations of the densities of the energy and the mass we write Consequently, we hae for n >, and ( ) ( ) P = + = + n, = n. B (3.4) P, = = P + n (3.5) 37

9 R. M. Yamalee, A. R. Rodríguez-Domínguez for n <. P +, = = P n From these formulae it follows that the eloities,, k related in ordinary way with the refratie index n. By analogy we may also define the group eloity V (3.6) k = admit an interpretation as the phase eloities dp V =, (3.7) dp satisfying always V. The group eloity is related to the phase eloity by the formula where an be either or. 4. Wae Mehanial Dynamis V = + (3.8) In the preious setion we hae obtained formulae for phase eloities as funtions of the densities of the energy, momentum and the mass. Consider lassial eletromagneti waes traelling aording to a salar wae equation of the simple form n ψ ψ =, (4.) t where ψ ( xyzt,,, ) represents any omponent of the eletri or magneti field and nxyz (,, ) is the (time independent ) refratie index of the medium. By assuming ψ = u r, ω exp iωt, (4.) with r ( xyz,, ) ( ) ( ) =, we get from (4.) the Helmholtz equation ( ) The solutions of the wae equation are looked of the form [5] π ω u+ nk u=, k = =. (4.3) λ ( ω) ( ω) ( φ( ω) ) u r, = R r, exp i r,, (4.4) with real R and φ funtions, whih represent respetiely, the amplitude and phase of the monohromati waes. The wae etor is defined as k = φ r, ω. (4.5) ( ) In the limit of geometrial optis, aording to the eikonal equation ( φ ) ( ) k nk, The ikonal equation in a geometrial wae theory has the form k ( W) = = n =. k (4.6) For the rays inside the medium with refratie index n defined ia P, by formulae (3.5)-(3.6) we write the eikonal equation of the form ( ) P + W = n =, n >, P 38

10 R. M. Yamalee, A. R. Rodríguez-Domínguez and P = =, <. P + ( ) W n n 5. Conlusions The onept of eloity for the radiation fields is a long stated problem. This problem of the eloity of the eletromagneti waes has been always onsidered as a definitely soled problem: these waes hae a eloity equal to = in the auum. From this formula, it follows that in the medium the eloity of the eletromagneti wae obeys atually the same formula =, whih is used also to be written as =, where n is the refratie index of the medium. To the present, n this expression has been onsidered phenomenologial in nature, it is howeer relatiisti. Here, it is lear that we are refering to the phase eloity of the waes, whih an be both smaller as well as larger than the speed of light. Howeer, we hae been able to relate this eloity to the true physial group eloity of the radiation whih is always present. We hae deeloped this study in a relatiisti formalism, onsidering the eloity of the eletromagneti radiation field as proposed by Zbigniew Oziewiz. This formalism is onstruted out of first priniples, beause it is based on the transformations of Lorentz group, in order to derie the formula for the eloity of the referene system where the Poynting etor is equal to zero. In this way we arried to the formula onneting the phase eloity with the densities of the energy and momentum. From the iewpoint of the theory, we hae been able to desribe the inertiality of the eletromagneti Field, we are able to assure that with the exeption of the propagation in auum, where the Poynting etor is a null four-etor, in a dieletri medium, the Poynting etor migrates into a time-like four-etor, whose norm is the inertial field density, formula (3.), that is why the rest frame of the propagation is reahable. To the present, it has been known to us only the massless quantum mehanial behaior of photons. We don t dare, for the moment, to desribe them, inside matter, as mutants in a lassial, massie state, but we just point out to the inertiality as an intrinsi hidden property of the eletromagneti field. How to quantify the inertial field density may surely be the subjet of future works. As we hae introdued the notion of inertial field density as a measure of the inertia of the eletromagneti radiation, in return, we hae formulated a new wae equation enoling the density of energy and the inertial field density. Referenes [] instein, A. (93) Fundamental Ideas and Problems of the Theory of Relatiity. Leture Deliered to the Nordi Assembly of Naturalists at Gothenburg, July 93. [] Kalb, M. and Ramond, P. (974) Physial Reiew D, 9, [3] Doeglazo, V. () Annales de la Fondation Louis de Broglie, 5, 8. [4] Malik, R.P. () Physis of Partiles and Nulei Letters, 8, [5] Rodríguez-Domínguez, A.R. and Martínez-González, A. () Journal of Nonlinear Mathematial Physis, 9, No. 4. [6] Keller, J. (986) International Journal of Theoretial Physis, 5, [7] Kholmetskii, A.L. (4) Annales de la Fondation Louis de Broglie, 9, [8] Oziewiz, Z. (6) letri Field and Magneti Field in a Moing Referene System (Abridged Abstrat). In: Adhikari, M.R., d., Physial Interpretations of Relatiity Theory Session, an International Symposium on Reent Adanes, Kolkata, [9] Oziewiz, Z. (8) Reiew Bulletin of Calutta Mathematial Soiety, 6, [] Barut, A.O. (98) letrodynamis and Classial Theory of Fields and Partiles. Doer Publiations, In., New York. [] Yamalee, R.M. (5) Annales de la Fondation Louis de Broglie, 3, [] Ryder, L.H. (985) Quantum Field Theory. Cambridge Uniersity Press. 39

11 R. M. Yamalee, A. R. Rodríguez-Domínguez [3] Keller, J. (993, 999) The Geometri Content of the letron Theory, Parts I and II. Adanes in Applied Clifford Algebras, 3, 5-5 and 9, [4] Rañada, A.F. (99) uropean Journal of Physis, 3, [5] Orefie, A., Gioanell, R. and Ditto, D. (5) Annales de la Fondation Louis de Broglie, 4, 95-. Submit or reommend next manusript to SCIRP and we will proide best serie for you: Aepting pre-submission inquiries through mail, Faebook, LinkedIn, Twitter, et. A wide seletion of journals (inlusie of 9 subjets, more than journals) Proiding 4-hour high-quality serie User-friendly online submission system Fair and swift peer-reiew system ffiient typesetting and proofreading proedure Display of the result of downloads and isits, as well as the number of ited artiles Maximum dissemination of your researh work Submit your manusript at: 33