Wave four-vector in a moving medium and the Lorentz covariance of Minkowski s photon and electromagnetic momentums

Transcription

1 Wave four-vecor in a moving meium an he Lorenz covariance of Minkowski s hoon an elecromagneic momenums Changbiao Wang ShangGang Grou, 7 uningon Roa, Aarmen, New aven, CT 65, USA In his aer, by analysis of a lane wave in a moving ielecric meium, i is shown ha he invariance of hase is a naural resul uner Lorenz ransformaions From he covarian wave four-vecor combine wih Einsein s lighquanum hyohesis, Minkowski s hoon momenum equaion is naurally resore, an i is inicae ha he formulaion of Abraham s hoon momenum is no comaible wih he covariance of relaiviy Fizeau running waer exerimen is re-analyze as a suor o he Minkowski s momenum I is also shown ha, ( here may be a seuo-ower flow when a meium moves, ( Minkowski s elecromagneic momenum is Lorenz covarian, (3 he covarian Minkowski s sress ensor may have a simles form of symmery, an (4 he moving meium behaves as a so-calle negaive inex meium when i moves oosie o he wave vecor a a faser-han-ielecric ligh see Key wors: invariance of hase, wave four-vecor, moving ielecric meium, ligh momenum in meium PACS numbers: 45Wk, 35e, 45-, 33+ I INTROUCTION Invariance of hase is a well-known funamenal conce in he relaivisic elecromagneic (EM heory owever he invariance of hase for a lane wave in a uniform ielecric meium has been quesione, because he frequency can be negaive when he meium moves a a faser-han-ielecric ligh see in he irecion oosie o he wave vecor [] To solve he robl, a new wave four-vecor base on an aiional assumion was roose: a lane wave ravels a a wave velociy ha is no necessarily arallel o he wave vecor irecion [,3] In his aer, by analysis of a lane wave in a moving ielecric meium, i is shown ha he invariance of hase is a naural resul uner Lorenz ransformaions, an i is a sufficien an necessary coniion for he Lorenz covariance of wave four-vecor When he negaive frequency aears, he moving meium behaves as a so-calle negaive inex meium, resuling in he change of roeries of wave roagaion Base on he symmery of hase funcion wih resec o all inerial frames, a symmeric efiniion of hase velociy an an invarian form of equi-hase lane equaion of moion are given The momenum of ligh in a meium is a funamenal quesion [4-8] Maxwell equaions suor various forms of momenum conservaion equaions [3,5,6] In a sense, he consrucion of EM momenum ensiy vecor is arificial, which is a kin of ineerminacy owever i is he ineerminacy ha resuls in he quesion of ligh momenum For examle, in he recen ublicaion [4], owerful argumens are rovie o suor boh Minkowski s an Abraham s formulaions of ligh momenum: one is canonical, an he oher is kineic, an hey are boh correc The above ineerminacy shoul be exclue by imosing hysical coniions, jus like he EM fiel soluions are require o saisfy bounary coniions Such a hysical coniion is he Lorenz covariance As we know, he hase funcion for a lane wave characerizes he roagaion of energy an momenum of ligh Since he hase funcion is symmeric wih resec o all inerial frames, he covariance of ligh momenum mus hol In his aer, i is also shown ha, Minkowski s ligh momenum is Lorenz covarian, while Abraham s is no Fizeau running waer exerimen is re-analyze as a suor o he Minkowski s momenum A simle examle is given o show: ( neiher Abraham s nor Minkowski s EM momenum can correcly escribe he real EM momenum when a seuomomenum exiss, an ( he aache-o-source fiel oes exhibi he Lorenz covariance of four-vecor when he seuomomenum isaears, which challenges he well-recognize viewoin in he communiy [8] The momenum of ligh in a meium can be escribe by a single hoon s momenum or by he EM momenum The covariance of hoon momenum is shown base on he invariance of hase an Einsein s ligh-quanum hyohesis, while he covariance of EM momenum is shown only base on he covariance of EM fiels The covariance of Minkowski s ligh momenum boies he consisence of Einsein s relaiviy an ligh-quanum hyohesis II INVARIANCE OF PASE Suose ha he frame X Y Z moves wih resec o he lab frame XYZ a a consan velociy of c, wih all corresoning coorinae axes in he same irecions an heir origins overlaing a, as shown in Fig The ime-sace Lorenz ransformaion is given by [9] γ x x + ( x + γ c, ( c γ ( c + x ( where c is he universal ligh see, an / γ ( is he ime ilaion facor The EM fiels E an B, an an resecively consiue a covarian secon-rank ani-symmeric α α ensor F ( E, B an G (,, of which he Lorenz ransformaions can be wrien in inuiive 3-vecor forms, given by [9-] E Bc γ E γ + γ, (3 c B B E c γ B γ γ, (4 wih E B, E ( B c,, an ( c as Lorenz scalars [] Suose ha he fiels in XYZ frame have a roagaion facor, namely ( E, B or (, ~ ex[ iψ( x, ] From Eqs (3 an (4, he hase funcion Ψ ( x, mus be inclue in ( E, B or (,, an Ψ ( x, Ψ( x,, wih ( x, in Ψ ( x, relace by ( x, in erms of Eqs ( an ( Thus he invariance is naural Of course, he invariance of hase oes

2 no necessarily mean ha he hase funcions mus have he same form For a lane wave in free sace, which is unienifiable, he hase funcions mus have he same form []; however, for a raiaion elecric iole [,,3,4] as a moving oin ligh source in free sace, he forms of he hase funcions mus be ifferen because he moving oin source has a referre frame [5] In conclusion, he invariance of hase is a naural resul uner Lorenz ransformaions Z' Y' c Z X' Fig Two inerial frames of relaive moion X Y Z moves wih resec o XYZ a c, while XYZ moves wih resec o X Y Z a c (no shown, wih III TE PROPAGATION IRECTION OF A PLANE WAVE In his secion, a symmeric efiniion of hase velociy for a lane wave is given The hase funcion, no maer in free sace or in a uniform ielecric meium, is given by Ψ ω n k x, where ω is he angular frequency, n k is he wave vecor, an n > is he refracive inex of meium Seing n ˆ k k or k k, we have Ψ ω n k x Geomerically, Ψ ω n k x consan is a lane, observe a he same ime, known as equi-hase lane or wavefron, wih n ˆ he uni normal or wave vecor Changing he sign of ω will no change he irecion of n ˆ, since n ˆ is suose o be known for a given EM robl All he equihase lanes are arallel Le us use c h o enoe he lane s moving velociy The moving velociy of a lane is efine as he changing rae of he lane s isance islacen over ime, namely c h lim[(ˆ n x ] wih ( x he isance islacen rouce wihin From he equi-hase-lane (wavefron equaion of moion ω n k x consan, we obain ( x ω ( n k, leaing o ˆ hc nω ( n k, known as hase velociy Seing h c ω ( n k, we have c c c h, from which an invarian form of equihase lane equaion of moion can be efine (Aenix A h ± h Therefore, a lane wave roagaes only arallel o he wave vecor, along he same or oosie irecion of he wave vecor, which is ifferen from he assumion inrouce by Gjurchinovski [] Noe: The efiniion of he hase velociy for a uniform lane wave given above is consisen wih he Ferma s rincile: ligh follows he ah of leas ime, an he absolue hase velociy is c for vacuum or c n for meium, ha is equal o he ou velociy an energy velociy owever, a isersive wave in a non-isersive meium, TE or TM moes in regular waveguies for examle [9], is mae u of a number of or even infiniy of lane waves; he suerosiion of hese lane waves forms an inerfering wavefron an is hase velociy is suerluminal an i is no he energy velociy The energy velociy in such a Y X case is he ou velociy ha is exacly equal o he rojecion velociy of he hase velociy of a uniform lane wave in he wave-guie irecion [4] Jus because of he comlexiy cause by he wave inerference an suerosiion, esecially in a srongly isersive meium [9], he jugmen of energy velociy is ofen conroversial [6-9] V STANAR LORENTZ COVARIANT WAVE FOUR- VECTOR IN A MOVING IELECTRIC MEIUM Suose ha he uniform ielecric meium wih a refracive inex of n > is a res in X Y Z fame From he invariance of hase, we have Ψ ( x, ω x ω n k x (5 where k ω c an k ω c I is seen from Eq (5 ha he hase funcion is symmeric wih resec o all inerial frames, ineenen of which frame he meium is fixe in From his we can conclue: (a K ( n k, ω c mus be Lorenz covarian, (b he efiniions of equi-hase lane an hase velociy shoul be ineenen of he choice of inerial frames By seing he ime-sace four-vecor X ( x, c an he wave four-vecor K ( n k, ω c, Eq (5 can be wrien in a ν covarian form, given by ( ω n wih he k x g ν K X ν meric ensor g g iag(,,, + [3] ν From Eqs ( an ( wih n x' an ω c c, we obain K ( n k, ω c Wih n ˆ, we have ω γ ( ω + n ω, (oler formula (6 γ ω n k ( + ( + γ (7 From Eq (6 an he Lorenz scalar equaion ( ( nk ω ω we obain he refracive inex in he lab frame, given by n ( + γ ( γ ( (8, wih ω > (9 where, wih c he lab frame s moving velociy wih resec o he meium-res frame I is seen from Eq (9 ha he moion of ielecric meium resuls in an anisoroic refracive inex IV SUPERLUMINAL MOTION OF TE MEIUM Suose he ielecric meium moves arallel o he wave vecor ( // Seing ±, from Eqs (6 an (7 we have ω ω γ ( + n, ( ω n k ( + γ, c ( ω c ω + c c h n k n ω + ( Noe: ω > was assume in geing Eqs (-(

3 For a meium moving a a faser-han-meium ligh see in he irecion oosie o he wave vecor n ˆ, ha is, < bu >, we have + < I is seen from Eq ( ha n k an n ˆ oin o he same irecion, namely From Eq( an Eq ( we fin ω < inee [], a negaive frequency, an he hase velociy c < h, he wave observe in he lab frame roagae in he (- n ˆ -irecion In such a case, he moving meium behaves as a negaive inex meium an he wave roagaion roeries are change (see Aenix B Because of he invariance of hase, he wave vecor n k mus be Lorenz covarian, as he sace-comonen of K ( n k, ω c Bu he hase velociy, efine by ˆ h c nω ( n k, is arallel o n k, which is a srong consrain Thus i is ifficul o se u a sanar Lorenz covariance for he hase velociy like he wave four-vecor, exce for in free sace (Aenix A VI IS TE ABRAAM-MINKOWSKI CONTROVERSY CLOSE? The momenum of ligh in a ransaren meium is a longsaning conroversy [4-8] Minkowski an Abraham roose ifferen forms of EM momenum-ensiy exressions, from which wo hoon-momenum formulaions resul For a uniform lane wave in he meium-res frame, wih he uni wave vecor n ˆ an roagaing in a meium wih a refracive inex, he elecric fiel E an magneic fiel B have he relaion B c Thus he wo EM momenum-ensiy exressions can be wrien as g B (, (3 M c g ( (4 A c c is he EM energy ensiy, roorional o he single hoon s energy h ω, while he EM momenum ensiy is roorional o a single hoon s momenum Thus from Eqs (3 an (4, we obain he hoon s momenum, given by h ω c M for Minkowski s an h ω ( c A for Abraham s The wo aarenly conraicing resuls are boh suore by exerimens [5,6,8,9]; esecially, he Abraham s momenum is srongly suore by a recen irec observaion [9], where a silica filamen fiber recoile as a laser ulse exie [,] I is ineresing o oin ou ha he Minkowski s hoon momenum also can be naurally obaine from he covariance of relaiviy, as shown below From he given efiniion of wave four-vecor, K (, ω c mulilie by he Planck consan h, we obain he hoon s momenum-energy four-vecor (see Aenix A, given by P ( h k, hω c (, c (5 In erms of he covariance, mus be he momenum; hus we have he hoon s momenum in a meium, given by c, ha is righ a Minkowski s hoon momenum (The Lorenz covariance of Minkowski s EM momenum is shown in Aenix B From he rincile of relaiviy, we have he invariance of hase, from which we have he covarian wave four-vecor From he wave four-vecor combine wih Einsein s lighquanum hyohesis, we have he Minkowski s hoon momenum, which srongly suors he consisency of Minkowski s EM-momenum exression wih he relaiviy In he classical elecroynamics, Fizeau running waer exerimen is usually aken o be an exerimenal evience of he relaivisic velociy aiion rule [9] In fac, i also can be aken o be a suor o he Minkowski s momenum To beer unersan his, le us make a simle analysis, as shown below Suose ha he running-waer meium is a res in he X Y Z frame Since he Minkowski s momenum-energy fourvecor ( hn k, hω c is covarian, we have Eqs (8 an (9 holing Seing (he waer moves arallel o he wave vecor, from Eq (9 we have he refracive inex in he lab frame, given by n + (6 + Thus he ligh see in he running waer, observe in he lab frame, is given by c c c +, for h n << (7 n n n which is he very formula confirme by Fizeau exerimen When he waer runs along (oosie o he wave vecor irecion, we have > ( < an he ligh see is increase (reuce The hoon-meium block hough exerimen [4] an he combinaion of Newon s firs law wih Einsein s mass-energy equivalence [-4] are all owerful argumens for Abraham s hoon momenum Bu Abraham s hoon momenum, smaller in ielecric han in vacuum, canno be obaine from he wave four-vecor in a covarian manner, an i is no consisen wih he relaiviy VII CONCLUSIONS AN REMARKS By analysis of a lane wave in a moving meium, we have shown ha he invariance of hase is a naural resul uner Lorenz ransformaions We also have shown ha he Minkowski s momenum is Lorenz covarian, while he Abraham s momenum is no comaible wih he relaiviy The covariance of hoon momenum P hk is shown base on he covariance of wave four-wave combine wih Einsein s ligh-quanum hyohesis, while he covariance of EM momenum P N ( B, E c is shown only base on he covariance of Maxwell equaions One migh ask: ( oes B always correcly enoe a EM momenum? ( Is here such a four-vecor for any EM fiels? The answer is no necessarily To unersan his, le us ake a look of a simle examle Suose ha here is a uniform saic elecric fiel E while B wih B in free sace, observe in he lab frame XYZ (confer Fig ; rouce by an infinie lae caacior, for examle [8] Observe in he moving frame X Y Z, from Eqs (3 an (4 we obain he EM fiels, given by E γ E ( γ ( E, (8 B γ E c c (9 Abraham s an Minkowski s momenum are equal, given by B ( c ( c, ( 3

4 which inclues wo erms, wih one in -irecion an he oher in -irecion owever he saic E-energy fixe in XYZ moves a observe in X Y Z, an he EM momenum shoul be arallel o From his we juge ha he erm ( c is a seuo-momenum Esecially, when // E an E < hol, leaing o he holing of E E an >, we have B [8], bu here shoul be a nonzero EM momenum observe in X Y Z, because he E-fiel has energy an mass, an i is moving Thus we conclue ha in such a case B oes no correcly reflec he EM momenum, wih N ( B c no Lorenz covarian owever, when E or E hols, leaing o he isaearing of he seuo-momenum, or ( c, we have he covariance of N ( B c coming back, wih B γ ( E c, an c γ E c A irec check inicaes ha N ( B c auomaically saisfies he four-vecor Lorenz ransformaion given by Eqs ( an ( wih N N γ Noe: E B oes no hol in such a siuaion I shoul be oine ou ha, however, he above caacior fiel is usually aken as a yical examle o show ha he aache-o-source EM fiel canno exhibi four-vecor behavior [8], alhough N ( B c oes show Lorenz covariance in our analysis when here is no seuomomenum exising Aenix A: Generalize Lorenz invariance of hase velociy an ou velociy As menione reviously, since he hase funcion is symmeric wih resec o all inerial frames, he efiniion of hase velociy shoul be symmeric Thus he hase velociy is given by ω c ω c h, (A n k n ω which saisfies he invarian form of equi-hase lane equaion of moion ω n k c (A h Base on he hase velociy, we can efine he ou velociy Traiionally, he ou velociy is efine as c ω ( n k, leaing o c c c h h, an we have c c h Thus he ou velociy is equal o he hase velociy I shoul be hasize ha, he form-invariance of Eq (A an (A is comleely base on he symmery of hase funcion, he ime-sace symmery, an he symmery of efiniion of hase velociy In a generalize sense, we can say ha hey are Lorenz invarian in form Suose ha he Planck consan h is a Lorenz scalar In he meium-res frame, he isersion equaion, irecly resuling from secon-orer wave equaion (insea of he invariance of hase, is given by [] ( ω c (, (A3 n which acually is he efiniion of refracive inex This isersion relaion is hough o be he characerizaion of he relaion beween EM energy an momenum, an he hoon s energy in a meium is suose o be n hω o kee a zero res energy [] owever i shoul be noe ha, alhough Eq (A3 is Lorenz invarian, ( k, ω c is no a Lorenz covarian four-vecor, since only K ( k, ω c is; exce for n If using ( hk, hω c o efine he hoon s momenum-energy four-vecor, hen i is no Lorenz covarian Thus hk ( h, hω c shoul be use o efine he hoon s momenum-energy four-vecor, because h K is covarian, wih Eq (A3 as a naural resul h K has an imaginary res energy, which can be exlaine o be an energy share wih he meium This (imaginary share energy behaves as a measurable quaniy only hrough he hoon s momenum APPENIX B: LORENTZ COVARIANCE OF MINKOWSKI S EM MOMENTUM FOR A PLANE WAVE IN A MOVING MEIUM In his aenix, we will show he Lorenz covariance of Minkowski s momenum by analysis of a lane wave in a moving ieal meium, which is he simles sric soluion o Maxwell equaions Suose ha he lane-wave soluion in he meium-res frame X Y Z is given by E,B,, (,B, exiψ, (B ( where Ψ ( ω k x, wih ω > assume, an ( E,B, are real consan amliue vecors ε an B hol, where ε > an > are he consan ielecric ermiiviy an ermeabiliy, wih ε n c Thus ( E, B, an (,, are, resecively, wo ses of righ-han orhogonal vecors, wih E ( c B an ( c, resuling from Maxwell equaions In he lab frame XYZ, he lane-wave soluion can be wrien as ( E, B,, ( E,B,, exiψ, (B where Ψ ( ω n k x, an ( E,B,, are obaine by insering ( E,B,, ino (reverse Eqs (3 an (4, given by E B γ ( γ n + γ n, (B3 γ B n γ B + γ (B4 n Noe: All quaniies are real in Eqs (B3 an (B4 an he ransformaions are synchronous ; for examle, E is exresse only in erms of E I is clearly seen from Eqs (B-(B4, he invariance of hase, Ψ Ψ, is a naural resul Because he meium is ieal, all fiel quaniies have he same hase facor, no maer in he meium-res frame or lab frame In he following, formulas are erive in he lab frame, because hey are invarian in forms in all inerial frames Alhough he fiels given by Eqs (B an (B are comlex, all equaions an formulas obaine are checke wih comlex an real fiels an hey are vali 4

5 Uner Lorenz ransformaions, he Maxwell equaions kee he same forms as in he meium-res frame, given by E B, ρ, (B5 J +, B, (B6 wih J an ρ for he lane wave From above, we have ω B n k E, an ω n k, (B7 leaing o E B, namely he elecric energy ensiy is equal o he magneic energy ensiy, which is vali in all inerial frames From Eqs (B3 an (B4, we obain some inuiive exressions for examining he anisoroy of sace relaions of EM fiels observe in he lab frame, given by an an, B B, (B8 E ( ( γ ( ( + γ ( γ ( + γ ( (, (B9, (B E B B,, (B E γ ( ( (, (B B γ ( ( B, (B3 n E γ ( n ˆ, (B4 B γ ( B, (B5 E ( Eˆ n h c B, (B6 ( + c (B7 h I is seen from above ha E B, B, an hol in he lab frame, bu E //, B //, E, B, E n ˆ, an usually o no hol any more Pseuo-ower flow cause by a moving meium From Eq (B7, we obain Minkowski s EM momenum an Poyning vecor, given by n B ( E E v, (B8 v [ v ( B (ˆ n B (ˆ n E ] (B9 where v c, wih v c an v, is he c n ou velociy, equal o he hase velociy c h, as efine in Aenix A Noe: The Minkowski s momenum B has he same irecion as he ou velociy mahaically, v while he Poyning vecor E has hree comonens: one in B -irecion, one in B-irecion, an one in -irecion; he laer wo are erenicular o he ou velociy v We can ivie he Poyning vecor ino wo ars, E S ower + S seu, which are erenicular each oher, given by S ower v ( B ( E v, (B S v [ (ˆ n B + (ˆ n E ] seu, (B where S ower is associae wih he oal EM energy ensiy ( E an he ou velociy v S ower carries he oal EM energy moving a v, an i is a real ower flow S seuo is erenicular o he ou velociy, an i is a seuo-ower flow, jus like he E vecor rouce by saic crosse elecric an magneic fiels, which canno be aken as a real ower flow Base on Eq (B, he energy velociy is suose o be equal o he ou velociy The essenial ifference in hysics beween S ower an S seu can be seen from he ivergence heor The ivergence of is given by S ower S [ E c] ower ( E [ cos ( ω-n k x] v, (B which means ha here is a ower flowing in an ou in he ifferenial box, bu he ime average < S > ower, meaning ha he owers going in an ou are he same in average, wih no ne energy lef in he box In conras, S seu hols for any ime resuling from B,, B k, an k, which means ha here is no ower flowing a any ime for any laces, an S seu is exacly like he E vecor rouce by saic crosse elecric an magneic fiels as menione above I is seen From Eqs (B9 an (B ha S seu vanishes in free sace, or when he ielecric meium moves arallel o he wave vecor ˆ From above analysis, we can see ha a Poyning vecor oes no necessarily enoe a real ower flowing owever such a henomenon ses neglece in he hysics communiy, in view of he fac ha he Abraham s momenum is efine hrough Poyning vecor wihou any coniions In summary, we can make some conclusions for he ower flowing ( The Minkowski s EM momenum B is arallel o he ou (hase velociy, observe in any inerial frames ( When a meium moves, he Poyning vecor E consiss of wo ars: one is arallel o he ou velociy, an i is a real ower flow; he oher is erenicular o he ou velociy, an i is a seuo-ower flow Lorenz covariance of Minkowski s EM momenum We have shown he Lorenz covariance of Minkowski s hoon momenum from he wave four-vecor combine wih Einsein s ligh-quanum hyohesis This covariance suggess us ha here shoul be a covarian EM momenum-energy four-vecor, given by P (, E c, (B3 where an E are, resecively, he EM momenum an energy er hoon, given by B, N E E (B4 N 5

6 wih N he hoon s ensiy The EM momenum-energy fourvecor an wave four-vecor are relae hrough P ( E ω K, wih E ω corresoning o h hysically The four-vecor P (, E c is require o follow a four-vecor Lorenz ransformaion given by Eqs ( an (, while he EM fiels mus follow a secon-rank ensor Lorenz ransformaion given by Eqs (3 an (4, or Eqs (B3 an (B4 for a lane wave From he four-vecor Lorenz ransformaion of E an ω given by Eq (, wih he invariance of hase Ψ Ψ consiere, we have N ω E (B5 N ω The above equaion has a clear hysical exlanaion ha he oler facor of EM energy ensiy is equal o he rouc of he oler facors of hoon ensiy an frequency From he secon-rank ensor Lorenz ransformaion of an E given by Eqs (B3 an (B4, we obain he ransformaion of EM energy ensiy, given by E [ γ ( ] γ, (B6 n namely Eq (B4 Comaring wih Eq (6, we know ha γ ( n ˆ is he frequency oler facor In free sace, he above equaion is reuce o Einsein s resul [3]: E γ ( Insering Eq (B6 ino Eq (B5, we obain he ransformaion of hoon ensiy, given by N γ N n (B7 So far we have finishe he roof of he covariance of P (, E c by resoring o a arameer, so-calle hoon ensiy Acually, we o no have o know wha he secific value of N or N is, bu he raio of N N, an P is ure classical, wihou Planck consan h involve owever, N mus be he hoon ensiy in he lab frame if N is aken o be in he meium-res frame Mahaically seaking, he exisence of his covariance is aaren P an K are jus ifferen by a facor of ( E ω which conains an inrouce arameer N o make he ransformaion hol In conclusion, Minkowski s EM Momenum B is Lorenz covarian, as he sace comonen of he EM momenum-energy four-vecor P N ( B, E c, jus like he Minkowski s hoon momen h n k which is Lorenz covarian, as he sace comonen of he hoon momenum-energy four-vecor P ( hn k, hω c Covariance an symmery of Minkowski s sress ensor As we know, in he erivaions of Minkowski s sress ensor in he meium-res frame, he linear relaions of ielecric maerial are loye, such as ε, so ha ( E ( hols an he ensor can be wrien in a symmeric form [3] As menione above, because of he anisoroy of a moving meium, such relaions o no hol in he lab frame Thus he sress ensor in he lab frame only can be erive base on he covariance of Maxwell equaions an he invariance of hase From he covarian Maxwell Eqs (B5 an (B6, wih J an ρ aken ino accoun we have [9] ( B ( E ( E + ( B B ( (B8 Since, B, B, an hol, we have ( B T, (B9 where he symmeric sress ensor is given by T I( E I( B, (B3 wih I he uni ensor Noe: Eq (B9 is vali for boh real an comlex fiels alhough only he real fiels are hysical Eq (B9 is he momenum conservaion equaion, an i is invarian in form in all inerial frames ogeher wih he Maxwell equaions Now le us ake a look of he alicable range of he sress ensor Consier he EM force exere on a given ielecric volume From Eq (B9, using ivergence heors for a ensor an a vecor, wih (, E (, E cos Ψ insere, we have ( B V S T ( E V E sin(ψ(n V (B3 ( k I is seen ha here is a ime-eenen force exerience by he volume along he wave vecor irecion, alhough is ime average is zero, namely < ( B V > owever he consrucion of sress ensor is flexible; in a sense, i is arificial For examle, ( E B hols for a lane wave, an he ensor can be re-wrien in a well-known form T E B + I ( E + B, (B3 which oes no affec he valiiy of Eq (B9 an Eq (B3, bu he local surface force S T is change I means ha S T oes no necessarily enoe a real EM force exerience by he surface elen S Fo r examle, from Eq (B3, we have S T S I ( E S( E No maer wha irecion S akes, here is a force owever, B is always arallel o he ou velociy or wave vecor as shown in Eq (B8 an Eq (B3, ineenenly of ime an locaions From above, we can see ha an alicaion of he sress ensor like Eq (B3 is vali, while if using i for calculaions of EM forces on arial local surfaces, he valiiy is quesionable Unconvenional henomenon for a suerluminal meium In aiion o he negaive-frequency effec reice by uang [], a Negaive Energy ensiy (NE may resul for a lane wave when he meium moves oosie o he wave vecor irecion a a faser-han-ielecric ligh see; his henomenon is calle NE zone for he sake of convenience The origin of negaive EM energy ensiy can be seen from Eq (B6 The energy ensiy oler facor is equal o he rouc of frequency s an hoon ensiy s The hoon ensiy oler facor is always osiive while he frequency s is negaive in he NE zone ( n ˆ n n >, leaing o he holing of E < when ω < In he NE zone, // E an // B are vali, an from Eqs (B3 an (B4 we have 6

7 E B REFERENCES ε <, (B33 [] Y-S uang, Eurohys Le 79, 6 (7 [] A Gjurchinovski, Eurohys Le 83, (8 [3] Y-S uang, Z Naurforsch 65a, 65 ( [4] S M Barne, Phys Rev Le 4, 74 ( < (B34 [5] R V Jones, Naure 67, 439 (95 [6] G B Walker, G Lahoz, an G Walker, Can J Phys 53, 577 (975 Because of ω < in he NE zone, from Eq (A we have [7] A Feigel, Phys Rev Le 9, 44 (4 c c h h wih h c < From Eqs (B6 an (B7 we [8] G K Cambell, A E Leanhar, J Mun, M Boy, E W Sree, have W Keerle, an E Prichar, Phys Rev Le 94, 743 (5 E h c B, (B35 [9] W She, J Yu, an R Feng, Phys Rev Le, 436 (8 [] I Brevik, Phys Rev Le 3, 93 (9 + c h (B36 [] M Mansuriur, Phys Rev Le 3, 93 (9 [] U Leonhar, Naure 444, 83 (6 I follows ha he lane wave in he NE zone is a lef-han [3] R N C Pfeifer, T A Niinen, N R eckenberg, an Rubinszein-unlo, Rev Mo Phys 79, 97 (7 wave: ( ( E, B, an (,, follow he lef-han rule, an [4] M Buchanan, Naure Phys 3, 73 (7 ( he hase or ou velociy is oosie o he wave vecor In oher wors, he moving meium in he NE zone behaves as a so-calle negaive inex meium [3-35], where he refracive inex is aken o be negaive [33], insea of he frequency here [] Noe: I is ifficul o efine a negaive refracive inex from he wave four-vecor wih he coninuiy of Lorenz ransformaion saisfie [confer Eq (9] From Eq (B8, B always carries he EM energy E moving a v In he NE zone, v is oosie o n k, an he minus energy ( E < moves along minus n k - irecion, which is equivalen o ha he osiive energy moves along lus nk -irecion; hus B always has he same irecion as n k, as escribe by he hoon momenum-energy four-vecor ( hn k, hω c : no maer wheher h ω ( E < or >, h n k ( B always akes he + n k - irecion ow o we unersan he ower flow Eq (B S ( E v ower in he NE zone? There are wo exlanaions ( The ower flow S ow er has he same irecion as he ou velociy v, because E iself inclues he minus sign mahaically ( S ower is oosie o v, since he energy ensiy mus be aken o be osiive raiionally [5] B A K, J Al Phys 9, ( [6] C Baxer an R Louon, J Mo O 57, 83 ( [7] T Ramos, G F Rubilar, an Y N Obukhov, Phys Le A 375, 73 ( [8] M Mansuriur, Phys Rev E 79, 668 (9 [9] J Jackson, Classical Elecroynamics, (John Wiley & Sons, NJ, 999, 3 r Eiion [] J A Kong, Theory of Elecromagneic Waves, (John Wiley & Sons, NY, 975 [] J A Sraon, Elecromagneic heory, (McGraw-ill, NY, 94 [] C Wang, Ann Phys (Berlin 53, 39 ( [3] W R Smyhe, Saic an ynamic elecriciy, (McGraw-ill, NY, 968, 3 r eiion [4] J Griffhs, Inroucion o Elecroynamics, (Prenice-all, NJ, 999, 3 r eiion [5] C Wang, h://asarxivorg/abs/798 [6] Mugnai, A Ranfagni, an R Ruggeri, Phys Rev Le 84, 483 ( [7] P Saari an K Reivel, Phys Rev Le 79, 435 (997 [8] J Salo, J Fagerholm, A T Friberg, an M M Salomaa, Phys Rev E 6, 46 ( [9] L J Wang, A Kuzmich, an A ogariu, Naure 46, 77 ( [3] W Rinler, Relaiviy: secial, general, an cosmological, (Oxfor, NY, 6, n eiion [3] A Einsein, Ann Phys (Leizig 7, 89 (95 [3] VG Veselago, Sov Phys Us, 59 (968 [33] PM Valanju, RMWalser, an A P Valanju, Phys Rev Le 88, 874 ( [34] N Garcia an M N Veserinas, Phys Rev Le 88, 743 ( [35] S A Ramakrishna, Re Prog Phys (5 7