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Home , Lorentz covariance, Wave vector

four-vector in a moving medium and the of Minkowski’s and electromagnetic momentums

Changbiao Wang ShangGang Group, 70 Huntington Road, Apartment 11, New Haven, CT 06512, USA In this paper, by analysis of a plane wave in a moving dielectric medium, it is shown that the invariance of phase is a natural result under Lorentz transformations. From the covariant wave four-vector combined with Einstein’s light- quantum hypothesis, Minkowski’s photon momentum equation is naturally restored, and it is indicated that the formulation of Abraham’s photon momentum is not compatible with the covariance of relativity. Fizeau running water experiment is re-analyzed as a support to the Minkowski’s momentum. It is also shown that, (1) there may be a pseudo-power flow when a medium moves, (2) Minkowski’s electromagnetic momentum is Lorentz covariant, (3) the covariant Minkowski’s stress may have a simplest form of , and (4) the moving medium behaves as a so-called “negative index medium” when it moves opposite to the wave vector at a faster-than-dielectric light speed.

Key words: invariance of phase, wave four-vector, moving dielectric medium, light momentum in medium PACS numbers: 42.50.Wk, 03.50.De, 42.25.-p, 03.30.+p

I. INTRODUCTION exhibit the Lorentz covariance of four-vector when the pseudo- Invariance of phase is a well-known fundamental concept in momentum disappears, which challenges the well-recognized the relativistic electromagnetic (EM) theory. However the viewpoint in the community [18]. invariance of phase for a plane wave in a uniform dielectric The momentum of light in a medium can be described by a medium has been questioned, because the can be single photon’s momentum or by the EM momentum. The negative when the medium moves at a faster-than-dielectric covariance of photon momentum is shown based on the light speed in the direction opposite to the wave vector [1]. To invariance of phase and Einstein’s light-quantum hypothesis, solve the problem, a new wave four-vector based on an while the covariance of EM momentum is shown only based on additional assumption was proposed: a plane wave travels at a the covariance of EM fields. The covariance of Minkowski’s “wave velocity” that is not necessarily parallel to the wave light momentum embodies the consistence of Einstein’s vector direction [2,3]. relativity and light-quantum hypothesis. In this paper, by analysis of a plane wave in a moving dielectric medium, it is shown that the invariance of phase is a II. INVARIANCE OF PHASE natural result under Lorentz transformations, and it is a Suppose that the frame X ′Y ′Z′ moves with respect to the lab sufficient and necessary condition for the Lorentz covariance of frame XYZ at a constant velocity of βc , with all corresponding wave four-vector. When the negative frequency appears, the coordinate axes in the same directions and their origins moving medium behaves as a so-called “negative index overlapping at t = t′ = 0 , as shown in Fig. 1. The time-space medium”, resulting in the change of properties of wave is given by [19] propagation. Based on the symmetry of phase function with γ −1 respect to all inertial frames, a symmetric definition of phase x = x′ + (β ⋅ x′)β + γ βct′ , (1) velocity and an invariant form of equi-phase plane equation of β 2 motion are given. The momentum of light in a medium is a fundamental ct = γ (ct′ + β ⋅ x′) . (2) question [4-18]. Maxwell equations support various forms of 2 −1/ 2 momentum conservation equations [13,15,16]. In a sense, the where c is the universal light speed, and γ = (1− β ) is the construction of EM momentum density vector is artificial, time dilation factor. The EM fields E and B, and D and H respectively constitute a covariant second-rank anti-symmetric which is a kind of indeterminacy. However it is the αβ αβ indeterminacy that results in the question of light momentum. tensor F (E,B) and G (D,H) , of which the Lorentz For example, in the recent publication [4], powerful arguments transformations can be written in intuitive 3D-vector forms, are provided to support both Minkowski’s and Abraham’s given by [19-20] formulations of light momentum: one is canonical, and the other is kinetic, and they are both correct. ⎛E′⎞ ⎛E⎞ ⎛ Bc ⎞ γ −1 ⎛E⎞ ⎜ ⎟ = γ ⎜ ⎟ + γβ × ⎜ ⎟ − β ⋅ ⎜ ⎟β , (3) The above indeterminacy should be excluded by imposing ⎜D′⎟ ⎜D⎟ ⎜H c⎟ β 2 ⎜D⎟ physical conditions, just like the EM field solutions are required ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ to satisfy boundary conditions. Such a physical condition is the ⎛ B′ ⎞ ⎛ B ⎞ ⎛E c⎞ γ −1 ⎛ B ⎞ Lorentz covariance. As we know, the phase function for a plane ⎜ ⎟ = γ ⎜ ⎟ − γβ × ⎜ ⎟ − β ⋅ ⎜ ⎟β , (4) ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 2 ⎜ ⎟ wave characterizes the propagation of energy and momentum of ⎝H′⎠ ⎝H⎠ ⎝ Dc ⎠ β ⎝H⎠ light. Since the phase function is symmetric with respect to all inertial frames, the covariance of light momentum must hold. with E′ ⋅ B′ , E′2 − (B′c) 2 , D′ ⋅ H′ , and (D′c) 2 − H′2 as Lorentz In this paper, it is also shown that, Minkowski’s light scalars [21]. momentum is Lorentz covariant, while Abraham’s is not. Suppose that the fields in XYZ frame have a propagation Fizeau running water experiment is re-analyzed as a support to factor, namely (E,B) or (D,H) ~ exp[iΨ(x,t)] . From Eqs. (3) the Minkowski’s momentum. A simple example is given to and (4), the phase function Ψ(x,t) must be included in (E′,B′) show: (1) neither Abraham’s nor Minkowski’s EM momentum or (D′, H′) , and Ψ′(x′,t′) = Ψ(x,t) , with (x,t) in Ψ(x,t) can correctly describe the real EM momentum when a pseudo- replaced by (x′,t′) in terms of Eqs. (1) and (2). Thus the momentum exists, and (2) the “attached-to-source field” does invariance is natural. Of course, the invariance of phase does

1 not necessarily mean that the phase functions must have the case is the that is exactly equal to the projection same form. For a plane wave in free space, which is velocity of the of a uniform plane wave in the unidentifiable, the phase functions must have the same form wave-guided direction [24]. Just because of the complexity [22]; however, for a radiation electric dipole [20,21,23,24] as a caused by the wave interference and superposition, especially in moving point light source in free space, the forms of the phase a strongly dispersive medium [19], the judgment of energy functions must be different because the moving point source has velocity is often controversial [26-29]. a [25]. In conclusion, the invariance of phase is a natural result under V. STANDARD LORENTZ COVARIANT WAVE FOUR- Lorentz transformations. VECTOR IN A MOVING DIELECTRIC MEDIUM Suppose that the uniform dielectric medium with a refractive Y' index of nd′ > 0 is at rest in X ′Y ′Z′ fame. From the invariance of phase, we have

Ψ′(x′,t′) = ω′t′ − n′d k′ ⋅ x′ = ωt − nd k ⋅ x . (5) Y where k′ = ω′ c and k = ω c . Z' X' It is seen from Eq. (5) that the phase function is symmetric ct with respect to all inertial frames, independent of which frame β the medium is fixed in. From this we can conclude: (a) µ K = (nd k,ω c) must be Lorentz covariant, (b) the definitions Z X of equi-phase plane and phase velocity should be independent of the choice of inertial frames. Fig. 1. Two inertial frames of relative motion. X ′Y ′Z′ moves with By setting the time-space four-vector X µ = (x,ct) and the respect to XYZ at βc , while XYZ moves with respect to X ′Y ′Z′ at µ wave four-vector K = (nd k,ω c) , Eq. (5) can be written in a β′c (not shown), with β′ = −β . µ ν covariant form, given by (ωt − nd k ⋅ x) = g µν K X with the metric tensor g = g µν = diag(−1,−1,−1,+1) [30]. III. THE PROPAGATION DIRECTION µν From Eqs. (1) and (2) with nd′k′ → x' and ω′ c → ct′ , we OF A PLANE WAVE µ obtain K = (nd k,ω c) . With nˆ′ = k′ k′ , we have In this section, a symmetric definition of phase velocity for a plane wave is given. The phase function, no matter in free ω = γ (ω′ + nd′ ω′ nˆ ′ ⋅ β) , (Doppler formula) (6) space or in a uniform dielectric medium, is given by γ −1 ⎛ ω′ ⎞ Ψ = ωt − n k ⋅ x , where ω is the angular frequency, n k is the n k = (n′k′) + (n′k′)⋅ββ + γβ⎜ ⎟ . (7) d d d d β 2 d c wave vector, and nd > 0 is the refractive index of medium. ⎝ ⎠ Setting nˆ = k k or k = nˆ k , we have Ψ = ωt − n k nˆ ⋅ x . d From Eq. (6) and the Lorentz equation Geometrically, Ψ = ωt − nd k nˆ ⋅ x = constant is a plane, observed at the same time, known as equi-phase plane or 2 2 ω′ ω wavefront, with nˆ the unit normal or wave vector. Changing ⎛ ⎞ 2 ⎛ ⎞ 2 ⎜ ⎟ − (n′d k′) = ⎜ ⎟ − (nd k) (8) the sign of ω will not change the direction of nˆ , since nˆ is ⎝ c ⎠ ⎝ c ⎠ supposed to be known for a given EM problem. All the equi- phase planes are parallel. we obtain the refractive index in the lab frame, given by Let us use β c to denote the plane’s moving velocity. The 2 2 2 ph (n′ −1) + γ (1− n′nˆ ′ ⋅ β′) moving velocity of a plane is defined as the changing rate of the d d nd = , with ω′ > 0 . (9) plane’s distance displacement over time, namely β phc ≡ γ (1− n′nˆ ′ ⋅ β′) lim[(nˆ ⋅ ∆x)nˆ ∆t] with (nˆ ⋅ ∆x)nˆ the distance displacement d produced within ∆t . From the equi-phase-plane (wavefront) where β′ = −β , with β′c the lab frame’s moving velocity with equation of motion ωt − nd k nˆ ⋅ x = constant, we obtain respect to the medium-rest frame. nˆ ⋅ (dx dt) = ω (nd k ) , leading to β ph c = nˆ ω (nd k ) , known as It is seen from Eq. (9) that the motion of dielectric medium phase velocity. Setting β ph c =ω (nd k ) , we have results in an anisotropic refractive index.

β ph c = β ph cnˆ = ± β ph c nˆ , from which an invariant form of equi- phase plane equation of motion can be defined (Appendix A). IV. SUPERLUMINAL MOTION OF THE MEDIUM Therefore, a plane wave propagates only parallel to the wave Suppose the dielectric medium moves parallel to the wave vector, along the same or opposite direction of the wave vector, vector (β // nˆ ′) . Setting β = β nˆ ′ = ± β nˆ ′ , from Eqs. (6) and which is different from the assumption introduced by (7) we have Gjurchinovski [2]. Note: The definition of the phase velocity for a uniform plane ω = ω′γ (1+ n′ β) , (10) wave given above is consistent with the Fermat’s principle: light d follows the path of least time, and the absolute phase velocity is ω′ nd k = γ (n′d + β ) nˆ′ , (11) c for vacuum or c nd for medium, that is equal to the group c velocity and energy velocity. However, a dispersive wave in a ω c ω 1 + n′ β non-dispersive medium, TE or TM modes in regular waveguides d β ph c = = = c . (12) for example [19], is made up of a number of or even infinity of nd k nd ω nd′ + β plane ; the superposition of these plane waves forms an interfering wavefront and its phase velocity is “superluminal” Note: ω′ > 0 was assumed in getting Eqs. (10)-(12). and it is not the energy velocity. The energy velocity in such a

2 For a medium moving at a faster-than-medium light speed in momentum, which strongly supports the consistency of the direction opposite to the wave vector nˆ′ , that is, β < 0 but Minkowski’s EM-momentum expression with the relativity.

β > 1 n′d , we have 1+ n′d β < 0 . It is seen from Eq. (11) that In the classical electrodynamics, Fizeau running water nd k and nˆ′ point to the same direction, namely nˆ = nˆ ′ . From experiment is usually taken to be an experimental evidence of Eq.(10) and Eq. (12) we find ω < 0 indeed [1], a negative the relativistic velocity addition rule [19]. In fact, it also can be frequency, and the phase velocity β phc < 0 , the wave observed taken to be a support to the Minkowski’s momentum. To better in the lab frame propagate in the (- nˆ′ )-direction. In such a case, understand this, let us make a simple analysis, as shown below. the moving medium behaves as a “negative index medium” and Suppose that the running-water medium is at rest in the the properties are changed (see Appendix B). X ′Y ′Z′ frame. Since the Minkowski’s momentum-energy four- vector ( n′k′, ω′ c) is covariant, we have Eqs. (8) and (9) Because of the invariance of phase, the wave vector nd k h d h must be Lorentz covariant, as the space-component of holding. Setting −β′ = β = β nˆ ′ (the water moves parallel to the µ K = (nd k,ω c) . But the phase velocity, defined by wave vector), from Eq. (9) we have the refractive index in the

β ph c = nˆ ω (nd k ) , is parallel to nd k , which is a strong lab frame, given by constraint. Thus it is difficult to set up a standard Lorentz covariance for the phase velocity like the wave four-vector, n′d + β n = . (16) except for in free space (Appendix A). d 1+ n′d β VI. IS THE ABRAHAM-MINKOWSKI Thus the light speed in the running water, observed in the lab CONTROVERSY CLOSED? frame, is given by The momentum of light in a transparent medium is a long- standing controversy [4-18]. Minkowski and Abraham c c ⎡ ⎛ 1 ⎞⎤ β c = ≈ ⎢1+ β ⎜n′ − ⎟⎥ , for n′ β << 1 (17) proposed different forms of EM momentum-density expressions, ph n n′ ⎜ d n′ ⎟ d from which two photon-momentum formulations result. For a d d ⎣⎢ ⎝ d ⎠⎦⎥ uniform plane wave in the medium-rest frame, with the unit which is the very formula confirmed by Fizeau experiment. wave vector nˆ ′ and propagating in a medium with a refractive When the water runs along (opposite to) the wave vector ′ ′ ′ index nd , the electric field E and magnetic field B have the direction, we have β > 0 (β < 0) and the light speed is ′ ′ ˆ ′ ′ relation B c nd = n × E . Thus the two EM momentum-density increased (reduced). expressions can be written as The photon-medium block thought experiment [4] and the n′ combination of Newton’s first law with Einstein’s -energy g′ = D′ × B′ = d (D′ ⋅ E′)nˆ ′ , (13) M c equivalence [12-14] are all powerful arguments for Abraham’s photon momentum. But Abraham’s photon momentum, smaller E′× H′ 1 ′ ′ ′ ˆ ′ in dielectric than in vacuum, cannot be obtained from the wave g A = 2 = (D ⋅ E )n . (14) c n′d c four-vector in a covariant manner, and it is not consistent with the relativity. D′ ⋅ E′ is the EM energy density, proportional to the single photon’s energy hω′ , while the EM momentum density is VII. CONCLUSIONS AND REMARKS proportional to a single photon’s momentum. Thus from Eqs. (13) and (14), we obtain the photon’s momentum, given by By analysis of a plane wave in a moving medium, we have ′ ′ ′ ′ ′ ′ shown that the invariance of phase is a natural result under pM = nd hω c for Minkowski’s and p A = hω (nd c) for Abraham’s. The two apparently contradicting results are both Lorentz transformations. We also have shown that the supported by experiments [5,6,8,9]; especially, the Abraham’s Minkowski’s momentum is Lorentz covariant, while the Abraham’s momentum is not compatible with the relativity. momentum is strongly supported by a recent direct observation µ µ [9], where a silica filament fiber recoiled as a laser pulse exited The covariance of photon momentum P = hK is shown [10,11]. based on the covariance of wave four-wave combined with Einstein’s light-quantum hypothesis, while the covariance of It is interesting to point out that the Minkowski’s photon µ −1 EM momentum P = N (D × B, D ⋅ E c) is shown only based momentum also can be naturally obtained from the covariance p of relativity, as shown below. on the covariance of Maxwell equations. One might ask: (1) From the given definition of wave four-vector, Does D×B always correctly denote a EM momentum? (2) Is ′µ ′ ′ ′ there such a four-vector for any EM fields? The answer is “not K = (nd k ,ω c) multiplied by the Planck constant h , we obtain the photon’s momentum-energy four-vector (see necessarily”. Appendix A), given by To understand this, let us take a look of a simple example. Suppose that there is a uniform static electric field E ≠ 0 while ′µ ′ ′ ′ ′ ′ B = 0 with D× B = 0 in free space, observed in the lab frame P = (hnd k ,hω c) = (p p , E p c) . (15) XYZ (confer Fig. 1); produced by an infinite plate capacitor, for

In terms of the covariance, p′p must be the momentum; thus we example [18]. Observed in the moving frame X ′Y ′Z′ , from have the photon’s momentum in a medium, given by Eqs. (3) and (4) we obtain the EM fields, given by

p′p = n′d E′p c , that is right a Minkowski’s photon momentum. (The Lorentz covariance of Minkowski’s EM momentum is E′ = γE − (γ −1)β −2 (β ⋅ E)β , (18) shown in Appendix B.) From the , we have the invariance of B′ = −γβ × E c = β′× E′ c . (19) phase, from which we have the covariant wave four-vector. From the wave four-vector combined with Einstein’s light- Abraham’s and Minkowski’s momentum are equal, given by quantum hypothesis, we have the Minkowski’s photon D′× B′ = (D′ ⋅ E′ c)β′ − (D′ ⋅ β′)E′ c , (20)

3 ′ ′ ′ ′ ′ ′ which includes two terms, with one in β -direction and the for nd = 1 . If using (nd hk , nd hω c) to define the photon’s other in E′ -direction. However the static E-energy fixed in momentum-energy four-vector, then it is not Lorentz covariant. XYZ moves at β′ observed in X ′Y ′Z′ , and the EM momentum ′µ ′ ′ ′ Thus hK = (nd hk ,hω c) should be used to define the ′ µ should be parallel to β . From this we judge that the term photon’s momentum-energy four-vector, because hK′ is ′ ′ ′ µ −(D ⋅β )E c is a pseudo-momentum. Especially, when β // E covariant, with Eq. (A3) as a natural result. hK′ has an and E ⋅ β < 0 hold, leading to the holding of E′ = E and imaginary rest energy, which can be explained to be an energy β′ ⋅ E′ > 0 , we have D′× B′ = 0 [18], but there should be a non- shared with the medium. This (imaginary) shared energy zero EM momentum observed in X ′Y ′Z′ , because the E-field behaves as a measurable quantity only through the photon’s has energy and mass, and it is moving. Thus we conclude that momentum. in such a case D′× B′ does not correctly reflect the EM −1 momentum, with N ′p (D′× B′, D′ ⋅ E′ c) not Lorentz covariant. APPENDIX B: LORENTZ COVARIANCE OF However, when β ⊥ E or β ⋅ E = 0 holds, leading to the MINKOWSKI’S EM MOMENTUM FOR A PLANE disappearing of the pseudo-momentum, or (D′ ⋅ β′)E′ c = 0 , we WAVE IN A MOVING MEDIUM −1 have the covariance of N ′p (D′× B′, D′ ⋅ E′ c) coming back, with D′× B′ = −γ 2 (D ⋅ E c)β , and E′ ⋅ D′ c = γ 2 E ⋅ D c . A In this appendix, we will show the Lorenz covariance of −1 Minkowski’s momentum by analysis of a plane wave in a direct check indicates that N ′ (D′× B′, D′ ⋅ E′ c) automatically p moving ideal medium, which is the simplest strict solution to satisfies the four-vector Lorentz transformation given by Eqs. Maxwell equations. (1) and (2) with N′p N p = γ . Note: D ⋅ E = B ⋅ H does not hold in such a situation. Suppose that the plane-wave solution in the medium-rest frame X ′Y ′Z′ is given by It should be pointed out that, however, the above capacitor field is usually taken as a typical example to show that the (E′,B′,D′,H′) = (E′0 ,B′0 ,D′0 ,H′0 )expiΨ′ , (B1) “attached-to-source” EM field cannot exhibit “four-vector −1 where Ψ′ = (ω′t′ − n′k′⋅x′) , with ω′ > 0 assumed, and behavior” [18], although N ′p (D′× B′, D′ ⋅ E′ c) does show d Lorentz covariance in our analysis when there is no pseudo- ( E′0 ,B′0 ,D′0 ,H′0 ) are real constant vectors. momentum existing. D′ = ε ′E′ and B′ = µ′H′ hold, where ε ′ > 0 and µ′ > 0 are the constant dielectric permittivity and permeability, with ε ′µ′ = n′2 c 2 . Thus (E′ ,B′ ,nˆ ′) and (D′,H′,nˆ ′) are, Appendix A: Generalized Lorentz invariance of d phase velocity and group velocity respectively, two sets of right-hand orthogonal vectors, with E′ = (c nd′ )B′ × nˆ ′ and H′ = nˆ′× (c nd′ )D′ , resulting from As mentioned previously, since the phase function is Maxwell equations. symmetric with respect to all inertial frames, the definition of In the lab frame XYZ , the plane-wave solution can be written phase velocity should be symmetric. Thus the phase velocity is as given by

ω c ω (E,B, D,H) = (E 0 ,B 0 , D0 , H 0 )expiΨ , (B2) β c = nˆ = nˆ , (A1) ph n k n ω d d where Ψ = (ωt − nd k ⋅ x) , and (E0 ,B 0 ,D0 ,H 0 ) are obtained by inserting (E′0 ,B′0 ,D′0 ,H′0 ) into (reversed) Eqs. (3) and (4), which satisfies the invariant form of equi-phase plane equation given by of motion

ω − nd k ⋅ β ph c = 0 . (A2) ⎛ E ⎞ ⎛ E′ ⎞ ⎜ 0 ⎟ = γ (1− n′nˆ′⋅β′)⎜ 0 ⎟ Based on the phase velocity, we can define the group velocity. ⎜H ⎟ d ⎜H′ ⎟ Traditionally, the group velocity is defined as ⎝ 0 ⎠ ⎝ 0 ⎠

β gr c = dω d(nd k ) , leading to β gr c = nˆ ⋅β phc = β phc , and we ⎛ γ −1 ⎞⎛ β′⋅E′ ⎞ ˆ 0 have β c = β c . Thus the group velocity is equal to the phase + ⎜γ n′d n′ − β′⎟⎜ ⎟ , (B3) gr ph ⎜ β 2 ⎟⎜β′⋅ H′ ⎟ velocity. ⎝ ⎠⎝ 0 ⎠ It should be emphasized that, the form-invariance of Eq. (A1) ⎛B ⎞ ⎛ 1 ⎞⎛B′ ⎞ and (A2) is completely based on the symmetry of phase ⎜ 0 ⎟ = γ ⎜1− nˆ ′ ⋅ β′⎟⎜ 0 ⎟ ⎜ ⎟ ⎜ ′ ⎟⎜ ′ ⎟ function, the time-space symmetry, and the symmetry of ⎝D 0 ⎠ ⎝ nd ⎠⎝D 0 ⎠ definition of phase velocity. In a generalized sense, we can say ⎛ 1 γ −1 ⎞⎛β′⋅B′ ⎞ that they are Lorentz invariant in form. + ⎜γ nˆ′ − β′⎟⎜ 0 ⎟ . (B4) Suppose that the Planck constant is a . In the ⎜ ′ 2 ⎟⎜ ′ ′ ⎟ h nd β ⎝β ⋅ D0 ⎠ medium-rest frame, the dispersion equation, directly resulting ⎝ ⎠ from second-order (instead of the invariance of Note: All quantities are real in Eqs. (B3) and (B4) and the phase), is given by [20] transformations are “synchronous”; for example, E0 is ′ 2 2 expressed only in terms of E0 . It is clearly seen from Eqs. (n′d ω′ c) − (n′d k′) = 0 , (A3) (B1)-(B4), the invariance of phase, Ψ = Ψ′ , is a natural result. Because the medium is ideal, all field quantities have the same which actually is the definition of refractive index. This phase factor, no matter in the medium-rest frame or lab frame. dispersion relation is thought to be the characterization of the In the following, formulas are derived in the lab frame, relation between EM energy and momentum, and the photon’s because they are invariant in forms in all inertial frames. energy in a medium is supposed to be n′ ω′ to keep a zero rest d h Although the fields given by Eqs. (B1) and (B2) are complex, energy [20]. However it should be noted that, although Eq. all equations and formulas obtained are checked with complex (A3) is Lorentz invariant, (n′d k′, n′d ω′ c) is not a Lorentz µ and real fields and they are valid. covariant four-vector, since only K′ = (nd′k′,ω′ c) is; except

4 Under Lorentz transformations, the Maxwell equations keep while the Poynting vector E× H has three components: one in the same forms as in the medium-rest frame, given by D×B -direction, one in B-direction, and one in D-direction; the

latter two are perpendicular to the group velocity v gr . ∇ × E = − ∂B ∂t , ∇ ⋅ D = ρ , (B5) We can divide the Poynting vector into two parts,

∇ × H = J + ∂D ∂t , ∇ × B = 0 , (B6) E × H = S power + S pseu , which are perpendicular each other, given by with J = 0 and ρ = 0 for the plane wave. From above, we S = v 2 (D × B) = (D ⋅ E) v , (B20) have power gr gr

ωB = nd k × E , and ωD = −nd k × H , (B7) S pseu = −vgr [ (nˆ ⋅ H)B + (nˆ ⋅ E) D ], (B21) leading to D⋅ E = B ⋅ H , namely the electric energy density is where S power is associated with the total EM energy density equal to the magnetic energy density, which is valid in all (D⋅E) and the group velocity v gr . S power carries the total EM inertial frames. energy moving at v gr , and it is a real power flow. S pseudo is From Eqs (B3) and (B4), we obtain some intuitive perpendicular to the group velocity, and it is a pseudo-power expressions for examining the of space relations of flow, just like the E× H vector produced by static crossed EM fields observed in the lab frame, given by electric and magnetic fields, which cannot be taken as a real power flow. Based on Eq. (B20), the energy velocity is D ⋅ nˆ = D′ ⋅ nˆ ′ = 0 , B ⋅ nˆ = B′ ⋅ nˆ ′ = 0 , (B8) supposed to be equal to the group velocity. The essential difference in physics between S power and S pseu 2 can be seen from the divergence theorem. The divergence of γ (nd′ −1)(E′ ⋅ β′) E ⋅ nˆ = , (B9) S power is given by 2 2 2 (nd′ −1) + γ (1− nd′nˆ ′ ⋅ β′) ∇ ⋅S power = ∇ ⋅[D⋅ E) β gr c] 2 2 γ (n′d −1)(H′ ⋅ β′) H ⋅ nˆ = , (B10) = (D0 ⋅ E 0 )[∇ cos (ωt-nd k ⋅ x)]⋅ v gr ≠ 0 , (B22) (n′2 −1) + γ 2 (1− n′nˆ ′ ⋅ β′) 2 d d which means that there is a power flowing in and out in the S 0 and differential box, but the time average < ∇ ⋅ power >= , meaning that the powers going in and out are the same in average, with

E ⋅ B = E′⋅ B′ = 0 , D⋅ H = D′⋅ H′ = 0 , (B11) no net energy left in the box. In contrast, ∇ ⋅ S pseu ≡ 0 holds for any time resulting from ∇ ⋅ B = 0 , ∇ ⋅ D = 0 , B ⊥ k , and 2 2 D ⊥ k , which means that there is no power flowing at any time E ⋅ H = γ (n′d −1)(β′ ⋅ E′ )(β′ ⋅ H′) , (B12) for any places, and S pseu is exactly like the E× H vector ⎛ 1 ⎞ produced by static crossed electric and magnetic fields as D ⋅ B = −γ 2 ⎜1− ⎟(β′ ⋅ D′ )(β′ ⋅ B′ ) , (B13 mentioned above. ⎜ n′2 ⎟ ⎝ d ⎠ It is seen From Eqs. (B9) and (B10) that S pseu vanishes in free space, or when the dielectric medium moves parallel to the wave 2 ⎛ 1 ⎞ ˆ D ⋅ E = γ (1− n′nˆ ′ ⋅ β′)⎜1− nˆ ′ ⋅ β′⎟D′ ⋅ E′ , (B14) vector n′ . d ⎜ n′ ⎟ From above analysis, we can see that a Poynting vector does ⎝ d ⎠ not necessarily denote a real power flowing. However such a ⎛ 1 ⎞ phenomenon seems neglected in the physics community, in 2 ′ ˆ ′ ′ ⎜ ˆ ′ ′⎟ ′ ′ B ⋅ H = γ (1− nd n ⋅ β ) 1− n ⋅ β B ⋅ H , (B15) view of the fact that the Abraham’s momentum is defined ⎜ n′ ⎟ ⎝ d ⎠ through Poynting vector without any conditions. and In summary, we can make some conclusions for the power E = (nˆ ⋅ E)nˆ − β cnˆ × B , (B16) flowing. (1) The Minkowski’s EM momentum D× B is ph parallel to the group (phase) velocity, observed in any inertial H = (nˆ ⋅ H)nˆ + β cnˆ × D . (B17) frames. (2) When a medium moves, the Poynting vector E× H ph consists of two parts: one is parallel to the group velocity, and it It is seen from above that E ⊥ B , B ⊥ nˆ , D ⊥ H and D ⊥ nˆ is a real power flow; the other is perpendicular to the group hold in the lab frame, but E // D , B // H , E ⊥ H , D ⊥ B , velocity, and it is a pseudo-power flow. E ⊥ nˆ , and H ⊥ nˆ usually do not hold any more. Lorentz covariance of Minkowski’s EM momentum. We Pseudo-power flow caused by a moving medium. From have shown the Lorentz covariance of Minkowski’s photon Eq. (B7), we obtain Minkowski’s EM momentum and Poynting momentum from the wave four-vector combined with Einstein’s vector, given by light-quantum hypothesis. This covariance suggests us that there should be a covariant EM momentum-energy four-vector, 2 given by ⎛ nd ⎞ D× B = ⎜ ⎟ (D ⋅ E) v gr , (B18) µ ⎝ c ⎠ P = ( pem , Eem c) , (B23) E × H = v [ v (D × B) − (nˆ ⋅ H)B − (nˆ ⋅ E) D ] . (B19) gr gr where pem and Eem are, respectively, the EM momentum and energy per photon, given by where v gr = β gr c , with vgr = β gr c ⋅ nˆ and vgr = c nd , is the group velocity, equal to the phase velocity β c , as defined in D× B D ⋅ E ph p = , E = . (B24) Appendix A. Note: The Minkowski’s momentum D×B has em em N p N p the same direction as the group velocity v gr mathematically,

5 with Np the photon’s density. The EM momentum-energy four- ∂(D × B) vector and wave four-vector are related through = (∇ ⋅ D)E − D × (∇ × E) µ µ ∂t P = (Eem ω)K , with Eem ω corresponding to h physically. µ + (∇ ⋅ B)H − B × (∇ × H) . (B28) The four-vector P = (pem , Eem c) is required to follow a four-vector Lorentz transformation given by Eqs. (1) and (2), while the EM fields must follow a second-rank tensor Lorentz Since ∇ ⋅ D = 0 , ∇ ⋅ B = 0 , B ⊥ nˆ , and D ⊥ nˆ hold, we have transformation given by Eqs. (3) and (4), or Eqs. (B3) and (B4) ∂(D× B) t for a plane wave. From the four-vector Lorentz transformation = −∇ ⋅ T , (B29) ∂t of Eem and ω given by Eq. (2), with the invariance of phase Ψ = Ψ′ considered, we have where the symmetric stress tensor is given by t t t N pω D 0 ⋅ E 0 T = I(D ⋅ E) = I(B ⋅ H) , (B30) = . (B25) t N ′pω′ D′0 ⋅ E′0 with I the unit tensor. Note: Eq. (B29) is valid for both real The above equation has a clear physical explanation that the and complex fields although only the real fields are physical. Doppler factor of EM energy density is equal to the product of Eq. (B29) is the momentum conservation equation, and it is the Doppler factors of photon density and frequency. invariant in form in all inertial frames together with the From the second-rank tensor Lorentz transformation of D Maxwell equations. 0 Now let us take a look of the applicable range of the stress and E0 given by Eqs. (B3) and (B4), we obtain the transformation of EM energy density, given by tensor. Consider the EM force exerted on a given dielectric volume. From Eq. (B29), using divergence theorems for a tensor and a vector, with (D,E) = (D ,E )cosΨ inserted, we D ⋅ E ⎡ ⎛ 1 ⎞⎤ 0 0 0 0 = []γ (1− n′nˆ ′ ⋅ β′) ⎢γ ⎜1− nˆ ′ ⋅ β′⎟⎥ , (B26) have D′ ⋅ E′ d ⎜ n′ ⎟ 0 0 ⎣⎢ ⎝ d ⎠⎦⎥ ∂ t (D× B)dV = − dS ⋅ T = − ∇(D ⋅ E)dV namely Eq. (B14). Comparing with Eq. (6), we know that ∂t ∫∫∫ ∫∫ ∫∫∫

γ (1− n′d nˆ ′⋅β′) is the frequency Doppler factor. In free space, the above equation is reduced to Einstein’s result [31]: = − (D ⋅ E )sin(2Ψ)(n k)dV . (B31) ∫∫∫ 0 0 d E = γ (1− nˆ ′ ⋅ β′) E′ . Inserting Eq. (B26) into Eq. (B25), we obtain the It is seen that there is a time-dependent force experienced by the transformation of photon density, given by volume along the wave vector direction, although its time average is zero, namely < ∂ ∂t ∫∫∫(D× B)dV > = 0 . ⎛ 1 ⎞ However the construction of stress tensor is flexible; in a N = γ ⎜1− nˆ ′ ⋅ β′⎟N ′ . (B27) p ⎜ n′ ⎟ p sense, it is artificial. For example, ∇ ⋅ (−DE − BH) = 0 holds for ⎝ d ⎠ a plane wave, and the tensor can be re-written in a well-known So far we have finished the proof of the covariance of form P µ = (p , E c) by resorting to a parameter, so-called t t 1 em em T = −DE − BH + I (D ⋅ E + B ⋅ H) , (B32) “photon density”. Actually, we do not have to know what the 2 specific value of N′ or N is, but the ratio of N N′ , and P µ p p p p which does not affect the validityt of Eq. (B29) and Eq. (B31), is pure “classical”, without Planck constant h involved. but the local surface force dS ⋅ T is changed. It means that ′ t However, N p must be the photon density in the lab frame if N p dS ⋅T does not necessarily denote a real EM force experienced is taken to be in the medium-rest frame. by the surface element dS . t Mathematically speaking, the existence of this covariance is For example, from Eq. (B30), we have dS ⋅ T = µ µ t apparent. P and K are just different by a factor of (Eem ω) dS ⋅ I(D ⋅ E) = dS(D⋅ E) . No matter what direction dS takes, which contains an introduced parameter N p to make the there is a force. However, D×B is always parallel to the group transformation hold. velocity or wave vector as shown in Eq. (B18) and Eq. (B31), In conclusion, Minkowski’s EM Momentum D×B is Lorentz independently of time and locations. covariant, as the space component of the EM momentum-energy µ −1 From above, we can see that an application of the stress tensor four-vector P = N p (D× B, D ⋅ E c) , just like the Minkowski’s like Eq. (B31) is valid, while if using it for calculations of EM photon moment hnd k which is Lorentz covariant, as the space forces on partial local surfaces, the validity is questionable. component of the photon momentum-energy four-vector P µ = ( n k, ω c) . Unconventional phenomenon for a superluminal medium. h d h In addition to the negative-frequency effect predicted by Huang Covariance and symmetry of Minkowski’s stress tensor. [1], a Negative Energy Density (NED) may result for a plane As we know, in the derivations of Minkowski’s stress tensor in wave when the medium moves opposite to the wave vector the medium-rest frame, the linear relations of dielectric material direction at a faster-than-dielectric light speed; this phenomenon are employed, such as D′ = ε ′E′ , so that D′× (∇′× E′) is called “NED zone” for the sake of convenience. = E′×(∇′× D′) holds and the tensor can be written in a The origin of negative EM energy density can be seen from symmetric form [13]. As mentioned above, because of the Eq. (B26). The energy density Doppler factor is equal to the anisotropy of a moving medium, such relations do not hold in product of frequency’s and photon density’s. The photon the lab frame. Thus the stress tensor in the lab frame only can density Doppler factor is always positive while the frequency’s be derived based on the covariance of Maxwell equations and is negative in the NED zone ( nd′nˆ ′⋅ β′ = nd′ β′ > 1 ), leading to the invariance of phase. the holding of D ⋅ E < 0 when ω < 0 . From the covariant Maxwell Eqs. (B5) and (B6), with J = 0 In the NED zone, D // E and H // B are valid, and from Eqs. and ρ = 0 taken into account we have [19] (B3) and (B4) we have

6 REFERENCES D 1− β′ nd′ = ε ′ < 0 , (B33) ′ ′ [1] Y.-S. Huang, Europhys. Lett. 79, 10006 (2007). E 1− nd β [2] A. Gjurchinovski, Europhys. Lett. 83, 10001 (2008). [3] Y.-S. Huang, Z. Naturforsch. 65a, 615 (2010). B 1− β′ n′d [4] S. M. Barnett, Phys. Rev. Lett. 104, 070401 (2010). = µ′ < 0 . (B34) H 1− n′ β′ [5] R. V. Jones, Nature 167, 439 (1951). d [6] G. B. Walker, D.G. Lahoz, and G. Walker, Can. J. Phys. 53, 2577 (1975). Because of ω < 0 in the NED zone, from Eq. (A1) we have [7] A. Feigel, Phys. Rev. Lett. 92, 020404 (2004). ˆ β phc = β phc n with β ph c < 0 . From Eqs. (B16) and (B17) we [8] G. K. Campbell, A. E. Leanhardt, J. Mun, M. Boyd, E. W. Streed, have W. Ketterle, and D. E. Pritchard, Phys. Rev. Lett. 94, 170403 (2005). E = −β cnˆ × B , (B35) [9] W. She, J. Yu, and R. Feng, Phys. Rev. Lett. 101, 243601 (2008). ph [10] I. Brevik, Phys. Rev. Lett. 103, 219301 (2009). H = +β cnˆ × D . (B36) [11] M. Mansuripur, Phys. Rev. Lett. 103, 019301 (2009). ph [12] U. Leonhardt, Nature 444, 823 (2006). It follows that the plane wave in the NED zone is a left-hand [13] R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Rev. Mod. Phys. 79, 1197 (2007). wave: (1) (E,B,nˆ) and (D,H,nˆ) follow the left-hand rule, and [14] M. Buchanan, Nature Phys. 3, 73 (2007). (2) the phase or group velocity is opposite to the wave vector. [15] B. A. Kemp, J. Appl. Phys. 109, 111101 (2011). In other words, the moving medium in the NED zone behaves as [16] C. Baxter and R. Loudon, J. Mod. Opt. 57, 830 (2010). a so-called “negative index medium” [32-35], where the [17] T. Ramos, G. F. Rubilar, and Y. N. Obukhov, Phys. Lett. A 375, refractive index is taken to be negative [33], instead of the 1703 (2011). frequency here [1]. Note: It is difficult to define a negative [18] M. Mansuripur, Phys. Rev. E 79, 026608 (2009). [19] J. D. Jackson, Classical Electrodynamics, (John Wiley & Sons, NJ, refractive index from the wave four-vector with the continuity rd of Lorentz transformation satisfied [confer Eq. (9)]. 1999), 3 Edition. From Eq. (B18), D× B always carries the EM energy D⋅ E [20] J. A. Kong, Theory of Electromagnetic Waves, (John Wiley & v v Sons, NY, 1975). moving at gr . In the NED zone, gr is opposite to nd k , and [21] J. A. Stratton, Electromagnetic theory, (McGraw-Hill, NY, 1941). the minus energy (D ⋅ E < 0) moves along minus nd k - [22] C. Wang, Ann. Phys. (Berlin) 523, 239 (2011). direction, which is equivalent to that the positive energy moves [23] W. R. Smythe, Static and dynamic electricity, (McGraw-Hill, NY, rd along plus nd k -direction; thus D× B always has the same 1968), 3 edition. direction as nd k , as described by the photon momentum-energy [24] D. J. Griffths, Introduction to Electrodynamics, (Prentice-Hall, NJ, 1999), 3rd edition. four-vector (hnd k,hω c) : no matter whether hω (→ D ⋅ E) [25] C. Wang, http://aps.arxiv.org/abs/1007.0980 . < 0 or > 0 , hnd k (→ D× B) always takes the +nd k - direction. [26] D. Mugnai, A. Ranfagni, and R. Ruggeri, Phys. Rev. Lett. 84, 4830 (2000). How do we understand the power flow Eq. (B20) [27] P. Saari and K. Reivelt, Phys. Rev. Lett. 79, 04135 (1997). S power = (D⋅ E) v gr in the NED zone? There are two [28] J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, Phys. explanations. (1) The power flow S power has the same direction Rev. E 62, 4261 (2000). as the group velocity v gr , because D⋅E itself includes the [29] L. J. Wang, A. Kuzmich, and A. Dogariu, Nature 406, 277 (2000). [30] W. Rindler, Relativity: special, general, and cosmological, “minus sign” mathematically. (2) S power is opposite to v gr , nd since the energy density must be taken to be positive (Oxford, NY, 2006), 2 edition. traditionally. [31] A. Einstein, Ann. Phys. (Leipzig) 17, 891 (1905). [32] V.G. Veselago, Sov. Phys. Usp. 10, 509 (1968). [33] P.M. Valanju, R.M.Walser, and A. P. Valanju, Phys. Rev. Lett. 88, 187401 (2002). [34] N. Garcia and M. N. Vesperinas, Phys. Rev. Lett. 88, 207403 (2002). [35] S. A. Ramakrishna, Rep. Prog. Phys. 68 449 (2005).

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