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Wave Four-Vector in a Moving Medium and the Lorentz Covariance of Minkowski's Photon and Electromagnetic Momentums 1

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Wave Four-Vector in a Moving Medium and the Lorentz Covariance of Minkowski's Photon and Electromagnetic Momentums 1 Wave four-vector in a moving medium and the Lorentz covariance of Minkowski’s photon 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 tensor may have a simplest form of symmetry, 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 frequency 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 Lorentz transformation 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 group velocity that is exactly equal to the projection same form. For a plane wave in free space, which is velocity of the phase velocity 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 preferred frame [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′ − nd′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 scalar 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′ ⎞ 2 ⎛ ω ⎞ 2 wavefront, with nˆ the unit normal or wave vector. Changing ⎜ ⎟ − (nd′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.
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