arXiv:0801.3149v2 [physics.optics] 25 Jan 2008 osatwhen constant rfieo h aet ayi pc iuodly namely sinusoidally, space in vary the choose to We ψ wave 7]. the 6, of [5, form profile wave harmonic a with bance n e fpae epniua otevector the to perpendicular planes of set a ing k rbe.I atclr h clr(o)poutbetween the product of (dot) vector analysis wave scalar the the the relativistically- in particular, of 4] In effect 3, problem. [2, the phase anisotropy of optical the ignorance induced of the non-invariance is apparent waves this of to Huang led transformation. 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ASnmes 22.p 03.30.+p 42.25.-p, numbers: PACS h esnta vnulyldteato farcn P lett EPL recent a of author the led eventually that reason The h oet-nain xrsinfrtepaeo ae sho waves of phase t the for in be expression Lorentz-invariant anisotropy has the waves optical of relativistically-induced phase of the effect of non-invariance the of conclusion nta ftewdl-sdepeso = Φ expression widely-used the of instead LORENTZ-INVARIANT? A A h naineo h hs fpaewvsaogieta fra inertial among waves plane of phase the of invariance The k k sin( .INTRODUCTION I. · tewvfotnra)adtevelocity the and normal) wavefront (the E r stepaeo ln ae rm-needn quantity? frame-independent a waves plane of phase the Is X k = t.CrladMtoisUiest,P .Bx12 00Skop 1000 162, Box O. P. University, Methodius and Cyril Sts. · const r .W bev that observe We ). k eateto hsc,Fclyo aua cecsadMath and Sciences Natural of Faculty Physics, of Department µ ( = hc sa qaindefin- equation an is which , k ω/c , ae ndifferen- on based ) k · r − ψ lkadrGjurchinovski Aleksandar ( ωt r Dtd aur 4 2008) 24, January (Dated: t , k ob a be to )is 0) = k The . · k r · − r − ωt r ecie yahroi aefnto nteform the in function wave harmonic a by velocity described a at moves disturbance ftewv,and wave, the of stepaeo h ae and wave, the of phase the is hc eoti ymrl replacing merely by obtain we which antd 2 magnitude a ment au of value velocity the to parallel necessarily not but tefi ieatratmoa period temporal a after time in itself nlt h lnsΦ= Φ at planes observer the to onal in loigteepeso o h hs ftewv in wave the of phase the for expression the allowing tion, u I.1 w-iesoa ktho ln aetaeigat traveling wave plane a velocity of sketch a two-dimensional A 1: FIG. nes fwihi h frequency the is which of inverse icuigvcu) h aevector wave the wave, vacuum), the (including of frequency by angular the denoted the of commonly as velocity to the referred and is vector wave, wave the between product quantity hc a iie validity. limited a has which − ftewv r aalladpitn ntesm direc- same the in pointing and parallel are wave the of eaayi ftepolm ti rudthat argued is It problem. the of analysis he u l etkni h omΦ= Φ form the in taken be uld t nλ h expression: The . nietfid–i steinrneo the of ignorance the is it – identified en ψ r[EPL er ntedrcinof direction the in | u ( ψ k r ∗ h aevector wave The . e sivsiae nsm details. some in investigated is mes ( t , r · r u t , = )rpasisl nsaeatradisplace- a after space in itself repeats 0) = π/λ | hc steaslt au ftedot the of value absolute the is which , = ) r 79 n 0 sa nee.I h rfieo h wave the of profile the If integer. an is 06(07]t spurious a to (2007)] 1006 , h aedisturbance wave the A n onigi h ieto orthog- direction the in pointing and = Φ sin e Macedonia je, ω ematics, k k ntecs fiorpcmedia isotropic of case the In . const · · r ( k r k k − where , − snra otewavefronts, the to normal is stewv etr having vector, wave the is o xdstationary fixed a For . k k u u · · t r u seFg ) tcnbe can it 1), Fig. (see = ) f − t r k λ T ftewv.The wave. the of k in A · stewavelength the is u u 2 = n h velocity the and i Φ sin ψ ψ /c ftewave. the of ( ( , r r π/ t , 0 , t , | k )with 0) = repeats ) · u | the , (2) (1) 2
Eq. (2) to be recasted in the following widely-used form: have a role of Lorentz scalar between different inertial reference frames. Φ= k · r − ωt. (3)
If the plane-wave disturbance in Eq. (1) is observed III. RELATIVISTIC TRANSFORMATIONS OF from a different reference frame, the phase of the wave Φ THE WAVE CHARACTERISTICS should remain invariant quantity. This claim is clarified by the fact that the elapsed phase of the wave is propor- Consider from a frame S a plane monochromatic wave tional to the number of wavecrests that have passed the propagating at a velocity u, whose wavefront normal is observer, and thus it must be frame-independent, and described by the wave vector k. With respect to the hence, a Lorentz scalar [7, 8]. Alternatively, the same ′ frame S which moves at a constant velocity V relative conclusion follows by considering optical interference ex- ′ ′ to S, the velocity of the wave is u , and k is the cor- periments from different inertial frames, where the phase responding wavefront normal. We assume that the cor- Φ is the quantity that determines the interference pattern ′ responding axes of S and S are parallel, and that the [9]. ′ velocity V of S in S is in an arbitrary direction. The In the aforementioned letter to EPL [1], Huang used invariance of the phase implies: the invariance of the phase Φ in Eq. (3) to Lorentz- ′ ′ ′ ′ ′ transform the wave characteristics between different in- k · r − k · ut = k · r − k · u t . (4) ertial frames. In this way, Huang obtained negative an- gular frequencies in the frame where the medium moves By applying the Lorentz transformation between S and at “superluminal” speeds against the wave. What has S′ [7]: not been taken into account in Huang’s analysis is that the dot product between the wave vector and the veloc- ′ (r · V) r = r + (γ − 1) V − γVt, (5) ity of the wave changes its sign from positive to neg- kVk2 ative when switching between the medium’s rest frame ′ r · V and the frame in which the medium moves “superlumi- t = γ t − , (6) c2 nally”. Let us clarify this point more explicitly. The −1 2 approach by using Eq. (3) as an expression for the phase kVk2 / γ = 1 − , (7) will work efficiently if the wave is propagating in vacuum. c2 In the case of vacuum, the wave vector and the velocity of the wave will remain parallel and unidirectional with into Eq. (4), we obtain: respect to any inertial frame. Hence, k · u = |k · u| = ω and k′ · u′ = |k′ · u′| = ω′, and the phase invariance k · r − k · ut = ′ ′ ′ ′ ′ ′ ′ ′ k · r − k · ut = k · r − k · u t between the frames S and ′ (k · V) (k · u ) ′ ′ ′ ′ ′ = k + V (γ − 1) + γ · r S would imply k · r − ωt = k · r − ω t . However, when kVk2 c2 ′ ′ the wave propagates in a material medium, the invari- − γk · (u + V)t. (8) ance of the phase given in Eq. (3) will generally not work even if the optical medium is isotropic in its rest frame. If the frames S and S′ are in standard configuration, This is due to the fact that an optical medium that is V k k′ ′ ′ ′ then = (V, 0, 0), = (kx, ky, kz), = (kx, ky, kz), optically isotropic in its rest frame S′ will possess an op- u u′ ′ ′ ′ r = (ux,uy,uz), = (ux,uy,uz), and = (x,y,z). tical anisotropy in the frame S in which it is moving at Substituting in Eq. (8) and comparing the terms for a constant velocity. It should be noted that this induced arbitrary x, y, z, and t, we obtain: optical anisotropy is of a purely relativistic origin, and it has nothing to do with the usual anisotropy in the crys- ′ V ′ ′ ′ ′ ′ ′ k = γ k + (k u + k u + k u ) , (9) tals [4]. Hence, while the wave vector and the velocity x x c2 x x y y z z of the wave are parallel and unidirectional with respect ′ ′ ky = ky, (10) to the rest frame S of the medium, this may not be the ′ case with respect to some reference frame S in which kz = kz, (11) the medium is in motion. In fact, it might happen that kxux + kyuy + kzuz = the dot product between the wave vector (the wavefront ′ ′ ′ ′ ′ ′ ′ = γ(kxV + kxux + kyuy + kzuz). (12) normal) and the velocity of the wave may have different signs with respect to different reference frames. In partic- The angular frequencies of the wave in the corresponding ′ ′ ′ ′ ′ ular, the phase invariance k · r − k · ut = k · r − k · u t frames are given by: may not imply k · r − ωt = k′ · r′ − ω′t′ in the latter case, ′ k′ u′ ′ ′ ′ ′ ′ ′ since it may happen that k · u = −|k · u| = −ω < 0 when ω = | · | = |kxux + kyuy + kzuz|, (13) k′ u′ k′ u′ ′ · = | · | = ω > 0. In this sense, the expression ω = |k · u| = |kxux + kyuy + kzuz|. (14) for the phase of the wave in Eq. (2) is more general (and thus, more correct) than the one in Eq. (3), and there- Furthermore, by putting Eqs. (9), (10) and (11) into Eq. k r k u ′ ′ fore, it is the expression Φ = · − · t that should (12), and comparing the terms for arbitrary kx, ky and 3
FIG. 2: A plane wave advancing in a stationary medium at a FIG. 3: A snapshot of the wave propagation in Fig. 2 with ′ ′ speed u in the direction of the positive y -axis. respect to the frame where the medium is moving at a con- stant speed V to the right. Notice that k and u are not parallel, which means that the light ray is not normal to the ′ kz, we obtain: wavefronts. ′ ux + V ux = ′ 2 , (15) 1+ uxV/c ′ uy/γ uy = ′ 2 , (16) 1+ uxV/c ′ uz/γ uz = ′ 2 , (17) 1+ uxV/c which are the relativistic velocity transformation formu- las for the wave. The set of equations (9)–(12) and (15)– (17) describe the relativistic transformation of the wave characteristics from S′ to S. The reverse transforma- tion is accomplished via V -reversal, that is, by replacing V with –V and interchanging the primed and unprimed quantities. We will demonstrate the effect of relativistically- FIG. 4: A plane wave propagates in a stationary medium at ′ ′ induced optical anisotropy by applying the above anal- a speed u in the direction of the positive x -axis. ysis to describe the transverse drag of light [3, 4]. In the rest frame S′ of the medium, a plane wave propa- gates in the direction of the positive y′-axis at a velocity u′ = (u′, 0, 0). The wave vector describing the wave- u′ = (0,u′, 0) (see Fig. 2). By assuming that the medium front normal is k′ = (k′, 0, 0), and the angular frequency is homogeneous, isotropic and nondispersive in S′, we of the wave is ω′ = |k′ · u′| = k′u′. Transforming to have k′ = (0, k′, 0). Evidently, the velocity of the wave the S frame in which the medium moves at a velocity and the wavefront normal in the S′-frame are parallel V = (V, 0, 0), we obtain k = [γk′(1 + Vu′/c2), 0, 0] and and unidirectional. Also, for the angular frequency of the u = [(u′ + V )/(1 + u′V/c2), 0, 0]. Consequently, the an- wave in S′, we have ω′ = |k′ · u′| = k′u′. With respect to gular frequency of the wave in S-frame is ω = |k · u| = the S-frame in which the medium is moving at a velocity γ|ω′ + k′V |. If the medium moves in the negative x- V = (V, 0, 0), from Eqs. (9)–(11) and Eqs. (15)–(17) we axis, the wave characteristics are obtained by replacing obtain k = (γV k′u′/c2, k′, 0) and u = (V,u′/γ, 0). Con- V with −V . Hence, k = [γk′(1 − Vu′/c2), 0, 0], u = sequently, k × u = [0, 0, k′V (u′2/c2 − 1)], which means [(u′ −V )/(1−u′V/c2), 0, 0] and ω = |k · u| = γ|ω′ −k′V |. that the velocity of the wave and the wavefront normal In the case of “superluminal” motion of the medium, ′ in the S-frame are generally not parallel (see Fig. 3), u
must take into consideration the effect of relativistically- induced optical anisotropy due to the motion of the medium. In this sense, the angle between the wave vec- tor (the wavefront normal) and the velocity of the wave is not Lorentz-invariant, and furthermore, the dot prod- uct between these vectors may have different signs with respect to different reference frames.
To employ the concept of phase invariance of the wave among inertial frames one should use the correct expres- sion for the wave four-vector kµ:
k u µ · FIG. 5: A snapshot of the wave in Fig. 4 with respect to the k = k, , (18) frame where the medium is moving at a “superluminal” speed c V to the left. In this case, although k and u are parallel, they are pointing in opposite directions. where k is the wave three-vector, and u is the velocity of the wave. Employing the less-general, but widely used IV. CONCLUSION expression for the four-vector in the form kµ = (k,ω/c) as in the recent EPL letter, one may be tempted into a When analyzing the wave propagation in a medium spurious conclusion that the invariance of the phase of with respect to different inertial reference frames, one waves among inertial frames is questionable.
[1] Y.-S. Huang, Europhys. Lett. 79, 10006 (2007). UP, Cambridge, 1999), 7th ed. [2] B. M. Bolotovskii and S. N. Stolyarov, Sov. Phys. Usp. [7] J. D. Jackson, Classical Electrodynamics (Wiley, New 17, 875 (1975). York, 1999), 3rd ed. [3] A. Gjurchinovski, Am. J. Phys. 72, 934 (2004). [8] W. Rindler, Relativity: Special, General and Cosmological [4] A. Gjurchinovski and A. Skeparovski, Eur. J. Phys. 28, (Oxford UP, Oxford, 2006), 2nd ed. 933 (2007). [9] H. Stephani, Relativity: An Introduction to Special and [5] E. Hecht, Optics (Addison-Wesley, Reading, 1990), 2nd General Relativity (Cambridge UP, Cambridge, 2004), 3rd ed. ed. [6] M. Born and E. Wolf, Principles of Optics (Cambridge