An Effective Photon Momentum in a Dielectric Medium: A Relativistic Approach. Abstract

Transcription

1 An Effetive Photon Momentum in a Dieletri Medium: A Relativisti Approah Bradley W. Carroll, Farhang Amiri, and J. Ronald Galli Department of Physis, Weber State University, Ogden, UT Dated: August 20, 2015) Abstrat We use a relativisti argument to define an effetive photon that travels through a transparent non-absorbing) nondispersive dieletri medium of index of refration n. If p is the momentum of the photon in a vauum, then the momentum of an effetive photon inside the medium may be of the form p eff = pn α and still reprodue the observed transverse relativisti drift when the medium is in motion. We employ an energy argument to determine the value of the exponent to be α = 1, so that the effetive photon momentum is p eff = p/n, whih is the Abraham momentum. 1

2 I. INTRODUCTION The refration of light as it passes from a vauum into a transparent dieletri medium and undergoes a speed hange is well-established and is inluded in standard eletromagnetism texts. 1 On a mirosopi level, it is understood that the propagation of a light ray through a medium is atually a ompliated proess that involves multiple satterings of the inident eletromagneti wave by the atoms that omprise the bulk of the medium. 2 The refrated and refleted rays are the result of the interferene of these sattered waves. The net result of this interferene is that the refrated ray moves with a redued speed of /n inside a medium with index of refration n. The wave s Poynting vetor S is direted along the path followed by the ray. The situation is more ompliated when the refrated ray is desribed from the point of view of its onstituent photons. First, we onsider the ase of a single photon that is inident upon a transparent dieletri blok at rest. The photon has a ertain probability of being transmitted or refleted from the blok s surfae. We will restrit our analysis to the ase when the photon is transmitted into the blok. Photons are massless partiles and, as suh, must travel at the speed of light; hene the refrated ray, moving at /n, annot be omposed of massless photons. Instead, we will define an effetive photon that follows the path of the refrated ray, transports the ray s momentum and energy, and moves with speed /n. The mass of the effetive photon is denoted by m, whih we take to be Lorentz-invariant. We define the effetive photon s energy by the usual relativisti expression E eff = γ v m 2 1) and the magnitude of its momentum by p eff = γ v mv, 2) where γ v is the Lorentz fator for the effetive photon s veloity, γ v = As usual, these may be ombined to yield 1 1 v2 2 = 1 1 1n 2. 3) E 2 eff p 2 eff 2 = m 2 4, 4) 2

3 whih expresses the invariant magnitude of the effetive photon s momentum-energy four vetor. Substituting v = /n for the effetive photon s speed, Eq. 2) beomes p eff = γ v m n, 5) and Eq. 1) shows that the effetive photon s momentum and energy are related by E eff = np eff 6) Substituting this into Eq. 4) shows that the mass of the effetive photon is m = p eff n2 1. 7) We assume that the momentum of the effetive photon inside a dieletri medium at rest is related to the momentum p of the inident momentum in vauum by p eff = n α p, 8) where α is a real number. The energy of the effetive photon is then, from Eq. 6), E eff = n α+1 p. 9) The value of α has not yet been determined. The two possibilities most often disussed are α = 1 so p eff = np the so-alled Minkowski momentum) and α = 1 so p eff = p/n the so-alled Abraham momentum). 3 II. THE TRANSVERSE RELATIVISTIC DRIFT To investigate the impliations of defining an effetive photon in this way, we will utilize the phenomenon of the aether drift, the hange in the veloity of light in a moving medium predited by A. Fresnel 4 in 1818 and first deteted experimentally by H. Fizeau 5 in This effet is now understood to be entirely due to the relativisti transformation of the veloity of light from its value in the rest frame of the medium. For this reason, we will refer to this hange in veloity as a relativisti drift. Consider a retangular blok of transparent nondispersive dieletri material of thikness T that is at rest relative to an inertial referene frame S, whih we all the rest frame. We restrit ourselves to the speial ase where the top and bottom faes of the blok are 3

4 parallel to the diretion of motion and perpendiular to the y axis. The frame S moves in the positive x-diretion with veloity u relative to another inertial referene frame S, whih we all the lab frame. Thus an observer at rest in the lab frame will see the dieletri blok moving in the positive x-diretion with veloity u see Fig. 1). We stipulate that the blok remains at rest in frame S when the photon enters beause the blok is suffiiently massive or is otherwise onstrained. Gjurhinovski 6 used the relativisti veloity transformations to show that the angles of inidene θ 1 ) and refration θ 2 ) of the ray in the moving blok are related in the lab frame S by n 2 1) u ) + 1 n2 u 2 sin θ tan θ 2 = γ 2 1 n 1 2 u ) 2 sin θ 1 sinθ 1 u ), 10) 2 where γ = 1/ 1 u 2 / 2. The 1 subsript denotes a value in the vauum below the dieletri blok, and the 2 subsript denotes a value within the dieletri blok.) If we set u = 0, it beomes equivalent to the familiar Snell s law, sin θ 1 = nsinθ 2. The signifiane of Eq. 10) is seen for the ase of a normally inident ray in the frame S. Setting θ 1 = 0 produes n 1 ) u n tan θ 2 = γ. 11) 1 u2 n 2 2 For non-zero u, this means that, in the lab frame S for normal inidene θ 1 = 0), the refrated ray deviates slightly in the positive x-diretion. If u/ 1, then to first order in u/ for a blok of thikness T, the ray experienes a transverse relativisti drift x of x = T tanθ 2 n 1 ) Tu n. 12) This is Fresnel s formula for his aether drift. A transverse relativisti drift has been experimentally onfirmed by Jones 7 9 and Leah et al.. 10 III. LORENTZ TRANSFORMATION OF THE WAVE FOUR-VECTOR To understand better the nature of light in a moving dieletri, we study the same phenomenon by performing a Lorentz transformation of the wave four-vetor, k x, k y, k z, ω/). 4

5 y y Frame S Frame S T θ 2 u T θ 2 blok at rest θ 1 θ 1 x x FIG. 1. The referene frames used in this paper. Relative to referene frame S the lab frame), the dieletri blok and referene frame S both move in the positive x-diretion with speed u. The blok remains at rest in frame S the rest frame). The situation shown is for n = 1.5 and u/ = 0.2. Speifially, as shown in Fig. 1, we will onsider a ray of light that is inident on the moving transparent nondispersive dieletri blok at a ertain angle of inidene θ 1. We will then transform to a referene frame in whih the blok is at rest. In that referene frame, we use Snell s law, sin θ 1 = nsinθ 2, 13) and well-established boundary onditions to desribe wave four-vetor after it enters the blok. Finally, we transform bak to the referene frame in whih the blok is moving, and alulate the refrated wave four-vetor in the moving blok as a funtion of the original angle of inidene. The relevant Lorentz transformations are k x = γ k x uω 2 ) 14) ω = γ ω uk x ). 15) We begin with a ray of light with wavenumber k 1 = 2π/λ 1 and angular frequeny ω 1 as observed in the lab frame S. The ray is inident upon the bottom surfae of the blok, making an angle θ 1 with the normal to the blok s surfae. The x-omponent of the wave vetor is k 1x = k 1 sinθ 1. 16) 5

6 We now use the Lorentz transformations to find the omponents of the inident wave vetor in frame S. Employing the fat that in the vauum ω 1 k 1 = ω 1 k 1 =, 17) we obtain k 1x = γ k 1x uω ) 1 = γ k 2 1 sin θ 1 uk ) 1 = γk 2 1 sinθ 1 u ) 18) ω 1 = γ ω 1 uk 1x ) = γ k 1 uk 1 sinθ 1 ) = γk 1 1 u ) sin θ 1. 19) Beause the dieletri blok is at rest in frame S, the wavelength λ 2 inside the blok is related to λ 1 by λ 2 = λ 1/n. Then the wave number k 2 within the blok at rest is k 2 = 2π λ 2 = 2π λ 1/n = nk 1. 20) Combining this with Snell s law, Eq. 13), we have ) 1 k 2 sinθ 2 = nk 1) n sinθ 1 = k 1 sinθ 1 21) or k 2x = k 1x = γk 1 sinθ 1 u ). 22) Beause the angular frequeny does not hange upon entering the blok in the rest frame S, we have Thus ω 2 = ω 1 = γk 1 1 u ) sinθ 1. 23) k 1 = ω 1 = γk 1 1 u ) sinθ 1. 24) The y -omponent of the wave vetor within the blok in frame S is obtained from k 2y k = 2 2 k 2 2x nk = 1 )2 [γk 1 sinθ 1 u )] 2. 25) Therefore k 2y = γk 1 n 2 1 u sinθ 1 ) 2 sinθ 1 u ) 2. 26) Finally, we use the inverse transformations, ) k 2x = γ k 2x + uω ) k 2y = k 2y 28) 6

7 to obtain the omponents of the wave four-vetor inside the moving blok as measured from the lab frame S. We are really interested in tan θ 2, tanθ 2 = k k 2x 2x + uω 2 = γ 2. 29) k 2y k 2y Thus sinθ 1 u ) + u tan θ 2 = γ n 2 1 u sinθ 1 1 u ) sinθ 1 ) 2 sinθ 1 u ) 30) 2 so that tan θ 2 = sinθ 1 γ n 1 2 u ) 2 sinθ 1 sinθ 1 u ). 31) 2 We thus find that the diretions of the light ray Eq. 10) and the wave vetor Eq. 31) in the moving blok are not the same. In partiular, for normal inidene θ 1 = 0) in the lab frame S, Eq. 31) shows that tanθ 2 = 0, so θ 2 = 0. That is, while the ray s veloity vetor shows a transverse relativisti drift Eq. 11), the wave vetor does not display a transverse relativisti drift. The ray s veloity vetor is in the same diretion as its Poynting vetor S, so we must onlude that the diretions of S and k diverge in the moving blok. This divergene of the diretions of S and k is aused by the diretional-dependene of the ray s veloity inside the moving blok. For the blok at rest in frame S, the omponents of the ray s veloity within the dieletri blok are v 2x = n sinθ 2 32) and v 2y = n os θ 2. 33) The veloity transformation equations provide the omponents of the ray s veloity inside the moving blok: and v 2y = γ v 2x = v 2x + u 1 + uv 2x 2 = v 2y 1 + uv 2x 2 ) = 7 n sinθ 2 + u 1 + u, 34) n sinθ 2 γ n os θ u ). 35) n sinθ 2

8 Obviously, the magnitude of the veloity of the ray in the moving blok is not onstant, but varies with diretion. IV. HUYGENS CONSTRUCTIONS The impliations of Eqs. 34) and 35) may be found by omparing the results of two Huygens onstrutions, one for a ray normally inident on a blok at rest, and the other for a ray normally inident on a moving blok. First we onsider a Huygens onstrution in the rest frame S of the blok. A wavefront is produed by a ray entering the blok at normal inidene, θ 1 = 0. If the soure of the expanding wavefront is plaed at the origin of frame S at time t = 0, the equation of the wavefront at a later time t is ) 2 x 2 + y 2 = t 2. 36) n An idential wavefront may be drawn with its soure displaed in the positive x -diretion. A horizontal line drawn tangent to the wavefronts is a line of onstant phase. As time passes, this line moves vertially upward, in the y -diretion. The wave vetor k is perpendiular to this line. The Poynting vetor, S, drawn from the soure to the point of tangeny, is learly parallel to the wave vetor, k. For the blok at rest, S and k are both in the same diretion, along the positive y -axis. Now onsider a Huygens onstrution in the lab frame S, in whih the blok is moving. A wavefront is produed by a ray entering the blok at normal inidene, θ 1 = 0. Assuming, as usual, that the origins of the frames S and S oinide at t = t = 0, the standard Lorentz transformations show that the oordinates of this wavefront in the moving blok in lab frame S must satisfy ) x ut) 2 + y 2 1 u2 = 1n t ux ) 2. 37) 2 2 For a given value of t, the oordinates x, y) are the same as those obtained by using x = v 2x t and y = v 2y t, where v 2x and v 2y are given by Eqs. 34) and 35), respetively. An idential wavefront may be drawn with its soure displaed in the positive x-diretion. Figure 2 shows two of these wavefronts. The horizontal line is tangent to the wavefronts, and so is a line of onstant phase. As time passes, this line moves vertially upward, in the y-diretion. The wave vetor k is perpendiular to this line, in the y-diretion, and so does not show a transverse relativisti drift. However the Poynting vetor S, drawn from the soure to 8

9 the point of tangeny, deviates in the positive x-diretion. The angle of deviation from the y-diretion of the line onneting the soure and tangent point is given by Eq. 10), 6 and so shows a transverse relativisti drift. This deviation of k and S is routinely found in anisotropi rystals; see, for example, Refs. 10 and 11. The onlusion is that the wave vetor k in the moving blok shows how the wavefronts move and does not show a transverse relativisti drift, while the Poynting vetor S in the moving blok shows how the momentum and move and does show a transverse relativisti drift. 3 2 wavefront k ym) 1 soure S soure S xm) FIG. 2. Two wavefronts spread out from two soures of light entering the bottom of the blok at normal inidene, as seen in the lab frame S. The horizontal line, the wavefront, is a line of onstant phase that moves vertially through the blok, in the diretion of the wave vetor k. The Poynting vetor S is direted from eah soure to the point of tangeny. The situation shown is for n = 1.5, u/ = 0.2, and t = 10 8 s. V. THE LORENTZ TRANSFORMATION OF THE MOMENTUM-ENERGY FOUR- VECTOR We now onsider the momentum of a photon in a transparent nondispersive dieletri medium. When a photon arrives at the blok s surfae, it is either ompletely refleted or ompletely transmitted into the blok. We need onsider only the transmitted photons. In the vauum in either frame, E = p, so E 1 = p 1 and E 1 = p 1. 38) 9

10 The momentum and energy of the refrated effetive photon are given by Eqs. 8) and 9), whih in the present notation are p 2 = nα p 1 39) and E 2 = nα+1 p 1 = nα+1 E 1. 40) The Lorentz transformations for the momentum-energy four-vetor are p x = γ p x ue ) 2 41) E = γ E up x ). 42) We begin with a photon with momentum p 1 and energy E 1 as observed in lab frame S. The ray is inident upon the bottom surfae of the blok, making an angle θ 1 with the normal to the blok s surfae. The x-omponent of the inident momentum is p 1x = p 1 sin θ 1. 43) We now use the Lorentz transformations to find the omponents of the inident photon momentum in frame S. Employing Eq. 38), we obtain p 1x = γ p 1x ue ) 1 = γ 2 p 1 sinθ 1 up ) 1 = γp 2 1 sinθ 1 u ) E 1 = γ E 1 up 1x ) = γ p 1 up 1 sin θ 1 ) = γp 1 44) 1 u ) sinθ 1. 45) Next, using Snell s law, Eq. 13) and Eq. 39) for the effetive momentum in the blok frame S, we have or ) 1 p 2 sinθ 2 = n α p 1) n sinθ 1 = n α 1 p 1 sinθ 1, 46) p 2x = n α 1 p 1x = n α 1 γp 1 sinθ 1 u ). 47) From Eq. 40) we have Thus E 2 = n α+1 E 1 = n α+1 γp 1 1 u ) sinθ 1. 48) p 1 = E 1 = γp 1 1 u ) sinθ 1. 49) 10

11 The y -omponent of the photon s momentum inside the blok in the rest frame S is then p 2y = p 2 2 p 2 2x = n α p 1) 2 [n α 1 γp 1 sin θ 1 u )] 2. 50) Thus so that p 2y = n 2α γ 2 p u 2 1) sinθ n 2α 2 γ 2 p 2 1 sinθ 1 u ) 2 51) p 2y = n α γp 1 1 u sinθ 1 ) 2 1 sinθ n 2 1 u ) 2. 52) Finally, we use the inverse Lorentz transformations, p 2x = γ ) p 2x + ue ) p 2y = p 2y 54) to obtain the omponents of the momentum-energy four-vetor inside the moving blok as measured from the lab frame S. As before, we are really interested in tan θ 2, Therefore tan θ 2 = γ A few lines of algebra produes the final result, tan θ 2 = p p 2x 2x + ue 2 = γ 2. 55) p 2y p 2y n α 1 sinθ 1 u ) + n α+1u 1 u ) sinθ 1 n α 1 u ) 2 sinθ 1 1 sinθ n 2 1 u ). 56) 2 n 2 1) u ) + 1 n2 u 2 sin θ tan θ 2 = γ 2 1 n 1 2 u ) 2 sin θ 1 sinθ 1 u ). 57) 2 This is idential to Eq. 10). It shows that regardless of the value of α, the effetive photon follows the trajetory of the ray and yields the transverse relativisti drift that is experimentally observed. This might have been expeted; it is ultimately a result of the relativisti self-onsisteny of our definition of the effetive photon. We therefore must turn to another argument to determine the value of α. 11

12 VI. PHOTON ENERGY CONSERVATION AND THE ABRAHAM MOMENTUM Let s return to onsidering the dieletri blok at rest. When a ray of idential monohromati photons is inident upon the surfae of the blok, the photons are either transmitted or refleted. For our non-dispersive medium, the energy of the photons is onserved; the inident energy is equal to the sum of the transmitted and refleted energies. The energy of the refleted photons is ertainly onserved beause neither the number of photons nor their frequeny hanges upon refletion. Beause none of the photons from the inident ray are absorbed, the energy of the transmitted photons must also be onserved. Therefore, sine the energy of the photon is onserved when it enters the blok at rest, Eq. 9) requires that α = 1. We must onlude that, to onserve energy, the effetive photon must have the Abraham momentum, p eff = p/n. The reader may be onerned that we may have, in our definition of an effetive photon, predetermined that its momentum must be the Abraham momentum. Might an alternative definition result in the effetive photon having the Minkowski momentum, p eff = np α = 1)? We will now show that suh an alternative definition of an effetive photon is unphysial. Speifially, we will assume that the Minkowski effetive photon has the Minkowski momentum, so p 2 = np 1, 58) and that the energy of the photon does not hange as it enters the blok at rest, E 2 = E 1 = p 1. 59) Note that we do not require that the Minkowski effetive photon move with speed /n. In this ase, we an solve for the speed of the Minkowski effetive photon in the blok at rest using Eqs. 1) and 2), p 2 E 2 = v. 60) Substituting for p 2 and E 2 shows that v = np 1 p 1 = n. 61) The Minkowski effetive photon must be superluminal, whih is unphysial. The mass of the Minkowski effetive photon omes from Eq. 4), E 2 2 p = m ) 12

13 This quikly yields m = p 1 1 n2, 63) whih is imaginary, and so is also unphysial. Nevertheless, if we insist on using this new formulation for the Minkowski effetive photon, another problem arises. Upon examining the form of the momentum four-vetor, p x, p y, p z, E/), and omparing it to the wave fourvetor, k x, k y, k x, ω/), we realize that the momentum and wave four-vetors transform in the same way under a Lorentz transformation. In the rest frame S we have p 2 = np 1 and k 2 = nk 1; E 1 = E 2 and ω 1 = ω 2; and in the vauum in both frames E 1 = p 1 and ω 1 = k 1. Indeed, we will arrive at Eq. 31) for tan θ 2, whih does not show a relativisti drift for normal inidene θ 1 = 0). Therefore we must onlude that the proposed definition of a Minkowski effetive photon is not onsistent with the experimentally observed relativisti drift. This eliminates the alternative definition of the Minkowski effetive photon. Suh an effetive photon would be superluminal, have an imaginary effetive mass, and not reprodue the observed transverse relativisti drift. This reinfores our onlusion that the effetive photon must have the Abraham momentum, p eff = p/n. Finally, we note in passing that our result disagrees with that of Wang 12, who uses a relativisti argument to onlude that the Abraham momentum and energy α = 1) do not onstitute a Lorentz four-vetor in a dieletri medium. However, Wang s analysis inorretly assumes that the momentum p and the wave-vetor k are in the same diretion in a moving blok. We have shown in Setion II that Wang s assumption is invalid. VII. CONCLUSIONS We have used relativisti arguments to define an effetive photon in a transparent nondispersive dieletri medium at rest that follows the path of the refrated ray, transports the ray s momentum and energy, and moves with speed /n. The effetive photon has a momentum given by Eq. 8) and an energy given by Eq. 9). Using a Lorentz transformation of the wave four-vetor, we have demonstrated that the Poynting vetor S and wave vetor k diverge and are not in the same diretion in a moving dieletri medium. For an effetive photon with its momentum in a dieletri medium given by Eq. 8), we alulated the transverse relativisti drift and showed that the transversedrift is independent of the value of α. However, imposing the onstraint of onservation of energy, we have shown 13

14 that, in a dieletri medium, α = 1 and that the effetive photon arries momentum p eff = p/n the Abraham momentum). Furthermore, we have shown that adopting the Minkowski momentum for an alternative definition of an effetive photon leads to results whih are unphysial and do not display the experimentally observed transverse relativisti drift. ACKNOWLEDGMENTS 1 David J. Griffiths, Introdution to Eletrodynamis, 4th Ed. Prentie-Hall, New Jersey, 2012). 2 Rihard Feynman, The Feynman Letures on Physis, Vol. II Addison Wesley, 1971). 3 Stephen M. Barnett, Resolution of the Abraham-Minkowski Dilemma, Phys. Rev. Lett. 104, p ). 4 A. Fresnel, Lettre d Augustin Fresnel à Franois Arago sur l influene du mouvement terrestre dans quelques phnomnes d optique, Annales de himie et de physique 9, pp ). 5 H. Fizeau, The Hypotheses Relating to the Luminous Aether, and an Experiment whih Appears to Demonstrate that the Motion of Bodies Alters the Veloity with whih Light Propagates Itself in their Interior, Philosophial Magazine 2, pp ). 6 Aleksandar Gjurhinovski, Aberration of Light in a Uniformly Moving Optial Medium, Am. J. Phys. 72 7), pp ). 7 R.V. Jones, Aberration of light in a moving medium, J. Phys. A. 4, pp. L1 L3 1971). 8 R.V. Jones, Fresnel aether drag in a transversely moving medium, Pro. R. So. Lond. A. 328, pp ). 9 R.V. Jones, Aether drag in a transversely moving medium, Pro. R. So. Lond. A. 345, pp ). 10 J. Leah, A.J. Wright, J.B. Götte, J.M. Girkin, L. Allen, D. Franke-Arnold, S.M. Barnett, and M.J. Padgett, Aether Drag and Moving Images, Phys. Rev. Lett. 100, p ). 11 Eugene Heht, Optis, 4th Ed. Addison Wesley, Boston, 2001). 12 Changbiao Wang, Can the Abraham Light Momentum and Energy in a Medium Constitute a Lorentz Four-Vetor? J. Mod. Phys. 4,8), pp ). 14