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New Physics: Sae Mulli (The Korean Physical Society), Volume 61, Number 3, 2011¸ 3 4, pp. 222∼226 Z DOI: 10.3938/NPSM.61.222 Relativistic Doppler Effect of Light and Matter Waves Keeyung Lee∗ Graduate School of Education, Inha University, Incheon 402-751 (Received 12 October 2010 : revised 6 December 2010 : accepted 17 January 2011) A relativistic Doppler effect on light waves is discussed based on the Lorentz transformation of the energy-momentum four vector of the photon. The transverse Doppler effect is discussed in detail. The Doppler effect on matter waves is also considered. In the limiting case of negligible rest mass, the matter wave Doppler effect is shown to converge to a light wave effect. PACS numbers: 01.40, 03.40, 03.75 Keywords: Relativity, Doppler effect, Matter wave I. INTRODUCTION where k is the wave vector. In this expression, k · r − ωt is the phase of the wave. Such representation of a light Doppler effect is introduced in classical physics for explaining the frequency shift of sound waves when the sound source and the observer are in relative motion. However light waves also show Doppler effect which is important in astronomy. Doppler effect is also important in the light emission process from atoms where spectral broadening due to recoil or collision of atoms can occur. For the light wave, Doppler effect should be treated relativistically although classical treatment is good enough for the sound wave. Relativistic Doppler effect is usually treated relying on the the four vector property of the (frequency, wave vector) combination [1–3]. In this work, the photon picture of light waves has been adopted to discuss the the Doppler effect of the light wave, using the relativistic energy momentum transformation. Especially, the transverse Doppler effect which is not treated often in textbooks is discussed in detail. In extension of such approach, Doppler effect of matter waves is also discussed. wave by the plane wave of single frequency is practically very useful and satisfactory in most cases, although a linear combination of many such plane waves is needed for exact representation. If a light wave of frequency ω and wave vector k is observed in a frame S, frequency ω 0 and wave vector k0 which are different from ω, k are usually observed in another frame S 0 . In typical treatments in textbooks, invariance property of the phase of a wave is used as the basis for Doppler effect discussion. In this argument, the fact that the physical point which corresponds to a certain point such as the zero or the maximum of electric field must be invariant independent of the observer is used to explain the relativistic invariance of the phase. After this fact is established, four-vector nature of the four component combination (ω, k) is argued as follows. In the relativity theory, time t and position coordinates r form a space-time four-vector xµ = (ict, x, y, z), which means that the four components of this vector transform according to the Lorentz transformation. An II. LIGHT WAVE DOPPLER EFFECT important property of the four-vector is that the scalar product of a four-vector by itself, or the “square” of a A light wave is usually represented as a propagating electromagnetic plane wave of a single angular frequency ω, with the electric field of the form E = E0 e(ik·r−ωt) , four-vector is an invariant quantity. The scalar product of any two four-vectors Aµ , Bµ is defined as Aµ Bµ = A0 B0 + A1 B1 + A2 B2 + A3 B3 , and can also be shown to ∗ E-mail: be an invariant [4]. Now, it could be noted that the phase [email protected] -222- Relativistic Doppler Effect of Light and Matter Waves – Keeyung Lee -223- −ωt + k · r represents the scalar product of a four component vector (iω/c, k) with the space-time four-vector xµ = (ict, x, y, z). This implies that the four component vector (iω/c, k) is also a four-vector. This is a rather long procedure, but once this fact is established, ( ω, k ) transformation from one frame to another can be easily done using the Lorentz transformation. Such approach is good enough for the Doppler effect discussion and is widely adopted in classical electrodyamics textbooks [1–3]. However, alternative more effective discussion could be based on the quantum concept of light waves. The photon concept of light waves can be suggested from the relativity theory. For the light wave, −(ω/c)2 + k 2 = 0 property is satisfied. This is of the same from as the −(E/c)2 +p2 = 0 property for the momentum four-vector pµ = (i(E/c), p) of a particle. This suggests association of ω and k of the light wave with the energy E and momentum p of a particle, which is the photon. To derive the Doppler effect formula using the photon picture, let us consider a photon with energy E and momentum p propagating along the direction n̂ in frame S [5]. Since the photon hac no rest mass, energy and momentum are related as E = pc. Therefore, the momentum four-vector of can be expressed as pµ = (~ω/c)(i, n̂). The energy E 0 of this photon in frame S 0 which moves with speed V along the positive x-axis direction can be expressed as E 0 = γ(E − V px ) from the Lorentz transp formation where γ = 1/ 1 − (V /c)2 . Therefore, the observed angular frequency ω 0 in frame S 0 can be written as ω 0 = γω(1 − V nx ) c (1) or, ω0 V = γ(1 − cos θ). ω c (2) In this relation, θ is the angle between the propagating direction of the photon and the x axis in frame S. Thus the Doppler effect formula for the light wave has been obtained effectively using the photon concept. The longitudinal Doppler effect formula can be obtained with the θ = 0 condition, in which case, Eq.(2) reduces to ω0 = ω s 1 − V /c 1 + V /c (3) Note that ω 0 is the frequency measured by the observer in frame S 0 , receding from the photon source if V > 0, and approaching toward the source if V < 0. An interesting point which can be noted from Eq.(2) is that there is a frequency shift even when θ = π/2, which corresponds to the situation when the photon propagation direction is perpendicular to the observer motion. Such effect which is known as the transverse Doppler effect does not exist in classical physics. The transverse Doppler effect is usually not treated in detail in textbooks, although it is sometimes discussed as a separate topic using the time dilation effect [6]. In some textbooks, it is discussed based on the formula of Eq.(2), but it is presented in a misleading way [3] or in a way difficult to understand [7]. When discussing the transverse effect, one may be tempted to put θ = π/2 in Eq.(2), which gives the frequency shift relation ω 0 /ω = γ [8]. This relation gives higher observed frequency than the emitted frequency, whereas, the correct transverse effect formula is known as ω 0 /ω = 1/γ, which gives lower observed frequency. Because of the symmetry principle in relativity, it is not uncommon to obtain a contradictory result such as this case, which is just the opposite of the correct result. Such problem comes up because proper definition of a physical quantity is not made [9]. For the transverse Doppler effect, a typical definition goes as, “The transverse Doppler effect applies to observations made at right angles to the direction of travel of the light source.” [10]. But such definition is not good enough as is explained below. In discussing the transverse effect, there exist two right angles to be considered. One situation is ‘when the light is emitted perpendicular to the observer motion’, and the other situation is ‘when the light is observed perpendicular to the observer motion’. Difference between these two situations may not look serious, but it turns out that they lead to exactly the opposite results. It is important to emphasize that the transverse Doppler effect is defined as the situation “when the light is observed perpendicular to the observer motion”. -224- Dz ô GÓ D to ü Æ < rt “D hÓ to ü ”, Volume 61, Number 3, 2011¸ 3 4 Z Thus it turns out that the the transverse effect formula cannot be obtained from Eq.(2) alone, unless the aberration effect is also considered. Rather than following such procedure, the inverse transformation relation E = γ(E 0 + p0x V ) of the energy component is practically more useful for discussing the transverse effect. It we put as p0x = (~ω 0 /c) cos θ0 , the following relation is obtained, Then the momentum transformation relation can be written as p0 cos θ0 = γ[p cos θ − (V /c2 )E] which gives the relation 1 ω0 = ω γ(1 + Vc cos θ0 ) (4) in which θ0 is the angle of the photon direction measured from the x-axis in the moving(observer) frame S 0 . When we put θ0 = π/2 in this equation, the correct relation ω 0 /ω = 1/γ can be obtained for the transverse Doppler effect. III. MATTER WAVE DOPPLER EFFECT Since the photon can be considered as a particle with zero rest mass, our discussion can be naturally extended to the Doppler effect of matter waves which corresponds to particles with nonzero rest mass. It is known that matter waves also have many typical properties of the light wave, such as the interference and diffraction effect. For example, it is well known that the interference pattern can be obtained in the two-slit interference effect experiment using the free electron beams. Even more interesting aspect of such experiment is that the same pattern is observed even when single electrons were used [11]. This shows that, single electrons behave reasonably well like plane waves with well defined wavelength. Let us assume that particles can be viewed as matter waves with well defined de Broglie wavelength. The relativistic energy-momentum relation is given as E 2 = (pc)2 + E02 , where E0 = mc2 and m is the rest mass of the particle. If the particle momentum is replaced by the de Broglie wavelength λ = h/p, energy E can now p be expressed as E = (hc/λ)2 + E02 . Let us first consider the x-component momentum transformation relation p0x = γ(px − (V /c2 )E . If θ and θ0 are the angles of particle direction measured from the x-axis direction in frame S and S 0 respectively, we can put as px = p cos θ and p0x = p0 cos θ0 , where p0 is the magnitude of particle momentum measured in frame S 0 . p λ 0 cos θ = γ[cos θ − (V /c) 1 + (aλ)2 ] λ0 (5) where λ0 = h/p0 , and a = E0 /hc. Now if the y-component momentum relation p0y = py can be expressed as p0 sin θ0 = p sin θ, which, with Eq. (5) gives the following aberration formula of matter waves, tan θ0 = 1 1 p γ cos θ − (V /c) 1 + (aλ)2 (6) If a = 0, which is the zero rest mass case, this reduces to the light wave aberration formula. The Doppler effect formula for matter waves could be obtained if we apply this relation in Eq.(5). However, it is more effective to use the energy transformation relation E 0 = γ(E − V px ) for this purpose. By squaring terms on each side of the energy relation 0 E = γ(E−V px ), the matter wave Doppler effect formula can be obtained as, λ =γ λ0 r p V V cos θ 1 + (aλ)2 + ( )2 (cos2 θ + (aλ)2 ) c c (7) When a = 0, which is the zero rest mass case, this formula reduces to the light wave Doppler effect formula. When θ = 0, the longitudinal Doppler effect formula is obtained as 1−2 p λ = γ[1 − (V /c) 1 + (aλ)2 ] λ0 (8) corresponding to the case of the receding observer from the particle source when V > 0. Note that unlike the light wave formula, λ0 and λ are not linearly related to each other in this case. Transverse Doppler effect also exists for matter waves. For example, if we put θ = 0 in Eq. (7), we obtain the transverse Doppler effect formula, λ = γ[1 + ((V /c)aλ)2 ] λ0 (9) which is good for the case when the matter wave is directed perpendicular to the observer motion. However, Relativistic Doppler Effect of Light and Matter Waves – Keeyung Lee to obtain the transverse Doppler effect formula which is good for the case when the matter wave is observed perpendicular to the observer motion, either the aberration effect or a reverse transformation relation from what we have obtained has to be considered. Let us consider the longitudinal Doppler effect formula only here and examine the nature of this formula for the two limiting cases. 1. Non-relativistic Limit In the non-relativistic limit, aλ >> 1 and γ ≈ 1 condition is satisfied, and the following relation is obtained,(when V < v), v±V λ ≈ λ0 v (10) with the (-) sign corresponding to the receding observer. In this formula, v denotes the particle speed. We believe this relation could be checked by experiments. For example in the two slit experiment using the electron source, ‘two slit + screen’ set could move toward or away from the electron source to measure the de Broglie wavelength shift. It is noted that this formula contains quantities which depends independently on the particle speed v and the observer speed V . This is due to the non-relativistic limit approximation. Any relativistic result should converge to the classical result at low speeds. It can be seen that the low speed limit formula of Eq.(9) is identical to the classical sound wave formula for the ‘source at rest and the observer moving’ situation. Unfortunately, the ‘source moving and the observer at rest’ situation formula for the sound wave cannot be obtained from our relativistic formula, since proper dispersion relation of the wave does not exist for this situation. 2. Relativistic Limit In the relativistic limit, (mc/h)λ << 1 condition is satisfied, and the following approximate relation for the matter wave Doppler effect is obtained, λ ≈ γ(1 ± V /c) λ0 (11) -225- with the (-) sign corresponding to the receding observer. This expression is of the same form as the light wave Doppler effect formula. This means that matter waves representing particles traveling close to the light speed behaves like light waves. There has been controversy over whether the elusive particle neutrino has rest mass or not, and it seems to have settled down in favor of non-zero rest mass. Whether the neutrino has negligible rest mass or not, our result shows that Doppler effect of neutrino matter waves should be similar to the light wave case. IV. CONCLUSION We have discussed the light wave Doppler effect using relativistic energy momentum transformation of the photon, discussing in detail on the transverse effect which can cause confusion if not carefully handled. We have also considered the matter wave Doppler effect and have shown that the matter wave Doppler effect formula converges to the light wave formula in the relativistic limit. ACKNOWLEDGEMENT This work was supported by the Inha University Research fund. REFERENCES [1] J. D. Jackson, 1975, Classical Electrodynamics, 2nd ed. (J. Wiley, New York). [2] W. K. H. Panofsky and M. Phillips, 1969, Classical Electricity and Magnetism (Addison-Wesley, London). [3] H. C. Ohanian, 1988, Classical Electrodynamics (Allyn and Bacon, London). [4] W. Rindler, 1991, Introduction to Special Relativity, 2nd ed. (Oxford University Press). [5] Conceptually a photon, which is a localized object, should be represented by a linear combination of many plane waves. But for practical purpose, a photon can be seen as a plane wave of single frequency which dominates over other plane wave components. -226- [6] A typical derivation of the transverse effect employs the time dilation effect. For such derivation, let us consider the situation where a source emits waves with frequency ω in frame S. Then let us consider another frame S 0 which is moving with speed V relative to the frame S. Let the emitted wave be assumed as a plane wave and the observer in S 0 travels perpendicular plane wave propagation direction(this point is very important in this discussion). From the Lorentz transformation the time in the two frames are connected as t02 = γ(t2 − (V /c2 )x2 and t01 = γ(t1 −(V /c2 )x1 . Therefore we get t02 −t01 = γ(t2 −t1 ), if x2 = x1 . It is important to note here that t2 − t1 is the time difference measured by a clock at rest in frame S. If the source frame is taken as the rest frame, 1 second in the source frame corresponds to γ seconds in the moving observer frame. Therefore, the moving observer counts ω number of wave crests per time interval γ which gives the observed frequency ω/γ. In this discussion, time interval relation in the two frames must be carefully handled. One could easily be lead a contradictory result if not careful on this point. [7] L. Landau and Lifshitz, 1975, The Classical theory of Fields (Pergamon Press, New York). [8] In the fine textbook of ref. [3], the author explains the transverse Doppler effect in detail, but in a misleading way. The author derives the formula as is Dz ô GÓ D to ü Æ < rt “D hÓ to ü ”, Volume 61, Number 3, 2011¸ 3 4 Z given in Eq.(2) and discuss the transverse effect as a special case when θ = π/2 which leads to the reversed result. To explain this reversed result, the author then assumes that ω 0 is the emitted frequency from a source at rest in frame S 0 , which is not right. [9] For example, let us consider the length contraction effect. When the frame S 0 is moving with speed V along the x-axis relative to the frame S, the x coordinates on each frame are related as x0 = γ(x − V t). Let the two end positions of a stick in the two frames are related as x02 = γ(x2 − V t) and x01 = γ(x1 − V t). From this, the difference of the two end positions can be found to have the relation x02 − x01 = γ(x2 − x1 ). At this stage, if x02 − x01 is defined as the length in frame S 0 , ‘length expansion’ effect is observed instead of the ‘length contraction’, which is a contradictory result. Such confusion arose because the ‘length’, which has to be defined as the difference of two positions measured simultaneously in relativity theory has not been defined properly. [10] C. Kittel, W. D. Knight, M. A. Ruderman Mechanics, Berkeley Physics Course vol. 1 (McGraw-Hill, New York). [11] A. Tonomura, J. Endo, T. Matsuda, T. Kawasaki and H. Ezawa, Amer. J. Phys. 57, 117 (1989).