Relativistic Doppler Effect of Light and Matter Waves

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New Physics: Sae Mulli (The Korean Physical Society), DOI: 10.3938/NPSM.61.222 Volume 61, Number 3, 2011¸ 3Z4, pp. 222∼226

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 ex- wave by the plane wave of single frequency is practically plaining the frequency shift of sound waves when the very useful and satisfactory in most cases, although a sound source and the observer are in relative motion. linear combination of many such plane waves is needed However light waves also show Doppler effect which is for exact representation. important in astronomy. Doppler effect is also important If a light wave of frequency ω and wave vector k is in the light emission process from atoms where spectral observed in a frame S, frequency ω0 and wave vector broadening due to recoil or collision of atoms can occur. k0 which are different from ω, k are usually observed in For the light wave, Doppler effect should be treated rel- another frame S0. In typical treatments in textbooks, ativistically although classical treatment is good enough invariance property of the phase of a wave is used as for the sound wave. Relativistic Doppler effect is usually treated relying on the the four vector property of the the basis for Doppler effect discussion. In this argument, (frequency, wave vector) combination [1–3]. the fact that the physical point which corresponds to a In this work, the photon picture of light waves has certain point such as the zero or the maximum of electric been adopted to discuss the the Doppler effect of the light field must be invariant independent of the observer is wave, using the relativistic energy momentum transfor- used to explain the relativistic invariance of the phase. mation. Especially, the transverse Doppler effect which After this fact is established, four-vector nature of the is not treated often in textbooks is discussed in detail. four component combination (ω, k) is argued as follows. In extension of such approach, Doppler effect of matter In the relativity theory, time t and position coordi- waves is also discussed. nates 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 four-vector is an invariant quantity. The scalar product electromagnetic plane wave of a single angular frequency (ik·r−ωt) of any two four-vectors Aµ, Bµ is defined as AµBµ = ω, with the electric field of the form E = E0e , A0B0 + A1B1 + A2B2 + A3B3, and can also be shown to ∗E-mail: [email protected] be an invariant [4]. Now, it could be noted that the phase -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 s ω0 1 − V/c x = (ict, x, y, z). This implies that the four component = (3) µ ω 1 + V/c vector (iω/c, k) is also a four-vector. This is a rather long procedure, but once this fact is Note that ω0 is the frequency measured by the observer established, ( ω, k ) transformation from one frame to in frame S0, receding from the photon source if V > 0, another can be easily done using the Lorentz transfor- and approaching toward the source if V < 0. mation. Such approach is good enough for the Doppler An interesting point which can be noted from Eq.(2) is effect discussion and is widely adopted in classical elec- that there is a frequency shift even when θ = π/2, which trodyamics textbooks [1–3]. corresponds to the situation when the photon propaga- However, alternative more effective discussion could tion direction is perpendicular to the observer motion. be based on the quantum concept of light waves. The Such effect which is known as the transverse Doppler photon concept of light waves can be suggested from the effect does not exist in classical physics. relativity theory. For the light wave, −(ω/c)2 + k2 = 0 The transverse Doppler effect is usually not treated in property is satisfied. This is of the same from as the detail in textbooks, although it is sometimes discussed −(E/c)2+p2 = 0 property for the momentum four-vector as a separate topic using the time dilation effect [6]. In pµ = (i(E/c), p) of a particle. This suggests association some textbooks, it is discussed based on the formula of of ω and k of the light wave with the energy E and Eq.(2), but it is presented in a misleading way [3] or in momentum p of a particle, which is the photon. a way difficult to understand [7]. To derive the Doppler effect formula using the photon When discussing the transverse effect, one may be picture, let us consider a photon with energy E and mo- tempted to put θ = π/2 in Eq.(2), which gives the fre- 0 mentum p propagating along the directionn ˆ in frame S quency shift relation ω /ω = γ [8]. This relation gives [5]. Since the photon hac no rest mass, energy and mo- higher observed frequency than the emitted frequency, mentum are related as E = pc. Therefore, the momen- whereas, the correct transverse effect formula is known as ω0/ω = 1/γ, which gives lower observed frequency. tum four-vector of can be expressed as pµ = (~ω/c)(i, nˆ). The energy E0 of this photon in frame S0 which moves Because of the symmetry principle in relativity, it is with speed V along the positive x-axis direction can be not uncommon to obtain a contradictory result such as 0 this case, which is just the opposite of the correct re- expressed as E = γ(E − V px) from the Lorentz transformation where γ = 1/p1 − (V/c)2. sult. Such problem comes up because proper definition Therefore, the observed angular frequency ω0 in frame of a physical quantity is not made [9]. For the transverse S0 can be written as Doppler effect, a typical definition goes as, “The transverse Doppler effect applies to observations made at right 0 V ω = γω(1 − nx) (1) angles to the direction of travel of the light source.” [10]. c But such definition is not good enough as is explained or, below. In discussing the transverse effect, there exist two right ω0 V = γ(1 − cos θ). (2) angles to be considered. One situation is ‘when the light ω c is emitted perpendicular to the observer motion’, and In this relation, θ is the angle between the propagating the other situation is ‘when the light is observed per- direction of the photon and the x axis in frame S. Thus pendicular to the observer motion’. Difference between the Doppler effect formula for the light wave has been these two situations may not look serious, but it turns obtained effectively using the photon concept. out that they lead to exactly the opposite results. It is The longitudinal Doppler effect formula can be ob- important to emphasize that the transverse Doppler ef- tained with the θ = 0 condition, in which case, Eq.(2) fect is defined as the situation “when the light is observed reduces to perpendicular to the observer motion”. -224- ôÇ²DGÓütoÆ

Thus it turns out that the the transverse effect for- Then the momentum transformation relation can be mula cannot be obtained from Eq.(2) alone, unless the written as p0 cos θ0 = γ[p cos θ − (V/c2)E] which gives aberration effect is also considered. Rather than follow- the relation ing such procedure, the inverse transformation relation 0 0 E = γ(E + pxV ) of the energy component is practically λ cos θ0 = γ[cos θ − (V/c)p1 + (aλ)2] (5) more useful for discussing the transverse effect. It we put λ0 as p0 = ( ω0/c) cos θ0, the following relation is obtained, 0 0 x ~ where λ = h/p , and a = E0/hc. 0 Now if the y-component momentum relation py = py ω0 1 can be expressed as p0 sin θ0 = p sin θ, which, with Eq. (5) = (4) ω V 0 γ(1 + c cos θ ) gives the following aberration formula of matter waves, in which θ0 is the angle of the photon direction measured from the x-axis in the moving(observer) frame S0. 0 1 1 tan θ = p (6) When we put θ0 = π/2 in this equation, the correct re- γ cos θ − (V/c) 1 + (aλ)2 0 lation ω /ω = 1/γ can be obtained for the transverse If a = 0, which is the zero rest mass case, this reduces Doppler effect. 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 III. MATTER WAVE DOPPLER EFFECT 0 use the energy transformation relation E = γ(E − V px) for this purpose. Since the photon can be considered as a particle with By squaring terms on each side of the energy relation zero rest mass, our discussion can be naturally extended E0 = γ(E−V p ), the matter wave Doppler effect formula to the Doppler effect of matter waves which corresponds x can be obtained as, to particles with nonzero rest mass. It is known that matter waves also have many typical properties of the r light wave, such as the interference and diffraction effect. λ V V = γ 1 − 2 cos θp1 + (aλ)2 + ( )2(cos2 θ + (aλ)2) For example, it is well known that the interference λ0 c c (7) pattern can be obtained in the two-slit interference effect When a = 0, which is the zero rest mass case, this experiment using the free electron beams. Even more formula reduces to the light wave Doppler effect formula. interesting aspect of such experiment is that the same When θ = 0, the longitudinal Doppler effect formula is pattern is observed even when single electrons were used obtained as [11]. This shows that, single electrons behave reasonably well like plane waves with well defined wavelength. λ Let us assume that particles can be viewed as mat- = γ[1 − (V/c)p1 + (aλ)2] (8) λ0 ter waves with well defined de Broglie wavelength. The corresponding to the case of the receding observer from relativistic energy-momentum relation is given as E2 = 2 2 2 the particle source when V > 0. Note that unlike the (pc) + E0 , where E0 = mc and m is the rest mass of light wave formula, λ0 and λ are not linearly related to the particle. If the particle momentum is replaced by each other in this case. the de Broglie wavelength λ = h/p, energy E can now p 2 2 Transverse Doppler effect also exists for matter waves. be expressed as E = (hc/λ) + E0 . For example, if we put θ = 0 in Eq. (7), we obtain the Let us first consider the x-component momentum 0 2 transverse Doppler effect formula, transformation relation px = γ(px − (V/c )E . If θ and θ0 are the angles of particle direction measured from the λ = γ[1 + ((V/c)aλ)2] (9) x-axis direction in frame S and S0 respectively, we can λ0 0 0 0 0 put as px = p cos θ and px = p cos θ , where p is the which is good for the case when the matter wave is di- magnitude of particle momentum measured in frame S0. rected perpendicular to the observer motion. However, Relativistic Doppler Effect of Light and Matter Waves – Keeyung Lee -225- to obtain the transverse Doppler effect formula which is with the (-) sign corresponding to the receding observer. good for the case when the matter wave is observed per- This expression is of the same form as the light wave pendicular to the observer motion, either the aberration Doppler effect formula. This means that matter waves effect or a reverse transformation relation from what we representing particles traveling close to the light speed have obtained has to be considered. behaves like light waves. Let us consider the longitudinal Doppler effect formula There has been controversy over whether the elusive only here and examine the nature of this formula for the particle neutrino has rest mass or not, and it seems two limiting cases. 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 1. Non-relativistic Limit waves should be similar to the light wave case.

In the non-relativistic limit, aλ >> 1 and γ ≈ 1 condition is satisfied, and the following relation is ob- IV. CONCLUSION tained,(when V < v), We have discussed the light wave Doppler effect using λ v ± V ≈ (10) relativistic energy momentum transformation of the pho- λ0 v ton, discussing in detail on the transverse effect which with the (-) sign corresponding to the receding observer. can cause confusion if not carefully handled. We have In this formula, v denotes the particle speed. We believe also considered the matter wave Doppler effect and have this relation could be checked by experiments. For exam- shown that the matter wave Doppler effect formula con- ple in the two slit experiment using the electron source, verges to the light wave formula in the relativistic limit. ‘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 ACKNOWLEDGEMENT which depends independently on the particle speed v and the observer speed V . This is due to the non-relativistic This work was supported by the Inha University Re- limit approximation. search fund. 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 classi- REFERENCES cal sound wave formula for the ‘source at rest and the observer moving’ situation. Unfortunately, the ‘source [1] J. D. Jackson, 1975, Classical Electrodynamics, 2nd moving and the observer at rest’ situation formula for ed. (J. Wiley, New York). the sound wave cannot be obtained from our relativis- [2] W. K. H. Panofsky and M. Phillips, 1969, Classical tic formula, since proper dispersion relation of the wave Electricity and Magnetism (Addison-Wesley, Lon- does not exist for this situation. don). [3] H. C. Ohanian, 1988, Classical Electrodynamics (Al- lyn and Bacon, London). 2. Relativistic Limit [4] W. Rindler, 1991, Introduction to Special Relativity, 2nd ed. (Oxford University Press). In the relativistic limit, (mc/h)λ << 1 condition is [5] Conceptually a photon, which is a localized object, satisfied, and the following approximate relation for the should be represented by a linear combination of matter wave Doppler effect is obtained, many plane waves. But for practical purpose, a photon can be seen as a plane wave of single frequency λ ≈ γ(1 ± V/c) (11) λ0 which dominates over other plane wave components. -226- ôÇ²DGÓütoÆ

[6] A typical derivation of the transverse effect employs given in Eq.(2) and discuss the transverse effect as the time dilation effect. For such derivation, let us a special case when θ = π/2 which leads to the re- consider the situation where a source emits waves versed result. To explain this reversed result, the au- with frequency ω in frame S. Then let us consider thor then assumes that ω0 is the emitted frequency another frame S0 which is moving with speed V rel- from a source at rest in frame S0, which is not right. ative to the frame S. Let the emitted wave be as- [9] For example, let us consider the length contraction sumed as a plane wave and the observer in S0 travels 0 perpendicular plane wave propagation direction(this effect. When the frame S is moving with speed V point is very important in this discussion). From the along the x-axis relative to the frame S, the x coor- Lorentz transformation the time in the two frames dinates on each frame are related as x0 = γ(x − V t). 0 2 0 are connected as t2 = γ(t2 − (V/c )x2 and t1 = Let the two end positions of a stick in the two frames 2 0 0 γ(t1 −(V/c )x1. Therefore we get t −t = γ(t2 −t1), 0 0 2 1 are related as x2 = γ(x2 − V t) and x1 = γ(x1 − V t). if x2 = x1. It is important to note here that t2 − t1 From this, the difference of the two end positions can is the time difference measured by a clock at rest 0 0 be found to have the relation x2 − x1 = γ(x2 − x1). in frame S. If the source frame is taken as the rest At this stage, if x0 − x0 is defined as the length in frame, 1 second in the source frame corresponds to γ 2 1 frame S0, ‘length expansion’ effect is observed in- seconds in the moving observer frame. Therefore, the moving observer counts ω number of wave crests per stead of the ‘length contraction’, which is a con- time interval γ which gives the observed frequency tradictory result. Such confusion arose because the ω/γ. In this discussion, time interval relation in the ‘length’, which has to be defined as the difference of two frames must be carefully handled. One could two positions measured simultaneously in relativity easily be lead a contradictory result if not careful theory has not been defined properly. on this point. [10] C. Kittel, W. D. Knight, M. A. Ruderman Mechan- [7] L. Landau and Lifshitz, 1975, The Classical theory ics, Berkeley Physics Course vol. 1 (McGraw-Hill, of Fields (Pergamon Press, New York). New York). [8] In the fine textbook of ref. [3], the author explains the transverse Doppler effect in detail, but in a mis- [11] A. Tonomura, J. Endo, T. Matsuda, T. Kawasaki leading way. The author derives the formula as is and H. Ezawa, Amer. J. Phys. 57, 117 (1989).