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Relativistic Doppler Effect of Light and Matter Waves

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Relativistic Doppler Effect of Light and Matter Waves 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 com- ponent 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 trans- formation 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 trans- verse 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- ôDzDGÓütoÆ<rt “DhÓüto”, Volume 61, Number 3, 2011¸ 3Z4 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 mea- sured 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.
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