Download Relativistic Doppler Effect of Light and Matter Waves

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
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Relativistic Doppler Effect of Light and Matter Waves – Keeyung Lee
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−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”.
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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)
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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.
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[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
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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).