Relativity and mathematical tools: Waves in moving media
Martin McCalla兲
Department of Physics, Imperial College London, United Kingdom SW7 2AZ
Dan Censor
Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, Israel
共Received 1 February 2007; accepted 14 July 2007兲
We discuss the electromagnetic wave equation and dispersion relation for moving media as an
example of the challenge of integrating vectors and dyadics and their covariant counterparts into the
teaching of special relativity. We discuss two methods of deriving the dispersion relation and its
associated wave equation. One is a direct approach for which the starting point is Maxwell’s
equations combined with the Minkowski constitutive relations in the laboratory frame ⌫ in which
the medium is moving uniformly. The second approach, starting in the medium’s rest frame ⌫⬘,
establishes the invariance properties of the dispersion relation, and then proceeds to derive its
counterpart in ⌫. Application to the calculation of the group velocity in each frame is also
discussed. © 2007 American Association of Physics Teachers.
关DOI: 10.1119/1.2772281兴
I. INTRODUCTION
Unlike most areas of physics such as mechanics and electricity and magnetism, which are amenable to simple laboratory demonstrations, learning special relativity is a voyage
into the abstract. No amount of hand waving can by itself
provide students with “understanding the physics.” Here the
vehicle into the abstract is the language of mathematics.
However, even students with a good background can become
lost and fail to integrate their mathematical knowledge with
the physics of interest.
This paper aims at closing the gap between elementary
introductory texts such as Ref. 1 and more advanced
presentations.2 As an example, we will discuss electromagnetic waves in simple media, and the relativistic invariance
of the dispersion relation and associated wave operator. The
prototypical student we have in mind has completed introductory courses in calculus, vector analysis, and matrix algebra, a course in electromagnetic theory at least at the level of
Maxwell’s equations, and preferably an introductory course
on modern physics.
II. DISPERSION RELATIONS IN MOVING MEDIA:
PROBLEM STRATEGY
We will discuss the dispersion relations G⬘共k⬘ , ␻⬘兲 = 0 and
G共k , ␻兲 = 0 in the frame where the dielectric medium is at
rest 共which we denote by ⌫⬘兲, and in which the medium is in
motion 共denoted by ⌫兲. The dispersion relations are scalar
algebraic expressions relating the frequency ␻ and the wave
vector k. The relativistic invariance of the dispersion relations, namely G⬘共k⬘ , ␻⬘兲 = G共k , ␻兲 = 0, is our main theme.
Dispersion relations can be used to solve wave problems
by tracing rays in the medium under consideration. A ray is
the locus whose tangent at any point is directed along the
group velocity vector.3,4 Tutorial texts on Hamiltonian ray
theory, wave packets, and group velocity are available.5,6
We show how the wave operator and the dispersion relations are related via Fourier transformation. The problem of
finding the allowed electromagnetic modes in a medium is
then reduced from solving the differential wave equation to
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Am. J. Phys. 75 共12兲, December 2007
http://aapt.org/ajp
solving the algebraic dispersion relation. Furthermore, the
dispersion relation is associated with the free-space Green’s
function, a subject usually discussed in advanced texts such
as Ref. 3. The present discussion provides a relatively simple
introduction to these ideas. To trace rays in a medium in
motion, the dispersion relation G共k , ␻兲 = 0 must be found in
the laboratory frame ⌫. The issue arises as to whether this
relation can be quickly determined from the dispersion relation G⬘共k⬘ , ␻⬘兲 = 0 in the medium’s rest frame ⌫⬘. It has been
shown7 that the dispersion relation is a scalar invariant in the
sense that G共k , ␻兲 = G⬘共k⬘ , ␻⬘兲 = 0 where k and ␻ are related
to k⬘ and ␻⬘ by the relativistic Lorentz transformations for
the spectral parameters, often referred to as the relativistic
Fresnel drag and Doppler effects, respectively.8
We consider two methods. In Sec. III the Minkowski constitutive relations5,9 prescribing the relations between the
electromagnetic field vectors in a moving medium are derived. In Sec. IV Maxwell’s equations and the Minkowski
constitutive relations facilitate the derivation of G共k , ␻兲 and
the associated scalar wave equation working directly in ⌫. It
is shown that the dispersion relation is factorizable, thus significantly simplifying the algebra. In Sec. V we start with
G⬘共k⬘ , ␻⬘兲 = 0 in ⌫⬘ and establish the invariance G共k⬘ , ␻⬘兲
= G⬘共k⬘ , ␻⬘兲 = 0. Finally we implement the spectral Lorentz
transformations to obtain G共k⬘ , ␻⬘兲 = G共k⬘关␻ , k兴 , ␻⬘关␻ , k兴兲,
the dispersion relation in ⌫ in terms of the wave parameters
in that frame. As expected, we obtain the same result for
G共k , ␻兲 = 0 using both methods. In Sec. VI we calculate the
group velocity from the dispersion relations. In ⌫ the medium is anisotropic due to the preferred direction prescribed
by the velocity. Nevertheless, the refractive indices are
shown to be degenerate, and in this sense there is no birefringence. Section VII provides an example for calculating
the velocity of a moving medium using the drag effect formula, based on the idea of the Fizeau experiment.10
In Sec. VIII we introduce the use of covariant methods,
appropriate for more advanced students. Using these methods we summarize the transformation properties of wave frequency and wave vector in the presence of media, and also
the dispersion invariance as applied to vacuum.
© 2007 American Association of Physics Teachers
1134
III. RELATIVISTIC ELECTRODYNAMICS AND THE
MINKOWSKI CONSTITUTIVE RELATIONS
We are interested in transformations between the frame ⌫,
in which a moving medium is observed and the frame ⌫⬘, in
which the same medium is at rest. Observed from ⌫, the
medium moves with velocity v. The transformation of coordinates between ⌫ and ⌫⬘ is facilitated by the Lorentz transformation for the space-time coordinates11
r⬘ = r⬘共r,t兲 = Ũ · 共r − vt兲,
共1a兲
t⬘ = t⬘共r,t兲 = ␥共t − v · r/c2兲,
共1b兲
where Ũ = Ĩ + 共␥ − 1兲v̂v̂, ␥ = 共1 − ␤2兲−1/2, ␤ = v / c, c−2 = ␮0⑀0, v̂
= v / v, and v = 兩v兩. The tilde denotes a dyadic, and c the speed
of light in vacuum. In Cartesian components the algebraic
product ã = bc is defined via
ã = bc = 共x̂bx + ŷby + ẑbz兲共x̂cx + ŷcy + ẑcz兲
共2a兲
=x̂x̂bxcx + x̂ŷbxcy + x̂ẑbxcz + ŷx̂bycx + ¯ + ẑẑbzcz .
共2b兲
The dyadic Ũ in Eq. 共1a兲 contains Ĩ, replicating the vector it
multiplies. In matrix notation we have
冢
axx
ayx
azx
axy
ayy
azy
冣冢冣
冢
axz
bx
ayz = by 共cx cy cz 兲
azz
bz
b xc x
= b ycx
b zc x
b xc y
bycy
b zc y
冣
b xc z
b ycz ,
b zc z
共3兲
or more simply aij = bic j, where i, j = x , y , z. The dyadic and
matrix operations are not commutative in general, so their
order must be preserved.
For regions that do not contain free charges and currents,
Maxwell’s equations in ⌫⬘ are
⳵ r⬘ ⫻ E ⬘ = − ⳵ t⬘B ⬘ ,
共4a兲
⳵ r⬘ ⫻ H ⬘ = ⳵ t⬘D ⬘ ,
共4b兲
⳵r⬘ · D⬘ = 0,
共4c兲
⳵r⬘ · B⬘ = 0.
共4d兲
The notation ⳵r⬘ instead of the symbol ⵜ emphasizes the
space coordinates involved.11 Einstein’s principle of
relativity12 postulates that Maxwell’s equations are form invariant so that in ⌫ we have
⳵r ⫻ E = − ⳵tB,
共5a兲
⳵r ⫻ H = ⳵tD,
共5b兲
⳵r · D = 0,
共5c兲
⳵r · B = 0.
共5d兲
The derivatives in Eqs. 共4兲 and 共5兲 are related according to
⳵r⬘ = ⳵r⬘共⳵r, ⳵t兲 = Ũ · 共⳵r + v⳵t/c 兲,
2
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Am. J. Phys., Vol. 75, No. 12, December 2007
共6a兲
⳵t⬘ = ⳵t⬘共⳵r, ⳵t兲 = ␥共⳵t + v · ⳵r兲.
共6b兲
Equations 共4兲–共6兲 determine that the fields transform according to
E⬘ = Ṽ · 共E + v ⫻ B兲,
B⬘ = Ṽ · 共B − v ⫻ E/c2兲
D⬘ = Ṽ · 共D + v ⫻ H/c2兲,
H⬘ = Ṽ · 共H − v ⫻ D兲
Ṽ = ␥Ĩ + 共1 − ␥兲ṽṽ,
共7a兲
共7b兲
共7c兲
where Ṽ in Eq. 共7兲 multiplies the field components perpendicular to v by ␥. In ⌫⬘, where the medium is at rest, the
simplest constitutive relations are chosen as
D ⬘ = ⑀ E ⬘,
B⬘ = ␮H⬘ ,
共8兲
with constant scalar parameters ⑀ and ␮. The substitution of
Eq. 共7兲 into Eq. 共8兲 yields the Minkowski constitutive
relations5,9
D + v ⫻ H/c2 = ⑀共E + v ⫻ B兲,
共9a兲
B − v ⫻ E/c2 = ␮共H − v ⫻ D兲.
共9b兲
Maxwell’s equations, Eq. 共5兲, together with Eq. 共9兲, facilitate
the derivation of the wave equation and the dispersion relation in ⌫ in which the medium is moving. In ⌫ the medium
is anisotropic because of the preferred direction introduced
by the medium’s velocity.
IV. DERIVATION OF THE DISPERSION RELATION
IN ⌫
If we assume all fields in ⌫ vary as exp i共k · r − ␻t兲, we
have from Eqs. 共4兲 and 共9兲
k ⫻ E = ␻B,
共10a兲
k ⫻ H = − ␻D,
共10b兲
D + v ⫻ H/c2 = ⑀共E + v ⫻ B兲,
共10c兲
B − v ⫻ E/c2 = ␮共H − v ⫻ D兲.
共10d兲
The switch between the spectral and spatiotemporal domain
is achieved via the substitutions
ik ⇔ ⳵r,
− i␻ ⇔ ⳵t ,
共11兲
and by appropriately interchanging spectral and spatiotemporal functions and their argument variables 关for example,
E共k , ␻兲 ⇔ E共r , t兲兴 via a 4-dim. Fourier transformation. Using
Eq. 共10兲 to eliminate D and B from Eq. 共9兲 yields
− k ⫻ H + ␻v ⫻ H/c2 = ⑀共␻E + v ⫻ 共k ⫻ E兲兲,
共12a兲
k ⫻ E − ␻v ⫻ E/c2 = ␮共␻H + v ⫻ 共k ⫻ H兲兲,
共12b兲
reducing the system to six scalar equations and six scalar
field component variables. In terms of dyadics, Eq. 共12兲 becomes
˜ · E,
− ˜␣ · H = ⑀␤
共13a兲
˜ · H,
˜␣ · E = ␮␤
共13b兲
where
Martin McCall and Dan Censor
1135
˜␣ = 共k − ␻v/c2兲 ⫻ Ĩ = k̄ ⫻ Ĩ,
共14a兲
k̄ = k − ␻v/c2 ,
共14b兲
˜␤ = ␻Ĩ + v ⫻ 共k ⫻ Ĩ兲 = ␻
¯ Ĩ + kv,
共14c兲
¯ = ␻ − v · k,
␻
共14d兲
v ⫻ 共k ⫻ Ĩ兲 = kv − 共v · k兲Ĩ,
共14e兲
where we have used the vector identity v ⫻ 共k ⫻ F兲
= k共v · F兲 − 共v · k兲F.
To proceed we must find the inverse of one of the dyadics
in Eq. 共13兲. We try
˜ −1 = aĨ + bkv,
␤
共15兲
˜ −1␤
˜ = Ĩ we obtain
and from the condition ␤
¯ −1,
a=␻
b = − a␻−1 .
共16兲
The equations for E and H in Eq. 共13兲 can then be written as
H = ˜␥ · E/␮ ,
共17a兲
E = − ˜␥ · H/⑀ ,
共17b兲
G共k, ␻兲 = det关G̃共k, ␻兲兴 = det关共Ĩ + iC⬘˜␥兲兴det关共Ĩ − iC⬘˜␥兲兴
共21兲
= 0,
yielding a scalar relation between k and ␻. In Eq. 共21兲 we
exploit the theorem that the determinant of a product equals
the product of the determinants.
The expression for k̄ in Eq. 共14b兲 indicates that in the
presence of a moving medium the wave vector is modified
by a vector component in the direction of the velocity. In the
following we will recognize this modification as the Fresnel
drag effect. Without reducing the generality of the analysis,
we now choose the directions as
v = vŷ,
k = kxx̂ + kyŷ.
We substitute Eq. 共22兲 into Eq. 共17c兲 and obtain
˜␥ = ␻
¯ −1关k̄ ⫻ Ĩ + ␻−1kv · 共k ⫻ Ĩ兲兴
共23a兲
¯ −1关共kx共ẑŷ − ŷx̂兲 + k̄y共x̂ẑ − ẑx̂兲兲
=␻
− ␻−1v共k2x x̂ẑ + kxkyŷẑ兲兴,
共23b兲
where k̄y = ky − ␻v / c2. Thus
¯ 兲2共k2x /␥2 − k̄2y 兲.
det关Ĩ + iC⬘˜␥兴 = det关Ĩ − iC⬘˜␥兴 = 1 − 共C⬘/␻
共24兲
where
˜␥ = ␻
¯ −1k̄ ⫻ Ĩ + ␻−1kv · 共k ⫻ Ĩ兲,
共17c兲
The relations in Eqs. 共17a兲 and 共17b兲 comprise six scalar
equations for six unknown scalar field components. We obtain the wave equations on the field vectors E, H from Eq.
共17兲 as
G̃ · E = 0,
共18a兲
G̃ · H = 0,
共18b兲
where
G̃共k, ␻兲 = 共Ĩ + iC⬘˜␥兲 · 共Ĩ − iC⬘˜␥兲,
共18c兲
where C⬘ = ⑀␮. The vector equations for both E and H are
thus identical. Each constitutes a system of three homogeneous algebraic equations for three scalar unknowns in terms
of k and ␻. Once E and H are determined, the associated D
and B can be computed using Eq. 共10兲.
To return to the spatiotemporal domain requires the substitution via Eq. 共11兲 so that G̃ in Eq. 共18兲 becomes G̃共⳵r , ⳵t兲,
and we have
−2
G̃共⳵r, ⳵t兲 · F共r,t兲 = 0,
共19兲
where F = E , H. In Eq. 共19兲 we now have vector wave equations for the fields E and H. Because an arbitrary wave function can be recast as a plane wave integral, F = F共r , t兲 now
represents an arbitrary solution of the wave equation, not
merely a plane wave. If v = 0, Eq. 共19兲 becomes
共⳵2t − C⬘2⳵2r 兲F共r,t兲 = 0,
共20兲
which is the Helmholtz wave equation for a simple medium
at rest.
The dispersion equation is obtained from Eq. 共18兲 according to
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共22兲
Am. J. Phys., Vol. 75, No. 12, December 2007
Finally we obtain the dispersion relation in the form
¯ 2 − C⬘2共k2x + ␥2k̄2y 兲 = 0.
G共k, ␻兲 = ␥2␻
共25兲
The substitution of Eq. 共11兲 into Eq. 共25兲 yields the scalar
wave operator as in Eq. 共19兲.
V. DISPERSION RELATION IN ⌫⬘
AND THE INVARIANCE PROPERTY
The other approach for deriving the dispersion relation
and the associated scalar wave operator is to start with the
wave equation in ⌫⬘. We assume that the fields in ⌫⬘ vary as
exp i共k⬘ · r⬘ − ␻⬘t⬘兲 so that analogous to Eqs. 共18兲 and 共19兲 we
have
2
关Ĩ⳵t⬘ + C⬘2⳵r⬘ ⫻ 共⳵r⬘ ⫻ Ĩ兲兴 · F⬘ = 共C⬘2k⬘2 − ␻⬘2兲F⬘ = 0,
共26兲
where F⬘ = E⬘ , D⬘ , H⬘ , B⬘ in the spatiotemporal and spectral
domains respectively. The fact that the Helmholtz wave operator in Eq. 共26兲 is a scalar implies that Eq. 共26兲 applies
individually to all Cartesian components of F⬘. If we substitute in Eq. 共26兲 F⬘ = E⬘ from Eq. 共7a兲 and B from Eq. 共10b兲,
we obtain
共C⬙2k⬘ ⫻ 共k⬘ ⫻ Ĩ兲 + ␻⬘2Ĩ兲 · Ṽv · 共Ĩ + v ⫻ 共k ⫻ Ĩ兲/␻兲 · E = 0.
共27兲
The same relation is found for H.
The result Eq. 共27兲 is of key importance. The fact that the
dyadic operator acts on a field measured in ⌫, indicates that
the dispersion relation will be relevant to ⌫, notwithstanding
the fact that the operator itself depends on the k⬘ , ␻⬘ variables of ⌫⬘. We again exploit the theorem for the determinant
Martin McCall and Dan Censor
1136
of a product of matrices, noting that in general both det关Ṽ兴
and det关Ĩ + v ⫻ 共k ⫻ Ĩ兲 / ␻兴 are nonzero, so that the dispersion
relation governing E and H in ⌫ is given by
G⬘共k⬘, ␻⬘兲 = G共k⬘, ␻⬘兲
= det关C⬘2共k⬘k⬘ − 共k⬘ · k⬘兲Ĩ兲 + ␻⬘2Ĩ兴 = 0,
共28兲
where the equality of G⬘ measured in ⌫⬘ with G measured in
⌫, means that the dispersion relation is a relativistic invariant. This invariance property does not, however, imply form
invariance as is the case for the Maxwell equations. Therefore from Eq. 共28兲 we have, after the substitutions of Eq. 共11兲
G⬘共⳵r⬘, ⳵t⬘兲F共r,t兲 = G共⳵r⬘, ⳵t⬘兲F共r,t兲,
共29兲
where F is either E or H. Note that ⳵k⬘ and ⳵t⬘ must be written
in terms of ⳵r and ⳵t using Eq. 共6兲 to act on F共r , t兲. If we
know the scalar wave operator G⬘共⳵r⬘ , ⳵t⬘兲 in ⌫⬘, then by
substituting from Eq. 共6兲 we can express G共⳵r⬘ , ⳵t⬘兲, the wave
operator in ⌫, in terms of the native coordinates of ⌫.
In order to express G共k⬘ , ␻⬘兲 = 0 in terms of unprimed
quantities, we need the spectral domain Lorentz transformations from k⬘ , ␻⬘ into k , ␻. For that we need the phase invariance principle k⬘ · r⬘ − ␻⬘t⬘ = k · r − ␻t from which it follows from Eq. 共6兲 that
k⬘ = k⬘共k, ␻兲 = Ũ · 共k − v␻/c2兲,
共30a兲
␻⬘ = ␻⬘共k, ␻兲 = ␥共␻ − v · k兲,
共30b兲
often referred to as the relativistic Fresnel drag effect and the
relativistic Doppler effect, respectively. Note that the substitution of Eq. 共11兲 in Eq. 共6兲 yields Eq. 共30兲 directly. By
substituting Eq. 共30兲 into Eq. 共28兲, and inserting Eq. 共22兲, we
obtain
det关C⬘2共k⬘k⬘ − 共k⬘ · k⬘兲Ĩ兲 + ␻⬘2Ĩ兴
共31a兲
=det关C⬘2共k2x x̂x̂ + ␥kxk̄yx̂ŷ + ␥kxŷx̂ + ␥2k̄2y ŷŷ
¯ 2Ĩ兴
− 共k2x + ␥2k̄2y 兲Ĩ兲 + ␥2␻
¯ 2共 ␥ 2␻
¯ 2 − C⬘2共k2x + ␥2k̄2y 兲兲2 = 0.
= ␥ 2␻
共31b兲
共31c兲
¯ ⫽ 0, we conclude that the dispersion relaBecause ␻⬘ = ␥␻
tion is
G = ␥2共␻ − vky兲2 − C⬘2关k2x + ␥2共ky − v␻/c2兲2兴
= ␻⬘2 − C⬘2k⬘2 = 0.
共32兲
The last equality in Eq. 共32兲 is the dispersion relation in the
medium at rest in ⌫⬘, obtained by applying the Helmholz
operator 共C⬘2⳵r⬘2 + ␻⬘2兲 to the assumed exp i共k⬘ · r⬘ − ␻⬘t⬘兲
field dependence.
To complete the verification of the relativistic invariance,
we need to show that the two methods yield the same expression for the dispersion relation. The substitution of Eq.
共30兲 in the form
k⬘y = ␥共ky − v␻/c2兲 = ␥k̄y ,
共33a兲
¯,
␻⬘ = ␥共␻ − vk兲 = ␥␻
共33b兲
into Eq. 共32兲 yields Eq. 共25兲. We note that when ky⬘ = 0, ky
⫽ 0. In other words, from the lab observer’s standpoint, the
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Am. J. Phys., Vol. 75, No. 12, December 2007
radiation has a component along the direction of relative
travel as a result of being “dragged” along with the moving
medium. This extra component inspires the name Fresnel
drag for Eq. 共33a兲. The Doppler shift described by Eq. 共33b兲
is well known as the frequency change due to relative motion
between source and observer.
VI. DISPERSION RELATION, GROUP VELOCITY,
AND REFRACTIVE INDEX
One reason for deriving the dispersion relation is to calculate the group velocity, facilitating ray tracing in various media. In the present case we are interested in the group velocity and ray tracing in the moving medium in ⌫. It is tempting
to think of rays as geometrical entities and assume that in a
moving medium we observe the rays of the medium at rest in
⌫⬘, except that the ray paths are now moving, subject to the
Lorentz transformation 共1兲. The idea that rays can be transformed in this way is a misconception. The moving medium
in ⌫ behaves as a different medium, and rays at rest in ⌫ will
be governed by the appropriate dispersion relation. The invariance of the dispersion relation facilitates the derivation
of the group velocity vg = ⳵k␻ in the moving medium in ⌫
from the knowledge of the dispersion relation in the medium
at rest in ⌫⬘. On the surface G = 0, dG = 关共⳵␻G兲共⳵k␻兲
+ ⳵kG兴 · dk = 0 for arbitrary dk in the surface. Hence the
group velocities in each frame are given by
vg = − ⳵kG/⳵␻G,
共34a兲
vg⬘ = − ⳵k⬘G⬘/⳵␻⬘G⬘ .
共34b兲
The invariance G = G⬘ in Eq. 共28兲 yields
vg = − ⳵kG共k, ␻兲/⳵␻G共k, ␻兲 = − ⳵kG⬘共k, ␻兲/⳵␻G⬘共k, ␻兲.
共35兲
Similar to Eqs. 共1兲 and 共6兲, we apply the chain rule to Eq.
共30兲 to derive the Lorentz transformations for the derivatives
in the spectral domain. Thus Eq. 共35兲 can be recast in the
form
vg = − 关Ũ · 共⳵k⬘ − v⳵␻⬘兲G⬘共k⬘, ␻⬘兲兴/关␥共⳵␻⬘
− v · ⳵k⬘/c2兲G⬘共k⬘, ␻⬘兲兴,
共36兲
completing the derivation of vg from the dispersion relation
in ⌫⬘.
One of the questions that arises when dealing with simple
moving media is anisotropy and birefringence. The moving
medium observed in ⌫ is anisotropic by virtue of the preferred direction prescribed by the velocity v. Can this preferred direction give rise to birefringence as observed in anisotropic crystals? This question is answered by Eq. 共36兲.
Inasmuch as in ⌫⬘ we have the simple dispersion relation Eq.
共26兲 for isotropic media, the operations in Eq. 共36兲 remain
independent of the polarization directions of the fields. Without such a dependence we cannot have birefringence.
The result of calculating the group velocity components
using Eqs. 共32兲 and 共36兲 is
冉 冊 冋冉
vgx = 1 −
冊 冉
C ⬘2
v2
v 2C ⬘2
2
− vk y 1 − 2
2 C⬘ kx ␻ 1 −
4
c
c
c
冊册
−1
,
共37a兲
Martin McCall and Dan Censor
1137
vgy =
冋冉
1−
冉
冊
C ⬘2
␻v + 共C⬘2 − v2兲ky
c2
− vk y 1 −
C ⬘2
c2
冊册
册冋 冉
␻ 1−
v 2C ⬘2
c4
冊
−1
.
共37b兲
Equation 共37兲 appears to imply a frequency dependence of
n=
the group velocity in ⌫. However, if we set kx = k sin ␪ and
ky = k cos ␪, where ␪ the angle between k and v, we see that
vgx and vgy depend only on ␻ / k = c / n, where n is the refractive index of the wave propagating in the ␪ direction. The
frequency independence of the group velocity is emphasized
by calculating n explicitly from the dispersion relation Eq.
共32兲. The result is
− c共1 − 共C⬘2/c2兲兲v cos ␪ ± c兵v2 cos2 ␪共1 − 共1 − 共C⬘2/c2兲兲兲2 + 关C⬘2 − v2共cos2 ␪ + 共1 − 共C⬘2/c2兲兲 sin2 ␪兲兴关1 − 共C⬘2v2兲/c4兴其1/2
.
关C⬘2 − v2共cos2 ␪ + 共C⬘2/c2兲 sin2 ␪兲兴
共38兲
The frequency independence shown in Eq. 共38兲 means that
the medium which is nondispersive in its rest frame ⌫⬘, retains this property in all inertial systems ⌫ from which the
medium is observed to be in uniform motion.
VII. AN EXAMPLE
In a chemical plant there is a process for which a highly
corrosive fluid is flowing at an extremely high speed through
a tube. The velocity of the fluid must be constantly monitored. Conventional methods were tried, such as immersing a
small mechanical propeller into the flow, but turned out to be
unreliable because the bearing on the axis of the propeller
quickly corroded. The fluid is nonconducting, and its refractive index n⬘ = c / C⬘ at rest is known. The scheme of Fig. 1,
based on the ideas of this paper, is proposed. A microwave
transmitter and receiver operating at a known frequency ␻
are connected to antennas immersed in the fluid. The distance L between the antennas is known. The transmitter creates a harmonic signal exp共−i␻t兲, which is also recorded at
the receiver, providing a reference phase −␻t. The signal
propagates downstream through the fluid arriving at the receiver as exp i共kL − ␻t兲. The phase difference kL is measured
electronically, which means that k is also determined. Hence
from the Doppler effect and Fresnel drag effect 关see Eq. 共30兲兴
we have k⬘ / ␻⬘ = n⬘ / c = 共k − v␻ / c2兲 / 共␻ − vk兲, and hence
v=
k − 共␻n⬘/c兲
.
共␻/c2兲 − 共kn⬘/c兲
This scheme is an electronic implementation of the Fizeau
interferometer8,10 in which the phase shift is measured electronically. The Fizeau experiment involves the interference
of two beams passing through oppositely directed flows. In
Fig. 1, instead of one of the beams, the necessary reference
phase is established by electronically connecting the transmitter output to the receiver. Students should explain how
the symmetrical arrangement of the original Fizeau experiment provides greater resolution by interfering two light
waves of almost identical length. Note that although the electronic comparison of phases in Fig. 1 is practical with the
electronic time resolution currently available, it yields a high
uncertainty in the resultant fluid speed measurement. The
optical interferometric method is much more accurate.
VIII. A BRIEF INTRODUCTION TO COVARIANT
METHODS
We add covariant tensor methods to the arsenal of vectordyadic and matrix methods that we have discussed. This subject is usually deferred to advanced mathematical physics
courses, but we believe that students should be introduced to
it at an early stage and acquire the flexibility of dealing with
various mathematical tools. The complexity of tensor methods requires that we refrain from explaining “the whole
truth,” including the introduction of the Minkowski metric.
The last requires a stronger basis of differential geometry
and the associated covariant and contravariant vectors. The
present introduction introduces the concept of dual vectors
with minimal extra tools.
In the covariant notation space and time are combined into
a single four-vector.11 Accordingly in ⌫ the spatiotemporal
coordinates are grouped into a quadruple
X␣ = 共ct,x,y,z兲,
Fig. 1. Geometry for fluid velocity measurement example. A transmitter/
receiver antenna pair is immersed in a fluid moving with velocity v. The
phase difference kL is determined electronically from which the fluid velocity is inferred from Eq. 共30兲.
1138
Am. J. Phys., Vol. 75, No. 12, December 2007
共39兲
共40兲
where in Eq. 共40兲 the superscript, or contravariant index ␣
runs from 0 to 3, so that X0 = ct, etc. The normalization ct
preserves the physical dimensionality as distance, on a par
with x, y, z. Similarly, the spectral domain parameters are
combined into a four-vector in the form
K␣ = 共− ␻c−1,kx,ky,kz兲,
Martin McCall and Dan Censor
共41兲
1138
In Eq. 共41兲 the subscript or covariant index ␣, together with
the factor −c−1, allows the phase of a plane wave to be written compactly as an inner product or contraction
X␣K␣ = − ␻t + kxx + ky y + kzz = k · r − ␻t.
共42兲
Here the Einstein summation convention is employed. In covariant notation summed indices always appear “one-up–
one-down,” and the signs in expressions such as Eq. 共42兲
take care of themselves.
The transformation of the components X␣ in ⌫ to X␣⬘ in
⌫⬘ is governed by the Lorentz transformation Eq. 共1兲, which
we write as
X ␣⬘ = ⌳ ␣⬘␤ X ␤ ,
⌳ ␣⬘␤ =
冤
␥
− ␥vx/c
− ␥vy/c
− ␥vz/c
共43a兲
冥
− ␥vx/c
− ␥vy/c
− ␥vz/c
␩v2x
␩ v xv y
1 + ␩v2y
␩v yv y
␩ v xv z
,
␩v yvz
1 + ␩vz2
1+
␩v yvx
␩ v zv x
共43b兲
where ␩ = 共␥ − 1兲 / v2 and v2 = v · v. Here v is the velocity of
the origin of ⌫⬘ as observed from ⌫, and its direction in
space is arbitrary. For the inverse transformation
X ␣ = ⌳ ␣␤⬘X ␤⬘ ,
共44兲
we need ⌳␣␤⬘, which is obtained from ⌳␣⬘␤ in Eq. 共43兲 by
replacing v by −v, that is, reversing the sign of all the velocity components. By substituting Eq. 共44兲 into the Eq. 共43a兲
we find
X ␣⬘ = ⌳ ␣⬘␤ X ␤ = ⌳ ␣⬘␤⌳ ␤␤⬘X ␤⬘ ,
共45兲
where ⌳␣⬘␤⌳␤␤⬘ = ␦␣⬘␤⬘, and ␦␣⬘␤⬘ is the Kronecker delta.
Interchanging primed and unprimed indices in Eq. 共45兲
yields ⌳␣␤⬘⌳␤⬘␤ = ␦␣␤. To obtain the Lorentz transformations
for K␣, given by Eq. 共30兲, we incorporate Eq. 共44兲 and its
inverse Eq. 共45兲 into the phase invariance principle X␣K␣
= X␤⬘K␤⬘ so that
X␣K␣ = 共⌳␣␤⬘X␤⬘兲K␣ = X␤⬘共K␣⌳␣␤⬘兲,
共46a兲
X␤⬘K␤⬘ = 共⌳␤⬘␣X␣兲K␤⬘ = X␣共K␤⬘⌳␤⬘␣兲.
共46b兲
and
Hence
K ␤⬘ = K ␣⌳ ␣␤⬘,
K ␣ = K ␤⬘⌳ ␤⬘␣ .
共47兲
Explicit calculations shows that Eq. 共47兲, with the definitions
in Eqs. 共41兲 and 共43兲, reproduce Eq. 共30兲. Note that the contravariant components of Eq. 共45兲 transform like the differential coordinates 关compare taking differentials of Eq. 共1兲兴,
whereas the covariant components of Eq. 共47兲 transform like
the derivative operators 关compare Eq. 共6兲兴.
In the covariant formulation the components of the electromagnetic fields E and B combine into the field tensor, an
array whose components in ⌫ are given by
1139
Am. J. Phys., Vol. 75, No. 12, December 2007
F p␴ =
冤
0
− Ex
− Ey
− Ez
Ex
0
cBz
− cBy
Ey
− cBz
0
cBx
Ez
cBy
− cBx
0
冥
共48兲
.
Similarly, the components of the fields D and H are combined as a tensor
G␣␤ =
冤
0
Dx
Dy
Dz
− Dx
0
cHz
− cHy
− Dy
− cHz
0
cHx
− Dz
cHy
− cHx
0
冥
.
共49兲
Both F␳␴ and G␣␤ are antisymmetric, that is, F␴␳ = −F␳␴ and
G␤␣ = −G␣␤.
The introduction of Eqs. 共48兲 and 共49兲 is partly motivated
by the fact that they can be used to write Maxwell’s equations very compactly. Consider Eq. 共49兲 and calculate derivatives according to
G␣␤,␤ = ⳵X␤G␣␤ = 0,
共50兲
where ,␤ indicates the differentiation of G with respect to
X␤. In Eq. 共50兲 we have to sum on the repeated index ␤. For
example, taking ␣ = 0 means that differentiating Eq. 共49兲
yields ⳵r · D = 0, i.e., Eq. 共5c兲. Taking ␣ through the indices
␣ = 1 , 2 , 3 and summing over ␤ yields the vector components
of ⳵r ⫻ H = ⳵tD, that is, Eq. 共5b兲. Now consider
F␣␤,␳ + F␳␣,␤ + F␤␳,␣ = 0,
共51兲
in Eq. 共48兲, where ␣, ␤, ␳, are any three of the four values 0,
1, 2, 3. For example, ␣ = 1, ␤ = 2, ␳ = 3 yields ⳵r · B = 0, that is,
Eq. 共5d兲. All other combination that include the index 0 contribute to the components of ⳵r ⫻ E = −⳵tB, that is, Eq. 共5a兲.
Because Maxwell’s equations are form invariant in inertial
systems, it follows that in ⌫ we have
G␣⬘␤⬘,␤⬘ = 0,
F␣⬘␤⬘,␳⬘ + F␳⬘␣⬘,␤⬘ + F␤⬘␳⬘,␣⬘ = 0.
共52兲
A deeper analysis of the second rank tensors F␳␴, G␣␤, Eqs.
共48兲 and 共49兲, respectively, reveals that the transformations
of Eq. 共7兲 now take the form
F␳⬘␴⬘ = F␳␴⌳␳␳⬘⌳␴␴⬘,
G␣⬘␤⬘ = ⌳␣⬘␣⌳␤⬘␤G␣␤ ,
共53兲
yielding the field components in ⌫⬘ from those in ⌫. Accordingly, a Lorentz transformation is applied to each of the indices and the summation convention occurs twice. Explicit
calculation of the component equations in Eq. 共53兲 confirms
that they are equivalent to Eq. 共7兲.
Up until this point the treatment has been general and
applicable to the transformation of the quantities, t, r, ␻, k,
E, B, D, and H between the frames ⌫ and ⌫⬘. To progress
further, the constitutive relation between the field tensors
must be specified. To keep this introduction to its bare minimum, we will skip the discussion of constitutive relations
involving fourth rank tensors.13 The discussion of dispersion
relations in the context of covariant methods will therefore
be restricted to vacuum. According to Eq. 共26兲, in ⌫⬘ all field
components appearing in Eqs. 共48兲 and 共49兲 individually satisfy the Helmholtz wave equation.
In covariant notation these can be written compactly as
⳵␺⬘⳵␺⬘F␳⬘␴⬘ = 0,
共54a兲
Martin McCall and Dan Censor
1139
⳵␺⬘⳵␺⬘G␣⬘␤⬘ = 0,
共54b兲
⳵␺⬘ = 共c−1⳵t⬘, ⳵x⬘⳵y⬘, ⳵z⬘兲,
共54c兲
⳵␺⬘ = 共− c−1⳵t⬘, ⳵x⬘, ⳵y⬘, ⳵z⬘兲,
共54d兲
in terms of the covariant ⳵␺⬘ and contravariant ⳵␺⬘ fourvectors, whose contraction yields the Helmholtz wave operator, which subject to Eq. 共6兲 can also be expressed in terms
of ⌫ native coordinates
⳵␺⬘⳵␺⬘ = ⳵r⬘ − c−2⳵t⬘ = ⳵␺⳵␺ = ⳵2r − c−2⳵2t .
2
2
共55兲
Hence the vacuum wave operator is a form invariant relativistic invariant. If we exploit Eq. 共55兲, we now have
⳵␺⬘⳵␺⬘F␳⬘␴⬘ = ⳵␺⬘⳵␺⬘F␳␴⌳␳␳⬘⌳␴␴⬘ = ⳵␺⳵␺F␳␴⌳␳␳⬘⌳␴␴⬘ = 0,
共56a兲
⳵␺⬘⳵␺⬘G␣⬘␤⬘ = ⳵␺⬘⳵␺⬘G␣␤⌳␣⬘␣⌳␤⬘␤ = ⳵␺⳵␺G␣␤⌳␣⬘␣⌳␤⬘␤ = 0.
共56b兲
A sufficient condition for the rightmost terms in Eq. 共56兲 to
vanish is that
⳵␺⳵␺F␳␴ = 0,
⳵␺⳵␺G␣␤ = 0,
共57兲
and because summation is on index ␺ only, Eq. 共57兲 is valid
for each entry of F␳␴ and G␣␤ individually. Because of the
primed, unprimed, fields in Eqs. 共54兲 and 共57兲, they refer to
measurements in ⌫⬘, ⌫, respectively.
Consider the special case when F␳⬘␴⬘ and G␣⬘␤⬘ are plane
waves. Subject to the phase invariance principle 共X␣K␣
= X␤⬘K␤⬘兲, we derive from Eqs. 共54兲–共57兲 the equalities
␤⬘
␤⬘
␣
⳵␺⬘⳵␺⬘共eiK␤⬘X 兲 = 共␻⬘2c−2 − k⬘x2兲eiK␤⬘X = ⳵␺⳵␺共eiK␣X 兲
共58a兲
␣
=共␻2c−2 − k2兲eiK␣X = 0.
共58b兲
Because the exponentials do not vanish, Eq. 共58兲 defines the
dispersion relation
G⬘共k⬘, ␻⬘兲 = ␻⬘2c−2 − k⬘2 = G共k, ␻兲 = ␻2c−2 − k2 = 0,
共59兲
which in the vacuum case is also form invariant. Similar to
the wave operator Eq. 共55兲, we could derive Eq. 共59兲 formally by the transformations Eq. 共30兲.
IX. CONCLUDING REMARKS
Tools such as vector and dyadic analysis, matrices, and
covariant 共tensor兲 methods are usually treated separately in
mathematics courses, and students are caught in the formalities without being able to comprehensively apply their
knowledge to problems of physics. We have chosen as an
example the problem of wave equations in simple media and
the relativistic invariance of the wave operators and the associated dispersion equations. The pedagogical innovation
claimed here is in the appropriate use of the ingredients.
1140
Am. J. Phys., Vol. 75, No. 12, December 2007
We started with a brief introduction to the relations needed
for discussing relativistic electromagnetics. Using vector and
dyadic methods, the Maxwell equations, relativistic transformations for coordinates and fields, and the Minkowski constitutive relations for simple media in motion were introduced. We then derived the dispersion relation in moving
media. By invoking the relativistic invariance of the wave
operator and the associated dispersion relation, the derivation
can be done in the medium at rest and then applied to the
moving medium.
As an application, we discussed the group velocity in media in motion and at rest. The group velocity describes the
motion of wave packets along trajectories called rays. The
correct way to derive ray trajectories is to calculate them
from the relevant dispersion in the medium at rest or in motion. We gave some physical insight to these considerations
via an example based on Fizeau’s interferometer.
Finally, a brief introduction to covariant methods was included, where we reviewed the transformation properties of
the fields in the presence of moving media. The details for
discussing the dispersion relation for material media are too
complicated for the present discussion, which was therefore
limited to gravity-free vacuum.
We have encountered some interesting facts: The use of
the Minkowski constitutive relations, Eq. 共9兲, without invoking Eq. 共30兲, and the use of Eq. 共30兲 without recourse to the
Minkowski constitutive relations yield the same dispersion
relations, and in this sense they are equivalent. We also concluded that a simple medium in motion is anisotropic, in the
sense that the velocity v introduces a preferred direction.
However, the results for the dispersion relation and group
velocity are independent of the directions of polarization of
the fields and therefore a simple medium in motion does not
become birefringent.
a兲
Author to whom correspondence should be addressed. Electronic mail:
[email protected]
1
Martin W. McCall, Classical Mechanics: A Modern Introduction 共Wiley,
Chichester, 2000兲.
2
Tomislav Ivezić, “True transformations: Relativity and electrodynamics,”
Found. Phys. 31, 1139–1183 共2001兲.
3
L. B. Felsen and Nathan Marcuvitz, Radiation and Scattering of Waves
共Prentice Hall, Englewood Cliffs, NJ, 1973兲.
4
John M. Kelso, Radio Ray Propagation in the Ionosphere 共McGraw-Hill,
New York, 1964兲.
5
Dan Censor, “Application-oriented ray theory,” Int. J. Electr. Eng. Educ.
15, 215–223 共1978兲.
6
Jonathan Molcho and D. Censor, “A simple derivation and an example of
Hamiltonian ray propagation,” Am. J. Phys. 54, 351–353 共1986兲.
7
Dan Censor, “Dispersion equations in moving media,” Proc. IEEE 68,
528–529 共1980兲.
8
Wolfgang Pauli, Theory of Relativity 共Pergamon, New York, 1958兲.
9
Arnold Sommerfeld, Electrodynamics 共Academic, New York, 1964兲.
10
Yehuda Ben-Shimol and D. Censor, “Wave propagation in moving chiral
media: Fizeau’s experiment revisited,” Radio Sci. 30, 1313–1324 共1995兲.
11
Dan Censor, “Application-oriented relativistic electrodynamics 共2兲,”
Prog. Electromagn. Res. 29, 107–168 共2000兲.
12
Albert Einstein, “Zur Elektrodynamik bewegter Körper,” Ann. Phys. 17,
891–921 共1905兲; English translation: “On the electrodynamics of moving
bodies,” in The Principle of Relativity 共Dover, New York, 1952兲.
13
E. J. Post, Formal Structure of Electromagnetics 共Dover, New York,
1997兲.
Martin McCall and Dan Censor
1140