Special Relativity and General Relativity

1. From First Principles
PART II – MODERN PHYSICS
May 2017 – R2.1
Maurice R. TREMBLAY
http://atlas.ch
Candidate Higgs
Decay to four
muons recorded by
ATLAS in 2012.
Chapter 2

2. Contents
PART II – MODERN PHYSICS
Charge and Current Densities
Electromagnetic Induction
Electromagnetic Potentials
Gauge Invariance
Maxwell’s Equations
Foundations of Special Relativity
Tensors of Rank One
4D Formulation of Electromagnetism
Plane Wave Solutions of the Wave
Equation
Special Relativity and
Electromagnetism
The Special Lorentz Transformations
Relativistic Kinematics
Tensors in General
The Metric Tensor
The Problem of Radiation in
Enclosures
Thermodynamic Considerations
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The Wien Displacement Law
The Rayleigh-Jeans Law
Planck’s Resolution of the Problem
Photons and Electrons
Scattering Problems
The Rutherford Cross-Section
Bohr’s Model
Fundamental Properties of Waves
The Hypothesis of de Broglie and Einstein
Appendix: The General Theory of
Relativity
References
2

3. 3
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1. Space is isotropic and homogeneous; that is, the equations of motion of mechanics
are unchanged by rotations and translations (or leaps) of the coordinate systems used to
describe the positions of particles, when the rotations and translations do not depend on
time as a parameter;
2. Time is independent of space; essentially, that is a parameter that measures the
separation between events as seen by an observer, and the observer’s location does not
affect the size of the standard intervals used to measure time;
3. Time is homogeneous; the equations of motion of mechanics are unchanged by a
displacements (or translations) in the time parameter t;
4. The equations of motion of mechanics are unchanged by spatial translations that
involve time as a linear parameter; that is, translations expressible in the form r→r−vt.
Frames of reference (i.e., a set of coordinate axes) related by such transformations are
called inertial frames, and the transformations are called Galilean transformations;
5. Time is independent of inertial reference frame (i.e., an observer’s state of inertial
motion does not affect the scale of time measurements).
Thus one has a picture of the world as a homogeneous and isotropic three-dimensio-
nal structure of space-points, with particles passing through these points according to
laws formulated in terms of a parameter t called time; an important property of the classi-
cal laws governing particle motion is that these laws are the same for all frames of refe-
rence moving with uniform velocity v relative to one another (i.e., classical mechanical
laws are the same for all frames related to one another by Galilean transformations).
Foundations of Special Relativity
Maxwell published his set of equations between 1861 and 1862. After that, and up until
1905, the following five statements enumerate some of the properties of space and time
as they were know by physicists at the time:

4. For the description of processes taking place in nature,* one must have a system of
reference. By a system of reference we understand a system of coordinates serving to
indicate the position of a particle in space, as well as clocks fixed in this system serving
to indicate the time.
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There exists a system of reference in which a freely moving body (i.e., a moving body
which is not acted upon by external forces) proceeds with constant velocity. Such
reference systems are said to be inertial.
* These next 3 slides are taken from L. Landau & E. Lifshitz, The Classical Theory of Fields, 4-th Ed., B-H (1975), pp. 1-2.
If two reference systems move uniformly relative to each other, and if one of them is
an inertial system, then clearly the other is also inertial (N.B., in this system too every
free motion will be linear and uniform). In this way one can obtain arbitrarily many inertial
systems of reference, moving uniformly relative to one another.
Experiment shows that the so-called principle of relativity is valid. According to this
principle all the laws of nature are identical in all inertial systems of reference. In other
words, the equations expressing the laws of nature are invariant with respect to
transformations of coordinates and time from one inertial system to another. This means
that the equation describing any law of nature, when written in terms of coordinates and
time in different inertial reference systems, has one and the same form.
The interaction of material particles is described in ordinary mechanics by means of a
potential energy of interaction, which appears as a function of the coordinates of the
interacting particles. It is easy to see that this manner of describing interactions contains
the assumption of instantaneous propagation of interactions.

5. For the forces exerted on each of the particles by the other particles at a particular
instant of time depend, according to this description, only on the positions of the
particles at this one instant. A change in the position of any of the interacting particles
influences the other particles immediately.
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However, experiment shows that instantaneous interactions do not exist in nature.
Thus a mechanics based on the assumption of instantaneous propagation of interac-
tions contains within itself a certain inaccuracy. In actuality, if any change takes place in
one of the interacting bodies, it will influence the other bodies only after the lapse of a
certain interval of time. It is only after this time interval that processes caused by the
initial change begin to take place in the second body. Dividing the distance between the
two bodies by this time interval, we obtain the velocity of propagation of the interaction.
We note that this velocity should, strictly speaking, be called the maximum velocity of
propagation of interaction. It determines only that interval of time after which a change
occurring in one body begins to manifest itself in another. If is clear that the existence of
a maximum velocity of propagation of interactions implies, at the same time, that
motions of bodies with greater velocities than this are in general impossible in nature.
For if such a motion could occur, then by means of it one could realize an interaction
with a velocity exceeding the maximum possible velocity of propagation of interactions.
Interactions propagating from one particle to another are frequently called ‘signals’,
sent our from the first particle and ‘informing’ the second particle of charges which
the first has experienced. The velocity of propagation of interaction is then referred
to as the signal velocity.

6. From the principle of relativity it follows in particular that the velocity of propagation of
interactions in the same in all inertial system of reference. Thus the velocity of
propagation is a universal constant. This constant velocity is also the velocity of light in
empty space. The velocity of light is usually designated by the letter c, and its numerical
value is:
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The large value of this velocity explains the fact that in practice classical mechanics
appears to be sufficiently accurate in most cases. The velocities with which we have
occasion to deal are usually so small compared with the velocity of light that the
assumption that the latter is infinity does not materially affect the accuracy of the results.
m/s8
10998.2 ×=c
The combination of the principle of relativity with the finiteness of the velocity of
propagation of interactions is called the principle of relativity of Einstein (formulated in
1905) in contrast to the principle of relativity of Galileo, which was based on an infinite
velocity of propagation of interactions.
The mechanics based on the Einsteinian principle of relativity is called relativistic. In
the limiting case when the velocities of the moving bodies are small compared to the
velocity of light we can neglect the effect of the motion of the finiteness of the velocity of
propagation of interactions; this mechanics is called Newtonian or classical. The limiting
transition from relativistic to classical mechanics can be produced formally by the
transition to the limit c→∞ in the formulas of relativistic mechanics.

7. The principle of Galilean invariance then asserts that the ‘laws of nature’ are the same
for two observers, i.e., that the form of the equations of motion is the same for both
observers. The equations of motion must therefore be covariant with respect to the
transformations r=r +vt and t =t. Unfortunately, this invariance principle applies only
in situations where the velocity v is much lower than that of light. We need more…
Experiments have also yielded the fact that space is isotropic so that the orientation in
space of an event is an irrelevant initial condition and this principle can be translated into
the statement that: ‘the laws of motion are invariant under spatial rotations.’ Newton’s
law of motion further indicated that the state of motion, as long as it is uniform with
constant velocity, is likewise and irrelevant initial condition. This is the principle of
Galilean invariance which assets that the laws of nature are independent of the velocity
of the observer, and more precisely, that the laws of motion of classical mechanics are
invariant with respect to Galilean transformations:
τ+→+→ ttandarr











 −
=





⇔
=
−=
tttt
t rvrvrr
10
1
The laws of nature are independent of the position of the observer or, equivalently, that
the laws of motion are covariant with respect to displacements in space and time, i.e.,
with respect to the transformations:
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7
We now look at the role played by invariance principles (i.e., changes in points of view
must not change the laws used to describe them) in the formulation of physical theories.

8. The acceleration of a particle is the time derivative of its velocity (i.e., ax =dux /dt, &c).
To find the Galilean acceleration transformations we differentiate the velocity transfor-
mations above (using the fact that t=t and v is considered a constant) to obtain:
The relationship between [ux,uy,uz] and [ux,uy,uz] is obtained from the time differen-
tiation of the Galilean coordinate transformation x ↔x. Thus, from r=x i and x =x −vt:
vuv
td
xd
td
td
td
vtxd
td
vtxd
td
xd
u xx −=





−=
−
=
−
== )1(
)()(
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where −v is the velocity of frame S relative to S. Altogether, the Galilean velocity transfor-
mations are:
zzyyxx uuuuvuu ==−= and,
zzyyxx aaaaaa === and,
Thus the measured acceleration components are the same for all observers moving with
uniform relative velocity. (N.B., That is why v was chosen to be a constant since it gives
us a uniform relativistic motion to deal with otherwise things get really complicated).
8
ˆ
In addition to the coordinate of an event, the velocity of a particle is of interest. Two ob-
servers, O and O, will describe the particle’s velocity by assigning three components to it,
with [ux,uy,uz] being the velocity components as measured by O in frame S and [ux,uy,uz]
being the velocity components as measured by O (i.e., relative to frame S).

9. Since, in elementary mechanics, longitudinal sound waves and transverse waves
along a string are familiar phenomena, one might expect that electromagnetic waves
should obey the same laws that govern such mechanical waves. Such a situation would
be very desirable, for then the physics of electromagnetism might be unified
conceptually with the physics of massive bodies. Let us therefore test the applicability of
classical mechanics to electromagnetic theory by carrying out a Galilean transformation
on the wave equation governing electromagnetic waves.
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We are interested in the telegrapher’s equation:
0),(
1
2
2
2
2
=








∂
∂
−∇ tf
tc
r
where f (r,t) is a scalar function of r ≡ri and t. Now, under a Galilean transformations,ri
→ri =ri −vit, and since f (r,t) is a scalar function, it must have the same numerical value
in both coordinate systems, although its form as a function of r and t may be different.
Thus, let:
]),,([),(),( ttgtgtf rrrr ==
where the last member demonstrates that g depends of t implicitly through the
dependence of r(r,t) on t, as well as explicitly.

10. Now we have:
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jj j
jj
k k
jk
k kj
k
j r
tg
r
tg
r
tg
r
tg
r
r
r
tf
∂
∂
=
∂
∂
=
∂
∂
=
∂
∂
∂
∂
=
∂
∂
∑∑∑=
),(),(),(),(),(
3
1
rrrrr
δδ
since δk j =∂rk /∂rj has the definition δkj =1 when k= j and δkj =0 when k≠ j, so that:
),(),( 22
tgtf rr ∇=∇
However:
t
tg
tg
t
tg
r
tg
t
r
t
tf
k k
k
∂
∂
+•=
∂
∂
+
∂
∂
∂
∂
=
∂
∂
∑=
),(
),(
),(),(),(
3
1
r
rv
rrr
∇∇∇∇
and:
444444 3444444 21
incouplingordersecondandFirst ∇∇∇∇
∇∇∇∇∇∇∇∇∇∇∇∇
•
••+
∂
∂
•+
∂
∂
=
∂
∂
v
rvv
r
v
rr
)],([
),(
2
),(),(
2
2
2
2
tg
t
tg
t
tg
t
tf
Therefore, the telegrapher’s equation transforms Galileanly according to the rule:
0),(
1
2),(
1
2
2
22
2
2
2
2
2
=








∂
∂
−
∂
∂
•−





•





•−∇=








∂
∂
−∇ tg
tctccc
tf
tc
r
vvv
r ∇∇∇∇∇∇∇∇∇∇∇∇
or
0),()()(2
1
),(
1 2
2
2
2
2
2
2
2
2
=
















•+
∂
∂
•+
∂
∂
−∇=








∂
∂
−∇ tg
ttc
tf
tc
rvvr ∇∇∇∇∇∇∇∇
Yikes!

11. Evidently, the equation governing electromagnetic wave propagation has a different
form in the bared reference frame, with coordinate ri, from that which it has in the ‘un-
bared’ reference frame, with coordinate ri. The form of the telegrapher’s equation is not
invariant under Galilean transformations. Since this equation represents the law of
electromagnetic wave propagation, one might say that the choice of Galilean reference
frame affects the laws of electromagnetism. Based on this argument alone, Galilean
invariance cannot be considered further.
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Classical mechanics and electromagnetism therefore cannot be welded together
without some changes. But which theory should we retain, if either? As it turned out,
both theories can be retained almost intact, but Newtonian mechanics has had to yield a
little bit by giving up, as an exact postulate, the invariance of the equations of motion
under Galilean transformations, together with the notion that time is independent of
reference frame. The only way this question could be answered, however, was to turn to
experiment and to try to determine physically whether the constant c that appears in the
telegrapher’s equation refers to the velocity of wave propagation in some preferred
reference frame (i.e., the frame of reference in which the “ether” or ‘propagating medium’
is a rest, presumably) so that Maxwell’s equations are strictly correct only in that
preferred frame, or whether some aspect of mechanics is a defective description of
nature. Since Maxwell’s equations were relative newcomers to physics, it is not
surprising that the main experimental effort was aimed at detecting a preferred frame of
reference to which electricity and magnetism always should be referred.

12. According to Coulomb’s law, fC(r)= (1/4πεo)ρ(r)∫∫∫V ρ(r)[(r−−−−r)/|r−−−−r|3]d3r, and the law of
Biot and Savart, fBS(r)= (µo/4π)J(r)××××∫∫∫V J(r)××××[(r−−−−r)/|r−−−−r|3]d3r, the total force f(r) per unit
volume exerted on a charge density ρ(r) and current density J(r) is:
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)()()()()()()( BSC rBrJrErrfrfrf ××××++++++++ ρ==
Now if the charge density vanishes everywhere except at a single point ro, and is so
large at ro that ∫∫∫V ρ(r)d3r=q is a finite number (if ro is in V of course), then we can write:
ˆ
)()( orrr −−−−δρ q=
that is, we have a point charge of magnitude q at ro. Similarly, if J(r) is generated by a
point charge of magnitude q moving with velocity v(r), have:
)()()()()( oo rrrvrvrrJ −−−−δρ q==
when the charge is at ro.
)]()([)()]()([
)]()()()([)(
ooo
33
rBrrrvrErr
rrBrJrErrrfF
××××−−−−++++−−−−
××××++++
δδ
ρ
qq
dd
VV
=
== ∫∫∫∫∫∫
so that we obtain:
)]()()([ ooo rBrvrEF ××××++++q=
With these expressions for the charge and current densities,the total force experienced
by the charge evidently is:
This is the Lorentz force law, which is in this case, as seen from an origin O, ro away.

13. The field vectors E(r) and B(r) can be written in terms of the scalar and vector
potentials, according to E(r)= −∇∇∇∇φ(r)−−−−∂A(r)/∂t and B(r)= ∇∇∇∇××××A(r), and for fields gene-
rated by a point charge q at r, the potentials given by φ(r)=(1/4πεo)∫∫∫V ρ(r)(1/|r−−−−r|)d3r
and A(r)=(µo/4π)∫∫∫V J(r)(1/|r−−−−r|)d3r] become:
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rr
rv
rA
rr
r
−−−−−−−−
)(
π4
µ
)(
1
επ4
)( o
o
qq
== andφ
where v(r) is the velocity of the source charge q. If the velocity v(r) is a constant, we
then have:
)(
π4
µ
)(
)(
επ4
)( 3
o
3
o
rv
rr
rr
rA0
rA
rr
rr
r ××××
−−−−
−−−−
××××∇∇∇∇
−−−−
−−−−
∇∇∇∇
q
t
q
−==
∂
∂
=− and,φ
Therefore the fields at point r arising from a uniformly moving charge q at r are:
3
o
3
o
)()(
π4
µ
)(
επ4
)(
rr
rrrv
rB
rr
rr
rE
−−−−
−−−−××××
−−−−
−−−− qq
== and
The force on charge q at ro due to these fields is:
)]}()([)(εµ){(
1
επ4
)]()()([ ooooo3
oo
ooo rrrvrvrr
rr
rBrvrEF −−−−××××××××++++−−−−
−−−−
××××++++
qq
q ==
where the first term is the electromagnetic force and the second is the magnetic
force. This is familiar from elementary physics of point charges.

14. Now the torque ττττ on the charge at ro about the point r is defined to be:
14
)]([
π4
µ
3
o
rvvr ××××××××××××ττττ
r
qq
=
Note that the torque arises entirely from magnetic effects!
)]}()([)({)(
π4
µ
)( ooo3
o
o
o rrrvrvrr
rr
Frr −−−−××××××××××××−−−−
−−−−
××××−−−−ττττ
qq
==
Consider the arrangement shown in the Figure, for which we take v(r)=v(ro)=v. Here
we have:
Schematic diagram of the arrangement of the
Trauton-Noble experiment.
If the angle between v and r is θ, the multiple cross-pro-
duct on the right-hand side can be rewritten in terms of θ as:
where r××××v ≡(r××××v)/|r××××v| is a unit vector in the direction of r××××v.
Hence the torque tending to restore the vector r to an
orientation perpendicular to v is (N.B., we use c2 =1/µoεo):
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where r=ro −−−−r.
vr
vr
vrrvrvrvrrvvr
××××
××××
××××−−−−××××××××××××××××
ˆ2sin
2
1
ˆcossin
))((])[()]([
22
22
2
θ
θθ
rv
rv
v
=
=
•=•=
ˆ
vr ××××ττττ ˆ2sin
επ4
1
2
o
θ





=
c
v
r
qq
v
r
v
q
Pivot
q
_
ro
r
O
v××××(v×××× r)
r××××vˆ
θ
2
π
θ −
2
π
ττττ

15. At this point, we must ask: With respect to what frame of reference are we to measure
v? If there were a fixed preferred frame of reference to which all electromagnetic effects
should be referred, then v certainly would be measured relative to that frame.
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In 1903, Trauton and Noble performed a series of experiments in which charged
condensers were moved through space at the velocity of the earth relative to the sun;
the apparatus was sufficiently sensitive to detect torques resulting from velocities
somewhat less than earth’s orbital velocity, the former presumably being measured
relative to some preferred reference frame. However, no torque relative to the preferred
frame, or the “ether”, than is the sun; thus, if one accepts the notion of an absolute
frame of reference for electromagnetic effects, one apparently is forced by this
experiment to reconsider the possibility of a Ptolemaic or geocentric theory of the
universe. A more reasonable alternative is to question the notion of a preferred
reference frame, and to reason that in the absence of such a preferred frame we cannot
make sense of ττττ=(1/4πεo)(qq/r)(v/c)2sin(2θ )r××××v until the meaning of the velocity v is
clarified (i.e., until the manner in which v is to be measured relative to an arbitrary frame
is specified).
ˆ
At about the same time, Michelson and Morley, and many others, carried out an
experiment using an interferometer; this interferometer produced circular fringes and
from these fringe patterns, they meant to measure an effect predicted by the notion of an
“ether” that sustains electromagnetic waves and that is stationary relative to the
preferred reference frame… None of them observed any significant phase shift!
_

16. It is a fundamental postulate of Physics that the laws of nature be expressed by
equations that are valid for all coordinate systems (i.e., locally inertial reference
frames). Alternatively, we say that the laws of nature are covariant, which means that
they have the same forms in all coordinate systems. A systematic method of investi-
gating the behavior of quantities that undergo a coordinate transformation is the subject
matter of tensor analysis. In developing the mathematical subject of absolute differential
geometry, Gauss, Riemann, and Christoffel (1829-1900) introduced the concept of a
tensor. The subject of absolute tensor calculus (i.e., tensor analysis) was introduced and
developed by Ricci (1853-1925) and his student Levi-Civita (1872-1941). Einstein (1879-
1955) made extensive use of tensors (i.e., technically called differential tensor calculus)
in his formulation of the general theory of relativity. Insofar, a tensor consists of a set of
quantities, called components, whose properties are independent of the coordinate
system used to describe them. The components of a tensor in two different coordinate
systems are related by the characteristic tensor transformation as discussed below.
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Now a word on notation. A collection of indices (subscripts and/or superscripts) is used
to make the mathematical development of tensor analysis compact. The superscripts,
contravariant indices, are used to denote the contravariant components of a tensor,
Tij.... The subscripts, covariant indices, are used to represent the covariant compon-
ents of a tensor, Tij.... The components of a mixed tensor are specified by indicating both
subscripts and superscripts (or a jumble of superscripts and subscripts), Ti
k
j
m
k
n
...
.... For
ever more will we will use this notation (Latin letters or Greek symbols; lower or upper
case; bold, italized or not, to denote the components of a tensor or the tensor itself.)
16
Tensors of Rank One

17. defines a new coordinate system specified by the mutually independent variables: x1, x2,
…, xn. The symbol φ i (e.g., a temperature distribution or field of some sort) are ass-
umed to be single-valuedreal functions of the coordinates with continuous derivatives.
The rank (order) of a tensor is the number (without counting an index which appears
once as a subscript and once as a superscript) of indices in the letter or symbol
representing a tensor (or the components of a tensor). Here a few examples of tensor
(and their rank) which will make an appearance very soon: S is a tensor of rank zero
(scalar – e.g., action); xi is a covariant vector of rank one (covariant vector – e.g., three-
dimensional Cartesian coordinate); Pµ is a contravariant tensor of rank one (contra-
variant vector – e.g., space-time momentum); Tµ
ν is a mixed tensor of rank two (e.g.,
energy-momentum-stress tensor) ; Gµν =Rµν +½gµνR is a (contravariant–e.g., Einstein’s
gravitational field tensor) tensor of rank two; Rµν ≡Σλ Rλ
µλν (note the contraction on the
index λ) is a tensor (e.g., Ricci curvature tensor) of rank two; Rµ
νρσ is a mixed tensor
(e.g., action Riemann curvature tensor) of rank four; and finally R ≡ΣµΣν gµν Rµν (i.e.,
another contrac-tion on both µ andν) is a tensor of rank zero (scalar – e.g., curvature
scalar). In an n-dimensional space, the number of components of a tensor of rank n is nr.
( )nixxxx nii
,,2,1),.,,( 21
KK == φ
Consider a ordered set of n mutually independent real variables x1,x2,…,xn =[xi],
called the coordinate of a point, Pn(x1,x2,…,xn). The collection of all such points
corresponding to all the sets of values [xi] forms an n-dimensional linear space (i.e., a
manifold – French for variété) which we specify by Vn. The set of n equations:
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17

18. On differentiating xi =φ i(x1, x2, …, xn) (N.B., xi for x is the same as xi for an index i) with
respect to x j, we obtain the following representation for an infinitesimal displacement in
the original coordinate system, xj, in terms of the new coordinate system, x i:
∑∑ == ∂
∂
=
∂
∂
=
∂
∂
++
∂
∂
+
∂
∂
=
n
j
j
j
in
j
j
j
i
n
n
iii
i
xd
x
x
xd
x
xd
x
xd
x
xd
x
xd
11
2
2
1
1
φφφφ
K
A set of components, Aj, which transform according to the law above, which is given by
the same process:
2017
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If det|∂xi/∂x j|≠0 (i.e., the Jacobian is non-zero) then the inverse transformation exists.
∑= ∂
∂
=
∂
∂
++
∂
∂
+
∂
∂
=
n
j
j
j
i
n
n
iii
i
A
x
x
A
x
x
A
x
x
A
x
x
A
1
2
2
1
1
K
forms the transformation law for the components of a contravariant tensor of
rank one.
As a quick example of its use, let us find the transformation for rotation in two
dimensions. In this case, we have: x1 =x1cosθ + x2sinθ and x2 =x2cosθ − x1sinθ. The
Jacobian of the transformation is given by:
1sincos)sin()(sincoscos
cossin
sincos
det 22
2
2
1
2
2
1
1
1
=+=−⋅−⋅=
−
==








∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
θθθθθθ
θθ
θθ
x
x
x
x
x
x
x
x
j
i
x
x
18

19. In this last equation (i.e., Ai =Σj (∂xi/∂xj)Aj) Aj is a function of the coordinates x j (i.e., Aj =
Aj(x j)) and Ai is a function of the coordinates xi (i.e., Ai =Ai(xi)), where x j and xi refer to
the old and new coordinate systems, respectively.
2017
MRT
where the Kronecker delta,δ k
j, is given by:
or changing the dummy index k to j:
∑= ∂
∂
=
n
i
i
i
j
j
A
x
x
A
1




≠
=
=
kj
kj k
kk
j
0
1
when
)for,(when δ
δ
e.g.
k
k
n
k
k
n
j
jk
j
n n kn
i
n
j
j
j
i
i
kn
i
n
j
j
j
i
i
kn
i
i
i
k
A
AA
AA
x
x
A
x
x
A
x
x
x
x
A
x
x
=
⋅=⋅=
=
∂
∂
=
∂
∂
=








∂
∂
∂
∂
=
∂
∂
∑
∑∑ ∑∑∑∑ ∑∑
=
=
== == == ==
1
1
11 11 11 11
)(
k
k
k
j
i j
j
j
i
i
x
x
δ
δδ
∂∂∂∂
∂∂∂∂
The equation Ai =Σj (∂xi/∂xj)Aj may be solved for Aj if it is multiplied by ∂xk/∂xi and
summed over i. In this case, we obtain:
19

20. A set of quantities Bk is called the components of a covariant tensor of rank one if:
2017
MRT
for an arbitrary contravariant tensor with components Ak. If Ak from Ak =Σi(∂xk/∂xi)Ai is
substituted into Σk Ak Bk =Σk AkBk, the result is:
∑∑∑ ∑∑∑ = == === ∂
∂
=








∂
∂
==
n
k
n
l
l
k
k
ln
k
k
n
i
i
i
kn
k
k
k
n
k
k
k
BA
x
x
BA
x
x
BABA
1 11 111
or
or
∑∑ ==
==
n
k
k
k
n
k
k
k
BABA
11
scalar)(ainvariant
0
1 1
=








∂
∂
−∑ ∑= =
n
k
n
l
lk
l
k
k
B
x
x
BA
∑= ∂
∂
=
n
l
lk
l
k B
x
x
B
1
since the Ak are arbitrary. Bk =Σl(∂xl/∂xk)Bl is the transformation law for the compo-
nents of a covariant tensor of rank one. An easy to remember mnemonic for the
placement of ∂xi is ‘CO BELOW’ for covariant components. Now for an application…
20

21. Maxwell’s equations (i.e., the foundation of the theory of electromagnetism)are equa-
tions that describe how electric field intensity (E) and magnetic field intensity (H) are
generated and altered by each other and by charge density (ρ) and current density (J):
2017
MRT
t
t
∂
∂
+=
=•
∂
∂
−=
=•
D
JH
B
B
E
D
××××∇∇∇∇
∇∇∇∇
××××∇∇∇∇
∇∇∇∇
0
ρ
In MKS units, the quantity B is the magnetic induction:
with εo (i.e., 8.854×10 −12 C2/N⋅m2) the permittivity of the vacuum.
HB oµ=
ED oε=
with µo (i.e., 4π×10−7 kg⋅m/C2) being the permeability of the vacuum and quantity D is the
electric displacement:
form the foundation of classical electrodynamics, classical optics (N.B., with isotropic
medium effects neglected), and electric circuits.
Coulomb’s Law:
Faraday’s Law:
Absence of free magnetic poles:
Ampère’s Law (∇∇∇∇•J=0):
21
These provide a set of partial differential equations that, together with the Lorentz
force law:
)( BvEF ××××++++q=
4D Formulation of Electromagnetism

22. In practice, the relation between the electric flux density D and the electric field E
depends on the electric properties of the medium! Similarly, the relation between the
magnetic flux density B and the magnetic field H depends on the magnetic properties of
the medium. Two equations help define these relations:
2017
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in which P is the polarization density and M is the magnetization density. In a dielectric
medium, the polarization density is the microscopic sum of the electrical dipole moments
that the electric field induces. The magnetization density is similarly defined. So, in free
space (i.e., P=M=0):
where:
EP χoε=
22
MHBPED ++++++++ oo µε == and
we recover the relations obtained earlier.
HBED oo µε == and
A medium in the simplest case is linear, nondispersive, homogeneous, and isotropic.
The vectors P and E at any position and time are parallel and proportional, so that:
where χ is a scalar constantcalled the electric susceptibility.Substituting P into D=εoE+P,
it follows that D and E are also parallel and proportional:
ED ε=
)1(εε o χ+=
is another scalar constant, the electric permittivity of the medium. The ratio ε/εo is the
relative permittivity or dielectric constant.

23. If we also consider a medium in which there are no free electric charges or currents,
(i.e., ρ=0 and J=0) Maxwell’s equations in free space with εo replaced by ε simplify to:
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23
Each of the components of E and H therefore satisfies the wave equation:
0
),(1
),( 2
2
2
2
=
∂
∂
−∇
t
tf
c
tf
r
r
tt ∂
∂
==•
∂
∂
−==•
D
HH
H
EE ε0µ0 o ××××∇∇∇∇∇∇∇∇××××∇∇∇∇∇∇∇∇ and,,
with a speed c =1/√(εµo). The different components of the electric and magnetic fields
propagate in the form of waves of speed:
n
c
c o
=
where:
oo
o
o µε
1
1
ε
ε
=+== cn andχ
The constant co is the speed of light in free space (e.g., a vacuum); the constant n is the
ratio of the speed of light in free space to that in the medium called the refraction index
of the medium – the refractive index is the square root of the dielectric constant!
In inhomogeneous dielectric media (e.g., graded index) the coefficients χ =χ(r) and
ε=ε(r) are functions of position. The refractive index n=n(r) is also position dependent.

24. In anisotropic media, the relation between the vector P and E depends on the direction
of the vector E, and these two vectors are not necessarily parallel. If the medium is
linear, nondispersive, and homogeneous, each component of P=[P1,P2, P3] is a linear
combination of the three components of E=[E1,E2, E3]:
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24




















=
++++++++=
++== ∑=
3
2
1
332313
322212
312111
o
333o223o113o332o222o112o331o221o111o
33o22o11o
3
1
o
ε
εεεεεεεεε
εεεε
E
E
E
EEEEEEEEE
EEEEP iii
j
jjii
χχχ
χχχ
χχχ
χχχχχχχχχ
χχχχ
where the indices i, j =1,2,3 denote the x, y, and z components. The dielectric properties
of the medium are described by an array [χij] of 3×3 constants known as the
susceptibility tensor.
Also for anisotropic media, a similar relation as above between D and E applies:
where [εij] are elements of the electric permittivity tensor.




















== ∑=
3
2
1
332313
322212
3121113
1 εεε
εεε
εεε
ε
E
E
E
ED
j
jjii

25. Now, considering an arbitrary fluid flow situation with the Lorentz (1853-1928)
condition:
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where the d’Alembertian operator, , or d’Alembertian for short, is defined by:
If this condition is satisfied, the basic equations for the vector and scalar potentials are:
With definitions for the vector A (i.e., the electromagnetic vector potential) and scalar
φ (i.e., the electromagnetic scalar potential) potentials, which are important parame-
ters when studying Lorentz invariance concepts and non-local phenomena, we obtain:
tt ∂
∂
−=•⇔=
∂
∂
+•
φφ
ΑΑ ∇∇∇∇∇∇∇∇ 0
JA oµ−=
oε
−=
ρ
φ
and:
2
2
2
2 1
tc ∂
∂
−∇=
t∂
∂
−−==
A
EAB φ∇∇∇∇××××∇∇∇∇ and
25

26. We began our discussion of special relativity by demonstrating that the demands of
Maxwell’s theory of electromagnetism were not consistent with the hypotheses of
Newtonian mechanics. Einstein (with Minkowski’s help) has shown us how to
reformulate physics, in terms of a rather elegant four-dimensional mathematical
framework, such that the two classical theories could be welded together into a unified
structure. We will now work out Maxwell’s theory into the same four-dimensional
framework. That this reformulation of electromagnetism will yield rather elegant and
apparently simple equations should be a gratifying reward for our efforts.
26
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vJ ρ=
In our original expression for the current density vector we had:
where ρ is the charge density for a distribution of charge moving with velocity v relative
to an observer. This suggests that the four-vector representation for the current density
might be:
τ
ρρ
µ
µµ
d
xd
uJ ==
where ρ is the invariant charge density (i.e., the charge density as seen by an observer
at rest relative to the charge involved). The components of the current density four-
vector are given by:
],[ vργργµ
cJ =
which is obtained by letting dτ =(1/γ )dt with γ =1/√(1−v2/c2) and using v=dx/dt.

27. This discussion immediately yields an interpretation for the temporal component of J µ.
Evidently, J 0 =γ ρc is just c times the charge density as seen by an observer relative to
whom the charges are moving with velocity v (i.e., J 0 is c times the charge density of the
charges having current density J=γ ρv). In the limit v/c<<1, we have γ ≅1, and:
27
2017
MRT
vJ ρρ ≅≅ andcJ 0
as the nonrelativistic approximations for the components of the current density four-
vector.
A particular pleasant result of this formulation of a four-vector current density is that the
equation of continuity takes a very simple form, namely:
03
3
2
2
1
1
0
0
3
0
=∂+∂+∂+∂=∂=
∂
∂
∑∑=
JJJJJ
x
J
µ
µ
µ
µ
µ
µ
and which, otherwise, would be written as:
0)(
)(1
=•+
∂
∂
vργ
ργ
∇∇∇∇
t
c
c
],,,[],[],,,[ 3210
zyxctctxxxxx ===≡ rµ
x
where we have set the following Cartesian coordinates to simplify things:

28. The definitions of the vector potential A and the scalar potential φ, namely:
28
2017
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and:
∫∫∫∫∫∫ ==
VV
dd r
rr
rvr
r
rr
rJ
rA 3o3o )()(
π4
µ)(
π4
µ
)(
−−−−−−−−
ρ
∫∫∫∫∫∫ ==
VV
dcd r
rr
r
r
rr
r
r 32o3
o
)(
π4
µ)(
επ4
1
)(
−−−−−−−−
ρρ
φ
suggests that we define a four-vector potential, Aµ, in terms of the current density J µ. To
generalize these formulas for A and φ directly to a four-vector potential, however, would
demand that we handle the denominators and the integrals in a covariant (relativistic)
manner. Instead, we simply note that in such a definition the temporal component would
be expected to behave like φ/c, since φ is equal to c times the constant µo/4π times an
integral over ρ(r)c, whereas the spatial component would be expected to behave like A,
since A is equal to the same constant, µo/4π, times a similar integral over ρ(r)v(r).

29. To check this, we recall the equations:
29
2017
MRT
vJA ργ
ργ
φ
µ
µ
µ
µ
µ
µ
oo
0
o
o
0
o
µµµ
εε
==∂∂===∂∂ ∑∑ andJc
c
J
we then have:
or, if we let:






=≡ A,
1
φν
c
AA
we have the relativistically covariant equation:
ν
µ
ν
µ
µ
JA oµ=∂∂∑
(N.B., Both sides of this equation transform like four-vectors under Lorentz transfor-
mations – of which we will discuss later – which is a fundamental requirement).
and note that the relativistic interpretation of the equations would require us to replace ρ
by γ ρ and to think of J as being equal to γ ρv instead of ρv. Since the d’Alembertian
takes on the four-dimensional form:
JA o2
2
2
2
o
2
2
2
2
µ
1
ε
1
−=








∂
∂
−∇−=








∂
∂
−∇
tctc
and
ρ
φ
∑∑∑ ∂∂−≡
∂
∂
∂
∂
−=
∂
∂
−
∂
∂
∂
∂
=
∂
∂
−∇=
== µ
µ
µ
µ
µ
µ
3
0
2
2
2
3
1
2
2
2
2 11
xxtcxxtc i ii

30. Previously we found that the fields E and B were given in terms of A and φ by the
following equations:
30
2017
MRT
ΑΑΑΑ××××∇∇∇∇∇∇∇∇ =
∂
∂
−−= B
A
E and
t
φ
thus:








∂
∂
−
∂
∂
−=
∂
∂
−
∂
∂
−=








∂
∂
−
∂
∂
−=
∂
∂
−
∂
∂
−=








∂
∂
−
∂
∂
−=
∂
∂
−
∂
∂
−=
∂
∂
−
∂
∂
=
∂
∂
−
∂
∂
−=
∂
∂
−
∂
∂
=
∂
∂
−
∂
∂
−=
∂
∂
−
∂
∂
=
∂
∂
−
∂
∂
−=
2
1
1
2
2
1
1
2
1
3
3
1
1
3
3
1
3
2
2
3
3
2
2
3
0
3
3
0
0
3
3
0
0
2
2
0
0
2
2
0
0
1
1
0
0
1
1
0
)()()()(
)()()()()()()()(
x
A
x
A
x
A
x
A
B
x
A
x
A
x
A
x
A
B
x
A
x
A
x
A
x
A
B
x
cA
x
cA
x
cA
x
cA
E
x
Ac
x
Ac
x
Ac
x
Ac
E
x
Ac
x
Ac
x
Ac
x
Ac
E
zy
xz
yx
and
,,
,,
This suggests that we define a new quantity:
µνµννµ
ν
µ
µ
ν
µν
FAA
x
A
x
A
F −=∂−∂=
∂
∂
−
∂
∂
≡
so that:
yxzzyx BFBFBFE
c
FE
c
FE
c
F −=−=−==== 132312302010 111
and,,,,

31. If one knows the components of the four-potential in one frame, one can make a
Lorentz transformation to a moving frame and, from the potentials, find the electric and
magnetic intensities. However, often one knows the field in one frame and would like to
find the fields in another frame without going through the potentials. This can be done
through the field tensor Fµν, given by:
31
2017
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where the four-vector operator ∂∂∂∂ is defined by:
44332211 ˆˆˆˆ
xxxx ∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
= eeee∂∂∂∂
with:
( )104
−==== ixitciwx
























∂
∂
−
∂
∂
∂
∂
−
∂
∂
∂
∂
−
∂
∂
∂
∂
−
∂
∂
∂
∂
−
∂
∂
∂
∂
−
∂
∂
∂
∂
−
∂
∂
∂
∂
−
∂
∂
∂
∂
−
∂
∂
∂
∂
−
∂
∂
∂
∂
−
∂
∂
∂
∂
−
∂
∂
==
0
0
0
0
2
3
3
2
1
3
3
1
0
3
3
0
3
2
2
3
1
2
2
1
0
2
2
0
3
1
1
3
2
1
1
2
0
1
1
0
3
0
0
3
2
0
0
2
1
0
0
1
x
A
x
A
x
A
x
A
x
A
x
A
x
A
x
A
x
A
x
A
x
A
x
A
x
A
x
A
x
A
x
A
x
A
x
A
x
A
x
A
x
A
x
A
x
A
x
A
F A××××∂∂∂∂µν

32. These quantities Fµν can be represented in the form of an antisymmetric matrix:
32
2017
MRT




















−
−
−
−−−
=
0
1
0
1
0
1
111
0
xyz
xzy
yzx
zyx
BBE
c
BBE
c
BBE
c
E
c
E
c
E
c
F µν
and they transform as the components of a tensor of rank two under Lorentz transforma-
tions (N.B., more on these Lorentz transformation just a bit later; but know that the point
I want to make here is that things will not change between two different observers:
∑∑ ΛΛ=
µ ν
νµσ
ν
ρ
µ
σρ
FF
where Fρσ are the components of the electromagnetic field tensor as seen by an
observer O in frame S. Fµν are the corresponding components as seen by an
observer O in frame S, and the Λµ
ν are the Lorentz transformation coefficients that
carry tensors in a frame S to a frame S. Evidently E and B cannot be generalized
directly as four-vectors. Rather, the components of E and B actually are components
of an antisymmetric tensor of rank two, Fµν. In fact, E and B do not transform as
would the spatial parts of four-vectors under Lorentz transformations. Let us check…

33. In matrix notation, we then have:
33
2017
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Note that the third matrix on the right hand side of the equation is the transpose of
the first (i.e., the matrix formed by interchanging rows and columns so that the
element in the µ-th row and ν -th column of the original matrix appears in the ν -th
row and µ-th column of the transpose matrix).














ΛΛΛΛ
ΛΛΛΛ
ΛΛΛΛ
ΛΛΛΛ




























ΛΛΛΛ
ΛΛΛΛ
ΛΛΛΛ
ΛΛΛΛ
=














ΛΛΛΛ
ΛΛΛΛ
ΛΛΛΛ
ΛΛΛΛ




























ΛΛΛΛ
ΛΛΛΛ
ΛΛΛΛ
ΛΛΛΛ
=
ΛΛ=














= ∑ ∑= =
3
3
3
2
3
1
3
0
2
3
2
2
2
1
2
0
1
3
1
2
1
1
1
0
0
3
0
2
0
1
0
0
33323130
23222120
13121110
03020100
3
3
3
2
3
1
3
0
2
3
2
2
2
1
2
0
1
3
1
2
1
1
1
0
0
3
0
2
0
1
0
0
3
3
3
2
3
1
3
0
2
3
2
2
2
1
2
0
1
3
1
2
1
1
1
0
0
3
0
2
0
1
0
0
33323130
23222120
13121110
03020100
3
3
3
2
3
1
3
0
2
3
2
2
2
1
2
0
1
3
1
2
1
1
1
0
0
3
0
2
0
1
0
0
3
0
3
0
33323130
23222120
13121110
03020100
][][][
FFFF
FFFF
FFFF
FFFF
FFFF
FFFF
FFFF
FFFF
F
FFFF
FFFF
FFFF
FFFF
F
T
T
µ ν
σ
ν
νµρ
µ
σρ

34. The field tensor in the S frame (moving at velocityβ = v/c in the x-direction) is therefore:
34
2017
MRT
where γ =1/√(1−v 2/c2) from which we obtain:


















−




















−−
−−
−−
−−−


















−
=
1000
0100
00
00
0
1
0
1
0
1
111
0
1000
0100
00
00
γγβ
γβγ
γγβ
γβγ
σρ
xyz
xzy
yzx
zyx
BBE
c
BBE
c
BBE
c
E
c
E
c
E
c
F
,,, )()( yzzzyyxx BvEEBvEEEE +=−== γγ






−=





+== yzzzyyxx E
c
v
BBE
c
v
BBBB 22
γγ ,,
and

35. These equations can be used to ascertain the effect of uniform motion upon the field of
a point charge. Let q be moving with speed u=|u|. We take the x-axis through the charge
and in the direction of its motion. Consider any point P not on the x-axis and let this point
and u establish the xy-plane. We choose an S frame in which q is at rest; in this frame
the field is purely electrostatic, and its components are:
35
2017
MRT
Since the S frame is moving with a velocity v=−u relative to S, the transformation
equations to the S frame corresponding to the above are:
0
επ4επ4 3
o
3
o
=== zyx E
r
yq
E
r
xq
E and,
)()( yzzzyyxx BuEEBuEEEE &−=+== γγ ,,






+=





−== yzzzyyxx E
c
u
BBE
c
u
BBBB 22
γγ ,,
and

36. Now B=0 in S, and so Bx =By =Bz =0. By the Lorentz transformations:
36
2017
MRT
Further:
yytux
cu
tux
x =−=
−
−
= and)(
1 22
γ
Upon making the substitutions in the equation above, we obtain:
0
)sin1(επ4
sin)1(
)sin1(επ4
cos)1(
232222
o
22
232222
o
22
=
−
−
=
−
−
= zyx E
cur
cuq
E
cur
cuq
E ,,
θ
θ
θ
θ
232222
o
22
)sin1(επ4
sin)1(
00
cur
cuuq
BBB zyx
θ
θ
−
−
=== ,,
and:
θθ sincos ryrtux ==− and








−=








+=+=+−=+=
2
22
22
2
2
22222222222222
sin
1
sin
cossincos)(
c
u
r
rrrytuxyxr
θ
γ
γ
θ
θγθθγγ
and

37. The fact that:
37
2017
MRTThe concentration of the electric flux of a moving charge into the equatorial plane; the decimals give the
fraction of the total flux lying within a cone with axis u. The inverse-square law applies at all speeds.
θ
θ
θ
θ
θ
θ
θ
tan
cos
sin
)sin1(επ4
cos)1(
)sin1(επ4
sin)1(
232222
o
22
232222
o
22
==
−
−
−
−
=
cur
cuq
cur
cuq
E
E
x
y
shows that the electric field points radially outward from the instantaneous location of q.
In spherical polar coordinates E is in the r direction and B in the ϕ direction in any
inertial frame with its origin instantaneously at q. However, the field of a moving charge
is concentrated into the equatorial plane (see Figure)).
0
u =0.9c
0
u =0.99c
01.0
0.9
0.5
β =0.9β =0 β =0.99
1.0 1.0
0.5
0.8
0.6 0.5
0.7
0.4
0.3
0.2
0.1

38. One of Maxwell’s equations involves ∇∇∇∇•E, and because of the manner in which Fµν is
constructed, this suggests that we calculate the four-divergence of Fµν (i.e., calculate
Σµ∂µ Fµν ), as a first attempt at obtaining the desired equations. Thus:
38
2017
MRT
∑∑∑∑ ∂∂−∂∂=∂−∂∂=∂
µ
µ
µ
ν
µ
νµ
µ
µ
µννµ
µ
µ
µν
µ AAAAF )()(
Recall that observable electromagnetic fields are unchanged by gauge transformations
of the second kind. Let us consider adding to Aµ another four-vector α µ(xν )=α µ:
µµµµ
α+=→ AAA
then we get:
µνµνµννµµννµµννµνµν
αα fFAAAAF +=∂−∂+∂−∂=∂−∂= )()(
where f µν =∂µαν −∂να µ. If f µν =0 (i.e., if α µ is such that ∂µαν −∂να µ =0), then α µ(xν ) is a
gauge transformation of the second kind, and it leaves Fµν unchanged. Because of this
freedom we have in choosing Aµ, we may always assume Σµ∂µ Aµ =0. If, for example, we
let Aµ →Aµ − Aµ +α µ, with am satisfying Σµ∂µα µ =−Σµ∂µ Aµ and ∂µαν =∂να µ =0, then
Σµ∂µ Aµ =0, and we may continue the discussion,replacing Aµ everywhere with Aµ =Aµ +α µ.
For instance, we may let α µ =∂µχ, where χ satisfies the inhomogeneous wave equation
Σµ∂µ∂µχ=−Σµ∂µ Aµ.

39. Taking Σµ∂µ Aµ =0, then, we get for the four-divergence of Fµν the result:
39
2017
MRT
ν
µ
νµ
µ
µ
µν
µ JAF oµ=∂∂=∂ ∑∑
where we have used Σµ∂µ∂µ Aν =µo Jν. If we now let ν =0, this equation becomes:
o
o
30
3
20
2
10
1
00
0
ε
1
µ
1 ργ
ργ
c
c
c
FFFF ==•=∂+∂+∂+∂ E∇∇∇∇
or:
oε
ρ
=•E∇∇∇∇
where ρ =γ ρ and which is the first Maxwell equation when the apparent increase in
charge density, arising from the motion of the charges making up ρ with velocity v
relative to the observer, is taken into account. Similarly, if we let ν =1, we get:
x
yzx
J
z
B
y
B
t
E
c
FFFF o2
31
3
21
2
11
1
01
0 µ
1
=
∂
∂
−
∂
∂
+
∂
∂
−=∂+∂+∂+∂
or:
t
E
c
J x
xx
∂
∂
+= 2o
1
µ)( B××××∇∇∇∇
which is the x-component of the fourth Maxwell equation (i.e.,∇∇∇∇××××B=µoJ +µoεo∂E/∂t). The
y- and z-components are obtain by calculating Σµ∂µ Fµ2 and Σµ∂µ Fµ3, so that we obtain:
tc ∂
∂
+=
E
JB 2o
1
µ××××∇∇∇∇

40. So, Σµ∂µ Fµν =Σµ∂µ∂µ Aν =µo Jν yields two of Maxwell’s equations. What of the other two
(i.e., ∇∇∇∇•B=0 and ∇∇∇∇××××E=−∂B/∂t). To obtain them, we utilize the ‘4D’ Levi-Civita symbol:
40
2017
MRT





−
+=
))3,2,1,0(),,,(1
)3,2,1,0(),,,(1
)3,2,1,0(),,,(0
ofnpermutatioanisif
ofnpermutatioanisif
ofnpermutatioaisunless
odd
even
σρνµ
σρνµ
σρνµ
εµνρσ
Clearly, εµνρνρνρνρσ =−εµρρρρννννσ; in general, a change of sign occurs every time a pair of adjacent
indices is interchanged. This tensor has 44 =256 elements, so we shall not write down all
of them. The nonzero elements are:
1021301320321301231203201210320312310130210231230 −============ εεεεεεεεεεεε
and:
1132012031032012302310312310230213210201321302301 +============ εεεεεεεεεεεε

41. Now we can define:
41
2017
MRT
That this divergence vanishes follows from the fact that εµνρσ is antisymmetric under the
interchange of indices, for Σµρενµρσ ∂µ∂ρ =−Σµρενρµσ ∂µ∂ρ and upon relabeling the dummy
indices µ and ρ on the right-hand side (i.e., certainly Σµν aµν aµν =Σνµaνµaνµ since all that
has been done here is to exchange the labels of the first and second indices), we obtain
Σµρενµρσ ∂µ∂ρ =−Σµρενµρσ ∂ρ ∂µ =−Σµρενµρσ ∂µ∂ρ which must vanish since the expression
on the far right of this relation is the negative of that on the far left. Hence we have:
∑∑∑ ==
σρ
σρ
σρνµ
ρ σ
σρ
σρνµµν εε FFG
2
1
2
1
Taking the divergence of this quantity, we obtain:
0
)(
2
1
)(
2
1
2
1
=
∂∂−∂∂−=
∂−∂∂=∂=∂
∑
∑ ∑∑∑
σρµ
ρσµσρµ
σρνµ
µ σρµ
ρσσρµ
σρνµ
σρ
σρ
σρνµ
µ
µ
µν
µ
ε
εε
AA
AAFG
0
2
1
2
1
=∂−=∂=∂ ∑∑∑ σρµ
ρσµ
σρµν
σρµ
ρσµ
σρνµ
µ
µν
µ
εε FFG

42. Letting ν =0 in this last equation yields:
42
2017
MRT
0=•B∇∇∇∇
or:
B•−=∂−∂−−∂=∂+∂+∂=
∂−∂+∂−∂+∂−∂−=∂−= ∑
∇∇∇∇zzyyxx BBBFFF
FFFFFFF
213132321
213123132312321231
0 )(
2
1
2
1
0
σρµ
ρσµ
σρµε
which is the second Maxwell equation. For ν =1, the result is:
t
Bx
x
∂
∂
−=)( E××××∇∇∇∇
or:






∂
∂
+





∂
∂
−
∂
∂
−=∂+∂+∂=
∂−∂+∂−∂+∂−∂=∂−= ∑
yz
x
E
cz
E
cyt
B
c
FFF
FFFFFFF
111
)(
2
1
2
1
0
023302320
203023032302320230
1
σρµ
ρσµ
σρµε
follows. This is the x-component of the third Maxwell equation (i.e., ∇∇∇∇××××E=−∂B/∂t); we get
the y- and z-components by letting ν =2 and 3, respectively, so we get:
which is the last Maxwell equations obtained from our four-dimensional formalism.
t∂
∂
−=
B
E××××∇∇∇∇

43. The equation Σµ∂µGµν =½Σµρσ ενµρσ ∂µFρσ =0 above is often written in the equivalent
form:
43
2017
MRT
0=∂+∂+∂ λνµµλννµλ
FFF
where (λ,µ,ν) is taken successively to be (0,1,2), (1,2,3), (2,3,0) and (3,0,1), this yielding
four equations equivalent to the vector equations ∇∇∇∇•B=0 and ∇∇∇∇××××E=−∂B/∂t.
Now onto electrodynamics and the introduction of the energy-momentum tensor. An
important relationship that is expressed economically in four-dimensional tensor form is
the law of forces in electrodynamics, namely:
∑=
λ
λ
λνν
JFf
where f ν is the force per unit volume exerted upon a current density Jλ =Σµηλµ Jµ by the
field Fλν. Thus:
)(
100
BvEFJE ××××++++ργ
λ
λ
λ
=≡•== ∑ i
f
c
JFf and
where J=γ ρv. The second equation summarizes the laws of Coulomb and of Biot-
Savart, whereas the first is equal to 1/c times the rate, per unit volume, at which the field
expends energy in accelerating the charge distribution ρ. All of this is implicit in the
fundamental law of electrodynamics as expressed in f ν =Σλ FνλJλ.

44. If we make use of Σµ∂µ Fµν =µo Jν and f ν =ΣµFνλJλ, we obtain;
44
2017
MRT
By interchanging indices in the last term of the above equation, we obtain:








∂−∂=∂= ∑∑∑ µλ
µλ
λνµ
µλ
µλ
λνµ
µλ
µλ
µλνν
FFFFFFf )()(
µ
1
)(
µ
1
oo
∑∑∑∑∑ ∂+∂=∂=∂−=∂=∂
µλ
µλ
µνλλνµ
µλ
µλ
µνλ
µλ
µλ
µνλ
µλ
λµ
µνλ
µλ
µλ
λνµ
FFFFFFFFFFF )(
2
1
)()()()(
Here we invoke ∂λFµν +∂νFλµ+∂µFνλ=0 above to convert the right-hand side of the
above equation into a single term, so that:
∑∑∑∑ ∂=∂−=∂−=∂
µλ
µλ
µλν
µλ
µλ
µλν
µλ
µλ
λµν
µλ
µλ
λνµ
FFFFFFFF
4
1
4
1
)(
2
1
)(
Substitution of this result in f ν above gives:
∑∑ ∑∑∑∑ ∂=
















−∂=








∂−∂=
µ
ν
µ
µ
µ λκ
λκ
λκν
µ
λ
µλ
λνµ
µλ
µλ
µλν
µλ
µλ
λνµν
δ TFFFFFFFFf
4
1
µ
1
4
1
)(
µ
1
oo
where:








−≡ ∑∑ λκ
λκ
λκν
µ
λ
µλ
λνν
µ δ FFFFT
4
1
µ
1
o
The force density f ν, then, can be expressed as the four-divergence of a mixed
tensor Tν
µ.

45. In order to evaluate the momentum and energy densities, let us calculate Tν
µ in terms
of E and B. The first term in Tν
µ =(1/µo)(Σλ FνλFµλ−¼δ ν
µΣκλ FκλFκλ) is proportional to:
45
2017
MRT






















++−−−−−−
−−++−−−−
−−−−++−−
−−−−
=




















−
−
−
−−−
−=












−
−
−




















−
−
−
−−−












−
−
−




















−
−
−
−−−
−=
−=−= ∑∑∑
222
222
2
222
22
22
222
2
2
2
111
)(
1
111
)(
1
111
)(
1
)(
1
)(
1
)(
11
0
1
0
1
0
1
111
0
1000
0100
0010
0001
0
1
0
1
0
1
111
0
1000
0100
0010
0001
0
1
0
1
0
1
111
0
yxzzyzyxzxzz
zyzyxzyyxyxy
zxzxyxyxzyxx
zyx
xyz
xzy
yzx
zyx
xyz
xzy
yzx
zyx
xyz
xzy
yzx
zyx
BBE
c
BBEE
c
BBEE
cc
BBEE
c
BBE
c
BBEE
cc
BBEE
c
BBEE
c
BBE
cc
cccc
BBE
c
BBE
c
BBE
c
E
c
E
c
E
c
BBE
c
BBE
c
BBE
c
E
c
E
c
E
c
BBE
c
BBE
c
BBE
c
E
c
E
c
E
c
FFFFFF
BE
BE
BE
BEBEBEE
××××
××××
××××
××××××××××××
λ
σµ
ρσ
λρ
λν
λ
λµ
λν
λ
µλ
λν
ηη

46. The last term in Tν
µ =(1/µo)(Σλ FνλFµλ−¼δ ν
µΣκλ FκλFκλ) is proportional to the trace (i.e.,
the sum of the diagonal elements) of the first term:
46
2017
MRT
or:






−=






++−++−++−−−=− ∑
22
2
222
2
222
2
222
2
2
2
1
2
1
1111
4
1
4
1
BE
E
c
BBE
c
BBE
c
BBE
cc
FF yxzxzyzyx
ν
µ
ν
µ
λκ
λκ
λκν
µ
δ
δδ






























−






−






−






−
=− ∑
22
2
22
2
22
2
22
2
1
2
1
000
0
1
2
1
00
00
1
2
1
0
000
1
2
1
4
1
BE
BE
BE
BE
c
c
c
c
FF
λκ
λκ
λκν
µδ

47. Using the two previous matrices into Tν
µ =(1/µo)(Σλ FνλFµλ−¼δ ν
µΣκλ FκλFκλ), we get:
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





























−+





−−−
−





−+





−−
−−





−+





−






+
−=
2222
222
2
2222
22
22
2222
2
22
2
o
2
1
2
1111
)(
1
1
2
1
2
111
)(
1
11
2
1
2
11
)(
1
)(
1
)(
1
)(
11
2
1
µ
1
BEBE
BEBE
BEBE
BEBEBEBE
zzzyzyxzxzz
zyzyyyyxyxy
zxzxyxyxxxx
zyx
BE
c
BBEE
c
BBEE
cc
BBEE
c
BE
c
BBEE
cc
BBEE
c
BBEE
c
BE
cc
cccc
T
××××
××××
××××
××××××××××××
ν
µ

48. Hence, according to our interpretation of −T0
0 as the energy density E of the field, we
have:
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We have ignored most of the components of Tµ
ν . Since the previous matrix
representing Tµ
ν is symmetric in its indices (i.e., Tµ
ν =Tν
µ), there are only 10, rather than
16, independent components of the tensor. We have discussed four of these
components; the other six define the so-called Maxwell stress-energy tensor, which is a
tensor under three-dimensional rotations. These do not play any important role in our
discussion of the four-dimensional formulation of electromagnetic theory, and we will not
consider them further.








+=−= 2
o
2
o
0
0
µ
1
ε
2
1
BETE
here we have used the relationship 1/c2 =εoµo. Furthermore, since −(1/c)T i
0, for i=1,2,3,
are the components of the momentum density P, we also have the result:
BE××××oε=P
Sometimes one writes P =(1/c2)[E××××(1/µo)B], and [E××××(1/µo)B] is called the Poynting
vector. It is important to note that the energy and momentum densities, E and P, are not
components of a four-vector, but rather are components of a tensor (i.e., the energy-
momentum – in this case mixed – tensor Tν
µ) of rank two, just as E and B are not
components of four-vectors but rather are components of rank-two tensors.

49. Soon we shall turn away from relativity theory and shall investigate another paradox in
physics, a paradox that arose shortly after the conflict between classical Newtonian-
Maxwellian theory and the experimental data of Michelson and Morley, Trauton and
Noble, &c., was discovered. The first moderately satisfactory solution of this problem
was found by Max Planck in 1900. The problem involved the distribution of energy
among electromagnetic waves contained within a closed cavity. It is appropriate that we
should turn to this problem now, since we have just completed deriving an expression for
the energy density of the electromagnetic field. However, a brief investigation of a
particular type of solution to the electromagnetic wave equation is in order before we go
on to consider the second paradox.
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0=∂∂∑µ
νµ
µ A
Recall that Σµ∂µ∂µ Aν =µo Jν is the inhomogeneous wave equation for the four-vector
potential Aν – inhomogeneous because of the term µo Jν on the right-hand side. We shall
consider solutions to this equation in regions of space where Jν =0; that is, we shall be
concerned only with solutions to the homogeneous wave equation:
or:
0),,,( 3210
23
2
22
2
21
2
20
2
=








∂
∂
−
∂
∂
−
∂
∂
−
∂
∂
xxxxA
xxxx
ν
Plane Wave Solutions of the Wave Equation

50. In particular, we shall consider solutions to this last equation that are confined to a
three-dimensional parallelepiped bounded by the coordinate axes and the planes xi =Li
(i=1,2,3), and confined within the fixed time interval 0≤x0/c=t≤T. With these restrictions,
our last equation can be solved by the method of separation of variables. In this method,
we assume that the solution for Aν (xµ) can be factored into four functions, each involving
only one of four space-time variables xµ; thus we assume:
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)()()()(),,,( 3
3
2
2
1
1
0
0
3210
xXxXxXxXxxxxA ννννν
=
Upon substituting this into our last equation and dividing the result by Aν (x0,x1,x2,x3), we
find:
0
1111
23
3
2
3
22
2
2
2
21
1
2
1
20
0
2
0
=
∂
∂
−
∂
∂
−
∂
∂
−
∂
∂
x
X
Xx
X
Xx
X
Xx
X
X
ν
ν
ν
ν
ν
ν
ν
ν
Each term on the left-hand side depends only upon one variable; in particular:
20
3
1
2
2
20
0
2
0
)(
11
k
x
X
Xx
X
X i
i
i
i
−=
∂
∂
=
∂
∂
∑=
ν
ν
ν
ν
where (k0)2 is a constant, because the left-hand member depends only upon x0 and the
middle member depends only on x. We use the symbol −(k0)2 for the constant because
it turns out that the desired solutions are arrived at more quickly when we take the
constant to be negative definite, and because k0 is to represent the temporal compo-
nent of a four-vector, as the left-hand member of the above equation suggests.

51. Now we may solve the spatial and temporal parts of the last equation independently. In
the first place, we have:
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where we have replaced partial derivatives with ordinary derivatives because X0
ν =X0
ν(x0)
depends upon only the one variable, x0. A more familiar form for this equation is:
20
20
0
2
0
)(
1
k
xd
Xd
X
−=
ν
ν
0)( 0
20
20
0
2
=+ ν
ν
Xk
xd
Xd
and as it is well-known, the most general solution for this second-order differential
equation can be written in the form:
0000
ee)( 00
0
0
xkixki
baxX −
+= ννν
The spatial portion can be solved in a similar way. First we separate the i=1 term from
the others in the sum, and write the equation in the form:
21
3
2
2
2
20
21
1
2
1
)(
1
)(
1
k
x
X
X
k
x
X
X i
i
i
i
−=








∂
∂
+−=
∂
∂
∑=
ν
ν
ν
ν
where (k1)2 again is a constant, since the left-hand member here is dependent upon
x1 only, whereas the middle member depends only upon x2 and x3.

52. The solution for X1
ν then can be written in the form:
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0)()()()( 220
3
1
220
=−=− ∑=
kkkk
i
i
Similarly, by continuing these arguments, we obtain:
1111
ee)( 11
1
1
xkixki
baxX −
+= ννν
33332222
ee)(ee)( 33
3
322
2
2
xkixkixkixki
baxXbaxX −−
+=+= νννννν
and
where k0, k1, k2, and k3 must satisfy:
This last equation must be satisfied in all coordinate frames because the equation that
we are solving (i.e., Σµ∂µ∂µ Aν =0) is relativistically covariant (i.e., it is valid in every
inertial frame of reference). Therefore (k0)2 −(k)2 must be an invariant quantity, and since
it has the form of the inner product of two four-vectors, we conclude that:
],[],,,[ 03210
kkkkkkk ==µ
is a contravariant four-vector. Similarly:
],[ 0
k−== ∑ kkk
ν
ν
νµµ η
is a covariant four-vector and ηµν is the metric.

53. Now ∂µ Aν must be a mixed tensor of rank two, and for this reason, only some of the
possible combinations of values of a0
ν,…,a3
ν, b0
ν,…,b3
ν represent manifestly acceptable
solutions for our system of electromagnetic waves in a box. Because of this restriction
that ∂µ Aν be a tensor, we must have a1
ν =a2
ν =a3
ν =0 whenever a0
ν ≠0 and b1
ν =b2
ν =b3
ν =
0 whenever b0
ν ≠0, so that only the invariant form Σµkµ xµ will appear in the exponents
that result when the X0
ν,…,X3
ν given by the equations above are multiplied together.
Thus the most general solution of this kind is:
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∑∑ −
−+ += µ
µ
µµ
µ
µ ννν xkixki
AAxxxxA ee),,,( 3210
where:
νννννννννν
32103210 aaabAbbbaA == −+ and
The invariant that appears in the exponentials above can be written:
xkxk •−=•−=∑ tckxkxk 000
µ
µ
µ
where if we let k0 =ω/c=2πν /c and let k=(2π/λ)k with k=k/|k| representing the direction
of the wave vector k (while |k| is called the reduced wave number |k|=k=2π/λ) we get:
ˆ ˆ







 •
−=•−=∑ λ
ν
µ
µ
µ
xk
xk
ˆ
π2ω ttxk
Note that if kµ, and hence ν and λ, are real, then ν represents an oscillation fre-
quency and λ represents a wavelength. Also ω=2πν =k0/c is an angular frequency.

54. Since:
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







−





=−





=−== ∑ 2
2
22
2
220 1
)π2(
ω
)()(0
λ
ν
µ
µ
µ
cc
kkk kk
or:
λ
ν
λ
ν c
c
±=⇒=−





0
1
2
2
we see that the familiar relationship between wavelength, frequency and the velocity of
propagation hold if we adopt the positive sign. Our interest will be confined to real values
of kµ, since we are concerned with the existence and properties of electromagnetic waves.
The solution given for Aν(x0,x1,x2,x3) above, then, represents waves with frequency ν
=ω/2π=k0c/2π and wavelength λ=2π/|k|. The equation Σµkµ xµ =ωt−k•x=2π(νt− k•x/λ2)
above indicates that the waves are traveling in direction k with velocity c=λν. These
waves are called plane waves, because at every point in a plane perpendicular to k, the
values of the phase Σµkµ xµ will be the same (i.e., the wave fronts of these waves are
planes perpendicular to k).
ˆ

55. Finally, let the amplitude of a light wave of frequency ν and wavelength λ be
represented by Acos(ωt−k•x), where ω=2πν and k=(2π/λ)k, k being the direction of
propagation of the wave. Since, according to special relativity, light travels with the same
speed in all inertial frames of reference, ωt−k•x must be an invariant under Lorentz
transformations:
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xkxk •−=•− tt ωω
where the bared quantities are in the S-frame moving with velocity v (along the z-axis,
say) relative to S. This suggests that we define:
ˆˆ






== kk ,
ω
],[ 0
c
kk µ
so that:
xk •−=∑ tkk ω
µ
µ
µ
as seen above so from the fact that Σµkµkµ is an invariant we can show that kµ is a true
four-vector.

56. Because of the freedom allowed by the principle that physical variables are invariant
under gauge transformations of the second kind, we always may impose the Lorentz
condition:
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0=∂∑µ
µ
µ A
as we have seen already. Incidentally, when one imposes this condition, he is said to be
working in the Lorentz gauge. Even when this condition is imposed, however, there is
still an infinite number of four-potentials Aµ that will give the same physical effects and
satisfy the Lorentz condition. This is because we are still free to add another four-vector
to Aµ, of the form ∂µλ and satisfy Σµ∂µ ∂µλ =0 without disturbing any physics and without
upsetting the Lorentz gauge. The existence of the special gauge transformation of the
second kind allows us to impose the additional condition that:
0=• A∇∇∇∇
in some convenient frame of reference. This gauge is unlike the Lorentz gauge in that it
is not Lorentz invariant. Therefore, once we have chosen a frame of reference in which
we wish to establish the radiation gauge (i.e., in which we wish to take ∇∇∇∇•A=0), we
must be careful to apply any conclusions drawn upon this condition only to the reference
frame. (N.B., In the Lorentz gauge, ∇∇∇∇•A=0 is equivalent to ∂0 A0 =0).

57. If we use our solution Aν (xµ)=A+
ν exp(iΣµkµ xµ)+ A−
ν exp(−iΣµkµ xµ), this means that we
may work in the radiation gauge only if:
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000
=•= AkandA
During the remainder of this discussion of solutions Aν to the homogeneous wave
equation we shall assume that a reference frame has been chosen, and that the
radiation gauge has been established in that frame; this is legitimate because one can
establish such a gauge in any reference frame he or she chooses.
otherwise the only solutions we shall obtain will be trivial ones with k=|k|=0. Since k is
the direction of the wave propagation, this condition implies that the vector A is
transverse to the wave vector k in the chosen reference frame.
ˆ

58. We now wish to study the energy and momentum densities of the fields specified by
the solutions Aν that we found for the inhomogeneous wave equation. If we use Fµν =
∂µAν −∂νAµ into Tν
µ =(1/µo)(Σλ FνλFµλ−¼δ ν
µΣκλ FκλFκλ), we obtain the following
expression for Tν
µ:
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We need not substitute our solution for Aν (xµ)=A+
ν exp(iΣµkµ xµ)+ A−
ν exp(−iΣµkµ xµ) into
all the terms above; the first terms are sufficiently general in character that the last three
terms can be treated as special cases.












∂∂−∂∂−




∂∂−∂∂−∂∂+∂∂=








∂−∂∂−∂−∂−∂∂−∂=
∑∑
∑ ∑∑∑
∑∑
λκ
λκ
κλ
λκ
λκ
λκν
µ
λ λ
µλ
λν
λµ
νλ
λ
µλ
νλ
λ
λµ
λν
λκ
κλλκ
κλλκν
µ
λ
µλλµ
νλλνν
µ
δ
δ
))(())((
2
1
))(())(())(())((
µ
1
))((
4
1
))((
µ
1
o
o
AAAA
AAAAAAAA
AAAAAAAAT

59. Thus, utilizing this solution for Aν, we get:
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We consider first the second term of the result for Tν
µ above, namely:






−





−−=∂∂
∑∑∑∑ −
−+
−
−+
ν
ν
νν
ν
νµ
µ
µµ
µ
µ
ωω
σσ
τ
ρ
ωτ
σρ xkixkixkixki
AAAAkkAA eeee))((
0eeee))(( =





−





−−=∂∂
∑∑∑∑ −
−+
−
−+∑∑ σ
σ
σσ
σ
σρ
ρ
ρρ
ρ
ρ
µµ
νν
λ
λ
λ
λ
µλ
νλ xkixkixkixki
AAAAkkAA
since Σλkλ kλ =(k0)2 −k2 =0. Next the third term of the result for Tν
µ above yields:
0)()(
eeee))((
=∂∂=






−





−−=∂∂
∑
∑ ∑∑∑∑ −
−+
−
−+
λ
λ
λν
µ
λλ
νν
µ
λ
λ
λµ
νλ σ
σ
σσ
σ
σρ
ρ
ρρ
ρ
ρ
AA
AAAAkkAA
xkixkixkixki
because we may impose the Lorentz condition Σλ∂λ Aλ =0. Now the special cases are:
0))(())(( =∂∂=∂∂ ∑∑ λκ
κλ
λκ
λκ
λκ
λκ
AAAA

60. We find, therefore, that only the first term on the right-hand side of our result for Tν
µ
survives; that is:
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whenever Aλ represents a solution to the inhomogeneous wave equation. From this we
find (recall the energy density of the field, E =½[εoE2+(1/µo)B2]):
∑ ∂∂=
λ
λµ
λνν
µ ))((
µ
1
o
AAT
∑ ∂∂−=−=
λ
λ
λ
))((
µ
1
0
0
o
0
0 AATE
and for i=1,2,3:
E
P
k
k
i
iii
xkixkixkixkii
ii
k
c
T
k
c
T
k
k
c
AA
k
k
c
AAAAkk
c
AA
c
T
c
1
11
))((
µ
1
eeee)(
µ
1
))((
µ
11
0
0
0
0
0
0
0
0o
0
o
0
o
0
=
==∂∂=






−





−−=
∂∂−=−=
∑
∑
∑∑∑∑ −
−+
−
−+
λ
λ
λ
λλ
λλ
λ
λ
λ
σ
σ
σσ
σ
σρ
ρ
ρρ
ρ
ρ

61. Writing this last result in three-vector form, we have:
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that is, the momentum density, P, of the wave field is in the direction of advancing
wavefronts, k=k/|k|, and its magnitude is equal to 1/c time the energy density, E.
k
k
k ˆ11
EEP
cc
==
ˆ
Comparing P =εoE××××B derived previously P =(1/c)(k/|k|)E above, we observe that E××××B
also is in the direction of k, and that:








+





=








+





=







+⋅== 2
2
2
2
oo
2
o
2
o
oo
1
2
1
ε2
11
ε
2
1
ε
1
ε
1
BEBEBEBE
c
c
cccc µµ
E××××
where we made use of E =½[εoE2+(1/µo)B2] again. This result can be written in the form:
which implies that |E××××B|=|E||B| so that E is perpendicular to B, and also that (1/c)|E|=|B|.
0
1
2
1 2
2
=+−





BBEE ××××
cc
In summary, k, E, and B, in that order or in any cyclic permutation of that order, form a
right-handed system of orthogonal vectors, and the magnitude of E is equal to c times
the magnitude of B; the energy density of the electromagnetic radiation field therefore is
E =εoE2. These are classical results for wave fields, and may be recalled from
elementary physics courses; here we have seen how they emerge from the four-
dimensional formalism.

62. If the radiation is confined to a finite region of space, the solutions of Aν(xµ) must
conform to restrictions imposed by conditions at the boundaries of the confining region.
We indicated earlier that we would consider radiation contained within a box having
edges with lengths Lm (m=1,2,3), parallel to the coordinate axes. That the sides of this
box are parallel to the coordinate planes curves serves to justify the separation of
variables, for unless the coordinate surfaces are parallel to the boundaries of the
system, the boundary conditions cannot be imposed properly.
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0ee)(0)0( 1111 3,2
1
3,2
11
3,2
1
3,2
1
3,2
1
3,2
1 =+==+= − LkiLki
baLXbaX and
The boundary conditions that we shall use here is equivalent to the assumption that
the walls of the container within which the radiation is confined have perfectly reflecting
surfaces. Thus we require that the components of Aν(xµ) parallel to a (perfectly
reflecting) wall must vanish at the surface of that wall. Since variables have been
separated when we have Aν(xµ)= X0
ν(x0)⋅X1
ν(x1)⋅X2
ν(x2)⋅X3
ν(x3) we may impose the
boundary conditions upon the solution, X1
ν(x1)=a1
ν exp(ik1x1)+b1
ν exp(−ik1x1), and ibid for
X2
ν(x2) and X3
ν(x3) separately. Thus, X1
2(x1) and X1
3(x1) must vanish when x1 =0 and
when x1 =L1; according to X1
ν(x1)=a1
ν exp(ik1x1)+b1
ν exp(−ik1x1), then:
Hence:
0)sin(2 11
3,2
1
3,2
1
3,2
1 =−= Lkaiab and

63. Nontrivial solutions to the last equation can occur only if k1 satisfies:
63
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)sin()()sin()( 332,1
3
22,1
3
221,3
2
21,3
2 xkAxXxkAxX == and
allowing negative values for n1 does not introduce any new solutions. Subject to this
condition on k1, the solutions X1
2,3(x1) are, then:
( )K,3,2,1,0
π
11
1
1
1
==−= nn
L
kk
where A1
2,3 = 2ia1
2,3. Similarly, we obtain:
)sin()( 113,2
1
13,2
1 xkAxX =
with:
( )KK ,3,2,1;,3,2,1,0
π
=== mnn
L
k mm
m
m
(N.B., Here we have reverted to the product form for the four-vector potential, and have
not used the manifestly covariant form of Aν (xµ)=A+
ν exp(iΣµkµ xµ)+ A−
ν exp(−iΣµkµ xµ). In
fact, a certain linear combination of this version of Aν is equal to the product version –
i.e., Aν(xµ)= X0
ν(x0)⋅X1
ν(x1)⋅X2
ν(x2)⋅X3
ν(x3) – used above).

64. The boundary condition, then restricts the values of the spatial components of kµ to
integral multiples of π divided by the corresponding dimension of the box. This gives the
result:
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







++=== 2
3
2
3
2
2
2
2
2
1
2
122220
π)()(
L
n
L
n
L
n
kk k
or:
2
3
2
3
2
2
2
2
2
1
2
10
π
L
n
L
n
L
n
kk ++==
with the nm all integers.

65. A particularly important consequence of this result is that the wavelength of a particular
spectral component of the radiation will increase in proportion to the linear dimension of
the enclosure to which it is confined. For simplicity, let L ≡ L1 =L2 =L3; then:
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If L is increased continuously by an amount that is small compared with λ, no sudden
jump in any of the numbers n1, n2, or n3 is to be expected, for this would imply that
sudden jumps in the wavelength should be observed as the volume is increased; in fact
such jumps are not observed. Hence a small change in L must lead to corresponding
small changes in λ. A spectral component with wavelength λ = 2L/√(n1
2+n2
2 +n3
2), then, is
characterized by the numbers n1, n2, and n3; for the integers n1, n2, n3 specify the essen-
tial features of the spectral component, whatever volume may be, whereas λ changes in
proportion to changes in the linear dimension of the enclosure.
2
3
2
2
2
1
ππ2
nnn
L
k ++==
λ
or:
2
3
2
2
2
1
2
nnn
L
++
=λ

66. Thus, since L3 =V, we may express the proportionality of λ to L=V 1/3 in the form:
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where λ1 is the wavelength of a spectral component when the volume is V1, and λ2 is the
wavelength of the same spectral component when the volume has been changed to V2.
This conclusion holds for enclosures other than cubes also. The entire analysis that
brought us to this conclusion can be confirmed by considering the reflection of radiation
from a moving wall of the container as the container is expanded; however, we shall not
pursue such an analysis here.
31
2
31
1
2
1
V
V
=
λ
λ

67. Let us try to figure out how Einstein saw stuff now that we have figured out how to
formulate Maxwell’s equations in a so called ‘relativistic’ form. Three considerations will
be studied now.
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Consider two equal point charges q moving with the same velocity. In a frame moving
with the charges, they are at rest (see Figure (a)) and experience only an electrostatic
repulsion, FE=qE. In our ‘laboratory’ frame, in which the charges q are moving at speed
v=|v| (see Figure (b)), each charge creates a magnetic field. The force between the
charges is therefore reduced by magnetic attraction, FB=qv××××B. The force between the
charges depends on the frame of reference employed (i.e., the observer’s point of view).
(a) In a frame in which two equal charges are at rest, they experience only electric repulsion. (b) In a
frame in which both charges have the same velocity, they also experience a magnetic
+ +q q
FE = qE
v
+ +q q
FB = q v××××B
v
(A) (B)
FE = qE
FE = qE
FE = qE
FB = q v××××B
67
Special Relativity and Electromagnetism

68. We already considered when a Galilean transformation is applied to Maxwell’s wave
equation, its form changes completely. So if Galilean transformation equations are
correct, Maxwell’s equations are valid in only one special frame – that of the ether. How-
ever, there is no evidence that Maxwell’s equations are restricted in this way. Consider a
short wire moving at constant velocity across the pole of a magnet. In the magnet’s
frame (see Figure (a)), the magnet is at rest and the wire moves at velocity +v. An
observer in the frame says that a charge q in the wire experiences a magnetic force. In
the wire’s frame (see Figure (b)), the wire is a rest and the magnet has velocity −v. Since
the charges are at rest in the wire’s frame, an observer in this frame would say that the
charge q is subject to an electric force. We know experimentally that it is only relative
motion of the wire and the source of the magnetic field that matters. Yet merely switching
from one inertial frame to another requires a change from magnetic field to electric field.
Even if both observers agree on the phenomenon, they use different laws to describe it.
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(a) The charges in a rod moving across the face of a magnet experiences a magnetic force. (b) If the rod
is at rest and the magnet moves in the opposite direction, the charges in the rod experience only an
electric force.
××××
××××
××××
××××
××××
FE =qEFB =q v××××B
××××
××××
××××
××××
××××
(a) (b)
××××
v
v
q q
68

69. As a student, Einstein was aware of these and other problems. Indeed, as a boy of 16,
he conceived of an intriguing question: What would one see if one travels with a beam of
light? One should see stationary sinusoidal variations in space and the electrical and
magnetic fields that constitute the wave. But this is not an acceptable solution of
Maxwell’s wave equation – which requires a wave moving at the speed of light, c. Could
the laws for the traveler be different from those for an observer at rest? Although by
1904 Einstein had found out about the Michelson-Morley experiment through the work of
Lorentz, this experiment did not lay a significant role in the formulation of his theory.
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Einstein had to make a choice. If the Galilean transformation and the laws of
mechanics are correct, then Maxwell’s equations had to be reformulated. If Maxwell’s
equations were correct, then the laws of mechanics were not exactly correct, even
though no exception had yet been encountered. The sticking success of Maxwell’s
theory made improbable that it was incorrect, so he decided that the Galilean
transformations and the laws of mechanics had to be modified. Einstein believed that
there must exist some powerful ‘universal principle’ that would guide him to the ‘true’
laws of physics.
69

70. In June 1905, in a paper entitled On the Electrodynamics of Moving Bodies, Einstein
introduced the special theory of relativity. Here is the opening passage:
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“It is known that Maxwell’s electrodynamics – as usually understood at
the present time – when applied to moving bodies, leads to asymmetries
which do not appear to be inherent in the phenomena. Take for example,
the reciprocal electrodynamic action of a magnet and a conductor. The
observable phenomenon here depends only on the relative motion of
the conductor and the magnet, whereas the customary view draws a
sharp distinction between the two cases in which either the one, of the
other of the bodies, is in motion. Examples of this sort, together with the
unsuccessful attempts to discover any motion of the Earth relative to
the ‘light medium,’ [n.d.l.r., the ‘ether’] suggest that the phenomenon of
electrodynamics as well as of mechanics possess no properties
corresponding to the idea of absolute rest.” (A. Einstein, Annalen der
Physik, 17, 891 (1905))
70

71. The fact that the speed of light, c, is an unattainable speed for a material particle
resolves Einstein’s boyhood question regarding what he would see if he were to ride
along with an electromagnetic wave. He would not see a stationary sinusoidal variation
of electric and magnetic fields because he could never catch up (e.g., with speedu) with
a light wave (i.e., traveling at the speed of light,c). The issue was raised in the quote
from Einstein’s 1905 paper above which recalls that Einstein was uneasy about the use
of an electric field or a magnetic field depending on one’s choice of reference frame (see
Figure).
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respectively.
71
O
y
x
O
y
x
+v
S
S
+v

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(a) A charge moving relative to a wire. The positive and negative charges in the wire are equally spaced
and have equal and opposite velocities. (b) In the frame of the charge q, the positive and negative
charges in the wire have different speeds. The different factors for length contraction mean that the
negative charge density is greater than the positive charge density.
Figure (a) shows a positive charge q moving at velocity u relative to a stationary wire
that carries a current I. For simplicity we assume that the current in the wire arises from
both positive and negative charges moving with opposite velocities, ±v . In the frame of
the wire, the charge q experiences a magnetic force toward the wire, but no net electrical
force. In the frame in which the charge q is at rest (see Figure (b)), it does not
experience and magnetic force. In this frame, the positive charges in the wire move
more slowly than v, where as the negative charges move faster than v. Electric charge is
invariant in special relativity. Hence, because of length contraction, the negative charge
density is greater than the positive charge density. The wire has a net negative charge in
the rest frame of charge q. We see that an electrostatic field in the rest frame of charge q
transforms into a magnetic field in another frame.
q
(a) (b)
+ u
FB = quB
+
−−−−
+
−−−−
+
−−−−
+
−−−−
+
−−−−
+
−−−− −−−−
+
−−−− −−−−
q +
FE = qE
v
v
I
72

73. The greatest impact of the special theory of relativity on electromagnetism is that by
starting with Coulomb’s law and special relativity, we can derive all the laws of
electromagnetism provided we assume the experimentally verified fact that the charge of
a moving particle is the same as when the particle is at rest, or the charge is invariant
with respect to the motion or under Lorentz transformation. Thus:
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The approach to be derived in this chapter can be used to show that the appearance
or nonappearance of the magnetic force between two moving charged particles depends
upon the reference frame of the observer and hence is a relativistic effect. For example,
suppose two charges qo and q are moving with velocities v and u, respectively, parallel to
the x-axis in the initial frame S. The charge q will feel a magnetic force FB =q(u××××B) where
B is the field produced by qo. Let us observe the situation from another frame, S. If S has
a velocity u, the velocity of the charge q will be zero and hence FB =0. If S has a velocity
v, the charge qo will be at rest in S and will not produce B, and, again, FB =0.
Electromagnetism= Coulomb’s Law + Special Relativity
From the above example, we may conclude that electric fields and magnetic fields do
not exist as separate identities, but are combined into a single concept of electromag-
netism. Whether an electromagneticfield will show up as a pure electric field, a pure mag-
netic field, or both will depend upon the reference frame. This leads to the conclusion
that we must have relations to transform different quantities from one reference frame to
another that are in relative motion. Thus we are concerned with 1) the transformation
of charge and current densities; and 2) the transformation equations for the fields.

74. Consider a wire of cross-sectional area Ao and length lo containing N electrons and
lying parallel to the x-axis in the frame S. The charge density, ρo, is Ne/lo Ao, and the
current density, Jo, is ρou=0, because the charges are at rest in S. Let us observe this
wire from the frame of reference S in which it is moving with a velocity u (see Figure).
Thus is the S frame the length of the wire will be lo√(1−u2/c2), while the cross-sectional
area Ao will be unchanged. The charge density in S will be ρ=Ne/lo Ao√(1−u2/c2), and the
current density, J=ρu. Replacing Ne/lo Ao by ρo, we get:
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2
2
o
2
2
o
11
c
u
u
J
c
u
−
=
−
=
ρρ
ρ and
A rod containing N electrons as viewed from two reference frames in relative motion.
x
y
z
x
y
z
oA
u
S
S
2
o 1 β−l
oA
ol

75. If we were dealing with the current-density vector J with components Jx, Jy, and Jz, we
would have obtained the following result:
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with of course, u2 =ux
2 +uy
2 +uz
2. There is an important significance in the equation ρ =
ρo/√(1−u2/c2)… Just as c2t2 −(x2 +y2 +z2) is equal to an invariant quantity c2τ 2, and m2c2 −
(px
2 +py
2 +pz
2) is equal to an invariant quantity mo
2c2; similarly, we can treat p2c2 −(Jx
2 +Jy
2 +
Jz
2) as an invariant quantity equal to ρo
2c2. This means that J and ρ transform exactly like
p and m, and hence if in a general case, S is moving with a velocity v along the x- and x-
axes, the quantities [ρ, Jx , Jy , Jz ] and [ρ, Jx , Jy , Jz ] are related by the transformation
equations:
2
2
o
2
2
o
2
2
o
2
2
o
1111
c
u
u
J
c
u
u
J
c
u
u
J
c
u
z
z
y
y
x
x
−
=
−
=
−
=
−
=
ρρρρ
ρ &,and
zzyy
x
xzzyy
x
x
xx
JJJJ
c
v
vJ
JJJJJ
c
v
vJ
J
c
v
J
c
v
c
v
J
c
v
SSSS
==
−
+
===
−
−
=
−
+
=
−
−
=
&,&,
toFromtoFrom
2
2
2
2
2
2
2
2
2
2
11
11
ρρ
ρ
ρ
ρ
ρ

76. As an example of the application of the above transformation equations we consider a
current-carrying wire at rest in the frame S (see Figure). The positive charges are at rest
while the electrons are moving to the right with a velocity u. Thus the net charge density
is:
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A current-carrying wire as viewed from two reference frames in relative motion.
++++
x
y
z
x
y
z
u
S
S
v
−v
u −v
̶
++++
++++
++++ ++++̶ ̶
̶
̶ ++++
++++ ++++̶
̶
̶
̶
where ne and −ne are the positive and negative charge densities, respectively. The
current density is:
0)( =−+=+= −+
enenρρρ
Because the charge density is zero (i.e., the wire is neutral) there is no electric field,
while there is a magnetic field because the current density is not zero!
uuJJJ ρρρ =+⋅=+= +−+
0

77. Let us view this wire from another reference frame, S, that is moving with a relativistic
velocity v along the x- and x-axes (see previous Figure). The total charge density in S is
given by:
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2
2
2
2
2
2
11
c
v
J
c
v
c
v
J
c
v
xx
−
−
+
−
−
=+=
−−++
−+
ρρ
ρρρ
But ρ+ =+ne, ρ− =−ne, Jx
+ =0, and Jx
− = ρ−u, we thus obtain:
2
2
2
1
c
v
c
uv
en
−
=ρ
The conclusion is that for an observer in S, the wire has E=0 but B≠0, while for an
observer in S both E≠0 and B≠0.

78. These are the basic Hypotheses of Special Relativity:
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1) The laws of physics are independent of the inertial frame of reference in which the
measurements are carried out, or in terms of which the mathematical description of the
laws is formulated;
2) The speed of light does not vary from one inertial frame to another (i.e., the speed of
light is a fixed quantity, having the same value in all inertial frames of reference);
3) Space and time measurements are best made by comparing properties of light
waves.
Since the laws of physics are, according to Hypothesis 2, independent of inertial
reference frame, and since the speed of light (i.e., c=1/√(µoεo) in a vacuum) also is
independent of reference frame, it seems natural to use the velocity of light as a
standard in comparing time and distance measurements carried out in different inertial
frames.
The Special Lorentz Transformations

79. Let us define the fundamental unit of time for our measurements to be the mean
period of electromagnetic waves emitted by a specific decay process from a designated
excited state of an atom, when measured by an apparatus at rest relative to the
decaying atom. We know that there is a well-defined mean frequency for the electro-
magnetic waves emitted by a certain kind of matter under specified conditions. We can
imagine many ways for measuring, or, more accurately, determine the fundamental time
interval τo. This standard interval now is to be recorded in time-recording devices and
compared with other time intervals when measures of the latter are desired.
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Because of Hypothesis 1 above, we expect that the atom will behave in the same way
according to all observers at rest relative to it, no matter how the atom and its observers
together are moving relative to any physical systems. However, if the atom were in
motion relative to the observer, the description of the time measuring process would
have to be changed, for the relationship between the observer and the observed is just
as important as any other part of the description of a measuring process, and without it
the description is incomplete. This suggests that a moving atom may not behave in the
same way as a stationary atom in terms of the measurements the observer carries out
on the two. However, the observer cannot properly carry out a time measurement on the
moving atom’s emitted electromagnetic waves with that of the stationary atom’s wave. If
someone wants to measure the oscillation period of the radiation from the moving atom,
then that someone must find a means to accelerate himself until he is at rest with the
atom before he can carry out an acceptable time measurement.

80. So the key conclusion is: measurements shall be considered acceptably defined only
when the measuring apparatus and the object whose properties are being measured
(e.g., the decaying atom discussion above) are at rest relative to one another.
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Having specified a method for measuring time intervals at one point in space, we can
specify a standard unit of length, again in terms of electromagnetic waves. We need to
know only that the velocity of light is finite, and to take into account Hypothesis 2, to
arrive at a satisfactory definition of a standard distance. Since the speed of light, c, is
independent of reference frame, and since Hypothesis 1 indicates that τo, if determined
according to our prescription, also is independent of reference frame, then the quantity
λo=cτo is also the same in all inertial frames. In addition, λo must be finite, because both
c and τo are finite. Obviously λo can be taken as a standard unit of length for all distance
measurements made in the frame of reference for which the atom defining τo is a rest.
In a nutshell: knowing λo and τo, we can calculate the speed of light c=λo/τo.
Of course it usually is not desirable nor even possible to build apparatus that is at rest
relative to the observer, or to accelerate an experimental set-up so that it will be at rest
relative to a particular observer. For example, decaying subnuclear particles, that are
among cosmic rays, often move very rapidly to Earth-bound observers, and it is quite out
of the question to observe their decays from reference frames relative to which they are
at rest. Thus it is necessary to find techniques for transforming the numerical values of
physical measurements from one frame of reference to another. This, in fact, is the
function performed by the mathematical formulation of relativity theory.

81. We now need to connect measurements made by an observer O along coordinate axes
in frame S with those of another observer O made in S. To do so, we will perform a mathe-
matical derivation that will yield the required transformation equations between frames.
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),,,( tzyxxx =
Because of the impossibility of synchronizing pairs of clocks in different reference
frames, the time coordinate and the space coordinates cannot be kept completely
separate for observers in different reference frames; the spatial separation between
clocks in one frame of reference makes these clocks seem nonsynchronous in any other
frame. Similarly, if two spatially separated events occur at different times in one frame of
reference, the time interval separating the events in that frame will affect the apparent
spatial separation between them as measured in any other coordinate frame. Thus if x,
y, z, and t are the space and time coordinates in S, we must expect that in a comparison
between the coordinate systems, x may depend on x, y, z, and t:
and similarly:
),,,(),,,(),,,( tzyxtttzyxzztzyxyy === and,
We therefore say that the comparison between coordinate systems may be accom-
plished by a coordinate transformation from [x,y,z,t] to [x,y,z,t], and the transformation is
determined by the functional dependence of the unbared coordinates upon the bared
coordinates (N.B., t is given the same status here as the spatial coordinates x, y, and z,
and also, because of the symmetry between S and S, a transformation in the reverse
direction, from coordinates of S to those of S, must exist viz x=x(x,y,z,t), &c.)

82. To make the notation more convenient, we introduce the four-coordinates:
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zxyxxxctx ==== 3210
and,,
which place space and time on the same footing (N.B., x0 has the dimensions of
distance). Similarly, we have:
zxyxxxtcx ==== 3210
and,,
Hence we seek transformations that can be written in such forms as:
)(),,,( 3210 νµµµµ
xxxxxxxxx == or
where µ,ν =0,1,2,3.
Now we apply the usual requirement of classical physics that space and time be
homogeneous; special relativity does not affect this assumption (i.e., if we change a
Cartesian coordinate system by adding a constant to each coordinate axis – displace the
origin by a constant vector – the coordinates in any other reference frame also will be
shifted by an additive constant). Thus, if aµ represents a set of four constants, and:
µµµ
axx +→
(i.e., ct→ct +cτ and xi →xi +ai where a0 =cτ and ai =a for i=1,2,3) then the effect of this
transformation on the S coordinate must be:
)()()()( νµνµννµνµ
aaxxaxxxx +=+→
since aµ must be an additive constant four-coordinate vector and therefore cannot
depend on x.

83. Taking the partial derivative of xµ with respect to xλ, then, we obtain:
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λ
µ
λ
µ
λ
µ
a
a
a
x
x
x
∂
∂
=
∂
∂
=
∂
∂
where µ,λ =0,1,2,3. Since ∂aµ/∂aλ cannot depend of xµ, then neither can ∂xµ/∂aλ,
because they are equal. Thus:
λ
νµ
µ
λ
x
xx
∂
∂
≡Λ
)(
must be a constant, and therefore:
0
2
=
∂∂
∂
=
∂
Λ∂
λν
µ
ν
µ
λ
xx
x
x
identically. It follows that all higher derivatives of xµ vanish, and hence that the Taylor’s
series expansion of xµ(xν ) in four dimensions reduces to the linear transformation:
∑=
Λ=
3
0
)(
ν
νµ
ν
νµ
xxx
where Λµ
ν =∂xµ(xλ)/∂xν as indicated above. By taking into account the symmetry
between S and S, we also obtain:
∑=
Λ=
3
0
)(
ν
νµ
ν
νµ
xxx
where Λµ
ν =∂xµ(xλ)/∂xν.

84. Thus the hypothesis that space and time are homogeneous has led us to the require-
ment that any transformation between coordinates in different frames of reference must
be a linear transformation (i.e., straight lines in one frame are transformed into straight
lines in another) as opposed to second order transformations (i.e., nonlinearities arise).
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zyxctzxzyxctyx
zyxctxxzyxcttcx
3
3
3
2
3
1
3
0
32
3
2
2
2
1
2
0
2
1
3
1
2
1
1
1
0
10
3
0
2
0
1
0
0
0
Λ+Λ+Λ+Λ==Λ+Λ+Λ+Λ==
Λ+Λ+Λ+Λ==Λ+Λ+Λ+Λ==
and
,,
To make clear the meaning of the xµ(xν )=Σν Λµ
ν xν equation above, let us write it out
explicitly by expanding the sum over ν =0,1,2,3, using [x0,x1,x2,x3]≡[ct,x,y,z], &c. Thus:
So, the xµ(xν )=Σν Λµ
ν xν and xµ(xν )=Σν Λµ
ν xν equations represent four equations each.
We can combine the xµ(xν )=Σν Λµ
ν xν and xµ(xν )=Σν Λµ
ν xν equations to get:
∑ ∑∑ ∑∑ 







ΛΛ=








ΛΛ=Λ=
λ
λ
ν
ν
λ
µ
ν
ν λ
λν
λ
µ
ν
ν
νµ
ν
µ
xxxx
Hence, since the magnitude of xµ is arbitrary, we must have:
µ
λ
ν
ν
λ
µ
ν δ=ΛΛ∑
where δ µ
λ=0 if µ ≠λ and δ µ
λ=1 if µ =λ so that, in fact:
µ
λ
λµ
λ
µ
δ xxx == ∑
while recalling that all the sums above run over ν,λ=0,1,2,3.

85. Now we introduce the hypothesis that the speed of light is independent of reference
frame. Let a light pulse be emitted from a source at point [a1,a2,a3] at time a0/c (i.e., at
the four-point [a0,a1,a2,a3]). After a time (x0/c)−(a0/c) the wave front will have traveled to
the surface of radius x0 −a0, and the equation for this surface is:
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00233222211
)()()( axaxaxax −=−+−+−
for x0 >a0, or:
0)()()()( 233222211200
=−−−−−−− axaxaxax
Since the velocity of light is the same in S as in S, the wave front of the pulse will have
reached a sphere of radius x0 −a0 in S after a time (x0 −a0)/c as measured by clocks at
rest in S, where aµ is the four-point in S at which the pulse is emitted. Hence the equation
for this surface is:
0)()()()( 233222211200
=−−−−−−− axaxaxax
the left-hand side of the last two equations above have the same form; hence the
equations for the surface representing the wave front of an electromagnetic wave are
form-invariant (i.e., covariant – the form of the equations is the same in all frames of
reference).
(1)
(2)

86. For our xµ(xν )=Σν Λµ
ν xν equation above we have that:
86
2017
MRT
∑ −Λ=−
ν
ννµ
ν
µµ
)()( axax
so that Eqs. (1) and (2) above yield:
∑
∑
∑
∑
∑
−−ΛΛ−ΛΛ−ΛΛ−ΛΛ=
−−ΛΛ−
−−ΛΛ−
−−ΛΛ−
−−ΛΛ=−−−−−−−
λν
λλνν
λνλνλνλν
λν
λλνν
λν
λν
λλνν
λν
λν
λλνν
λν
λν
λλνν
λν
))(()(
))((
))((
))((
))(()()()()(
33221100
33
22
11
00233222211200
axax
axax
axax
axax
axaxaxaxaxax
(3)
or:
∑∑ −−ΛΛ=−
νµλ
λλννµ
λ
µ
ν
µ
µµ
))(()( 2
axaxax

87. At this point it is convenient to introduce several innovations in our notation. First, let a
quantity ηµν be defined by:
87
2017
MRT





==−
==+
≠
=
3211
01
0
or,if
if
if
νµ
νµ
νµ
ηµν
Thus ηµν can be written in matrix form as follows:












−
−
−
=
1000
0100
0010
0001
µνη
With this we can define xµ =Σνηµν xν. Thus in matrix language:














−
−
−
=


























−
−
−
=












3
2
1
0
3
2
1
0
3
2
1
0
1000
0100
0010
0001
x
x
x
x
x
x
x
x
x
x
x
x
or:
zxxyxxxxxctxx −=−=−=−=−=−=== 3
3
2
2
1
1
0
0 and,,