Journal of Modern Physics, 2012, 3, 1458-1464
http://dx.doi.org/10.4236/jmp.2012.310180 Published Online October 2012 (http://www.SciRP.org/journal/jmp)
Classical Derivation of Auxiliary Lorentz Transforms:
Their Relations with Special Relativity
Sankar Hajra
Indian Physical Society, Calcutta, India
Received July 2, 2012; revised September 2, 2012; accepted September 8, 2012
ABSTRACT
In this paper we have given a direct deduction of the auxiliary Lorentz transforms from the consideration of Maxwell.
In the Maxwell’s theory, if c is considered to be the speed of light in ether space, his equations should be affected on the
surface of the moving earth. But curiously, all electromagnetic phenomena as measured on the surface of the moving
earth are independent of the movement of this planet. To dissolve this problem, Einstein (1905) assumes that Maxwell’s
equations are invariant to all measurers in steady motion which acts as the foundation of Special Relativity. This as-
sumption of Einstein is possible when all four auxiliary Lorentz transforms are real. There is not a single proof that
could properly justify Einstein’s assumption. On the contrary it is now known that classical electrodynamics could eas-
ily explain all relativistic phenomena rationally.
Keywords: Auxiliary Lorentz Transforms; Real Lorentz Transforms; Classical Electrodynamics; Special Relativity
1. Introduction
Thomson’s auxiliary co-ordinate transforms [1] along
with the auxiliary time transform of Lorentz (1904) [2]
are customarily called auxiliary Lorentz transforms.
Auxiliary Lorentz Transforms are some mathematical
constructs invented to solve electrodynamic problems
classically. Those are made in such a way that Maxwell’s
equations are invariant in the auxiliary get-up too.
This paper deals with the nature of Auxiliary Lorentz
transforms, their classical derivation and their relations
with the real Lorentz Transforms extensively used in
special relativity.
In Sections 2 and 3 of this paper, we have shown how
Thomson (1889) invented auxiliary coordinate trans-
forms to solve potential problems of steadily moving
electromagnetic bodies electrostatically from the consid-
eration of Maxwell [1] and how the auxiliary time trans-
form of Lorentz could be used ingeniously to solve clas-
sically the radiating problems of a system of steadily
moving radiating bodies in static format.
In Section 4, we have deduced the auxiliary time
transform of Lorentz classically using auxiliary coordi-
nate transforms of Thomson and Maxwell’s equations
directly.
In Sections 5 and 6, we have described the inherent
reasons why Lorentz assumed that the co-ordinate trans-
forms of Thomson are real and why Einstein assumed
that all four auxiliary Lorentz Transforms are real.
In Sections 7 and 8, we have described the construc-
tion of the Special Relativity of Einstein that justifies that
the auxiliary Lorentz Transforms are real and the conse-
quences of such a justification.
Our goal is to show that 1) mathematics used in Spe-
cial Relativity is exactly the same as that of Classical
Electrodynamics and 2) the contention of the Special
Relativity is not based on appropriate experimental evi-
dence.
2. Auxiliary Coordinate Transformation
Equations of Thomson


0
The potential

of a system of charges stationary in
ether space is determined by the Poisson’s equation
2
00
00
d
,4πr




  


(1)

But, the scalar potential  and the induced vector
potential


*
A
of this system of charges when steadily
moves in the OX direction with a velocity u in ether
space are governed by D’Alembert’s equation:
2
0

(2)



2** *
2
0
,0,0
xyz
u
AAA
c



 (3)

is the charge density of the system, 0 where

is the
permittivity and 0

is the permeability of ether space
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opyright © 2012 SciRes. JMP

S. HAJRA 1459
such that 00 1c

 

,,
, and x, y, z are the Cartesian co-
ordinates introduced in ether space.
In such a situation when the system of charges steadily
moves in ether space in the OX direction, the potentials
at the point

x
yz

d, ,
at the instant t and those at the
point

x
utyz

dtt
at the instant

in ether space
will be the same. Therefore,
dd
x
tut

 
t

 (4)
,u
tx
 


22
2
22
u
tx
 


(5)
Similarly,
**
x
x
A
A
u
tx



,
2* 2*
2
22
x
x
A
A
u
tx



 (6)
Equations (5) and (6) are customarily called “steady
state operators”.
By the use of Equation (5), Equation (2) could be re-
placed by

22
22
22
1uc xy
2
2
0
z


 




 (7)
and by the use of Equation (6), Equation (3) could be
replaced by the Equations:

2* 2*
22
22
1xxx
AAA
uc 2*
2 2
0
u
x
yz c




0*0
z
A


 (8)
*
y
A, (9)
Comparing Equation (7) with Equation (8), we have,
*2
x
A
uc

(10)
Therefore, to determine E and B, we are only to de-
termine .
Now construct an auxiliary system (invented by
Thomson) where the system of charges is stationary such
that

22
,,
1,1
x
xy
ku
yz z
c k





dddxyz 



k
 (11)
and let charges remain the same in this system such
that
ddQ  


 (12)
In this system Poisson’s equation should be valid.
Therefore, in this auxiliary system, using Equations
(11) and (12) we have, for auxiliary charge density


 (13)
The Poisson’s Equation being still valid in this auxil-
iary system, we have,
0,






0
d
4πr





 
 (14)

is the auxiliary potential of the auxiliary system and



2222
rrx y z


 is the auxiliary distance of the
origin from the auxiliary position of the point where the
field is to be determined.
The Equations (11) transforms Equation (7) to
2
0



 (15)
Comparing Equation (15) with Equation (14), we
have,



 (16)
Thus we see that the potential of a moving system of
charges is not connected to the potential of the same sys-
tem at rest. That potential is

times of the potential of
the stationary auxiliary system in which all the coordi-
nates parallel to OX, OY and OZ have been changed in
the ratio determined by Equation (11). From Equation
(16) we could very easily deduce many electrodynamic
formula classically as given in [3].
For the potential

of a steadily moving point
charge Q we have from Equation (16),

12
22
00
4π4π1sin
QQ
rru
  

(17)

being the angle between the OX axis and the radius
vector r of the point P where the potential is to be deter-
mined at the instant when the moving point charge is at
the origin.
For the potential of a charge uniformly distributed
over the surface of a sphere of radius R moving with a
velocity u in ether space in OX direction,
22
22
0
ln
8π
pl p Q
pplp



  
22
,pa b ,aR
(18)


(19) bR
l being the positive root of the equation
222
21
xyz
l pl
 (20)



 should be replaced by
x
where in final calculation

x

.

here is the potential of a stationary ellipsoid of
revolution in the auxiliary coordinate notation which
could be constructed by the elongation of OX axis of the
sphere by the factor

such that its semi-axis = a is in
the direction of motion and the equatorial radius = b.
3. Auxiliary Time Transformation Equation
of Lorentz
The problem that Thomson addressed was to model the

S. HAJRA
1460
potentials for charges having a constant translational ve-
locity in ether space. He solved this problem by trans-
forming the D’Alembert’s equation in an invariant form
with Poisson’s in the auxiliary system. Lorentz’s prob-
lem was to model the radiation from steadily moving
radiating bodies. He solved this problem by transforming
Maxwell’s equations for the moving radiating body to
the same form for the auxiliary system, which correlates
between the static and dynamic radiation states.
Suppose that a dipole stationary in ether space is emit-
ting a train of plane monochromatic light wave of unit
amplitude from the origin of a co-ordinate system

,,
x
yz. The rays are chosen in
x
y
0
plane, making an
angle

with the OX axis. Let us concentrate on the
point

,,

x
yz such that

12
222
yz
2
00E
rx
The radiation equation should read,
(21)
E0 being the radiating electric field at a point in ether
space due to the system of charges inside the dipole sta-
tionary in ether space which implies a wave propagation
in the form
00
r
cos
P
t
c




 


(22)
Putting 00 2π

 corx and 00 s siny



 where
0

is the radian frequency, 0

is the frequency and
0

is the wave length of the propagating wave, an ex-
pression describing that propagation could be constructed
from the propagation Equation (22) as
00
0
sin t 




 



20E
0
0
cos
cos 2πxy P










(23)
Now suppose that the dipole is moving with a velocity
u in the OX axis. What will be angle between the emitted
ray and the direction of motion of the dipole?
When the dipole steadily moves, the electric and
magnetic fields of the radiation fields of the dipole must
change and thereby the ray if at all emitted must continue
to move in a different direction.
We know that Heaviside’s fields obey Maxwell’s
equations just like Coulomb’s fields do [4]. Therefore, if
the stationary dipole radiates, it must radiate, too, in
steady motion [Verification could be found in Appendix
A].
Therefore, when the above dipole steadily moves, the
Maxwell’s radiation equation should read
(24)
E being the radiating electric field at a point in ether
space due to the system of charges inside the dipole
moving steadily in ether space.
The expression describing this propagation should read
cos sin
cos 2πxy

t







P



(25)


x
y
The rays are too in

plane, making an angle

with the OX axis. In this case, the dynamic field
E
is
not connected with the stationary field 0
E
by any direct
relation. Therefore,

could not be calculated in terms
of 0

by any direct equation.
E
is connected with
0
E
through the auxiliary field
E
constructed by the
Thomson’s auxiliary coordinate transformation equation.
Therefore we are to calculate

in terms of 0 via


.

Now following Thomson, let us construct an auxiliary
system defined by the following general coordinate
transformation equations to study the potential problems
of the steadily moving systems of electrodynamic bodies
at These are as follows:
0.t


,,

x
xut yyzz 



0t
(26)
[The Equation (11) is used to study the situation at the
time

when the moving charge is at the origin of
the frame fixed with the ether space. At other instants the
changing electric and magnetic fields will look the same,
although translated to the right by an amount ut. There-
fore, in that case, the Equations (11) will be transformed
to these equations of (26)].

Now, if the E-field is transformed to the
E
-field (the
auxiliary field of Thomson) by the use of Thomson’s
auxiliary coordinate transformation equations (26), Max-
well’s equations must not be valid in that system which
one can easily verify by direct substitution. Thomson
(1889) found that the auxiliary Equations (11) could
transform the D’Alembert’s equation of dynamic poten-
tial to the static Poisson’s format in the auxiliary system
which could facilitate the calculation of dynamic scalar
and vector potentials correctly from the consideration of
Maxwell. Lorentz (1904) found that if he would con-
struct an auxiliary time transformation equation such that


2
ttuxc 

20E
20
(27)
the dynamic radiation equation will be trans-
formed in the Thomson’s Auxiliary system in the format

 E
22222
(28)
(Vide the deduction in Section 4) and this, too, facili-
tate the radiation calculation of steadily moving radiating
bodies correctly from the consideration of Maxwell as
given below.
Equation (28) implies

x
yzct    (29)
From Equation (28) we get,
xcos sin
cos 2πyt











P




(30)



S. HAJRA 1461
Now, Equation (25) and (30) are the same.
Relation between dynamic and auxiliary radiation
quantities
Comparing Equation (25) and (30) we have,

kuc sin
tan cos









(31)

1 cos1ccos

 

   

 (32)
Relation between auxiliary and stationary radiation
quantities
Now from the definition of the auxiliary system and
the stationary system we have,
00
222222
sin, sin, cos
2
2
0
2
, cos,
1sin
y
yx
rr
rxyz r
 






x
rr
u
c










(33)
Relation b etween dynamic and stat ionary rad iation
quantities
Using Equations (31)-(33), we have for the angle



of the ray with OX axis when the radiating system is
steadily moving

12
222
00
inuc 



sinsin1 sk 



 (34)
0
coscos


 (35)
Using Equations (34) and (35), we get from Equation
(31),
2
0
0
sin
cos
k 
tan 





(36)
Similarly, for the calculation of frequency

of the
steadily moving radiating bodies we have from the last of
Equation (33),
0




(37)
Using Equations (35) and (37), we have from Equation
(32)
00
2
cos 


22222
  (38)
4. Deduction of Lorentz’ Auxiliary Time
Transform
When radiating body steadily moves in ether space
Maxwell’s Equation (24) implies
x
yzct
22222
(39)
To make the Maxwell’s radiation equation invariant
for the auxiliary system of Thomson, it is now required
x
yzct
 

 (40)
Solving Equations (39) and (40) using Equation (26),
we get,


2
ttuxc 
 (41)
This is the famous auxiliary time transform of Lorentz
which along with the co-ordinate transforms of Thomson
could recast Auxiliary radiation equation of Thomson in
the format of Maxwell’s radiation equation.
Thus we see that equations dealing with radiation from
steadily moving radiating bodies could be readily de-
duced classically and correctly using the Auxiliary Lor-
entz transforms. Auxiliary Lorentz Transforms are clas-
sical. Therefore, results deduced by the use of those aux-
iliary equations are classical.
(An interesting feature of the Auxiliary Lorentz trans-
forms is that the inverse of the Auxiliary Lorentz trans-
forms are correct too. This means if



2
,,,xut yyzzttuxc 

 
x
(42)
are correct then the inverse equations i.e. ,



2
,,xut yyzzttuxc 
 
 
,,,
x
(43)
are also correct which one could easily verify. This sim-
ple observation was used by A. Einstein in his Special
Theory of Relativity.)
Auxiliary Lorentz transforms are the equations which
express the relation between auxiliary
x
yzt



,,x
and
real x, y, z, t in Classical Electrodynamics. The first three
of the Auxiliary Lorentz Transforms could be used to
study the scalar and vector potentials of any system of
steadily moving charges and line current [3] but not to
study the scalar and vector potentials of the similarly
moving surface or volume currents. Auxiliary Lorentz
Transforms conjointly are applicable only for point
charge electrodynamics but not for large charge electro-
dynamics. These are applicable to study the cases when
the system moves in ether space but the measurer is at
rest. These could not be used to study the cases when the
measurer moves but the system is at rest in ether space.
5. Assumption of the Reality of the
Thomson’s Auxiliary System
y
z
and Its Consequences
To explain the null result of the Michelson-Morley Ex-
periment, Fitzgerald in 1889 assumed that all bodies
contract towards its direction of motion in ether space by
an amount depending on the square of the ratio of their
velocity to that of light.
According to Classical Electrodynamics when a charged
sphere of radius R stationary in ether space moves in that
space


,,
x
yz in a steady motion, the shape of the
charged sphere remains the same, but the potential of this
moving charged sphere should be

times of the auxil-
iary potential of an oblate ellipsoid of revolution with the

S. HAJRA
1462
axes , in the auxiliary coordinate notations

::RRR 

,,
x
yz


,,
to be finally replaced by real coordinate nota-
tions
x
yz R [

being in the direction of motion,
vide Equation (18)]. This means that relative to the sta-
tionary auxiliary system, the moving electromagnetic
system is contracted with the k factor towards its direc-
tion of motion.
All measurements we use in electrodynamics are per-
formed on the surface of the moving earth and these are
real. Therefore the dynamic system in electrodynamics is
real. But the stationary ether system is illusive.
To match electrodynamic principles with Fitzgerald’s
assumption, Lorentz assumed with novelty that Auxiliary
coordinate transforms of Thomson are real (effective/
true). This means that when an electrodynamic body (as
well as any mechanical body) steadily moves in ether
space, it contracts with k factor towards its direction of
motion and the auxiliary quantities of the stationary aux-
iliary system too, act as real quantities as in the real sta-
tionary system. This somehow explains the null result of
the Michelson-Morley Experiment with some distortion
and obscurity of Classical Electrodynamics.
We measure electrodynamic quantities on the surface
of the moving earth and these quantities are real. Elec-
trodynamic quantities relating to the electromagnetic
system at rest in ether space is illusive. Lorentz assumed
the effectiveness (reality) of the Thomson’s Auxiliary
system which lessened the importance of ether space in
his electrodynamics.
According to Lorentz’ contraction hypothesis the afo-
resaid spherical charge changes to an contracted ellipsoid
while in steady motion and its scalar poten-
tial becomes

RRR



::

k

times of the scalar potential of a
stationary charged sphere of radius R in the auxiliary
notations to be finally transformed to dynamic notation
notations as given in Equation (17).
The calculation and interpretation on the effects of a
moving point charge are same to both Classical Electro-
dynamics and Lorentz’ contraction hypothesis; as the
geometry of the point charge remains the same in the
auxiliary system. But the calculation for steadily moving
charged big bodies are different as per these two different
interpretations.
Lorentz always advocated for his moving contracted
ellipsoidal electron against spherical electron of Abra-
ham. He tried to establish that his electron had advantage
over Abraham’s which was his bias.
Lorentz considered his time as local time. t

t
t
6. Assumption of the Reality of Lorentz’
Auxiliary Time
In the Maxwell’s theory, if c is considered to be the
speed of light in ether space, Maxwell’s equations are
then valid in ether space where the earth is obviously
moving with an appreciable velocity. Therefore, the
Maxwell’s equations should be affected on the surface of
the moving earth. But curiously, all electromagnetic
phenomena as measured on the surface of the moving
earth are independent of the movement of this planet.
To dissolve this problem, Einstein (1905) assumes that
Maxwell’s equations are invariant to all measurers in
steady relative motion which abolishes absolute space
and acts as the foundation of Special Relativity. This
assumption of Einstein is possible when all four auxiliary
quantities in Lorentz transforms (which are deduced to
make Maxwell’s equation invariant in the auxiliary sys-
tem as shown in Section 4) are real to those measurers,
which abolishes the polemical preferred frame wherein
physical laws were considered to preserve their simplest
forms. In this interpretation, auxiliary time is real too
along with auxiliary coordinates.
Then Einstein somehow tried to justify the reality of
those auxiliary quantities and their reciprocity in the fol-
lowing way.
7. Construction of Special Relativity
Lorentz transforms could be deduced algebraically from
many sets of arbitrary equations. Einstein [5] chose such
two sets of equations, the first set viz.,



1) ,,
x
xut yyzz 





,, 2)
x
xut yyzz 




[From the Standard & inverse Lorentz transforms Eq-
uations (42) and (43)]

being unknown, and the sec-
ond set, viz.,
22222
x
yzct
22222
1)

2)
x
yzct
 

[From real dynamic radiation Equation (24) and its
auxiliary (28)] from which Lorentz Transforms could be
recovered.
Now, Einstein assumed that both the sets of above eq-
uations are real (which means that auxiliary state in clas-
sical electrodynamics is itself real) and therefore, Lorentz
Transforms deduced from those two sets of equations are
themselves real. The reality of the Lorentz transforms to
the measurers in steady relative motion implies that dis-
tance and time are relative.
Einstein, however, liked to describe physics in terms
of principles (assumptions). He stated his first set of as-
sumption with certainty: the laws of physics are the same
in all inertial frames. No preferred inertial frame exists.
This is the First Principle which means that his first set of
equations is real to measurers in steady relative motion.
The first principle is customarily called “the principle of
relativity”.
Similarly, he stated his second set of assumption with

S. HAJRA 1463
the same certainty: the speed of light is the same “c” in
all inertial frames. That means that his second set of eq-
uations is real to measurers in steady relative motion.
The second principle is customarily called the “the prin-
ciple of the constancy of the speed of light”.
Now, if those two principles are real, then Lorentz
Transforms are real too. This conclusion constitutes spe-
cial relativity theory which contradicts Classical Physics.
In such an interpretation, all the auxiliary equations as
given in Sections 2, 4-6 are real. In Section 3, instead of
Classical Equations (36) and (38), the auxiliary Equa-
tions (31) and (32) are accepted as correct calculations as
per Special Relativity.
Real Lorentz transforms as per Einstein are the equa-
tions which express the relation between real ,,,
x
yzt


,,, and real
x
yzt. According to him the equations are
valid universally in any case of relative steady motion.
8. Consequences of the Assumptions of the
Reality of Lorentz’ Auxiliary System
In his explanation, A. Einstein has tried to remove the
question on the real contraction of moving electrons ad-
vocated by Lorentz. According to this interpretation, the
shape of the electron remains the same whereas different
measurers with different relative velocities should meas-
ure the shape of the electron differently.
To Einstein all auxiliary Lorentz transforms including
the auxiliary time transformation equation are real, abso-
lutely exact and applicable to every field of physics.
Thus when auxiliary Lorentz transforms are transcended
as reality to all realm of physics, length contracts, inertial
mass varies or converts into energy and more interest-
ingly time dilates with velocity!
It has been proved with certainty that the speed of light
is the same c on the surface of the moving earth which
possesses appreciable gravity on its surface. However,
there is not a single experiment to show that the speed of
light is the same c in an inertial frame (with very negligi-
ble or no gravity) steadily moving in ether space. There-
fore, the second principle does not have any physical
foundation.
Similarly, Maxwell’s radiation equation is seen to be
invariant on the moving earth but it is not proved by any
experiment that the equation preserves its invariant form
in an inertial frame with no or negligible gravity steadily
moving in ether space. Therefore, the first principle, too,
lacks in scientific rigor for which the theory boast.
In the domain of Classical electrodynamics, Auxiliary
Lorentz transforms are actually the relation of co-ordi-
nates and time between the imaginary auxiliary state of
Classical Electrodynamics and real dynamic state. Rela-
tivists make an effort to justify the reality from the as-
sumption that auxiliary state of Classical Electrodynam-
ics itself is real.
But from the consideration of Classical Electrody-
namics, the reality of such assumption has not been
proved by any experiments.
For point charges both the calculations are the same.
But for large charge electrodynamics both the calcula-
tions markedly differ. So, the efforts of the relativists
could be successful if they cite the results of large charge
experiments conforming only to their calculations which
they avoid.
To justify reality of auxiliary state beyond Classical
electrodynamics, relativists takes the advantage of the
situation that for bodies like planets and satellites,
2
11uc and so they concludes that both the re-


sults will be same, for which they are not eager to verify
their conclusion by experiment.
There is not a single experiment to prove that moving
bodies really contract with the same factor k.
As per classical electrodynamics, when a radiating
source having frequency

moves away in ether space
with a velocity u from a stationary receiver with an elec-
tromagnetic clock, the frequency of the source will de-
crease to k

([3], Equation (80)) and the stationary
receiver will measure the frequency of the source as


1;kuc 

but in case the source is at rest and the
receiver similarly moves in opposite direction, the fre-
quency of the source remains the same, the receiver’s
one second will increase to

seconds ([3], Equation
(83)) and the moving receiver will measure the frequency
as


1,uc


 both of which are the same. But if
the receiver measures the frequency with a mechanical
clock such that the ticking of the clock does not change
with velocity, the moving receiver will measure the fre-
quency as


1.uc 
 Therefore, Ives-Stilwell ex-
periment could not be used to uphold relativistic
space-time as well as equivalence principle. The issue
could only be settled by the use of mechanical clocks
which do not change ticking with velocity.
Reality of the auxiliary quantities in the Auxiliary Lo-
rentz transforms has not been proved by the relativists.
Therefore, till this day, in the true sense, the theory of
relativity could not claim its validity experimentally. The
theory is still now a proposition only.
9. Conclusions
This paper concludes that the Special Relativity Theory
stands only on the proposition that the auxiliary Lorentz
transformation equations of Classical Electrodynamics
are real.
But the reality of those equations has not been proved
by any experiments. All electrodynamic phenomena
could easily and rationally be explained from classical
physics only replacing ether space by free space where

S. HAJRA
1464
[2] H. A. Lorentz, “Electromagnetic Phenomena in a System
Moving with Any Velocity Smaller than That of Light,”
Proceedings of the Royal Netherlands Academy of Arts
and Sciences, Vol. 6, 1904, pp. 809-831.
there is negligible or no gravity to disturb electromag-
netic fields which are real and subject to gravitation [3].
10. Acknowledgements [3] S. Hajra, “Classical Interpretations of Relativistic Phe-
nomena,” Journal of Modern Physics, Vol. 3, No. 2, 2012,
pp. 187-199. doi:10.4236/jmp.2012.32026
I thank Alak Bandyopadhyay for his kind help to prepare
the manuscript.
[4] P. Lorrain and D. Corson, “Electromagnetic Fields and
Waves,” 2nd Edition, CBS Publishers and Distributors,
Delhi, 1986, pp. 271-276.
REFERENCES
[1] J. J. Thomson, “On the Magnetic Effects Produced by
Motion in the Electric Field,” Philosophical Magazine,
Vol. 28, No. 170, 1889, pp. 1-14.
doi:10.1080/14786448908619821
[5] A. Einstein, “On the Electrodynamics of Moving Bod-
ies,” Annalen der Physik, Vol. 17, 1905, pp. 891-921.
doi:10.1002/andp.19053221004
Appendix A
It has been shown in [4] that Heaviside’s fields obey
Maxwell’s equations just like Coulomb’s fields do.
Therefore, if a stationary dipole radiates in ether space, it
will also radiate while moving in ethers pace at constant
translational velocity which we may verify as follows:
Let us consider an oscillating dipole whose centre is
stationary in ether space. Let us further consider that the
direction of oscillation is OZ.
The components of the electric field


0
E
and the
magnetic field due to this dipole are as follows:

0
B
 
  
22
000000
33 2
00
00
,,
,,0
xyz
xyz
x
zzyxy
AAA
rr r
AA
yx
cr cr

 


0
EEE
BBB
c
(1)
As expected 00
E
B00 0EB and which satis-
fies the radiation condition.
Now suppose that the dipole is steadily moving in the
OX direction.
Then the components of the auxiliary fields are as fol-
lows:
22
000
33 2
00
,, ;
,,0.
xyz
xyz
x
zzyxy
AAA
rr r
AA yx
cr cr




 
 





EE E
BBB
(2)
Following the Equations (27) and (28) of [3], the
components of the dynamic fields are as follows [first of
the Equations (27) in [3] should be corrected to
x
x

E
E:

00
33
22
02
22
0
022
022
,,
;
,,
.
xy
z
xy
z
xz zy
AA
rr
xyx
Ar
r
Ayxuxy
A
crcrcr
uyz u
Ac
cr






 

  



 



 


 



 


EE
E
BB
B
c
(3)
E
B0 and
E
It is trivial to show that B which
satisfies the radiation condition which is in conformity
with the proposition.