Physics of aberration rather than special relativity
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Physics of aberration rather than special relativity
Yong Gwan Yi
arXiv:physics/0006005v55 [physics.gen-ph] 5 May 2019
May 7, 2019
Abstract A phenomenological explanation is presented We need to rethink some of the established thought and
for the physics of aberration, which is in contrast with review the understanding of special relativity physics.
special relativity physics. The effect of relativity is identi-
fied with an effect due to the velocity of observation being
affected by the velocity of a moving particle. In contrast
2 Ether drift
to the currently accepted view, it is demonstrated that
the classical concepts of time and simultaneity are natu- The Michelson-Morley experiment was undertaken to
ral for describing relativistic phenomena. investigate the possible existence of an ether drift [1]. In
principle, it consisted merely of observing whether there
Keywords Ether drift, Twin paradox, Time dilation, was any shift of the fringes in the Michelson interferome-
Superluminal motion, Aberration of starlight, Aberration ter when the instrument was turned through an angle of
of field, Liénard-Wiechert potential, Magnetic frequency. 90◦ . Observations showed that the shift is at most but a
small fraction of the predicted value. The negative result
was explained as demonstrating the absence of the ether
rift. However, it could have been due to the experiment
itself being incapable of demonstrating the ether drift.
1 Introduction Fizeau performed an experiment to determine whether
the speed of light in a material medium is affected by mo-
Einstein’s theory of special relativity has become a tion of the medium relative to the source and observer.
commonplace in modern physics, as taken for granted The experiment is much in the same way as the Rayleigh
as Newton’s law of classical mechanics or the Maxwell refractometer except the tubes containing water flowing
equations of electromagnetism. However, it was resisted rapidly between the source and observer. An alteration of
for many years because of the second postulate on which the speed of light was observed in the Fizeau experiment,
the theory is based. The second postulate, which states which was in reasonable agreement with the value given
that the speed of light is independent of the motion of its by Fresnel’s dragging formula. In the Michelson-Morley
source, destroys the concept of time as a universal vari- experiment, it is assumed that the ether is in uniform mo-
able independent of the spatial coordinates. It forces on tion through the source and observer. As viewed from the
us a radical rethinking of our ideas about time and space. Fizeau experiment, the ether drift cannot be assumed in
Many attempts were made to invent theories that would this arrangement. The circumstances are the same as for
explain all the observed facts without this assumption. the Earth, whose motion cannot be defined without an
Our changed concept of time is the result of its gradual extraterrestrial reference. Even if the Michelson-Morley
establishment through experiments in violent controversy. experiment is performed in water flowing rapidly in one
This work is another such attempt. In contrast with direction, the null result is expected since the velocity
previous works, I tried to pick out an essential physical of the water flow cannot be defined in this arrangement.
point in the relativistic formalism. Attention was focused In the case of sound under the same circumstances, no
on the Lorentz condition which led to the formulation of change of pitch is to be expected as remarked by Rayleigh
special relativity. In this attempt, I have come to see a about Doppler’s principle [2].
physics behind the aberration of starlight. In this paper, We should mention the Michelson-Morley experiment
I present a phenomenological explanation for the physics performed with an extraterrestrial light source. Appar-
of aberration. This is in contrast with the relativistic ently, the motion of the light source relative to the half-
explanation of special relativity physics. It begins by rea- silvered mirror is ineffective in changing the interference
soning a physical origin of relativistic phenomena, leading pattern. As shown in the Michelson interferometer, only
to the relativistic form of equations on the basis of clas- the motion of the half-silvered mirror relative to one of
sical physics. There is no need to make an assumption. the other two mirrors can give rise to an effect on the
1interference fringes. It is clear that the point of splitting by intention. The four-vector velocity cannot be defined
into two beams plays the role of an effective source in by the Lorentz time dilation; they are alternative concep-
that interferometer. The experiment using sunlight dif- tually. In fact, in that definition has the path dilation
fers from the original only by taste rather than coverage. been disregarded. The mean free path measured in the
experiment is not the distance of its proper lifetime but
that multiplied by the γ factor. Once the Lorentz time
dilation is taken into account, there is no room for the
3 Twin paradox
four-vector velocity formulation. This is what we observe.
Lorentz obtained transformation equations by using a Either the time dilation or the four-velocity can be consis-
covariant condition which preserves the speed of light in tent with the experimental result. From the experiment it
all uniformly moving systems. Einstein showed that the is evident that the time dilation and the four-velocity are
transformation equations with the covariant condition re- alternative. To see the definite result, the mean lifetime
quire revision of the usual concepts of time and simultane- of a rapidly moving π-meson beam must be determined
ity, leading to the result that a moving clock runs more by direct measurement in experiment. The mean lifetime
slowly than a stationary clock. Such a concept of time thus obtained will be the same as the data measured in
gives rise to the twin paradox, however. In mechanics, it the rest system of π-mesons if the twin paradox is the cor-
is impossible by means of any physical measurements to rect argument. Such an experiment has never been done
label a coordinate system as intrinsically “stationary” or in the past. Nevertheless, we can infer the result from a
“uniformly moving”; one can only infer that the two sys- comparison with astronomical observations.
tems are moving relative to each other. According to this A series of observations by a new technique between
fundamental postulate, like velocity and distance, time 1968 and 1970 indicated that the components making up
must also be symmetric with respect to the two systems. the nucleus of radio source 3C279 were in motion [5]. The
This is what the twin paradox points out. activity, which occurs on a scale of milliseconds of arc,
We consider the experiments performed to verify the could not have been detected with the techniques avail-
phenomenon of time dilation. The mean lifetime of π- able before the early 1970s. Surprisingly, the speed of
mesons was determined using the decay of π-mesons at the components was estimated to be about ten times the
rest in a scintillator [3]. In this method, the mean lifetime speed of light. The mysterious phenomenon received sci-
of π-mesons was determined by a direct measurement of entific attention, immediately. Some other quasars such
the time required to decay. In order to investigate the as 3C273 also turned out to be superluminal sources.
phenomenon of time dilation, an attempt to measure the From direct observations of the distances travelled and
mean lifetime of a rapidly moving π-meson beam was un- the times required it is reported that their nuclei contain
dertaken [4]. An experiment of this nature was arranged components apparently flying apart at speeds exceeding
to measure the attenuation in flight of a π-meson beam of the speed of light. The concept of the speed of light as a
known lifetime using a scintillation counter telescope of limiting speed of material particles, which has been con-
a variable length. The measured mean free path was di- firmed in physics, has been questioned in astronomy.
vided by the mean velocity to get the mean lifetime. The It seems that the π-meson experiment and the ob-
mean lifetime thus obtained, when the Lorentz time dila- servation of superluminal motion are equivalent. The
tion was taken into account, was in fair agreement with only difference would be in their explanations. In phys-
the data measured in the rest system of π-mesons. It is ical meaning, the observation of superluminal motion is
generally recognized that these experiments have verified equivalent to an experiment that has measured directly
the phenomenon of time dilation. the mean lifetime of a rapidly moving π-meson beam. It
However, those experiments have an ambiguous bear- is certain therefore without requiring an explicit experi-
ing on the phenomenon of time dilation. In the latter ment that the mean lifetime of a rapidly moving π-meson
experiment, the relativistic correction was made directly beam obtained by direct measurement is equivalent to
in the mean lifetime, keeping the particle velocity intact. the mean lifetime in the π-rest system. Their equivalence
This is otherwise without example in high-energy physics, leads us to the conclusion that a particle velocity itself
where the relativistic correction has been made in the appears dilated to the observer, keeping time intact. It
form of four-vector velocity. is then only natural to predict an equal ageing of twins
In special relativity, the four-vector velocity is used in relative motion, by which the twin paradox is resolved
as the relativistic velocity corrected by the Lorentz time naturally. The Lorentz time dilation is nothing more than
dilation. The space components are defined as the rate of a merely mathematical relation. The phenomenon of time
change of the path of a particle with respect to its proper dilation is nothing but a physical misconception of it. As
time, the time component being defined as that of a light. pointed out by the twin paradox, the concept of time di-
Such a definition is a result of confusion, however, unless lation violates the relativity of uniformly moving systems.
24 Aberration of light the distance to the solar system is R, the distance to the
Earth is γR. Regardless of whether the Earth is at rest
The Bradley observation of the aberration of starlight or in motion, consequently, the time required for light to
seems to be even more important to modern physics than reach the Earth is R/c. In the relativistic explanation,
previously thought. This is because the aberration effect the velocity of the Earth and the velocity of light relative
can physically be interpreted as expressing an equation to it are respectively v and c, whereas the velocity of light
which is in contrast with the Lorentz condition leading to relative to the solar system at rest is assumed to be c/γ
the formulation of special relativity. I would like to show in the Earth’s frame [6]. The time required to reach the
a physics behind the aberration which is in contrast with Earth is here γR/c. Although explanations are different,
special relativity physics. the same relations are given for the angle of aberration.
In 1727, Bradley discovered an apparent motion of For the Michelson-Morley experiment, however, they are
star which he explained as due to the motion of the Earth different. In contrast to the relativistic explanation, the
in its orbit. This effect, known as aberration, is quite null result is expected from the present interpretation.
distinct from the well-known displacements of the nearer Having revealed the hidden nature of the aberration
stars known as parallax. Bradley’s explanation of this of starlight, we are going to examine its effect on the
effect was that the apparent direction of the light reaching equations of motion in Newtonian mechanics. From the
the Earth from a star is altered by the motion of the Earth vector difference between c′ and v′ for the velocity of
during propagation. The reason for this is much the same light, a derivative with respect to time gives the equation
as that involved when a little girl walking in the rain must of corresponding accelerations
tilt her umbrella forward to keep the rain off her feet.
Let the vector v represent the velocity of the Earth dc′ dv′ dc
− = = 0. (2)
relative to a system of coordinates fixed in the solar sys- dt dt dt
tem, and c that of the light relative to the solar system. The scalar product of the accelerations in this equation
Then the velocity of the light relative to the Earth has with the corresponding velocity vectors is written
the direction of c′ , which is the vector difference between
c and v. This is the direction in which the telescope must dc′ dv ′ d(γc) d(γv)
c′ − v′ = 0, so c −v = 0. (3)
be pointed to observe the star image on the axis of the dt dt dt dt
instrument. When the Earth’s motion is perpendicular
Equation (3) can also be obtained by differentiating the
to the direction of the star, the relation c′2 − v 2 = c2
Bradley relation c′2 − v ′2 = c2 with respect to time. The
follows from the vector difference. If we set c′ = kc, we
kinetic energy T is defined in general to be such that the
see that the observation is performed at speed c′ greater
scalar product of the force and the velocity is the time
than when the Earth is at rest. Keeping in mind that
rate of change of T . In comparing (3) with the definition
the speed of light can be a measure of speed, the altered
of T , the relativistic expression for kinetic energy is seen
speed of observation may give rise to the same effect as
to be T = γmc2 [7]. In the present discussion, the mass
would be the case if the velocity scale were altered at the
has been treated as a constant [8]. The Bradley relation
moment of observation. Accordingly, the velocity of the
c′2 − v ′2 = c2 can then be expressed in terms of kinetic
Earth is supposed to be v ′ = kv in relation to the obser-
energy and momentum, which is the covariant energy-
vation. Taking this velocity of the Earth, the “Bradley”
momentum equation with T 2 /c2 −p2 = m2 c2 . There is no
relation becomes c′2 − v ′2 = c2 . The velocity scale can
difficulty in obtaining the relativistic form of energy and
then be written in the closed form k = 1/(1 − v 2 /c2 )1/2 .
momentum equations along the physical line of thought
This is just the γ factor in special relativity. As a result,
in the framework of classical mechanics.
the angle of aberration α is given by
Because the aberration effect is ascribed to a change
sin α = β, cos α = 1/γ, and tan α = γβ, (1) in the velocity of observation due to the motion of an
observer, it is thought that relativistic phenomena would
where β = v/c. The appearance as the velocity scale appear due to the measurement velocity being affected
shows that the γ factor is of an optic nature at the speed by a particle velocity. It is just like a vector difference
of observation. This means that the relativistic effect is between velocities. This illustrates why relativistic phe-
in nature an optical phenomenon. nomena appear more pronounced as the velocity of par-
After this consideration, mention may be made of the ticles approaches the velocity of light. The idea becomes
difference between the present interpretation and the rel- clear. Is the relativistic effect just an effect due to the
ativistic explanation. In the present interpretation, the velocity of measurement being affected by the velocity of
velocity of the Earth and the velocity of light relative to a particle? Understood as such, special relativity physics
it are respectively assumed to be γv and γc, while the is identified itself as denoting the branch of physics which
velocity of light relative to the solar system at rest is c. If takes into consideration even the measurement velocity
3In the covariant form this gives the fourth coordinate as
star star time. But it is given for the length of the path of propa-
✓✁ ✓✁ gation of light in terms of time. The Lorentz condition is
✓✁ ✓✁ a geometric relation. It has no bearing on the two points
✓✁ ✓✁ in relative motion. With this very reason, the Lorentz
✓✁ ✓✁ transformation equations turn out to be the result of an
✓ ✁ ✓ ✁ ill-conceived marriage.
✓ ✁ ✓ ✁
✓ ✁ ✓ ✁ Seeing the Doppler effect, there is no doubt that the
✓ ✁ ✓ ✁ velocity of light is not independent of the motion of its
✓ ✁ ✓ ✁ source. The invariance of the velocity of light in all uni-
c′ t ✓ ct ✁ ct′ ✓ ct ✁ formly moving systems, which plays so decisive a role in
✴✓
✓ ✁☛✁ ✴✓
✓ ☛✁
✁ the Lorentz transformation, has an ambiguous bearing
x′ x O x′ x O on the experimental facts. To be consistent with observa-
tion for the aberration of starlight, the Doppler shift, and
the Michelson-Morley experiment, the second postulate
Figure 1: The aberration effect and the Lorentz condition should be replaced by the restricted, but more accurate,
postulate that the velocity of light appears the same in
all uniformly moving systems if and only if the source and
as affected by the particle velocity. This makes clear why the observer are both in a given system.
the velocity of light appears in the equations of motion While a pulse of light propagates to the Earth, the
of a material particle. In this regard, a particle speed motion of the Earth displaces its position: x′ = x − vt. In
as fast as or faster than light, apart from the possibility the same manner as derived the Lorentz transformation
of existence, is unobservable because such a particle goes equations we can obtain an expression for the propagation
beyond the limit of observation. path of starlight to the Earth. The aberration of starlight
We suppose that the Earth is uniformly moving with expressed in (4) can equally be solved to give
velocity v with respect to the solar system. For simplicity,
let the origins of the coordinates of the Earth and the c′ t = γ(ct − vx/c) or c′ = γc(1 − β cos θ). (5)
solar system be coincident at time t = 0, at which time
Since the ratio between x and ct is the direction cosine
the star emits a pulse of light. If this pulse of light reaches
the solar system at a time t, the propagation paths of the of the propagation path of starlight with respect to v, it
light to the solar system and the Earth are respectively can be expressed in the more familiar form of the Doppler
given by R = ct and R′ = c′ t. Let x and x′ be the shift formula. It is of interest to see that the aberration
respective projections of R and R′ along the direction of of starlight gives a general derivation of the relativistic
v. By the Pythagoras theorem, then, the geometric figure formula for the Doppler shift. This leads us ultimately
of aberration gives us the expression to consider the transverse Doppler shift as due to the
aberration effect and thus as observed in the direction
c2 t2 − x2 = c′2 t2 − x′2 . (4) inclined at the angle of aberration toward the direction
of motion of a moving source.
The general form of expression for aberration stands We can give a general derivation of the expression for
in contrast with the Lorentz condition which led to the the angle of aberration. As shown in the geometric figure,
transformation equations. It suggests taking c′ t in place the ratio between the propagation path of starlight and
of ct′ as used in the Lorentz condition. They can be il- the path of the Earth is a direction cosine. We obtain
lustrated by the geometry of the Pythagoras theorem. In
form, they correspond to an orthogonal transformation in cos θ − β x′ γ(x − vt)
cos θ′ = from = . (6)
a four-dimensional space consisting of the path of prop- 1 − β cos θ c′ t γ(ct − vx/c)
agation of light and the three coordinates of space. It is
important to notice their difference. This is the same expression as given by considering the
The aberration of starlight shows the simultaneous ar- transformation of the phase of light wave, by Einstein [9].
rival of light signals starting from the star at the two It has been shown algebraically that two successive
′
points x and x in relative motion. The effect gives a phys- transformations with velocity parameters β1 and β2 are
ical interpretation for the four-dimensional space, which equivalent to a single Lorentz transformation of param-
includes the observation in the description of motion. The eter β = (β 1 + β 2 )/(1 + β1 β2 ). This also follows from
Lorentz condition finds its explanation in a spreading the ratio in (6), in consequence of the interpretation of
spherical wave with time, which starts from the star and x/ct as the velocity parameter of a particle in the rest
′ ′
′
reaches the point x at time t and the point x at time t . ′ system and x /c t as the velocity parameter of observer
4in the laboratory. The formula for the addition of ve- and magnitude from that when the Earth is at rest. In the
locities comes from the inverse transformation equations. geometric figure the difference is shown to be an acceler-
The inverse equations differ only by a change in the sign ation that the moving Earth has during the propagation.
of v. Note that the γ factor is symmetric with respect to The spatial variation in propagation of the gravitational
two systems in relative motion, the physics of relativity. field may be expressed in the form
It is misleading to introduce the relation ∆x′ /γ = ∆x as
GM GM d(γv)
a basis for the Lorentz-FitzGerald contraction hypothesis n ⇒ (n − β) + .
from ∆x′ = γ∆x given as a consequence of the Lorentz R2 t−R/c γ 2 R2 (1 − β · n)2 dt t
transformation equations. (10)
This equation shows that the gravitational field acting
on a moving system must be balanced by an acceleration
the system would have during propagation. Total gravi-
5 Aberration of fields tational effects observed at a moving system will thus be
the same, regardless of how fast it moves. This makes the
Newton’s gravitational force is a static force. There
gravitational field invariant in the covariant form.
is no notion of propagation, an action at a distance. In
Following the same line of reasoning, the Coulomb
modern physics, it is required that a force be transmitted
potential produced by a moving electron can be expressed
with a velocity. If the gravitational field propagates with
in the form of (8) by replacing the gravitational charge
the velocity of light instead of instantaneously, the gravi-
GM by the electronic charge e. The Coulomb field thus
tational field must suffer aberration, just as light does. It
obtained is in formal agreement with the electric field of
is then realized that the aberration of starlight expresses
an electron in uniform motion in electrodynamics. We can
the aberration of the gravitational field of star.
make a comparison with the Liénard-Wiechert potential
Let R be the radius vector from a star to the retarded
in terms of the retarded and present times:
position of the Earth. If the star is in a direction perpen-
dicular to the motion of the Earth, the path of propaga- e e
, . (11)
tion of starlight to the Earth is given by R/ cos α = γR. R(1 − β · n) t−R/c γR(1 − β · n) t
The gravitational potential of the star can then be written
as Since the relation of the retarded position to the present
GM GM position of a moving electron is not, in general, known,
and . (7)
R t−R/c γR t the Liénard-Wiechert potential ordinarily permits only
the evaluation of the field in terms of retarded position
We may infer this form of gravitational potential from the
and velocity of the electron. In the present approach, the
aberration of starlight. It shows that the gravitational po-
unknown effect occurring during the propagation is as-
tential at the point of observation at time t is determined
sumed to be an aberration effect on the field attributed
by the state of motion of the Earth at the retarded time
to its finite propagation velocity. As applied to a moving
t − R/c, for which the time of propagation of light from
source of light, the aberration effect on the propagation
the star to the observation point just coincides with R/c.
of light to the observer yields an expression equal to the
We can extend this to the case where the star is not
relativistic formula for the Doppler shift. This furnishes
in a direction perpendicular to the motion of the Earth.
support for that assumption. The unknown effect occur-
The propagation path of starlight to the Earth is then
′ ring during the propagation would be the aberration of
given by (5) as R = γR(1 − β · n), where n is a unit
the Coulomb field produced by a moving electron.
vector in the direction of R. The gravitational potential
The electric field of a moving electron divides itself
can thus be written in the general form
into a velocity field and an acceleration field [11]. In the
GM
. (8) present approach, the Coulomb potential alone induces
γR(1 − β · n) the velocity field. Thus to make this approach agree with
If we define the gravitational field by the gradient of po- the electric field of a moving electron, the vector potential
tential, then we obtain from the gravitational potential should be deduced solely from the acceleration field. On
the expression the assumption that the γ factor is cancelled out by the
relativistic correction to velocity, this deductive reasoning
GM
(n − β), (9) leads to the following expressions for the vector potential:
γ 2 R2 (1 − β · n)2
e v e v − (v · n)n
where we have used ∇R(1 − β · n) = n − β [10]. , . (12)
c R(1 − β · n) t−R/c c R(1 − β · n) t
It is thought possible to express in a covariant form
the aberration effect on the gravitational field. The grav- This shows that the vector potential is evaluated by the
itational field acting on the Earth is different in direction component of velocity perpendicular to n. When we view
5the vector potential in this way, we realize that the com- Insight into the relativistic velocity of an electron can
ponent of velocity parallel to n has been incorporated in be provided by considering the mechanism by which the
the velocity of field propagation. This makes it reason- velocity of an electron is determined. An electrostatic
able to expect the form of (12). Actually, it is true that spectrograph to determine the velocity of an electron con-
its perpendicular component appears to be the velocity sists in balancing the magnetic and electric deflections
of source relative to the velocity of propagation. against each other [12]. The electron moving in a uni-
The Liénard-Wiechert potentials are to be evaluated form magnetic field H, perpendicularly to H, describes a
at the retarded time. For derivatives, thus, we make use of circular path of radius RH :
transformations obtained by differentiating R = c(t − t0 ):
mv 2 RH /RH
2
= ev/c × H. (18)
∂ 1 ∂ n ∂
= , ∇ = ∇R − . (13) If this electron moves in a radial electric field E, it can
∂t (1 − β · n) ∂t0 c(1 − β · n) ∂t0 describe a circular path of radius RE given by
When viewed from the present point, however, the aberra- mv 2 RE /RE
2
= eE. (19)
tion effect should be taken into consideration. In passing,
we remark that the effect requires the vector potential to The equation of motion for the electron moving in the
be transverse, satisfying the radiation gauge. In addition, fields H and E applied simultaneously is then given by
the effect requires to evaluate the vector potential with balancing the centrifugal force arising from the magnetic
respect to the path of propagation, c′ dt in place of cdt. deflection against the centrifugal force due to the electric
In the radiation gauge, then, the electric field is given by deflection, by
1 ∂A e n × (v × n) e n × {(n − v) × v̇} eERE = ev/c × HRH . (20)
E=− ⇒ 2 2
+ 2 .
c ∂t c γR (1 − β · n) c γR(1 − β · n)3 Taking into account the aberration occurring in the form
(14) of the vector difference between v and H, the angle be-
The first term is a result of differentiating R by noting tween v and H is tilted at an angle of aberration toward
here R = ct. The second term is in agreement with the the direction of motion of the moving electron. Thus,
acceleration field except the γ factor. As shown by (14),
in form, the time derivative is equivalent to the differential cERE = vHRH sin(π/2 − α). (21)
operator. In the intuitive form, therefore, the magnetic
The velocity of the electron is found to be γcERE /HRH ,
induction may be evaluated in terms of the electric field:
where β = ERE /HRH . In this regard, cERE /HRH is
n ∂ seen to be the intrinsic velocity the electron would have
B=∇×A=− × A = n × E. (15) if the velocity of propagation of the fields were infinite,
c ∂t
thereby not suffering aberration. This elucidates why a
The aberration effect on the potential fields lends itself particle velocity itself appears dilated to the observer.
to incorporation in the classical theory of radiation. The speed of high-energy particles of γv can easily be su-
We now consider the motion of an electron in a uni- perluminal phenomenologically. It should be noted that
form magnetic field H. If the electron has no velocity the apparent speed of high-energy particles is ascribed to
component along the field, it moves along a circle in the the aberration of uniform magnetic field. The relativistic
plane perpendicular to the field. The electron moving in velocity is identified with the apparent velocity of which
the field satisfies the equation the γ factor arises out of the effect of aberration.
mv 2 r/r2 = ev/c × H. (16)
There would be an aberration of uniform magnetic field 6 Covariant Maxwell equations
because of its finite propagation velocity. The physics of
the situation is reminiscent of the aberration of starlight, We consider the electromagnetic fields seen by an ob-
where the field replaces starlight and the electron replaces
server in the system S when a point charge q moves by in
the Earth in its orbit. The angle between v and H must a straightline path along the x direction with a velocity
be π/2 − α, instead of being π/2. The equation is written v. Let S ′ be the moving coordinate system of q. The
charge is at rest in this system. But when viewed from
mv 2 /r = (evH/c) sin(π/2 − α). (17) the system S, the charge represents a current J = qv
in the x direction. The electromagnetic fields are then
From the relation in (1), we find the magnetic frequency
related through Ampère’s law:
to be eH/γmc. We can find a complete derivation of the
relation for the magnetic frequency from the point of view
4π 1 ∂E 1 ∂E
∇×B= J + = ∇ × B = . (22)
of aberration. The γ factor must be the aberration effect. c c ∂t S c ∂t S ′
6Ampère’s law keeps its form invariant with respect to space, as Minkowski addressed [13], the forms in which
the two systems in relative motion. They are related at the equations of motion are displayed gain in intelligibil-
the same time, and so are the equations: t = t′ . In the ity. The Lorentz transformation equations were obtained
covariant form, nonetheless, it is instructive to write the by applying a covariant condition to two systems in rel-
equations of transformation between S and S ′ in terms ative motion. However, two systems in relative motion
of t and t′ . For that purpose, instead of using ct and c′ t, cannot be covariant in time and space. In the relative mo-
we use here the Lorentz transformation equations. tion of two systems it is assumed that time is the same in
Let us apply to the equation the Lorentz transforma- both systems. The Lorentz condition can be satisfied by
tion of coordinates with [γ(ct − βx), γ(x − vt), y, z]S = an equation for motion of a system or relative motion of
[ct, x, y, z]S ′ . The y and z components are homogeneous two systems, providing a geometry in time and space for
equations. The transformation of these components is motion. But two systems in relative motion must not be
straightforward. The x component is an inhomogeneous confused with a relative motion of two systems. As noted
equation. Its transformation does not seem to be so. by Sommerfeld [14], the fourth coordinate is not t but ct.
By Coulomb’s law ∇ · E = 4πq, the equation can be In the case of a moving source of light, furthermore, it is
written as the velocity of light that appears dilated to the observer.
We can find in the aberration effect a phenomenologi-
∂Bz ∂By v 1 ∂Ex
− = (∇ · E) + . (23) cal explanation of special relativity physics. This reflects
∂y ∂z c c ∂t
that the physical origin of relativistic phenomena lies in
If we multiply the γ factor and use the inverse equations, the aberration of starlight. In contrast to the currently
we can transform the equation into the form accepted view, it is demonstrated that the classical con-
cepts of time and simultaneity are natural for describing
∂ v ∂ v 1 ∂Ex relativistic phenomena. It has shown that this alternative
γ B z − Ey − γ B y + Ez = .
∂y ′ c ∂z ′ c c ∂t′ point of view constructs a new way of deriving familiar
(24) results. This leads us to an understanding of relativistic
We may start with Faraday’s law. In exactly the same phenomena using a physical reasoning. Einstein’s argu-
manner, we use the relation ∇ · B = 0 to obtain the equa- ment is in essence a mathematical explanation based on
tions of transformation. This completes the transforma- the transformation equations. The resulting equations of
tion of electromagnetic fields from the Maxwell equations. Einstein’s theory were proved to be correct, contributing
greatly to modern physics. However, the correct result
does not always warrant the correctness of assumption.
7 Concluding remarks In the past controversy, the incorrect argument is not
in opponents’ minds but in Einstein’s theory assuming
We are taught special relativity in such a way that the dilation of time scales. The concept of time dilation
the phenomenon of time dilation is daily verified in high- makes no sense physically; time is an independent vari-
energy physics laboratories. But the verification is not able and motion is relative to each other.
so explicit; one can only infer the lifetime dilation from
the mean free path for the π-meson decay measured in
the experiment. Nor are we unanimous in accepting or Appendix: Remark on the superluminal motion
interpreting the concept of time dilation. The superlumi-
nal motion is by no means mysterious. The astronomical There has been a precision measurement of the neu-
observation has shown us that a particle velocity itself trino velocity at 17 GeV with the OPERA detector at
appears dilated to the observer phenomenologically. Had the underground Gran Sasso Laboratory [15]. The neu-
the time been measured directly, the π-meson experiment trino speed is measured by passing through about 730
would have shown essentially the same. Not only exper- km of the Earth’s crust from the CERN, showing values
iment but also theory is incomplete. The aberration of equivalent to the light speed within experimental errors.
uniform magnetic field has been overlooking in modern Intense debate on the experiment increases our interest
physics. The aberration of field gives rise to the γ factor in the OPERA result [16]. At much higher energy, the
of velocity, which disproves the phenomenon of time di- amazing result is compatible with earlier measurement of
lation. For the relativistic mass, likewise, the aberration the neutrino velocity at 3 GeV from the Fermilab NuMI
disproves the experimental result. The effect of relativity beam with the MINOS detector [17].
is due to the γ factor of velocity. In the early 1970s, we were aware of a superluminal
From special relativity we learn that the equations of motion from the observation of radio source 3C279. Most
motion should be covariant in the mathematical structure astronomers could not believe the motion to be the case
of time and space. By identical treatment of time and because the superluminal velocity cannot be accepted by
7the theory of special relativity. The current explanation References
given in astronomy must be reasonable, but it cannot be
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suggests their speed for why neutrinos could not be de- [13] Reference 9, p. 75
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This is because the four-momentum equation is given for (2011); ICARUS Collaboration, arXiv:1110.3763 (2011)
the motion of a particle which is observable at the speed [17] MINOS Collaboration, arXiv:0706.0437 (2007)
of light. Neutrino physics is a new physics beyond the [18] B. G. Piner et al., arXiv:astro-ph/0301333 (2003)
observation and description of motion.
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