Relativistic Transverse Doppler Effect
The Complete Correlation of the Lorentz Effect to the Doppler Effect in Relativistic Physics
Copyright © 2005 Joseph A. Rybczyk
Copyright © Revised 2006 Joseph A. Rybczyk
Abstract
An in-depth analysis is presented that succeeds in establishing what appears to be the true
relationship of the relativistic Lorentz transformations to the relativistic transverse Doppler effect
for electromagnetic energy propagating space. The purpose of the investigation was to
determine the precise underlying nature of these two effects in order to clear away the confusion
presently associated with their relationship to the constancy of light speed. In the process of
successfully accomplishing these initial objectives a revised version of a previously introduced
formula for the relativistic transverse Doppler effect is fully validated and a previously identified
discrepancy in the special relativity formulas is confirmed along with other irregularities in the
present treatment. The final outcome of this latest research is a new understanding of relativistic
motion mechanics and energy propagation at the most fundamental levels of physics. This
improved understanding suggests new approaches to problem solving regarding not only the
kinetics of motion but also the treatment of time and space in future investigations.
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Relativistic Transverse Doppler Effect
The Complete Correlation of the Lorentz Effect to the Doppler Effect in Relativistic Physics
Copyright © 2005 Joseph A. Rybczyk
Copyright © Revised 2006 Joseph A. Rybczyk
1. Introduction
In attempting to correlate the prevailing scientific evidence regarding light propagation
into a theoretical explanation, Albert Einstein started a controversy that has continued for an
entire century. The special theory of relativity1 of which we are mostly concerned with here has
two fundamental underlying postulates. The speed of light in empty space has the same value c
in all inertial systems regardless of the speed of the source, and when properly formulated, the
laws of physics have the same form in all inertial systems. From these two founding principles
arise the abstract concepts of distance contraction, time dilation, increase in mass with increase
in speed, interchangeability of matter and energy and other changes of views incorporated into
the modern science of physics. In this present work we are primarily concerned with the
constancy of light speed and its most fundamental relationship to the Lorentz2 transformations
and the relativistic Doppler effect.
If we accept the evidence obtained through the various Michelson and Morley type
interferometer experiments3 conducted over the years in conjunction with the results of countless
other experiments or observations involving light from moving stars in space and moving
subatomic particles in the laboratory, there appears to be three different factors involved in light
propagation and subsequently all electromagnetic energy propagation in empty space. Light
travels at a constant speed c, and is unaffected by either the combined unified motion of the
source and receiver through space, or the motion of the source relative to the receiver through
space. These three different considerations, it will be shown, can alternately be expressed as the
correlation of the Doppler effect with both, the null results of the Michelson and Morley
experiments and the null results of the de Sitter binary star observations4. What will be shown is
that the Lorentz contraction factor suggested by the Michelson and Morley experiments is not
needed for the application that suggested it, but for a different application implicit in the de Sitter
findings. More precisely, it is seen that the null results of the Michelson and Morley experiments
are explainable through the application of the classical Doppler effect in conjunction with the
principles of relative motion and the constancy of light speed alone, whereas inclusion of the de
Sitter findings necessitates the need for a Lorentz correction also. When all of these
considerations are properly correlated and accounted for, it appears that light propagation and all
electromagnetic propagation are clearly relativistic inertial frame related phenomena as presently
believed, or more specifically, phenomena that involve transition between two different inertial
frames, each associated with a different object or body of matter. It will also be shown that there
is most likely available a method to resolve the conflict involving light speed constancy once and
for all; a method involving a modified form of de Sitter’s binary star experiments. But most
important of all, it will be shown that our understanding of relativistic transverse motion is not
entirely correct as it presently stands and requires modifications that lead to a revised approach
in its treatment and application.
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2. The Initial Source of the Confusion
Under close analysis it is seen that the Lorentz transformation formula, although devised
by Lorentz to explain the null results of the Michelson and Morley experiments, is not actually
needed to correct for any observed effects in the received light under the conditions of the
experiments specified. This is because, as will be elaborated upon later, there was no relative
motion between the source and receiver. Discounting, for the moment, this second
consideration, (the absence of motion between the source and receiver) we have already
encountered an important distinction between the derivation process involving the Doppler effect
and that involving the Lorentz effect. In apparent diametric opposition to the Doppler effect that
was adopted to explain an observed effect on the frequency and wavelength of light from a
moving source, the Lorentz effect, at least in its initial application, was devised to explain the
absence of an expected effect on the speed of light regarding a fixed source and receiver
configuration moving relative to an assumed medium of light travel. More concisely stated the
Doppler effect was needed to explain an observed effect while the Lorentz effect was needed to
explain the absence of such an effect. Nonetheless, upon progressing with the analysis it will be
shown that from an application standpoint, when properly understood, the two effects are not as
different as they may first appear to be. Yet, in spite of this, the very nature of the observations
leading to their required derivations started a chain of confusing adaptations that perpetuated as
the Lorentz effect was extrapolated by Einstein into an expanded form in the development of his
comprehensive theory of relativity.
3. Lorentz Effect in the Michelson and Morley Experiments
Rather than waste time rehashing all of the alleged inconsistencies of the Lorentz
transformations regarding the correlation of light speed from one frame to another, let us
immediately begin the process of clarification regarding the true nature of the Lorentz effect as
applicable to its use in the theory of relativity. First of all, when properly understood and
applied, the Lorentz effect fundamentally supplements the Doppler effect. That is, it will be
shown that the same conditions that produce a Doppler effect simultaneously produce a Lorentz
effect, and subsequently if there is no detectable Doppler effect, there is no detectable Lorentz
effect. To understand why this is so, let us briefly revert back to the Michelson and Morley
experiments. What the experimenters were originally looking for was a Doppler shift in the
wavelength and frequency of the transmitted light depending on its direction of travel relative to
the Earth’s path of travel through the medium believed to be necessary for light propagation.
There was no such shift detected and the Lorentz effect was subsequently devised to explain
why. In this use then, the Lorentz effect was devised to explain the absence of a Doppler shift,
or more concisely, the absence of an effect rather than it presence. Unfortunately, this reverse
order in the reasoning process starts the confusion from the very beginning between the Lorentz
effect and the Doppler effect since it is in exact opposition to the rationale used for the earlier
defined Doppler effect. Moreover, in its expanded use in the theory of relativity, the Lorentz
effect takes on a different meaning entirely than that of its original use regarding the Michelson
and Morley experiments. In this new and final use, the effect is associated with the relative
motion between the source and receiver and subsequently occurs simultaneously with the
Doppler effect. With some thought based on this final adaptation of the effect it becomes
apparent that only the derivation procedures are at odds, not the actual results. That is, with
regard to the Michelson and Morley experiments, neither a Doppler shift nor a Lorentz shift of
any kind was detected. In other words, a Lorentz shift in the final context of its meaning5 would
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be nothing more than a difference in Doppler shifts between the light traveling in one direction
as opposed to a second, right angle, direction in the interferometer used. In the relativity case,
however, the Doppler shifts were expected as a result of a difference in light speed. This, of
course is not the original intended purpose of the Lorentz effect regarding the Michelson and
Morley Experiment. The intent was that the Lorentz effect explained the absence of a frequency
and wavelength shift in one direction compared to the other by showing that distance in one
direct contracted to counteract the Doppler shift. In that case, it may be argued that the very
absence of a Doppler shift gives evidence of a Lorentz shift. Whereas this may be true, it is not
the kind of effect we normally have in mind when we think of the Lorentz effect in relation to
Einstein’s employment of it in the theory of relativity. That aspect of the Lorentz effect is
discussed next.
4. Einstein Effect in Special Relativity
Unlike the case of the Michelson and Morley experiments where the light source and
receiver move together as a unit through space, in the special relativity (SR) treatment Einstein
modified and expanded the use of the Lorentz factor to explain primarily what happens when
there is relative motion between the light source and receiver, or as alternately stated, when there
is relative motion between the source and observer. In this new application there are many
different things that are assumed to be affected and all are detectable in one manner or another.
These include an increase in mass and energy for the object in motion and also a contraction of
distance in the direction of motion in the frame of the object along with a slowing of time in that
frame. Of these, the most controversial is the slowing of time between frames in motion relative
to each other and that aspect is therefore the focus of our attention here.
As can already be seen, Einstein did not use the Lorentz effect in the manner initially
suggested by Lorentz but in a different manner entirely. In this expanded use, no attention is
given to the motion of a light source or observer through space, but rather the attention is
redirected to the motion of the light source relative to the observer. Or stated another way, the
focus is on the motion of a uniform motion frame (UF) of reference relative to a stationary frame
(SF) of reference. This use of the Lorentz effect has more relevance to the de Sitter binary star
observations than it does to the Michelson and Morley experiments and in fact, along with Ritz’s
emission theory6 was the actual motivation behind de Sitter’s observations. Unlike the
Michelson and Morley case, however, in this case there are detectable affects. Actually, in this
case it is the effects predicted by Ritz’s emission theory7 that are not observed and not those
expected in accordance with the principles of relativity. In accordance with the principles of
relativity, if the speed of light is constant and does not take on the speed of the source, the
observed motion of an orbiting star viewed edgewise will follow the geometry prescribed by
Kepler and that appears to be the accepted scientific position. (This subject involving the light
from orbiting binary stars is discussed further in Section 14.)
Before proceeding it will be helpful to take a moment to officially, at least within the
framework of the millennium theory8, designate this new and expanded use of the Lorentz
formula, the Einstein effect, or the Einstein transformation effect, or the Einstein transformation
formula, etc., in a long overdue tribute to Einstein’s contribution to our understanding of the
physical laws of Nature. From this point forward then, the terms Lorentz effect and associated
variations will be used in reference only to the original meanings of such terms as associated
with the Michelson and Morley interferometer experiments. In similar manner the terms,
Einstein effect and associated variations will be use only in reference to the modified meanings
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of these terms as defined in Einstein’s theory of relativity. This long overdue distinction
between the two different uses of the Lorentz formula will help reduce the confusion associated
with the use of the formula as we proceed with the analysis. Let us continue now with our
discussion from the preceding paragraph.
Although the de Sitter observations were directly related to the theory of relativity, it is
not specifically this aspect of the evidence supporting the theory that causes most of the
controversy. In fact it is not the evidence at all, but actually the misunderstood relationship of
the Einstein effect with regard to the constancy of light speed that causes the controversy. That
is, since the original intention of the Lorentz effect was to explain that the reason light speed is
unaffected by the motion of the source and receiver through a medium is because distance in the
direction of motion contracts by an amount specified by the Lorentz factor, it is a common
misconception that the Einstein factor must somehow, from a mathematical standpoint, directly
nullify or cancel the effects of the speed of recession or approach of the observed light source in
order to maintain the constancy of light speed. This, however, is not the case, as will be
discussed next.
5. The Fundamental Relationships Underlying the Constancy of Light Speed
From a straightforward albeit simplistic point of view it can be stated that the Doppler
effect by itself provides the complete explanation for why the speed of light is unaffected by the
speed of the source. For example, in its simplest form, straight line-of-sight recession or
approach, the light moves at speed c directly toward the observer while the source moves in
either the opposite direction, (recession) or in the same direction, (approach) at speed v.
Subsequently the wavelength of the emitted light is stretched over distance c + v in the first case
and compressed into distance c – v in the second case. Thus, the speed of light remains constant
and the wavelength of the emitted light is precisely accounted for using only the classical line-ofsight Doppler effect formula. There are, however, many unaddressed issues in this simple
explanation. For example, quite obviously there are countless bodies in the universe moving at
an infinite variety of speeds in every conceivable direction relative to the light source under
discussion. How does light managed to maintain the same constant speed c relative to each of
these bodies simultaneously, and what after all is the role of the Einstein transformation effect in
all of this? This more representative example of light-speed constancy will be discussed next.
A more realistic reason that light does not take on the speed of the source relative to the
observer is perhaps a bit more complicated than we would wish. Yet, if we follow the evidence
it appears that light has a previously unrecognized relative property in addition to its constant
property involving speed c. Or, more precisely stated, light has a speed c relative to the body it
is referenced to. Thus, if light speed c has a constant value and this value is the same for all
observation bodies regardless of their motion relative to the same source, then something must
be different about these bodies relative to each other. According to the evidence, however, the
laws of physics must not vary from one body to the next. From this we derive the following
rationale: If the laws of physics are the same in all inertial systems, it follows then, that the
atomic activity of one body is shifted relative to that of another body if there is relative motion
between the two bodies. Thus, events on each body as measured by atomic activity on that body,
or the distance traveled by light relative to that body, or through the application of any law of
physics relative to that body will be shifted relative to the same events on the opposite body if
there is motion between the two bodies. From this conjecture arises a profound revelation
regarding the true nature of motion. Whereas two objects in inertial (non-accelerated) motion
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relative to each other are in identical rest states, these rest states are not identical relative to each
other. They are shifted. When an object undergoes acceleration, the forces acting on the body
impose permanent changes on the underlying atomic structure. The g forces that a body
experiences while under acceleration cause internal changes that are retained when the
acceleration discontinues. (This is rather obvious when one thinks about it. If the activity in
atoms involves motion of any kind, the forces acting on the atoms will permanently affect that
motion.) More correctly stated, if acceleration discontinues at any instant, the changes brought
about by the acceleration are retained. The g forces experienced during the acceleration are
related, not to what the body has already gone through, but what it is currently going through.
During the acceleration process the body is changing its relationship to all other bodies in the
universe and when the acceleration stops, the body has identical internal atomic attributes to
those of other bodies sharing the same inertial frame, but shifted relative to bodies in different
inertial frames depending on the relative speed between them. To perform identical physical
activities on any body, one needs simply to apply the same laws of physics on that body as on
any other. To compare results, however, one must rely on the Einstein transformation formula to
account for the shift that is present if the bodies are in motion relative to each other.
In summarizing the main points of the discussion it can be stated that the classical
Doppler effect alone accounts for the constancy of light speed between a moving source and
stationary observer whereas the Einstein effect is needed to transform values internal to each of
the individual bodies. Since these shifted values include time itself, the period of the source is
affected and thereby also the frequency and wavelength of the emitted light relative to the
reference frame of the receiving body to which such relative motion is referenced. Thus, in the
final analysis an additional shift in frequency and wavelength, beyond that accounted for in the
classical Doppler effect, must be taken into consideration involving light received from a moving
source. When such additional effect is included, the resulting total effect is then referred to as
the relativistic Doppler effect.
Although most of the confusion concerning the reconciling of the constancy of light
speed with the Einstein effect involves straight line-of-sight motion between the source and
observer, additional issues arise to compound this confusion when motion transverse to the lineof-sight is included in the investigation. Since the area involving transverse motion is far more
complex than the straight line motion under immediate discussion, we will wait until the
mathematical treatment of the straight line motion is completed before going into the transverse
aspects in any significant manner. Let us now begin the mathematical treatment of straight lineof-sight motion between the source and observer.
Referring to Figure 1, the light emitted by a star in relative motion is illustrated. To
avoid giving two separate examples the star is shown moving in a straight line between two
observers A and B that are at rest relative to each other. As the star moves away from A and
toward B at speed v, the light given off at the half way point O reaches A and B at the same
instant. Since light does not take on the speed of the star, a principle condition of relativistic
theory as previously discussed, the speed of the received light at both, points A and B is c
relative to each of the observers. If there were no Einstein effect to consider, the received light
would contain only a Doppler shift of wavelength and frequency to account for speed v of the
star relative to each of the observers, A and B. Given the direction of motion of the star relative
to each observer, for observer A the Doppler effect would have the forms
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o  e
cv
c
(1) Classical Doppler Effect Recession Formula for Wavelength
where λo and λe are the observed wavelength and emitted wavelength respectively and
fo  fe
c
cv
(2) Classical Doppler Effect Recession Formula for Frequency
where fo and fe are the observed frequency and emitted frequency respectively. For observer B
the Doppler effect would have the forms
o  e
cv
c
(3) Classical Doppler Effect Approach Formula for Wavelength
where λo and λe are the observed wavelength and emitted wavelength respectively and
fo  fe
c
cv
(4) Classical Doppler Effect Approach Formula for Frequency
where fo and fe are the observed frequency and emitted frequency respectively.
Light Emitted at Point O
A
O
Star
B
v
c
c
FIGURE 1 Star Receding from A while Approaching B
If these were the only considerations involving line-of-sight motion, our analysis would
be complete at this point. But they aren’t because they do not account for the Einstein effect also
supported by the evidence. Although as demonstrated by the Michelson and Morley experiments
the Lorentz effect can be ignored in the frame of the star if the motion is reasonably uniform, the
Einstein effect cannot be ignored in the frame of the observers when such relative motion of the
star is present. This latter effect as previously discussed is an additional (relativistic) effect that
must be properly accounted for along with the classical Doppler effect just covered. Referring
then to Figure 2 let us first evaluate this relativistic consideration separately before determining
its specific relationships to the classical Doppler related factors just covered.
In Figure 2 given in abbreviated form is the light-clock example used by Einstein to
illustrate the use of the Einstein transformation in correlating time in the moving frame, shown
here as sphere S2 with time in the stationary frame, shown as sphere S1. Although observers A
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and B are still shown in their original positions because their positions relative to the star plays a
role in the classical Doppler effect, such consideration plays no role in the Einstein effect. Since
as originally stipulated, they share the same inertial frame, the Einstein effect applies to both
equally. The only thing that matters is that they are stationary relative to point O in the
stationary frame. In short, the Einstein effect involves relative motion between frames only and
for purposes of correlation is concerned specifically with the light in the moving frame that is
perpendicular to the path of motion of the source. That is, it involves only the point of origin of
the emitted light (i.e. the source itself), and the point O in the stationary frame where both the
light under evaluation was emitted and to which the motion of the source is referenced. The
specific locations of the observers and their respective directions from the source are concerns
only in regard to the classical Doppler effect already discussed but plays no specific role in the
Einstein effect which involves relative speed only and not direction between the source frame
and the stationary frame to which the speed is referenced. If in the given examples, the direction
of the star is reversed, the Doppler effect for the two observers will also be reversed, but the
Einstein effect will be unchanged. Let us now proceed with the development of the Einstein
transformation formulas from the relationships given in Figure 2.
Y
S1
cT
A
O
S2
ct
Star
B
vT
FIGURE 2 Einstein Relationship between Inertial Frames of Star and Observers
Whereas the back and forth motion of the emitted light along a path perpendicular to the
path of motion of the source was used by Einstein in apparent reference to the Michelson and
Morley experiments and in apparent recognition of the fact that it provides the means to compare
the emitted light pulse with the reflected light pulse, it is actually unnecessary from the
theoretical standpoint of deriving the Einstein transformation formulas. More specifically, in
Figure 2, where c is the speed of light, v is the speed of the star relative to the stationary frame of
the observers, and t is the time interval in the moving frame of the star that transpires during
stationary frame time interval T during which the distances are traveled, a Pythagorean
relationship exists that can be exploited to derive the Einstein transformation formulas directly.
Since by definition, distance ct is perpendicular to distance vT, and the same point of light that
travels distance ct in the moving frame simultaneously travels distance cT, in the stationary
frame, a Pythagorean relationship exists that directly relates these distances to each other and
consequently provides the relationship between time interval t of the moving frame and time
interval T of the stationary frame. Thus, where t is the moving inertial frame time interval, T is
the corresponding stationary frame time interval, c is the speed of light in a vacuum, and v is the
speed of the star relative to the stationary frame we derive
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(ct ) 2  (cT ) 2  (vT ) 2
(5) Pythagorean Relationship of Distances
(ct ) 2  c 2T 2  v 2T 2
(6)
(ct ) 2  T 2 (c 2  v 2 )
(7)
ct  T c 2  v 2
t T
c2  v2
c
(8)
(9) Einstein Time Transformation for t Simplified
where the final resulting formula is a simplified form of the Einstein time transformation formula
giving the relationship of time interval t to time interval T. The inverse relationship of T to t is
than given by
T t
c
c2  v2
(10) Einstein Time Transformation for T Simplified
where T is the time interval in the stationary frame.
For those who feel more comfortable with the special relativity versions, from Eq. (9) we
obtain
v2
t  T 1 2
c
t T
c2  v2
c2
(11)
t T
c2 v2

c2 c2
(12)
(13) Einstein’s Time Transformation for t
for the relationship of t to T. Similarly, the relationship of T to t is then given by
T t
1
v2
1 2
c
(14) Einstein’s Time Transformation for T
where T is the stationary frame time interval.
Before continuing with the primary purpose of this work it will be helpful to first get the
distance contraction proposition of special relativity behind us. Here we are obligated by the
evidence to take a slight excursion from Einstein’s treatment. According to special relativity,
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distances in the direction of the source’s travel are contracted in the moving frame by an amount
given by the Einstein factor whereas distances in the perpendicular direction are unaffected. The
illustration in Figure 2 that is in strict agreement with the evidence, however, shows this not to
be the case. As can be seen, perpendicular distance ct in the moving frame represents the
distances traveled by light relative to the source in all directions in that frame and although
contracted relative to the corresponding distance cT in the stationary frame, is contracted in all
directions equally. This representation of distances in the moving frame is consistent with the
Michelson and Morley findings upon discounting the unnecessary introduction of the Lorentz
factor that was devised to explain how it could be true if the source and receivers were moving as
a fixed unit through a medium believed to be necessary for light propagation. In the final
analysis it is universally agreed that no experiment conducted on a body in uniform motion will
give evidence of that motion. Considering the size of the Earth’s orbit compared to the small
distance the light traveled in the interferometer used, the motion of the interferometer during the
test period was essentially uniform. Thus, when all factors are considered, the light in the
interferometer traveled the same distance in all directions in the inertial frame of the experiment.
With respect to distances that traverse the two frames, however, the analysis is quite
different. In relation to the interferometer experiments it is the equivalent of having the source
and the mirror that is perpendicular to the path of motion fixed in the moving frame while the
mirror along the axis of travel is fixed at the location of either observer A or B located at a point
on the surface of the stationary frame sphere. (In the abbreviated example it would be the
detectors, not mirrors.) In either case, consider the example given in Figure 2 where the light
given off by the source at point O in the stationary frame reaches point Y in the moving frame at
the same time it reaches the observers at points A and B. Since point O is the center of sphere
S1, distance cT from point O to point Y in the stationary frame is the same as distances point O
to point A, or point O to point B. Thus distance cT in the stationary frame is representative of all
distances from the point of origin O to all points along the surface of sphere S1 and not just the
distance to the observers. The same is true of the moving frame where the distance ct from the
center of the star to point Y is representative of all distances from the center of the star to all
points along the surface of sphere S2. Thus the comparison of perpendicular distance ct in the
moving frame, to path of motion distance cT in the stationary frame, is the equivalent of
comparing all represented distances in all directions in the moving frame to all represented
distances in all directions in the stationary frame. With this insight it is apparent then that
relativistic distance contraction applies equally to all distances in the moving frame, and not just
those that are perpendicular to the path of motion. With this issue behind us we can now show in
rather novel form a simple and direct way of deriving the distance contraction formula from the
information given in Figure 2.
Let d represent distance ct and therefore all corresponding distances in the moving frame
while D represents distance cT and therefore all associated distances in the stationary frame.
This gives us
d  ct
D  cT
(15) Moving Inertial Frame Distance
(16) Corresponding Stationary Frame Distance
where d is the contracted distance in the moving inertial frame that corresponds to distance D in
the stationary frame.
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If the relationship between moving frame distances and stationary frame distances is
(ct ) 2  (cT ) 2  (vT ) 2
(5) Pythagorean Relationship of Distances
as already shown in Equation (5), and if the relationship of time t in the moving frame to time T
in the stationary frame is
t T
c2  v2
c
(9) Einstein Time Transformation for t Simplified
as already shown in Equation (9), we can then claim that
c2  v2
dD
c
(17) Einstein Distance Contraction for d Simplified
since by substitution from Equations (15) and (16) we obtain
c2  v2
ct  cT
c
(18)
which simplifies to
t T
c2  v2
c
(9) Einstein Time Transformation for t Simplified
of which hopefully we are already in agreement on.
6. Validating the Pythagorean Relationship
A far more important issue here is this. Why the Pythagorean relationship at all? What is
so special about this relationship between the two frames of reference that gives it priority over
some other geometric/mathematical model? The answer is as simple, or as complicated as we
wish it to be. If we accept the evidence and not concern ourselves about issues that might be
beyond our present abilities to understand, we can draw certain conclusions from the evidence
supported relationships given in Figure 2. If light does not take on the speed of the source
relative to the stationary frame, then a pulse of light will propagate outward at the same speed in
all directions relative to the stationary frame from the stationary point at which it was emitted by
the moving source. The resulting sphere, S1 in Figure 2, can be thought of as a frame of
reference that is valid to the stationary observer in that frame. Since according to the laws of
physics nothing dependent on the physical resources available in this frame can exceed the speed
of light relative to this frame, this expanding sphere of light contains the limits of reality in this
frame. Only those things that do not exceed the speed of this expanding sphere adhere to the
known laws of physics relative to this frame of reference. This is not to say that it might not be
possible for something using resources from another frame of reference that has motion relative
to this stationary frame to exceed the speed of light relative to this frame. It is only to say that if
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so, such occurrence is outside the presently known laws of physics relevant to this frame. The
same is true of the moving frame in regard to the expanding sphere of light centered over the
light source, sphere S2 in Figure 2. With that being the case, only an activity that is totally
confined within these limits in both frames can be used to translate values from one frame to the
other. In Einstein’s light clock example it was the correlation of the moving frame clock time
with the times of the stationary frame clocks. In Figure 2 a similar comparison is possible by
simply using the same propagated particle of light to compare times in the different frames. To
be valid, however, the chosen particle must be completely confined within the limits of both
reference spheres along its entire path of travel. With some analysis it soon becomes apparent
that only a particle along the circle of contact between the two expanding spheres meets this
criterion. Since all of the particles along the circle of contact are equal in this regard, any can be
chosen. For the purpose of convention the author has chosen a particle along the positive Y axis.
Referring now to Figure 3, the interface circle of contact between the two spheres is
illustrated by a dotted line.
Y
S1
S2
cT
A
ct
O
Star
B
vT
Contact Interface Circle
FIGURE 3 Interface Circle between Expanding Light Spheres
As can be readily seen, rotating the right triangle about the path of motion axis between points A
and B will not change the relationship the triangle represents between spheres S1 and S2. And
since distance O-Y is the radius of stationary frame sphere S1, it will always be completely
enclose in sphere S1. The same is true for the distance from the center of the star to point Y in
the moving frame. Since it is the radius of the moving frame sphere it will always be completely
enclosed in sphere S2. Thus, the same particle that travels any of these paths in the one frame
simultaneously travels a corresponding path in the opposite frame and can therefore be used to
correlate time and distances between the two frames.
Now let us evaluate the role of the observer with regard to the Einstein effect. As
previously stated, whereas the position of the observer relative to a moving light source plays an
important role in regard to the observed Doppler effect, the same is not true in regard to the
Einstein effect. With regard to a moving source, the observer plays the role of the light detector
in the stationary frame in a similar manner to point Y of the moving frame. Since, however, all
distances are affected equally in all directions from the point of emission in the stationary frame;
the Einstein effect is the same for all observers regardless of direction or distance from the point
of emission. This includes the 90 degree transverse direction (any point along the interface circle
of contact) along with those in the forward direction (direction of approach) of the moving
source in addition to those in the opposite direction (direction of recession) and any position in
12
between these points. And since point Y in the moving frame is a surface point in the sphere S2
representing that frame, it represents any and all points along the surface of that sphere and thus
all distances to the center of the source simultaneously in that sphere. In other words, the
Einstein effect in the final analysis involves any point along the surface of the moving frame
sphere S2 and its time/distance relationship to any point along the surface of the stationary frame
sphere S1 in an equal and indiscriminate manner. It is simply the distance traveled by light in
each frame that is being compared, and although different in the moving frame in comparison to
the stationary frame, it is the same in all directions in each frame separately. Thus the Einstein
effected time and distance in the stationary frame is the same for all observers regardless of
location. This, of course, is not true regarding the Doppler effect, in which case, as is already
known, the position of the observer does matter. It should be understood here that for
comparison purposes, since both spheres are expanding proportionally over time, the distance of
the observer from the point of emission in the stationary frame also does not matter in regard to
either the Doppler effect or the Einstein effect, as long as the proportional relationships of the
distances in the two frames are maintained.
Once the Einstein effect is properly understood an alternate and direct comparison with
the original Michelson and Morley interferometer experiment is possible. If the interferometer is
constructed with one arm increased in length by the distance traveled by the source compared to
the distance traveled by light during the same period in the frame of the experiment, the observed
interference fringes is representative of the Einstein shift between the moving frame and the
stationary frame. Since the entire modified interferometer is operated in a single frame, there is
no need to include the difference in distance traveled by light between the two frames
represented. As already concluded from the 90 degree angle used in the original Michelson and
Morley experiments and as further elaborated upon in this work, the angular position of one arm
to the other is inconsequential. In an actual experiment, because of the effects of gravity,
however, both arms must still be maintained in a plane tangential to the gravitational field of the
experiment.
7. The Transformation Effect for Wavelength and Frequency
Once the fundamentals of relativistic transformations between inertial frames is
established it is a simple task to derive the formulas needed to transform the wavelength and
frequency of the emitted light to values that are valid in the stationary frame. These are not the
final values that will be observed in the stationary frame, but rather intermediate values that have
been shifted only in accordance with the Einstein effect, and not the Doppler effect. Starting
then with the formula for wavelength, and the acknowledgement that a wavelength represents a
distance, we can use distance transformation Equation (17) to determine the transformed value of
the emitted wavelength that would be experienced in the stationary frame. Rearranging Equation
(17) to give the stationary frame distance that corresponds to a moving frame distance we obtain
Dd
c
c  v2
2
(19) Einstein Distance Transformation for D Simplified
where D is the stationary frame distance that corresponds to moving frame distance d. By direct
substitution of λx and λe for distances D and d respectively we obtain
13
x  e
c
c2  v2
(20) Wavelength Transformation UF to SF
where λx is the transformed (increased) wavelength experienced in the stationary frame that
corresponds to the emitted wavelength λe of the moving frame. The classical formula that relates
wavelength to frequency
f 
c

(21) Classical Wavelength to Frequency Relationship
where f is frequency and λ is wavelength, can then be used to derive the corresponding
frequencies for the wavelengths given by Equation (20). Thus, we obtain
fe 
c
e
(22) Emitted Wavelength to Emitted Frequency Relationship
where fe is the moving frame emitted frequency that corresponds to moving frame emitted
wavelength λe and
fx 
c
x
(23) Transformed Wavelength to Transformed Frequency Relationship
where fx is the transformed frequency in the stationary frame that corresponds to the transformed
wavelength λx. Rearranging Equations (22) and (23) to give wavelength instead of frequency
provides us with
e 
c
fe
(24) Emitted Frequency to Emitted Wavelength Relationship
and
x 
c
fx
(25) Transformed Frequency to Transformed Wavelength Relationship
the right sides of which can be substituted into Equation (20) to obtain
c
c

f x fe
c
c2  v2
(26)
which can then be solved for the transformed frequency fx in terms of emitted frequency fe. This
gives
14
c
c2

f x fe c2  v2
(27)
f x fe c2  v2

c
c2
(28)
and
f x  fe
c2  v2
c
(29) Frequency Transformation UF to SF
where fx is the transformed (reduced) frequency experienced in the stationary frame that results
from the emitted frequency fe of the moving frame. In view of these findings, Equation (20) for
wavelength transformation and Equation (29) for frequency transformation, it is clear that the
Einstein transformation effect is a recessionary effect. I.e. the transformed wavelength will
always be longer and the frequency will always be lower than the emitted wavelength and
frequency prior to application of the Doppler effect. (Could this explain the increase in
expansion of the universe recently detected?) We are now ready to integrate the earlier derived
formulas for the classical line-of-sight Doppler effect with the just derived formulas for
wavelength and frequency transformation to arrive at the complete formulas for the relativistic
line-of-sight Doppler effect.
8. The Complete Relativistic Line-of-Sight Doppler Effect
By combining the earlier given line-of-sight Doppler Wavelength Equations, (1) for
recession, and (3) for approach, we obtain
o  e
cv
c
(30) Classical Line-of-Sight Doppler Effect for Wavelength (+R, –A)
where λo is the observed wavelength, λe is the emitted wavelength, and v, the speed of the
source, is + for recession and – for approach. Doing the same for the line-of-sight Doppler
Frequency Equations, (2) for recession, and (4) for approach, we, in a similar manner obtain
fo  fe
c
cv
(31) Classical Line-of-Sight Doppler Effect for Frequency (+R, –A)
where fo is the observed frequency, fe is the emitted frequency, and v, the speed of the source, is
+ for recession and – for approach. To relativize the above equations we need to first change the
emitted variables, λe and fe to the transformation variables, λx and fx respectively, because these
are the actual values that will be experienced in the observation frame. This gives
o  x
cv
c
(32) Relativistic Line-of-Sight Doppler Effect for Wavelength (+R, –A)
15
and
fo  f x
c
cv
(33) Relativistic Line-of-Sight Doppler Effect for Frequency (+R, –A)
where λx and fx are the respective transformed wavelength and frequency provided by Equations
(20) and (29) respectively. By substituting the right sides of Equations (20) and (29) in place of
the respective variables, λx and fx in Equations (32) and (33) we then obtain
o  e
cv
c2  v2 c
c
(34)
that simplifies to
o  e
cv
c2  v2
(35) Relativistic Line-of-Sight Doppler Effect for Wavelength (+R, –A)
and
fo  fe
c2  v2 c
c
cv
(36)
that simplifies to
fo  fe
c2  v2
cv
(37) Relativistic Line-of-Sight Doppler Effect for Frequency (+R, –A)
where λo and fo are the observed wavelength and frequency respectively, λe and fe are the emitted
wavelength and frequency respectively, and v, the speed of the source, is + for recession and –
for approach where indicated.
9. The Relativistic Transverse Doppler Effect for Initial Wavelengths and Frequencies
The analysis to be presented next is but a small fraction of the total analysis conducted by
the author in the preparation of this work. During the supporting research, every reasonable
approach was investigated including a thorough analysis of the resulting formulas. This included
a direct mathematical comparison of the derived formulas with the transverse Doppler effect
formulas of special relativity. It was finally concluded that the transverse Doppler effect formula
originally introduced in the millennium theory of relativity is in fact correct, but only in most
restricted sense regarding the initial wavelength emitted by the source. Once this initial process
is properly understood, however, the principles can be easily expanded upon to derive a modified
version of the original formula that is applicable to wavelength propagation beyond the initial
wavelengths. This, as will be seen, is accomplished simply by following the relativistic
postulates previously discussed. Specifically, that light propagates in all directions at speed c
from the SF point of emission without taking on the speed of the source.
16
Referring now to Figure 4, we will begin the analysis by first establishing the correct
definition for the classical transverse Doppler effect involving the emission of the initial
waveform at the source. In Figures 4-A through 4-D, the arrow designated c depicts the speed,
distance and direction of light relative to the start point of emission s in the stationary frame of
reference where the light was emitted. In a similar manner, the arrow designated v depicts the
speed, distance, and direction of the source from that same point s, and the arrow designated D
depicts the resultant Doppler effected distance and its relative direction from the source to the
observer. The dotted line then indicates the projected path of the source under the conditions
specified, assuming no new or unknown factors. Since all of the depicted distances are
stationary frame distances, the time interval T, such as in the products cT, vT, or DT, is not
needed for the analysis because in such use it represents a proportionality constant and therefore
has no affect on the outcome of the mathematical derivations that will be conducted.
c
D
c
D
θr
s
θa
v
c
θr θa
s
v
s
D
θr
v
c
θr
θa
s
θa
D
v
Recession
Recession
90 Degrees
Approach
A
B
C
D
FIGURE 4 Classical Transverse Doppler Effect Defined
Of greatest importance in this analysis is to understand that the Doppler effected distance D is
valid only in the direction depicted by the arrow representing D, and not by the arrow
representing c that defines the unaffected direction of light travel. (Although perhaps not yet
apparent, this stipulation is in strict conformance to all relevant relativistic principles and
scientific doctrine.) Thus in view 4-A if D represents a single wavelength of light, it is stretched
as indicated by the associated arrow in a direction generally opposite the direction of source’s
motion and subsequently represents the condition of recession with regard to the motion of the
light source. In view 4-B then, even though the light path from the start point of emission s in
the stationary frame is at a right angle to the path of travel of the source, that right angle is not
valid for the Doppler effected distance, (or wavelength) and direction, which is again defined by
the arrow designated as D. In fact, in all of the views shown in Figure 4, the angle of recession,
or approach with regard to the source’s direction of travel relative to an observer is given by the
designators θr and θa respectively. Thus, only in view 4-C is the 90 degree transverse Doppler
effect properly illustrated for the initial wavelength conditions being discussed. Under these
conditions then, there is an actual classical Doppler effect at the 90 degree transverse direction to
the path of travel of the source. More specifically, from the classical perspective the wavelength
of the emitted light is contracted, blueshifted, when observed at a 90 degree angle to the path of
travel of the source. Before elaborating on this extremely important issue let us first conclude
the main points of the present discussion by referring to view 4-D where the transverse condition
of a source in approach relative to the direction of the observer is illustrated. In this view as in
the others of Figure 4, and in regard to this work in general, θr and θa are supplementary angles
17
relative to the actual and projected path of travel of the source. Thus from a mathematical
perspective the relationship between the two angles is given by
 a  180 deg   r
(38) Relationship of Angle of Recession θr to Angle of Approach θa
 r  180 deg   a
(39) Relationship of Angle of Approach θa to Angle of Recession θr
and
where θa is the transverse angle of approach and θr is the transverse angle of recession between
the source and observer. As should be rather obvious, from a purely technical standpoint,
considering also semantics, each of these angles is valid for their intended purpose only for the
range 0 – 90 degrees. Yet, with some thought it will be apparent that both angles define the
same identical direction away from the source and therefore also the same direction to the
observer. In other words, in properly derive formulas for wavelength and frequency, the two
different angles will provide identical results. Let us now discuss in detail the underlying
fundamentals of the presented theory that ultimately show with regard only to the classical
transverse Doppler effect involving the initial waveforms of emitted light, there is a blueshift
(shortening) of wavelengths at the 90 degree transverse angle of observation. Later, in Sections
12 and 13 it will be shown how in regard to the initial wavelength this effect at 90 degrees is
exactly cancelled out by the Einstein effect for any speed v of the source.
There are two important propositions that must be understood in order to develop a full
understanding of the transverse Doppler effect. First, regardless of how we perceive light
emission and subsequent propagation, whether as a wave, or wavelet, a photonic particle of some
sort, or an energy packet, there is a beginning and end to the emission process at such
fundamental level. That is, in the final analysis, nothing is absolutely instantaneous. Along that
premise then, the s designator used in Figure 4 was chosen to indicate the start of an emission
process that has a duration and therefore also an end. And this in turn brings us to the second
proposition; that involving the consistency and limitations of the concepts used. The same
fundamental relationships that apply to the distances involving the initial wavelength, apply also
to the experimental distances of the laboratory and even generally to the astronomical distances
involving light propagation in space. Thus, the diagrams given in Figure 4 could be expanded
upon to include the theoretical propagation relationships involving the observation of light from
a distant star, in addition to the theoretical relationships involving the emission of a single
wavelength of light at the surface of the star. In either case in accordance with the principles of
relativity, light does not take on the speed of the source and therefore emanates from the point in
the observation frame at which it was emitted. In accordance with our previous discussion
involving Figure 4, however, and in regard to the Doppler effect in general, it is important to
understand that the principles of propagation are not restricted to just a single point at which a
single wavelength of light begins the emission process but apply to an infinite number of such
points up to and including the end point for that single wavelength of light. Moreover, it is rather
obvious that the end point of one wavelength is actually the start point of the next wavelength
should the propagation continue. This revelation regarding the true nature of these points is
crucial to a proper understanding of the propagation process because ultimately it is the start
point and end point for the initial wavelength of the emitted light that is most important in
determining the transverse Doppler effect. And of these two points, it is the end point that plays
18
the defining role in determining the basis for the angle of observation. This, as will be shown
later, is in opposition to the special theory of relativity that uses neither of these points, (start, or
end) but rather the center of a wavelength in determining the angle of observation. A choice that
substantially affects the angle referenced for distances very near the source. Whereas, given the
length of a single wavelength of light, the two points, start point and end point of the single
wavelength, when viewed from a distance in space, or even at the laboratory scale, are virtually
the same, they are substantially different when viewed at the wavelength scale near the point of
emission and subsequently greatly affect the angle of observation. Moreover, the distance
between these points, start and end, is dramatically increased for the longer waves at the lower
end of the electromagnetic spectrum thereby increasing any related effects even more. Thus,
even that aspect of wave propagation must be taken into consideration.
We are now ready to begin the mathematical derivation process for the initial wavelength
of light during the period in which it is emitted into space. When finished, we can apply our
findings to the principles of extended propagation. Interestingly, the formula to be derived is
actually the same formula originally introduced in the millennium theory of relativity. This time,
however, a direct method is used that was not uncovered during the original derivation process.
As a result, a more thorough analysis and much deeper understanding of the transverse Doppler
effect is possible.
In a similar manner to that of the line-of-sight mathematical derivations there are two
different factors to consider in the transverse motion derivations, one involving the classical
Doppler effect, and the other involving the relativistic Einstein effect. Since the Einstein effect
is speed dependent only, and not direction dependent, it is the same for the relativistic transverse
Doppler effect as it was for the relativistic line-of-sight Doppler effect. Therefore, the Einstein
factors already derived for the line-of-sight treatment can be used again in the transverse
treatment. All that is needed at this point then is a mathematical treatment that will provide a
classical solution to the previously discussed triangles of Figure 4 that give the relationships
between the source and emitted waveforms. Such a triangle is illustrated separately in Figure 5.
As can be seen, the triangle is essentially the same triangle used previously in Figure 4-A to
illustrate the relationships involved in recession.
θp
D
c
θs
s
θr
v
θa
source
FIGURE 5 Classical Transverse Doppler Effect Relationships
In this case, however, the two remaining angles have been identified with appropriate variables
to complete the labeling process. Thus, we now have θp for the projection angle, θs for the start
angle, θr for the recession angle, θa for the approach angle, v for the side that represents the
distance and direction traveled by the source at speed v, c for the side that represents the distance
and direction traveled by light at speed c, and D for the Doppler effected distance and direction
19
that results along the remaining side. As stated previously, since all of these distances are
stationary frame distances using stationary frame time, the time interval will be the same for all
of the distances and can therefore be omitted from the analysis. I.e. under the conditions
specified, stationary frame time interval T becomes nothing more than a proportionality constant
and will therefore have no effect on the final derivation. Also to be understood is the fact that
the greater length of the Doppler effected side of the triangle does not mean that the effected
light travels that distance faster than c. This is nothing more than an extension of the principle
used earlier in the line-of-sight analysis. While the leading edge of the signal travels toward the
observer at speed c, the trailing edge is stretched in the opposite direction by the motion of the
source. Quite obviously then, the illustrated triangle represents recession, or the conditions
regarding a receding source. Nonetheless, it could just as easily have been shown leaning to the
right, as in the case of Figure 4-D, in which case it would show distance D being compressed as
a result of source’s motion and therefore represent the conditions of approach, or an approaching
source. In either case, or any other that might be represented by the triangle, the variance in the
length of side D has nothing to do with the speed of the emitted waveform. Moreover, as it turns
out, the resulting triangles, regardless of which of its many forms are used, are easily solved
using the Law of Cosines Theorem. Interestingly, as mentioned earlier, it just so happens that
the main original formula used in the Millennium Theory of Relativity was unknowingly an
indirect derivation involving the Law of Cosines Theorem.
Continuing now, still referring to Figure 5 for the present derivation, involving initial
wavelengths, we need simply to express the appropriate Law of Cosines equation containing the
angle of recession θr and then solve it for side D. This gives us
c 2  v 2  D 2  2vD cos  r
(40) From Law of Cosines Theorem – Initial Wavelength
from which we use Mathcad to derive
D  v cos  r  v 2 cos 2  r  c 2  v 2
(41) Doppler Effected Initial Wavelength
the right side of which is placed over c in order to give D in terms of c to derive
D( fi )
v cos  r  v 2 cos 2  r  c 2  v 2

c
(42) Classical Transverse Doppler Effect Factor for Initial
Wavelength
where D(fi) is defined as the classical transverse Doppler effect factor for initial wavelengths
from a receding source. This is the required mathematical factor for the classical transverse
Doppler effect involving initial wavelengths of light emitted by a receding source, or for that
matter, even an approaching source if the angle of observation θr is greater than 90 degrees. The
complete classical transverse Doppler effect for initial wavelengths emitted by a receding source
in uniform motion then is
r  e
v cos  r  v 2 cos 2  r  c 2  v 2
c
(43) Classical Transverse Doppler Effect for Initial Wavelength
20
where λr is the observed initial wavelength from a receding source, λe is the emitted wavelength,
v is the speed of the source, θr is the angle of recession, and c is the speed of light in empty
space. (Note: In referring to any of the author’s Mathcad files it should be understood that due to
program design the mathematical convention for trigonometric expressions such as cos2 θr must
be expressed as cos(θr)2 or similar variants.)
Given the fact that this just determined formula was derived from the Law of Cosines, it
gives the correct initial wavelength for the entire range of angles from θr = 0 – 180 degrees. Of
course, at 90 degrees the result is neither that of recession nor approach, and at greater than 90
degrees the result is that of approach. Thus, with one formula and no need to change signs,
every initial wavelength can be determined for any source at any speed from 0 to c from any
direction whether approaching or receding, and at any angle from 0 to 180 degrees. If, however,
we wish to have a separate formula for approach, since θr and θa are supplementary angles we
can simply substitute θa for θr in Equation (43) and then change the sign for v in the first term at
the top of the fraction to obtain
a  e
 v cos  a  v 2 cos 2  a  c 2  v 2
c
(44) Classical Transverse Doppler Effect for Initial Wavelength
where the observed initial wavelength λa is based on the angle of approach θa instead of the angle
of recession θr. As before, the results are valid over the whole range of the previously described
conditions with the exception that results for angles greater than 90 degrees now apply to the
condition of recession and not approach. (It is very important to keep in mind here that, if
supplementary angles for the same direction to an observer are used, both Equations, (43) and
(44) will give the same result. For example, θr = 60 degrees and θa = 180 – 60 = 120 degrees.
These are after all, two different descriptions of the same path to an observer.)
For convenience, Equations (43) and (44) can be combined into a single formula for
initial wavelengths. This gives
o  e
 v cos  o  v 2 cos 2  o  c 2  v 2
c
(45) Classical Transverse Doppler Effect for Initial Wavelength
where λo is the observed initial wavelength, λe is the emitted initial wavelength, θo is the angle of
observation, c is the speed of light in a vacuum, and v, the speed of a source in uniform motion is
+ for recession and – for approach where indicated. The only problem with the just derived
formulas other than their limited use relative to the initial wavelength emitted by the source is
that theoretically it is not the emitted wavelength that is operated on by the classical transverse
Doppler effect factor represented by the fraction to the right, but rather the transformed
wavelength given by Equation (20) earlier. The first step in correcting the formulas for the
relativistic effect then is to substitute the variable λx for the variable λe in the just derived
formulas, since, as with the line-of-sight formulas, the classical versions of the transverse
Doppler effect formulas actually operate on the transformed wavelength λx and not the emitted
wavelength λe. For the relativistic derivation then, this gives us
21
v cos  r  v 2 cos 2  r  c 2  v 2
r  x
c
(46) Classical Factor for Transverse Doppler Effect, Initial
Wavelength
into which we substitute the right side of the previously derived Equation (20)
x  e
c
c  v2
2
(20) Wavelength Transformation UF to SF
in place of the transformation variable λx to obtain
r  e
v cos  r  v 2 cos 2  r  c 2  v 2
c
c2  v2
c
(47)
that simplifies to
r  e
v cos  r  v 2 cos 2  r  c 2  v 2
c2  v2
(48) Relativistic Transverse Doppler Effect Recession, Initial
Wavelength
for the formula for the relativistic transverse Doppler effect for initial wavelengths from a
receding source. From this initial variation we derive the additional forms
a  e
 v cos  a  v 2 cos 2  a  c 2  v 2
c2  v2
(49) Relativistic Transverse Doppler Effect Approach, Initial
Wavelength
for approach and
o  e
 v cos  o  v 2 cos 2  o  c 2  v 2
c2  v2
(50) Relativistic Transverse Doppler Effect Observed, Initial
Wavelength (+R, –A)
for the main formula, where λo is the observed wavelength and θo is the angle of observation for
either approach or recession and where v is + for recession and – for approach where indicated.
The important thing to keep in mind when using these formulas, as will become apparent
with some experimentation, is that any single version of the formula will provide the same range
of results for the same speed v, over the entire range of observation angles from 0 to 180 degrees.
In plain words, depending on the version used, the wavelength will increase, or decrease from
some initial value to some final value as the angle changes from 0 to 180 degrees, or from 180 to
0 degrees. The decreasingly lower values to one side of the 90 degree point are obviously the
result of an approaching source and the increasingly higher values to the other side of the 90
degree point are obviously the result of a receding source. According to the theory presented
22
here, there is no relativistic Doppler effect for the initial wavelength at an angle of 90 degrees.
That is, at 90 degrees, the Einstein effect exactly cancels the classical Doppler effect for the
initial wavelength and therefore the observed wavelength will equal the emitted wavelength as
will be discussed in detail later.
In the same manner that the frequency-to-wavelength relationships were used earlier to
derive the line-of-sight formulas for frequency they can be used again to derive the main
transverse Doppler effect formula for frequency. That is, given
o 
c
fo
(51) Observed Frequency to Observed Wavelength Relationship
and the previous
e 
c
fe
(24) Emitted Frequency to Emitted Wavelength Relationship
for the frequency to wavelength relationships, we can substituting the right sides of these
equations in place of λo and λe respectively in Equation (50) to obtain
2
2
2
2
c
c  v cos  o  v cos  o  c  v

fo fe
c2  v2
(52)
fo fe
c2  v2

c
c  v cos  o  v 2 cos 2  o  c 2  v 2
(53)
that can be simplified to
and
fo  fe
c2  v2
 v cos  o  v 2 cos 2  o  c 2  v 2
(54) Relativistic Transverse Doppler Effect Observed, Initial
Frequency (+R, –A)
for the observed initial frequency formula where fe is the initial emitted frequency in the moving
frame and fo is the observed frequency in the stationary frame. Now that we have defined the
process for the relativistic transverse Doppler effect regarding the initial wavelength and
frequency emitted by a source in uniform motion we are in a position to apply the defined
principles to the extended wave propagation process.
10. The Complete Relativistic Transverse Doppler Effect for Wavelengths and Frequencies
Accepting the evidence that light propagation exhibits wave-like characteristics in
addition to the propositions that the trailing edge of one wave is the leading edge of the next
wave and that light does not take on the speed of the source but propagates from the point in the
23
stationary observation frame at which it was emitted, we can systematically define the extended
propagation process. Beginning in the same manner used previously for the initial wavelength,
we will first derive the classical transverse Doppler effect separately. Then we can add the
Einstein transformation effect for the complete relativistic derivation.
Referring now to Figure 6, the relationships previously defined in regard to the initial
wavelength have been extended beyond the initial propagation process following the same rules
used in the initial wavelength analysis and just repeated above in the previous paragraph.
θp
D = λo
S1
S2
xc
s is start point of emission
S1 is leading edge of wave
xc-c
λo
λe = c θp
c+v
θs θr θa
source
s v e
c-v
e is end point of emission
S2 is trailing edge of wave
xc = radius of S1
xc-c = radius of S2
x = whole number
Figure 6 Extended Relationships - Classical Transverse Doppler Effect
In strict adherence to the proposition that light does not take on the speed of the source, the
leading edge of the light wave propagates outward in all directions at speed c from the SF point
of emission depicted as point s in the illustration. The leading edge of the wave can therefore be
represent as sphere S1 centered over point s in the SF as it expands in all directions at speed c
relative to that SF point. Whereas for the initial wavelength shown as the small triangle enclosed
in the larger triangle, the radius of the initial sphere S1 was defined as c, the new expanded radius
is defined as xc where the variable x is a whole number as will be explained shortly. Since the
source had moved to point e in the SF at the instant the trailing edge of the wave, synonymous
here with the leading edge of the next wave, would then be propagated, there will be a later
emitted wave represented by the smaller sphere S2 centered over point e in the SF. In other
words, as the source continues its motion along the dashed line path to the present location
illustrated in Figure 6, both spheres expand at speed c to the sizes shown. Whereas sphere S2 did
not exist when the source was at point e along its path of travel, it now has a radius
mathematically defined as xc – c.
To understand what is happening here, let us revert back for a moment to the initial
wavelength analysis represent by the small enclosed triangle in Figure 6. As the source moved
24
along its path of travel from point s to point e, the leading edge of the wave emitted at point s
propagated away from that point in all directions at speed c. In the illustration then, c represents
the unaffected wavelength of the emitted wave λe and subsequently the radius of the expanding
sphere S1 at that instant in time. The Doppler effected wavelength λo (previously given as
distance D) is then represented as the distance from the point of intersection of c and λo (forming
projection angle θp) and point e in the SF occupied at that instant by the source. If we wish to
understand what happens as the propagation continues, we need only to multiply distance c with
a whole number to represent the increasing distance in terms of whole wavelengths as the source
continues along its path of travel. In so doing, we can then use the variable x to represent the
number of whole waves contained in that distance and apply the resulting term as a standard by
which the distance traveled by the next successive wave can be expressed. This then results in a
radius of xc for the sphere S1 (representing the leading edge of the emitted wave) centered over
point s in the SF as the source moves to the location shown. The leading edge of the next wave,
(synonymous with the trailing edge of the present wave) emitted later at point e then results in an
expanding sphere S2 with a radius of xc – c as illustrated. This sphere, as previously stated, also
expands at speed c but is centered over point e and not point s as is the case with sphere S1. To
understand what an observer in the path of these expanding waves will see, we simply have to
project the path of each wave from its SF emission point to the SF point of observation, i.e. the
intersecting observation point defined by projection angle θp. For the leading edge of the wave,
the path is simply xc. For the trailing edge of the wave the path is (xc – c + D) where D is the
new Doppler effected distance for the observed wavelength λo.
Given the just completed analysis combined with the observation that the angles θr and θp
are obviously changed in relation to the extended propagation (large) triangle over what they
were in relation to the initial wave (small) triangle, it is reasonable to assume that the affected
wavelength λo represented by distance D is also changed from what its value was in relation to
the initial wave. If true, this would mean that the distance to the observer is also a factor in
regard to the classical transverse Doppler effect. At least in regard to distances very near the
source as might be encountered in laboratory experiments involving the transverse Doppler
effect. With some thought it also becomes clear that even though the distance between the points
of emission s and e may be very small for light waves, as initially mentioned in Section 9, the
distance varies not only with the speed of the source but also with the frequency of the emitted
waves and therefore can be very significant for other waves that make up the electromagnetic
spectrum. This in turn extends the distance at which a predicted effect will be observed and
therefore must be taken into consideration regarding any transverse Doppler effect applications
or experiments.
To derive new classical transverse Doppler effect formulas for the extended propagation
relationships associated with the larger triangle just discussed, we again use the law of cosines
theorem for the relationships associated with angle of recession θr. Using the mathematical
terms previously assigned, this gives us
( xc) 2  v 2  ( xc  c  D) 2  2v( xc  c  D) cos  r
(55) From Law of Cosines Theorem
where x is a whole number, c is the speed of light, v is the speed of the source, D is the Doppler
effected distance representing the observed wavelength λr and θr is the angle of recession.
Solving for the expression (xc – c + D) in a math program (i.e. using a single variable for the
stated expression) we obtain
25
( xc  c  D)  v cos  r  v 2 cos 2  r  ( xc) 2  v 2
(56)
that when solved for D gives
D  c  xc  v cos  r  v 2 cos 2  r  ( xc) 2  v 2
(57) Doppler Effected Distance D
where D is the Doppler effected distance representing the observed wavelength λr. By placing
the right side of the resulting equation over c we can convert it into the more general form of a
factor that can be applied to any propagated wavelength. Thus, where Df is the classical
transverse Doppler effect factor for all wavelengths, we have
Df 
c  xc  v cos  r  v 2 cos 2  r  ( xc) 2  v 2
c
(58) Classical Transverse Doppler Effect Factor for
Wavelength
as the final Equation for the classical transverse Doppler effect factor for all wavelengths. The
complete classical transverse Doppler effect formula for wavelengths is then
c  xc  v cos  r  v 2 cos 2  r  ( xc) 2  v 2
r  e
c
(59) Classical Transverse Doppler Effect for
Wavelength
where λr is the observed wavelength for angle of recession θr and λe is the emitted wavelength of
the source. By simply changing the sign for v to – where shown, and substituting λa for λr and θa
for θr we obtain
c  xc  v cos  a  v 2 cos 2  a  ( xc) 2  v 2
a  e
c
(60) Classical Transverse Doppler Effect for
Wavelength
where λa is the observed wavelength for angle of approach θa and λe is the emitted wavelength of
the source. Combining the two equations then gives
o  e
c  xc  v cos  o  v 2 cos 2  o  ( xc) 2  v 2
c
(61) Classical Transverse Doppler Effect for
Wavelength
where λo is the observed wavelength, λe is the emitted wavelength, θo is the angle of observation,
and v is + for recession and – for approach where indicated.
In the same manner that the classical initial wavelength formulas (43), (44) and (45) were
converted to their relativistic forms in Section 9, Equations (59), (60) and (61) can now be
converted to their relativistic forms using the wavelength transformation formula (20). This
gives
26
r  e
c  xc  v cos  r  v 2 cos 2  r  ( xc) 2  v 2
c2  v2
(62) Relativistic Transverse Doppler Effect
Recession, Wavelength
and
a  e
c  xc  v cos  a  v 2 cos 2  a  ( xc) 2  v 2
c2  v2
(63) Relativistic Transverse Doppler Effect
Approach, Wavelength
and
o  e
c  xc  v cos  o  v 2 cos 2  o  ( xc) 2  v 2
c2  v2
(64) Relativistic Transverse Doppler Effect
Observed, Wavelength (+R, –A)
as the various forms of the complete formula for the relativistic transverse Doppler effect for
wavelengths. Although in regard to Equations (62) and (63) the angles of observation, θr for
recession, and θa for approach, can be from 0 to 180 degrees, it is customary to limit each to the
range 0 – 90 degrees consistent with the true meaning of the terms recession and approach. This
is also true for the angle of observation θo in regard to Equation (64) where v is + for recession
and – for approach where indicated.
Applying the same method of derivation to the just derived wavelength Equation (64)
that was used earlier on Equation (50) to derive the initial frequency formula (54) gives
fo  fe
c2  v2
c  xc  v cos  o  v 2 cos 2  o  ( xc) 2  v 2
(65) Relativistic Transverse Doppler Effect
Observed, Frequency (+R, –A)
where fe is the emitted frequency of the source, fo is the observed frequency in the stationary
frame, and speed v of the source is + for recession and – for approach where indicated.
Now that we have completed a thorough analysis of the underlying fundamental
principles supporting the relativistic transverse Doppler effect formulas originally introduced in
the millennium theory of relativity and now updated in this present work to correctly predict the
effects of extended propagation we can expand our investigation to include the special relativity
formulas for wavelength.
11. The Special Relativity Relativistic Transverse Doppler Effect Formulas for Wavelength
If one wishes to become totally confused regarding the transverse Doppler effect one
need simply to seek out the special relativity formulas and make an attempt to apply them.
Which apparently almost nobody does, because as far as the author can determine even the
original formula given in Einstein’s original 1905 paper, On the Electrodynamics of Moving
Bodies, is in error in more ways than one. Putting aside for the moment the real issues that will
27
be treated shortly in this present work, it seems that the original formula given in Einstein’s
paper actually gives the observed wavelength and not the observed frequency as stated. The
author has verified the English translation of the formula derivation from two reputable sources9
and they both give the same identical translation that is repeated here.
Einstein, having just previously given the equation
    1  lv / c 
in his paper, On the Electrodynamics of Moving Bodies, goes on to state
“From the equation for ω` it follows that if an observer is moving with velocity v relatively to an
infinitely distant source of light of frequency f, in such a way that the connecting line “source–
observer” makes the angle ø with the velocity of the observer referred to a system of coordinates, which is at rest relative to the source of light, the frequency f ` of the light perceived
by the observer is given by the equation
f f
1  cos   v / c
1  v
2
/ c2 
.
This is Doppler’s principle for any velocities whatever.” (Note: the variable ν that was actually
used for frequency in the translated works was changed to f in this quoted version to help reduce
some of the confusion.)
Whether this is an oversight that has since been cleared up in subsequent works is of little
importance here, because it is never mentioned in regard to the original paper. In short, the
author has found several incorrect versions of the special relativity formulas from a variety of
sources, especially on the internet. And even those who use the formula correctly do not seem to
be aware that they are using it differently from that initially given in Einstein’s paper. With this
issue having now been addressed, we will proceed with the presentation of the three different
versions of the special relativity formulas used by the author in the investigation.
The special relativity formula used for comparison purposes in the original Millennium
Theory of Relativity paper (see p 43 of that work) is
o  e
c  cos  o v
c
c
c2  v2
(66) SR Relativistic Transverse Doppler Effect Observed Wavelength
where the author substituted the equivalent millennium relativity version of gamma for the
special relativity version in that previous work. Since this formula gives precisely the results
claimed for special relativity, right out to v = 0.9999999999999999c, it was used for comparison
purposes in the original millennium theory of relativity work and also in this present work. The
author is no longer aware of its original source. Also, the convention for variables has been
changed to agree with this present work. The actual versions of the above formula used for
comparison in the present work are
28
r  e
c  v cos  r
c2  v2
(67) SR Relativistic Transverse Doppler Effect Recession Wavelength
and
a  e
c  v cos  a
c2  v2
(68) SR Relativistic Transverse Doppler Effect Approach Wavelength
where λr and θr are respectively the observed wavelength and angle of observation for recession,
and λa and θa are respectively the observed wavelength and angle of observation for approach.
These two formulas give precisely the effect claimed for Einstein’s treatment, namely that there
will be only an Einstein effect at the 90 degrees angle of observation for a source in transverse
motion. This is contrary to the author’s original claim that there will be neither a Doppler effect
nor an Einstein effect at the 90 degrees angle of transverse motion. The confusion surrounding
this claim has been resolved in this most recent work. Not only is this claim again verified by
the analysis, albeit in a more restricted sense than originally given in the millennium theory of
relativity work, but now the author can show precisely how, why, and to what degree the
Einstein claim is incorrect. But first, let us give the remaining SR formulas used in the analysis.
The next SR formula is actually the formula from Einstein’s 1905 paper but now used
correctly for wavelengths and not frequencies. Thus we have
r  e
v
cos  r
c
2
v
1  
c
(69) SR Relativistic Transverse Doppler Effect Recession Wavelength
v
cos  a
c
2
v
1  
c
(70) SR Relativistic Transverse Doppler Effect Approach Wavelength
1
and
a  e
1
where the variables for wavelength and angle of observation are the same as those previously
given. These formulas give identical results to the next two formulas that follow, but as already
mentioned, not as precise as the two previous formulas for values of speed v that very closely
approach c.
In finishing now, the final two formulas used in the comparison are frequency formulas
that are in this case correctly stated. The obtained results are then converted to wavelengths for
comparison with the other formulas used. Thus we have
r 
v
c
29
(71)
and
1  r
fr  fe
1   r cos  r
2
(72) SR Relativistic Transverse Doppler Effect Recession Frequency
and
r 
c
fr
(73) SR Relativistic Transverse Doppler Effect Recession Wavelength
for the first set of formulas, where fr is the observed frequency for recession, fe is the emitted
frequency, βr is as defined, and the other variables are the same as given previously. And finally
we have
a 
v
c
(74)
and
1  a
fa  fe
1   a cos  a
2
(75) SR Relativistic Transverse Doppler Effect Approach Frequency
and
a 
c
fa
(76) SR Relativistic Transverse Doppler Effect Approach Wavelength
for the remaining set of formulas, where fa is the observed frequency for approach, fe is the
emitted frequency, βa is as defined, and the other variables are again the same as given
previously. It should be understood here that all of the combinations involving the change of
sign for speed v (where indicated) and the change of the angle of observation from one
supplementary angle to the other as previously discussed in association with the millennium
formulas are still a factor of consideration even in regard to the special relativity formulas. And,
considering the fact that changing only one of these two factors has a different effect than
changing both, it is quickly apparent that care must be taken when conducting mathematical
operations and analyses involving this aspect of light propagation.
12. The Contradiction in Einstein’s Treatment Revealed
As was the previous case with velocity composition, the author has again uncovered a
contradiction in Einstein’s special theory of relativity. Unlike the previous case, however, in this
case the evidence against Einstein’s treatment and in favor of the author’s treatment already
exists. That is, it is well known that past attempts to detect an Einstein effect at a 90 degree
transverse angle relative to a moving source, as predicted by Einstein’s formula, have failed.
30
Many different reasons have been given for the inability to detect this effect, mostly relating to
the difficulty of the experiment, but the fact remains that there is no reliable evidence supporting
this aspect of Einstein’s theory. Yet, an analysis of any of the special relativity formulas for the
transverse Doppler effect will show that they all reduce to the formula for the Einstein effect at
that angle. In other words, they predict that although there will be no classical Doppler effect at
the 90 degree angle of observation, the Einstein effect, since it is speed dependent only, will still
be evident at that point if there is relative motion of the source. This is a perfectly reasonable
claim that, on the surface at least, is difficult to argue against if one supports the validity of
relativistic principles to begin with. Thus, some see the failure to obtain clear evidence of this
effect as proof that Einstein’s theory is invalid in general. Although the author obtained different
results for the millennium theory of relativity by simply following the evidence, even the author
was troubled by this apparent contradiction of logic. (It would seem, from a purely logical point
of view, that there should be no classical Doppler effect at an angle of 90 degrees and therefore
only the Einstein effect should be detected.) Only now, after five years of investigative analysis
is the author finally able to show that contrary to expert opinion, the expected effect is actually
based on a contradiction of relativistic principles, even those of the millennium theory. In fact, it
was this overall concern regarding the implication to the validity of the millennium theory that
motivated the author to finally resolve this issue once and for all.
Referring now to Figure 7 let us analyze further the relationships involved in the
relativistic transverse Doppler effect.
D1
Recession
c
D2
Approach
c
c
θs
θr
v
E
source
FIGURE 7 The 90 Degree Relativistic Transverse Doppler Effect
As in some of the earlier illustrations, Figure 7 shows the relationship of a moving light source
relative to its expanding sphere of emitted light propagating in the stationary frame of the
observer. Included also is the previously discussed triangular relationships associated with the
transverse Doppler effect for initial wavelengths, as discussed in Section 9, in addition to another
that shows the Einstein effect. Thus, all of the important relationships involving the 90 degree
aspects of light propagation are included in a single illustration. With regard to the 90 degree
transverse Doppler effect, we can immediately discount the largest of the triangles at the extreme
left that includes side D1 because it gives the 90 degree angle for the unaffected light indicated
by the arrow designated c at the start of the emission process and not for the Doppler effected
31
light as previously discussed. The next triangle shown divided into two equal sections at the
middle of the group is the triangle represented by the special relativity formulas. This is the
triangle that results when the start angle θs and the angle of recession θr are equal as illustrated.
Very compelling is the fact that the Doppler effected side D2 equals side c that represents the
unaffected light. Thus, if we divide this triangle into two equal parts by projecting a vertical line
through its center as shown, we can then correctly claim that this vertical line forms a 90 degree
angle with the path of motion of the source. And since the Doppler effected side equals the
unaffected side for normal light propagation, we can rightfully claim that there is no classical
Doppler effect at this point. Thus, there will only be an Einstein effect as represented by the
final triangle to the far right that contains side E. Since this final triangle is speed dependent
only, it will always give the same effect for the motion of the source independent of the direction
of the observer. This is the effect predicted by the special relativity formulas and, for that matter,
even by the millennium relativity formulas, and therefore the author has no real disagreement
against it. The problem is, however, that in the millennium theory this effect is given for the
angle of recession, θr in the illustration and not the devised 90 degree angle at the center of the
middle triangle discussed previously. The question becomes, how can the effected length of side
D2 have the same value in a direction different from that in which the arrow defining D2 is
pointed? Quite obviously we cannot have different rules for different aspects of the same
behavior. When we devise rules, or laws of physics and mathematics to define various aspects of
behavior, we cannot at our convenience change the rules to suit other divergent beliefs. And that
brings us to the issue involving Einstein’s solution. It completely contradicts the principles set
forth in special relativity to explain the Einstein effect.
As is well known, when Einstein applied the Lorentz transformation to the moving frame
of the source, it was stipulated that the angle involved was that which was perpendicular to the
path of motion of the source, or a 90 degree angle in that respect as depicted by side E of the
triangle to the far right in Figure 7. This angle is thus used to establish a common relationship
between the two frames by which the propagation of light in the one will correspond to the
propagation of light in the other. Otherwise stated, this angle defines the 90 degree angle for
light propagated relative to the path of motion of the source in the direction indicated by the
arrow designated E in the illustration. We cannot now devise a second definition for a 90 degree
angle between the two frames as is done in the case of the 90 degree angle associated with the
arrows indicating the directions of sides c and D2 in the middle triangle. More precisely, we
cannot claim that side E of the triangle representing the Einstein effect is at a 90 degree angle to
the path of motion of the source with regard to the stationary frame and is therefore unaffected
by the motion of the source while at the same time insisting that side D2 of the middle triangle
whose direction relative to the path of motion in the stationary frame is given by recession angle
θr is also somehow at a 90 degree angle regarding its relationship to that same stationary frame.
Such argument is absolutely indefensible.
As alluded to earlier, this contradiction in Einstein’s theory was not fully understood by
the author at the time the millennium formula was originally derive in the earlier millennium
theory of relativity work. Only now can it be truly understood how the millennium results
predicted by the millennium formula for the transverse Doppler effect for initial wavelengths
actually comes about. That is, unlike Einstein’s theory, in the millennium theory when the
classical Doppler effect for initial wavelengths is given for 90 degrees, its representative triangle
is superimposed over the triangle representing the Einstein effect that is also valid for 90 degrees.
Subsequently there is a corresponding blueshift (shortening of wavelengths due to the 90 degree
32
transverse Doppler effect) that is exactly cancelled out by the corresponding redshift
(lengthening of wavelengths due to time dilation) of the Einstein effect. Only now after five
years of analysis since the initial millennium work was completed, is it finally clear how the
relativistic transverse Doppler effect correlates the principles of the classical transverse Doppler
effect with the principles of the Einstein transformation effect. And only now is the
contradiction in Einstein’s treatment finally revealed.
Yet, in spite of this violation of scientific principles, it is found that unless the distance
between the source and observer is extremely small (i.e. on the order of less than 104
wavelengths regarding the value assigned to x in the given equations) or the speed of the source
is very low (i.e. less than a significant fraction of light speed) Einstein’s formula will give
approximately correct results. This is not true for distances of a single wavelength, however, in
which case Einstein’s formula will give incorrect results for any speed. At that minimum
distance, only the millennium theory formulas given in this work will produce correct results for
the entire range of speeds from 0 to virtually light speed c.
Conversely, the narrower the triangle representing the classical Doppler effect as it
extends in the direction of the observer, the more accurate Einstein’s formula becomes. Thus, as
the distance to the observer increases, Einstein’s formula becomes more accurate. On the other
hand, since the base of the triangle increases with the source speed, the higher the speed v of the
source, the less accurate Einstein’s formula becomes. Nonetheless, as the distance to the
observer increases beyond approximately 17 meters, for light waves, the results given by
Einstein’s formula are relatively accurate and increase in accuracy as the distance continues to
increase even for speeds approaching light speed c. And since the new formula, (64) becomes
erratic at such distance (approximately 17 meters, for light waves) due to its mathematical
structure, Einstein’s formula becomes the preferred formula in spite of the fact that it is
technically incorrect. Such is not the case with regard to the longer wavelengths associated with
Infrared, Microwaves, and Radio and TV waves, however, in which case the useful range of the
new formulas extend beyond 300,000,000 km. For example, 107.5 wavelengths x Long Radio
waves at 104 m in length = 316,227,766 km. It should be obvious therefore, that the full
implications and overall impact of these findings will take some time to determine with any real
degree of certainty.
This discovery regarding the inaccuracy of Einstein’s formula in relation to small
distances may at least help to explain the previously discussed difficulty experienced in past
attempts to verify the predictions of Einstein’s formula6 as the angle of observation increases
from 0 to 90 degrees. As already mentioned, according to Einstein’s formula, there is no
classical Doppler effect at the 90 degree angle of observation and consequently there will only be
the redshift predicted by the Lorentz factor. Also, as already mentioned, to the author’s
knowledge, no such shift has ever been directly detected in laboratory experiments. This failure
to detect any Doppler shift at the 90 degree angle of observation is fundamentally in agreement
with the predictions of Equation (64) for distances that approach a single wavelength and thus
provides support for the findings presented here. Nonetheless, for the reasons previously given,
as the distance increases beyond a single wavelength, the classical Doppler effect at an
observation angle of 90 degrees will quickly drop to zero and only the Lorentz shift will be
observed. Even so, because of the generalized nature of Einstein’s formula, only the results
given by the millennium theory formulas presented in this work are truly accurate within the
useful range of distances for which they will properly function.
33
13. Comparing Results of the Millennium Formulas with Those of Special Relativity
Prior to making a direct comparison of the results obtained using the millennium theory
formulas to those obtained using the special relativity formulas; it will be helpful to first
establish a direct correlation between the two different sets of formulas for the special relativity
90 degree angle of observation. This can be easily done by simply correlating the condition of
equality of the two millennium theory defined angles, θs and θr with the associated 90 degree
observation angle of special relativity. It can then be shown mathematically that for initial
wavelengths, the results obtained with the millennium theory formulas when the start angle θs is
equal to the angle of recession θr are the same as the results obtained with the special relativity
formulas when the angle of observation is 90 degrees.
Since we already know from the middle triangle depicted in Figure 7 that side D will
equal side c when angle θs equals angle θr, we can take the previously given law of cosines
formula for initial wavelengths that contains θr
c 2  v 2  D 2  2vD cos  r
(40)
and by substituting c for D, obtain
c 2  v 2  c 2  2vc cos  r
(77)
that can be solved for θr to obtain
v 2  2vc cos  r  0
2vc cos  r  v 2
v2
cos  r 
2vc
cos r 
v
2c
(78)
(79)
(80)
(81)
and finally
 v 

 2c 
 r  a cos
(82) Corresponding MR Angle of Recession for SR 90 Degree Angle – Initial Wavelength
where the angle given for the angle of recession θr can be used in a millennium theory formulas
for initial wavelengths to obtain the same results given for 90 degrees in the corresponding
special relativity formula. Thus, though use of this formula at the level of the initial wavelength
we can quickly determine the millennium theory angle of recession θr that corresponds to the
special relativity theory 90 degree angle for any source speed v. Conversely, we can solve
Equation (81) for speed v to obtain
34
v  2c cos  r
(83) Corresponding Speed v where MR Angle θr = SR 90 Degree Angle – Initial Wavelength
where for any millennium theory angle of recession θr involving the initial wavelength we can
find source speed v that gives the same results for the special relativity theory at an angle of 90
degrees.
To adapt the just derived formulas for use with the expanded principles related to
Equations (62), (63), (64), and (65) we need only to replace the variable c in Equations (82) and
(83) with the term xc that was substituted for c in the extended analysis. This gives us
 v 

 2 xc 
 r  a cos
(84) Angle of Recession for Null Classical Transverse Doppler Effect - All Wavelengths
and
v  2 xc cos  r
(85) Speed v for Null Classical Transverse Doppler Effect – All Wavelengths
where x can be any whole number from 1 to approximately 107.5. (Note: as previously discussed,
for values of x greater than approximately 107.5 the triangle becomes too narrow for the new
relativistic transverse Doppler effect formulas to function properly.) This completes the
mathematical analysis and derivation process and leaves only the direct comparison of results
obtained by the opposing formulas of the two theories. Such comparison is provided in the form
of Appendixes A-1 through A-3.
Given in Appendixes A-1 through A-3 is a summarized copy of one of the many Mathcad
models devised by the author for comparing the results of the millennium theory formulas to the
special relativity theory formulas. For those readers who may not be familiar with the Mathcad
convention, the following should be helpful in interpreting the results shown: The, equal sign
preceded by a colon, symbol := is an assignment operator that assigns the quantity or formula
given to the right of it to the variable or constant given to the left of it. A normal equal sign such
as = without the colon : is a fully functional operator that gives the (previously computed) results
shown to the right for the variable, constant or formula shown to the left. A normal equal sign
that is shown in bold type (if encountered) is a limited function assignment operator that allows
Mathcad to solve the associated equation for a selected variable or constant if possible, but that
does not give the results computed as in the case of the non-bolded equal sign previously
discussed.
14. Proposal for a Modified Approach to Binary Star Observations
In the original attempts by de Sitter to obtain visual evidence of the effects predicted by
Ritz’s emission theory, it was determined that multiple images of the secondary star would be
observed if the light from the star took on the speed of the star. As is well known, visual binary
star systems with orbits edgewise to the Earth were chosen for the observations. The
understanding was that if the light from the secondary star took on the star’s orbital speed, the
faster light from the star when it was moving toward the Earth during one half of its orbit, if
viewed from a great enough distance, would overtake the slower light emitted later when the star
was moving away from the Earth and subsequently the star would appear to be in several
locations of its orbit at the same time. The problem with this method is the distance required for
the faster light to overtake the slower light, assuming Ritz’s theory is correct. From what the
35
author has been able to determine, the distance would have to be so great that it would not be
possible to optically resolve the images of the two stars that make up the system. It is apparent
to the author, however, that it should not be necessary to be so far away as to see multiple images
of the star. All that is really required is to be far enough away to be able to detect abnormal
orbital behavior of the star. If the light actually does take on the speed of the star, as the distance
to the star increases, the secondary star will appear to move either faster, or slower than normal
in one half of its orbit as compared to the other half. Therefore by correlating the speed of the
star along its orbital path to the distance the star moves first to one side of the primary star and
then to the other, it might be possible to distinguish whether the star is obeying Kepler’s laws of
planetary motion or alternately, the apparent motion predicted by Ritz’s emission theory. I.e. the
distance the secondary star moves to each side of the primary star defines the circularity, or
eccentricity of the orbit. Any variation of the orbital speed of the star along its orbital path
should then either agree with or depart from Keplerian defined motion.
It should be understood here that the author is not certain that such observations and
associated analyses have not already been conducted. If so, however, it does not appear that
much effort has gone into making the details of such findings available to the general scientific
community.
15. Conclusion
The analysis presented in this work resolves an area of conflict and confusion that has
continued perhaps for as long as the special theory of relativity has been around. Because the
principles involved, the Doppler effect, the Einstein effect, and the constancy of light speed, not
only form the basis of the special theory of relativity, but also the basis of the millennium theory
of relativity, both theories were in jeopardy if this issue could not be properly resolved. As a
result of this work, correctly formulated relativistic principles as given in the millennium theory
are on firmer grounds than ever before. The same, of course, cannot be claimed for the special
theory of relativity. The discrepancy uncovered in this work involving the special relativity
transverse Doppler effect treatment is the second major discrepancy uncovered by this author
regarding the special theory of relativity. The first involved the incorrect treatment of velocity
composition where in a similar manner as to what was found here, the derived formula violates
the very principles of the theory upon which it is supposedly founded. It is also interesting to
note that the area covered in this work is one in which reliable evidence in support of the special
relativity predictions has never been obtained. To the contrary, the evidence actually supports
the findings of the millennium theory in that there does not appear to be any detectable Einstein
effect at the 90 degree transverse angle of observation. In view of this latest effort, the scientific
establishment will find it evermore difficult to defend the special theory of relativity.
Appendixes
The following Appendixes are included as part of this paper:
Appendix A-1 – Relativistic Transverse Doppler Effect (Overall Analysis)
Appendix A-2 – Relativistic Transverse Doppler Effect (Constant Speed – Range of Angles)
Appendix A-3 – Relativistic Transverse Doppler Effect (Constant Angle – Range of Speeds)
Appendixes A-4 and A-5 – Derivation of Formulas
36
Appendix A-1
Relativistic Transverse Doppler Effect
February 1, 2006 Joseph A. Rybczyk
Relativistic Transverse Doppler Effect Wave Theory.mcd
9
c  299792458
e  550 10
fe 
c
2
14
fe  5.451  10
e
v  c .5
v
60
1000
8
 5.396  10
Speed
km/hr
MR Relativistic Transverse Doppler Effect - r is Angle of Recession, a is Angle of Approach, and o is Angle of Observation in deg.
 r  90 deg
r  e
 a  180 deg   r
 
 
2
c  x c  v  cos  r 
v  cos  r
2
c v
a  e
 
2 2
 x c  v
x  10
x e
D
 
v  cos  a
2
c v
2 2
 x c  v
c
2
2
x  e
lr  e
cv
2
c v
2
c v
2
 ( x c) 2  v 2  ( x c  c  D) 2

2 x c v


c
2
 ( x c) 2  ( x c  c  D) 2  v 2 

2 x c ( x c  c  D)


 p  acos 
2
7
Angle of Start (0-180 deg)
 p  30deg
 s  59.9999999999999deg
Emitted
Angle of Recession (0-180 deg)
 r  90deg
Transformed
7
Angle of Approach (0-180 deg)
 a  90deg
Observed Rec
7
e  5.5  10
7
la  3.175  10
lr  9.526  10
Angle of Projection (0-0-0 deg)
x  6.351  10
7
x e  5.5  10m
 s  acos 
cv
la  e
2
 10 km
 5.5  10
 0.866
c v
Line-of- Sight
given at right
 2  x2 c2  v2
v  cos  r
Distance to observer
2
1000
2
2
D  c  x c  v  cos  r 
2
2
c  x c  v  cos  a 
2
 
0
 a  90deg
Observed App
7
7
r  5.5  10
a  5.5  10
MR formulas for finding r when v is given, or for finding vwhen r is given to make r = s and to make side (xc-c+d) = side xc
Set r to x to make r = s
Set v to vx to make r = s
 x  acos 
v 

 2 x c 
 
 x  75.5224878140701deg
vx
v x  2 x c cos  r
0
c
Speical Relativity Formulas ----------------------------------------------------------------------------------------- Formulas good from 0 to 180 deg.
Special Relativity formula referenced in MTR
SR Recession ( r = 0-180) SR Approach ( a = 0-180)
Er  e
 
c  v  cos  r
2
c v
1
Er2  e
v
c
2
 
 cos  r
v

c
 r 
 a 
c
v
c
 
c  v  cos  a
2
c v
1  
v
Ea  e
1
Ea2  e
2
v
c
 r  90deg
2
7
Er  6.351  10
 
 cos  a
1  
v

c
fEr3  fe
fEa3  fe
 
1   r cos  r
1  a
14
c
Er3 
fEr3
14
c
Ea3 
fEa3
fEr3  4.721  10
2
 
1   a cos  a
7
Er2  6.351  10
2
2
1  r
 a  90deg
fEa3  4.721  10
37
7
Ea  6.351  10
7
Ea2  6.351  10
7
Er3  6.351  10
7
Ea3  6.351  10
Appendix A-2
Relativistic Transverse Doppler Effect
February 2, 2006 Joseph A. Rybczyk
Relativistic Transverse Doppler Effect using Angle of Recession
Speed/Souce
x Factor
Dist/Observer m
cm
3
4
v  c .9
x  10
x e  5.5  10
x e 100  0.55
MR Emitted e
c
x  e
2
c v
2
Recession Angle r
7
For v = 80,000km/s use .2669
MR formula
MR Transformed x
6
e  5.5  10
SR Formula
MR Observed
o
x  1.262  10
SR Observed Ea & Er
 r  0 deg
o  e
 
 2  x2 c2  v2
2
c  x c  v  cos  r 
v  cos  r
2
c v
2
Er  e
 
c  cos  r  v
2
c v
2
 
 2  x2 c2  v2
2
c  x c  v  cos  r 
v  cos  r
2
c v
2
Er  e
 
c  cos  r  v
2
c v
2
o  e
 
 
2
c  x c  v  cos  r 
v  cos  r
2
c v
2
2 2
 x c  v
2
Er  e
2
 
c  cos  r  v
2
c v
2
 
 2  x2 c2  v2
2
c  x c  v  cos  r 
v  cos  r
2
c v
2
Er  e
 
c  cos  r  v
2
c v
2
 
 2  x2 c2  v2
2
c  x c  v  cos  r 
v  cos  r
2
c v
2
Er  e
 
c  cos  r  v
2
c v
2
o  e
 
 2  x2 c2  v2
2
c  x c  v  cos  r 
v  cos  r
2
c v
2
Er  e
 
c  cos  r  v
2
c v
2
o  e
 
 2  x2 c2  v2
2
c  x c  v  cos  r 
v  cos  r
2
c v
2
Er  e
 
c  cos  r  v
2
c v
2
 
 
2
c  x c  v  cos  r 
v  cos  r
2
c v
2
2
2 2
 x c  v
2
Er  e
 
c  cos  r  v
2
c v
2
38
6
Er
6
 0.999979051723495
6
o  1.262  10
Er  1.262  10
o
Er
7
 0.999959499999408
7
o  6.939  10
Er  6.94  10
o
Er
7
 0.999944772724393
7
o  2.783  10
Er  2.783  10
o
Er
7
 0.999954097684429
7
o  1.262  10
Er  1.262  10
o
Er
7
 0.999999999993485
7
o  1.262  10
Comparison o /Er
 0.999994309952512
Er  1.83  10
o
Comparison o /Er
 r  180 deg
o  e
6
o  1.83  10
Comparison o /Er
 r  180 deg
6
Er
Comparison o /Er
 r  150 deg
 1.00000000000034
Er  2.245  10
o
Comparison o /Er
 r  120 deg
o  e
6
o  2.245  10
Comparison o /Er
 r  90 deg
o  e
Er
Comparison o /Er
 r  60 deg
6
Er  2.397  10
o
Comparison o /Er
 r  30 deg
o  e
6
o  2.397  10
Er  1.262  10
o
Er
 0.999999999993485
Appendix A-3
Relativistic Transverse Doppler Effect
February 2, 2006 Joseph A. Rybczyk
MR Emitted e
MR Transformed x
 7 Different for each v. Given
3
e  5.5  10
by SR for r=90 deg, x=1
x e 100  1.739  10
Relativistic Transverse Doppler Effect using Angle of Recession
Recesson Angle r
x Factor
Dist/Observer m
cm
 r  90 deg
Source Speed
7.5
x  10
x e  17.393
MR Formula
SR Formula
MR Observed
o
SR Observed Ea & Er
v  c 0
o  e
 
 
2
c  x c  v  cos  r 
v  cos  r
2
c v
2
2 2
 x c  v
2
Er  e
2
 
c  cos  r  v
2
c v
2
7
o  5.5  10
7
Er  5.5  10
v  c .1
o  e
 
 
2
c  x c  v  cos  r 
v  cos  r
2
c v
2
2 2
 x c  v
2
2
Er  e
 
c  cos  r  v
2
c v
2
7
o  5.528  10
7
Er  5.528  10
v  c .2
o  e
 
 2  x2 c2  v2
2
c  x c  v  cos  r 
v  cos  r
2
c v
2
Er  e
 
c  cos  r  v
2
c v
2
7
o  5.613  10
7
Er  5.613  10
v  c .3
o  e
 
 2  x2 c2  v2
2
c  x c  v  cos  r 
v  cos  r
2
c v
2
Er  e
 
c  cos  r  v
2
c v
2
7
o  5.766  10
7
Er  5.766  10
v  c .4
o  e
 
 2  x2 c2  v2
2
c  x c  v  cos  r 
v  cos  r
2
c v
2
Er  e
 
c  cos  r  v
2
c v
2
7
o  6.001  10
7
Er  6.001  10
v  c .5
o  e
 
 2  x2 c2  v2
2
c  x c  v  cos  r 
v  cos  r
2
c v
2
Er  e
 
c  cos  r  v
2
c v
2
7
o  6.351  10
7
Er  6.351  10
v  c .6
o  e
 
 2  x2 c2  v2
2
c  x c  v  cos  r 
v  cos  r
2
c v
2
Er  e
 
c  cos  r  v
2
c v
2
7
o  6.875  10
7
Er  6.875  10
v  c .7
o  e
 
 2  x2 c2  v2
2
c  x c  v  cos  r 
v  cos  r
2
c v
2
Er  e
 
c  cos  r  v
2
c v
2
7
o  7.702  10
7
Er  7.702  10
v  c .8
o  e
 
 2  x2 c2  v2
2
c  x c  v  cos  r 
v  cos  r
2
c v
2
Er  e
 
c  cos  r  v
2
c v
2
7
o  9.167  10
7
Er  9.167  10
v  c .9
o  e
 
 2  x2 c2  v2
2
c  x c  v  cos  r 
v  cos  r
2
c v
2
Er  e
 
c  cos  r  v
2
c v
2
6
6
o  1.262  10
Er  1.262  10
o  29.148
Er  29.148
v  c .9999999999999999
o  e
 
 2  x2 c2  v2
2
c  x c  v  cos  r 
v  cos  r
2
c v
2
Er  e
 
c  cos  r  v
2
c v
39
2
Appendix A-4
Relativistic Transverse Doppler Effect
Derivation of formula for classical transverse Doppler distance d using Law of Cosines ------------------------------------------------------
 
 r  90 deg
cos  r  0
v  c .9
( x c  c  D)
Where d
 
1


2
 v  cos    v 2 cos  2  x2 c2  v2  




r
r

 


1 

2 
 v  cos    v 2 cos  2  x2 c2  v 2  
 r 
 r

 
( x c)
 
2
d  v  cos  r 
v  cos  r
 
( x c  c  D)
2
2
2 2
 x c  v
 
v  cos  r
 
2
 
2
v  d  2 v  d  cos  r
2
v  cos  r 
c  x c  v  cos  r 
D
2
 
v  cos  r
2
2
2
2 2
 x c  v
2 2
 x c  v
2
2
D
Df
c
Derivation of formula for classical transverse Doppler recession using Law of Cosines ------------------------------------------------------
Df
 
2
c  x c  v  cos  r 
D
 
v  cos  r
c
2
2 2
 x c  v
2
r
c
e
 
 
2
c  x c  v  cos  r 
v  cos  r
2
2 2
 x c  v
2
c
Derivation of formula for relativistic transverse Doppler recession using Law of Cosines ------------------------------------------------------
r
e
 
 
2
c  x c  v  cos  r 
v  cos  r
2
2 2
 x c  v
c
2
c

2
c v
r
2
e
 
 
2
c  x c  v  cos  r 
v  cos  r
2
c v
2
2 2
 x c  v
2
2
Derivation of formulas for classical and relativistic transverse Doppler approach using Law of Cosines --------------------------------Change sign of v to - where shown and change r to a and r to a
a
e
 
c  x c  v  cos  a 
 
2
v  cos  a
2
2 2
 x c  v
2
a
c
e
 
 
2
c  x c  v  cos  a 
v  cos  a
2
c v
2
2 2
 x c  v
2
2
Derivation of formula for classical and relativistic transverse Doppler Start Angle using Law of Cosines -------------------------------( x c  c  D)
2
 
2 ( x c)  v  cos  s
2
 
2
( x c)  v  2 ( x c)  v  cos  s
2
2
( x c)  v  ( x c  c  D)
2
 
2
2
2
2 ( x c)  v  cos  s  ( x c  c  D)
 
cos  s
2
( x c)  v  ( x c  c  D)
2
2
s
acos 
( x c)  v
2 ( x c)  v
 ( x c) 2  v2  ( x c  c  D) 2 

2 ( x c)  v


Derivation of formula for classical and relativistic transverse Doppler Projection Angle using Law of Cosines -----------------------v
2
2
 
2 ( x c)  ( x c  c  D)  cos  p
p
 
2
( x c)  ( x c  c  D)  2 ( x c)  ( x c  c  D)  cos  p
2
2
( x c)  ( x c  c  D)  v
2
 
2 ( x c)  ( x c  c  D)  cos  p  v
 
cos  p
 ( x c) 2  ( x c  c  D) 2  v 2 

 2 ( x c)  ( x c  c  D) 
acos 
40
2
2
2
( x c)  ( x c  c  D)  v
2 ( x c)  ( x c  c  D)
2
( x c)  ( x c  c  D)
2
2
Appendix A-5
Relativistic Transverse Doppler Effect
Derivation of formula for Angle of Recessionr to equal Start Angle s ----------------------------------------------------------------------Substituting c for xc, and d for (xc-c+D) we have from the Law of cosines,
If
s
r
Then
d
2
2
c v c
2
2
2
2
c  v  2 c v  cos s
c
Therefore the above can be restated as
2 c v  cos s
d
cos s
v
c
2
2
2
cos s
2 c v
Substituting xc back in place of c gives
2
v
which can be solved for o to obtain
c  v  2 c v  cos o
v
giving
2 c
 
2 x c cos  s
and
 
2 c cos  s
v
s
acos 
v


 2 x c 
and
s
acos 


 2 c 
v
for final versions.
REFERENCES
1
Albert Einstein, Special Theory of Relativity, (1905), originally published under the title, On the Electrodynamics
of Moving Bodies, in the journal, Annalen der Physik.
2
The mathematical factor devised by Dutch physicist H. A. Lorentz in 1895 to explain the null results of the
Michelson and Morley interferometer experiments of 1887.
3
The 1887 experiments conducted by American physicists Michelson and Morley that showed light speed to be
unaffected by the Earth’s motion through a supposed light carrying medium called ether.
4
The 1913 binary star observations by astronomer Willem de Sitter that supposedly showed light speed to be
unaffected by the motion of the source.
5
The original meaning of the Lorentz formula is different from that involving its use in special relativity. To avoid
continued confusion regarding its two different meanings, in its new and predominant use within the framework of
relativity it will be subsequently redefined as the Einstein effect, or the Einstein transformation as appropriate.
6
Walter Ritz, Emission Theory of Electricity, (1908) A theory of electricity where electromagnetic waves travel
with speed c relative to the source of these waves.
7
The interpretation of these data is still controversial. See J. G. Fox, Am. J. Phys. 33,1 (1965), and K. Brecher,
Phys. Rev. Lett. 39, 1051 (1977).
8
Joseph A. Rybczyk, Millennium Theory, (2001 - 2005) a series of ten principal research papers and many
supplementary papers beginning with the, Millennium Theory of Relativity, (2001).
9
Stephen Hawkin, On the Shoulders of Giants, ISBN 0-7624-1348-4 (Running Press Book Publishers, 2002) p
1183, and A. Einstein, On the Electrodynamics of Moving Bodies, (
http://www.fourmilab.ch/etexts/einstein/specrel/www/)
41
Relativistic Transverse Doppler Effect
The Complete Correlation of the Lorentz Effect to the Doppler Effect in Relativistic Physics
Copyright © 2005 Joseph A. Rybczyk
Copyright © Revised 2006 Joseph A. Rybczyk
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