relativistic transverse doppler effect pdf version transverse... · 2012-11-27 · applied, the...

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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

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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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-of-sight 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 line-of-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 line-of-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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c

vceo

(1) Classical Doppler Effect Recession Formula for Wavelength

where λo and λe are the observed wavelength and emitted wavelength respectively and

vc

cff eo

(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

c

vceo

(3) Classical Doppler Effect Approach Formula for Wavelength

where λo and λe are the observed wavelength and emitted wavelength respectively and

vc

cff eo

(4) Classical Doppler Effect Approach Formula for Frequency

where fo and fe are the observed frequency and emitted frequency respectively.

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

Light Emitted at Point O

vc c

B A Star O

FIGURE 1 Star Receding from A while Approaching B

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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.

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

Y

vT

B A Star O

FIGURE 2 Einstein Relationship between Inertial Frames of Star and Observers

cTct

S1 S2

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222 )()()( vTcTct (5) Pythagorean Relationship of Distances

22222)( TvTcct (6)

)()( 2222 vcTct (7)

22 vcTct (8)

c

vcTt

22 (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

22 vc

ctT

(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

2

22

c

vcTt

(11)

2

2

2

2

c

v

c

cTt (12)

2

2

1c

vTt (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

2

2

1

1

c

vtT

(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

ctd (15) Moving Inertial Frame Distance

cTD (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

222 )()()( vTcTct (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

c

vcTt


as already shown in Equation (9), we can then claim that

c

vcDd

22 (17) Einstein Distance Contraction for d Simplified

since by substitution from Equations (15) and (16) we obtain

c

vccTct

22 (18)

which simplifies to

c

vcTt


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.

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

B

FIGURE 3 Interface Circle between Expanding Light Spheres

Y

A

S1 S2

vT

Star O

cTct

Contact Interface Circle

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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

22 vc

cdD

(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

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22 vc

cex

(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

c

f (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

ee

cf

(22) Emitted Wavelength to Emitted Frequency Relationship

where fe is the moving frame emitted frequency that corresponds to moving frame emitted wavelength λe and

xx

cf

(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

ee f

c (24) Emitted Frequency to Emitted Wavelength Relationship

and

xx f

c (25) Transformed Frequency to Transformed Wavelength Relationship

the right sides of which can be substituted into Equation (20) to obtain

22 vc

c

f

c

f

c

ex (26)

which can then be solved for the transformed frequency fx in terms of emitted frequency fe. This gives

15

22

2

vcf

c

f

c

ex (27)

2

22

c

vcf

c

f ex (28)

and

c

vcff ex

22 (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

c

vceo

(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

vc

cff eo

(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

c

vcxo

(32) Relativistic Line-of-Sight Doppler Effect for Wavelength (+R, –A)

16

and

vc

cff xo

(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

c

vc

vc

ceo

22 (34)

that simplifies to

22 vc

vceo

(35) Relativistic Line-of-Sight Doppler Effect for Wavelength (+R, –A)

and

vc

c

c

vcff eo

22

(36)

that simplifies to

vc

vcff eo

22

(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.

17

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.

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

θr θa

FIGURE 4 Classical Transverse Doppler Effect Defined

Recession Recession 90 Degrees Approach

θr θr θr θa θa θa c

c c c

s v

D D D

D

s v s v s v

A B C D

18

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

ra deg180 (38) Relationship of Angle of Recession θr to Angle of Approach θa

and

ar deg180 (39) Relationship of Angle of Approach θa to Angle of Recession θr

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

19

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.

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

θs θr s v source

D

c

θp

FIGURE 5 Classical Transverse Doppler Effect Relationships

θa

20

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

rvDDvc cos2222 (40) From Law of Cosines Theorem – Initial Wavelength from which we use Mathcad to derive

2222 coscos vcvvD rr (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

c

vcvvD rr

fi

2222

)(

coscos

(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

c

vcvv rrer

2222 coscos

(43) Classical Transverse Doppler Effect for Initial Wavelength

21

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

c

vcvv aaea

2222 coscos


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

c

vcvv ooeo

2222 coscos


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

22

c

vcvv rrxr

2222 coscos

(46) Classical Factor for Transverse Doppler Effect, Initial

Wavelength into which we substitute the right side of the previously derived Equation (20)

22 vc

cex

(20) Wavelength Transformation UF to SF

in place of the transformation variable λx to obtain

c

vcvv

vc

c rrer

2222

22

coscos

(47)

that simplifies to

22

2222 coscos

vc

vcvv rrer

(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

22

2222 coscos

vc

vcvv aaea

(49) Relativistic Transverse Doppler Effect Approach, Initial

Wavelength for approach and

22

2222 coscos

vc

vcvv ooeo

(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

23

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

oo f

c (51) Observed Frequency to Observed Wavelength Relationship

and the previous

ee f

c (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

22

2222 coscos

vc

vcvv

f

c

f

c oo

eo

(52)

that can be simplified to

2222

22

coscos vcvv

vc

c

f

c

f

oo

eo

(53)

and

2222

22

coscos vcvv

vcff

oo

eo

(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

24

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.

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

source

λo

xc

λe = c θr

v

xc-c

D = λo

c+v c-v

S2

S1

s is start point of emission S1 is leading edge of wave 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

e s θs θa

θp

θp

25

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

rDcxcvDcxcvxc cos)(2)()( 222 (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

26

2222 )(coscos)( vxcvvDcxc rr (56)

that when solved for D gives

2222 )(coscos vxcvvxccD rr (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

c

vxcvvxccD rr

f

2222 )(coscos

(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

vxcvvxcc rrer

2222 )(coscos

(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

vxcvvxcc aaea

2222 )(coscos


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

c

vxcvvxcc ooeo

2222 )(coscos


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

27

22

2222 )(coscos

vc

vxcvvxcc rrer

(62) Relativistic Transverse Doppler Effect

Recession, Wavelength and

22

2222 )(coscos

vc

vxcvvxcc aaea


Approach, Wavelength and

22

2222 )(coscos

vc

vxcvvxcc ooeo


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

2222

22

)(coscos vxcvvxcc

vcff

oo

eo


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

28

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

clv /1 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 co-ordinates, 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

./1

/cos122 cv

cvff

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

22

cos

vc

c

c

vc oeo

(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

29

22

cos

vc

vc rer

(67) SR Relativistic Transverse Doppler Effect Recession Wavelength

and

22

cos

vc

vc aea

(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

2

1

cos1

c

v

c

vr

er

(69) SR Relativistic Transverse Doppler Effect Recession Wavelength

and

2

1

cos1

c

v

c

va

ea

(70) SR Relativistic Transverse Doppler Effect Approach Wavelength

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

c

vr (71)

30

and

rr

rer ff

cos1

1 2

(72) SR Relativistic Transverse Doppler Effect Recession Frequency

and

rr f

c (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

c

va

(74)

and

aa

aea ff

cos1

1 2

(75) SR Relativistic Transverse Doppler Effect Approach Frequency

and

aa f

c (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.

31

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.

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

θs θr

FIGURE 7 The 90 Degree Relativistic Transverse Doppler Effect

Recession Approach

E

D1

D2c

c c

v source

32

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

33

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.

34

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

rvDDvc cos2222 (40) and by substituting c for D, obtain

rvccvc cos2222 (77) that can be solved for θr to obtain

0cos22 rvcv (78)

2cos2 vvc r (79)

vc

vr 2

cos2

(80)

c

vr 2

cos (81)

and finally

c

var 2

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

35

rcv cos2 (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

xc

var 2

cos (84) Angle of Recession for Null Classical Transverse Doppler Effect - All Wavelengths

and

rxcv cos2 (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

36

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

37

Appendix A-1 Relativistic Transverse Doppler Effect

r 90deg a 90deg

Emitted Transformed Observed Rec Observed App

e 5.5 107

x 6.351 107

r 5.5 107

a 5.5 107

MR formulas for finding r when v is given, or for finding v when 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 acosv

2 x c

x 75.5224878140701deg vx 2 x c cos r vx

c0

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)

r 90deg a 90degEr e

c v cos r

c2

v2

Ea3 6.351 107

Ea3c

fEa3fEa3 4.721 10

14fEa3 fe

1 a2

1 a cos a a

v

c

Er3 6.351 107

Er3c

fEr3fEr3 4.721 10

14fEr3 fe

1 r2

1 r cos r r

v

c

Ea2 6.351 107

Er2 6.351 107

Ea2 e

1v

ccos a

1v

c

2

Er2 e

1v

ccos r

1v

c

2

Ea 6.351 107

Er 6.351 107

Ea e

c v cos a

c2

v2

Relativistic Transverse Doppler Effect Wave Theory.mcd February 1, 2006 Joseph A. Rybczyk

Speedkm/hrc 299792458 e 550 10

9 fe

c

e fe 5.451 10

14 v c .5 v

602

1000 5.396 10

8

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 a 180 deg r a 90deg x 100

D c x c v cos r v2

cos r 2 x2

c2

v2

Distance to observer

r e

c x c v cos r v2

cos r 2 x2

c2

v2

c2

v2

s 59.9999999999999degp 30deg

Angle of Projection (0-0-0 deg) Angle of Start (0-180 deg) Angle of Recession (0-180 deg) Angle of Approach (0-180 deg)

la 3.175 107

lr 9.526 107

la ec v

c2

v2

lr ec v

c2

v2

Line-of- Sightgiven at right

p acosx c( )

2x c c D( )

2 v

2

2 x c x c c D( )

x e

c

c2

v2

a e

c x c v cos a v2

cos a 2 x2

c2

v2

c2

v2

s acosx c( )

2v

2 x c c D( )

2

2 x c v

D

c0.866

m x e 5.5 107

km x e

10005.5 10

10

38


o 2.783 107

Er e

c cos r v

c2

v2

o e

c x c v cos r v2

cos r 2 x2

c2

v2

c2

v2

o

Er0.999944772724393Comparison o/Er r 150 deg

Er 6.94 107

o 6.939 107

Er e

c cos r v

c2

v2

o e

c x c v cos r v2

cos r 2 x2

c2

v2

c2

v2

o


Er 1.262 106

o 1.262 106

Er e

c cos r v

c2

v2

o e

c x c v cos r v2

cos r 2 x2

c2

v2

c2

v2

o

Er0.999999999993485Comparison o/Er

Er 1.262 107

o 1.262 107

Er e

c cos r v

c2

v2

o e

c x c v cos r v2

cos r 2 x2

c2

v2

c2

v2

o


Er 1.262 107

o 1.262 107

Er e

c cos r v

c2

v2

o e

c x c v cos r v2

cos r 2 x2

c2

v2

c2

v2

o


Er 2.783 107

Er 2.397 106

o 2.397 106

Er e

c cos r v

c2

v2

o e

c x c v cos r v2

cos r 2 x2

c2

v2

c2

v2

r 0 deg

Recession Angle r MR formula SR Formula MR Observed o SR Observed Ea & Er

x 1.262 106

e 5.5 107

.2669For v = 80,000km/s usex e

c

c2

v2

MR Emitted e MR Transformed x

x e 100 0.55x e 5.5 103

x 104

v c .9

Relativistic Transverse Doppler Effect using Angle of Recession February 2, 2006 Joseph A. RybczykSpeed/Souce x Factor Dist/Observer m cm

o


Er 1.83 106

o 1.83 106

Er e

c cos r v

c2

v2

o e

c x c v cos r v2

cos r 2 x2

c2

v2

c2

v2

o


Er 2.245 106

o 2.245 106

Er e

c cos r v

c2

v2

o e

c x c v cos r v2

cos r 2 x2

c2

v2

c2

v2

o


39


Er 6.001 107

v c .5

o e

c x c v cos r v2

cos r 2 x2

c2

v2

c2

v2

Er e

c cos r v

c2

v2

o 6.351 107

Er 6.351 107

v c .6

o e

c x c v cos r v2

cos r 2 x2

c2

v2

c2

v2

Er e

c cos r v

c2

v2

o 6.875 107

Er 6.875 107

v c .7

o e

c x c v cos r v2

cos r 2 x2

c2

v2

c2

v2

Er e

c cos r v

c2

v2

o 7.702 107

Er 7.702 107

v c .8

o e

c x c v cos r v2

cos r 2 x2

c2

v2

c2

v2

Er e

c cos r v

c2

v2

o 9.167 107

Er 9.167 107

v c .9

o e

c x c v cos r v2

cos r 2 x2

c2

v2

c2

v2

Er e

c cos r v

c2

v2

o 1.262 106

Er 1.262 106

v c .9999999999999999

o e

c x c v cos r v2

cos r 2 x2

c2

v2

c2

v2

Er e

c cos r v

c2

v2

o 29.148 Er 29.148

Relativistic Transverse Doppler Effect using Angle of Recession February 2, 2006 Joseph A. RybczykRecesson Angle r x Factor Dist/Observer m cm MR Emitted e MR Transformed x

Different for each v. Givenby SR for r=90 deg, x=1r 90 deg e 5.5 10

7

x 107.5

x e 17.393 x e 100 1.739 103

Source Speed MR Formula SR Formula MR Observed o SR Observed Ea & Er

v c 0

o e

c x c v cos r v2

cos r 2 x2

c2

v2

c2

v2

Er e

c cos r v

c2

v2

o 5.5 107

Er 5.5 107

v c .1

o e

c x c v cos r v2

cos r 2 x2

c2

v2

c2

v2

Er e

c cos r v

c2

v2

o 5.528 107

Er 5.528 107

v c .2

o e

c x c v cos r v2

cos r 2 x2

c2

v2

c2

v2

Er e

c cos r v

c2

v2

o 5.613 107

Er 5.613 107

v c .3

o e

c x c v cos r v2

cos r 2 x2

c2

v2

c2

v2

Er e

c cos r v

c2

v2

o 5.766 107

Er 5.766 107

v c .4

o e

c x c v cos r v2

cos r 2 x2

c2

v2

c2

v2

Er e

c cos r v

c2

v2

o 6.001 107

40


r e

c x c v cos r v2

cos r 2 x2

c2

v2

c2

v2

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 v2

cos a 2 x2

c2

v2

c a e

c x c v cos a v2

cos a 2 x2

c2

v2

c2

v2

Derivation of formula for classical and relativistic transverse Doppler Start Angle using Law of Cosines --------------------------------

x c c D( )2

x c( )2

v2

2 x c( ) v cos s 2 x c( ) v cos s x c c D( )2

x c( )2

v2

2 x c( ) v cos s x c( )2

v2

x c c D( )2

cos s x c( )2

v2

x c c D( )2

2 x c( ) vs acos

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 ------------------------

v2

x c( )2

x c c D( )2 2 x c( ) x c c D( ) cos p 2 x c( ) x c c D( ) cos p v

2 x c( )2

x c c D( )2

2 x c( ) x c c D( ) cos p x c( )2

x c c D( )2

v2

cos p x c( )2

x c c D( )2

v2

2 x c( ) x c c D( )

p acosx c( )

2x c c D( )

2 v

2

2 x c( ) x c c D( )

Derivation of formula for classical transverse Doppler distance d using Law of Cosines ------------------------------------------------------

r 90 deg cos r 0 v c .9 Where d x c c D( ) x c( )2

v2

d2 2 v d cos r

d v cos r v2

cos r 2 x2

c2

v2

v cos r v2

cos r 2 x2

c2

v2

1

2

v cos r v2

cos r 2 x2

c2 v

2

1

2

x c c D( ) v cos r v2

cos r 2 x2

c2

v2

D c x c v cos r v2

cos r 2 x2

c2

v2

DfD

c

Derivation of formula for classical transverse Doppler recession using Law of Cosines ------------------------------------------------------

DfD

c

c x c v cos r v2

cos r 2 x2

c2

v2

cr e

c x c v cos r v2

cos r 2 x2

c2

v2

c

Derivation of formula for relativistic transverse Doppler recession using Law of Cosines ------------------------------------------------------

r e

c x c v cos r v2

cos r 2 x2

c2

v2

c

c

c2

v2

41


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/)

for final versions.s acosv

2 x c

and v 2 x c cos s Substituting xc back in place of c gives

s acosv

2 c

and v 2 c cos s giving cossv

2 ccoss

v2

2 c v2 c v coss c

2v

2 c

2

which can be solved for o to obtain c2

c2

v2

2 c v cosoTherefore the above can be restated as

d cThen s rIf

d2

c2

v2

2 c v cossSubstituting c for xc, and d for (xc-c+D) we have from the Law of cosines,

Derivation of formula for Angle of Recession r to equal Start Angle s ------------------------------------------------------------------------

42




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