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Millennium Theory of Relativity

Copyright © 2001 Joseph A. Rybczyk Abstract

The Millennium Theory of Relativity is a fundamental theory in relativistic physics. Through methodical analysis of the evidence, strong and convincing arguments are developed in opposition to those of special relativity. Specifically, the evidence supporting the constancy of light speed is accepted, and with it the self-evident proposition that the Laws of physics are the same in all inertial systems. Yet, it is shown that the apparent relationship of these two entities cannot be properly reconciled without the introduction of spherical reference frames. Only then, can a correct understanding of relativistic principles be achieved, and this understanding is substantially different from that previously arrived at. The evolving principles of the new theory will clearly demonstrate that many of the assumptions of the presiding theory are untenable.

Although it will be shown that time is affected by relative motion, other associated effects are different from those of currently held beliefs. Distances in space for example, and the size of objects it contains, are not really affected by such motion. In fact, the principle of time variance is contingent upon precisely that condition. Whereas actual distances are unaffected, however, the distance light travels in an interval of time, is affected, and this effect on distance is not limited to the direction of motion. The shrinking distance traveled by light within a moving frame of reference occurs equally in all directions. This change of view, is consistent with the prevailing evidence, yet has a profound bearing on the way the physical laws of nature are perceived.

Another consequence of the undertaken analysis is the realization that the Lorentz formulas are not direct representations of the principles of relativistic behavior. Even though these formulas yield correct mathematical results, by the very nature of their indirectness they tend to be misleading. It is contended, that these formulas together with the rectangular reference frames they are normally associated with, led to the misunderstandings embodied in special relativity. It is further argued, that only by use of the spherical reference frames and the directly derived equations developed in the presented analysis, is a proper understanding possible.

A final consequence of the ensuing analysis in this present work is a somewhat different view of the relativistic, transverse, Doppler effect. Comparison of the different results reveals yet another possible flaw in the presiding theory. This is not to say that additional flaws might not surface in a future supplemental work involving mass and energy.

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Millennium Theory of Relativity Contents 1. Introduction 4 2. The Underlying Postulate 4 3. The Speed of Light 6 4. Constancy of Light Speed between Natural Systems 6 5. The Transformation Process 7 6. Developing the Transformation Formulas 9 7. The Time Transformation Formulas 10 8. The Distance Transformation Formulas 12 9. The Flaw in Special Relativity 21 10. Proving the Fundamentals 24 11. Effects on Wavelength and Frequency 27 a. The Transformation Effect 28 b. The Doppler Effect 30 c. The Combined Transformation and Doppler Effect Equations 32 12. Doppler Effect for Transverse Motion 36 13. Correlation of All Effects 43 14. Conclusion 45 References 46 Equations 1. Time in the moving frame 11 2. Time in the stationary frame 12 3. Distance in the moving frame 15 4. Radius of stationary sphere 16 5. Radius of moving sphere 17 6. Distance in the stationary frame 18 7. Equality of distance ratios 18 8. Frequency to wavelength relationship 28 9. Wavelength to frequency relationship 28 10. Wavelength transformation, MF to SF 29 11. Wavelength transformation, SF to MF 29 12. Frequency transformation, MF to SF 30 13. Frequency transformation, SF to MF 30 14. Doppler effect – Observed wavelength 31 15. Doppler effect – Transformed wavelength 31 16. Doppler effect – Observed frequency 32 17. Doppler effect – Transformed frequency 32 18. Observed wavelength 33 19. Observed frequency 33 20. Emitted wavelength 33

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21. Emitted frequency 33 22. Observed wavelength, Transverse motion 39 23. Observed frequency, Transverse motion 39 24. Observed wavelength, Transverse motion, Alternate equation 42 25. Observed frequency, Transverse motion, Alternate equation 42 Figures 1. The Transformation Process 8 2. Developing the Transformation Formulas 9 3. The Distance Transformation Formulas 13 4. Shows how all distances are affected by motion 19 5. Unveils the flaw in special relativity 22 6. Proof of the Fundamentals 25 7. The Direct Method 25 8. The Doppler Effect 30 9. Combined Transformation Process and Doppler Effect 35 10. Doppler Effect, Recession 37 11. Math Definition of Doppler Recession 38 12. Doppler Effect, Approach 40 13. Math Definition of Doppler Approach 41 14. Correlation of All Effects 43

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Millennium Theory of Relativity

Copyright © 2001 Joseph A. Rybczyk 1. Introduction

The Millennium Theory of Relativity is the result of more than four years of intense research involving light-speed phenomena. It is an independent effort, relying only on the analysis presented here, and that experimental and observational evidence that shows light to have a finite value c, that is constant relative to a stationary frame of reference, and does not appear to be affected by the motion of the source. Conversely, it does not rely on any theoretical assumptions, or theories of the past or present. This includes Einstein’s special theory of relativity1 and any theoretical assumptions upon which that theory is founded. Specifically, the theory presented here, accepts the evidence of the Michelson and Morley2 experiments of 1887, de Sitter’s3 experiments involving binary stars, and experiments showing that light given of by subatomic particles moving at near light-speed is unaffected by the motion of such particles4. Also considered is the apparent increase in the life expectancy of μ mesons traveling at near light speed in the earth’s atmosphere5. Of course, all experimental evidence establishing the finite value of c is also accepted, and acknowledgement is given to the many contributors involved6. Aside from these acknowledgements, the theory presented here relies solely on theoretical assumptions arrived at through the theory itself. Although exclusivity of some of these assumptions cannot be claimed, it is claimed that only within the presented theory are these assumptions properly understood and correctly applied. Consequently, it will also be shown that the special theory of relativity is flawed to the extent of invalidating itself. The evidence to that effect is clear and overwhelming and is presented here in the order it was discovered.

In summary, the Millennium Theory of Relativity provides a new way of viewing the physical laws of nature. It is a theory that makes sense out of natural laws that previously defied true understanding. In so doing, it provides clearer insights into the workings of those natural laws, thereby opening the door to new opportunities of discovery.

It is important to note here, that the Millennium Theory of Relativity begins where the Theory of Natural Motion, a previous paper by this same author, leaves off. The Theory of Natural Motion, copyrighted in 1996, and published on MightyWords.com in October of 1999, though not a prerequisite to this present work, will nonetheless benefit the reader in understanding many of its concepts.

A final note: The Millennium Theory of Relativity is presented here in the same order in which it was developed. Initially an attempt is made to understand the nature of light travel and subsequent, related light-speed phenomena. Shortly after beginning, the principles for the present theory begin to become apparent and are seized upon to develop the final comprehensive theory on its own. 2. The Underlying Postulate

As previously discovered in the Theory of Natural Motion, the laws of nature are simple, predictable, and few, and the truth lies inward, not outward. Such appears to be the case involving light-speed phenomena. Whereas in the previous work it was found that Newton’s three laws of motion could be combined into a single comprehensive law, it is found here that a single postulate is sufficient to explain all physical phenomena related to light speed. This is in contrast to the two postulates put forth in Einstein’s special relativity. Essentially, what is

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discovered here is that the speed of light relative to the observer (defined in Einstein’s first postulate) is not an independent consideration, but rather a consequence of the laws of physics applicable to all inertial systems (treated in Einstein’s second postulate). That is, we are not postulating here that the speed of light in empty space has the same value c in all inertial systems. We are accepting the evidence derived through scientific observations and experiments that seem to verify such, and then proceed with the process of determining how the physical laws of Nature cause this to be apparently true. The postulate upon which the present theory is founded is simply this:

Postulate: The laws of physics are the same in all natural systems.

We refrain from the temptation to continue use of the phrase “inertial systems” in the

above postulate because it was discovered in the Theory of Natural Motion that these terms in that usage are technically incorrect. Nevertheless, the phrase, “natural system” refers to a, “natural frame of reference” or simply, “natural frame” and can be thought of as having the same general meaning as the phrase, “inertial system” or, “inertial frame of reference”. Thus, a natural frame of reference is defined as a frame of reference attached to an object or body in a state of rest, or a state of uniform motion, or more correctly, an object or body not under acceleration. (As pointed out in the previous theory, a state of rest and a state of uniform motion are one and the same thing.)

The principle issue here is that the above postulate means that the characteristics of all physical phenomena directly associated with one natural system, will be identical, under the same conditions, to the characteristics of the same physical phenomena associated with any other natural system. Put simply, under the same conditions, any scientific experiment will yield the same results in any natural system, and any observation, or relationship found valid in one natural system will be found valid in any other natural system. This includes the speed of light, or any other form of electromagnetic propagated energy. Such view is consistent with the proposition that no experiment conducted on an object in uniform motion will give evidence of that motion.

Whereas the characteristics defined above are the same in all natural systems, this does not mean that such characteristics are directly transferable from one natural system to another. As we proceed, it will be shown that these characteristics are affected by several factors. Those factors include the relative motion between frames, gravity, and medium of light travel. Of these, the relative motion between frames is the most important because it affects the results of many experiments and calculations, yet is not covered by the qualifying stipulation, “under the same conditions”. Gravity and medium of light travel, however, though possibly having effects similar to that of relative motion, are the principle reasons for the qualifying stipulation. On the one hand, relative motion will affect how physical phenomena in one natural system is perceived in another natural system regardless of the fact that the two systems are identical. On the other hand, gravity and medium of light travel can also have an effect on how physical phenomena is perceived, but is handled differently than relative motion. For example, gravity will affect how physical phenomena is related between two large bodies of different masses, widely separated in space, (stars and planets for instance) even if they reside in the same natural system (are not moving relative to each other). Moreover, since the speed of light is known to be affected by the medium of travel, gases and other properties in space can also have an effect that must be taken into consideration.

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3. The Speed of Light

It is universally accepted that the speed of light, designated by the symbol c has a finite speed in empty space of 299792485 m/s. The qualifier here is, empty space, for it is also universally accepted that the speed of light is affected by the medium through which it travels, and additionally in a manner by gravity. The important consideration here, however, is that under the same conditions, the speed of light has the same value in all natural systems. These two factors, finiteness and constancy relative to all natural systems make it possible to use light as a means of measuring distances within any natural system. This characteristic of light will be made use of later in this paper. 4. Constancy of Light Speed between Natural Systems

Not as universally accepted is the proposition that light in empty space has a constant speed c in all inertial systems regardless of the speed of the source. (Einstein’s first postulate in special relativity) Although this proposition appears to be true, and has general acceptance throughout the scientific community, there are nonetheless, those that disagree. It has often been said of special relativity that the problem lays not so much in being able to understand it, but in being able to believe it. After having struggled with it for over four years, my conclusion is, there is truth on both fronts. Perhaps the biggest failing of special relativity is its inability to explain how the speed of light can be constant relative to the observer, in a manner that can be both understood and believed. And, since this is the basis on which the principles of time dilation, length contraction, and mass and energy increases are founded, those principles are equally difficult to understand and believe. With this paper, I hope to change that.

The first problem with special relativity is the proposition that light does not take on the speed of the source. In the Theory of Natural Motion, it is shown that physical objects do not experience motion. Motion, it was pointed out, is not something an object does, but rather, a relationship between objects in different natural states. A natural state was accordingly defined as neither a state of rest, nor a state of motion, but actually a state in which there are no acceleration forces working on the object. Moreover, acceleration was shown not to be a form of motion at all. Since an object does not experience motion, it is meaningless to speculate that light either does, or doesn’t take on the speed of the source. The source has no speed. Speed is a relational consideration between masses in different natural states.

The above definitions notwithstanding, it is often convenient to attribute a state of motion or rest to an object, or frame of reference, in order to simplify and make easier to understand, that which is being conveyed. Such liberties will be taken throughout this paper. From the frame of reference of the source then, light must take on its speed, otherwise the speed of light would be affected by the speed of the source, and we know from the Michelson and Morley experiments of 1887, and many other experiments conducted since, that this is not true. (If a source of light were thought to be moving through space, then the light would have to move along with it to maintain a constant speed c relative to that source.) From the point of view of a stationary observer (an observer in a frame that the source is moving relative to) the light coming toward him or her, does not appear to have taken on the speed of the source, otherwise the light would be slower or faster than c depending on the direction of motion of the source. This characteristic of light speed is also supported by evidence, including that based on observations of binary and other stars, and also on measurements made on light from subatomic particles that are accelerated to near light speed in particle accelerators.

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Since light has a constant speed c relative to a moving source, yet somehow has that same speed relative to a stationary observer, it is incorrect to say that the speed of light is unaffected by the relative motion between the source and the observer. It must be affected, otherwise as stated previously it would appear to have taken on the speed of the source relative to the observer in the stationary frame of reference. That is, light appears to somehow adjust its speed in inverse proportion to the speed of the relative motion between the source and observer.

Put simply, while light appears to maintain a constant speed c relative to the source, as viewed by an observer in the reference frame of the source, it appears to change its speed relative to the source, as viewed by an observer in a stationary frame of reference. To this stationary observer, the light must have either speeded up or slowed down in inverse proportion to the speed of the source, in order to be received by that observer at speed c. To resolve this apparent contradiction, let us now examine the entire process in detail. 5. The Transformation Process

Referring to figure 1A, assume that points A, B and C, at the three corners of the triangle, are points in a stationary frame of reference. Assume further that a point light source, moving at a uniform speed v, gives off a pulse of light at Point A as it moves to point B. And finally, assume that a point of this light moving at speed c in a perpendicular direction relative to the path of travel of the moving source, reaches point C at the instant the source reaches point B. In unison, this same point of light will travel from point A to point C in the stationary frame of reference.

Since light does not take on the speed of the source in the stationary frame of reference, it emanates outward in all directions from point A at speed c. Thus, it forms a perfect sphere centered on point A with distance A-C as its radius as illustrated in figure 1B. The light in the moving frame of reference, however, emanates outward from the moving source at speed c, with the source as its center as illustrated in figure 1C. This perfect sphere of light has the distance B-C as its radius relative to the moving frame of reference.

Accepting the evidence that nothing can exceed the speed of light, the light source in the above example can never escape from within the sphere of light (sphere 1) emanating from point A in the stationary frame of reference. That is, the sphere will always be growing at a faster rate than the source can travel. At any instant in time then, the source will always be located at a point along the path of travel within the sphere of light evident in the stationary frame of reference. (The fact that the smaller sphere (sphere 2) intersects point A is coincidental. A slower or faster moving source would provide a larger or smaller sphere that would not intersect point A.)

Of significance in figure 1C is the fact that the point of intersection of the two spheres exactly coincides with the perpendicular axis of the sphere of light centered on the source. We can now draw on this geometrical relationship of the two spheres and formalize that relationship mathematically. Before proceeding, however, it will be useful to take a moment to clear up a few areas of potential confusion.

It was intentional that a stationary observer was not used in the above explanation. While use of such an observer is often helpful with regard to explaining the concepts just covered, the location of the observer can often cause confusion yet has no bearing on the result. In fact, the observer can assume any position at all within the stationary frame of reference without affecting the outcome. That is to say, it doesn’t matter if the source is moving away from the observer, passing by the observer, or moving toward the observer. The only thing that matters is that the

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observer occupies the same frame of reference as points A and B, in order for the results to be valid with regard to that observer. It is the relationships the source establishes with points along the path of travel in the stationary frame that matter, and not the location of an observer within that frame. It should also be noted that the concepts just discussed and those still to be covered apply to all other frames of reference simultaneously. It is only the specifics that will vary, not the fundamental concepts. I.e., the source simultaneously establishes similar relationships in all other frames of reference. Subsequently, if the source has a different speed v relative to each different frame, the specific results will vary from one frame to another.

It should also be noted, that there are not actually two spheres of light as illustrate in the preceding figure. There is only one sphere of light, but it exists in a different form for each different frame of reference. Thus, the relationships being discussed involve comparing the form of the sphere as it appears in the moving frame to the form as it appears in the stationary frame. It must be emphasized here, that only the sphere of light centered on point A is experienced in the stationary frame. Thus, in the stationary frame, sphere 2 can be arrived at only through mathematics. The converse is true for the moving frame. Only sphere 2 is experienced in that

Light Source v

Sphere 1, Light in Stationary Frame

c

B

B A

C

c

A

c

v Light Source A B

C

FIGURE 1 The Transformation Process

v

Sphere 1, Light in Stationary Frame

c

C

BA

C

cSphere 2, Light inMoving Frame

Light Source

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frame of reference, and sphere 1 would have to be arrived at mathematically. In this case, the relationship is reversed, and sphere 2 is the larger of the two spheres.

One final point needs to be addressed here. Since the speed of the light in the stationary frame has a speed c as verified by the evidence, it is reasonable to assume that along the path A-B it has a speed c + v relative to the source as experienced in the stationary frame. This gives evidence that speed c in the moving frame is slower than speed c in the stationary frame. This is also confirmed by the fact that sphere 2 is smaller than sphere 1. Our postulate and common sense, however, dictate that speed c at the source must equal speed c in the stationary frame of reference. This gives rise to the assumption that some sort of transformation must exist between the two frames, an issue to be addressed in the next section. 6. Developing the Transformation Formulas

We are now ready to develop the mathematical relationships that define the transformation process. Referring to figure 2, we can see that a Pythagorean relationship exists between the two forms of the light sphere being discussed. This relationship can be exploited and integrated with another relationship involving time, distance and rate to arrive at mathematical equations that fully define the transformation process.

In accordance with the equation that relates time to distance and rate, we obtain,

that gives d = rt for distances, where t = time, d = distance, and r = rate of speed. Thus, in figure 2, we can express the distance traveled by the source between points A and B in the stationary frame as d1 = vt, where v = the speed of the source in the stationary frame. And since the light emanating from point A has a speed c in the stationary frame, the distance the light traveled to

r

dt

d2 = ct

d1 = vt

d`

A B

C

22 )()(` vtctd

FIGURE 2 Developing the Transformation Formulas

Sphere 1 Sphere 2

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point C during the same period in the stationary frame can be expressed as d2 = ct. It will be recognized that the distance between point A and point C represents both the radius of sphere 1, and the hypotenuse of the right triangle that includes the horizontal line A-B, and the vertical line B-C. It will also be recognized, that since the vertical line defines the distance traveled by the light in the moving frame, it represents the radius of sphere 2 in addition to one side of the right triangle. These facts have great significance that will be discussed later. For now, it is enough to know that the triangle is a valid right triangle because the line B-C represents light traveling in a perpendicular direction from the path of motion of the source.

Since we now have a way of relating distances in the stationary frame to distances in the moving frame, we can define the distance along the vertical line in the moving frame in terms that are valid in the stationary frame. Thus, the distance d` in the moving frame can be expressed as,

Before proceeding further, we must decide what it is we hope to find. Referring to figure 2, we can see that the light travels a greater distance in the stationary frame than it does in the moving frame. Moreover, since it has been shown, and is reasonable to assume, that both spheres were formed simultaneously, it follows that the speed the light traveled in the moving frame must be slower than the speed it traveled in the stationary frame. This of course has already been borne out in our analysis. It was determined that, c relative to the source is a slower speed than c relative to the observer. This is a very profound observation, since it implies that time relative to a moving source must be slower than time relative to a stationary observer.

Bear in mind, that if our postulate is correct, and if an observer were to travel to the source, he would find no differences there. Any measurements he makes with equipment brought along, would show, distances, time and the speed of light there to be the same as he experienced in the stationary frame. Any differences that were experienced in the stationary frame then must be a result of the relative motion of the source. It follows therefore, that if the source’s speed of light is slower, everything else on the source must be slower. This can only mean one thing; time is slower in the reference frame of the source. We now know what we should be looking for. We need a way to determine the relationship of time in the source’s frame to time in the stationary frame. 7. The Time Transformation Formulas

Going back to the time, rate, and distance formula, and using the convention that symbols marked prime represent values in the moving frame we get,

Having already established that d` can be represented by,

,)()(` 22 vtctd

.'

''

r

dt

.)()(` 22 vtctd

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and given, that c has the same value in the moving frame that it does in the stationary frame, by way of substitution in the time, rate, and distance formula, we get,

The resulting equation can now be simplified into a more usable form as follows:

Using only simple algebra we have independently arrived at a time transformation equation that provides the exact same answers as the equation used in special relativity, without relying on either that theory, or the Lorentz formula. The resultant equation relies only on presently accepted scientific evidence and the premises of the presented theory.

Obviously, if this equation gives the exact answers as Relativity’s equation, it must be another form of that equation. Starting again with,

and some additional manipulation we get,

.)()(

`22

c

vtctt

2

222 )()(`

c

vtctt

2

22

2

222`

c

tv

c

tct

2

2222`

c

tvtt

2

2222`

c

vttt

c

tvtct

2222

`

c

vctt

)(`

222

c

vctt

22

`

Eq. (1) Time in the moving frame.

,)()(

`22

c

vtctt

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Anyone familiar with special relativity will immediately recognize the final form of this equation. It is the time transformation equation used in that theory, but arrived at here indepent of any considerations involving that theory. At this point, however, our only contribution is that of presenting a much clearer, and believable argument that time is affected by motion. That will change soon enough, but first we will take a moment to develop the transformation equation for t.

Starting with equation 1, by manipulation we get,

Doing similar with the other form of the equation gives us,

2

222 1`

c

vtt

2

22 1`

c

vtt

2

2

1`c

vtt

.

1

1`

2

2

c

vtt

2

2

1

`

c

v

tt

2

2

1`c

vtt

c

vc

tt

22

`

22`

vc

ctt

Eq. (2) Time in the stationary frame.

c

vctt

22

`

Eq. (1)

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This of course is the alternate form of the relativity equation, but arrived at here independently. 8. The Distance Transformation Formulas

Before proceeding, it might be helpful to verify that distance d` in the preceding example has the same value in both frames of reference. Referring to figure 3, it can be seen that when distance d` is expressed in terms related to the source, the distance is expressed as r`t`. In the preceding example we expressed this distance in terms related to the stationary frame. We will now reverse the process and show that d` = d.

Starting with the equation,

we can substitute c for r` because we know that this is the speed of light relative to the moving source in the moving frame of reference. And since,

we can substitute the right side of this equation for t` in the first equation to get,

d2 = ct

d1 = vt

d & d`

A B

C

22 )()( vtctd

FIGURE 3 The Distance Transformation Formulas

Sphere 1 Sphere 2

``` trd

.`22

c

vcctd

c

vctt

22

`

Eq. (1)

`,`` trd

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This in turn can be simplified and manipulated to get,

which is the same expression we had for the stationary frame. Doing similar using the more familiar alternate equation,

leads of course, to the same result.

Again, we arrive at the same expression we had for the stationary frame. This result is expected since it is based on the transformation formula which was arrived

at in similar fashion. The main point of the exercise was to emphasize that the distance B-C in figure 3 is the same for both, the stationary frame of reference, and the moving frame of reference. This is always true, regardless of the speed of the source. That is, this distance will vary from ct to 0 in direct relation to the source’s speed of 0 to c respectivly, in both frames of

,1`2

2

c

vtt

2

2

1`c

vctd

2

22 1)(`

c

vctd

2

222 )(

)(`c

vctctd

2

2222)(`

c

vtcctd

22 )()(` vtctd

22` vctd

)(` 222 vctd

2222` vtctd

,)()(` 22 vtctd

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reference. We must also remember that motion is relative, and either frame in our example can be thought of as being at rest. From that perspective, the radius of the smaller sphere, is equal to the radius of the larger sphere when each are measured in the frame to which they belong. Under those conditions, each frame is considered to be at rest, and distance B-C in figure 3, will equal distance A-C. This places in question the special relativity assumption that only horizontal distances in the direction of motion are affected by that motion. Moreover, as we continue it will be shown that even the very nature of such effects appears to be different from that defined in special relativity. For now, let us proceed with the process of developing the equations that relate distances in the moving frame to those of the stationary frame.

Referring back to figure 3, we might be tempted to define distance B-C as a vertical distance in the moving frame of reference. Such, however, is not the case. While distance B-C is certainly a vertical distance in the stationary frame, it represents the radius of sphere 2, and thus all distances in the moving frame. Likewise is true of distance A-C relative to the stationary frame. Realizing this has many implications, beyond the obvious that all dimensions are equally affected by relative motion. It also implies that reference frames are more correctly defined as spheres, and not rectangular frames as is the present convention. When one compares the two different spherical frames of reference that result from relative motion, the relationships between the two frames becomes exceedingly clear. Developing this concept further, let us define distance d` in the moving frame as representing any distance in the moving frame as it relates to the same distance in the sationary frame. Starting again with the equation, d` = r`t` and substituting c, the speed of light in the moving frame, for r`, we get,

And since we now know that,

we can substitute the right side of this equation for t` in the preceding equation to get,

Interestingly, we passed through this very equation in our last example. If we now take the time to observer that ct = the radius of the larger sphere, the same distance represented by d` in the smaller sphere, we can replace ct in the above equation with d, where d represents the same distance in the large sphere that d` represents in the small sphere. This give us,

Again, doing similar using the more familiar alternate equation,

c

vctt

22

`

Eq. (1)

.`22

c

vcctd

c

vcdd

22

`

Eq. (3) Distance in the moving frame.

`.` ctd

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we get,

Once again, anyone familiar with special relativity will immediately recognize this equation as that dealing with length contraction. But here we are in disagreement with special relativity in a manner that has far reaching consequences. In special relativity, length contraction applies only to horizontal distances in the direction of motion. In the presented theory, this transformation formula applies to all distances. Also, it will soon become apparent that it is the standard of measure that is affected and not actually the distances themselves. These are profound distinctions.

Specifically, d and d` in the just arrived at equations represent the radii of the stationary frame sphere and the moving frame sphere respectively. But since all distances in either sphere are effected in direct proportion to the size of the respective sphere, the equations can be used in a general sense to apply to any distance in either sphere. In that usage then, d represents any distance in the stationary frame sphere while d` represents the corresponding distance in the moving frame sphere. This broader definition is the preferred definition.

At this point one might conclude that a moving particle, object, or body of mass will shrink in size relative to a stationary frame of reference as the relative speed increases. As stated previously, however, it is the standard of measure that shrinks and not the distances themselves. This shrinkage which also affects the measure of distances traveled by any electromagnetic energy propogated by the mass, will occur proportionally in all directions and only become significant at very high speeds that are in themselves a significant fraction of the speed of light. In other words, all objects and propagated energy fields, or more correctly all distances, even those in space will be subject to a standard of measure that shrinks in direct proportion to the size of the moving light sphere defined previouslly. (This concept will be covered in detail in the next section.)

Although the distance contraction equation can be used to find the radius of the moving sphere if the stationary sphere is known, and can be reformulated to find the radius of the stationary sphere if the moving sphere is known, it would be both useful and instructive to develop equations that can be used to find these radii independent of each other. For the stationary sphere the equation is simply,

2

2

1`c

vtt

2

2

1`c

vctd

.1`2

2

c

vdd

,ctdr Eq. (4) Radius of stationary sphere.

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where dr = the radius of the stationary sphere. For dr` = the radius of the moving sphere, we can substitute dr` for d` and ct for d in the

distance contraction equation 3 and simplify the equation as follows:

Thus, we have the equation for finding dr` independent of dr. This of course, could also have been accomplished using the more familiar alternate distance contraction equation as follows:

c

vcdd

22

`

Eq. (3)

c

vcctdr

22

`

22` vctdr Eq. (5) Radius of moving sphere.

2

2

1`c

vctdr

2

22 1)(`

c

vctdr

2

2

1`c

vdd

2

222 )(

)(`c

vctctdr

2

2222)(`

c

vtcctdr

222)(` vtctdr

2222` tvtcdr

)(` 222 vctdr

22` vctdr Eq. (5)

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Thus, we again have the desired equation for finding the radius of the moving sphere independent of the radius of the stationary sphere. We will conclude by taking a moment to reformulate the distance equations for finding d when d` is known. Starting with equation 3,

and some manipulation, we obtain,

And now to finish up, we will do the same with the more familiar version of the equation. Starting with,

we get,

Now that we have defined the important variables involving distance contraction we can combine them in a manner that expresses that distances in both spheres change in direct proportion to the size of the sphere and that the ratio of distance to the sphere radius of one sphere always equals the corresponding ratio of the other sphere. Thus we have,

c

vcdd

22

`

Eq. (3)

c

vc

dd

22

`

22`

vc

cdd

Eq. (6) Distance in the stationary frame.

2

2

1`c

vdd

.

1

1`

2

2

c

vdd

2

2

1

`

c

v

dd

`

`

rr d

d

d

d Eq. (7) Equality of distance ratios.

19

Referring now to figure 4, we will finalize our argument that, contrary to special relativity, all distances (in the context of the presented theory) are affected by relative motion. For this example, assume that a light source emits a pulse of light at point A while traveling along the path A-B at a speed v of .9c/s as illustrated in figure 4A. Further assume that the distance A-B was traveled in one second relative to the stationary frame of reference, represented by sphere 1. And finally, assume that the light pulse reaches point C in the stationary frame at the instant the source reaches point B in the stationary frame. With such being the case, distance A-C in the stationary frame = ct and distance A-B in the stationary frame = vt. Distance B-C in the stationary frame can then be calculated as follows: Where d = distance B-C,

(Note: All results are rounded in these computations.)

Now, according to both, special relativity and the presented theory, distances along the path of motion shrink in the moving frame of reference. Distance A`-B` in the moving frame, as shown in figure 4B, would therefore shrink by an amount given by the distance contraction equation. This distance then, will be,

81.19.1)()( 222 vtctd = distance B-C .436.19. c

or, alternately,

81.19.1

9.19.1'

2

2

2

c

vdd = distance A`-B` c392.19.9.

81.19.1

9.19.`

222

c

vcdd = distance A`-B` c392.19.9.

A B

C

A` B`

C`

Sphere 1 Sphere 1

Sphere 2 Sphere 2

A B

FIGURE 4 Shows how all distances are affected by motion

v = .9c/s

20

where d`= distance A`-B`. However, it can be readily seen that distance A`-C` has also shrunk in the moving frame.

According to the present theory, this distance is,

where dr`= distance A`-C`. This result is to be expected, since both distances, B-C and A`-C` represent the radius of sphere 2.

We can now calculate distance B`-C` in the moving frame. Where d`` = distance B`-C`, we get,

It can be readily seen that,

is the same result for comparing distance A-B to distance A-C in the stationary frame, that we get when we compare distance A`-B` to distance A`-C` in the moving frame as shown below:

Obviously, the same holds true when we compare distance B-C to distance A-C in the stationary frame and compare that result to that of the corresponding distances in the moving frame.

Of course we could have obtained the same answers for the moving frame distances by simply using the distance contraction equations. For example,

81.19.1` 222 vctdr = distance A`-C` c436.19.

,9.1

9.c

CA

BA

d

d

r

cCA

BA

d

d

r

9.436.

392.

``

``

`

`

cCA

CB

d

d

r

436.1

436.

cCA

CB

d

d

r

436.436.

19.

``

``

`

``

2222 392.436.```` ddd r = distance B`-C`c19.036.154.19.

1

9.11

222

c

vcddistance A`-C` ,436.19.81.1 c

c

vcd

22 and distance B`-C` ,19.19.436. c

21

While the exercise just completed may have been instructional, it was superfluous with regard to our argument that distance contraction applies to all distances, or dimensions, within the moving frame of reference. The true spherical nature of light propagation, devoid of external influences, seems unquestionable. Likewise, the Pythagorean relationship between two such reference frames brought about by a moving source of light, while conforming to the evidence that the speed of light is constant, and the proposition that the laws of nature are the same in all reference frames. On these bases the concepts introduced in the preceding paragraphs seem self-evident and beyond reproach. Special relativity, on the other hand, appears to be seriously flawed as previously noted. The nature of this flaw is the topic of our next discussion. 9. The Flaw in Special Relativity

Although the math in special relativity appears to be correct, the interpretation of the results is fundamentally incorrect and subsequently misleading. The underlying problem appears to have resulted from the manner in which the transformation equations were arrived at, specifically undue reliance upon rectangular coordinate systems. This reliance is apparent even in the well-known light clock example used to demonstrate special relativity in present day physics books. The example that uses a light source pulse traveling between two vertically displaced mirrors while all as a unit move in the horizontal direction. Although the results found using the example seem to agree with the transformation equations, closer inspection, in view of what has been shown here, reveals the example to be flawed. In fact, the example does not faithfully represent nature. The fault lies in the proposition that the distance between the mirrors is not affected in the same manner as horizontal distances in the direction of motion. According to special relativity the distance between the mirrors is unchanged whereas the horizontal distances in the direction of motion shrink in relation to the motion. In the presented theory it is agreed, and will be shown that the distance between the mirrors (in our case the distance between the light source and the mirror) is unchanged as a result of relative motion. However, it will also be shown that distances in the direction of motion, or any other direction for that matter, are also unchanged as a result of the motion. In other words, contrary to special relativity, it will be shown that distances do not actually change at all. What does change is the standard by which distances are measured. This in turn has the effect of causing the size of physical objects and distances in space to be experienced in the stationary frame as expanding rather than shrinking as a result of relative motion.

To demonstrate, let us perform an exercise involving the described apparatus. To get to the point quickly, we will take a few liberties in presenting the problem. For one, we will use inordinate distances for the purpose of simplicity and subsequent ease of understanding. (Of course, these distances could be scaled down to any practical size necessary without changing

Or alternately,

.19.19.436. cand distance B`-C` 2

2

1c

vd

1

9.111

2

2

2

c

vddistance A`-C` ,436.19.81.1 c

22

the results, if one were determined to do an actual experiment.) For two, and again for the sake of simplicity, we will use an apparatus that does not employ a mirror at the location of the light source. And finally, we will only travel half the distance normally given in textbooks on special relativity. (Again, if one were determined to perform an actual experiment, traveling the whole distance will not affect the results.)

Referring now to figure 5, assume that points A through G are located in the stationary frame of reference. You should also make note that, to facilitate understanding, a more descriptive form of notation is introduced. The symbol dr is now used to define the radius of the light sphere in the stationary frame of reference. The r may be thought of as symbolizing radius, or alternately, reference distance, since it is the speed of light and the distance light travels in an interval of time that other speeds or distances are referenced to. The prime mark is still used to signify a corresponding value in the moving frame of reference, thus the symbol dr` designates the radius of the light sphere in the moving frame of reference, or, reference distance, in the moving frame.

At this point, it is important to remember that a radius may be drawn in any direction from the center of the sphere. Therefore, in viewing figure 5, it should be understood that distance dr could be shown a second time as a vertical distance, (although this would be redundant) and thus

c

A B C

E F G

Light Source

Mirror

Sphere 1

Sphere 2

Ddp = dr`

dv

dr

FIGURE 5 Unveils the flaw in special relativity

dp

23

the relationship to dr` becomes apparent. Additionally, it now becomes necessary to designate distance C-G (or alternately distance B-D) in the stationary frame with its own symbol, dp. The p signifies this to be a distance perpendicular to the path of motion in the stationary frame of reference. This distinction becomes necessary because dr` represents the radius of the moving sphere and therefore distances in other directions besides perpendicular. Whereas, dp and dr` are equal in value, they are not equal in function. The need for this distinction will become clearer as we proceed. In finishing, the symbol dv signifies the distance traveled relating to relative motion in the stationary frame of reference and the symbol dv` when used, would indicate the corresponding distance in the moving frame of reference. We can now proceed with the problem.

Assume as illustrated in figure 5 that a light source and opposing mirror share an arrangement that keeps them a fixed distance of 1c apart along a vertical axis such as that defined by points A-E. Assume next that the light source gives off a pulse of light at point A as it moves at a uniform speed of v = .9 c/s along path A-B-C. If the light source trip were limited to 1 second of stationary frame time t, the light source will only reach point B and the pulse moving in the vertical direction along path B-D will only reach point D. This is because the distance, dr the light travels along path A-D-G in the stationary frame is limited to tc = 1c, the distance from A to D. That is, distance A-D must equal 1c, the same distance between the mirror and the light source. Therefore, distance B-D that is the same in both the stationary frame and the moving frame cannot equal 1c, and the pulse will not reach the mirror at point F. In other words, if speed v = .9 c/s and t = 1s, then distance dv = .9c and distance dr = 1c. Distance dp from point B to point D is then found to be,

Since dp is a vertical distance, this result is in agreement with both special relativity and the presented theory. However, it must be remembered that dp always equals dr`, the radius of the moving sphere, and therefore all distances in the moving frame of reference. This gives evidence in support of the proposition that actual distances in the stationary frame are unchanged in the motion frame. Therefore, since the distance from the light source to the mirror will not change as a result of relative motion, the light pulse will not reach the mirror until time t` in the moving frame reaches an interval of 1 second. Using equation 1 to find t`, we can see that this was not the case in the previous example.

In order for t` to reach an interval of 1s, the time interval for the trip in the stationary frame must be increased. The required time interval t can be determined using equation 2. Thus, for a time interval of t`= 1s, we obtain,

.4358899.9.1 222 cddd vrp

sc

vctt 4358899.

1

9.11`

222

svc

ctt 2941574.2

9.1

11`

222

24

for the new time interval t. Referring back to figure 5, the new distance traveled by the light source moving at a

uniform speed of v = .9 c/s is found to be dv = vt = .9 x 2.2941574 = 2.0647417c. This is represented as distance A-C in the illustration. The new distance traveled by the light pulse along path A-D-G is then found to be dr = ct = 2.2941574c. This new distance is represented as distance A-G in the illustration. The new distance dp is then obtained in the same manner as before. Thus,

giving the same distance for dp as the 1c distance between the source and mirror now located at points C and G respectively. This result is provable through experimentation and is supported by any and all evidence that supports the proposition that vertical distances are unaffected by relative motion as specified in special relativity. It has been shown in the presented theory, however, that distance dp always equals distance dr`, the radius of the moving frame sphere, and thus all distances in the moving frame. This proposition is also supported by the evidence and is consistent with all of the foregoing analysis that in turn is consistent with our postulate and all of the propositions stated in the introduction. If real physical distances are not actually affected by relative motion while at the same time the distance light travels shrinks in accordance with the contraction equations, one must conclude the following: Physical objects and the distances between them in space relative to a moving frame of reference are perceived to be expanding in relation to a stationary frame of reference. This in itself may be sufficient to explain why it takes greater and greater amounts of energy to accelerate an object to speeds approaching a significant fraction of the speed of light. 10. Proving the Fundamentals

Although the principles presented up to this point seem to be overwhelmingly supported by the evidence, they fall short of being proven in the theoretical sense. The main question that still remains to be resolved is this: Why should a point of light moving in the perpendicular direction to the motion of the source be given precedence over any other point of light moving in another direction when formulating the relationship between reference frames? The answer to this question is simple. Use of any other point of light gives results that are inconsistent with the evidence. To demonstrate, let us go through the following exercise: If the Laws of Nature are the same for all frames of reference the following must also be true:

1. If the speed of light has a constant value in one frame of reference it must, under the

same conditions, have that same constant value in all frames of reference. 2. If time is invariant in one frame of reference it must, under the same conditions, be

invariant in all frames of reference. 3. If the speed of light has the same constant value from one reference frame to another,

time must vary from one reference frame to another.

,10647417.22941574.2 22 c22vrp ddd

25

Now, referring to figure 6, assume that points A through E are points in a stationary frame of reference. Assume further, that a light source, now located at point B gave off a pulse of light at point A while moving away from Point A at a uniform rate of speed v. In the moving frame of reference, light moving away from the source in all directions forms sphere 2 with radius B-D. If c is the speed of light, and t` is the time interval in the moving frame, then ct` is the distance the light traveled from the source during that interval. If we now randomly select a point of light along sphere 2, other than from the perpendicular direction represented by point D, point E for example, we can see that it traveled path A-E in the stationary frame of reference. Since light must have a constant speed in the stationary frame of reference, path A-E represents the radius of sphere 1A showing the distance light would travel in all directions in the stationary frame at that same speed. The distance the light travels in the stationary frame is then, ct, where c is the speed of light and t is the time interval in the stationary frame. For consistency, however, we must now also select a point of light moving in the opposite direction at the same angle to the path of motion that the original point was moving in the forward direction. This point of light is at point C and while traveling the same distance in the moving frame as point E during interval t`, travels a different distance than point E in the stationary frame. Its distance is A-C and forms the radius for sphere 1B in the stationary frame. Obviously, the shorter distance A-C cannot be represented as the same distance ct that represents the longer distance A-E. If c is constant in the stationary frame, then t would have to vary in order to validate the selection of these two points of light. Such selection than violates the principle that time is invariant within all frames of reference. This contradiction is avoided only when the point of light selected from the moving frame is one that is traveling away from the source in a direction perpendicular to the path of travel. Such point of light then represents the travel distance of all points of light in the moving frame and the corresponding distance traveled in the stationary frame as previously shown.

When we accept the evidence that speed c is constant, and time varies from one frame to another, the transformation equations can be arrived at simply and directly. In figure 7 we show

Sphere 1A

Sphere 1B

Sphere 2 A B

CD

E

vt

ctct`

ct`

ct`ct

FIGURE 6 Proof of the Fundamentals

26

the now familiar right triangle that represents the relationship between the moving frame and the stationary frame.

Now, however, where v represents the speed of the source, c represents the speed of light, and t represents the time interval in the stationary frame, t` is used directly to represent the time interval in the moving frame. Therefore, where vt represents the distance traveled by the source in the stationary frame, and ct represents the radius of the stationary frame, ct` directly represents the radius of the moving frame.

Given that the triangle is a right triangle, we can immediately derive,

Substituting the symbols d` for ct` and d for ct gives,

22 )()(` vtctct

2222` tvtcct

)(` 222 vctct

22` vctct

Step 1

Step 2

Step 3

Step 4

c

vctt

22

`

Step 5 Eq. (1) Time in the moving frame.

c

vcctct

22

`

Step 6

c

vcdd

22

`

Step 7 Eq. (3) Distance in the moving frame.

vt

ctct`

FIGURE 7 The Direct Method

27

Thus in a handful of steps we have arrived at all of the transformation equations developed previously.

Whereas the first part of our analysis validates the underlying fundamentals used in the theory, the second part validates the mathematics employed. Taken together, these examples provide compelling evidence as to the correctness of the presented theory. On the other hand, the indirect methods used in special relativity lead to misinterpretation of evidence and erroneous conclusions. It has been shown here convincingly that all distances are equally affected by relative motion. Such affect, however, is indirect, thus not affecting actual distances but rather the shrinking standard by which such distances are measured. This in turn leads to the conclusion stated earlier that real distances, although unchanged in the moving frame, are perceived to be expanding in the stationary frame. When one considers the role motion plays in the energy to mass relationship, such change in perspective can have profound consequences. Does mass really increase as its speed increases, or is it that energy becomes less effective? An interesting question, but beyond the scope of this present work. 11. Effects on Wavelength and Frequency

It is well established, and supported by a preponderance of evidence, that the wavelength and frequency of light are affected by the relative motion of the source. Specifically, the observed wavelength will increase and the observed frequency will decrease if the source is receding, and conversely the observed wavelength will decrease and the observed frequency will increase if the source is approaching. Let us then, formulate mathematically the effects that

22` vctdr Step 8 Eq. (5) Radius of the moving sphere.

(From Step 4) By substituting the symbol dr` for ct` we get,

ctdr Step 9 Eq. (4) Radius of the stationary frame.

And of course,

22`

vc

ctt

Step 10 Eq. (2) Time in the stationary frame.

The corollary of equation 1 is,

22`

vc

cdd

Step 11 Eq. (6) Distance in the stationary frame.

The corollary of equation 3 is,

28

relative motion has on the wavelength and frequency of light. We can do this by analyzing separately two different processes that are involved. First we must determine what effects the transformation process, between the moving frame and the stationary frame, has on these properties of light, and then we must determine how the results of that process are affected by the relative motion in the stationary frame alone. After determining each of these processes separately, we can combine the results into a single set of equations that will give us the total of all the effects for any situation.

Before proceeding with the transformation process it is necessary to understand the relationship between the wavelength and frequency of light emitted by the source. The first point that must be understood is this: Whenever we refer to the emitted wavelength or frequency of light, we are referring to their values in the moving frame while it is at rest. In other words, these values are identical to those that result from a source of light at rest in the stationary frame. This stipulation is consistent with the postulate that the laws of physics are the same in all frames of reference. Therefore, when we consider the values of the emitted wavelength and frequency of the source, we are setting the time t` at the source equal to the unit value of 1s, the same value t has in the stationary frame when we compare light from the moving source to light in the stationary frame. Remember, the symbol t` designates the time in the moving frame, and that time in accordance with the time transformation equation (Eq. 1) can have a value of 0s to ts depending on the speed of the source. In this case it has the value of t because we are referring to the values for wavelength and frequency as they are emitted in the frame of the source. In fact t is normally omitted from the equations that express the wavelength to frequency relationship as will be shown below. A final point, the relationships to be developed are valid in any frame of reference.

Now, since the speed of light is c, and since wavelengths are distances, the number of successive waves contained in the distance light travels in an interval of time, is the frequency for that interval. Where f is the frequency, λ is the wavelength, c is the speed of propagation, and t is the time interval, we get,

By dividing the expression ct/λ by t we assure that the resulting relationship will always represent a time interval of 1. Therefore, through simplification of the above equation we arrive at,

equation 8, giving the wavelength to frequency relationship for any frame of reference. The corollary equation for wavelengths is then,

.t

ct

f

c

f Eq. (8) Frequency to wavelength relationship.

t

ctf

1

29

By way of substitution into the just arrived at equations, the frequency and wavelength of light emitted from a moving source is then expressed as,

where f` is the emitted frequency, λ` is the emitted wavelength, and c is the speed of propagation.

a. The Transformation Effect Having now established the formulas defining the wavelength and frequency of the

emitted light we can determine what effects the transformation process will have on their values. Starting with the wavelength and acknowledging that since wavelengths are distances as stated previously, these distances will be shorter in the moving frame than they are in the stationary frame. Therefore, by using equation 6 that defines the relationship between distances in each of these frames, we can substitute the wavelengths for those distances and derive an equation for transforming wavelengths from the moving frame to the stationary frame. Thus, where λx is the wavelength in the stationary frame, and λ` is the wavelength in the moving frame, by way of substitution into equation 6, we get,

The corollary equation for finding wavelength λ` in the moving frame is then,

(The subscript x chosen to identify transformed values may be thought of as short for transformed, i.e., xformed. The more obvious choice, t, proved to be confusing because of its use in identifying time intervals.)

We are now ready to develop the transformation equations for frequency. Drawing again on general equation 9, where λx is the transformed wavelength and fx is the transformed frequency, we have,

c

vcx

22

`

Eq. (11) Wavelength transformation, SF to MF.

22`

vc

cx

Eq. (10) Wavelength transformation,

MF to SF.

22`

vc

cdd

Eq. (6)

f

c Eq. (9) Wavelength to frequency relationship.

``

c

f `

`f

cand

30

Having already established the same for the emitted values as repeated below,

we can substitute the right sides of these equations into equation 10 to get,

With some manipulation we obtain,

And of course, the corollary equation for finding the emitted frequency is,

b. The Doppler Effect We are now ready to proceed to step two and develop the equations needed to define the

Doppler effect relative motion has on the wavelength and frequency of light in the stationary frame of reference. Referring to figure 8, a source of light is shown emitting light at points along the path of travel as it moves from point A to point B in the stationary frame of reference. As illustrated by the spheres, in the stationary frame of reference each succeeding light wave emanates outward at speed c from the point along the path of travel where it was emitted by the source. This causes the waves in the direction of approach to be crowded together while those in the direction of recession are spread farther apart. Since the light is traveling in all directions at speed c, and the source is traveling at speed v, the combined distance traveled between the source and the outer sphere in the direction of recession during an interval t, is ct + vt. During

xx f

c

``

f

c

22`

vc

cx

Eq. (10)

.` 22 vc

c

f

c

f

c

x

c

vc

c

f

c

fx22`

c

vcffx

22

`

Eq. (12) Frequency transformation, MF to SF.

22`

vc

cff x

Eq. (13) Frequency transformation,

SF to MF.

31

the same interval, the distance between the source and the outer sphere in the direction of approach is, ct – vt. By taking these distances and dividing them by the distance ct that light

travels in the same period, we can express these distances in terms of ct. Therefore, where ct is the distance traveled by light in the stationary frame, and vt is the distance traveled by the source along the same axis in the stationary frame, we get,

that can be used as a factor to determine the change in the length of a light wave caused by relative motion in the stationary frame. This is true because the length of the waves in the stationary frame of reference are affected in direct proportion to the amount that the distances are affected by the relative motion. This effect must not be confused with the shrinking of distances in the moving frame. Here, the distances can both shrink and expand. Thus, the wavelength and frequency of the light can be affected in either manner depending on the location of the observer in the stationary frame of reference.

Now, to continue -- through simplification of the preceding expression, we get,

Thus, where c is the speed of light, v is the speed of the source, λx is the transformed wavelength, and λo is the observed wavelength we have,

ct

vtct

c

vc

ct

vct

ct

vtct

)(A form of the Doppler effect formula.

BvtA

SourceApproach Recession

ct ct

FIGURE 8 The Doppler Effect

32

for finding the observed wavelength when v and the transformed wavelength are known. The corollary equation for finding the transformed wavelength is,

Having established the equations needed for the Doppler effect involving wavelengths, we can now do the same for the effects involving frequency. Again in accordance with equation 9, where, λo is the observed wavelength and fo is the observed frequency we can derive,

And, having previously derived,

where λx is the transformed wavelength, and fx is the transformed frequency, we can substitute the right sides of these equations into equation 14 to arrive at a Doppler effect equation for frequency. Thus,

Then, with some manipulation we obtain,

c

vcxo

Eq. (14) Doppler effect, Observed wavelength.

(v is + for recession – for approach)

vc

cox

Eq. (15) Doppler effect, Transformed wavelength. (v is + for recession – for approach)

c

vcxo

Eq. (14)

.c

vc

f

c

f

c

xo

cf

vcc

f

c

xo

)(

)( vcc

cf

c

f xo

.o

o f

c

xx f

c

33

And of course the corollary equation is,

We now have all of the equations needed to determine the total effect that relative motion of a source along the line of sight has on the wavelength and frequency of the observed light in the stationary frame of reference. These equations can now be combined into their final form.

c. The Combined Transformation and Doppler Effects Equations Since equation 10 gives the transformation value λx of the emitted wavelength λ`, and the

Doppler effect equation, 14, derives the observed wavelength λo from the transformed wavelength, the two equations can be combined into a single equation that derives the observed wavelength directly from the emitted wavelength. Therefore, by substituting the right side of equation 10 in place of λx in equation 14, and rearranging the result we get,

Notice that v must use the appropriate sign in the Doppler portion only. The sign for v in the transformation portion of the equation does not change. Doing similar now, with the remaining equations gives us,

Eq. (17) Doppler effect, Transformed frequency. (v is + for recession - for approach) c

vcff ox

vc

cff x

o

vc

cff xo

Eq. (16) Doppler effect, Observed frequency. (v is + for recession - for approach)

22`

vc

cx

Eq. (10)

c

vcxo

Eq. (14)

Eq. (18) Observed wavelength. (v + for recession – for approach) 22

`vc

vco

22`

vc

c

c

vco

or

Eq. (19) Observed frequency. (v + for recession – for approach) vc

vcffo

22

`c

vc

vc

cffo

22

`

or

c

vc

vc

co

22

`

or Eq. (20) Emitted wavelength.

(v + for recession – for approach) vc

vco

22

`

34

If we were now to use these equations in a typical application we will discover something very interesting. For example, assume that a star is receding directly away from earth at a speed of .3c/s. An emitted wavelength of 434 nanometers would have an emitted frequency of,

where w/s is waves per second. The observed wavelength on earth would be,

where m is meters, and the observed frequency would be,

If we now reverse the example and have the star approaching earth at the same speed and under the same conditions, we get,

for observed wavelength and,

for observed frequency. If we now compare the percent change in wavelength and frequency for both cases we find they are identical as shown below:

swc

f /10908.610434

299792485

`` 14

9

%277.13610010434

104.591100

` 9

9

o

Recession

22`

vc

c

c

vcff o

Eq. (21) Emitted frequency.

(v + for recession – for approach) or 22

`vc

vcff o

22`

vc

c

c

vco

m9

2

9 104.5913.1

1

1

3.110434

c

vc

vc

cffo

22

`

./10069.5

1

3.1

3.1

110908.6 14

214 sw

22`

vc

c

c

vco

m9

2

9 105.3183.1

1

1

3.110434

c

vc

vc

cffo

22

`

./10414.9

1

3.1

3.1

110908.6 14

214 sw

35

Obviously we expected the change in frequency to match the change in wavelength in each of the separate cases, but somewhat surprising is the fact that the percentage change for approach also matches the percentage change for recession. This implies that the equations used for recession are the reciprocal of the equations used for approach. Yet analysis of the equations and the manner in which they were arrived at does not make that readily apparent. For example, the transformation equations give the same results regardless of the direction of the sources’ motion. It is only the Doppler effect equations that change, and the change is of signs, not of inversion. We will end our analysis by giving an example that the equations are in fact reciprocals of each other.

The question really is, are the expressions,

reciprocals of each other? If they are, then either one in its inverted form will equal the other. to resolve this question we will take a moment and perform just such an analysis.

%277.13610010069.5

10908.6100

`14

14

of

f

%277.136100105.318

10434100

`9

9

o

%277.13610010908.6

10414.9100

` 14

14

f

fo

Recession

Approach

Approach

22 vc

c

c

vc

22 vc

c

c

vc

and

22

22

vc

c

c

vc

c

vc

vc

c

22

22 )(

)( vcc

vcc

vcc

vcc

)())(( 2222 vccvcvcc

)()( 222222 vccvcvcvcc

)()( 222222 vccvcc

36

As can be seen, these expressions are indeed reciprocals of each other. Nature loves symmetry, therefore the very fact that the equations give identical results for recession and approach are a good indication that the equations are correct.

Before concluding this part of our analysis, it might prove instructive to observe the graphical relationship between the transformation process and the Doppler process in a single illustration. Figure 9 shows a light source emitting light as it moves along the arrowed line from point 0 to point 6 at a uniform speed of .9c/s.

Each of the equally spaced points along the path of travel are chosen samples from an infinite number of points along that path. The concentric spheres centered over point 0 represent a single growing sphere of light in the stationary frame of reference. This sphere growing at speed c reaches the size indicated by a number that corresponds with the same number along the travel path of the source. In other words, the sphere emitted at point 0, grows to size 1 through 6 as the source reaches those points respectively. A similar sphere is emitted at every successive point along the path of travel each reaching a successively smaller size in any instant in time because it is emitted at a later time. Showing these spheres, even for the few points represented along the travel path would unduly clutter the illustration and therefore are omitted. For each of the points

0 6 1 2 3 4 5

6

1

2

3

4

5

0

1

2

3

4

5

Stationary Frame Sphere

Moving FrameSphere

Doppler Spheres

Source

FIGURE 9 Combined Transformation Process and Doppler Effect

37

shown, however, such spheres would reach the size of the solid line Doppler spheres centered over each point of the travel path. These spheres in the stationary frame are in fact the Doppler spheres, each centered over the point at which it was emitted and growing at speed c as the source travels along the defined path. As can be seen, the Doppler spheres are numbered in correspondence to the point on the travel path at which they were emitted and each successive sphere reaches one size smaller then its predecessor during any instant in time. In the illustration, the solid line Doppler spheres thus represent the size each sphere has reached at the instant the source reaches point 6.

Associated with the growing sphere at point 0 is a correspondingly growing right triangle that correlates the growth of that sphere with the growth of the moving frame sphere labeled as such. Whereas the Doppler spheres as illustrated can be thought of as multiple spheres, this is not true for the spheres centered over point 0 or for the moving frame spheres. These dotted line spheres are intended only to show how the moving frame sphere grows in unison with the stationary frame sphere centered over point 0. This representation of correlated growth between the stationary frame sphere and the moving frame sphere is not unique to the sphere centered on point 0. However, showing the similar relationships for all of the other points would again simply clutter the illustration and make it undecipherable. Nonetheless, from what has been shown here it can be readily appreciated how easily the relationship between the transformation process and the Doppler effect can be graphically represented. Such graphical representation of relativistic principles provides us with another powerful analytical tool not available in special relativity. (Important – Refer to Doppler Note at end of Document before proceeding.) 12. Doppler Effect for Transverse Motion

When the motion of the source is across the field of vision rather than along the line of sight it is said to have transverse motion relative to the observer. Such transverse motion must now be analyzed to determine what affect it will have on the observed wavelength and frequency of the received light. Referring now to figure 10, we will conduct just such an analysis.

Figure 10 shows the path of motion of the source from point (a) to point (b) in the stationary frame of reference. The outer sphere represents light that was emitted at point (a) that as can be seen is Doppler affected relative to the position of the source. The inner Doppler spheres and the moving frame sphere are given only as visual aids and are not essential for analyzing the Doppler effect illustrated by the path of motion and position of the source within the outer sphere. To an observer located anywhere around the top of the outer sphere from the 0 degree point to the ninety degree point, the source is defined as receding. This of course is true for the bottom of the sphere as well but is redundant with regard to the analysis.

As can be seen in the illustration, light traveling from the source at point (b) to point (c) in the direction of the observer forms angle A with the line that represents the path of travel of the source away from the observer. Knowing distance (b-c) that results from the light traveling toward the observer during the same time interval that the source moves from point (a) to point (b) and the light traveling from point (a) takes to reach point (c), we can determine the Doppler effect the observer will observe. The triangle formed by the intersection of the three distances just discussed is shown in the illustration and is representative of any other such triangle that would result if the observer were located anywhere within the quadrant between the 0 and 90 degree points. Of course the triangle would disappear if the observer were along the 0 degree path, and would become congruent with the right triangle that defines the relationship between the outer sphere and the moving sphere if the observer were at the 90 degree location.

38

With the information given in the illustration we can solve any such triangle and thus determine the Doppler effect the observer would observe.

Referring now to figure 11, a representative triangle of the kind just discussed is evaluated mathematically. Our objective is to define the length of the line along the top of the triangle (Line of sight of observer to source) and put it in terms of c. Such an expression can then be used to define the Doppler effect of the observed wavelength. We start by dividing the triangle into two right triangles by extending a line from the line that represents the observer’s line of sight to the point where line ct intersects line vt. Trigonometrically, the cosine of angle A (the angle between the line of sight of the observer and the path of motion of the source) is given as,

.,,

cosvt

sideadjacent

hypotonuse

sideadjacentA

FIGURE 10 Doppler Effect, Recession

Stationary Frame/Doppler Spheres

0º

90º Moving FrameSphere

Source A

To observer

Recession

ct

vta b

c

39

The adjacent side of the right triangle containing angle A is then,

The side of this triangle opposite angle A is then defined as,

And the long side of the second right triangle then becomes,

We now have enough information to define the entire distance of the line of sight line. Our next objective is to simplify the resulting expression and express it in terms of c.

Starting with,

and some manipulation, we get,

).(cos, Avtsideadjacent

.)(cos)( 22 Avtvt

222 ))(cos()()( Avtvtct

222 ))(cos()()()(cos AvtvtctAvt

222 ))(cos()()()(cos AvtvtctAvt

22222 )(cos()(cos AvvctAvt

A

ct

vt

vt(cosA)

22 )(cos)( Avtvt

222 ))(cos()()( Avtvtct

FIGURE 11 Math Definition of Doppler Recession

40

We can now place the resulting expression over ct and simplify it to its final form. Thus we get,

As it turns out, the resulting expression when applied to the transformed wavelength of the source gives the Doppler effect for any angle of recession or approach. By combining it with the wavelength transformation equation (Eq. 10) we have the complete equation for transverse motion involving recession or approach of the source. Thus, where A is the angle of view, v is the speed of the source, c is the speed of light, λ` is the emitted wavelength, and λo is the observed wavelength we have,

The corresponding equation for frequency is then,

As stated previously, the above equations work for either recession or approach. Although another set of equations, as will be shown, was developed for an approaching source, they were found to give results identical to those of the previous equations. Consequently, the alternate equations can also be used in both applications. However, the above equations are the preferred equations for both uses.

2222 )(cos)(cos AvvctAvt

2222 )(cos)(cos AvvcAvt

ct

AvvcAvt 2222 )(cos)(cos

.)(cos)(cos 2222

c

AvvcAv

22

2222 )(cos)(cos`

vc

c

c

AvvcAvo

Eq. (22) Observed wavelength Transverse motion (v + for recession – for approach)

c

vc

AvvcAv

cffo

22

2222 )(cos)(cos`

Eq. (23) Observed frequency Transverse motion (v + for recession – for approach) *

(* Due to some sort of mathematical, or possible computer or software aberration, the above frequency equation loses accuracy when used for approach. A similar experience in the past proved to be a computer/software application problem. When dealingwith very small numbers such as those defining wavelengths, the accuracy of older desktop computers and software can sometimes cause problems. In the previous case numbers arrived at during computation sometimes contained a first significant digit that fell outside of the fifteen-digit limit of the software application. This allowed division by zero without flagging it as an error. This in turn led to erroneous answers similar to those encountered here. Needless to say, caution must be used during these types of computations.)

41

Referring now to figure 12, it can be seen that a triangle can also be formed at the approach side of the source. This triangle consisting of the path of the source, line (a-b), the path of light, line (a-c), and the line of sight to the observer, line (b-c) can be used to develop an alternate equation for the transverse Doppler effect. Like the previous triangle, this triangle varies according to the position of the observer. It disappears if the observer is along the 0 degree line, and it becomes congruent with the triangle used for the transformation effect if the observer is along the 90 degree light path. Again, angle A is the angle between the path of motion and the line of sight to the observer.

In figure 13 a triangle like that of figure 12 has its three angles identified by the symbols a, b, and c. Additionally, the unknown side has been identified with the symbol x. It is the length of this side, the path to the observer, that determines the wavelength and frequency the observer will observe. We can determine the length of this side by using the trigonometric Law of Sines.

b

FIGURE 12 Doppler Effect, Approach

90º

Stationary Frame/Doppler Spheres

Source

Approach

vta0º

Moving FrameSphere

A

To observer

ctc

42

Therefore, if external angle A is the angle of observation, then angle c = 180º – A. Using the Law of Sines, we then get,

Using the left side of the first equation for a second time, we get,

And now, solving this equation for x, we get,

.)180sin()sin(

ct

A

x

a

)180sin()sin( A

ct

a

x

)180sin(

)sin(

A

ctax

vt

b

ct

A )sin()180sin(

v

b

c

A )sin()180sin(

c

Avb

)180sin()sin(

c

Avab

)180sin(sin

)180()180sin(

sin180 Ac

Avaa

c

AvaAa

)180sin(sin

xct

vt

Ac

b

a

FIGURE 13 Math Definition of Doppler Approach

43

We can now substitute the expression found for (a) previously, in place of (a) in this equation. This gives us,

If we now place the right side of this equation over ct, and simplify, we arrive at the desired Doppler factor. Thus,

Using this expression in place of the earlier arrived at Doppler factor in equation 22, give us,

As stated previously, these are not the preferred equations for determining the transverse Doppler effects. Additionally, the frequency equations should be used with caution. In fact, it may be better to rely on the preferred wavelength equation and arrive at the frequency by dividing the observed value for wavelength into c. Nonetheless these equations are superior to the preferred equations presently used in special relativity for transverse motion. For example, the equation,

.)180sin(

)180sin(sinsin

A

ctc

AvaA

x

ctA

ctc

AvaA

)180sin(

)180sin(sinsin

)180sin(

)180sin(sinsin

A

c

AvaA

22)180sin(

)180sin(sinsin

`vc

c

A

c

AvaA

o

Eq. (24) Observed wavelength Transverse motion Alternate equation

c

vc

c

AvaA

Affo

22

)180sin(sinsin

)180sin(`

Eq. (25) Observed frequency Transverse motion * Alternate equation

(* Use with caution. See previous comments.)

22

)cos(`

vc

c

c

vAc

44

though simpler than the equations arrive at here, gives erratic and obviously erroneous results. For one thing, this equation does not give the emitted wavelength when used for 90 degrees transverse motion. In fact, it gives the emitted wavelength twice for the same relative speed of the source. For example, when the source is receding at a speed of .2c/s, the above equation gives the emitted frequency of 434 nanometers for both recession and approach. This emitted wavelength is given for 96 degrees of recession (an impossible situation), and 89.9 degrees of approach. At no time does it seem to give the emitted frequency at 90 degrees. According to the present theory, there is no Doppler effect at 90 degrees, or more correctly it is nullified by the transformation effect at that point, so one would expect the observed wavelength and frequency to equal the emitted wavelength and frequency. (More will be said on this later.) 13. Correlation of All Effects

Illustrated in figure 14 is a summary of all of the effects presented in the new theory. These include time and distance changes and the effects they have on wavelength and frequency in addition to the separate effects brought about by Doppler considerations.

Referring to the center of the moving frame, it can be seen how the distances represented by the small solid line sphere are related via the right triangle to the large stationary sphere. Since time in the moving sphere lags behind time in the stationary sphere, the size of the moving sphere increases at a slower rate than the size of the stationary sphere. If the size of the stationary sphere shown represents 1s of stationary time t, then the larger dashed line moving

dr` and t`

dr, t and Emitted λ` and f`

dr, t and Xformed λx and fx

Recession Observed λo and fo

Transverse RecessionObserved λo and fo

Transverse Approach Observed λo and fo

Approach Observed λo and fo

λ and f

Stationary Frame Moving Frame

dr, t and Standard λ and f

FIGURE 14 Correlation of All Effects

45

sphere will reach that exact same size in 1s of moving sphere time t`. Obviously, since the stationary sphere continues growing at a faster rate, it will be proportionally larger than the size illustrated in figure 14 at that moment.

Consider now, light waves, represented by the sine waves shown in the illustration, emanating outward from the source in all directions. A sample of these waves is shown, for the purpose of discussion, leaving the source in a vertical direction only. These waves are the emitted light waves λ` and in 1s of time t` reach a distance illustrated by the large dashed line sphere in the moving frame. This is the emitted wavelength that is inputted into the wavelength transformation equations. These waves are identical to the same wavelength waves λ emitted in the stationary frame, shown emanating vertically from the center of the stationary frame. In practice it is these standard waves emitted in the stationary frame that the moving frame waves are compared to when observed in the stationary frame. However, as we have seen, it is not a one-for-one comparison.

Because of the time and distance differences between frames, only that portion of the moving frame waves that is contained in the smaller moving frame sphere are transformed into the stationary frame sphere during any 1s period of stationary frame time, t. In other words it takes fewer of the emitted waves to fill the stationary frame. Thus, during 1s of stationary frame time, the waves λ` that fill radius dr` of the smaller moving frame sphere, also fill radius dr of the larger stationary frame sphere. That is, the waves λ` spread along the vertical side of the right triangle in the moving frame, through the process of transformation become waves λx spread a greater distance alone the hypotenuse of that triangle in the stationary frame. Although the transformation process affects the wavelength and frequency of the received light in the stationary frame, unlike the Doppler effect, it is a one-way change. The transformed wavelength λx will always be longer, and the transformed frequency fx will always be lower in direct relation to the speed of the source. However, since the Doppler effect occurs concurrently with the transformation process, these transformed values are never separately observed.

If the observer is directly along the path of recession, these Doppler affected observable wavelengths λo will be even longer and the frequency fo even lower than the transformed values. On the other hand, if the observer is directly along the path of approach, these wavelengths will be shorter and the frequencies higher than the transformed values, as illustrated. Should transverse motion be involved, depending on the position of the observer, the observed wavelengths and frequencies will be somewhere between these two extremes. And finally, if the observer is located directly along the 90-degree location, the observed wavelength and frequency will be exactly equal to the emitted wavelength and frequency. At this location, the Doppler effect exactly cancels out the transformation effect (another indicator of Nature’s love for symmetry and order, and nothing short of amazing when one considers how different the processes by which the transformation and Doppler equations are arrived at). This does not mean that the time and distances changes between the two frames of reference are cancelled. It is only the wavelength and frequency changes that are nullified. With some thought, it will become apparent that the relative sizes of the two spheres, does not change as a result of the observer’s location. The Doppler effect is simply a result of the relative position of the source to the stationary sphere, and the relative position of the observer to the source. This has no effect on the time and distance changes between the two reference frames.

Before ending here, we will take a moment to revisit the questions raised regarding the transverse motion equation used in special relativity. If one looks closely at the equation it is apparent that if the angle of observation is 90 degrees, the equation reduces to a unit value of 1.

46

This is true even when the equation is used in its reciprocal form. The effect then, when used in conjunction with the transformation equation is to allow only the transformation values to prevail at an observation angle of 90 degrees. Earlier it was shown that the none-transverse Doppler equation, used in both the presented theory and special relativity, has the effect of removing the one-way bias inherent in the transformation process. It was shown that the same percentage change results regardless of the direction of observation, recession or approach. In other words, the Doppler effect offsets the bias of the transformation process to produce equal changes in both directions. It is reasonable to assume therefore that this leveling effect should continue throughout the range of observation angles, including the 90 degree mid point between recession and approach. On that basis, it seems inconsistent and even contrived that the special relativity transverse motion equation gradually increases the bias as 90 degrees is approached, and allows 100 percent of it as that point is reached. The questions to be raised here are these, “Is this more than just an inaccurate equation? Is this in fact another flaw is special relativity?” It would seem that if such bias is justified by evidence, then it should result throughout the entire range of values including those at the very extremes. 14. Conclusion

As stated at the beginning, the theory presented here is based only on evidence and a single assumption; the evidence that the speed of light is finite, and constant regardless of the motion of the source, and the assumption that the laws of physics are the same in all natural frames of reference. The assumption itself is founded on the belief that it is unreasonable to assume uniqueness to any particular frame of reference. Nowhere does the presented theory rely upon the assumptions of other theories. As was seen, if the transformation process and the Doppler effect had not been previously discovered, they would still have been discovered here. By simply following a path of unbiased reasonableness, and consistency in seeking out the truth, what was sought after was indeed found.

The single most important contribution of the presented theory is the discovery of how different reference frames are truly related to each other. Once this relationship is uncovered, and correctly defined, the remaining concepts introduced here are readily arrived at. Other contributions include the introduction of spherical reference frames, graphical analytical capabilities, more concise transformation equations, and more accurate Doppler effect equations. Other contributions are evident such as simplification and believability of previously unfathomable concepts, and still others will become apparent as the theory gains acceptance. Hopefully, such acceptance will be swift and universal.

47

REFERENCES 1 Albert Einstein, Special Theory of Relativity, (1905) published in the journal Annalen der Physik, as presented in Physics for Scientists and Engineers, second edition, (Ginn Press, MA, 1990) and Exploration of the Universe, third edition, (Holt, Rinehart and Winston, NY, 1975)

2 Albert Michelson and Edward Morely, (1887 experiment to detect ether), as presented in Physics for Scientists and Engineers, second edition, (Ginn Press, MA, 1990) and Exploration of the Universe, third edition, (Holt, Rinehart and Winston, NY, 1975)

3 Willem de Sitter, (1913 observation of visual binary stars) Observed light from twenty-seven visual binary star systems and concluded that light does not take on the speed of the source.

4 Physics for Scientists and Engineers, second edition, (Ginn Press, MA, 1990) 5 Physics for Scientists and Engineers, second edition, (Ginn Press, MA, 1990) and Exploration of the Universe, third edition, (Holt, Rinehart and Winston, NY, 1975)

6 Physics for Scientists and Engineers, second edition, (Ginn Press, MA, 1990) and Exploration of the Universe, third edition, (Holt, Rinehart and Winston, NY, 1975)

Doppler Note

Important Update on the Relativistic Transverse Doppler Effect

March 14, 2006

Five years of analyses beginning with that used to derive the transverse Doppler effect given next in the Millennium Theory of Relativity work (completed Feb 5, 2001) led to the recent discovery that the effect is far more complex than currently realized and therefore the original treatment is incomplete and subsequently not entirely correct. In short, the original treatment given next in Sections 12 and 13 was found to be valid only during the initial emission process involving the initial waves emitted by the source. As the emitted waves propagate at speed c relative to the stationary frame, changes continue to take place that are not accounted for in the present formulas. This problem has now been resolved and the complete treatment was added to the web site on March 5, 2006 as a separate work under the title, Relativistic Transverse Doppler Effect. The corresponding Equation list was also added to the equations section of the site at that time. The author has decided to leave the existing treatment in Sections 12 and 13 of the Millennium Theory of Relativity work unchanged except for this cautionary note, because the treatment is part of the discovery process that led to the final version of the referenced relativistic transverse Doppler effect work completed February 9, 2006 and posted on the Millennium Relativity web site March 5, 2006.

The author believes the updated formulas for the relativistic Doppler effect contained in the new work are the most accurate formulas in existence when used within their useful range as explained in the new paper. It should also be noted that the author has since completed two additional works on the Doppler effect and is in the process of trying to get them published in science journals. In addition, the author submitted a new proposal to the U.S. National Science Foundation on February 1, 2007 based on these latest findings. The proposal was accepted for consideration and sent out for external peer review the next day. The process can take as long as six month to complete, however, so there is nothing more to comment on at the present time.

Joseph A. Rybczyk

48

Millennium Theory of Relativity

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