Relativistic Transverse Doppler Effect Simplified pdf Version Transverse Doppler Effect...

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Relativistic Transverse Doppler Effect Simplified

Complete Treatment for Distances from Zero to Infinity

Copyright © 2007 Joseph A. Rybczyk

Abstract A complete simplified explanation of the relativistic transverse Doppler effect is presented for detection distances from zero to infinity. 1. Introduction The relativistic transverse Doppler effect is presented in as brief a manner as practical for the entire range of detection distances from zero to infinity. 2. Important Definitions

a. Relativistic Doppler Effect – A motion related shift in the frequency and wavelength of electromagnetic waves based on the proposition that the speed of propagation does not take on the speed of the source. The relativistic Doppler effect has two components, the Einstein redshift effect1 and the classical Doppler effect2. b. Redshift and Blueshift – References to the change in wavelength and frequency of the detected electromagnetic waves. A redshift is a shift to the red end of the spectrum indicated by a reduced frequency and increased wavelength. A blueshift is a shift to the blue end of the spectrum indicated by an increased frequency and reduced wavelength. c. Einstein Redshift Effect – The redshift effect on emitted electromagnetic waves resulting from time dilation due to the relative motion of the source. Although the time dilation effect is determined using a point of propagated light detected in the stationary frame at a 90° angle to the point of emission along the path of motion of the source, the resulting redshift effect on electromagnetic waves is valid for all angles of detection in the stationary frame, including 0° and 180°. In brief, the Einstein Effect is always a redshift effect whose magnitude is dependent only on the relative speed of the source and not on the angle of detection. d. Classical Doppler Effect – The shift in the frequency and wavelength of the propagated waves resulting from the relative motion of the source, the angle of detection in the stationary frame, and if the angle of detection is transverse, also the distance between the detector and the stationary frame points of emission along the path of motion of the source. The classical Doppler effect can be either redshifted or blueshifted. It is redshifted for a signal received from a receding source and blueshifted for a signal received from an approaching source. It is now known that the effect at a transverse angle of detection, including at an angle of exactly 90°, depends not only on speed of the source and the detection angle, but also on the emitted wavelengths related distances between the point of detection and the points of emission in the stationary frame3.

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e. Longitudinal Effect – In the classical context, the longitudinal effect is the redshift effect or blueshift effect on a signal detected from a moving source at a 0° or 180° angle of detection relative to the path of motion of the source. In the relativistic context, the longitudinal effect is the net effect of the combined Einstein redshift effect and the classical redshift or blueshift effect just described. f. Transverse Effect – In the classical context, the transverse effect is the redshift or blueshift effect on a signal detected from a moving source at an angle of detection in the range (0° < angle < 180°) relative to the stationary frame point of emission along the path of motion of the source. In the relativistic context, the transverse effect is the net effect of the combined Einstein redshift effect and the classical redshift or blueshift effect just described.

3. The Einstein Redshift Effect Given the millennium relativity4 simplified version of the Lorentz/Einstein transformation factor5 and its reciprocal we have

c

vc 22 and

22 vc

c

Millennium Relativity Transformation Factors

that transpose into the special relativity factors

2

2

1c

v and

2

2

1

1

c

v

Special Relativity Transformation Factors

respectively, where v is the speed of the source, or other uniform motion object, and c is the speed of light in the vacuum of space. Using either version of these factors, it follows from the principles of relativity that time transformation has the relationship

c

vcTt

22 (1) and

22 vc

ctT

(2) Time Transformation

where t can be either the time unit or time interval in the moving frame and T is the corresponding time unit or time interval in the stationary frame. If we factor both sides of these equations by c, we obtain

c

vccTct

22 and

22 vc

cctcT

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where ct is the distance traveled at speed c in the moving frame during moving frame time interval t and cT is the corresponding distance traveled in the stationary frame at speed c during stationary frame time interval T. Substituting d for ct and D for cT we obtain

c

vcDd

22 (3) and

22 vc

cdD

(4) Distance Transformation

for our final formulas for distance contraction where d is the contracted distance in the uniform motion frame that corresponds to distance D in the stationary frame. Now, since a wavelength can be viewed as simply the distance between the leading edges of two successive waves, we can use Equation (4) to show that by way of substitution

22 vc

cex

(5) Einstein Redshift Effect for Wavelength

where λe is the wavelength emitted in the moving frame of the source and λx is the redshifted wavelength experienced in the stationary frame due to the relativistic transformation process. This is the effect referred to as the Einstein redshift effect for wavelengths. Then, given the standard formulas for the wavelength to frequency relationships

c

f (6) and f

c (7) Wavelength to Frequency Relationship

where f is the corresponding frequency for wavelength λ in the same reference frame, we can use Equation (7) by way of substitution into Equation (5) to obtain

22 vc

c

f

c

f

c

ex

that simplifies to

c

vcff ex

22 (8) Einstein Redshift Effect for Frequency

where fe is the frequency emitted in the moving frame and fx is the redshifted frequency experienced in the stationary frame due to the Einstein redshift effect. In analyzing Equation (5) it can be easily seen that as relative source speed v increases from 0 to c for any wavelength λe the redshifted wavelength λx will increase from λe to ∞. In a similar manner, if we analyze Equation (8) it is easily seen that as v increases from 0 to c the redshifted frequency fx will decrease from fe to 0. In referring to the millennium relativity and special relativity transformation factors given at the beginning of this section it can be seen that by substituting the latter for the former in Equations (5) and (8) we obtain the special relativity versions of those two Equations.

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Of greatest importance here is the consideration that the predicted redshift in wavelength and frequency due to the Einstein redshift effect is dependent only on the relative speed of the source and in the absence of any classical effect will be the only effect detected. Since according to the previously given definitions of Section 2 the classical effect is redshifted for a receding source and blueshifted for an approaching source, it stands to reason that somewhere between these two extremes there will be no classical effect at all and therefore only the Einstein redshift effect will be detected. To determine the angle at which this will happen6, we need to examine the classical effect in detail as will be accomplished next and then go on to develop all of the formulas necessary to predict the associated effects for any angle of detection. 4. The Classical Doppler Effect a. The Classical Longitudinal Effect In the classical context, the longitudinal Doppler effect is given in terms of a redshift or blueshift effect on a signal detected at an angle of 0° or 180° to the path of motion of the source. If the signal from a source travels toward the detector from the point of emission in the stationary frame at speed c while the source moves away from the point of emission in a direction away from the detector or toward the detector at speed v, then the effect on the wavelength of the emitted signal is given by the factor

cT

vTcT or simply

c

vc Classical Factor for Longitudinal Waves

where cT is the distance traveled by the signal and vT is the distance traveled by the source during stationary frame interval T. In the given factors the sign for v is + for a receding source and – for an approaching source with regard to the point of detection. If we apply the final form of the factor to the wavelength of the emitted wave λe, we obtain

c

vceo

(9) Classical Formula for Longitudinal Wavelength

for the classical longitudinal effect where λo is the detected wavelength, c is the speed of light and speed v of the source is + for recession and – for approach where shown. Again using the wavelength to frequency relationship given by Equation (7) we obtain

vc

cff eo

(10) Classical Formula for Longitudinal Frequency

where fe is the emitted frequency, fo is the observed frequency, c is the speed of light and v the speed of the source is + for recession and – for approach. b. The Classical Transverse Effect In the classical context, the transverse effect is the effect that the motion of the source has on the signal detected in the stationary frame at an angle in the range of 0° < angle < 180° with regard to the point of emission along the source’s path of travel. Since the full value of source speed v is used in an additive manner for recession and in a subtractive manner for approach,

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intuition suggests (though not entirely correctly as will be shown later in Section 7) that the value of v must reduce toward 0 as the angle of observation approaches the midway point between the two angular extremes, or more specifically, the 90° point. Thus, what is needed is a factor to apply to the value of v that will vary from 1 to 0 as the angle of detection varies from 0° to 90°. A suitable factor supported by the evidence is found in the form of the trigonometric cosine function. Applying this function to the longitudinal Formulas (9) and (10) gives in respective order

c

vc oeo

cos (11) Classical Formula for Transverse Wavelength

and

oeo vc

cff

cos (12) Classical Formula for Transverse Frequency

where λe and fe are the emitted wavelength and frequency respectively, λo and fo are the observed wavelength and frequency respectively, θo is the angle of detection with regard to the point of emission along the source’s path of travel, and source speed v is + for recession and – for approach where indicated. At this point it is important to note that with the availability of transverse Equations (11) and (12), the longitudinal Equations (9) and (10) are no longer actually needed. This is because Equations (11) and (12) will give the same results as Equations (9) and (10) respectively for the angles of observation of 0° and 180°. From this it is valid to claim that the longitudinal effect is simply a special case of the transverse effect where the angle of observation is either 0° or 180°. 5. The Relativistic Doppler Effect a. The Relativistic Longitudinal Effect According to the evidence the complete Doppler effect contains both the Einstein redshift effect of Section 3 and the classical Doppler effect of Section 4. Stated mathematically it is really the transformed, or redshifted, wavelength and frequency λx and fx respectively and not the emitted wavelength and frequency λe and fe respectively that the classical effects are applied to. For the complete relativistic longitudinal effect then, we can simply rewrite Equation (9) as

c

vcxo

(13) Relativistic Formula for Longitudinal Wavelength

where λx is substituted for λe and by way of substitution using the right side of Equation (5) for the Einstein redshift effect in place of λx in the above formula obtain

c

vc

vc

ceo

22 (14) Relativistic Formula for Longitudinal Wavelength

that simplifies to

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22 vc

vceo

(15) Relativistic Formula for Longitudinal Wavelength

for the final version of the formula for the complete relativistic longitudinal Doppler effect for wavelengths. As before, speed v of the source is + for recession and – for approach. Again using the wavelength to frequency relationship of Equation (7) but this time on the just derived Equation (15), we obtain

vc

vcff eo

22

(16) Relativistic Formula for Longitudinal Frequency

for the complete relativistic longitudinal Doppler effect for frequency. And again, speed v is + for recession and – for approach where indicated. b. The Relativistic Transverse Effect Repeating the procedure just used for the longitudinal effect, but now for the transverse effect given in Equation (11), we rewrite Equation (11) to obtain

c

vc oxo

cos (17) Relativistic Formula for Transverse Wavelength

where λx replaces λe. Then by way of substitution using the right side of Equation (5) for the Einstein redshift effect we obtain

c

vc

vc

c oeo

cos22

(18) Relativistic Formula for Transverse Wavelength

and through simplification

22

cos

vc

vc oeo


as the complete formula for the relativistic transverse Doppler effect for wavelength. As previously, λe is the emitted wavelength, λo is the detected wavelength, θo is the angle of detection and v is + for recession and – for approach where indicated. Repeating the procedure for deriving the frequency version of the wavelength formula on the just derived wavelength formula using the wavelength to frequency relationship of equation (7) we obtain

oeo vc

vcff

cos

22

(20) Relativistic Formula for Transverse Frequency

where fe is the emitted frequency, fo is the detected frequency, θo is the angle of detection and v is + for recession and – for approach where indicated.

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c. The Special Relativity Doppler Effect Formula If the special relativity transformation factor for the Einstein redshift effect given at the beginning of Section 3 had been used in Equation (5) instead of the millennium relativity factor, then when Equation (5) was applied to Equation (17) repeated below

c

vc oxo

cos (17) Relativistic Formula for Transverse Wavelength

in deriving the equation for wavelengths, we would have obtained

c

vc

c

vo

eo

cos

1

1

2

2


for the transverse wavelength formula. Simplifying then gives

2

2

1

cos

c

vc

vc oeo


2

2

1

cos

c

vc

vc o

eo


and finally

2

2

1

cos1

c

vc

vo

eo

(24) Special Relativity Doppler Effect for Wavelength

that is the formula given in Einstein’s original 1905 paper, “On the Electrodynamics of Moving Bodies” but correctly written here. Following the wavelength to frequency conversion procedure using Equation (7) this then gives

o

eo

c

vc

v

ffcos1

12

2

(25) Special Relativity Doppler Effect for Frequency

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for the frequency formula where in both cases, Equations (24) and (25) v is + for recession and – for approach where shown. 6. The Relativistic Doppler Effect in the Proximity of the Source Close analysis of the underlying fundamentals relating to the transverse Doppler effect reveals a flaw in the just covered treatments. Specifically, the treatments just covered are generalized treatments that work properly only at detection distances of approximately 107.5 emitted wavelengths or greater from the point of emission along the path of motion of the source. The reason for this is quite simple to explain. Since in the transverse relationship the source is moving across the line of sight, the angle of observation will be different for the two successive waves that determine the wavelength of the detected signal. Whereas the change in angle will be too small to detect for a distance of 107.5 emitted wavelengths or greater between the initial emission point and the detector, as the distance reduces toward zero emitted wavelengths the difference in angles can be substantial. This is a problem directly associated with transversely detected waves and does not involve longitudinally detected waves. For the longitudinal Doppler effect, the angle does not change from 0° or 180° regardless of how close the detector is to the nearest emission point. A simple solution to this problem for transversely detected waves where the distance between the detector and the point of emission of the first of two successive waves is in the range λe ≤ distance ≤ 107.5λe lies in use of the trigonometric law of cosines theorem. In Figure 1 an example of the triangular relationship between the two wave propagation paths and the emission points along the source’s path of travel is shown. In the illustration two successive waves whose leading edges are represented by sphere λe1 centered over point 1 along the source’s path of travel, and sphere λe2 centered over point 2 along the source’s path of travel, propagate outward in all directions as the source moves to the right as shown. Thus, the time interval between the two successive emissions represents the period of the source.

Since wavelength, represented by the variable D in the illustration, is the distance between the two leading edges of the emitted waves, the wavelength varies from its greatest value along the

θr θa

xc xc - c

D

1 v 2 source 1 v 2 source

λe1

λe2 D

FIGURE 1 Classical Transverse Doppler Effect Relationships

2

3 1

2

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source’s longitudinal path of travel at the left to its least value along the source’s longitudinal path of travel at the right. Thus, depending on the location of the point of detection around sphere λe1, the wavelength will have a different value. On the basis that the wavelength is determined by the distance between the two waves, it follows that it is the distance measured along the propagation path of the second wave centered over point 2 along the source’s path of travel. This distance is represented by the dashed lines directed toward detection points 1, 2 and 3 along the circumference of sphere λe1 in the illustration. Although any of the three triangles shown, or any other similar triangles we wish to construct, can be used to derive a formula for the transverse wavelength D, the triangle directed at detection point 2 will be used. Repeated in the view to the right of the illustration the dimensional components that make up the triangle have been identified. Since the source speed v represents the side of the triangle between emission points 1 and 2 along the source’s path of travel and the propagated waves travel at speed c away from these two points, the values of the two different speeds can be used to determine the proportional dimensions of the triangle. Thus, if v gives the side of the triangle along the source’s path of travel for a single propagation period of the source, then c gives the side for a single emitted wavelength λe propagating toward the detection point and therefore xc gives the total distance for the wave centered over point 1 as the sources continues its travel to its illustrated position at the right. Although in the original work3 it was stated that x is a whole number, it can actually have any positive value in the range 1 ≤ x ≤ 107.5. If c represents a single emitted wavelength λe, and the wave λe2 centered over emission point 2 was emitted a single period after the wave λe1 centered over emission point 1, then the distance traveled by the second wave λe2 is xc – c as illustrated. Therefore, the total length of the side of the triangle between emission point 2 and detection point 2 is given by xc – c + D. The final information needed to derive the classical transverse Doppler formula for wavelengths in the proximity of the source is the angle of detection. In the illustration these are given as θr for the angle of recession, and θa for the angle of approach. Since these are supplementary angles, their relationship to each other is given by

ra 180 (26) Relationship of Angle of Recession θr to Angle of Approach θa

and

ar 180 (27) Relationship of Angle of Approach θa to Angle of Recession θr

where for purposes of brevity either can be represented by the symbol θo for angle of observation. Continuing now with the Law of Cosines Theorem, the triangle given to the right in Figure 1 can be represented as

rDcxcvDcxcvxc cos)(2)()( 222 (28) From the Law of Cosines Theorem

where x is any positive number in the range 1 ≤ x ≤ 107.5, c is the speed of light, v is the speed of the source, D is the Doppler effected distance representing the detected wavelength λr and θr is the angle of recession. Solving for the expression (xc – c + D) in a math program (i.e. using a single variable for the stated expression) we obtain

10

2222 )(coscos)( vxcvvDcxc rr (29)

that when solved for D gives

2222 )(coscos vxcvvxccD rr (30) Doppler Effected Distance D

where D is the Doppler effected distance representing the detected wavelength λr. By placing the right side of the resulting equation over c we can convert it to the more general form of a factor that can be applied to any propagated wavelength. Thus, where Df is the classical transverse Doppler effect factor for all wavelengths, we have

c

vxcvvxccD rr

f

2222 )(coscos

(31) Classical Transverse Doppler Effect Factor for

Wavelength as the final Equation for the classical transverse Doppler effect factor for all wavelengths. The complete classical transverse Doppler effect formula for wavelengths is then

c

vxcvvxcc rrer

2222 )(coscos

(32) Classical Transverse Doppler Effect for

Wavelength where λr is the detected wavelength for angle of recession θr and λe is the emitted wavelength of the source. By simply changing the sign for v to – where shown, and substituting λa for λr and θa for θr we obtain

c

vxcvvxcc aaea

2222 )(coscos


Wavelength where λa is the detected wavelength for angle of approach θa and λe is the emitted wavelength of the source. Combining the two equations then gives

c

vxcvvxcc ooeo

2222 )(coscos


Wavelength where λo is the detected wavelength, λe is the emitted wavelength, θo is the angle of detection, and v is + for recession and – for approach where indicated. In the same manner that the classical transverse wavelength formula (11) was converted to its relativistic form in Section 5.b, Equations (32), (33) and (34) can now be converted to their relativistic forms using the Einstein redshift effect of Equation (5). This gives

11

22

2222 )(coscos

vc

vxcvvxcc rrer

(35) Relativistic Transverse Doppler Effect

Recession, Wavelength and

22

2222 )(coscos

vc

vxcvvxcc aaea


Approach, Wavelength and

22

2222 )(coscos

vc

vxcvvxcc ooeo


Observed, Wavelength (+R, –A) as the various forms of the complete formula for the relativistic transverse Doppler effect for wavelengths in the proximity of the source. Although in regard to relativistic Equations (35) and (36) and also the earlier classical Equations (32) and (33) the angles of detection, θr for recession, and θa for approach, can be from 0 to 180 degrees, it is customary to limit each to the range 0° – 90° consistent with the true meaning of the terms recession and approach. This is also true for the angle of detection θo in regard to relativistic Equation (37) and the earlier classical Equation (34) where v is + for recession and – for approach where indicated. Applying the same method of derivation to the just derived wavelength Equation (37) that was used earlier on Equation (19) to derive frequency Equation (20) gives

2222

22

)(coscos vxcvvxcc

vcff

oo

eo


Observed, Frequency (+ R, – A) where fe is the emitted frequency of the source, fo is the detected frequency in the stationary frame, and speed v of the source is + for recession and – for approach where indicated. 7. The Two Flaws in the Special Relativity Treatment In Figure 2 the leading edge of the wave that was emitted at point 1 along the source’s path of travel reaches detection point 1 along the surface of the expanding propagation sphere λe1 at the exact instant that the source reaches point 2 along the path of travel where the next wave λe2 will be emitted. Thus, to reach the same detection point 1 just reached by wave λe1, this second wave will travel the dashed line distance D. Obviously then, distance D is the Doppler effected distance between the two successive waves, λe1 and λe2 for detection point 1. If, as shown in the illustration, angle θs that path c toward detection point 1 forms with the source’s path of travel is equal to the angle of recession θr formed by path D with the source’s path of travel, then the resultant c, D, v triangle is an isosceles triangle where side D = side c. In that case we have the null condition of the classical Doppler effect at a single emitted wavelength

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distance between emission point 1 and detection point 1. It should be rather obvious that as long as the two angles θs and θr are equal, sides c and D (the paths to detection point 1) will also be equal regardless of the distance to detection point 1. Of course, as previously shown, for distances greater than a single wavelength, side c will be defined in terms of xc and side D will be defined in terms of xc – c + D. Since, in the case under discussion, in the latter of the just given expressions, distance D = c, we actually have the condition xc – c + D = xc. In other words, these are the actual conditions in which there will be the null classical Doppler effect defined in special relativity.

As distance xc increases to the point where x = 107.5 emitted wavelengths, angles θs and θr, as long as they remain equal to each other, will both approach 90°, and the c, D, v triangle will become so narrow that sides xc and xc – c + D will in effect merge into the single line used in the special relativity treatment. At closer detection distances, however, angles θs = θr are not 90° angles and therefore it is quite clear that the null classical effect does not really occur at a 90° angle to the point of emission along the path of travel of the source. Yet, it is this null classical effect that is defined in the special theory of relativity in regard to which only the Einstein redshift effect is supposedly detected. To emphasize the contradiction being discussed here, it must be noted that the Einstein redshift effect, defined by the second triangle c, E, v shown in the illustration, is based on a true 90° angle that side E of that triangle forms with the path of travel of the source. Nonetheless, since at distances where x ≥ 107.5 emitted wavelengths, the angle of recession θr (under the condition θr = θs) essentially equals 90°, the Einstein redshift effect can be factored along with the classical null effect to show the net Einstein redshift-only effect as predicted in special relativity. At closer distances, however, this is not the case, and is the second flaw in the special relativity treatment. I.e. the first flaw is the use of a single line between the source’s path of travel and the point of detection even at distances in close proximity to the source where such approach has been shown to be incorrect since each of the two successive waves propagates from different points along the source’s path of travel. The second flaw involves combining two different effects where the angles are in conflict with each other. (In actuality it is the 90° angle, with respect to the path of travel of the source, formed by the line

θs θr

FIGURE 2 The 90 Degree Relativistic Transverse Doppler Effect

1 Null Classical Effect

2 Einstein Redshift Effect

E

Dc c

1 v 2 source

λe1

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that dissects the classical triangle between emission points 1 and 2 that is used in the special theory of relativity and not the true angle of recession or approach for the detected signal.) To summarize, it should be noted that in the millennium relativity treatment, when the two different effects are combined for the 90° angle of detection for the single emitted wavelength distance shown in Figure 2, the classical effect triangle is superimposed over the Einstein redshift effect triangle. Thus, at a distance of a single emitted wavelength, there is a classical blueshift effect that exactly cancels the Einstein redshift effect resulting in no Doppler effect at all for the transverse angle of 90°. This is, as previously stated, in contrast to the special relativity prediction of an Einstein, redshift-only effect at that 90° detection angle. Nonetheless, as the distance between the emission points and the detection point approaches 107.5 emitted wavelengths, the angle of detection under the conditions discussed, approach the 90° angle of the Einstein redshift effect and therefore, though technically incorrect, the special relativity treatment is accurate from that distance assumedly on to infinity. At the same time, the millennium relativity treatment is no longer usable because the triangle is too narrow to compute and essentially merges with the single line of the special relativity treatment which then takes precedence. 8. The Relativistic Doppler Effect at Fractional Wavelength Distances a. The Classical Transverse Doppler Effect at Fractional Wavelength Distances Although the earlier derived proximity of the source formulas of Section 6 will give the correct relativistic transverse Doppler effect for distances down to a single emitted wavelength between the detector and the initial point of emission, they do not cover the less than a wavelength distances used in some experiments. To derive such formulas we can draw upon the previously completed analysis of Section 6 and with appropriate modifications arrive at the correct solution. As was true in the previous case, any of the triangles illustrated in Figure 1, or any other similar triangles can be used in the derivation process. For the present derivation we will use a triangle similar to the one directed at detection point 3 in the Figure 1 illustration. Since, however, this analysis is valid only for distances equal to or less than a single emitted wavelength, for the same speed v = 0.5c used for Figure 1, the triangle used in Figure 3 of the present analysis will be less narrow than that of Figure 1. In fact, in this case the triangle shown represents a single initial wavelength emitted by the source at propagation point 1, and subsequently the source is shown at the propagation point 2 position along its path of travel that the next wave will be emitted from should the propagation continue. To begin the analysis consider that a wave emitted at point 1 along the source’s path of travel to the right in Figure 3 has just reached the full initial wavelength detection point 3 at the exact instant that the source reaches point 2 along its path of travel. Since the source would emit its next sequential wave from its present location at point 2 as it continues its travel to the right, the dashed line distance labeled D is the Doppler effected wavelength that will be detected at point 3. In other words, this is the distance between the leading edge of the single wave that just propagated from point 1 to point 3, and the leading edge of the next wave that will now propagate from point 2 to point 3. With that being the case, the distance from point 1 to point 3 is the radius of the propagation sphere centered over point 1 and also the wavelength of the signal emitted from point 1. To locate a point anywhere along the line extending from point 1 to point 3, other than at point 3 itself, then is to identify a point that is less than an emitted wavelength away from point 1. This new point then is the fractional wavelength distance that

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the signal from point 2 will be measured against. The point selected along propagation path 1-3 in Figure 3, shown as point o, just happens to be located where the intersecting propagation path from point 2 is at a 90° angle to the path of motion of the source. The angle of this new detection path relative to the path of motion of the source is identified as θrf for the fractional distance angle of recession, and as θaf for the supplementary fractional distance angle of approach. As can be readily deduced from the illustration, if the fractional distance angle of recession θrf reduces toward 0°, the fractional distance 1-o along the path 1-3 reduces toward 0 and distance d approaches the value of distance 1-2, or distance v. The length of this section of path 1-3 is defined as f2. If, on the other hand, the fractional distance angle of approach θaf reduces toward 0°, then the fractional distance f2 approaches the full initial emitted wavelength distance 1-3 and distance d approaches the value of distance D.

Given the fact that distance 1-3 = c, in this latter case if the angle of approach θaf reaches 0°, then distance d = distance D = c – v, or the condition of classical longitudinal approach. If, on the other hand, we allow point o to coincide with point 3, and reduce the angle of recession θrf to 0°, we then have the opposite condition of classical longitudinal recession where distance d = distance D = c + v. And lastly, if the angle of recession θrf and the angle of approach θaf remain at 90°, distance d can vary from D at detection point 3 all the way down to 0 at emission point 2. Therefore, depending on these two angles and also on the position of detection point o along propagation path 1-3, every condition for the classical transverse Doppler effect is covered for a detection distance f2 equal to a single emitted wavelength all the way down to zero wavelengths. The only remaining questions are, what are the classical Doppler effects for all of these conditions? That we deal with next. Before proceeding it is important to remember that the fractional wavelength distance actually being referred to in this treatment is the distance from point 1 along the source’s path of travel to point o along the propagation path of the first wave to detection point 3. In other words, distance f2 in Figure 3. It is this distance that can vary from 0 emitted wavelengths to the single emitted wavelength limit represented by detected point 3. Still referring to Figure 3, consider what it is that is actually detected at fractional wavelength detection point o. At the instant shown in the illustration, the leading edge of the wave from point 1 has already been detected at point o and has since traveled an additional distance defined as f1. At that same exact instant, the leading edge of the next wave to be emitted at point 2 will begin to travel in all directions from point 2 including along path d to detection point o. Given that both signals, the signal from point 1 traveling distance f1 and the signal from

f2

f1

Dd

full wavelengthdetection point

fractional wavelength detection point

o

3

1 v 2 (source)

θrf θaf

FIGURE 3 Fractional Wavelength Detection Distances

c = f1 + f2

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point 2 that will travel distance d to point o have the same speed c, one thing should be pretty obvious. The distance they travel is proportional to the time that they traveled those distances, and conversely, the time that they travel is proportional to the distance they travel during that time. In plan words, it doesn’t matter that the two signals travel in different directions at different times. The fact that the one distance traveled takes place directly after the other means that they can be added together both spatially and chronologically. In short, the classical transverse Doppler effect detected at point o for the two signals is the direct sum of the two distances traveled, or the two time intervals during which the travel occurred. All we need to do then, is to add up these distances and we will have the classical Doppler effected wavelength of the signals detected at point o. More specifically, the wavelength detected at point o, when the wave from point 2 travels distance d, is given by the formula

dfo 1 (39) Detected Wavelength at Point o

where λo is the wavelength detected at point o for fractional wavelength distance f2. Thus, if distance d is given we need only to determine the fractional wavelength distance f1 to find the wavelength λo that will be detected at point o. Distance f1, however, is related to distance f2 by the formula

21 ffc (40) Relationship of Fractional Wavelength Distances f1 and f2

or

21 fcf (41) Fractional Emitted Wavelength Distance f1

and since distance f2 is more easily determined from the information given, it makes logical sense to first develop a formula to find distance f2. Assuming for the moment that distance d along with source speed v and angle of detection θrf are known, let us proceed then to develop a formula for finding distance f2. Resorting again to the Law of Cosines Theorem we have

rfvddvf cos22222 (42) From Law of Cosines Theorem

giving

rfvddvf cos2222 (43) Fractional Emitted Wavelength Distance f2

where f2 is the fractional wavelength distance between point 1 and point o, v is the speed of the source, d is the distance between the source and the point of detection o, and θrf is the angle of detection. Referring back to Equation (41) we can now state by way of substitution

rfvddvcf cos2221 (44) Fractional Emitted Wavelength Distance f1

giving

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1fdDx (45) Total Wavelength Distance Detected at Point o

where Dx is the total wavelength distance detected at point o. Through substitution with Equation (44) this gives

rfx vddvcdD cos222 (46) Total Wavelength Distance Detected at Point o

where distance Dx can be converted to a factor in terms of c, giving

c

vddvcdD

rf

xf

cos222 (47) Classical Transverse Doppler Effect Factor for Fractional

Wavelength Distances where Dxf is the required factor for the classical transverse Doppler effect for fractional wavelength distances. For the classical transverse Doppler effect for fractional wavelength distances this then gives

c

vddvcd rf

er

cos222 (48) Classical Transverse Doppler Effect for Fractional

Wavelength Distances where λr is the detected wavelength at point o from a receding source, λe is the emitted initial wavelength, d is the distance to the source, v is the speed of the source, θrf is the angle of recession, and c is the speed of light in empty space. Expanding on this as we did for the wavelength formulas derived earlier in Section 6 gives

c

vddvcd af

ea


Wavelength Distances and

c

vddvcd of

eo


Wavelength Distances where λa is the initial wavelength detected at point o at angle of approach θaf, and λo is the initial wavelength detected at point o at angle of detection, θof. As with the earlier wavelength formulas, a change of sign is required in these last three equations involving detection at fractional wavelength distances. In this case, however, the signs are reversed; i.e. – for recession and + for approach and take place inside the radical as shown.

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b. The Relativistic Transverse Doppler Effect at Fractional Wavelength Distances Since it is not actually the emitted wavelength that is operated on by the classical transverse Doppler effect factor we again have to substitute the transformed wavelength λx for the emitted wavelength λe in the just arrived at formulas. And since, as shown previously

22 vc

cex

(5) Einstein Redshift Effect for Wavelength

we need to simply factor Equations (48) (49) and (50) with the millennium version of the Einstein redshift factor to arrive at

22

22 cos2

vc

vddvcd rf

er

(51) Relativistic Transverse Doppler Effect for Fractional


22

22 cos2

vc

vddvcd af

ea



22

22 cos2

vc

vddvcd of

eo


Wavelength Distances where λr, λa, and λo are the respective wavelengths detected at point o from a receding, approaching, or either type of source, λe is the emitted wavelength in the source’s frame of reference, d is the distance between the source and detector at the instant of emission of the second wave, θrf, θaf, and θof are the respective angles of detection, v is the speed of the source, and c is the speed of light in a vacuum. (Note sign change in Equation (53); – for recession and + for approach where indicated) Again, although any angle from 0° to 180° can be used in any these formulas, it is customary to use the formula where the angle falls in the range 0° ≤ θ ≤ 90°. For the frequency version of the formula we can, as done previously for the earlier wavelength formulas, apply the relationships given by equation (7) to equation (53) to arrive at

of

eovddvcd

vcff

cos222

22


Wavelength Distances

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where fo is the frequency detected at point o from a receding or approaching source, fe is the emitted frequency, and the term inside the radical containing θof is – for recession and + for approach as discussed previously. 9. The 90° Relativistic Transverse Doppler Effect at Fractional Wavelength Distances For the 90° transverse Doppler effect used in many experiments conducted at fractional wavelength distances from propagation point 1, a much simpler formula is available using the Pythagorean Theorem. Figure 4 shows a simplified version of the initial wavelength previously illustrated in Figure 3. Now, however, the angle of detection between the source and detection point o is limited to 90° as shown. Thus, the fractional wavelength distance previously labeled f2 is now directly defined by the Pythagorean Theorem in terms of source speed v and distance d.

Proceeding as before, the classical transverse Doppler effect for detection point o is given by

1fdDx (45) Total Wavelength Distance Detected at Point o

where Dx is the total wavelength distance detected at point o. Since, in accordance with Figure 4

221 dvfc (55) Initial Wavelength between Points 1 and 3

and therefore

221 dvcf (56) Fractional Emitted Wavelength Distance f1

by way of substitution of the right side of this equation into equation (45) we obtain

22 dvcdDp (57) Total Wavelength Distance Detected at Point o

where Dx, now relabeled Dp to avoid confusion with its other use, is the wavelength detected at point o. Division of the right side of the resulting equation (57) by c then gives

c

dvcdDpf

22 (58) 90° Classical Transverse Doppler Effect Factor for Fractional Wavelength

Detection Distances

FIGURE 4 90° Fractional Wavelength Detection Distances

f1

Dd

1 v 2 (source)

o

c = f1 + 3

22 dv

22 dv

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where Dpf is the required factor for the 90° classical transverse Doppler effect for fractional wavelength detection distances. Applying this factor to the emitted wavelength when the source is at a 90° angle to detection point o gives

c

dvcdeo

22 (59) 90° Classical Transverse Doppler Effect Wavelength for Fractional

Wavelength Detection Distances where λo is the wavelength detected at point o. Factoring this equation with the millennium version of the Einstein redshift factor as demonstrated previously then gives

22

22

vc

dvcdeo

(60) 90° Relativistic Transverse Doppler Effect Wavelength for Fractional

Wavelength Detection Distances for the relativistic version of the 90° formula. Applying the Equation (7) relationships to the just derive equation then gives

22

22

dvcd

vcff eo

(61) 90° Relativistic Transverse Doppler Effect Frequency for Fraction Wavelength

Detection Distances for the frequency version of the fractional wavelength distances formula. This completes the analysis. 10. Summary of Important Findings

a. Not only source speed and angle of detection, but also distance is a factor in the relativistic Doppler effect. Thus, it is now apparent that three different treatments are required to determine the relativistic Doppler effect. These are as follows: 1. The original special relativity treatment for distances in the range of approximately

107.5 emitted wavelengths to infinity. 2. The millennium relativity proximity of the source treatment for distances in the range

of one emitted wavelength to approximately 107.5 emitted wavelengths. 3. The millennium relativity fractional distance treatment for distances in the range of

zero emitted wavelengths to one emitted wavelength. b. The special relativity prediction of no classical effect for the transverse condition of 90° and therefore a net effect equal to the Einstein redshift effect at that angle is invalid at distances less than approximately 107.5 emitted wavelengths from the point of emission of the first of the two waves. In accordance with the findings of the millennium relativity proximity of the source treatment there will be a classical Doppler blueshift effect at 90° that will partially cancel the Einstein redshift effect resulting in a net redshift effect less than predicted by special relativity. This counter effect will increase as the distance

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decreases and at a distance of exactly one wavelength the two effects are equal and opposite and therefore cancel each other out. Thus, for any source speed from v = 0 to c, there is no effect at all for a detection angle of 90° relative to the point of emission of the second of two waves along the path of the source at a distance of one wavelength to the point of emission of the first of two successive waves. This appears to be a fundamental law of Nature. c. The generalized special relativity treatment of the transverse Doppler effect conceals the fact that the 90° angle of detection is not actually 90° and therefore incorrect on that basis alone and also inconsistent with the correctly defined 90° angle used in the special relativity derivation of the Einstein redshift effect. Nonetheless, at distances in which the treatment works properly, the result of the incorrect Doppler effect angle is negligible and even though technically incorrect, it is no longer a concern. d. The 107.5 emitted wavelength distance in which the special relativity treatment is invalid ranges from virtually zero meters at the gamma ray end of the electromagnetic spectrum to approximately 13 – 22 meters for visible light waves (depending on wavelength) to approximately 32,000,000,000,000 km for long radio waves at the other end of the spectrum. Quite obviously, the interpretation of the results of all previous relativistic transverse Doppler effect experiments are called into question. This is especially true for laboratory experiments conducted at or near the transverse angle of 90°. Of equal concern is the ramification of these findings in regard to an indeterminable number of practical applications where high accuracy is of critical importance. On that basis alone, one would think there would be a sense of urgency to make these findings as broadly disseminated as practical. e. Although the Einstein redshift effect, even though predicated on a 90° angle between reference frames, can be experimentally verified at any angle, even near or at 0° or 180° such as in the Ives – Stilwell experiment7, the relativistic classical Doppler effect at a 90° angle can only be experimentally verified at 90°. f. Following is a complete list of affected distances by wave type in which the special relativity formula is invalid and the proximity of the source formulas must be used: Wave Type Wavelength in m Range of Affected Doppler Distances Gamma Rays 10-17 – 10-10 0.000,000,000,3 m 0.003 m X-Rays 10-11.4 – 10-7.5 0.000,1 m 1 m Ultraviolet Rays 10-8.5 – 0.4 x 10-6 0.1 m 13 m Visible Light 4 x 10-7 – 7 x 10-7 13 m 22 m Infrared Rays 0.7 x 10-6 – 10-3.4 22 m 13 km Microwaves 10-3.4 – 0.6 13 km 19,000 km Short Radio Waves 10-3.4 – 102 13 km 3,000,000 km Radar 10-1.7 – 103 631 km 32,000,000 km TV, FM 3 – 50 95,000 km 1,600,000 km AM Radio 102 – 103 3,000,000 km 32,000,000 km Long Radio Waves 103 – 109 32,000,000 km 32,000,000,000,000 km

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11. Conclusion The millennium relativity findings regarding the discrepancies and limitations of the special relativity transverse Doppler effect are beyond dispute. Thus, not only are the results of all previous experiments regarding the transverse effect brought into question, but all practical applications involving the effect, especially those where a high degree of accuracy is critical, are in need of review. Moreover, due to our present inability to accurately determine the full impact of these findings, it is of utmost importance that they be taken seriously and treated with an appropriate degree of urgency. REFERENCES 1 Albert Einstein, Special Theory of Relativity, (1905), originally published under the title, On the Electrodynamics of Moving Bodies, in the journal, Annalen der Physik. 2 In 1842 Christian Doppler (1803-1853) pointed out that if a light source is approaching or receding from the observer, the light waves will be, respectively, crowded together or spread out. The principle, known as the Doppler principle or Doppler effect was based on the belief that light traveled through a medium called aether. The aether medium was discarded in the special theory of relativity. 3 First discovered in, “Joseph A. Rybczyk, Relativistic Transverse Doppler Effect, (2005 – 2006)”. 4 Joseph A. Rybczyk, Millennium Theory of Relativity, (2001). 5 The mathematical factor devised by Dutch Physicist H. A. Lorentz in 1895 to explain the null results of the Michelson and Morley interferometer experiments of 1887. The application of the factor was expanded upon by Einstein to include time dilation and other relativistic effects in the Special Theory of Relativity. 6 Contrary to the findings of the special relativity Doppler treatment, there will be a classical Doppler blueshift effect at the 90° angle of detection in the proximity of the source. Therefore the prediction that there will be only an Einstein redshift effect at that angle is incorrect. This discrepancy in the special relativity treatment was first revealed in, “Joseph A. Rybczyk, Relativistic Transverse Doppler Effect, (2005 – 2006), Section 12.” It was demonstrated again in, “Joseph A. Rybczyk, The Theoretical Evidence against the Special Relativity Transverse Doppler Effect, (2006), Section 8”, and also in “Joseph A. Rybczyk, Einstein’s Transverse Doppler Effect Proven Wrong, (2006), Section 9”. 7 One of the most credible, convincing, and well known experiments confirming the Einstein redshift effect was the Ives-Stilwell experiment of 1938. This experiment which has been conducted many times since then, used light waves detected at angles very close to 0° and 180°. The very reason this experiment used the indirect method of detecting essentially longitudinal waves instead of transverse waves was because of the known difficulty and unreliability of experiments designed to directly detect 90° transverse waves. It is important to note here, that although the Ives-Stilwell experiment may have convincingly showed the presence of the special relativity predicted Einstein redshift effect resulting from time dilation, it does not show the conditions that prevail at the transverse angle of 90°. Only a test conducted at the 90 angle can be used for that purpose.

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Relativistic Transverse Doppler Effect Simplified

Complete Treatment for Distances from Zero to Infinity

Copyright © 2007 Joseph A. Rybczyk

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in whole or in part in any form without permission.

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