Einstein s Transverse Doppler Effect Proven Wrong. The Complete Proof. Copyright 2006 Joseph A. Rybczyk

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1 Einstein s Tansese Dopple Effet Poen Wong The Complete Poof Copyight 006 Joseph A. Rybzyk Abstat Stit adheene to the sientifi method in oelating the piniples of light popagation with the piniples of elatiisti physis leads to the deiation of a new fomula fo the elatiisti tansese Dopple effet. In the poess it is found that Einstein s tansese Dopple effet fomula is atually a genealized solution that iolates the piniples of the theoy upon whih it is pupotedly founded. The speifi eo in Einstein s fomula is peisely identified and the extent of its effet on theoetially oet peditions is unambiguously pesented. A stak lesson in sientifi piniples esults fom the ealization that a fomula shown to be tehnially inoet is still needed to supplement the newly deied theoetially oet fomula due to mathematial limitations of the latte. 1. Intodution As is often the ase in the disoey poess, the undelying piniples of the elatiisti tansese Dopple effet 1 ae so simple and staightfowad when finally esoled that it is quite supising that it would take so long to aomplish. Pehaps thee is at least one saing gae in that egad. Wheeas an elegant solution is often seen as a defining sign of undelying tuth, the speial elatiity fomula will be seen to be of ultimate elegane, yet tehnially inoet. The new fomula though oet itually beyond dispute is anything but elegant o een selfsuffiient fo that matte. Ionially, due to inheent limitations of the new fomula, both fomulas, at least fo the pesent time, ae needed in spite of the fat that the peious fomula is obiously wong.. The Fundamental Piniples of Light Popagation Aoding to the piniples of speial elatiity, light popagates though the auum of empty spae at a onstant speed and does not take on the speed of the soue elatie to the stationay obsee. Thus, as is well undestood and ageed upon by those who suppot elatiisti piniples, light fom a moing soue will popagate at speed in all dietions fom the stationay fame (SF) point at whih it was emitted by the soue. Also ageed upon by those same authoities ae the popositions that nothing (inluding the soue) an exeed the speed of light, and that light exhibits the dual haateistis of both a wae and a patile depending on the appliation. With egad to light taeling though spae it is geneally aeptable to teat it as popagating waes. In fat, suh popagation is nomally efeed to in tems of fequeny and waelengths in that egad and suh will be the ase hee. If we teat light popagation as waes, measuable in tems of waelength and fequeny it is easonable to onlude that the end of one waelength is the beginning of the next. In othe wods, the beginning of a wae and the end of a wae ae atually the same thing. It s simply a matte of the ode of emission. Thus, it is not only easonable but in fat ditated though stit adheene to sientifi piniples that any laws of physis that apply to the stat of a wae, apply 1

2 equally to the end of the wae, synonymous hee with the stat of the next wae. With all of these piniples in mind let us now efe to Figue 1. In Figue 1 a point light soue is shown taeling though spae at a unifom speed while emitting a steady steam of light waes along its path of tael in the stationay fame of efeene. Fo laity only the last thee waes ae shown. Sphee S 1 epesents the leading edge of the wae that was emitted at point 1 along the soue s path of tael, depited by the dashed line aow, wheeas sphee S epesents the next wae emitted at point, and sphee S 3 epesents the wae emitted at point 3. The next wae will be emitted fom the point pesently oupied by the soue, should the emission poess ontinue. The dotted line epesents the oeall path of tael fom the left fom whee the soue ame, and the pojeted path of tael to the ight whee the soue will ontinue its jouney baing any outside influenes. Sine light popagates in all dietions at speed fom the stationay fame point of emission, eah sphee expands at that speed while emaining enteed oe the stationay fame point at whih it was emitted. And sine nothing an exeed the speed of light, the soue an nee pass any of the waes being emitted and thus will always be enlosed in the expanding sphees of emitted light as shown. (Fo infomation puposes, though not equied fo the mathematial deiation poess, the soue is shown moing at half the speed of light in the illustation. Thus, the adius of eah sphee is twie the distane that the soue has taeled sine the epesented wae was emitted.) S 1 S S soue Figue 1 Wae Popagation fom Point of Emission Sine, as peiously stated, the leading edge of eah sueeding wae is the tailing edge of the peeding wae, sphee S 3 epesents the tailing edge of the wae whose leading edge is epesented by sphee S, and sphee S epesents the tailing edge of the wae whose leading edge is epesented by sphee S 1. Thus, as also peiously stated, eah sueeding wae must

3 follow the same laws of physis as the wae that peeded it, een though it may be iewed as the tailing edge of that peious wae. 3. The Relatiisti Tansese Dopple Effet Refeing next to Figue, the elationships shown in Figue 1 ae epeated with the exeption that sphee S 3 has been omitted fo the pupose of laity. To omplete the definition poess we need to show the position of the stationay obsee elatie to the popagating waes peiously emitted by the soue. Suh position an be anywhee along the sufae of sphee S 1 without effeting the deiation poess. Fo the pupose of late ompaisons, the position hosen is dietly oe point e pependiula to the soue s path of motion. With the obsee s position now epesented in elation to the popagating waes, we an finish the definition poess. Point s along the soue s path of tael epesents the point of emission of the wae whose leading edge is epesented by sphee S 1 wheeas point e along the same path of tael epesents the point of emission fo the next sueeding wae whose leading edge is epesented by sphee S. Thus, point s is defined as the stat point of emission fo a wae that has a leading edge epesented by sphee S 1, and a tailing edge epesented by sphee S. Point e is then defined as the end point of emission fo a wae that has a leading edge epesented by sphee S 1 and a tailing edge epesented by sphee S, synonymous hee with the leading edge of the next suessie wae. obsee S 1 D S x θ p x θ s θ θ a s e soue Figue Classial Tansese Dopple Effet In iew of what has been disussed so fa, thee should be no question that point s is the ente of sphee S 1, point e is the ente of sphee S, and both points ae stationay elatie to the stationay fame obsee. Thus, if we pojet a staight line fom eah of these points to the 3

4 obsee, we ae in theoy defining the path of eah wae fom the espetie point of emission to the obsee. With that being the ase, the angles elating to the esultant tiangle fomed by these two paths in onjuntion with the path of the soue an be defined as pojetion angle θ p, stat angle θ s, angle of eession θ, and angle of appoah θ a, whee angle θ a is the supplementay angle of angle θ. (I.e. θ a = 180 deg θ and θ = 180 deg θ a ) The only thing left to do now with egad to the definition poess is to define the sides of the tiangle. Nomally we would elate the time inteals that the taels took plae in with the speeds inoled to define the distanes taeled. Sine all of the distanes ou duing the same stationay fame time inteal, howee, the inteal beomes nothing moe than a popotionality onstant when used in that manne and theefoe has no effet on the deiation poess othe than making it moe ompliated than it needs to be. Wheeas the time inteal does still play a ole in the analysis, it is unlea at this ealy stage of disoey as to how it is best handled. Thus, we will poeed dietly to the final definitions and deal with it as it beomes a fato of onsideation duing subsequent eseah. If the emitted waes popagate at speed while the soue taels at speed, it is undestood then that as the soue taels the distane fom point s to point e, the wae emitted at point s taels outwad in all dietions fom point s, inluding in the dietion of the obsee at speed. Sine points s and e ae both stationay, howee, neithe hanges its elationship to the stationay obsee o to eah othe fo that matte, as the light ontinues its tael outwad fom point s. Thus, if distane esulted as the soue taeled fom point s to point e whee the next wae will be emitted, distane effetiely epesents the unaffeted waelength of the emitted light. In othe wods, if the soue wee not in motion, the next wae would be emitted at that instant fom point s just as the peious wae was. Thus the distane between the two waes would epesent a waelength unaffeted by the motion of the soue. If we theefoe multiply the distane epesented by by a fato x as shown in the illustation, we an define the distane fom point s to the obsee in tems of whole unaffeted waelengths of emitted light. And sine the wae emitted at point e will hae the same speed away fom that point in all dietions, inluding in the dietion of the obsee, that distane to the obsee an also be expessed in tems of x. Howee, the fist wae aleady taeled distane befoe the seond wae was emitted; theefoe the seond wae taels distane x as shown in the illustation. With some thought, it should also be appaent hee, that x is atually a time inteal, e.g. the distane taeled at speed T whee T is the stationay fame time inteal in tiny fations of a seond. It affets only the distane of the popagated light, howee, and not the distane between points, s and e. Also appaent hee, is the fat that the pesent loation of the soue has no eleane to the elationships being disussed. One the soue has emitted the two waes being defined, it no longe eally mattes what the soue does. It ould hange its ate of speed, o dietion, o een ease to exist fo that matte. It no longe has any beaing on what happens to the waes peiously emitted along its path of tael. Now that we hae defined the distanes taeled by the emitted light along the defined paths fom points, s and e to the obsee, we ae left with only one emaining distane to define. That distane is designated distane D in the illustation and epesents the emaining distane to the obsee fo the tailing edge of the wae, (o the leading edge of the next suessie wae). Thus, this distane epesents the Dopple effeted waelength of the waes eahing the obsee. In iewing Figue, it should be eadily appaent that the illustated tiangle ould be otated to the left o ight to define the elationships to othe obseation points along the sufae of sphee S 1 in addition to the one illustated. Thus it should be ey appaent that distane D 4

5 an hae a alue that aies fom + along the path of motion of the soue to the left of the illustation, to along the path of motion of the soue to the ight of the illustation. Subsequently, distane D epesents the obseed waelength fom a eeding soue fo any point along the sufae of the sphee between the depited obsee and the path of motion of the soue to the left of the depited obsee. Conesely, distane D epesents the obseed waelength fom an appoahing soue fo any point along the sufae of the sphee between the depited obsee and the path of motion of the soue to the ight of the depited obsee. Late in the analysis it will be shown that in defining a 90 degee angle of obseation between the obsee and the path of motion of the soue othe than that shown in Figue is a iolation of the most fundamental piniples of speial elatiity. 4. Deiing the Classial Tansese Dopple Effet Fomula Still efeing to Figue we an use the law of osines theoem to sole the depited tiangle fo distane D based on the angle of eession θ. This gies ( x) ( x D) ( x D) os (1) Fom Law of Cosines Theoem whee x is a whole numbe, is the speed of light, is the speed of the soue, D is the Dopple effeted distane epesenting the obseed waelength, λ fom a eeding soue, and θ is the angle of eession. Soling fo the expession (x + D) in a math pogam we obtain ( x D) os os ( x) () that when soled fo D gies D x os os ( x) (3) Dopple Effeted Distane D whee D is the Dopple effeted distane epesenting the obseed waelength λ. By plaing the ight side of the esulting equation oe we an onet it into the moe geneal fom of a fato that an be applied to any popagated waelength. Thus, whee D f is the lassial tansese Dopple effet fato fo all waelengths, we hae D f x os os ( x) Waelengths (4) Classial Tansese Dopple Effet Fato fo as the final equation fo the lassial tansese Dopple effet fato fo all waelengths. The omplete lassial tansese Dopple effet fomula fo waelengths is then x os os ( x) e (5) Classial Tansese Dopple Effet fo Waelengths 5

6 whee λ is the obseed waelength fo angle of eession θ, and λ e is the emitted waelength of the soue. By simply hanging the sign fo to whee shown, and substituting λ a fo λ and θ a fo θ we obtain x os a os ( x) a a e (6) Classial Tansese Dopple Effet fo Waelengths whee λ a is the obseed waelength fo angle of appoah θ a, and λ e is the emitted waelength of the soue. Combining the two peious equations then gies x os o os ( x) o o e (7) Classial Tansese Dopple Effet fo Waelengths whee λ o is the obseed waelength, λ e is the emitted waelength, θ o is the angle of obseation, and is + fo eession and fo appoah whee indiated. 5. Deiing the Relatiisti Tansfomation Fomulas In ode to onet the just deied fomulas fo the lassial tansese Dopple effet to thei elatiisti fom, we need to fist deie the Loentz tansfomation fomulas. This an be aomplished dietly using a simplified fom of the speial elatiity light lok method. In that egad, Einstein s deiation of the Loentz tansfomations fo time and distane an be gaphially epesented as shown in Figue 3. S o T T t soue FIGURE 3 Einstein s Definition of the Loentz Tansfomation Using this simplified example of the speial elatiity light lok we an eiew the piniples undelying Einstein s deiation of the Loentz tansfomation fato. Gien o as the point of oigin in the stationay fame whee a pulse of light is emitted by the soue as the soue taels at a unifom speed along the path shown, we an laim that sine light does not take on the speed of the soue it will tael outwad fom point o in all dietions at speed as depited by 6

7 sphee S in the illustation. Thus, if t is the unifom motion fame (UF) time inteal that oesponds to stationay fame time inteal T, and is also the speed of light in the moing fame of the soue, and the laws of physis ae the same in all inetial systems, the podut t an be used to epesent the moing fame distane taeled by a point of the same light pulse in the moing fame of the soue in a dietion pependiula to the soue s path of tael in the stationay fame. In a simila manne, T and T epesent the espetie distanes taeled by light and the soue in the stationay fame. Subsequently, sine the distane t is pependiula to the path of tael of the soue, we hae the Pythagoean elationship defined by the esultant ight tiangle in Figue 3 that an be expessed as, t ( T) ( T) (8) Pythagoean Relationship of Distanes that when soled fo t gies t T (9) Simplified Fomula fo Time Tansfomation whee t is the moing fame time inteal oesponding to stationay fame time inteal T. The deied fomula tanslates to the moe familia speial elatiity fom by way of t T (10) t T (11) and t T 1 (1) Speial Relatiity Time Tansfomation whee again, t is the moing fame time inteal that oesponds to stationay fame time inteal T. If we wish to expess the tansfomation in tems of distane instead of time, we an do so ey simply. If D is the distane taeled by light in the stationay fame duing the same time inteal that light taels distane d in the moing inetial fame of the soue, we an state that and D T (13) d t (14) 7

8 wheeby multiplying both sides of Equation (1) by gies t T 1 (15) and though substitution using Equations (13) and (14) gies d D 1 (16) Speial Relatiity Distane Tansfomation (SF to UF) fo the elationship of stationay fame distane D to unifom motion fame distane d. The inese of these distanes is then gien by 1 D d (17) Speial Relatiity Distane Tansfomation (UF to SF) 1 whee D is the ineased distane obseed in the stationay fame of the obsee. The deied fomula an then be oneted to its simplified fom by multiplying the top and bottom of the fation in the tem to the ight of the = sign by. This gies D d (18) 1 and D d (19) Speial Relatiity Distane Tansfomation Simplified (UF to SF) whee, as befoe, D is the ineased distane obseed in the stationay fame of the obsee. 6. The Complete Relatiisti Tansese Dopple Effet fo Waelengths Sine, in aodane with the piniples of elatiity, time and distane in the moing fame of the soue ae tansfomed in the stationay fame, the distane epesented by the emitted waelength λ e in the moing fame of the soue is edshifted (ineased in length) when obseed in the stationay fame of the obsee. Thus, whee λ e is the emitted waelength in the fame of the soue, though the distane tansfomation poess gien by equation (19) we hae x e (0) Waelength Tansfomation (UF to SF) 8

9 whee λ x is the ineased waelength obseed in the stationay fame of the obsee. Sine it is eally λ x and not λ e that is opeated on by the peiously deied equations fo the lassial tansese Dopple effet (equations 5, 6 and 7), λ x an be substituted in plae of λ e in those equations to onet them to thei elatiisti foms. Fo example, substituting λ x in plae of λ e in equation (5) gies x os os ( x) x (1) Relatiisti Tansese Dopple Effet fo Waelengths wheeby substituting the ight side of the peiously deied equation (0) in plae of λ x in the just deie equation gies that simplifies to x os os ( x) e () x os os ( x) e (3) Relatiisti Tansese Dopple Effet fo Waelengths fo the final fom of the elatiisti tansese Dopple effet fomula fo the obseed waelength fom a eeding soue. Doing simila to equations (6) and (7) gies x os a os a ( x) a e (4) Relatiisti Tansese Dopple Effet fo Waelengths and x os o os o ( x) o e (5) Relatiisti Tansese Dopple Effet fo Waelengths as the final foms of the omplete fomulas fo the elatiisti tansese Dopple effet fo waelengths fom an appoahing soue o fom any soue in the latte ase whee speed is + fo eession and fo appoah whee indiated. Although in egad to equations (3) and (4) the angles of obseation, θ fo eession, and θ a fo appoah, an be fom 0 to 180 degees, it is ustomay to limit eah to the ange 0 90 degees onsistent with the tue meaning of the tems eession and appoah. This is also tue fo the angle of obseation θ o in egad to equation (5) whee is + fo eession and fo appoah whee indiated. 9

10 7. The Genealized Deiation of the Speial Relatiity Fomula A most definite sign that one should poeed with a degee of aution and a bit of skeptiism aises immediately upon seeing how easily the speial elatiity fomula an be deied. Refeing bak to Figue we ae eminded that distane D an ay fom + to as the illustated tiangle is otated to the left o to the ight as peiously disussed. It an be logially onluded theefoe that the midpoint between these extemes is that point whee the obsee is at an exat 90 degee angle to the path of motion of the soue. This assumption an then be ealuated in elation to the well eified fomula fo the staight-line-of-sight lassial Dopple effet fo waelengths to aie at anothe similaly unsientifi onlusion. That is, gien o e (6) Classial Line-of-Sight Dopple Effet fo Waelengths fo the staight-line-of-sight Dopple effet, whee λ e is the emitted waelength, λ o is the obseed waelength, and soue speed is + fo eession and fo appoah we an onlude the following: If has a alue of zeo elatie to the 90 degee midpoint between the two extemes gien by the staight-line-of-sight fomula, then simply by aying the alue of inesely with the angle of obseation, we an appoximate the Dopple effet on the obseed waelength. It just so happens that the tigonometi osine funtion lends itself ideally to this line of easoning sine it aies fom 0 to 1 as the angle aies fom 90 degees to 0 degees. Thus, by modifying equation (6) in the manne suggested, we deie o os o e (7) Speial Relatiity Classial Tansese Dopple Effet fo Waelengths whee, as in the ase of the peious fomulas, (5), (6), and (7), λ x must be substituted fo λ e to aie at the elatiisti esion. Thus we obtain o os o x (8) Speial Relatiity Relatiisti Tansese Dopple Effet fo waelengths fo the speial elatiity elatiisti tansese Dopple effet fo waelengths. Substituting λ x and λ e in plae of D and d espetiely in speial elatiity distane tansfomation fomula (17) gies 1 x e (9) Speial Relatiity Waelength Tansfomation (UF to SF) 1 fo the speial elatiity esion of the waelength tansfomation fomula. Substituting the ight side of this fomula in plae of λ x in equation (8) then gies 10

11 1 os o o e (30) 1 that simplifies to os o o e (31) 1 and finally 1 os o o e (3) Speial Relatiity Relatiisti Tansese Dopple Effet fo Waelengths 1 whee λ o is the obseed waelength, λ e is the emitted waelength, θ o is the angle of obseation, and the sign is + fo eession and fo appoah whee indiated. Upon seeing how easily the speial elatiity fomula an be deied, one an t help but wonde just how alid the fomula eally is. As will be seen late, suh skeptiism is well justified. 8. Final Consideations Een afte an in-depth analysis of all of the assoiated fatos, it an still be ey unlea as to what onstitutes a alid mathematial definition fo the tansese Dopple effet. In fat, the two opposing mathematial solutions deied in this wok, though quite aied in appeaane ae not eally as diffeent as they may fist appea to be. In the final analysis, howee, the oet solution depends on just exatly what onstitutes the oet angle of obseation. To detemine this haateisti in as unambiguous a manne as possible, it is neessay to ealuate the emission poess at the leel of the initial waelength emitted by the soue. In the ase of the new fomulas deied in this wok, i.e. equations (5, 6, and 7) and thei elatiisti foms (3, 4, and 5), it is the equialent of assigning a alue of unity to the aiable x. Refeing now to Figue 4, onside what happens as a soue moing at unifom speed emits a single waelength of light along its stationay fame path of tael while taeling fom point s to point e. If the emission poess begins at the instant the soue eahes point s while moing towad point e, the leading edge of the emitted wae will popagate away fom point s in all dietions at speed. Thus, enteed oe point s is the expanding sphee S epesenting the size of the popagating wae font at the instant the soue eahes point e, its pesent loation in Figue 4. In analyzing the oeall natue of the popagating wae and its elationships to aious points of obseation in the path of its expansion, shown as points 1 though 7 in the illustation, we an aie at the following onlusions: Beginning with point 1, whih has an obseation angle of zeo elatie to the path of motion of the soue, we hae the Dopple ondition 11

12 identified as staight-line-of-sight eession. In this ondition the Dopple effet on the popagated wae is popotional to alue of + (the ombined distane taeled by the light and the soue) as shown. At point, the angle of obseation is the small angle between the line-of-sight to the soue, designated by the aow D, and the path of motion of the soue, designated by the aow. In this ase, the Dopple effet, epesented by the aow D is somewhat less than +, but still a ondition of eession. At obseation point 3, the angle of obseation between the soue and the path of motion ineases a bit moe, and subsequently the Dopple effet epesented by the aow D is somewhat less eessionay than it was at point. At point 4, the angle of obseation has ineased onsideably oe what it was at point 3, and the Dopple effeted distane D is signifiantly less than it was at point 3, but still a ondition of eession. At point 5, the angle of obseation between the soue and its path of motion is at exatly 90 degees. Sine epesents the unaffeted distane taeled by the popagating wae, and in this ase the aow epesenting D is shote than the aow epesenting, the ondition would appea to be one of appoah. But in the stitest tehnial sense, that is not the ase, beause, at the instant the wae font eahes point 5, the soue is neithe eeding fom no appoahing that point. In fat, it is this point (pependiula to the soue along its path of tael in the stationay fame) that is used by Einstein in detemining the Loentz tansfomation fomulas in speial elatiity. With some thought, it should be lea that thee is no atual Dopple effet until the tailing edge of the wae eahes the obsee. Sine the tailing edge of the wae will popagate fom point e, it will not eah point 5 until the soue has moed on in its tael in a geneal dietion away fom point 5 thus suppoting the poposition that its pesent loation elatie to point 5 is neithe that of appoah no eession. 1 4 S D 3 5 D D D 6 D 7 s e (soue at e) + FIGURE 4 Classial Tansese Dopple Effet fo Initial Waelength At point 6 the ondition is undeniable that of appoah sine the wae font eahes that point befoe the soue passes it along its path of tael. In this ase, the angle of appoah between the pojeted path of motion of the soue and the Dopple effeted distane D is less than 90 degees wheeas the angle of eession between the path of motion of the soue and distane D is geate than 90 degees. And finally, at point 7 we hae the ondition of staight-line-of-sight appoah sine the angle of obseation between that point and the pojeted path of motion of 1

13 the soue is zeo degees. In this final ase the Dopple effet is popotional to the alue of sine the soue is appoahing the point of obseation. 9. The Disepany in Einstein s Teatment Reealed Now that we hae oeed all of the piniples inoled, it is a athe simple task to show the disepany in Einstein s mathematial teatment of the tansese Dopple effet. Refeing to Figue 5, the tiangle shown to the ight is fundamentally the same as that shown in Figue 3 and also fo point 5 in Figue 4 just disussed. It is the tiangle epesenting the Loentz tansfomation and subsequently establishes the definition of a 90 degee angle elatie to the path of motion of the soue, i.e. the angle between side L and soue path as shown. Yet when defining the 90 degee angle of obseation fo the tansese Dopple effet, Einstein, fo some eason, deided that it is the angle fomed between the obsee and the ente of the emitted wae along the path of emission tael by the soue. Still efeing to Figue 5, this is neithe the stat point of emission designated by point s, no the end point of emission e designated by the pesent loation of the soue, but at the midpoint between these loations. It is defined by extending a staight line dietly fom the point of obseation (whee and D inteset) to the path of motion of the soue, exatly midway between point s and point e pesently oupied by the soue. This is the point of intesetion between the dashed line, shown in the middle of the lage tiangle, and the path of motion of the soue. As an be eadily undestood, this line foms a 90 degee angle with the path of motion of the soue only when side D of the lage tiangle equals side, and subsequently angle of eession θ, fo the emitted wae, equals stat angle θ s. Reession D θ s θ s soue Appoah L FIGURE 5 Einstein s 90 Degee Relatiisti Tansese Dopple Effet Let it be lealy undestood hee, that thee is no doubt in the autho s mind that side D does in fat equal side unde the onditions shown, and subsequently thee will be no lassial Dopple effet at the indiated point of obseation, (the point whee and D inteset). In the autho s mathematial teatment of the tansese Dopple effet, howee, this ondition ous only when angle θ equals angle θ s, duing the emission poess of the initial wae and not unde the onditions inheent in Einstein s teatment. Quite lealy, angles θ and θ s annot be equal and still equal 90 degees unless thee is no motion of the soue. And quite lealy, thee annot 13

14 be two diffeent and onfliting definitions as to what onstitutes a 90 degee angle egading the light eeied in the stationay fame elatie to the path of motion of the soue. In spite of this disepany in Einstein s teatment, the esultant fomula still pefoms well if the distane to the obsee is suffiiently geate than that of the initial waelength. Thee ae two easons fo this. The fist inoles distane D. As the distane is ineased beyond that of the initial waelength, Dopple effeted distane D no longe makes up the entie side of the tiangle. And, the geate the distane to the obsee beomes, the less signifiant distane D beomes egading the total length of the espetie side. The seond eason inoles the width of the tiangle. If the soue is fa enough away, o its speed is low enough, the tiangle beomes so naow that side (x + D) is essentially paallel to side x, and subsequently the dashed line used in Einstein s teatment is also essentially paallel to side (x + D). Thus the angles fomed between eah of the two lines and the path of motion of the soue will be itually the same, een at the 90 degee angle of obseation point. As an be seen then, although appaently not peiously ealized, the distane to the soue is a fato egading the tansese Dopple effet. The poblem, howee, is a lot moe inoled than een this as will be disussed next. 10. Compaing the Results of the Two Diffeent Mathematial Teatments As just disussed, as the obseation tiangle beomes naowe, its two sides essentially mege to beome a single line tansending the distane between the obsee and the ente point at whih the obseed wae was emitted in the stationay fame along the path of motion of the soue. At that point, whee the two lines essentially mege, the esults gien by the new fomulas pesented in this wok mege with those gien by Einstein s fomulas. Unfotunately, the tansition between the two teatments is not smooth. Beyond the point whee the two teatments mege and the new fomulas gie itually idential esults to Einstein s fomulas, the new fomulas beome somewhat eati. This disadantage of the new fomulas is a diet esult of thei dependeny on a tiangle that is now beoming indisenible fom a staight line. Thus, Einstein s fomulas ae equied afte that point is eahed. In the othe dietion, howee, whee the distane to the soue deeases, it is Einstein s fomulas that beome uneliable, and the shote the distane the less eliable the esults. In egad to the naowing of the tiangle, not only is the distane to the obsee a fato, but as stated peiously, so is the speed of the soue. That is, the lowe the speed, the naowe the tiangle and the highe the speed, the wide the tiangle. Thus, the new fomulas beome moe auate as the speed ineases and Einstein s fomula beomes moe auate as the speed edues. But een this is not the end of the onsideations inoling the auay of the two diffeent appoahes. Sine waelengths effet the oeall size of the tiangle, the emitted waelength is also a fato in the auay of the fomulas as will be disussed next. Fo any gien speed of the soue, the fequeny of the emitted waes will detemine how fa the soue will tael befoe a single waelength is ompletely emitted. Sine the leading edge of the wae ontinues to tael outwad fom the point of emission s at speed, it will tael a geate distane towad the obsee as the soue eates a wide distane between points s and e due to its ineased tael time. Thus the oeall effet is a tiangle whose size is dietly popotional to the waelength and inesely popotional to the fequeny of the emitted waes. Theefoe, the useful ange of the new fomulas fo waelengths is also dietly popotional to the waelengths and inesely popotional to the fequeny of the emitted waes. And of ouse the inese is tue fo Einstein s fomulas with egad to the useful ange of distanes being disussed. 14

15 Fo light waes, een at speeds lose to the speed of light, this defining distane is ey small. In fat it is appoximately 17 metes fom the soue. This is the basi distane that esults when the aiable x in the new fomulas is assigned a alue of And fo waes at the shote end of the eletomagneti spetum this distane is shote yet. But fo waes at the longe end of the eletomagneti spetum, those inoling Infaed, Miowaes, and Radio and TV waes, the distane an be substantial. The exteme in this dietion inoles Long Radio waes whee the useful ange fo the new fomulas is extended beyond 300,000,000 km. Fo example, x Long Radio waes at 10 4 m in waelength = 316,7,766 km. Thus, the full impliations of these findings ae not yet ompletely undestood and it is unlea as to how long it will be befoe they will be known with any eal degee of etainty. This eelation egading the inauay of Einstein s fomula may at least help to explain the well known diffiulties expeiened in past attempts to eify the peditions of Einstein s fomula as the angle of obseation ineases fom 0 to 90 degees. Aoding to Einstein s fomula, thee is no lassial Dopple effet at the 90 degee angle of obseation and onsequently thee will only be the edshift pedited by the Loentz fato. To the autho s knowledge, no suh shift has ee been dietly deteted in laboatoy expeiments 3. This failue to detet any Dopple shift at the 90 degee angle of obseation is fundamentally in ageement with the peditions of Equation (5) fo distanes that appoah a single waelength and thus poides suppot fo the findings pesented hee. In aodane with the theoy pesented hee, at the 90 degee angle of obseation (whee angle of eession θ and angle of appoah θ a both = 90 deg.) theoetially thee will be a lassial Dopple effet in the fom of a blueshift. Sine, howee, at that 90 degee angle of obseation inoling the initially emitted waelength, the tiangle epesenting the lassial Dopple effet is exatly supeimposed oe the tiangle epesenting the Loentz edshift effet, the two opposing effets exatly anel eah othe out. Thus, unlike speial elatiity s pedition of a tansfomation edshift, at the 90 degee angle of obseation, aoding to the findings pesented hee, thee will be no shift at all at a distane of one waelength and the obseed waelength will theefoe equal the emitted waelength of the soue. Nonetheless, as the distane to the obsee ineases beyond a single waelength, fo the easons peiously disussed the lassial Dopple effet at an obseation angle of 90 degees will quikly dop to zeo and only the Loentz edshift will be obseed. Een so, beause of the genealized natue of Einstein s fomula, only the esults gien by the new fomulas intodue in this wok ae tuly auate within the useful ange of distanes peiously disussed. To do a diet ompaison between the two diffeent fomulas teated in this wok it will be helpful to hae a means of ompaing them at the point whee the angle of obseation θ of the new fomulas is equal to the stat angle θ s. This allows diet ompaison to Einstein s fomula at its impopely defined 90 degee angle of obseation. The equied fomulas fo detemining this point (whee θ = θ s ) fo the new fomulas ae easily deied using the law of osines theoem. Refeing bak to Figue, the tiangle epesenting the lassial tansese Dopple effet will hae the angle of eession θ equal to stat angle θ s when side (x + D) equals side x. Thus, using the law of osines fomula that ontains angle of eession θ we an easily deie fomulas that will tell us at what angle θ, (when is known) o soue speed, (when θ is known) will side (x + D) equal side x. That is, by taking peiously deied Equation (1) and substituting x fo (x + D), we obtain ( x) ( x) ( x) os (33) 15

16 that an be soled fo θ to obtain ( x) os 0 (34) ( x) os (35) os ( x) (36) os (37) x and finally aos x (38) Fomula fo Angle of Reession θ = Stat Angle θ s fo Waelengths whee the angle gien fo the angle of eession θ an be used in waelength Equation (5) to obtain the ondition whee θ = θ s, and subsequently (x + D) = x. In soling Equation (37) fo speed we obtain x os (39) Speed fo Angle of Reession θ = Stat Angle θ s fo Waelengths whee fo any angle of eession θ we an detemine the soue speed at whih θ = θ s, and (x + D) = x. Thus, with these two fomulas we an easily detemine the alues needed in waelength Equation (5) to obtain the esults that ae to be ompaed to those of Einstein s fomula fo its impopely defined 90 degee angle of obseation. This ompletes the analysis. Appendix A is inluded at the end of the pape showing a aiety of diet ompaisons between the new tansese Dopple effet fomulas and those of speial elatiity. Many suh models wee used by the autho in eifying the laims and findings pesented in this wok. 11. Conlusion The newly deied fomulas fo the elatiisti tansese Dopple effet pesented in this wok wee shown to be in stit ageement with the fundamental piniples of speial elatiity. It is inoneiable theefoe that suh findings ould be hallenged without hallenging the fundamental basis of the speial theoy of elatiity itself. At the same time it was shown in equally onining tems that Einstein s teatment fo the tansese Dopple effet annot be entiely alid. Hee the esulting fomula iolates the piniples upon whih the Loentz tansfomations ae pediated and subsequently the speial theoy of elatiity is founded. In spite of this, it was shown that the esulting fomula etains a alid basis in suppot of its ontinued use as long as its limitations ae aknowledged and not exeeded. Ionially the new fomulas whih ae shown to be in full ageement with the piniples of elatiity ae also shown to hae patial limitations that goen thei ange of alid appliations. Subsequently, it 16

17 appeas that both teatments ae needed to popely oe the entie ange of appliations that exist fo the tansese Dopple effet. Although obiously speulatie in tems of a sientifi poposition, pehaps the eal disoey hee is one onening the oelation of quantum theoy with elatiisti theoy. Sine the new fomulas eplae the uent fomulas at the initial waelength sale, they may poide links between the two theoetial appoahes that wee peiously unknown. Thus, this would appea to be an aea of inteest fo futue inestigations. Also of inteest would be the deiation of a single unified fomula fo the elatiisti tansese Dopple effet to eplae the two that ae needed as a esult of this uent wok. But een moe impotant than these onsideations ae those inoling the uent use of the tansese Dopple effet fomulas in egad to ountless patial and sientifi appliations. Viewing what was shown hee in the pespetie of the entie eletomagneti spetum it is unlea what the total impat might be. Theefoe it would appea that a etain amount of ugeny egading the distibution of these findings is in ode. Appendix Inluded as Appendix A is a opy of one of the mathematial models eated in Mathad by the autho and used to eify the findings pesented in this wok. REFERENCES 1 Joseph A. Rybzyk, Relatiisti Tansese Dopple Effet, (005), Reised, (006) Unpublished Wok Albet Einstein, Speial Theoy of Relatiity, (1905), oiginally published unde the title, On the Eletodynamis of Moing Bodies, in the jounal, Annalen de Physik 3 Hatwig W. Thim, Absene of the Relatiisti Dopple Shift at Miowae Fequenies, (00), Einstein s Tansese Dopple Effet Poen Wong The Complete Poof Copyight 006 Joseph A. Rybzyk All ights eseed inluding the ight of epodution in whole o in pat in any fom without pemission. 17

18 Appendix A Einstein s Tansese Dopple Effet Poen Wong Relatiisti Tansese Dopple Effet Wae Theoy.md Febuay 1, 006 Joseph A. Rybzyk Speed e f e f e km/h e 1000 MR Relatiisti Tansese Dopple Effet - is Angle of Reession, a is Angle of Appoah, and o is Angle of Obseation in deg. 90 deg a 180deg a 90deg x 10 0 D x os os x x os e x os a a e Line-of- Sight gien at ight l e os x os a x la e D x e x e 1000 l Distane to obsee km x m e ( x ) ( x D) s aos x ( x ) ( x D) p aos x ( x D) la Angle of Pojetion (0-0-0 deg) Angle of Stat (0-180 deg) Angle of Reession (0-180 deg) Angle of Appoah (0-180 deg) p deg s deg 90deg a 90deg Emitted Tansfomed Obseed Re Obseed App e x a MR fomulas fo finding when is gien, o fo finding when is gien to make = s and to make side (x-+d) = side x Set to x to make = s Set to x to make = s x aos x x deg x x os x 0 Speial Relatiity Fomulas Fomulas good fom 0 to 180 deg. Speial Relatiity fomula efeened in MTR SR Reession ( = 0-180) SR Appoah (a = 0-180) os E e 1 os E e 1 os a Ea e 1 os a Ea e 1 90deg a 90deg E E Ea Ea f E3 f e 1 os f E E3 f E3 E a 1 a f Ea3 f e 1 a os a f Ea Ea3 f Ea3 Ea

19 Appendix A Einstein s Tansese Dopple Effet Poen Wong Relatiisti Tansese Dopple Effet using Angle of Reession Speed/Soue x Fato Dist/Obsee m m.9 x e x 10 4 Fo = 80,000km/s use x e x e Febuay, 006 Joseph A. Rybzyk MR Emitted e MR Tansfomed x e x Reession Angle MR fomula SR Fomula MR Obseed o SR Obseed Ea & E 0deg x os o e 30 deg x os o e 60 deg x os o e 90 deg x os o e 10 deg x os o e 150 deg x os o e 180 deg x os o e 180 deg x os o e os x os x os x os x os x os x os x os x os E e os E e os E e os E e os E e os E e os E e os E e o Compaison o / E o Compaison o / E o Compaison o / E o Compaison o / E o Compaison o / E o Compaison o / E o Compaison o / E o Compaison o / E E o E E o E E o E E o E E o E E o E E o E E o E 19

20 Appendix A Einstein s Tansese Dopple Effet Poen Wong Relatiisti Tansese Dopple Effet using Angle of Reession Febuay, 006 Joseph A. Rybzyk Reesson Angle x Fato Dist/Obsee m m MR Emitted e MR Tansfomed x 90 deg Diffeent fo eah. Gien e x x by SR fo =90 deg, x=1 e x e Soue Speed MR Fomula SR Fomula MR Obseed o SR Obseed Ea & E 0 x os os x os o e E e o E x os os x os o e E e o E E x os os x os o e E e o E x os os x os o e E e o E x os os x os o e E e o E x os os x os o e E e o E x os os x os o e E e o E x os os x os o e E e o E x os os x os o e E e o E x os os x os o e E e o E x os os x os o e E e o

21 Appendix A Einstein s Tansese Dopple Effet Poen Wong Deiation of fomula fo lassial tansese Dopple distane d using Law of Cosines deg os 0.9 Whee d ( x D) ( x ) d dos os os os x os x 1 1 d os ( x D) os D x os x os os x os x D D f Deiation of fomula fo lassial tansese Dopple eession using Law of Cosines D f D x os os x x os e os x Deiation of fomula fo elatiisti tansese Dopple eession using Law of Cosines x os e os x x os e os x Deiation of fomulas fo lassial and elatiisti tansese Dopple appoah using Law of Cosines Change sign of to - whee shown and hange to a and to a x os a os a x x os a a e a e os a x Deiation of fomula fo lassial and elatiisti tansese Dopple Stat Angle using Law of Cosines ( x D) ( x ) ( x ) os s ( x ) os s ( x D) ( x ) ( x ) os s ( x ) ( x D) os s ( x ) ( x D) ( x ) s ( x ) ( x D) aos ( x ) Deiation of fomula fo lassial and elatiisti tansese Dopple Pojetion Angle using Law of Cosines ( x ) ( x D) ( x ) ( x D) os p ( x ) ( x D) os p ( x ) ( x D) ( x ) ( x D) os p ( x ) ( x D) os p ( x ) ( x D) ( x ) ( x D) p ( x ) ( x D) aos ( x ) ( x D) 1

22 Appendix A Einstein s Tansese Dopple Effet Poen Wong Deiation of fomula fo Angle of Reession to equal Stat Angle s Substituting fo x, and d fo (x-+d) we hae fom the Law of osines, d os s If s Then d Theefoe the aboe an be estated as os o whih an be soled fo o to obtain os s os s os s giing os s and s aos Substituting x bak in plae of gies x os s and s aos x fo final esions. Einstein s Tansese Dopple Effet Poen Wong The Complete Poof Copyight 006 Joseph A. Rybzyk All ights eseed inluding the ight of epodution in whole o in pat in any fom without pemission. Note: If this doument was aessed dietly duing a seah, you an isit the Millennium Relatiity web site by liking on the Home link below: Home