Millennium Theory of Relativity

Transcription

1 Millennium Theory of Relaiiy Copyrigh 001 Joseph A. Rybzyk Absra The Millennium Theory of Relaiiy is a funamenal heory in relaiisi physis. Through mehoial analysis of he eiene, srong an onining argumens are eelope in opposiion o hose of speial relaiiy. Speifially, he eiene supporing he onsany of ligh spee is aepe, an wih i he self-eien proposiion ha he Laws of physis are he same in all inerial sysems. Ye, i is shown ha he apparen relaionship of hese wo eniies anno be properly reonile wihou he inrouion of spherial referene frames. Only hen, an a orre unersaning of relaiisi priniples be ahiee, an his unersaning is subsanially ifferen from ha preiously arrie a. The eoling priniples of he new heory will learly emonsrae ha many of he assumpions of he presiing heory are unenable. Alhough i will be shown ha ime is affee by relaie moion, oher assoiae effes are ifferen from hose of urrenly hel beliefs. Disanes in spae for example, an he size of objes i onains, are no really affee by suh moion. In fa, he priniple of ime ariane is oningen upon preisely ha oniion. Whereas aual isanes are unaffee, howeer, he isane ligh raels in an ineral of ime, is affee, an his effe on isane is no limie o he ireion of moion. The shrinking isane raele by ligh wihin a moing frame of referene ours equally in all ireions. This hange of iew, is onsisen wih he preailing eiene, ye has a profoun bearing on he way he physial laws of naure are pereie. Anoher onsequene of he uneraken analysis is he realizaion ha he Lorenz formulas are no ire represenaions of he priniples of relaiisi behaior. Een hough hese formulas yiel orre mahemaial resuls, by he ery naure of heir inireness hey en o be misleaing. I is onene, ha hese formulas ogeher wih he reangular referene frames hey are normally assoiae wih, le o he misunersanings emboie in speial relaiiy. I is furher argue, ha only by use of he spherial referene frames an he irely erie equaions eelope in he presene analysis, is a proper unersaning possible. A final onsequene of he ensuing analysis in his presen work is a somewha ifferen iew of he relaiisi, ranserse, Doppler effe. Comparison of he ifferen resuls reeals ye anoher possible flaw in he presiing heory. This is no o say ha aiional flaws migh no surfae in a fuure supplemenal work inoling mass an energy. 1

2 Millennium Theory of Relaiiy Conens 1. Inrouion 4. The Unerlying Posulae 4 3. The Spee of Ligh 6 4. Consany of Ligh Spee beween Naural Sysems 6 5. The Transformaion Proess 7 6. Deeloping he Transformaion Formulas 9 7. The Time Transformaion Formulas The Disane Transformaion Formulas 1 9. The Flaw in Speial Relaiiy Proing he Funamenals Effes on Waelengh an Frequeny 7 a. The Transformaion Effe 8 b. The Doppler Effe 30. The Combine Transformaion an Doppler Effe Equaions 3 1. Doppler Effe for Transerse Moion Correlaion of All Effes Conlusion 45 Referenes 46 Equaions 1. Time in he moing frame 11. Time in he saionary frame 1 3. Disane in he moing frame Raius of saionary sphere Raius of moing sphere Disane in he saionary frame Equaliy of isane raios Frequeny o waelengh relaionship 8 9. Waelengh o frequeny relaionship Waelengh ransformaion, MF o SF Waelengh ransformaion, SF o MF 9 1. Frequeny ransformaion, MF o SF Frequeny ransformaion, SF o MF Doppler effe Obsere waelengh Doppler effe Transforme waelengh Doppler effe Obsere frequeny Doppler effe Transforme frequeny Obsere waelengh Obsere frequeny Emie waelengh 33

3 1. Emie frequeny 33. Obsere waelengh, Transerse moion Obsere frequeny, Transerse moion Obsere waelengh, Transerse moion, Alernae equaion 4 5. Obsere frequeny, Transerse moion, Alernae equaion 4 Figures 1. The Transformaion Proess 8. Deeloping he Transformaion Formulas 9 3. The Disane Transformaion Formulas Shows how all isanes are affee by moion Uneils he flaw in speial relaiiy 6. Proof of he Funamenals 5 7. The Dire Meho 5 8. The Doppler Effe Combine Transformaion Proess an Doppler Effe Doppler Effe, Reession Mah Definiion of Doppler Reession Doppler Effe, Approah Mah Definiion of Doppler Approah Correlaion of All Effes 43 3

4 Millennium Theory of Relaiiy Copyrigh 001 Joseph A. Rybzyk 1. Inrouion The Millennium Theory of Relaiiy is he resul of more han four years of inense researh inoling ligh-spee phenomena. I is an inepenen effor, relying only on he analysis presene here, an ha experimenal an obseraional eiene ha shows ligh o hae a finie alue, ha is onsan relaie o a saionary frame of referene, an oes no appear o be affee by he moion of he soure. Conersely, i oes no rely on any heoreial assumpions, or heories of he pas or presen. This inlues Einsein s speial heory of relaiiy 1 an any heoreial assumpions upon whih ha heory is foune. Speifially, he heory presene here, aeps he eiene of he Mihelson an Morley experimens of 1887, e Sier s 3 experimens inoling binary sars, an experimens showing ha ligh gien of by subaomi pariles moing a near ligh-spee is unaffee by he moion of suh pariles 4. Also onsiere is he apparen inrease in he life expeany of μ mesons raeling a near ligh spee in he earh s amosphere 5. Of ourse, all experimenal eiene esablishing he finie alue of is also aepe, an aknowlegemen is gien o he many onribuors inole 6. Asie from hese aknowlegemens, he heory presene here relies solely on heoreial assumpions arrie a hrough he heory iself. Alhough exlusiiy of some of hese assumpions anno be laime, i is laime ha only wihin he presene heory are hese assumpions properly unersoo an orrely applie. Consequenly, i will also be shown ha he speial heory of relaiiy is flawe o he exen of inaliaing iself. The eiene o ha effe is lear an oerwhelming an is presene here in he orer i was isoere. In summary, he Millennium Theory of Relaiiy proies a new way of iewing he physial laws of naure. I is a heory ha makes sense ou of naural laws ha preiously efie rue unersaning. In so oing, i proies learer insighs ino he workings of hose naural laws, hereby opening he oor o new opporuniies of isoery. I is imporan o noe here, ha he Millennium Theory of Relaiiy begins where he Theory of Naural Moion, a preious paper by his same auhor, leaes off. The Theory of Naural Moion, opyrighe in 1996, an publishe on MighyWors.om in Oober of 1999, hough no a prerequisie o his presen work, will noneheless benefi he reaer in unersaning many of is oneps. A final noe: The Millennium Theory of Relaiiy is presene here in he same orer in whih i was eelope. Iniially an aemp is mae o unersan he naure of ligh rael an subsequen, relae ligh-spee phenomena. Shorly afer beginning, he priniples for he presen heory begin o beome apparen an are seize upon o eelop he final omprehensie heory on is own.. The Unerlying Posulae As preiously isoere in he Theory of Naural Moion, he laws of naure are simple, preiable, an few, an he ruh lies inwar, no ouwar. Suh appears o be he ase inoling ligh-spee phenomena. Whereas in he preious work i was foun ha Newon s hree laws of moion oul be ombine ino a single omprehensie law, i is foun here ha a single posulae is suffiien o explain all physial phenomena relae o ligh spee. This is in onras o he wo posulaes pu forh in Einsein s speial relaiiy. Essenially, wha is 4

5 isoere here is ha he spee of ligh relaie o he obserer (efine in Einsein s firs posulae) is no an inepenen onsieraion, bu raher a onsequene of he laws of physis appliable o all inerial sysems (reae in Einsein s seon posulae). Tha is, we are no posulaing here ha he spee of ligh in empy spae has he same alue in all inerial sysems. We are aeping he eiene erie hrough sienifi obseraions an experimens ha seem o erify suh, an hen proee wih he proess of eermining how he physial laws of Naure ause his o be apparenly rue. The posulae upon whih he presen heory is foune is simply his: Posulae: The laws of physis are he same in all naural sysems. We refrain from he empaion o oninue use of he phrase inerial sysems in he aboe posulae beause i was isoere in he Theory of Naural Moion ha hese erms in ha usage are ehnially inorre. Neerheless, he phrase, naural sysem refers o a, naural frame of referene or simply, naural frame an an be hough of as haing he same general meaning as he phrase, inerial sysem or, inerial frame of referene. Thus, a naural frame of referene is efine as a frame of referene aahe o an obje or boy in a sae of res, or a sae of uniform moion, or more orrely, an obje or boy no uner aeleraion. (As poine ou in he preious heory, a sae of res an a sae of uniform moion are one an he same hing.) The priniple issue here is ha he aboe posulae means ha he haraerisis of all physial phenomena irely assoiae wih one naural sysem, will be ienial, uner he same oniions, o he haraerisis of he same physial phenomena assoiae wih any oher naural sysem. Pu simply, uner he same oniions, any sienifi experimen will yiel he same resuls in any naural sysem, an any obseraion, or relaionship foun ali in one naural sysem will be foun ali in any oher naural sysem. This inlues he spee of ligh, or any oher form of eleromagnei propagae energy. Suh iew is onsisen wih he proposiion ha no experimen onue on an obje in uniform moion will gie eiene of ha moion. Whereas he haraerisis efine aboe are he same in all naural sysems, his oes no mean ha suh haraerisis are irely ransferable from one naural sysem o anoher. As we proee, i will be shown ha hese haraerisis are affee by seeral faors. Those faors inlue he relaie moion beween frames, graiy, an meium of ligh rael. Of hese, he relaie moion beween frames is he mos imporan beause i affes he resuls of many experimens an alulaions, ye is no oere by he qualifying sipulaion, uner he same oniions. Graiy an meium of ligh rael, howeer, hough possibly haing effes similar o ha of relaie moion, are he priniple reasons for he qualifying sipulaion. On he one han, relaie moion will affe how physial phenomena in one naural sysem is pereie in anoher naural sysem regarless of he fa ha he wo sysems are ienial. On he oher han, graiy an meium of ligh rael an also hae an effe on how physial phenomena is pereie, bu is hanle ifferenly han relaie moion. For example, graiy will affe how physial phenomena is relae beween wo large boies of ifferen masses, wiely separae in spae, (sars an planes for insane) een if hey resie in he same naural sysem (are no moing relaie o eah oher). Moreoer, sine he spee of ligh is known o be affee by he meium of rael, gases an oher properies in spae an also hae an effe ha mus be aken ino onsieraion. 5

6 3. The Spee of Ligh I is uniersally aepe ha he spee of ligh, esignae by he symbol has a finie spee in empy spae of m/s. The qualifier here is, empy spae, for i is also uniersally aepe ha he spee of ligh is affee by he meium hrough whih i raels, an aiionally in a manner by graiy. The imporan onsieraion here, howeer, is ha uner he same oniions, he spee of ligh has he same alue in all naural sysems. These wo faors, finieness an onsany relaie o all naural sysems make i possible o use ligh as a means of measuring isanes wihin any naural sysem. This haraerisi of ligh will be mae use of laer in his paper. 4. Consany of Ligh Spee beween Naural Sysems No as uniersally aepe is he proposiion ha ligh in empy spae has a onsan spee in all inerial sysems regarless of he spee of he soure. (Einsein s firs posulae in speial relaiiy) Alhough his proposiion appears o be rue, an has general aepane hroughou he sienifi ommuniy, here are noneheless, hose ha isagree. I has ofen been sai of speial relaiiy ha he problem lays no so muh in being able o unersan i, bu in being able o beliee i. Afer haing sruggle wih i for oer four years, my onlusion is, here is ruh on boh frons. Perhaps he bigges failing of speial relaiiy is is inabiliy o explain how he spee of ligh an be onsan relaie o he obserer, in a manner ha an be boh unersoo an beliee. An, sine his is he basis on whih he priniples of ime ilaion, lengh onraion, an mass an energy inreases are foune, hose priniples are equally iffiul o unersan an beliee. Wih his paper, I hope o hange ha. The firs problem wih speial relaiiy is he proposiion ha ligh oes no ake on he spee of he soure. In he Theory of Naural Moion, i is shown ha physial objes o no experiene moion. Moion, i was poine ou, is no somehing an obje oes, bu raher, a relaionship beween objes in ifferen naural saes. A naural sae was aoringly efine as neiher a sae of res, nor a sae of moion, bu aually a sae in whih here are no aeleraion fores working on he obje. Moreoer, aeleraion was shown no o be a form of moion a all. Sine an obje oes no experiene moion, i is meaningless o speulae ha ligh eiher oes, or oesn ake on he spee of he soure. The soure has no spee. Spee is a relaional onsieraion beween masses in ifferen naural saes. The aboe efiniions nowihsaning, i is ofen onenien o aribue a sae of moion or res o an obje, or frame of referene, in orer o simplify an make easier o unersan, ha whih is being oneye. Suh liberies will be aken hroughou his paper. From he frame of referene of he soure hen, ligh mus ake on is spee, oherwise he spee of ligh woul be affee by he spee of he soure, an we know from he Mihelson an Morley experimens of 1887, an many oher experimens onue sine, ha his is no rue. (If a soure of ligh were hough o be moing hrough spae, hen he ligh woul hae o moe along wih i o mainain a onsan spee relaie o ha soure.) From he poin of iew of a saionary obserer (an obserer in a frame ha he soure is moing relaie o) he ligh oming owar him or her, oes no appear o hae aken on he spee of he soure, oherwise he ligh woul be slower or faser han epening on he ireion of moion of he soure. This haraerisi of ligh spee is also suppore by eiene, inluing ha base on obseraions of binary an oher sars, an also on measuremens mae on ligh from subaomi pariles ha are aelerae o near ligh spee in parile aeleraors. 6

7 Sine ligh has a onsan spee relaie o a moing soure, ye somehow has ha same spee relaie o a saionary obserer, i is inorre o say ha he spee of ligh is unaffee by he relaie moion beween he soure an he obserer. I mus be affee, oherwise as sae preiously i woul appear o hae aken on he spee of he soure relaie o he obserer in he saionary frame of referene. Tha is, ligh appears o somehow ajus is spee in inerse proporion o he spee of he relaie moion beween he soure an obserer. Pu simply, while ligh appears o mainain a onsan spee relaie o he soure, as iewe by an obserer in he referene frame of he soure, i appears o hange is spee relaie o he soure, as iewe by an obserer in a saionary frame of referene. To his saionary obserer, he ligh mus hae eiher speee up or slowe own in inerse proporion o he spee of he soure, in orer o be reeie by ha obserer a spee. To resole his apparen onraiion, le us now examine he enire proess in eail. 5. The Transformaion Proess Referring o figure 1A, assume ha poins A, B an C, a he hree orners of he riangle, are poins in a saionary frame of referene. Assume furher ha a poin ligh soure, moing a a uniform spee, gies off a pulse of ligh a Poin A as i moes o poin B. An finally, assume ha a poin of his ligh moing a spee in a perpeniular ireion relaie o he pah of rael of he moing soure, reahes poin C a he insan he soure reahes poin B. In unison, his same poin of ligh will rael from poin A o poin C in he saionary frame of referene. Sine ligh oes no ake on he spee of he soure in he saionary frame of referene, i emanaes ouwar in all ireions from poin A a spee. Thus, i forms a perfe sphere enere on poin A wih isane A-C as is raius as illusrae in figure 1B. The ligh in he moing frame of referene, howeer, emanaes ouwar from he moing soure a spee, wih he soure as is ener as illusrae in figure 1C. This perfe sphere of ligh has he isane B- C as is raius relaie o he moing frame of referene. Aeping he eiene ha nohing an exee he spee of ligh, he ligh soure in he aboe example an neer esape from wihin he sphere of ligh (sphere 1) emanaing from poin A in he saionary frame of referene. Tha is, he sphere will always be growing a a faser rae han he soure an rael. A any insan in ime hen, he soure will always be loae a a poin along he pah of rael wihin he sphere of ligh eien in he saionary frame of referene. (The fa ha he smaller sphere (sphere ) inerses poin A is oinienal. A slower or faser moing soure woul proie a larger or smaller sphere ha woul no inerse poin A.) Of signifiane in figure 1C is he fa ha he poin of inerseion of he wo spheres exaly oinies wih he perpeniular axis of he sphere of ligh enere on he soure. We an now raw on his geomerial relaionship of he wo spheres an formalize ha relaionship mahemaially. Before proeeing, howeer, i will be useful o ake a momen o lear up a few areas of poenial onfusion. I was inenional ha a saionary obserer was no use in he aboe explanaion. While use of suh an obserer is ofen helpful wih regar o explaining he oneps jus oere, he loaion of he obserer an ofen ause onfusion ye has no bearing on he resul. In fa, he obserer an assume any posiion a all wihin he saionary frame of referene wihou affeing he ouome. Tha is o say, i oesn maer if he soure is moing away from he obserer, passing by he obserer, or moing owar he obserer. The only hing ha maers is ha he 7

8 obserer oupies he same frame of referene as poins A an B, in orer for he resuls o be ali wih regar o ha obserer. I is he relaionships he soure esablishes wih poins along he pah of rael in he saionary frame ha maer, an no he loaion of an obserer wihin ha frame. I shoul also be noe ha he oneps jus isusse an hose sill o be oere apply o all oher frames of referene simulaneously. I is only he speifis ha will ary, no he funamenal oneps. I.e., he soure simulaneously esablishes similar relaionships in all oher frames of referene. Subsequenly, if he soure has a ifferen spee relaie o eah ifferen frame, he speifi resuls will ary from one frame o anoher. Sphere 1, Ligh in Saionary Frame C C A B Ligh Soure A Ligh Soure B A Sphere 1, Ligh in Saionary Frame B C A Ligh Soure B Sphere, Ligh in Moing Frame C FIGURE 1 The Transformaion Proess I shoul also be noe, ha here are no aually wo spheres of ligh as illusrae in he preeing figure. There is only one sphere of ligh, bu i exiss in a ifferen form for eah ifferen frame of referene. Thus, he relaionships being isusse inole omparing he form of he sphere as i appears in he moing frame o he form as i appears in he saionary frame. I mus be emphasize here, ha only he sphere of ligh enere on poin A is experiene in he saionary frame. Thus, in he saionary frame, sphere an be arrie a only hrough mahemais. The onerse is rue for he moing frame. Only sphere is experiene in ha 8

9 frame of referene, an sphere 1 woul hae o be arrie a mahemaially. In his ase, he relaionship is reerse, an sphere is he larger of he wo spheres. One final poin nees o be aresse here. Sine he spee of he ligh in he saionary frame has a spee as erifie by he eiene, i is reasonable o assume ha along he pah A- B i has a spee + relaie o he soure as experiene in he saionary frame. This gies eiene ha spee in he moing frame is slower han spee in he saionary frame. This is also onfirme by he fa ha sphere is smaller han sphere 1. Our posulae an ommon sense, howeer, iae ha spee a he soure mus equal spee in he saionary frame of referene. This gies rise o he assumpion ha some sor of ransformaion mus exis beween he wo frames, an issue o be aresse in he nex seion. 6. Deeloping he Transformaion Formulas We are now reay o eelop he mahemaial relaionships ha efine he ransformaion proess. Referring o figure, we an see ha a Pyhagorean relaionship exiss beween he wo forms of he ligh sphere being isusse. This relaionship an be exploie an inegrae wih anoher relaionship inoling ime, isane an rae o arrie a mahemaial equaions ha fully efine he ransformaion proess. Sphere 1 C Sphere = ( ) ( ) A 1 = B FIGURE Deeloping he Transformaion Formulas In aorane wih he equaion ha relaes ime o isane an rae, we obain, ha gies = r for isanes, where = ime, = isane, an r = rae of spee. Thus, in figure, we an express he isane raele by he soure beween poins A an B in he saionary frame as 1 =, where = he spee of he soure in he saionary frame. An sine he ligh emanaing from poin A has a spee in he saionary frame, he isane he ligh raele o r 9

10 poin C uring he same perio in he saionary frame an be expresse as =. I will be reognize ha he isane beween poin A an poin C represens boh he raius of sphere 1, an he hypoenuse of he righ riangle ha inlues he horizonal line A-B, an he erial line B-C. I will also be reognize, ha sine he erial line efines he isane raele by he ligh in he moing frame, i represens he raius of sphere in aiion o one sie of he righ riangle. These fas hae grea signifiane ha will be isusse laer. For now, i is enough o know ha he riangle is a ali righ riangle beause he line B-C represens ligh raeling in a perpeniular ireion from he pah of moion of he soure. Sine we now hae a way of relaing isanes in he saionary frame o isanes in he moing frame, we an efine he isane along he erial line in he moing frame in erms ha are ali in he saionary frame. Thus, he isane in he moing frame an be expresse as, ( ) ( ). Before proeeing furher, we mus eie wha i is we hope o fin. Referring o figure, we an see ha he ligh raels a greaer isane in he saionary frame han i oes in he moing frame. Moreoer, sine i has been shown, an is reasonable o assume, ha boh spheres were forme simulaneously, i follows ha he spee he ligh raele in he moing frame mus be slower han he spee i raele in he saionary frame. This of ourse has alreay been borne ou in our analysis. I was eermine ha, relaie o he soure is a slower spee han relaie o he obserer. This is a ery profoun obseraion, sine i implies ha ime relaie o a moing soure mus be slower han ime relaie o a saionary obserer. Bear in min, ha if our posulae is orre, an if an obserer were o rael o he soure, he woul fin no ifferenes here. Any measuremens he makes wih equipmen brough along, woul show, isanes, ime an he spee of ligh here o be he same as he experiene in he saionary frame. Any ifferenes ha were experiene in he saionary frame hen mus be a resul of he relaie moion of he soure. I follows herefore, ha if he soure s spee of ligh is slower, eeryhing else on he soure mus be slower. This an only mean one hing; ime is slower in he referene frame of he soure. We now know wha we shoul be looking for. We nee a way o eermine he relaionship of ime in he soure s frame o ime in he saionary frame. 7. The Time Transformaion Formulas Going bak o he ime, rae, an isane formula, an using he onenion ha symbols marke prime represen alues in he moing frame we ge, ' '. r' Haing alreay esablishe ha an be represene by, ( ) ( ), 10

11 11 an gien, ha has he same alue in he moing frame ha i oes in he saionary frame, by way of subsiuion in he ime, rae, an isane formula, we ge, The resuling equaion an now be simplifie ino a more usable form as follows: Using only simple algebra we hae inepenenly arrie a a ime ransformaion equaion ha proies he exa same answers as he equaion use in speial relaiiy, wihou relying on eiher ha heory, or he Lorenz formula. The resulan equaion relies only on presenly aepe sienifi eiene an he premises of he presene heory. Obiously, if his equaion gies he exa answers as Relaiiy s equaion, i mus be anoher form of ha equaion. Saring again wih, an some aiional manipulaion we ge,. ) ( ) ( ) ( ) ( ) ( Eq. (1) Time in he moing frame., ) ( ) (

12 1 Anyone familiar wih speial relaiiy will immeiaely reognize he final form of his equaion. I is he ime ransformaion equaion use in ha heory, bu arrie a here inepen of any onsieraions inoling ha heory. A his poin, howeer, our only onribuion is ha of presening a muh learer, an belieable argumen ha ime is affee by moion. Tha will hange soon enough, bu firs we will ake a momen o eelop he ransformaion equaion for. Saring wih equaion 1, by manipulaion we ge, Doing similar wih he oher form of he equaion gies us, Eq. () Time in he saionary frame. Eq. (1)

13 This of ourse is he alernae form of he relaiiy equaion, bu arrie a here inepenenly. 8. The Disane Transformaion Formulas Before proeeing, i migh be helpful o erify ha isane in he preeing example has he same alue in boh frames of referene. Referring o figure 3, i an be seen ha when isane is expresse in erms relae o he soure, he isane is expresse as r. In he preeing example we expresse his isane in erms relae o he saionary frame. We will now reerse he proess an show ha =. Sphere 1 C Sphere = & ( ) ( ) A 1 = B r Saring wih he equaion, FIGURE 3 The Disane Transformaion Formulas r, we an subsiue for r beause we know ha his is he spee of ligh relaie o he moing soure in he moing frame of referene. An sine, Eq. (1) we an subsiue he righ sie of his equaion for in he firs equaion o ge,. 13

14 14 This in urn an be simplifie an manipulae o ge, whih is he same expression we ha for he saionary frame. Doing similar using he more familiar alernae equaion, leas of ourse, o he same resul. Again, we arrie a he same expression we ha for he saionary frame. This resul is expee sine i is base on he ransformaion formula whih was arrie a in similar fashion. The main poin of he exerise was o emphasize ha he isane B-C in figure 3 is he same for boh, he saionary frame of referene, an he moing frame of referene. This is always rue, regarless of he spee of he soure. Tha is, his isane will ary from o 0 in ire relaion o he soure s spee of 0 o respeily, in boh frames of, 1 1 ) 1 ( ) ( ) ( ) ( ) ( ) ( ) (, ) ( ) (

15 referene. We mus also remember ha moion is relaie, an eiher frame in our example an be hough of as being a res. From ha perspeie, he raius of he smaller sphere, is equal o he raius of he larger sphere when eah are measure in he frame o whih hey belong. Uner hose oniions, eah frame is onsiere o be a res, an isane B-C in figure 3, will equal isane A-C. This plaes in quesion he speial relaiiy assumpion ha only horizonal isanes in he ireion of moion are affee by ha moion. Moreoer, as we oninue i will be shown ha een he ery naure of suh effes appears o be ifferen from ha efine in speial relaiiy. For now, le us proee wih he proess of eeloping he equaions ha relae isanes in he moing frame o hose of he saionary frame. Referring bak o figure 3, we migh be empe o efine isane B-C as a erial isane in he moing frame of referene. Suh, howeer, is no he ase. While isane B-C is erainly a erial isane in he saionary frame, i represens he raius of sphere, an hus all isanes in he moing frame. Likewise is rue of isane A-C relaie o he saionary frame. Realizing his has many impliaions, beyon he obious ha all imensions are equally affee by relaie moion. I also implies ha referene frames are more orrely efine as spheres, an no reangular frames as is he presen onenion. When one ompares he wo ifferen spherial frames of referene ha resul from relaie moion, he relaionships beween he wo frames beomes exeeingly lear. Deeloping his onep furher, le us efine isane in he moing frame as represening any isane in he moing frame as i relaes o he same isane in he saionary frame. Saring again wih he equaion, = r an subsiuing, he spee of ligh in he moing frame, for r, we ge, An sine we now know ha,. Eq. (1) we an subsiue he righ sie of his equaion for in he preeing equaion o ge,. Ineresingly, we passe hrough his ery equaion in our las example. If we now ake he ime o obserer ha = he raius of he larger sphere, he same isane represene by in he smaller sphere, we an replae in he aboe equaion wih, where represens he same isane in he large sphere ha represens in he small sphere. This gie us, Eq. (3) Disane in he moing frame. Again, oing similar using he more familiar alernae equaion, 15

16 1 we ge, 1 1. One again, anyone familiar wih speial relaiiy will immeiaely reognize his equaion as ha ealing wih lengh onraion. Bu here we are in isagreemen wih speial relaiiy in a manner ha has far reahing onsequenes. In speial relaiiy, lengh onraion applies only o horizonal isanes in he ireion of moion. In he presene heory, his ransformaion formula applies o all isanes. Also, i will soon beome apparen ha i is he sanar of measure ha is affee an no aually he isanes hemseles. These are profoun isinions. Speifially, an in he jus arrie a equaions represen he raii of he saionary frame sphere an he moing frame sphere respeiely. Bu sine all isanes in eiher sphere are effee in ire proporion o he size of he respeie sphere, he equaions an be use in a general sense o apply o any isane in eiher sphere. In ha usage hen, represens any isane in he saionary frame sphere while represens he orresponing isane in he moing frame sphere. This broaer efiniion is he preferre efiniion. A his poin one migh onlue ha a moing parile, obje, or boy of mass will shrink in size relaie o a saionary frame of referene as he relaie spee inreases. As sae preiously, howeer, i is he sanar of measure ha shrinks an no he isanes hemseles. This shrinkage whih also affes he measure of isanes raele by any eleromagnei energy propogae by he mass, will our proporionally in all ireions an only beome signifian a ery high spees ha are in hemseles a signifian fraion of he spee of ligh. In oher wors, all objes an propagae energy fiels, or more orrely all isanes, een hose in spae will be subje o a sanar of measure ha shrinks in ire proporion o he size of he moing ligh sphere efine preiouslly. (This onep will be oere in eail in he nex seion.) Alhough he isane onraion equaion an be use o fin he raius of he moing sphere if he saionary sphere is known, an an be reformulae o fin he raius of he saionary sphere if he moing sphere is known, i woul be boh useful an insruie o eelop equaions ha an be use o fin hese raii inepenen of eah oher. For he saionary sphere he equaion is simply, r, Eq. (4) Raius of saionary sphere. 16

17 where r = he raius of he saionary sphere. For r = he raius of he moing sphere, we an subsiue r for an for in he isane onraion equaion 3 an simplify he equaion as follows: Eq. (3) r r Eq. (5) Raius of moing sphere. Thus, we hae he equaion for fining r inepenen of r. This of ourse, oul also hae been aomplishe using he more familiar alernae isane onraion equaion as follows: 1 r 1 ( ) 1 r r ( ) ( ) r ( ) r ( ) r r ( ) r Eq. (5) 17

18 Thus, we again hae he esire equaion for fining he raius of he moing sphere inepenen of he raius of he saionary sphere. We will onlue by aking a momen o reformulae he isane equaions for fining when is known. Saring wih equaion 3, Eq. (3) an some manipulaion, we obain, Eq. (6) Disane in he saionary frame. An now o finish up, we will o he same wih he more familiar ersion of he equaion. Saring wih, 1 we ge, Now ha we hae efine he imporan ariables inoling isane onraion we an ombine hem in a manner ha expresses ha isanes in boh spheres hange in ire proporion o he size of he sphere an ha he raio of isane o he sphere raius of one sphere always equals he orresponing raio of he oher sphere. Thus we hae, r r Eq. (7) Equaliy of isane raios. 18

19 Referring now o figure 4, we will finalize our argumen ha, onrary o speial relaiiy, all isanes (in he onex of he presene heory) are affee by relaie moion. For his example, assume ha a ligh soure emis a pulse of ligh a poin A while raeling along he pah A-B a a spee of.9/s as illusrae in figure 4A. Furher assume ha he isane A-B was raele in one seon relaie o he saionary frame of referene, represene by sphere 1. An finally, assume ha he ligh pulse reahes poin C in he saionary frame a he insan he soure reahes poin B in he saionary frame. Wih suh being he ase, isane A-C in he saionary frame = an isane A-B in he saionary frame =. Disane B-C in he saionary frame an hen be alulae as follows: Where = isane B-C, ( ) ( ) = isane B-C (Noe: All resuls are roune in hese ompuaions.) Sphere 1 Sphere 1 C Sphere Sphere C A =.9/s B A B A B FIGURE 4 Shows how all isanes are affee by moion Now, aoring o boh, speial relaiiy an he presene heory, isanes along he pah of moion shrink in he moing frame of referene. Disane A-B in he moing frame, as shown in figure 4B, woul herefore shrink by an amoun gien by he isane onraion equaion. This isane hen, will be, = isane A-B or, alernaely, ' = isane A-B 19

20 where = isane A-B. Howeer, i an be reaily seen ha isane A-C has also shrunk in he moing frame. Aoring o he presen heory, his isane is, r = isane A-C where r= isane A-C. This resul is o be expee, sine boh isanes, B-C an A-C represen he raius of sphere. We an now alulae isane B-C in he moing frame. Where = isane B-C, we ge, r = isane B-C I an be reaily seen ha, r A B A C.9 1.9, is he same resul for omparing isane A-B o isane A-C in he saionary frame, ha we ge when we ompare isane A-B o isane A-C in he moing frame as shown below: r A B A C Obiously, he same hols rue when we ompare isane B-C o isane A-C in he saionary frame an ompare ha resul o ha of he orresponing isanes in he moing frame. r B C A C r BC A C Of ourse we oul hae obaine he same answers for he moing frame isanes by simply using he isane onraion equaions. For example, 1.9 isane A-C , 1 an isane B-C , 0

21 Or alernaely,.9 isane A-C , 1 an isane B-C While he exerise jus omplee may hae been insruional, i was superfluous wih regar o our argumen ha isane onraion applies o all isanes, or imensions, wihin he moing frame of referene. The rue spherial naure of ligh propagaion, eoi of exernal influenes, seems unquesionable. Likewise, he Pyhagorean relaionship beween wo suh referene frames brough abou by a moing soure of ligh, while onforming o he eiene ha he spee of ligh is onsan, an he proposiion ha he laws of naure are he same in all referene frames. On hese bases he oneps inroue in he preeing paragraphs seem selfeien an beyon reproah. Speial relaiiy, on he oher han, appears o be seriously flawe as preiously noe. The naure of his flaw is he opi of our nex isussion. 9. The Flaw in Speial Relaiiy Alhough he mah in speial relaiiy appears o be orre, he inerpreaion of he resuls is funamenally inorre an subsequenly misleaing. The unerlying problem appears o hae resule from he manner in whih he ransformaion equaions were arrie a, speifially unue reliane upon reangular oorinae sysems. This reliane is apparen een in he well-known ligh lok example use o emonsrae speial relaiiy in presen ay physis books. The example ha uses a ligh soure pulse raeling beween wo erially isplae mirrors while all as a uni moe in he horizonal ireion. Alhough he resuls foun using he example seem o agree wih he ransformaion equaions, loser inspeion, in iew of wha has been shown here, reeals he example o be flawe. In fa, he example oes no faihfully represen naure. The faul lies in he proposiion ha he isane beween he mirrors is no affee in he same manner as horizonal isanes in he ireion of moion. Aoring o speial relaiiy he isane beween he mirrors is unhange whereas he horizonal isanes in he ireion of moion shrink in relaion o he moion. In he presene heory i is agree, an will be shown ha he isane beween he mirrors (in our ase he isane beween he ligh soure an he mirror) is unhange as a resul of relaie moion. Howeer, i will also be shown ha isanes in he ireion of moion, or any oher ireion for ha maer, are also unhange as a resul of he moion. In oher wors, onrary o speial relaiiy, i will be shown ha isanes o no aually hange a all. Wha oes hange is he sanar by whih isanes are measure. This in urn has he effe of ausing he size of physial objes an isanes in spae o be experiene in he saionary frame as expaning raher han shrinking as a resul of relaie moion. To emonsrae, le us perform an exerise inoling he esribe apparaus. To ge o he poin quikly, we will ake a few liberies in presening he problem. For one, we will use inorinae isanes for he purpose of simpliiy an subsequen ease of unersaning. (Of ourse, hese isanes oul be sale own o any praial size neessary wihou hanging 1

22 he resuls, if one were eermine o o an aual experimen.) For wo, an again for he sake of simpliiy, we will use an apparaus ha oes no employ a mirror a he loaion of he ligh soure. An finally, we will only rael half he isane normally gien in exbooks on speial relaiiy. (Again, if one were eermine o perform an aual experimen, raeling he whole isane will no affe he resuls.) Referring now o figure 5, assume ha poins A hrough G are loae in he saionary frame of referene. You shoul also make noe ha, o failiae unersaning, a more esripie form of noaion is inroue. The symbol r is now use o efine he raius of he ligh sphere in he saionary frame of referene. The r may be hough of as symbolizing raius, or alernaely, referene isane, sine i is he spee of ligh an he isane ligh raels in an ineral of ime ha oher spees or isanes are referene o. The prime mark is sill use o signify a orresponing alue in he moing frame of referene, hus he symbol r esignaes he raius of he ligh sphere in he moing frame of referene, or, referene isane, in he moing frame. Sphere 1 E F G Mirror Sphere r D p = r p Ligh Soure A B C FIGURE 5 Uneils he flaw in speial relaiiy A his poin, i is imporan o remember ha a raius may be rawn in any ireion from he ener of he sphere. Therefore, in iewing figure 5, i shoul be unersoo ha isane r oul be shown a seon ime as a erial isane, (alhough his woul be reunan) an hus

23 he relaionship o r beomes apparen. Aiionally, i now beomes neessary o esignae isane C-G (or alernaely isane B-D) in he saionary frame wih is own symbol, p. The p signifies his o be a isane perpeniular o he pah of moion in he saionary frame of referene. This isinion beomes neessary beause r represens he raius of he moing sphere an herefore isanes in oher ireions besies perpeniular. Whereas, p an r are equal in alue, hey are no equal in funion. The nee for his isinion will beome learer as we proee. In finishing, he symbol signifies he isane raele relaing o relaie moion in he saionary frame of referene an he symbol when use, woul iniae he orresponing isane in he moing frame of referene. We an now proee wih he problem. Assume as illusrae in figure 5 ha a ligh soure an opposing mirror share an arrangemen ha keeps hem a fixe isane of 1 apar along a erial axis suh as ha efine by poins A-E. Assume nex ha he ligh soure gies off a pulse of ligh a poin A as i moes a a uniform spee of =.9 /s along pah A-B-C. If he ligh soure rip were limie o 1 seon of saionary frame ime, he ligh soure will only reah poin B an he pulse moing in he erial ireion along pah B-D will only reah poin D. This is beause he isane, r he ligh raels along pah A-D-G in he saionary frame is limie o = 1, he isane from A o D. Tha is, isane A-D mus equal 1, he same isane beween he mirror an he ligh soure. Therefore, isane B-D ha is he same in boh he saionary frame an he moing frame anno equal 1, an he pulse will no reah he mirror a poin F. In oher wors, if spee =.9 /s an = 1s, hen isane =.9 an isane r = 1. Disane p from poin B o poin D is hen foun o be, p r Sine p is a erial isane, his resul is in agreemen wih boh speial relaiiy an he presene heory. Howeer, i mus be remembere ha p always equals r, he raius of he moing sphere, an herefore all isanes in he moing frame of referene. This gies eiene in suppor of he proposiion ha aual isanes in he saionary frame are unhange in he moion frame. Therefore, sine he isane from he ligh soure o he mirror will no hange as a resul of relaie moion, he ligh pulse will no reah he mirror unil ime in he moing frame reahes an ineral of 1 seon. Using equaion 1 o fin, we an see ha his was no he ase in he preious example s 1 In orer for o reah an ineral of 1s, he ime ineral for he rip in he saionary frame mus be inrease. The require ime ineral an be eermine using equaion. Thus, for a ime ineral of = 1s, we obain, s 1.9 3

24 for he new ime ineral. Referring bak o figure 5, he new isane raele by he ligh soure moing a a uniform spee of =.9 /s is foun o be = =.9 x = This is represene as isane A-C in he illusraion. The new isane raele by he ligh pulse along pah A-D-G is hen foun o be r = = This new isane is represene as isane A-G in he illusraion. The new isane p is hen obaine in he same manner as before. Thus, p r , giing he same isane for p as he 1 isane beween he soure an mirror now loae a poins C an G respeiely. This resul is proable hrough experimenaion an is suppore by any an all eiene ha suppors he proposiion ha erial isanes are unaffee by relaie moion as speifie in speial relaiiy. I has been shown in he presene heory, howeer, ha isane p always equals isane r, he raius of he moing frame sphere, an hus all isanes in he moing frame. This proposiion is also suppore by he eiene an is onsisen wih all of he foregoing analysis ha in urn is onsisen wih our posulae an all of he proposiions sae in he inrouion. If real physial isanes are no aually affee by relaie moion while a he same ime he isane ligh raels shrinks in aorane wih he onraion equaions, one mus onlue he following: Physial objes an he isanes beween hem in spae relaie o a moing frame of referene are pereie o be expaning in relaion o a saionary frame of referene. This in iself may be suffiien o explain why i akes greaer an greaer amouns of energy o aelerae an obje o spees approahing a signifian fraion of he spee of ligh. 10. Proing he Funamenals Alhough he priniples presene up o his poin seem o be oerwhelmingly suppore by he eiene, hey fall shor of being proen in he heoreial sense. The main quesion ha sill remains o be resole is his: Why shoul a poin of ligh moing in he perpeniular ireion o he moion of he soure be gien preeene oer any oher poin of ligh moing in anoher ireion when formulaing he relaionship beween referene frames? The answer o his quesion is simple. Use of any oher poin of ligh gies resuls ha are inonsisen wih he eiene. To emonsrae, le us go hrough he following exerise: If he Laws of Naure are he same for all frames of referene he following mus also be rue: 1. If he spee of ligh has a onsan alue in one frame of referene i mus, uner he same oniions, hae ha same onsan alue in all frames of referene.. If ime is inarian in one frame of referene i mus, uner he same oniions, be inarian in all frames of referene. 3. If he spee of ligh has he same onsan alue from one referene frame o anoher, ime mus ary from one referene frame o anoher. 4

25 Now, referring o figure 6, assume ha poins A hrough E are poins in a saionary frame of referene. Assume furher, ha a ligh soure, now loae a poin B gae off a pulse of ligh a poin A while moing away from Poin A a a uniform rae of spee. In he moing frame of referene, ligh moing away from he soure in all ireions forms sphere wih raius B-D. If is he spee of ligh, an is he ime ineral in he moing frame, hen is he isane he ligh raele from he soure uring ha ineral. If we now ranomly sele a poin of ligh along sphere, oher han from he perpeniular ireion represene by poin D, poin E for example, we an see ha i raele pah A-E in he saionary frame of referene. Sine ligh mus hae a onsan spee in he saionary frame of referene, pah A-E represens he raius of sphere 1A showing he isane ligh woul rael in all ireions in he saionary frame a ha same spee. The isane he ligh raels in he saionary frame is hen,, where is he spee of ligh an is he ime ineral in he saionary frame. For onsiseny, howeer, we mus now also sele a poin of ligh moing in he opposie ireion a he same angle o he pah of moion ha he original poin was moing in he forwar ireion. This poin of ligh is a poin C an while raeling he same isane in he moing frame as poin E uring ineral, raels a ifferen isane han poin E in he saionary frame. Is isane is A-C an forms he raius for sphere 1B in he saionary frame. Obiously, he shorer isane A-C anno be represene as he same isane ha represens he longer isane A-E. If is onsan in he saionary frame, hen woul hae o ary in orer o aliae he seleion of hese wo poins of ligh. Suh seleion han iolaes he priniple ha ime is inarian wihin all frames of referene. This onraiion is aoie only when he poin of ligh selee from he moing frame is one ha is raeling away from he soure in a ireion perpeniular o he pah of rael. Suh poin of ligh hen represens he rael isane of all poins of ligh in he moing frame an he orresponing isane raele in he saionary frame as preiously shown. Sphere 1A Sphere 1B A C D B E Sphere FIGURE 6 Proof of he Funamenals When we aep he eiene ha spee is onsan, an ime aries from one frame o anoher, he ransformaion equaions an be arrie a simply an irely. In figure 7 we show 5

26 he now familiar righ riangle ha represens he relaionship beween he moing frame an he saionary frame. FIGURE 7 The Dire Meho Now, howeer, where represens he spee of he soure, represens he spee of ligh, an represens he ime ineral in he saionary frame, is use irely o represen he ime ineral in he moing frame. Therefore, where represens he isane raele by he soure in he saionary frame, an represens he raius of he saionary frame, irely represens he raius of he moing frame. Gien ha he riangle is a righ riangle, we an immeiaely erie, ( ) ( ) Sep 1 Sep ( ) Sep 3 Sep 4 Sep 5 Eq. (1) Time in he moing frame. Sep 6 Subsiuing he symbols for an for gies, Sep 7 Eq. (3) Disane in he moing frame. 6

27 (From Sep 4) By subsiuing he symbol r for we ge, r Sep 8 Eq. (5) Raius of he moing sphere. An of ourse, r Sep 9 Eq. (4) Raius of he saionary frame. The orollary of equaion 1 is, Sep 10 Eq. () Time in he saionary frame. The orollary of equaion 3 is, Sep 11 Eq. (6) Disane in he saionary frame. Thus in a hanful of seps we hae arrie a all of he ransformaion equaions eelope preiously. Whereas he firs par of our analysis aliaes he unerlying funamenals use in he heory, he seon par aliaes he mahemais employe. Taken ogeher, hese examples proie ompelling eiene as o he orreness of he presene heory. On he oher han, he inire mehos use in speial relaiiy lea o misinerpreaion of eiene an erroneous onlusions. I has been shown here oniningly ha all isanes are equally affee by relaie moion. Suh affe, howeer, is inire, hus no affeing aual isanes bu raher he shrinking sanar by whih suh isanes are measure. This in urn leas o he onlusion sae earlier ha real isanes, alhough unhange in he moing frame, are pereie o be expaning in he saionary frame. When one onsiers he role moion plays in he energy o mass relaionship, suh hange in perspeie an hae profoun onsequenes. Does mass really inrease as is spee inreases, or is i ha energy beomes less effeie? An ineresing quesion, bu beyon he sope of his presen work. 11. Effes on Waelengh an Frequeny I is well esablishe, an suppore by a preponerane of eiene, ha he waelengh an frequeny of ligh are affee by he relaie moion of he soure. Speifially, he obsere waelengh will inrease an he obsere frequeny will erease if he soure is reeing, an onersely he obsere waelengh will erease an he obsere frequeny will inrease if he soure is approahing. Le us hen, formulae mahemaially he effes ha 7

28 relaie moion has on he waelengh an frequeny of ligh. We an o his by analyzing separaely wo ifferen proesses ha are inole. Firs we mus eermine wha effes he ransformaion proess, beween he moing frame an he saionary frame, has on hese properies of ligh, an hen we mus eermine how he resuls of ha proess are affee by he relaie moion in he saionary frame alone. Afer eermining eah of hese proesses separaely, we an ombine he resuls ino a single se of equaions ha will gie us he oal of all he effes for any siuaion. Before proeeing wih he ransformaion proess i is neessary o unersan he relaionship beween he waelengh an frequeny of ligh emie by he soure. The firs poin ha mus be unersoo is his: Wheneer we refer o he emie waelengh or frequeny of ligh, we are referring o heir alues in he moing frame while i is a res. In oher wors, hese alues are ienial o hose ha resul from a soure of ligh a res in he saionary frame. This sipulaion is onsisen wih he posulae ha he laws of physis are he same in all frames of referene. Therefore, when we onsier he alues of he emie waelengh an frequeny of he soure, we are seing he ime a he soure equal o he uni alue of 1s, he same alue has in he saionary frame when we ompare ligh from he moing soure o ligh in he saionary frame. Remember, he symbol esignaes he ime in he moing frame, an ha ime in aorane wih he ime ransformaion equaion (Eq. 1) an hae a alue of 0s o s epening on he spee of he soure. In his ase i has he alue of beause we are referring o he alues for waelengh an frequeny as hey are emie in he frame of he soure. In fa is normally omie from he equaions ha express he waelengh o frequeny relaionship as will be shown below. A final poin, he relaionships o be eelope are ali in any frame of referene. Now, sine he spee of ligh is, an sine waelenghs are isanes, he number of suessie waes onaine in he isane ligh raels in an ineral of ime, is he frequeny for ha ineral. Where f is he frequeny, λ is he waelengh, is he spee of propagaion, an is he ime ineral, we ge, f. By iiing he expression /λ by we assure ha he resuling relaionship will always represen a ime ineral of 1. Therefore, hrough simplifiaion of he aboe equaion we arrie a, 1 f f Eq. (8) Frequeny o waelengh relaionship. equaion 8, giing he waelengh o frequeny relaionship for any frame of referene. The orollary equaion for waelenghs is hen, 8

29 f Eq. (9) Waelengh o frequeny relaionship. By way of subsiuion ino he jus arrie a equaions, he frequeny an waelengh of ligh emie from a moing soure is hen expresse as, f an f where f is he emie frequeny, λ is he emie waelengh, an is he spee of propagaion. a. The Transformaion Effe Haing now esablishe he formulas efining he waelengh an frequeny of he emie ligh we an eermine wha effes he ransformaion proess will hae on heir alues. Saring wih he waelengh an aknowleging ha sine waelenghs are isanes as sae preiously, hese isanes will be shorer in he moing frame han hey are in he saionary frame. Therefore, by using equaion 6 ha efines he relaionship beween isanes in eah of hese frames, we an subsiue he waelenghs for hose isanes an erie an equaion for ransforming waelenghs from he moing frame o he saionary frame. Thus, where λ x is he waelengh in he saionary frame, an λ is he waelengh in he moing frame, by way of subsiuion ino equaion 6, we ge, Eq. (6) x Eq. (10) Waelengh ransformaion, MF o SF. The orollary equaion for fining waelengh λ in he moing frame is hen, x Eq. (11) Waelengh ransformaion, SF o MF. (The subsrip x hosen o ienify ransforme alues may be hough of as shor for ransforme, i.e., xforme. The more obious hoie,, proe o be onfusing beause of is use in ienifying ime inerals.) We are now reay o eelop he ransformaion equaions for frequeny. Drawing again on general equaion 9, where λ x is he ransforme waelengh an f x is he ransforme frequeny, we hae, 9