Undulating Relativity ABSTRACT

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1 Unlaing Relaiiy h: lfe ias Meia Gaia -ail: STRCT The Speial They f Relaiiy aes s esls ha pesenly ae nsiee inepliable any enne sieniss, n: -The ilaain f ie, an -The nain f he enz engh. The slin hese hae ien he ah he eelpen f he Unlaing Relaiiy (UR) hey, hee he Tepal aiain is e he iffeenes n he e f he ligh ppagain an he lenghs ae nsans beeen lanas in nif elaie een. The Unlaing Relaiiy pies ansfains beeen he lanas ha iffes f he ansfains f enz f: Spae (,y,z), Tie (), Spee ( ), eleain ( a ), negy (), Men ( p ), e ( ), leial iel ( ), Magnei iel ( ), igh eqeny ( y ), leial Cen ( J ) an leial Chage (ρ ). he analysis f he eelpen f he Unlaing Relaiiy, he flling an be synhesize: - I is a hey ih piniples pleely n physis; - The ansfains ae linea; - eeps nhe he liian piniples; - Cnsies he Galile s ansfain isin n eah efeenial; - Ties he Spee f igh an Tie a niqe phenenn; - The enz fe an be aaine by isin ypes f ile es, an - Wih he absene f he spaial nain f enz, eah he sae lassial esls f he speial elaiiy ning is n neessay as nle n he pple effe. h, he Unlaing Relaiiy an he Speial Relaiiy f lbe insein eplain he epeiene f Mihel- Mley, he lngiinal an ansesal pple effe, an spplies ealy ienial flain : beain f zenih gα / esnel s fla ( ) n n.. Mass ( ) ih eliy ( ) [esing ass ( )]/... Men p.. Relain beeen en (p) an negy ().. p. Relain beeen he elei fiel ( ) an he agnei fiel ( V ). µο.i i-saan s fla µ..π.r is e glie s ae eqain ψ(,) a.si nπγ ; /

2 lng ih he eqains f ansfains beeen efeenes f he UR, e ge he inaiane f shape Maell s eqains, sh as: ρ i ; i. ο i. R. R µο. j ο.µο. ; R ο.µο.. We als ge he inaiane f shape he eqain f ae an eqain f niniy ne iffeenial shape: y z ρ.j. Ohe Ws: 9 plaining he Sagna ffe ih he Unlaing Relaiiy plaining he epeiene f Ies-Silell ih he Unlaing Relaiiy Tansfain f he pe f a lins ay beeen efeenials in he Speial They f Relaiiy ineaiy Riha C. Tlan. Veliies psiin 5 Inaiane 6 Tie an eqeny 7 Tansfain f. enz 8 The Mihelsn & Mley epeiene 9 Regessin f he peihelin f Mey f 7, 9 ane f Mey s peihelin f.79 Ineia ane f Mey s peihelin f.79 allae ih he Unlaing Relaiiy /

3 h: lfe ias Meia Gaia. e-ail: Unlaing Relaiiy Tansla: Rlf Mas Venâni e-ail: Tansfain spae an ie The Unlaing Relaiiy (UR) eep he piniple f he elaiiy an he piniple f Cnsany f ligh spee, ealy lie lbe insein s Speial Relaiiy They efine: a) The las, ne hih he sae f physis syses ae hange ae he sae, eihe hen efee a eeine syse f inaes any he ha has nif anslain een in elain he fis. b) ny ay f ligh es in he esing inaes syse ih a eeine eliy, ha is he sae, haee his ay is eie by a esing by by a by in een (hih eplains he epeiene f Mihel-Mley). e s iagine fis ha bsees O an O (in a), ing in nif anslain een in elain eah he, ha is, he bsee n ae elaiely eah he. In his ay, he bsee O gehe ih he ais, y, an z f a syse f a eangle Caesian inaes, sees he bsee O e ih eliy, n he psiie ais, ih he espeie paallel ais an sliing alng ih he ais hile he O, gehe ih he, y an z ais f a syse f a eangle Caesian inaes sees O ing ih eliy, in negaie iein as he ais ih he espeie paallel ais an sliing alng ih he ais. The bsee O eases he ie an he O bsee eases he ie ( ). e s ai ha bh bsees se hei ls in sh a ay ha, hen he iniene f he igin f he inae syse happens ze. In he insan ha, a ay f ligh is pjee f he n igin bh bsees. fe he ie ineal he bsee O ill nie ha his ay f ligh ha silanesly hi he inaes pin (, y, z) ih he ay f he O bsee ih eliy an ha he igin f he syse f he O bsee has n he isane alng he psiie ay f he ais, nling ha: y z... The sae ay afe he ie ineal he O bsee ill nie ha his ay f ligh silanesly hi ih he bsee O he inae pin (, y, z ) ih eliy an ha he igin f he syse f he bsee O has n he isane n he negaie ay f he ais, nling ha: y z... Maing. eqal. e hae y z y z..5 ease f he syey y y en z z, ha siplify.5 in..6 T he bsee O (.) ha applie in.6 spplies ( ) f hee..7 T he bsee O (.) ha applie in.6 spplies ( ) f hee /

4 ..8 Table I, ansfains he spae an ie.. y y.. y y.. z z.. z z he eqain syse fe by. an. e fin (nsieing > e >).9 ha ensaes he inaiane f he spae in he Unlay Relaiiy. he eqain syse fe by.7 an.8 e fin... If in. hen, ha applie in. spplies,... If in. an hen... T he bsee O he piniple f ligh spee nsany gaanees ha he pnens, y an z f he ligh spee ae als nsan alng is ais, hs y y z z, y, z. an hen e an ie.. Wih he se f.7 an.9 an. e an ie..5 iffeeniaing.9 ih nsan an, else, nly he ie aying e hae,.6 b f.5 hen eing an nsans, he eazns..7 an / in.5 s als be nsan bease f his he iffeenial f s be eqal ze f hee e nle, ha is ealy he sae as..

5 T he bsee O he piniple f Cnsany f eliy f ligh gaanees ha he pnens, y, an z f eliy f ligh ae als nsan alngsie is ais, hs y y z z, y, z,.8 an ih his e an ie,..9 Wih he se f.8,.9, an.9 e an ie.. iffeeniaing.9 ih an nsan, ha is, nly he ie aying e hae,. b f. hen eing an nsan he iisins.. an in. als hae be nsan bease f his he iffeenial f s be eqal ze f hee e nle, ha is ealy lie.8. Replaing. an.9 in. e hae... T he bsee O he e psiin f he pin f inaes (,y,z) is R i yj z,. an he e psiin f he igin f he syse f he bsee O is R..5 R i j i T he bsee O, he e psiin f he pin f inaes (,y,z ) is R i y j z,.6 an he e psiin f he igin f he syse f he bsee O is R i j R i..7 e.9,.5, an.7 e hae, R R..8 s. is eqal.5 pls.6 e hae R R R R R R..9 pplying.8 in.9 e hae, R R R.. 5/

6 6/ T he bsee O he e eliy f he igin f he syse f he bsee O is i j i R.. T he bsee O he e eliy f he igin f he syse f he bsee O is i j i R...5,.,., an. e fin he flling elains beeen an... Obseain: in he able I he flas.,.., an.. ae he pnens f he e.9 an he flas.,.., an.. ae he pnens f he e.. a f eliy ansfains an iffeeniaing.9 an iiing i by.7 e hae R R R.. iffeeniaing. an iiing i by. e hae R R R.. Table, a f eliy ansfains an.... y y.. y y.. z z.. z z

7 7/ Mliplying. by iself e hae..7 If in.7 e ae hen as i is eqie by he piniple f nsany f eliy f ligh. Mliplying. by iself e hae..8 If in.8 e ae hen as i is eqie by he piniple f nsany f eliy f ligh. If in. e ae hen as i is eqie by he piniple f nsany f eliy f ligh. If in. e ae hen as i is eqie by he piniple f nsany f eliy f ligh. Reeling.7 an.8 e hae..9.. The ie elains beeen he ies an eliies f pins in spae an be baine ih he eqaliies ing f., ha applie in.7,.,., an.5 spply,.,.,...

8 beain f he zenih T he bsee O alng ih he sa, y an z, an he bsee O alng ih he ah e hae he njn., y, z, y y z ealy as feseen by he piniple f elaiiy. T he bsee O he ligh ppagaes in a iein ha aes an angle ih he eial ais y gien by / angα.5 y. ha is he abeain fla f he zenih in he speial elaiiy. If e inee he bsees e l hae he njn., ( ) y y, z, y z ( ) / angα.6 y. ha is eqal.5, ih he negaie sign iniaing he nay iein f he angles. Cnsieing in., elain he appaas hen esnel s fla / n he eliy f ligh elaiily he ae, ill be he eliy f ligh elaiely he labay / n / n / n n Igning he e / e hae n n n n n an igning he e / n n n n n n n he eliy f ae in n e hae he esnel s fla n..7 8/

9 Maing y z an ( ) ( ) as pple effe eplaing hen, hen ( ) ( ) e ill efine y z in.5 e hae an.7 e fin ( ) ( ) hee aen he piniple f elaiiy.8 Resling in he epessin ( ) syei an inaiable beeen he bsees. T he bsee O an epessin in he fla f ψ(,) f ( ).9 epesens a e ha ppagaes in he iein f R. T he bsee O an epessin in he fla f ψ, f. epesens a e ha ppaes in he iein f pplying in.8 λ λ e R. π π,,.,.9,.,.5, an.6 e hae λ λ λ λ,. ha applie in yλ y λ spply, y y an y y.. Cnsieing he elain f Plan-insein beeen enegy ( ) an feqeny ( y ), e hae he bsee O hy an he bsee O hy ha eplae in. spply an.. If he bsee O ha sees he bsee O ing ih eliy in a psiie ay he ais, eis aes f feqeny y an eliy in a psiie ay he ais hen, aing. an bsee O ill ease he aes ih eliy an feqeny ha is ealy he lassi fla f he lngiinal pple effe. he y y,. If he bsee O ha sees he bsee O ing ih eliy in he negaie ay f he ais, eis aes f feqeny y an eliy, hen he bsee O aing. an ill ease aes f feqeny y an eliy in a pepenila plane he een f O gien by γ γ,.5 ha is ealy he fla f he ansesal pple effe in he Speial Relaiiy. iffeeniaing. an iiing i by.7 e hae Tansfains f he aeleains a an a / / a a ( ) a ( ).. iffeeniaing. an iiing i by. e hae / / a a ( ) a ( ).. 9/

10 Table, ansfains f he aeleains a an a a a a a a ( ). a ( ). a a a a a ( ). a ( ). ay a a y a a y y.. ay y az a a z a a z z.. az z a a a.8 a he ables an e an nle ha if he bsee O hen i is als he bsee O an is pepenila.a ze an y z,.a ze an y z, hs is pepenila a a as he es hey eqies. iffeeniaing.9 ih he eliies an he ies hanging e hae,, b nsieing.6 e hae,.7 Whee eplaing.5 an iiing i by.7 e hae, We an als eplae. in.7 an iie i by. eing a a..8 a a..9 The ie elains beeen he les f he aeleains a an a f pins in spae an be baine ih he a ing f., ha applie in.8 an.9 spply a a a an a a a.. Tha an als be ee f. an. if e se he sae eqaliies ing f.. a Tansfains f he Mens p an p efine as p an p hee an ( ) We ill hae he elains beeen an ( ),. syblizes he fnin asses f he les f eliies an. an he esing ass, analyzing he elasi llisin in a plane beeen he sphee s ha f he bsee es alngsie he ais y ih eliy y an he sphee s ha f he bsee O es alngsie he ais y ih eliy y -. The sphees hile bsee in elaie esing ae ienial an hae he ass. The nsiee llisin is syei in elain a paallel line he ais y an y passing by he ene f he sphees in he en f. Cllisin. efe an afe he llisin he sphees hae eliies bsee by O an O aing he flling able gen f able /

11 Sphee Obsee O Obsee O efe s s ze, ys s Cllisin s s, fe s ze Cllisin s s, ys, y s s ze, y s s, ys s ys, y s s ze, y s T he bsee O, he piniple f nseain f ens esablishes ha he ens py y p an, f he sphees s an s in elain he ais an y, eain nsan befe an afe he llisin hs f he ais e hae ( s ys ) s ( s ys ) s ( s ys ) s ( s ys )s hee eplaing he ales f he able e hae f hee e nle ha, an f he ais y ( s ys ) ys ( s ys ) ys ( s ys ) ys ( s ys )ys hee eplaing he ales f he able e hae,, siplifying e hae,, hee hen bees b is eqal he esing ass hs, ih a elaie eliy ha applie in. spplies p. Wih he sae pees e l hae f he O bsee.., /

12 ( ). p an Siplifying he siblgy e ill ap an ( ).... ha siplify he ens in p an p.. pplying. an. in.9 an. e hae an efining fe as Nen e hae inei enegy (, ) as p an.. p ( ), ih his e an efine hen.r.r ( ). ( ), an ( ).R.R ( ). ( ) Reeling. an. an iffeeniaing e hae. an, ha applie in he flas f inei enegy spplies an hee an,.5.6 ae he al enegies as in he speial elaiiy an he esing enegy..7 pplying.6 in. e hae ealy...6,.,., an. e fin p an p.8 ienial elains he Speial Relaiiy. Mliplying. an. by e ge /

13 an p p p p Table, ansfains f ens p an p p p.9 p p. p p. p p. p y py.. py p y.. p z pz.. pz p z..... ( ) p.8 p.8 Wae eqain f is e glie The bsee O assiaes a esing paile in is igin he flling ppeies:.9.. -Resing ass -Tie -Resing negy -eqeny -Wae fnin y h h ψ asenπ y ih a nsan. The bsee O assiaes a paile ih eliy he flling: -Mass -Tie -negy (f. hee ) an ) (f. ih an ) (f.7 ih /

14 -eqeny y y /h (f. ih -isane (f. ih ) -Wae fnin -Wae lengh an y y ) π ih ψ asen y asenπ y asenπ y ) yh h yλ λ (f.9 ih p p p p p T g ba he O bsee efeenial hee, e ill nsie he flling aiables: -isane (f. ih ) -Tie -eqeny -Veliy (f.8 ih ) y y (f. ih ) (e.) ha applie he ae fnin spplies ψ asenπ y asenπ y asen y π, b as an y y hen ψ ψ. 5 Tansfains f he es an iffeeniaing.9 an iiing by.7 e hae p p iffeeniaing. an iiing by. e hae (.) p p he syse fe by 5. an 5. e hae.. (. ) , 5. ha is an inaian beeen he bsees in he Unlaing.Relaiiy. /

15 Table 5, ansfains f he es an (.) 5.. (.). 5. y y/ 5.. y y / 5.5. z z/ 5.. z z / Tansfains f he ensiy f hageρ, ρ an ensiy f en J an J q Mliplying. an. by he ensiy f he esing elei hage efine as ρ e hae ρ ρ ρ ρ ρ ρ J J ρ ρ ρ ρ an ρ ρ ρ J J ρ Table 6, ansfains f he ensiy f hagesρ, ρ an ensiy f en J an J J J ρ 6. J J ρ 6. J Jρ 6. J J ρ 6. J y Jy 6.. Jy J y 6.. J z Jz 6.. Jz J z 6.. J ρ 6.5 J ρ 6.6 ρ ρ ρ ρ ρ ρ 6.9 ρ ρ Κ he syse fe by 6. an 6. e ha 6.9 an Tansfain f he elei fiels, an agnei fiels, pplying he fes f enz q( ) an q( ) q( ) q( ) [ q( ). ] an q( ) q( ) [ ] q in 5. an 5. e hae, ha siplifie bee ( ) ( ) (.) an ( ). f hee e ge he inaiane f. beeen he bsees as a nseqene f 5. an he flling pnens f eah ais 5/

16 yy zz y z z y yz zy 7. y z z [ y z z] 7.. z y y [ z y y ] 7.. y y z z yzzy y z z y 7. y z z [ y z z ] 7.. z y y [ z y y ] 7.. T he njn 7. an 7. e hae slins esibe in he ables 7 an 8. Table 7, ansfains f he elei fiels, an agnei fiels e y z y 7.. y z y 7.. z y z z y 7.. z y y z 7.5. y y z 7.6. z z y 7.5. z z y 7.6. y y 7.7 y y 7.8 z z 7.7. z z 7.8. y z 7.9 y z 7. z y 7.9. z y 7.. Table 8, ansfains f he elei fiels, an agnei fiels e (.) y ( y z) 7.. y ( y z ) 7.. z ( z y) 7.. z ( z y ) y y 7.. y y 7.. z z 7.. z z 7.. 6/

17 Relain beeen he elei fiel an agnei fiel If an elei-agnei fiel has he bsee O he nagh agnei pnen ze an he elei pnen. T he bsee O his fiel is epesene ih bh pnens, being he agnei fiel esibe by he njn 7.5 an hás as pnens ze, y ha ae eqialen z, y z, la f i-saa The bsee O assiaes a esing elei hage, nifly isibe alngsie is ais he flling elei-agnei ppeies: -inea ensiy f esing elei hage ρ q -Nagh elei en I ze -Nagh agnei fiel ze ze ρ -Raial eleial fiel f le y z a any pin f ais R πr he pnen ze. y z ih T he bsee O i elaes an elei hage nifly isibe alngsie is ais ih eliy hih i assiaes he flling elei-agnei ppeies: -inea ensiy f he elei hage ρ ρ (f 6.7 ih ) -lei en I ρ ρ -Raial eleial fiel f le (aing he njns 7. an 7.5 ih ze ze an ) -Magnei fiel f pnens ze, z y y, z an le ρ µ I hee µ, being in he eial f πr πr µ I πr 7.7 hee is a niay e pepenila he eleial fiel an angen he ifeene ha passes by he pin f ais R y z bease f he njn 7. an 7.6. ze. 7/

18 8 Tansfains f he iffeenial peas Table 9, iffeenial peas y y z z y y z z he syse fe by 8., 8., 8., an 8. an ih.5 an. e nly fin he slins / an /. 8.5 hee e nle ha nly he fnins ψ (.9) an ψ (.) ha spply he niins ψ / ψ an ψ / ψ, 8.6 an epesen he ppagain ih eliy in he Unlaing Relaiiy iniaing ha he fiel ppagaes ih efinie eliy an ih isin being applie. an.8. ease f syey e an als ie he he ais ψ y y / ψ ψ y y / ψ, an ψ z / ψ z ψ z z / ψ,. 8.7 he ansfains f spae an ie f he Unlay Relaiiy e ge Jab s hee J an (,y,z,) (, y,z,) J, 8.8 (, y,z,) (, y,z,) aiables ih an as a nseqene f he piniple f nany f he ligh eliy b ae eqal ais J J an ill be eqal ne J J hen. The ae eqain he bsee O is Inaiane f he ae eqain y z ze hee applying he flas f ables 9 an. e ge f hee e fin y z ze 8/

19 9/ ze z y ha siplifying spplies ze z y 6 hee eeing he es e fin ze z y 8.9 b f 8.5 an. e hae ze / ha applie in 8.9 spplies he ae eqain he bsee O ze z y. 8. T en he efeenial f he bsee O e ill apply 8. he flas f ables 9 an.8, geing ze z y f hee e fin ze z y ha siplifying spplies ze z y 6 hee eeing he es e fin ze z y b f 8.5 an.8 e hae ze / ha eplae in he eee eqain spplies he ae eqain he bsee O. Inaiane f he Cniniy eqain The niniy eqain in he iffeenial f he bsee O is ze z Jz y Jy J ρ ze.j ρ 8. hee eplaing he flas f ables 6, 9, an. e ge ze z Jz y Jy ρ J ρ

20 aing he peains e fin ρ ρ ρ ρ J J ρ ρ Jy y Jz z ze ha siplifying spplies ρ ρ J J Jy Jz y z ze hee applying J ρ ih nsan e ge ρ ρ J ( ρ) Jy Jz ρ J Jy Jz ze y z y z ze 8. ha is he niniy eqain in he iffeenial f he bsee O. T ge again he niniy eqain in he iffeenial f he bsee O e ill eplae he flas f ables 6, 9, an.8 in 8. geing J y J z ρ ( J ρ ) ze y z aing he peains e fin ρ ρ ρ ρ J J ρ ρ J y y J z z ze ha siplifying spplies ρ ρ J J J y J z y z hee applying J ρ ih nsan e ge ρ ρ J ( ρ ) J y J z y z ze ρ J J y J z ze y z ha is he niniy eqain in he iffeenial f he bsee O. Tha in he iffeenial f ae ien his ay Inaiane f Maell s eqains ze Wih eleial hage T he bsee O T he bsee O y z ρ y 8. y z y y z y z y z y z y y z z y z y µ Jz µ y z z z ρ y z y z y z y z y y z z y z y µ J z µ y z /

21 z y µ J µ y z z µ Jy µ z y z y z y z z Wih eleial hage ρ ρ ze an J J ze T he bsee O T he bsee O y z y z y z y z y z y z y y z z y z y z µ y z y µ y z z y µ z µ µ J µ µ J y µ y z y z y z y z y z y z y y z z y z y z µ y z y µ y z z y µ z y We ensae he inaiane f he a f Gass in he iffeenial f ha f he bsee O is y z ρ 8. y z hee eplaing he flas f he ables 6, 7, 9, an.8, an nsieing nsan, e ge z z y y y ρ Κ aing he ps, sing an sbaing he e z z y y z z y z z z z ρ, e fin y y z y y y ha eeing esls z y y z y z y z ρ /

22 hee he fis paenheses is 8.5 an bease f his eqal ze, he sen blan is eqal ρ ( µ J) µ ρ gen f 8.5 an 8.5 esling in y z ρ y z y z ρ f hee e ge y z ha is he a f Gass in he iffeenial f he bsee O. ρ ρ T ae he inese e ill eplae in 8. he flas f he ables 6, 7, 9, an., an nsieing nsan, e ge z z y y y ρ Κ aing he ps, aing an sbaing he e z z y z ha eeing esls in z z z y z z z y y y z y z, e ge y y y y ρ y z ρ y z hee he fis blan is 8.5 an bease f his eqals ze, he sen blan is eqal ρ ( µ J ) µ ρ gen f 8.6 an 8.5 esling in y y f hee e ge bsee. z z y y ρ z ρ z ρ ρ 8. ha is he a f Gass in he iffeenial f he O Peeing his ay e an pe he inaiane f f f all he he eqains f Maell. 9 plaining he Sagna ffe ih he Unlaing Relaiiy We s ansf he saigh een f he bsees O an O se in he ein f he Unlaing Relaiiy in a plain ila een ih a nsan ais. e s iagine ha he bsee O sees he bsee O ning an ih a angenial spee in a lise ay (C) eqals he psiie se f he ais f UR an ha he bsee O sees he bsee O ning an ih a angeial spee in a nlise ay (U) eqals he negaie se f he ais f he UR. In he en ze, he bsee O eis ays f ligh f he n igin bh bsees, ne in a nlise ay f a U an anhe in a lise ay f a C, heefe U C an U C, bease is he spee f he nsan ligh, an U an C he ie. /

23 In he en ze he bsee O als eis ays f ligh f he n igin bh bsees, ne in a nlise ay (seless) f a U an anhe ne in a lise ay f a C, hs U C an U C bease is he spee f he nsan ligh, an U an C he ie. Reiing he eqains.5 an. f he Unlaing Relaiiy (UR):..5.. Maing ( ay f ligh pjee alngsie he psiie ais ) an spliing he eqains e hae: When he igin f he bsee O ees he nlise ay f he bsee O, ill be a he isane f he bsee O an silanesly ill ee is lise ay f ligh a he sae pin f C U he bsee O, in a syei psiin he iaee ha ges hgh he bsee O bease U C U C an U C U C, flling he f eqains abe e hae: U πr C πr C 9.5 C πr U πr C 9.6 When he igin f he bsee O ees he lise ay f he bsee O, silanesly ill ee is n lise ay an ill be a he isane f he bsee O, hen flling he eqains,, an abe e hae: C U C πr πr C C 9.7 C πr πr C 9.8 The ie iffeene he bsee O is: πr πr πr C C 9.9 The ie iffeene he bsee O is: C C πr πr πr ( ) 9. Replaing he eqains 5 in e pe ha hey nfi he ansfains f he Unlaing Relaiiy. /

24 plaining he epeiene f Ies-Silell ih he Unlaing Relaiiy We shl eie he eqains (.) he ae lengh in he Unlaing Relaiiy: λ λ an λ λ,. Maing ( Ray f ligh pjee alngsie he psiie ais ), e hae he eqains: an λ λ λ λ,. If he bsee O, h sees he bsee O ging aay ih he eliy in he psiie ay f he ais, eis aes, penien f a esing se in is igin ih eliy an ae lengh λ in he psiie ay f he ais, hen aing he eqain. he bsee O ill ease he aes ih eliy an he ae lengh λ aing he flas: λ λ an λ λ,. If he bsee O, h sees he bsese O ging aay ih eliy in he negaie ay f he ais, eis aes, penien f a esing se in is igin ih eliy an he ae lengh λ in he psiie ay f he ais, hen aing he eqain. he bsee O ill ease aes ih eliy an ae lengh λ aing he flas: λ λ an λ λ,. The esing ses in he igin f he bsees O an O ae ienial hs λ λ. We allae he aeage ae lengh λ f he ease aes (,λ )., he lef sie in eah eqain: λ λ λ λ λ λ λ λ λ sing he eqains. an λ We allae he iffene beeen he aeage ae lengh λ an he eie ae lengh by he ses λ λ λ : λ λ λ λ λ λ λ λ λ λ /

25 λ λ λ λ. Refeene hp://.babin.ne/physis/faaj7.h Ies-Silell (ninain) The pple s effe ansesal he Unlaing Relaiiy as baine in he as flls: If he bsee O, ha sees he bsee O, es ih he spee in a negaie ay he ais, eis aes ih he feqeny y an he spee hen he bsee O aing. an ill ease aes f feqeny y an spee in a pepenila plane he een f O gien by y y.5 e ill hae ze an y y an he se feqeny y y lie his y he ansesal feqeny y ih his e an ie he elain beeen.5 Wih y λ y λ e hae he elain beeen he lengh f he ansesal ae he se ae λ λ λ an he lengh f λ.6 The aiain f he lengh f he ansesal ae in he elain he lengh f he se ae is: λ λ λ λ λ λ λ.7 ha is he sae ale gen in he They f Speial Relaiiy. pplying.7 in. e hae λ λ Wih he eqains. an. e an ge he elains.9,., an. esibe as flls λ λ.9 n f his e hae he fla f spee λ λ.8. λ λ λ λ λ. pplying. an. in.6 e hae λ λ λ λ. λ 5/

26 .8 an. e nle ha λ λ λ λ λ.. S ha e he ales f λ an λ baine f he Ies-Siell epeiene e an ealae λ, an nle hehe hee is n he spae efain peie in he They f Speial Relaiiy. Tansfain f he pe f a lins ay beeen efeenials in he Speial They f Relaiiy The elainship ihin he pe eelpe by he fes beeen efeenials is ien in he Speial They f he Relaiiy in he flling ay:.. The efiniin f he pnen f he fe alng he ais is: p. a lins ay, he piniple f ligh spee nsany gaanees ha he pnen f he ligh spee is als nsan alng is ais, hs The fla f enegy is nsan, ensaing ha in he efiniin f enegy e hae pplying 5 in e hee: λ,. ze. f hee e hae.. ha applying in an e hae..5. (. ). hee e fin ha...6 esl eqal 5. f he Unlaing Relaiiy ha an be epeienally pen, nsieing he Sn as he se. ineaiy The They f Unlaing Relaiiy has as is fnaenal ai he neessiy ha ineial efeenials be nae elsiely as hse nes in hih a ay f ligh eie in any iein f is igin speas in a saigh line, ha is aheaially esibe by he flae (.,.8, 8.6 e 8.7) f he Unlaing Relaiiy: y y z z y, z,. y y z z, y, z.8 Wlea Vig e in.887 he linea ansfain beeen he efeenials s he bsees O e O in he flling ay: 6/

27 .. Wih he espeie inee eqains:.. Whee,, an ae nsans an bease f he syey e n nsie he es ih y, z an y, z. We n ha an ae pjeins f he ays f lighs an ha spea ih Cnsan spee (e he nsany piniple f he Ray f ligh), eie in any iein f he igin f he espeie ineials efeenial a he en in hih he igins ae inien an a he en hee: ze.5 bease f his in he eqain. a he en hee ze e s hae ze s ha e als hae ze, e an asse ha hen ze, als be eqal ze, bease if he speaing happens in he plane y z e ill hae ze pls ze. We shl eie he ee eqains ( ze):.6.7 Wih he espeie ee inee eqains:.8.9 If he speaing happens in he plane y z e hae ze an iiing.6 by.7 e hae:. hee is he le f he spee in hih he bsee O sees he efeenial f he bsee O ing alngsie he ais in he psiie ay bease he sign f he eqain is psiie. If he speaing happens in he plane y z e hae ze an iiing.8 by.9 e hae:. hee is he le f he spee in hih he bsee O sees he efeenial f he bsee O ing alngsie he ais in he negaie ay bease he signal f he eqain is negaie. The eqain.6 esibes he nsany piniple f he spee f ligh ha s be asse by he eqains.6.9:.6 pplying.6 an.7 in.6 e hae: ( ) hee e hae: 7/

28 ( ) hee aing in he baes in a an in he saigh baes e hae he eqaliy beeen bh sies f he eqal signal f he eqain. ppllying in e hae. ppllying in. e hae. Tha applie in. splies: (, ). as (, ) is eqal he fnin epening f he aiables an. pplying.8 an.9 in.6 e hae: hee e hae: hee aing in he bae in a an in he saigh bae e hae he eqaliy beeen bh sies f he eqal signal f he eqain. (,). pplying an. in e hae: as (, ) is eqal he fnin epening n he aiables an. We s ae he flling naing aing.5 an.6:..5 s he eqain (, ) f. an (, ) f. s be eqal, e hae: 8/

29 .6 Ths: aly eqal...7 Reiing he eqains.6,.7,.8 an.9 aing he fnin f, an e hae:.6.7 Wih he espeie inee ee eqains:.8.9 We hae he eqains.6,.7,.8 an.9 finals eplaing by he espning flae:.6.7 Wih he espeie inee final eqains:.8.9 Tha ae ealy he eqains f he able I s an hen he elains beeen an ae..8 We ill ansf (.) fnin f he eleens,, an f (.) fnin f he eleens, an, eplaing in. he eqains.8,.9 an.8: ( ) ( ) 9/

30 Tha is ealy he eqain.. We ill ansf (.) fnin f he eleens,, an f (.) fnin f he eleens, an, eplaing in. he eqains.6,.7 an.8: ( ) Tha is ealy he eqain.. We hae allae he al ifeenial f (, ) (.): as: an.9 e hae: hee applying.8 e fin:. hee e nle ha fnin f an is a nsan. We hae allae he al ifeenial f (, ) (.): as: an. e hae:. /

31 hee applying. e fin: hee e nle ha fnin f an is a nsan. The eqains. an.8 epesen he bsees O an O he piniple f nsany f he ligh spee ali f infiniely sall he infiniely big an ean ha in he Unlaing Relaiiy he spae an ie ae ease silanesly. They shln be inepee ih a epeneny beeen spae an ie. The ie has is n inepeain ha an be nes if e analyze a eeine bsee he eissin f ays f ligh f he insan ze. If e a he ies e ge, f eah ay f ligh, e ill ge a esl ih any se f he physis. If in he insan ze, he bsee O eis ays f ligh, ne alngsie he ais an he he alngsie he ais y, afe he ineal f ie, he ays hi f he bsee O, silanesly, he pins an y he isane f he igin, alhgh f he bsee O, he pins n be hi silanesly. bh ays f lighs be silanes bh bsees, hey s hi he pins ha hae he sae ais in elain he ais an ha pie he sae ie f bh bsees ( an ), hih eans ha nly ne ay f ligh is neessay he he ie beeen he efeenials. ing, bh efeenials f he bsees O an O ae ineial, hs he ligh speas in a saigh line aing ha is eane by he fnaenal ai f he Unlaing Relaiiy, bease f his, he iffeene in eliies an is e nly a iffeene in ie beeen he efeenials... We an als elae na ineial efeenial f hih he ligh spea in a saigh line aing ha is eane by he fnaenal ai f he Unlaing Relaiiy, ih an aeleae ing efeenial f hih he ligh spea in a e line, nsieing ha in his ase he iffeene an isn e nly he iffeene f ie beeen he efeenials. ing, if he bsee O a he insan ze, eis a ay f ligh f he igin f is efeenial, afe an ineal f ie, he ay f ligh his he pin ih inaes (, y, z, ) he isane f he igin f he bsee O, hen e hae: fe hiing he pin he ay f ligh sill spea in he sae iein an in he sae ay, afe an ineal f ie, he ay f ligh his he pin ih inaes (, y y, z z, ) he isane he pin, hen e hae: an ih his e ge: ( ) ( ) ( ) ( ) The geey f spae an ie in he Unlaing Relaiiy is saize in he fige bel ha an be epane n pins an seeal bsees. /

32 In he fige he angles hae a elain ψ an ae eqal he flling segens: O O O is eqal O O O ( O O ) O O is eqal O O ( O O ( ) ( ) O O O O ) n ae paallel he flling segens: O is paallel O O is paallel O X X is paallel X X The sine f he angles f inlinain an he ays f he bsees O an O aing. an. ae: s / s sen n ih his e hae: sen s s / s / s. s /. s s / s.5

33 n ih his e hae sen sen.6 The sine f he angle ψ ih inesein f ays eqal : s s s ψ.7 n ih his e hae: The inaiane f he Unlaing Relaiiy. sen sen sen ψ.8 s ψ shs he hany f all ape hypheses f spae an ie in he The aials s s s ψ is eqal he Jabians f he ansfains f he spae an ie f he pie I, hee he an J i j ae nsiee aiables an ae eie. (,y,z, ) (,y,z, ) / ψ 8.8 J l / (,y,z, ) (,y,z, ) ψ 8.8 Riha C. Tlan The Tansfains f he Mena f Unlaing Relaiiy as eelpe base n he epeiene ne by eis an Tlan, aing he efeene []. Whee he llisin f sphees peseing he piniple f nseain f enegy an he piniple f nseain f ena, shs ha he ass is a fnin f he eliy aing : hee is he ass f he sphee hen in esing psiin an / he le f is spee. nalyzing he llisin beeen ienial sphees hen in elaie esing psiin, ha f he bsee O ae nae S an S ae ing alng he ais in he nay ay ih he flling eliies befe he llisin: Table sphee S sphee S y ze y ze z ze z ze he bsee O he sae sphees ae nae S an S an hae he eliies,, y z ze befe he llisin allae aing he able as flls: i i

34 The eliy f he sphee S is eqals :. The ansfain f aing. f Table is:. Tha applie in spplies: The eliy f he sphee S is eqal : ( ) Table Sphee S Sphee S ze y ze y ze z ze z ze ze he bsees O an O he sphees hae he sae ass hen in elaie esing psiin. n f he bsee O he sphees llie ih eliies f eqal le an ppsie iein bease f his he ena ( p p ) nll heseles ing he llisin, fing f a bief ie ( ) nly ne by f ass. ing he piniple f nseain f ena f he bsee O e ill hae ipse ha he ena befe he llisin ae eqal he ena afe he llisin, hs: ( ) Whee f he bsee O, is he abiay eliy ha sppsely f a bief ie asses nie ( ) ing. s he asses i ill als see he hae iffeen eliies an he asses ay aing hei n eliies, his eqain ann be siplifie algebaially, haing his aiain f asses: T he lef sie f he eqal sign in he eqain e hae: /

35 5/ ze ze T he igh sie f he eqal sign in he eqain e hae: pplying in he eqain f nseain f ena e hae:. hee e hae: s f he bsee O he asses nie ln e enaily alngsie he bsee O hih is neiable if e nsie ha he insans ae iffeen hee sppsely he asses l be in a esing psiin f he pin f ie f eah bsee an ha he ass aing ih eliy is bigge han he ass in esing psiin. If e peae ih hese aiables in line e l hae:.

36 hee e nle ha hih s be eqal he peis ale f, ha is: elain beeen an ha is baine f Table hen he eliy aing e he sphee in esing psiin. Refeene Millenni Relaiiy Veliies psiin UR: hp://.elaiiy.ne/miefs/vcp_si_sab_way.h ha espns f he bsee O e s ie he ansfains f eni. enz f spae an ie in he Speial They f Relaiiy:.a.a y y.b y y.b z z. z z... he e bain he eqains f eliy ansfain: y z y.5a.5b z.5 y z y.6a.6b z.6 e s nsie ha in elain he bsee O an bje es ih eliy: 5,5. / s(,5 ). 6/

37 n ha he eliy f he bsee O in elain he bsee O is: 5,5. / s(,5 ). The eliy f he bje in elain he bsee O s be allae by he fla.6a: 5 5,5.,5. 5,5..,5. 5 (,. ) 5 5,. / s(,8 ). 5 Whee e se,. / s(, ). Cnsieing ha he bje has e ing ne sen in elain he bsee O (,s ) e an hen ih. allae he ie passe he bsee O :,5., 5.,. 5 (,. ) 5 (,5. ) 5 (,. ) 5,6, 75,69s T he bsee O he bsee O is aay he isane gien by he fla: 5 5,5..,,5.. T he bsee O he bsee O is aay he isane gien by he fla: 5.,6 5,5.,9.., 75 T he isane f he bje (, ) in elain he bsees O an O is gien by he flae: O O 5 5,..,,.. O 5,6 5,5..,9.., 75 O T he bsee O he isane beeen he bje an he bsee O is gien by he fla: 5 5 5,.,5.,9.. O T he bsee O he eliy f he bje in elain he bsee O is gien by: 5,9.,9. / s,s 5 (,) Relaing he ies an sing he fla is nly pssible an elsiely hen an ze ha isn he ase abe, ae i pssible nesan his e ie he eqains. an. in he fla bel:. s. s. 7/

38 Whee s an s. The eqains abe an be ien as: f (,) e f (, ).7 In eah efeenial f he bsees O an O he ligh ppagain eaes a sphee ih ais an ha ineep eah he fing a ifeene ha ppagaes ih eliy. The ais an an he psiie ay f he ais an f he angles an nsan beeen he efeenials. If f he sae pai f efeenials e angles ee aiable he ie l be alleay an l bee seless f he Physis. In he eqain f (,) e hae ienial fnin f an, if e hae in i nsan an aies aing e ge he n elain beeen he ies an beeen efeenials, hee if e hae nsan an aies aing e ill hae f eah ale f f,. ne ale f an beeen iffeen efeenials, an his analysis is als ali f iiing.5a by e hae: s s s Whee s an s Islaing he eliy e hae:..8. ( ss ) ( ss ).9 hee e nle ha e s hae angles an nsan s ha e hae he sae eliy beeen he efeenials. This ean f nsan angles beeen he efeenials s sle he nesies f ebe ingle. 5 Inaiane The ansfains he spae an ie f able I, gp. pls.7, in he ai f is ien lie his: y y z z 5. Tha ien in he f bel epesens he sae inae ansfains: / y y z z 5. We all as: i y z /, α αij, j y z 8/ 5.

39 Tha ae he fnins i i j i i ( ) (,,, ) (, y, z, ) 5. Tha in he sybli f is ien: α. in he inee f i α j ij j i α ij j 5.5 Whee e se insein s s nenin. The ansfains he spae an ie f able I, gp. pls.8, in he ai f is ien: y y z z 5.6 Tha ien in he f bel epesens he sae inae ansfains: / y y z z 5.7 Tha e all as: y z /, α α l, l y z l Tha ae he fnins ( ) (,,, ) (, y, z, ) Tha in he sybli f is ien: α. in he inee f α l l l α eing (.7), (.8) an. (.). l l 5. The ansfain aies α α ij an α l α hae he ppeies: / / α. α αijα l αijα jl I δ j α α α jiα l α jiα i I δ i / / i l j Whee α α ji is he anspse ai f α ij δ is he nee s ela. α an α α l is he anspse ai f α α l an / / α. α α l αij α l α lj I δ j l 5. 9/

40 α α α l α ji α l α i I δ / / l i 5. Whee α α l is he anspse ai f α l an δ is he nee s ela. Obseain: he aies an α α. ij The paial eiaies l α ij an i i j elae aing ( ) nsiee nsan an eqal : l α an α α ji is he anspse ai f α α ij α α α ae inese f ne anhe b ae n hgnal, ha is: ji l i i i j f he al iffeenial f he inae pnens ha j j, hee in he ansfain ai α α ij he aial is Table, paial eiaies f he inae pnens: i j j i j j i j j i j j The al iffeenial f he inaes in he ai f is eqal : / 5.5 Tha e all as: i, i j i j /, j 5.6 i i Then e hae j j j i i j j 5.7 The paial eiaies elae aing nsiee nsan an eqal : l f he al iffeenial f he inae pnens ha l l l, hee in he ansfain ai α α l he aial is /

41 Table paial eiaies f he inae pnens: l l l l l l l l The al iffeenial f he inaes in he ai f is eqal : / 5.8 Tha e all as:, l l /, l 5.9 Then e hae: l l l l l 5. The Jabians f he ansfains 5.5 an 5.8 ae: J J l i j (,,, ) (,,, ) (,,, ) (,,, ) / / Whee (.5), (.6) an. (.). The aies f he ansfain an aies α an α. als hae he ppeies 5., 5., 5. an 5. f he l l he fnin ( ) [ ( )] hee he inaes elae in he f ( ) hae esibe as: l l e /

42 / Tha in he ai f an ih pesening he fnin bees: l Whee eplaing he ies bel: Obseain: his las elain shs ha he ie aies in an eqal f beeen he efeenials. We ge: l Tha is he gp 8. pls 8. f he able 9, iffeenial peas, in he ai f. he fnin [ ] j i i hee he inaes elae in he f j i i e hae j i i j esibe as:

43 / i i i i i i i i Tha in he ai f an ih pesening he fnin bees: j Whee eplaing he ies bel: Obseain: his las elain shs ha he ie aies in an eqal f beeen he efeenials. We ge: j Tha is he gp 8. pls 8. f he able 9, iffeenial peas in he ai f. pplying 8.5 in 8. an in 8. e siplify hese eqains in he flling ay:

44 Table 9, iffeenial peas ih he eqains 8. an 8. siplifie: ze ze The able 9, in he ai f bees: 5. / 5. / The sqae aies f he ansfains abe ae anspse f he aies an. Inaiane f he Tal iffeenial In he bsee O efeenial he al iffeenial f a fnin ( ) is eqal : ( ) 5.5 l Whee he inaes elae ih he nes f he bsee O aing ( ) ansfains 5. an 5.8 an ih pesening he fnin e hae:, eplaing he / 5.6 / The lipliain f he ile aies spplies: / / / / 5.7 Resl ha an be iie in aies: / / / / 5.8 /

45 Tha applie he al iffeenial spplies: / 5.9 / eing he peains f he sen e e hae: / Whee applying 8.5 e hae: / ( ) ze Then e hae: / / ze 5. Wih his esl e hae in 5.9 he inaiane f he al iffeenial: l l In he bsee O efeenial he al iffeenial f a fnin ( i ) is eqal : 5. i i ( ) 5. i i i j Whee he inaes elae ih he nes f he bsee O efeenial aing ( ), eplaing he ansfains 5. an 5.5 an ih pesening he fnin e hae: / i 5. i / The lipliain f he ile aies spplies: / / / / 5. Resl ha an be iie in aies: 5/

46 6/ / / / / 5.5 Tha applie he al iffeenial spplies: / / i i 5.6 eing he peains f he sen e e hae: / / Whee applying 8.5 e hae: ze Then e hae: ze / / 5.7 Wih his esl e hae in 5.6 he inaiane f he al iffeenial: j j i i 5.8 Inaiane f he Wae qain The ae eqain he bsee O is eqal : 5.9 Whee applying 5. an he anspse f 5. e hae:

47 5. The lipliain f he hee ile aies spplies: 5. Resl ha an be iie in aies: 5. Tha applie in he ae eqain spplies: 5. eing he peains f he sen e e hae: eing he peains e hae: ( ) Whee applying 8.5 e hae: ( ) Then e hae: ( ) ze ( ) 7/

48 8/ ze 5. Wih his esl e hae in 5. he inaiane f he ae eqain: 5.5 The ae eqain he bsee O is eqal : 5.6 Whee applying 5. an he anspse f 5. e hae: 5.7 The lipliain f he hee ile aies spplies: 5.8 Resl ha an be iie in aies: 5.9 Tha applie in he ae eqain spplies:

49 9/ 5.5 eing he peains f he sen e e hae: eing he peains e hae: Whee applying 8.5 e hae: ze Then e hae: ze 5.5 Then in 5.5 e hae he inaiane f he ae eqain: 5.5 Inaiane f he eqains 8.5 f linea ppagain Replaing., 8., 8. in 8.5 e hae: ze eing he peains e hae: ze

50 Tha siplifie spplies he inaiane f he eqain 8.5: ze Replaing., 8., 8. in 8.5 e hae: ze eing he peains e hae: ze Tha siplifie spplies he inaiane f he eqain 8.5: ze The able in a ai f bees: p / p p p p 5.5 p / / p / p p p p 5.5 p / / The able 6 in a ai f bees: J / J J J J J ρ ρ J / J J J J J ρ ρ Inaiane f he Cniniy qain The niniy eqain he bsee O is eqal : J ρ J J J ρ. J J ze J ρ 5.57 Whee eplaing 5. an 5.56 e hae: / J ρ. J J ze J / ρ /

51 The p f he ansfain aies is gien in 5.7 an 5.8 ih his: ρ.j / / J J J ρ 5.59 eing he peains f he sen e e hae: / ρ / J J J J ρ Whee eplaing J an 8.5 e hae: ( ) ρ ze ρ ρ Then e hae: ρ ρ / / J J J ze ρ 5.6 Wih his esl e hae in 5.59 he inaiane f he niniy eqain: ρ.j J J ρ.j J ρ 5.6 The niniy eqain he bsee O is eqal : J ρ J J J ρ. J J ze J ρ Whee eplaing 5. an 5.55 e hae: / J ρ. J J ze J / ρ The p f he ansfain aies is gien in 5. an 5.5 hen e hae: ρ J / / J J J ρ. 5.6 eing he peains f he sen e e hae: 5/

52 / J J J ρ J / ρ ρ Whee eplaing J an 8.5 e hae: ( ) ρ ze ρ ρ ρ Then e hae: / J J ze J / ρ 5.65 Wih his esl e hae in 5.6 he inaiane f he niniy eqain: ρ.j J J ρ.j J ρ 5.66 Inaiane f he line iffeenial eleen: Tha he bsee O is ien his ay: ( s ) ( ) ( ) ( ) ( ) [ ] Whee eplaing 5.8 an he anspse f 5.8 e hae: 5.67 s 5.68 [ ] The lipliain f he hee enal aies spplies: 5.69 Resl ha an be iie in aies: 5.7 Tha applie in he line iffeenial eleen spplies: 5/

53 s 5.7 [ ] eing he peains f he sen e e hae: [ ] ze Then e hae: [ ] ze 5.7 Wih his esl e hae in 5.7 he inaiane f he line iffeenial eleen: 5.7 ( s ) [ ] ( ) ( ) ( ) ( ) ( s ) T he bsee O he line iffeenial eleen is ien his ay: ( s ) ( ) ( ) ( ) ( ) [ ] Whee eplaing 5.5 an he anspse f 5.5 e hae: 5.7 s 5.75 [ ] The lipliain f he hee enal aies spplies: 5.76 Resl ha an be iie in aies: /

54 Tha applie in he line iffeenial eleen spplies: s 5.78 [ ] eing he peains f he sen e e hae: [ ] ze Then e hae: [ ] ze 5.79 Wih his esl e hae in 5.78 he inaiane f he line iffeenial eleen: 5.8 ( s ) [ ] ( ) ( ) ( ) ( ) ( s) In 7 as a nseqene f 5. e ha he inaiane f.. hee n applying 7.., 7.., 7.., 7.. an he eliy ansfain flae f able e hae ne elains beeen an isin f 7. an 7. an ih he e eie he able 7 in he f bel: Table 7 y y y y z z 7.. z z y y z y y z z z y z z y y z y z z y z y Wih he ables 7 an 9 e an hae he inaiane f all Maell s eqains. 5/

55 Inaiane f he Gass a f he eleial fiel: y z y z ρ 8. Whee applying he ables 6, 7 an 9 e hae: y z ρ ( /) y z Whee siplifying an eplaing 8.5 e hae: Tha eee spplies: y z ρ ( / ) y z y z ρ ( / ) y z Tha siplifie spplies he inaiane f he Gass a f he eleial fiel. Inaiane f he Gass a f he agnei fiel:. y z ze y z 8.6 Whee applying he ables 7 an 9 e hae: y y Tha eee spplies: z z z y z z y y z y z y Whee he e in paenhesis is he aaay-eny s a (8.9) ha is eqal ze f hee e hae he inaiane f he Gass a f he agnei fiel. Inaiane f he aaay-eny s a: y z y 8.8 Whee applying he ables 7 an 9 e hae: y y ( /) z Tha siplifie an liplie by ( /) e hae: y z y y Whee eeing he ps an eplaing 7.9. e hae: 55/

56 y z y y y s he e in paenhesis is he eqain 8.5 ha is eqal ze hen e hae he inaiane f he aaay-eny s a. Inaiane f he aaay-eny s a: z y y z 8. Whee applying he ables 7 an 9 e hae: z y y z Tha siplifie spplies he inaiane f he aaay-eny s a. Inaiane f he aaay-eny s a: z y z 8. Whee applying he ables 7 an 9 e hae: ( / ) z y z Tha siplifie an liplie by ( /) e hae: z z z y Tha siplifying an aing he peains e hae: z y z Whee applying 7.9 e hae: z y z z y z z. z z s he e in paenhesis is he eqain 8.5 ha is eqal ze hen e hae he inaiane f he aaay-eny s a. Inaiane f he pee-maell s a: y y z J z µ µ 8. Whee applying he ables 6, 7 an 9 e hae: y z µ Jz µ y z Tha siplifying an aing he peains e hae: 56/

57 57/ z y z z z z Jz y y µ µ Whee siplifying an applying 7.9 e hae: z z z z Jz y y µ µ Tha eganize spplies z z z Jz y y µ µ s he e in paenhesis is he eqain 8.5 ha is eqal ze hen e hae he inaiane f he pee-maell s a: Inaiane f he pee-maell s a: J z y y z µ µ 8.6 Whee applying he ables 6, 7 an 9 e hae: / J z y z y z y µ ρ µ Maing he peains e hae: / z z y y J z y y z µ ρ µ µ Replaing in he fis paenhesis he Gass a an liplying by e hae: J z y y z J z y y z µ µ µ Whee eplaing J ρ, 7.9., 7.9 an 8.5 e hae: z z y y J z y y z ρ µ µ µ Tha siplifie spplies: z z y y J z y y z ρ µ µ µ Replaing in he fis paenhesis he Gass a e hae: J z y y z µ µ Tha eganize aes: J z y y z µ µ

58 s he e in paenhesis is he eqain 8.5 ha is eqal ze hen e hae he inaiane f he pee-maell s a: Inaiane f he pee-maell s a: z y µ J y µ 8.8 z Whee applying he ables 6, 7 an 9 e hae: z z Maing he peains e hae: y µ Jy µ y z y µ Jy µ z y Whee siplifying an applying 7.9. e hae: z y µ Jy µ z Tha eganize aes: y z y y y µ Jy µ z y y y y z s he e in paenhesis is he eqain 8.5 ha is eqal ze hen e hae he inaiane f he pee-maell s a: Inaiane f he Gass a f he eleial fiel ih eleial hage: y y z ze y z 8. Whee applying he ables 7 an 9 e hae: ( / ) y z y z ze Whee siplifying an eplaing 8.5 e hae: ( / ) Tha eganize aes: ( / ) y z ze y z y z ze y z. Tha siplifie spplies he Gass a f he eleial fiel ih eleial hage. 58/

59 Inaiane f he pee-maell s a ih eleial hage: y y z µ 8. Whee applying he ables 7 an 9 e hae: y z µ y Maing he peains e hae: z y z µ y z Whee siplifying an applying 7.9 e hae: y z µ y Tha eganize aes: z y z z z y µ z z z z y s he e in paenhesis is he eqain 8.5 ha is eqal ze hen e hae he inaiane f he pee-maell s a ih eleial hage: Inaiane f he pee-maell s a ih eleial hage: z y y z µ z 8. Whee applying he ables 7 an 9 e hae: z y y y z z µ ( / ) Maing he peains e hae: z y y z y z y z µ ( / ) Replaing in he fis paenhesis he Gass a ih eleial hage an liplying by ( / ) hae: z y µ y z z y y z Whee eplaing 7.9, 7.9. an 8.5 e hae: z y µ y z Tha siplifie spplies: y z y z e 59/

60 6/ z z y y z y y z µ Replaing in he fis paenhesis he Gass a ih eleial hage e hae: z y y z µ Tha eganize aes: J z y y z µ µ s he e in paenhesis is he eqain 8.5 ha is eqal ze hen e hae he inaiane f he pee-maell s a ih eleial hage: Inaiane f he pee-maell s a ih eleial hage: y z z µ 8. Whee applying he ables 6, 7 an 9 e hae: y y z z µ Maing he peains e hae: y z y y y y z z µ Whee siplifying an applying 7.9. e hae: y y y y z z µ Tha eganize aes: y y y z z µ s he e in paenhesis is he eqain 8.5 ha is eqal ze hen e hae he inaiane f he pee-maell s a ih eleial hage: 5 Inaiane (ninain) fnin f f θ.9 Whee he phase is eqal θ 5.8 In e epesen an nlaing een ha ges n in ne abiay iein s ply ih he ae eqain an bease f his e hae: ze f f z y f z y θ θ θ θ θ θ 5.8 Tha esn ee ih he ae eqain bease he las eleens ge nle b he fis ne esn.

61 In e ee his pble e eflae he phase θ f he fnin in he flling ay. niay e sh as n s i sαj s β 5.8 hee s, y y s α, z z s β 5.8 has he le eqal n n n.n s s α s β Maing he p y z ( si sαj sβ )(. i yj z ) s sαy sβz n.r e hae n.r s sαy s βz Φ ( ) ( n.r ) ( s sαy s βz ) ha applie he phase θ spplies a ne phase ih he sae eaning f he peis phase θ Φ. Replaing phase in he f n.r s sαy s βz e in he phase θ liplie by e als ge anhe Φ s sαy s βz 5.88 θ. ih he sae eaning f he peis phase Φ Ths e an ie a ne fnin as: f s sαy sβz ( Φ) f 5.89 Tha eplae in he ae eqain ih he ie sine nsiee nsan spplies: f Φ ( Φ) f ( Φ) f ( Φ) f ( Φ) s s α s β ze 5.9 Φ Φ Φ ha siplifie ees he ae eqain. The psiie esl f he phase Φ in he ae eqain is an elsie nseqene f he ie sines being nsan in he paial eiaies shing ha he ae eqain eans he ppagain hae ne seay iein in he spae (plane ae). he bsee O a se lae in he igin f is efeenial pes in a an pin lae a he isane y z f he igin, an eleial fiel esibe by: i yj z 5.9 6/

62 Whee he pnens ae esibe as: y z y z. f. f. f ( Φ) ( Φ) ( Φ) 5.9 Tha applie in spplies: f f ( Φ) i f ( Φ) j f ( Φ) [ i j ] f ( Φ) ( Φ) y z ih le eqal ( ) ( ) ( ).f ( Φ). f ( Φ) eing y z y z i j 5.95 y z The ai aplie e Cnsan ih he pnens, y, z 5.96 n le y z 5.97 eing f ( Φ) a fnin ih he phase Φ eqal eiing he pnen in elain an e hae: ( Φ) Φ f ( Φ) f ( Φ) f ( Φ) f Φ Φ Φ ( Φ) Φ f ( Φ) f ( Φ) ( ) f Φ Φ Φ Φ ha applie in 8.5 spplies ( Φ) Φ / f ( Φ) Φ f ( Φ) / f Φ / Φ ze ze ze Φ Φ Φ ( Φ) f Φ / Φ Φ / Φ ze ze Φ ensaing ha i is he phase Φ ha s ply ih 8.5. Φ as / Φ ze ( ) / ( ) hen plies ih 8.5. ze / 5. ( ) ze ze s he phase is he sae f he pnens y an z hen hey als ply ih 8.5. s he phases f he bsees O an O ae eqal ( ) ( ) bsee O als ply ih 8.5. / ( ) / ( ) hen he pnens f he ze 5. 6/

63 The pnens elaiely he bsee O f he eleial fiel ae ansfe f he efeenial f he bsee O aing he ables 7, 7 an 8. pplying in 8.5 a ae fnin ien in he f: Ψ e ( ) iφ e ( ) i ( ) Φ i Φ i s sin s sin 5. hee i. eiing e hae: Ψ senφ is Φ en Ψ senφ i s Φ 5. Ψ e iφ an Ψ e iφ 5. Tha applie in 8.5 spplies: Ψ / Ψ ze / ( senφ i sφ) ( senφ i sφ) ze ha is eqal : i sin Φ i sφ ze Ψ / Ψ iφ / ze e e ze hee e s hae he effiiens eqal ze s ha e ge na ieniy, hen: ze i i ze iφ / iφ ( e ) ( e ) ze iφ Whee applying e hae: Then ee ih he eqain 8.5 e s hae a ae ppagain alng he ais ih he spee. If e apply an e hae:. esl als gen f he is e glie s ae eqain. 6/

64 Cnsieing he pple effe as a la f physis. 6 Tie an eqeny We an efine a l as any eie ha pes a feqeny f ienial eens in a seies pssible be enlise an ae in sh a ay ha a an een n f a eie ill be ienial any een in he seies f eens pe by a eplia f his eie hen he eens ae pae in a elaie esing psiin. The ylial een f a l in a esing psiin aing he bsee O efeenial ses he ie in his efeenial an he ylial een f he as f a l in a esing psiin aing he bsee O ses he ie in his efeenial. The flas f ie ansfain.7 an.8 elae he ies beeen he efeenials in elaie een hs, elae eens in elaie een. The elaie een beeen he ineial efeenials pes he pple effe ha pes ha he feqeny aies ih eliy an as he feqeny an be inepee as being he feqeny f he ylial een f he as f a l hen he ie aies in he sae ppin ha aies he feqeny ih he elaie een ha is, i is engh eplae he ie an in he flas.7 an.8 by he feqenies y an y ge he flas f feqeny ansfain, hen: y y.7 bees. y y.8 bees. The Galile s ansfain f eliies beeen ineial efeenials pesens ininsially hee efes ha an be esibe his ay: a) The Galile s ansfain f eliy he ais is. In ha ne if e hae hen an if e hae hen. s bh esls ae n silanesly pssible else e hae hen he ansfain esn all ha a ay f ligh be silanesly bsee by he bsees O an O ha shs he piilege f an bsee in elain he he bease eah bsee an nly see he ay f ligh nning in is n efeenial (ininsi efe he lassi analysis f he Sagna s effe). b) I ann als ply Nen s fis la f ineia bease a ay f ligh eie paallel he ais f he igin f he espeie ineial efeenials a he en ha he igins ae inien an a he en in hih ze ill hae by he Galile s ansfain he eliy f ligh alee by ± he efeenials, n he nay f he ineial la ha ln all he eisene f a aiain in eliy bease hee is n eenal ain aing n he ay f ligh an bease f his bh bsees shl see he ay f ligh ih eliy. ) s i nsies he ie as a nsan beeen he efeenials i esn pe he epal aiain beeen he efeenials in een as i is eqie by he pple effe. The piniple f nsany f ligh eliy is nhing b a eqieen f he Nen s fis la, he ineia la. Nen s fis la, he ineia la, is ine in Galile s ansfain hen he piniple f nsany f ligh eliy is applie in Galile s ansfain piing he eqain f ables an f he Unlaing Relaiiy ha esn hae he hee efes esibe. The ie an eliy eqains f ables an an be ien as: s.7.5 s 6/

65 s.8. s The isane beeen he efeenials is eqal he p f eliy by ie his ay:.9 I esn epen n he ppagain angle f he ay f ligh, being elsiely a fnin f eliy an ie, ha is, he ppagain angle f he ay f ligh, nly ales beeen he ineial efeenial he ppin beeen ie an eliy, eeping he isane nsan in eah en, any ppagain angle. The eqains abe in a fnin f ae ien as: (,) e (, ) e.9 (,, ) f.7 (, ) (,, ) g.5 f.8 (, ) g. Then e hae ha he isane is a fnin f aiables, he ie a fnin f hee aiables an he eliy a fnin f aiables. he efiniin f en. an enegy.6 e hae: p 6. The eleae he pe f spplies: p 6. leaing he pe f he enegy fla e hae: Whee applying 6. e hae: p p.8 hee e nle ha if he ass in esing psiin f a paile is nll ze he paile enegy is eqal p /

66 Tha applie in 6. spplies: p p 6. ( p) hee e nle ha he een f a paile ih a nll ass in esing psiin alays be a he eliy f ligh. ze ill pplying in p he elains yh an yλ e hae: yh yλp p h an in he sae ay λ p h λ 6.5 qain ha elaes he en f a paile ih a nll ass in esing psiin ih is n ay lengh. leaing he pe f he fla f en ansfain (.9) e hae: p p p p p Whee applying p an p ps p e fin: ( p) p p p p p p p p 6.6 Whee applying 6.5 esls in: h h λ λ p p λ inee λ. λ λ Whee applying yλ an yλ e hae: y y inee y y. In e hae he eqains. an. applying he piniple f elaiiy he ae phase. 7 Tansfain f. enz bsees in a elaie een, he eqain ha epesens he piniple f nsany f ligh spee f a an pin is: 7. y z y z In his eqain aneling he syei es e hae: Nesa anelan s es siéis bes: 7. Tha e an ie as: ( )( ) ( )( ) 7. If in his eqain e efine he ppin fas η an µ as: ( ) η( ) ( ) µ ( ) 66/ 7.

67 67/ hee e s hae µ η. ply 7.. The eqains 7. hee fis gen by lbe insein. When a ay f ligh es in he plane y z he bsee O e hae ze an an sh niins applie he eqain 7. spplies: 7.5 This esl ill als be spplie by he eqains an f he gp 7. ne he sae niins: µ η 7.6 hse e hae: η an µ 7.7 Whee e hae pen ha µ η.. he gp 7. e hae he Tansfains f. enz: η µ µ η 7.8 µ η η µ 7.9 µ η µ η 7. µ η µ η 7. Inees eqains µ η, η µ an µ η : µ η µ η 7. η µ η µ 7. µ η µ η 7.

68 Sagna effe When bh bsees igins ae eqal he ie is zee ( ze) in bh efeenials an ays f ligh ae eie f he n igin, ne in he psiie iein (lise ine ) f he ais an ih a ae fn an anhe in he negaie iein (ne-lise ine ) f he ais an ih a ae fn. The ppagain niins abe applie he enz eqains spply he ables an bel: Table qain Clise ay () qain Cne-lise ay () S f ays Resl Resl Cniin Cniin Table µ η µ η µ 7.9 η µ η µ η qain Clise ay () qain Cne-lise ay () S f ays Resl Resl Cniin Cniin 7. η 7. µ η µ η µ 7. η 7. µ η µ We bsee ha he ables an ae inese ne anhe. When e f he gp f he s eqains f he ays f ables an : µ η η µ 7.5 Whee he bsee O is he isane beeen he fn aes an an hee he bsee O is he isane beeen he fn aes an. f he In he eqains 7.5 abe, e he ispy f spae an ie an he fn aes ays f ligh being he sae f bh bsees, he s f ays f ligh e ies s be inaiable beeen he bsees, hih e an epess by: 7.6 This esl ha geneaes an eqain f ispy f spae an ie an be alle as he nseain f spae an ie piniple. The hee hyphesis f ppagain efine as flls ill be applie in 7.5 an ese pe he nseain f spae an ie piniple gien by 7.6: 68/

69 yphesis : If he spae an ie ae ispi an hee is n een ih n piilege f ne bsee nsiee e he he in an epy spae hen he ppagain geey f ays f ligh an be gien by: an 7.7 This hyphesis applie he eqain f he gp 7.5 plies he spae an ie nseain piniple gien by 7.6. The hyphesis 7.7 applie he ables an esls in: µ a η a yphesis : η µ C 7.8 If he spae an ie ae ispi b he bsee O is in an absle esing psiin in an epy spae hen he geey f ppagain f he ays f ligh is gien by: 7.9 Tha applie he able an esls in: a µ η η a µ µ η Sing an in 7. e hae: C η µ η µ i This esl esn ply ih he nseain f spae an ie piniple gien by 7.6 an as esls in a siain f f ays f ligh, eah bsee, an eah ay f ligh ih is espeie inepenen fn ae f he hes. yphesis C: If he spae an ie ae ispi b he bsee O is in an absle esing psiin in an epy spae hen he ppagain geey f he ays f ligh is gien: 7. Tha applie he ables an esls in: a a µ η η µ C 7. η µ /

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