Undulating Relativity ABSTRACT

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1 Unling Reliiy h: lfe is Mei Gi -il: STRCT The Speil They f Reliiy es s esls h pesenly e nsiee ineplible ny enne sieniss, n: -The ilin f ie, n -The nin f he enz engh. The slin hese he ien he h he eelpen f he Unling Reliiy (UR) hey, hee he Tepl iin is e he iffeenes n he e f he ligh ppgin n he lenghs e nsns beeen lns in nif elie een. The Unling Reliiy pies nsfins beeen he lns h iffes f he nsfins f enz f: Spe (,y,z), Tie (), Spee ( ), elein ( ), negy (), Men ( p ), e ( ), leil iel ( ), Mgnei iel ( ), igh eqeny ( y ), leil Cen ( J ) n leil Chge ( ρ ). he nlysis f he eelpen f he Unling Reliiy, he flling n be synhesize: - I is hey ih piniples pleely n physis; - The nsfins e line; - Keeps nhe he liin piniples; - Cnsies he Glile s nsfin isin n eh efeenil; - Ties he Spee f igh n Tie niqe phenenn; - The enz fe n be ine by isin ypes f ile es, n - Wih he bsene f he spil nin f enz, eh he se lssil esls f he speil eliiy ning is n neessy s nle n he pple effe. h, he Unling Reliiy n he Speil Reliiy f lbe insein eplin he epeiene f Mihel- Mley, he lngiinl n nsesl pple effe, n spplies ely ienil flin : bein f zenih gα / esnel s fl ( ) n n.. Mss ( ) ih eliy ( ) = [esing ss ( )]/... Men p.. Relin beeen en (p) n negy ().. p. Relin beeen he elei fiel ( ) n he gnei fiel ( V ). μο.i i-sn s fl μ..π.r is e glie s e eqin ψ(,).si nπγ ; /8

2 Ohe Ws: 9 plining he Sgn ffe ih he Unling Reliiy. plining he epeiene f Ies-Silell ih he Unling Reliiy. Tnsfin f he pe f lins y beeen efeenils in he Speil They f Reliiy. ineiy. Rih C. Tln. Veliies psiin. 5 Inine. 6 Tie n eqeny. 7 Tnsfin f. enz. 8 The Mihelsn & Mley epeiene. 9 Regessin f he peihelin f Mey f 7,. 9 ne f Mey s peihelin f.79. Inei. Inei (lifiins) ne f Mey s peihelin f.79 lle ih he Unling Reliiy. Spil efin. Spe n Tie en. Viinl Piniple. 5 gihi Spil. 6 Mey Peihelin ne f.99". 7 neen f Peihelin f Mey f.99 "n Cniins". 8 Siplifie Peielli ne. 9 Y Penil negy Cninin. negy Cninin Clifiins Siple n Mehnis ein f in Shöinges qins Reliisi Vesin f in Sh inge qin /8

3 h: lfe is Mei Gi. e-il: Unling Reliiy Tnsl: Rlf Ms Venâni e-il: Tnsfin spe n ie The Unling Reliiy (UR) eep he piniple f he eliiy n he piniple f Cnsny f ligh spee, ely lie lbe insein s Speil Reliiy They efine: ) The ls, ne hih he se f physis syses e hnge e he se, eihe hen efee eeine syse f ines ny he h hs nif nslin een in elin he fis. b) ny y f ligh es in he esing ines syse ih eeine eliy, h is he se, hee his y is eie by esing by by by in een (hih eplins he epeiene f Mihel-Mley). e s igine fis h bsees O n O (in ), ing in nif nslin een in elin eh he, h is, he bsee n e eliely eh he. In his y, he bsee O gehe ih he is, y, n z f syse f engle Cesin ines, sees he bsee O e ih eliy, n he psiie is, ih he espeie pllel is n sliing lng ih he is hile he O, gehe ih he, y n z is f syse f engle Cesin ines sees O ing ih eliy, in negie iein s he is ih he espeie pllel is n sliing lng ih he is. The bsee O eses he ie n he O bsee eses he ie ( ). e s i h bh bsees se hei ls in sh y h, hen he iniene f he igin f he ine syse hppens = = ze. In he insn h = =, y f ligh is pjee f he n igin bh bsees. fe he ie inel he bsee O ill nie h his y f ligh h silnesly hi he ines pin (, y, z) ih he y f he O bsee ih eliy n h he igin f he syse f he O bsee hs n he isne lng he psiie y f he is, nling h: + y + z =. =.. The se y fe he ie inel he O bsee ill nie h his y f ligh silnesly hi ih he bsee O he ine pin (, y, z ) ih eliy n h he igin f he syse f he bsee O hs n he isne n he negie y f he is, nling h: + y + z =. = +.. Ming. eql. e he + y + z = + y + z..5 ese f he syey y = y en z = z, h siplify.5 in =..6 T he bsee O = (.) h pplie in.6 spplies = ( ) f hee..7 T he bsee O = + (.) h pplie in.6 spplies ( + ) = f hee /8

4 ..8 Tble I, nsfins he spe n ie =. = +. y = y.. y = y.. z = z.. z = z...7 he eqin syse fe by. n. e fin.8 = (nsieing > e >).9 h enses he inine f he spe in he Unly Reliiy. he eqin syse fe by.7 n.8 e fin. =.. If in. = hen =, h pplie in. spplies,. =.. If in. = n = hen... T he bsee O he piniple f ligh spee nsny gnees h he pnens, y n z f he ligh spee e ls nsn lng is is, hs y y z z, y, z. n hen e n ie =.. Wih he se f.7 n.9 n. e n ie = =..5 iffeeniing.9 ih nsn n, else, nly he ie ying e he,.6 b f.5 hen..7 eing n nsns, he ezns iffeenil f he se s.. n /8 in.5 s ls be nsn bese f his he s be eql ze f hee e nle, h is ely

5 T he bsee O he piniple f Cnsny f eliy f ligh gnees h he pnens, y, n z f eliy f ligh e ls nsn lngsie is is, hs y y z z, y, z,.8 n ih his e n ie, =..9 Wih he se f.8,.9, n.9 e n ie = =.. iffeeniing.9 ih n nsn, h is, nly he ie ying e he,. b f. hen.. eing n nsn he iisins iffeenil f ely lie.8. n in. ls he be nsn bese f his he s be eql ze f hee e nle Repling. n.9 in. e he, h is. =.. T he bsee O he e psiin f he pin f ines (,y,z) is R i yj z,. n he e psiin f he igin f he syse f he bsee O is R i j R i..5 T he bsee O, he e psiin f he pin f ines (,y,z ) is R i y j z,.6 n he e psiin f he igin f he syse f he bsee O is R i j R i..7 e.9,.5, n.7 e he, R s. is eql.5 pls.6 e he R R R R R R R pplying.8 in.9 e he, R R R.. 5/8

6 6/8 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,.,., n. e fin he flling elins beeen n... Obsein: in he ble I he fls.,.., n.. e he pnens f he e.9 n he fls.,.., n.. e he pnens f he e.. f eliy nsfins n iffeeniing.9 n iiing i by.7 e he K R R R.. iffeeniing. n iiing i by. e he K R R R.. Tble, f eliy nsfins n K. K. K. K. K y y.. K y y.. K z z.. K z z.. K.5 K. K.5 K.6

7 7/8 Mliplying. by iself e he..7 If in.7 e e = hen = s i is eqie by he piniple f nsny f eliy f ligh. Mliplying. by iself e he..8 If in.8 e e = hen = s i is eqie by he piniple f nsny f eliy f ligh. If in. e e = hen s i is eqie by he piniple f nsny f eliy f ligh. If in. e e = hen s i is eqie by he piniple f nsny f eliy f ligh. Reeling.7 n.8 e he =..9 =.. The ie elins beeen he ies n eliies f pins in spe n be bine ih he eqliies ing f., h pplie in.7,.,., n.5 spply,.,.,.

8 .. bein f he zenih T he bsee O lng ih he s =, y = n z =, n he bsee O lng ih he h e he he njn. y, y, z =, y z ely s feseen by he piniple f eliiy. T he bsee O he ligh ppges in iein h es n ngle ih he eil is y gien by / ngα.5 y. h is he bein fl f he zenih in he speil eliiy. If e inee he bsees e l he he njn., y y, z =, y z / ngα.6 y. h is eql.5, ih he negie sign iniing he ny iein f he ngles. Cnsieing in., elin he pps hen esnel s fl / n he eliy f ligh eliily he e, ill be he eliy f ligh eliely he lby / n / n / n n n n n Igning he e / e he n n n n n n igning he e / n e he he esnel s fl n n n n he eliy f e in n..7 8/8

9 Ming y z n s pple effe epling hen, hen e ill efine y z in.5 e he n.7 e fin hee en he piniple f eliiy.8 Resling in he epessin syei n inible beeen he bsees. T he bsee O n epessin in he fl f ψ, f.9 epesens e h ppges in he iein f R. T he bsee O n epessin in he fl f ψ, f. epesens e h ppes in he iein f pplying in.8 λ λ e K R. π π,,.,.9,.,.5, n.6 e he λ λ λ λ, K. h pplie in yλ y λ spply, y y K n y y K.. Cnsieing he elin f Pln-insein beeen enegy ( ) n feqeny ( y ), e he he bsee O hy n he bsee O hy h eple in. spply K n K.. If he bsee O h sees he bsee O ing ih eliy in psiie y he is, eis es f feqeny y n eliy in psiie y he is hen, ing. n he bsee O ill ese he es ih eliy n feqeny h is ely he lssi fl f he lngiinl pple effe. y y,. If he bsee O h sees he bsee O ing ih eliy in he negie y f he is, eis es f feqeny y n eliy, hen he bsee O ing. n ill ese es f feqeny y n eliy in pepenil plne he een f O gien by γ γ,.5 h is ely he fl f he nsesl pple effe in he Speil Reliiy. iffeeniing. n iiing i by.7 e he Tnsfins f he eleins n / K / K K.. K K K K iffeeniing. n iiing i by. e he / K K / K K K K K.. 9/8

10 Tble, nsfins f he eleins n.. K K K K.. K K K K y y y y K K.. y y K K z z z z K K.. z z K K.8.9 K K K.5 K he bles n e n nle h if he bsee O hen i is ls he bsee O n is pepenil. ze n s he es hey eqies.. ze n y z, hs is pepenil y z, iffeeniing.9 ih he eliies n he ies hnging e he,, b nsieing.6 e he,.7 Whee epling.5 n iiing i by.7 e he, We n ls eple. in.7 n iie i by. eing K K K K The ie elins beeen he les f he eleins n f pins in spe n be bine ih he ing f., h pplie in.8 n.9 spply n.. Th n ls be ee f. n. if e se he se eqliies efine s hee ing f.. Tnsfins f he Mens p n p p n p n n,. syblizes he fnin sses f he les f eliies n. We ill he he elins beeen n he esing ss, nlyzing he elsi llisin in plne beeen he sphee s h f he bsee es lngsie he is y ih eliy y = n he sphee s h f he bsee O es lngsie he is y ih eliy y = -. The sphees hile bsee in elie esing e ienil n he he ss. The nsiee llisin is syei in elin pllel line he is y n y pssing by he ene f he sphees in he en f. Cllisin. efe n fe he llisin he sphees he eliies bsee by O n O ing he flling ble gen f ble /8

11 Sphee Obsee O Obsee O efe s s ze, ys s, y s Cllisin s s, ys s ze, y s fe s s ze, ys s, y s Cllisin s s, ys s ze, y s T he bsee O, he piniple f nsein f ens esblishes h he ens py y p n, f he sphees s n s in elin he is n y, ein nsn befe n fe he llisin hs f he is e he s ys s s ys s s ys s s ys s hee epling he les f he ble e he f hee e nle h, n f he is y s ys ys s ys ys s ys ys s ys ys hee epling he les f he ble e he,, siplifying e he,, hee hen bees b is eql he esing ss hs, ih elie eliy. h pplie in. spplies p Wih he se pees e l he f he O bsee.., /8

12 . n p.. Siplifying he siblgy e ill p. n. h siplify he ens in p n p.. pplying. n. in.9 n. e he K n K efining fe s Nen e he inei enegy n, s p n.r.r... p.r.r. Reeling. n. n iffeeniing e he,., ih his e n efine hen n, h pplie in he fls f inei enegy spplies n hee n,.5.6 e he l enegies s in he speil eliiy n he esing enegy..7 pplying.6 in. e he ely...6,.,., n. e fin p n p.8 ienil elins he Speil Reliiy. Mliplying. n. by e ge /8

13 n p p p p.. Tble, nsfins f ens p n p p p.9 p p. p p. p p. p y py.. py p y.. p z pz.. pz p z.. K. K... K. K p p.8 We eqin f is e glie The bsee O ssies esing pile in is igin he flling ppeies:.9 -Resing ss -Tie -Resing negy -eqeny -We fnin y h h ψ senπ y ih = nsn. The bsee O ssies pile ih eliy he flling: -Mss -Tie (f. hee ) (f.7 ih n ) /8

14 -negy -eqeny (f. ih n ) y /h y (f. ih n y y ) -isne = (f. ih = ) -We fnin -We lengh ih ψ sen y senπ y senπ y yh h yλ λ (f.9 ih p p p p p ) T g b he O bsee efeenil hee, e ill nsie he flling ibles: -isne = (f. ih = ) -Tie -eqeny (f.8 ih ) y y (f. ih ) -Veliy (e.) h pplie he e fnin spplies ψ senπ y senπ y sen y b s n y y hen ψ ψ. iffeeniing.9 n iiing by.7 e he p 5 Tnsfins f he es n p K K K K iffeeniing. n iiing by. e he,. p p K K K K he syse fe by 5. n 5. e he , 5. h is n inin beeen he bsees in he Unling.Reliiy. /8

15 Tble 5, nsfins f he es n. K 5.. K. K. K 5. y y/ K 5.. y y / K 5.5. z z/ K 5.. z z / K Tnsfins f he ensiy f hge ρ, ρ n ensiy f en J n J q Mliplying. n. by he ensiy f he esing elei hge efine s ρ e he ρ ρ ρ ρ ρ ρ J J ρ 6. ρ ρ ρ n ρ ρ ρ J J ρ. 6. Tble 6, nsfins f he ensiy f hges ρ, ρ n ensiy f en J n J J J ρ 6. J J ρ 6. J J ρ 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 K 6.9 ρ ρ Κ 6. he syse fe by 6. n 6. e h 6.9 n Tnsfin f he elei fiels, n gnei fiels, pplying he fes f enz q n q q q q. K K n q q q in 5. n 5. e he, h siplifie bee. K n. K f hee e ge he inine f. beeen he bsees s nseqene f 5. n he flling pnens f eh is 5/8

16 y z z y yy zz yz zy K 7. y z z y z z K 7.. z y y z y y K 7.. yzzy y y z z y z z y K 7. y z z y z z K 7.. z y y z y y K 7.. T he njn 7. n 7. e he slins esibe in he bles 7 n 8. Tble 7, nsfins f he elei fiels, n gnei fiels e K K y z y K K 7.. y z y K K 7.. z y z y z K 7.. z K K K y y z 7.5. y y z z z y 7.5. z z y y y K 7.7 y y K z z K z z K y z 7.9 y z z y 7.9. z y Tble 8, nsfins f he elei fiels, n gnei fiels e. K. K 7. y y z 7.. y y z K K 7.. z z y 7.. z z y K K y y 7.. y y 7.. z z 7.. z z 7.. 6/8

17 Relin beeen he elei fiel n gnei fiel ze n he. T he bsee O his fiel is epesene ih bh pnens, being he If n elei-gnei fiel hs he bsee O he ngh gnei pnen elei pnen gnei fiel esibe by he njn 7.5 n hás s pnens z ze, y h e eqilen, y z, l f i-s The bsee O ssies esing elei hge, nifly isibe lngsie is is he flling elei-gnei ppeies: -ine ensiy f esing elei hge ρ -Ngh elei en I ze -Ngh gnei fiel ze ze ρ -Ril eleil fiel f le y z ny pin f is R πεr he pnen ze. q y z ih T he bsee O i eles n elei hge nifly isibe lngsie is is ih eliy hih i ssies he flling elei-gnei ppeies: -ine ensiy f he elei hge -lei en I ρ ρ -Ril eleil fiel f le ze ze n ) ρ ρ (f 6.7 ih = ) -Mgnei fiel f pnens ze (ing he njns 7. n 7.5 ih z, y ρ I πε R R hee I R, y z n le, being in he eil f 7.7 hee is niy e pepenil he eleil fiel n ngen he ifeene h psses by he pin f is R y z bese f he njn 7. n 7.6. ze. 7/8

18 8 Tnsfins f he iffeenil pes Tble 9, iffeenil pes y y z z K K y y z z K K he syse fe by 8., 8., 8., n 8. n ih.5 n. e nly fin he slins / n /. 8.5 hee e nle h nly he fnins ψ (.9) n ψ (.) h spply he niins ψ / ψ n ψ / ψ, 8.6 n epesen he ppgin ih eliy in he Unling Reliiy iniing h he fiel ppges ih efinie eliy n ih isin being pplie. n.8. ese f syey e n ls ie he he is ψ y y / ψ ψ y y / ψ, n ψ z / ψ z ψ z z / ψ,. 8.7 he nsfins f spe n ie f he Unly Reliiy e ge Jb s hee J,y,z,, y,z, K n J, y,z,, y,z, K, 8.8 ibles ih n s nseqene f he piniple f nny f he ligh eliy b e eql is J J n ill be eql ne J J hen. The e eqin he bsee O is Inine f he e eqin y z ze hee pplying he fls f bles 9 n. e ge f hee e fin y z K K ze 8/8

19 9/8 ze z K y K K h siplifying spplies ze z K y K K 6 hee eeing he es e fin ze z K y K K 8.9 b f 8.5 n. e he ze / h pplie in 8.9 spplies he e eqin he bsee O ze z y. 8. T en he efeenil f he bsee O e ill pply 8. he fls f bles 9 n.8, geing ze K K z y f hee e fin ze z K y K K h siplifying spplies ze z K y K K 6 hee eeing he es e fin ze z K y K K b f 8.5 n.8 e he ze / h eple in he eee eqin spplies he e eqin he bsee O. Inine f he Cniniy eqin The niniy eqin in he iffeenil f he bsee O is ze z Jz y Jy J ρ ze.j ρ 8. hee epling he fls f bles 6, 9, n. e ge ze z Jz y Jy ρ J K ρ K K

20 ing he peins e fin ρ ρ ρ ρ J J ρ ρ Jy y Jz z ze h siplifying spplies ρ ρ J J Jy Jz y z ze hee pplying ρ ρ J ρ ih nsn e ge J Jy Jz y z ρ J Jy Jz ze y z ze 8. h is he niniy eqin in he iffeenil f he bsee O. T ge gin he niniy eqin in he iffeenil f he bsee O e ill eple he fls f bles 6, 9, n.8 in 8. geing K K ρ K J y y J z z J ρ ze ing he peins e fin ρ ρ ρ ρ J J ρ ρ J y y J z z ze h siplifying spplies ρ ρ J J J y J z y z hee pplying J ρ ih nsn e ge ρ ρ J J y J z y z ze ρ J J y J z ze y z h is he niniy eqin in he iffeenil f he bsee O. Th in he iffeenil f e ien his y Inine f Mell s eqins ze Wih eleil hge T he bsee O T he bsee O y z y 8. y z y y z y z y z y z y y z z y z y Jz y z z z y z y z y z y z y y z z y z y J z y z /8

21 z y J y z z Jy z y z y z y z z J J y Wih eleil hge ze n J J ze T he bsee O T he bsee O y z y z 8.9 y z y z y z y z 8. y z y z y z y z 8. y y z y z y 8.5 y z y z z y z y 8.7 z z y z y z 8.9 y y z y z y 8. y z y z z y z y 8. z z 8.5 y We ense he inine f he f Gss in he iffeenil f h f he bsee O is y z 8. y z hee epling he fls f he bles 6, 7, 9, n.8, n nsieing nsn, e ge z z K K y K y y K ρ Κ ε ing he ps, sing n sbing he e z z y y z z y z z z z K ρk, e fin y y z y y y h eeing esls z y y z y z y z ρk /8

22 hee he fis penheses is 8.5 n bese f his eql ze, he sen bln is eql J gen f 8.5 n 8.5 esling in y z ρ y z y z f hee e ge y z h is he f Gss in he iffeenil f he bsee O. ρ ρ T e he inese e ill eple in 8. he fls f he bles 6, 7, 9, n., n nsieing nsn, e ge z z K K y K y y K ρ Κ ε ing he ps, ing n sbing he e z y z z h eeing esls in y y z z z z z y z z z K y y y z y z ρ K, e ge y y y y ρ K hee he fis bln is 8.5 n bese f his eqls ze, he sen bln is eql J gen f 8.6 n 8.5 esling in y y f hee e ge bsee. z ρ z y z y z ρ ρ 8. h is he f Gss in he iffeenil f he O Peeing his y e n pe he inine f f f ll he he eqins f Mell. 9 plining he Sgn ffe ih he Unling Reliiy We s nsf he sigh een f he bsees O n O se in he ein f he Unling Reliiy in plin il een ih nsn is. e s igine h he bsee O sees he bsee O ning n ih ngenil spee in lise y (C) eqls he psiie se f he is f UR n h he bsee O sees he bsee O ning n ih ngeil spee in nlise y (U) eqls he negie se f he is f he UR. In he en = = ze, he bsee O eis ys f ligh f he n igin bh bsees, ne in nlise y f U n nhe in lise y f C, heefe U = C n U = C, bese is he spee f he nsn ligh, n U n C he ie. /8

23 In he en = = ze he bsee O ls eis ys f ligh f he n igin bh bsees, ne in nlise y (seless) f U n nhe ne in lise y f C, hs U = C n U = C bese is he spee f he nsn ligh, n U n C he ie. Reiing he eqins.5 n. f he Unling Reliiy (UR): =..5 =.. Ming = = ( y f ligh pjee lngsie he psiie is ) n spliing he eqins e he: When he igin f he bsee O ees he nlise y f he bsee O, ill be he isne f he bsee O n silnesly ill ee is lise y f ligh he se pin f C U he bsee O, in syei psiin he iee h ges hgh he bsee O bese U C U C n U C U C, flling he f eqins be e he: U R C R C 9.5 C R U R C 9.6 When he igin f he bsee O ees he lise y f he bsee O, silnesly ill ee is n lise y n ill be he isne,, n be e he: f he bsee O, hen flling he eqins C U C R R C C 9.7 C R R C 9.8 The ie iffeene he bsee O is: R R R C C 9.9 The ie iffeene he bsee O is: R R R C C 9. Repling he eqins 5 in e pe h hey nfi he nsfins f he Unling Reliiy. /8

24 plining he epeiene f Ies-Silell ih he Unling Reliiy We shl eie he eqins (.) he e lengh in he Unling Reliiy: n,. Ming = = ( Ry f ligh pjee lngsie he psiie is ), e he he eqins: n,. If he bsee O, h sees he bsee O ging y ih he eliy in he psiie y f he is, eis es, penien f esing se in is igin ih eliy n e lengh in he psiie y f he is, hen ing he eqin. he bsee O ill ese he es ih eliy n he e lengh ing he fls: n,. If he bsee O, h sees he bsese O ging y ih eliy in he negie y f he is, eis es, penien f esing se in is igin ih eliy n he e lengh in he psiie y f he is, hen ing he eqin. he bsee O ill ese es ih eliy n e lengh ing he fls: n,. The esing ses in he igin f he bsees O n O e ienil hs. We lle he ege e lengh f he ese es, sing he eqins. n., he lef sie in eh eqin: We lle he iffene beeen he ege e lengh n he eie e lengh by he ses : /8

25 . Refeene hp://.bbin.ne/physis/fj7.h Ies-Silell (ninin) The pple s effe nsesl he Unling Reliiy s bine in he s flls: If he bsee O, h sees he bsee O, es ih he spee in negie y he is, eis es ih he feqeny y n he spee hen he bsee O ing. n ill ese es f feqeny y n spee in pepenil plne he een f O gien by y y.5 e ill he ze n y n he se feqeny y y lie his he nsesl feqeny y y y ih his e n ie he elin beeen.5 Wih y y e he he elin beeen he lengh f he nsesl e he se e 5/8 n he lengh f.6 The iin f he lengh f he nsesl e in he elin he lengh f he se e is:.7 h is he se le gen in he They f Speil Reliiy. pplying.7 in. e he.8 Wih he eqins. n. e n ge he elins.9,., n. esibe s flls.9 n f his e he he fl f spee.. pplying. n. in.6 e he.

26 .8 n. e nle h.. S h e he les f n bine f he Ies-Siell epeiene e n ele, n nle hehe hee is n he spe efin peie in he They f Speil Reliiy. Tnsfin f he pe f lins y beeen efeenils in he Speil They f Reliiy The elinship ihin he pe eelpe by he fes beeen efeenils is ien in he Speil They f he Reliiy in he flling y:... The efiniin f he pnen f he fe lng he is is: p. lins y, he piniple f ligh spee nsny gnees h he pnen f he ligh spee is ls nsn lng is is, hs The fl f enegy is ze. f hee e he. nsn, ensing h in he efiniin f enegy e he pplying 5 in e hee:. h pplying in n e he..5,... hee e fin h...6 esl eql 5. f he Unling Reliiy h n be epeienlly pen, nsieing he Sn s he se. ineiy The They f Unling Reliiy hs s is fnenl i he neessiy h ineil efeenils be ne elsiely s hse nes in hih y f ligh eie in ny iein f is igin spes in sigh line, h is heilly esibe by he fle (.,.8, 8.6 e 8.7) f he Unling Reliiy: y y z z y, z,. y y z z, y, z.8 Wle Vig e in.887 he line nsfin beeen he efeenils s he bsees O e O in he flling y: 6/8

27 .. Wih he espeie inee eqins:.. Whee,, n e nsns n bese f he syey e n nsie he es ih y, z n y, z. We n h n e pjeins f he ys f lighs n h spe ih Cnsn spee (e he nsny piniple f he Ry f ligh), eie in ny iein f he igin f he espeie ineils efeenil he en in hih he igins e inien n he en hee: = = ze.5 bese f his in he eqin. he en hee = ze e s he = ze s h e ls he = ze, e n sse h hen = ze, ls be eql ze, bese if he speing hppens in he plne y z e ill he = ze pls ze. We shl eie he ee eqins ( = ze):.6.7 Wih he espeie ee inee eqins:.8.9 If he speing hppens in he plne y z e he = ze n iiing.6 by.7 e he:. hee is he le f he spee in hih he bsee O sees he efeenil f he bsee O ing lngsie he is in he psiie y bese he sign f he eqin is psiie. If he speing hppens in he plne y z e he = ze n iiing.8 by.9 e he:. hee is he le f he spee in hih he bsee O sees he efeenil f he bsee O ing lngsie he is in he negie y bese he signl f he eqin is negie. The eqin.6 esibes he nsny piniple f he spee f ligh h s be sse by he eqins.6.9:.6 pplying.6 n.7 in.6 e he: hee e he: 7/8

28 hee ing = in he bes in n in he sigh bes e he he eqliy beeen bh sies f he eql signl f he eqin. ppllying = in e he. ppllying = in. e he. Th pplie in. splies:,. s (, ) is eql he fnin epening f he ibles n. pplying.8 n.9 in.6 e he: hee e he: hee ing = in he be in n in he sigh be e he he eqliy beeen bh sies f he eql signl f he eqin. pplying = n. in e he:,. s (, ) is eql he fnin epening n he ibles n. We s e he flling ning ing.5 n.6: K. K.5 K K s he eqin (, ) f. n (, ) f. s be eql, e he: 8/8

29 .6 Ths: K K ly eql...7 Reiing he eqins.6,.7,.8 n.9 ing he fnin f, n e he:.6.7 Wih he espeie inee ee eqins:.8.9 We he he eqins.6,.7,.8 n.9 finls epling by he espning fle:.6.7 Wih he espeie inee finl eqins:.8.9 Th e ely he eqins f he ble I s n hen he elins beeen n e..8 We ill nsf (.) fnin f he eleens,, n f (.) fnin f he eleens, n, epling in. he eqins.8,.9 n.8: 9/8

30 Th is ely he eqin.. We ill nsf (.) fnin f he eleens,, n f (.) fnin f he eleens, n, epling in. he eqins.6,.7 n.8: Th is ely he eqin.. We he lle he l ifeenil f (, ) (.): s: K n K.9 e he: K K hee pplying.8 e fin:. K K hee e nle h fnin f n is nsn. We he lle he l ifeenil f (, ) (.): s: K n K. e he: K K. /8

31 hee pplying. e fin: K K hee e nle h fnin f n is nsn. The eqins. n.8 epesen he bsees O n O he piniple f nsny f he ligh spee li f infiniely sll he infiniely big n en h in he Unling Reliiy he spe n ie e ese silnesly. They shln be inepee ih epeneny beeen spe n ie. The ie hs is n inepein h n be nes if e nlyze eeine bsee he eissin f ys f ligh f he insn =ze. If e he ies e ge, f eh y f ligh, e ill ge esl ih ny se f he physis. If in he insn = = ze, he bsee O eis ys f ligh, ne lngsie he is n he he lngsie he is y, fe he inel f ie, he ys hi f he bsee O, silnesly, he pins n y he isne f he igin, lhgh f he bsee O, he pins n be hi silnesly. bh ys f lighs be silnes bh bsees, hey s hi he pins h he he se is in elin he is n h pie he se ie f bh bsees ( = n = ), hih ens h nly ne y f ligh is neessy he he ie beeen he efeenils. ing, bh efeenils f he bsees O n O e ineil, hs he ligh spes in sigh line ing h is ene by he fnenl i f he Unling Reliiy, bese f his, he iffeene in eliies n is e nly iffeene in ie beeen he efeenils... We n ls ele n ineil efeenil f hih he ligh spe in sigh line ing h is ene by he fnenl i f he Unling Reliiy, ih n elee ing efeenil f hih he ligh spe in e line, nsieing h in his se he iffeene n isn e nly he iffeene f ie beeen he efeenils. ing, if he bsee O he insn = = ze, eis y f ligh f he igin f is efeenil, fe n inel f ie, he y f ligh his he pin ih ines (, y, z, ) he isne f he igin f he bsee O, hen e he: fe hiing he pin he y f ligh sill spe in he se iein n in he se y, fe n inel f ie, he y f ligh his he pin ih ines ( +, y + y, z + z, + ) he isne he pin, hen e he: n ih his e ge: The geey f spe n ie in he Unling Reliiy is size in he fige bel h n be epne n pins n seel bsees. /8

32 In he fige he ngles he elin n e eql he flling segens: O O O is eql O O O O O O O is eql O O O O n e pllel he flling segens: O O O O O is pllel O O is pllel O X X is pllel X X The sine f he ngles f inlinin n he ys f he bsees O n O ing. n. e: s / s s s / s K. sen n ih his e he: sen K. s / s s s / s K.5 /8

33 n ih his e he sen sen K.6 The sine f he ngle ih inesein f ys eql : s s s K K K K.7 n ih his e he: The inine f he Unling Reliiy. sen sen sen K K.8 s shs he hny f ll pe hypheses f spe n ie in he The s is eql he Jbins f he nsfins f he spe n ie f he pie I, hee he ils s s n K K J i j e nsiee ibles n e eie.,y,z,,y,z, K K / K 8.8 J l K,y,z,,y,z, K K / K K 8.8 Rih C. Tln The Tnsfins f he Men f Unling Reliiy s eelpe bse n he epeiene ne by eis n Tln, ing he efeene []. Whee he llisin f sphees peseing he piniple f nsein f enegy n he piniple f nsein f en, shs h he ss is fnin f he eliy ing : hee is he ss f he sphee hen in esing psiin n he le f is spee. nlyzing he llisin beeen ienil sphees hen in elie esing psiin, h f he bsee O e ne S n S e ing lng he is in he ny y ih he flling eliies befe he llisin: Tble sphee S sphee S y ze y ze z ze z ze /8

34 he bsee O he se sphees e ne S n S n he he eliies,, yi zi ze The eliy f he sphee S is eqls : befe he llisin lle ing he ble s flls:. The nsfin f ing. f Tble is:. Th pplie in spplies: The eliy f he sphee S is eql : Tble Sphee S Sphee S ze y ze y ze z ze z ze ze he bsees O n O he sphees he he se ss hen in elie esing psiin. n f he bsee O he sphees llie ih eliies f eql le n ppsie iein bese f his he en p p nll heseles ing he llisin, fing f bief ie nly ne by f ss. ing he piniple f nsein f en f he bsee O e ill he ipse h he en befe he llisin e eql he en fe he llisin, hs: Whee f he bsee O, is he biy eliy h sppsely f bief ie ill ls see he sses nie ing. s he sses i he iffeen eliies n he sses y ing hei n eliies, his eqin nn be siplifie lgebilly, hing his iin f sses: T he lef sie f he eql sign in he eqin e he: /8

35 5/8 ze ze T he igh sie f he eql sign in he eqin e he: pplying in he eqin f nsein f en e he:. hee e he: s f he bsee O he sses nie ln e enily lngsie he bsee O hih is neible if e nsie h he insns e iffeen hee sppsely he sses l be in esing psiin f he pin f ie f eh bsee n h he ss ing ih eliy is bigge hn he ss in esing psiin. If e pee ih hese ibles in line e l he:.

36 hee e nle h hih s be eql he peis le f, h is: elin beeen n h is bine f Tble hen he eliy ing e he sphee in esing psiin. Refeene Millenni Reliiy Veliies psiin UR: hp://.eliiy.ne/miefs/vcp_si_sb_wy.h h espns f he bsee O e s ie he nsfins f eni. enz f spe n ie in he Speil They f Reliiy:.. y y.b y y.b z z. z z... he e bin he eqins f eliy nsfin: y z y.5.5b z.5 y z y.6.6b z.6 e s nsie h in elin he bsee O n bje es ih eliy: 5,5. / s,5. 6/8

37 n h he eliy f he bsee O in elin he bsee O is: 5,5. / s,5. The eliy f he bje in elin he bsee O s be lle by he fl.6: 5 5,5.,5. 5,5..,5. 5,. 5 5,. / s,8. 5 Whee e se,. / s,. Cnsieing h he bje hs e ing ne sen in elin he bsee O hen ih. lle he ie psse he bsee O : 5,5..,.,,. 5,5. 5,. 5 5,6, 75,69s T he bsee O he bsee O is y he isne gien by he fl: 5 5,5..,,5.. T he bsee O he bsee O is y he isne gien by he fl: 5.,6 5,5.,9.., 75.,s e n T he isne f he bje, in elin he bsees O n O is gien by he fle: O O O O 5 5,..,,.. 5,6 5,5..,9.., 75 T he bsee O he isne beeen he bje n he bsee O is gien by he fl: 5 5 5,.,5.,9.. O T he bsee O he eliy f he bje in elin he bsee O is gien by: 5,9.,9. / s (=,),s 5 Reling he ies n sing he fl is nly pssible n elsiely hen n ze h isn he se be, e i pssible nesn his e ie he eqins. n. in he fl bel: s. s. 7/8

38 Whee s n s. The eqins be n be ien s: f, e f,.7 In eh efeenil f he bsees O n O he ligh ppgin ees sphee ih is n h ineep eh he fing ifeene h ppges ih eliy. The is n n he psiie y f he is n f he ngles n nsn beeen he efeenils. If f he se pi f efeenils e ngles ee ible he ie l be lley n l bee seless f he Physis. In he eqin f, e he ienil fnin f n, if e he in i nsn n ies ing e ge he n elin beeen he ies n beeen efeenils, hee if e he nsn n ies ing e ill he f eh le f f,. ne le f n beeen iffeen efeenils, n his nlysis is ls li f iiing.5 by e he: s s s Whee s n s Isling he eliy e he:..8. ss ss.9 hee e nle h e s he ngles n beeen he efeenils. nsn s h e he he se eliy This en f nsn ngles beeen he efeenils s sle he nesies f ebe ingle. 5 Inine The nsfins he spe n ie f ble I, gp. pls.7, in he i f is ien lie his: y z y z K 5. Th ien in he f bel epesens he se ine nsfins: / y y z z K 5. We ll s: i y z /, ij, K j y z 8/8 5.

39 Th e he fnins i i j i i,,,, y, z, 5. Th in he sybli f is ien:. in he inee f i j ij j i ij j 5.5 Whee e se insein s s nenin. The nsfins he spe n ie f ble I, gp. pls.8, in he i f is ien: y z y z K 5.6 Th ien in he f bel epesens he se ine nsfins: / y y z z K 5.7 Th e ll s: y z /, l, K l y z l Th e he fnins,,,, y, z, Th in he sybli f is ien:. in he inee f l l l eing K (.7), K (.8) n K. K (.). l l 5. The nsfin ies ij n l he he ppeies: / / i. ij l ij jl I l j K K ji l ji i I i / K / K j Whee ji is he nspse i f ij is he Knee s el. n l is he nspse i f l n / /. l ij l lj I j l K K 5. 9/8

40 l ji l i I / K / K l i 5. Whee l is he nspse i f l n is he Knee s el. Obsein: he ies n. ij The pil eiies l ij n i i j ele ing nsiee nsn n eql : l n ji is he nspse i f ij e inese f ne nhe b e n hgnl, h is: ji l i i i j f he l iffeenil f he ine pnens h j j, hee in he nsfin i ij he il K is Tble, pil eiies f he ine pnens: i j j i j j i j j i j j K The l iffeenil f he ines in he i f is eql : / K 5.5 Th e ll s: i, / i i j j, K j 5.6 i Then e he j i j j i i j j 5.7 The pil eiies l l ele ing nsiee nsn n eql : f he l iffeenil l f he ine pnens h l, hee in he nsfin i l he il K is /8

41 Tble pil eiies f he ine pnens: l l l l l l l l K The l iffeenil f he ines in he i f is eql : / K 5.8 Th e ll s:, / l l, K l 5.9 Then e he: l l l l l 5. The Jbins f he nsfins 5.5 n 5.8 e: J J l i j,,,,,,,,,,,, / K / K K K Whee K (.5), K (.6) n K. K (.). The ies f he nsfin n ies n. ls he he ppeies 5., 5., 5. n 5. f he l l he fnin hee he ines ele in he f he l l esibe s: e /8

42 /8 Th in he i f n ih pesening he fnin bees: l K K Whee epling he ies bel: K K K K Obsein: his ls elin shs h he ie ies in n eql f beeen he efeenils. We ge: l K K Th is he gp 8. pls 8. f he ble 9, iffeenil pes, in he i f. he fnin j i i hee he ines ele in he f j i i e he j i i j esibe s:

43 /8 i i i i i i i i Th in he i f n ih pesening he fnin bees: j K K Whee epling he ies bel: K K K K Obsein: his ls elin shs h he ie ies in n eql f beeen he efeenils. We ge: j K K Th is he gp 8. pls 8. f he ble 9, iffeenil pes in he i f. pplying 8.5 in 8. n in 8. e siplify hese eqins in he flling y:

44 Tble 9, iffeenil pes ih he eqins 8. n 8. siplifie: K ze K ze The ble 9, in he i f bees: 5. / K 5. / K The sqe ies f he nsfins be e nspse f he ies n. Inine f he Tl iffeenil In he bsee O efeenil he l iffeenil f fnin is eql : 5.5 l Whee he ines ele ih he nes f he bsee O ing nsfins 5. n 5.8 n ih pesening he fnin e he:, epling he / 5.6 / K K The lipliin f he ile ies spplies: / / K K / / 5.7 Resl h n be iie in ies: / / / / 5.8 /8

45 Th pplie he l iffeenil spplies: / 5.9 / eing he peins f he sen e e he: / Whee pplying 8.5 e he: / ze Then e he: / / ze 5. Wih his esl e he in 5.9 he inine f he l iffeenil: l l In he bsee O efeenil he l iffeenil f fnin i is eql : 5. i i 5. i i i j Whee he ines ele ih he nes f he bsee O efeenil ing, epling he nsfins 5. n 5.5 n ih pesening he fnin e he: / i 5. i / K K The lipliin f he ile ies spplies: / / / K K / 5. Resl h n be iie in ies: 5/8

46 6/8 / / / / 5.5 Th pplie he l iffeenil spplies: / / i i 5.6 eing he peins f he sen e e he: / / Whee pplying 8.5 e he: ze Then e he: ze / / 5.7 Wih his esl e he in 5.6 he inine f he l iffeenil: j j i i 5.8 Inine f he We qin The e eqin he bsee O is eql : 5.9

47 Whee pplying 5. n he nspse f 5. e he: 5. K K The lipliin f he hee ile ies spplies: K K 5. Resl h n be iie in ies: 5. Th pplie in he e eqin spplies: 5. eing he peins f he sen e e he: eing he peins e he: Whee pplying 8.5 e he: ze 7/8

48 8/8 Then e he: ze 5. Wih his esl e he in 5. he inine f he e eqin: 5.5 The e eqin he bsee O is eql : 5.6 Whee pplying 5. n he nspse f 5. e he: K K 5.7 The lipliin f he hee ile ies spplies: K K 5.8 Resl h n be iie in ies: 5.9

49 9/8 Th pplie in he e eqin spplies: 5.5 eing he peins f he sen e e he: eing he peins e he: Whee pplying 8.5 e he: ze Then e he: ze 5.5 Then in 5.5 e he he inine f he e eqin: 5.5 Inine f he eqins 8.5 f line ppgin Repling., 8., 8. in 8.5 e he: ze K K

50 5/8 eing he peins e he: ze Th siplifie spplies he inine f he eqin 8.5: ze Repling., 8., 8. in 8.5 e he: ze K K eing he peins e he: ze Th siplifie spplies he inine f he eqin 8.5: ze The ble in i f bees: / p p p K / / p p p 5.5 / p p p K / / p p p 5.5 The ble 6 in i f bees: J J J K / J J J 5.55 J J J K / J J J 5.56 Inine f he Cniniy qin The niniy eqin he bsee O is eql : ze J J J ρ J J J ρ J. 5.57

51 Whee epling 5. n 5.56 e he: / J ρ. J J J ze / K K 5.58 The p f he nsfin ies is gien in 5.7 n 5.8 ih his: ρ.j / / J J J 5.59 eing he peins f he sen e e he: / / J J J J Whee epling J n 8.5 e he: ze Then e he: / / J J ze J 5.6 Wih his esl e he in 5.59 he inine f he niniy eqin: ρ.j J J ρ.j J 5.6 The niniy eqin he bsee O is eql : J ρ J J J ρ. J J J ze Whee epling 5. n 5.55 e he: / J ρ. J J ze J / K K /8

52 The p f he nsfin ies is gien in 5. n 5.5 hen e he: ρ J / / J J J. 5.6 eing he peins f he sen e e he: / J J J J / Whee epling J n 8.5 e he: ze Then e he: / J J ze J / 5.65 Wih his esl e he in 5.6 he inine f he niniy eqin: ρ.j J J ρ.j J 5.66 Inine f he line iffeenil eleen: Th he bsee O is ien his y: s Whee epling 5.8 n he nspse f 5.8 e he: 5.67 s 5.68 K K The lipliin f he hee enl ies spplies: K K /8

53 Resl h n be iie in ies: 5.7 Th pplie in he line iffeenil eleen spplies: s 5.7 eing he peins f he sen e e he: ze Then e he: ze 5.7 Wih his esl e he in 5.7 he inine f he line iffeenil eleen: 5.7 s s T he bsee O he line iffeenil eleen is ien his y: s Whee epling 5.5 n he nspse f 5.5 e he: 5.7 s 5.75 K K 5/8

54 The lipliin f he hee enl ies spplies: K K 5.76 Resl h n be iie in ies: 5.77 Th pplie in he line iffeenil eleen spplies: s 5.78 eing he peins f he sen e e he: ze Then e he: ze 5.79 Wih his esl e he in 5.78 he inine f he line iffeenil eleen: 5.8 s s In 7 s nseqene f 5. e h he inine f.. hee n pplying 7.., 7.., 7.., 7.. n he eliy nsfin fle f ble e he ne elins beeen n isin f 7. n 7. n ih he e eie he ble 7 in he f bel: 5/8

55 Tble 7 y y K K K y y K z z K 7.. z z K y y z y y z z z y z z y y z y z z y z y 7.9. Wih he bles 7 n 9 e n he he inine f ll Mell s eqins. Inine f he Gss f he eleil fiel: y z y z 8. Whee pplying he bles 6, 7 n 9 e he: K y K z K K / y z Whee siplifying n epling 8.5 e he: Th eee spplies: y z / y z y z / y z Th siplifie spplies he inine f he Gss f he eleil fiel. Inine f he Gss f he gnei fiel:. y z ze y z 8.6 Whee pplying he bles 7 n 9 e he: y y z z z y 55/8

56 Th eee spplies: y z z y y z y z Whee he e in penhesis is he y-eny s (8.9) h is eql ze f hee e he he inine f he Gss f he gnei fiel. Inine f he y-eny s : y z y 8.8 Whee pplying he bles 7 n 9 e he: y K y K / K z Th siplifie n liplie by / e he: y z y y Whee eeing he ps n epling 7.9. e he: y z y y y s he e in penhesis is he eqin 8.5 h is eql ze hen e he he inine f he y-eny s. Inine f he y-eny s : z y y z 8. Whee pplying he bles 7 n 9 e he: z y K K K y z Th siplifie spplies he inine f he y-eny s. Inine f he y-eny s : z y z 8. Whee pplying he bles 7 n 9 e he: K / z K K y z Th siplifie n liplie by / e he: z z z y z 56/8 z

57 Th siplifying n ing he peins e he: z y z Whee pplying 7.9 e he: z y z z y z z. s he e in penhesis is he eqin 8.5 h is eql ze hen e he he inine f he y-eny s. Inine f he pee-mell s : y z J z 8. y Whee pplying he bles 6, 7 n 9 e he: y z Jz y K z K Th siplifying n ing he peins e he: y z Jz y z Whee siplifying n pplying 7.9 e he: y z Jz y Th egnize spplies z y z z z Jz y z z z z y s he e in penhesis is he eqin 8.5 h is eql ze hen e he he inine f he pee-mell s : Inine f he pee-mell s : z y J 8.6 y z Whee pplying he bles 6, 7 n 9 e he: z z y y z J y z K K / Ming he peins e he: z y y z J y z y z / 57/8

58 58/8 Repling in he fis penhesis he Gss n liplying by e he: J z y y z J z y y z Whee epling J, 7.9., 7.9 n 8.5 e he: z z y y J z y y z Th siplifie spplies: z z y y J z y y z Repling in he fis penhesis he Gss e he: J z y y z Th egnize es: J z y y z s he e in penhesis is he eqin 8.5 h is eql ze hen e he he inine f he pee-mell s : Inine f he pee-mell s : y y J z z 8.8 Whee pplying he bles 6, 7 n 9 e he: K y K Jy y z z Ming he peins e he: y z y y y y Jy z z Whee siplifying n pplying 7.9. e he: y y y y Jy z z Th egnize es: y y y Jy z z s he e in penhesis is he eqin 8.5 h is eql ze hen e he he inine f he pee-mell s :

59 Inine f he Gss f he eleil fiel ih eleil hge: y z ze y z 8. Whee pplying he bles 7 n 9 e he: K y K z K / y z ze Whee siplifying n epling 8.5 e he: y z ze / y z Th egnize es: y z ze. / y z Th siplifie spplies he Gss f he eleil fiel ih eleil hge. Inine f he pee-mell s ih eleil hge: y z 8. y Whee pplying he bles 7 n 9 e he: y z y Ming he peins e he: K z K y z y z Whee siplifying n pplying 7.9 e he: y z y Th egnize es: z y z z z y z z z z y s he e in penhesis is he eqin 8.5 h is eql ze hen e he he inine f he pee-mell s ih eleil hge: Inine f he pee-mell s ih eleil hge: z y 8. y z Whee pplying he bles 7 n 9 e he: z 59/8

60 6/8 / K K z y z y z y Ming he peins e he: / z z y y z y y z Repling in he fis penhesis he Gss ih eleil hge n liplying by / e he: z y y z z y y z Whee epling 7.9, 7.9. n 8.5 e he: z z y y z y y z Th siplifie spplies: z z y y z y y z Repling in he fis penhesis he Gss ih eleil hge e he: z y y z Th egnize es: J z y y z s he e in penhesis is he eqin 8.5 h is eql ze hen e he he inine f he pee-mell s ih eleil hge: Inine f he pee-mell s ih eleil hge: y z z 8. Whee pplying he bles 6, 7 n 9 e he: K y K y z z Ming he peins e he: y z y y y y z z Whee siplifying n pplying 7.9. e he: y y y y z z

61 Th egnize es: z y y y z s he e in penhesis is he eqin 8.5 h is eql ze hen e he he inine f he pee-mell s ih eleil hge: fnin f f 5 Inine (ninin).9 Whee he phse is eql 5.8 In e epesen n nling een h ges n in ne biy iein s ply ih he e eqin n bese f his e he: y z f f f y z ze 5.8 Th esn ee ih he e eqin bese he ls eleens ge nle b he fis ne esn. In e ee his pble e efle he phse f he fnin in he flling y. niy e sh s n si sj s 5.8 hee s, s y y, s z z 5.8 hs he le eql n n n.n s s s Ming he p y z n.rsi sj s. i yj z s sy sz e he n.r s sy s z h pplie he phse spplies ne phse n.r s sy s z ih he se ening f he peis phse. Repling n.r s sy s z phse in he f e in he phse liplie by e ls ge nhe s sy s z 5.88 ih he se ening f he peis phse Ths e n ie ne fnin s:. 6/8

62 f s sy sz f 5.89 Th eple in he e eqin ih he ie sine nsiee nsn spplies: f f f f s s s ze 5.9 h siplifie ees he e eqin. The psiie esl f he phse in he e eqin is n elsie nseqene f he ie sines being nsn in he pil eiies shing h he e eqin ens he ppgin he ne sey iein in he spe (plne e). he bsee O se le in he igin f is efeenil pes in n pin le he isne y z f he igin, n eleil fiel esibe by: i yj z 5.9 Whee he pnens e esibe s: y z y z.f. f.f 5.9 Th pplie in spplies: f i f j f i j f f y z ih le eql. f. f eing y z y z i j 5.95 y z The i plie e Cnsn ih he pnens, y, z 5.96 n le y z 5.97 eing f fnin ih he phse eql eiing he pnen in elin n e he: f f f f f f f h pplie in 8.5 spplies / ze / f f f ze / ze 6/8

63 f / / ze ze 5. ensing h i is he phse h s ply ih 8.5. / / ze ze s hen plies ih 8.5. / ze ze s he phse is he se f he pnens y n z hen hey ls ply ih 8.5. s he phses f he bsees O n O e eql bsee O ls ply ih 8.5. / / hen he pnens f he ze 5. The pnens eliely he bsee O f he eleil fiel e nsfe f he efeenil f he bsee O ing he bles 7, 7 n 8. pplying in 8.5 e fnin ien in he f: e i e s isin s isin i 5. hee i. eiing e he: sen is en sen i s 5. e i n e i 5. Th pplie in 8.5 spplies: / ze / sen is sen i s ze h is eql : i sin i s ze / i / ze e e ze hee e s he he effiiens eql ze s h e ge n ieniy, hen: ze i i ze i 6/8

64 i / i e e Whee pplying ze e he: Then ee ih he eqin 8.5 e s he e ppgin lng he is ih he spee. If e pply n e he:. esl ls gen f he is e glie s e eqin. Cnsieing he pple effe s l f physis. 6 Tie n eqeny We n efine l s ny eie h pes feqeny f ienil eens in seies pssible be enlise n e in sh y h n een n f eie ill be ienil ny een in he seies f eens pe by epli f his eie hen he eens e pe in elie esing psiin. The ylil een f l in esing psiin ing he bsee O efeenil ses he ie in his efeenil n he ylil een f he s f l in esing psiin ing he bsee O ses he ie in his efeenil. The fls f ie nsfin.7 n.8 ele he ies beeen he efeenils in elie een hs, ele eens in elie een. The elie een beeen he ineil efeenils pes he pple effe h pes h he feqeny ies ih eliy n s he feqeny n be inepee s being he feqeny f he ylil een f he s f l hen he ie ies in he se ppin h ies he feqeny ih he elie een h is, i is engh eple he ie n in he fls.7 n.8 by he feqenies y n y ge he fls f feqeny nsfin, hen: K y y K.7 bees. K y y K.8 bees. The Glile s nsfin f eliies beeen ineil efeenils pesens ininsilly hee efes h n be esibe his y: ) The Glile s nsfin f eliy he is is. In h ne if e he hen n if e he hen. s bh esls e n silnesly pssible else e he hen he nsfin esn ll h y f ligh be silnesly bsee by he bsees O n O h shs he piilege f n bsee in elin he he bese eh bsee n nly see he y f ligh nning in is n efeenil (ininsi efe he lssi nlysis f he Sgn s effe). b) I nn ls ply Nen s fis l f inei bese y f ligh eie pllel he is f he igin f he espeie ineil efeenils he en h he igins e inien n he en in hih = = ze ill he by he Glile s nsfin he eliy f ligh lee by he efeenils, n he ny f he ineil l h ln ll he eisene f iin in eliy bese hee is n eenl in ing n he y f ligh n bese f his bh bsees shl see he y f ligh ih eliy. ) s i nsies he ie s nsn beeen he efeenils i esn pe he epl iin beeen he efeenils in een s i is eqie by he pple effe. 6/8

65 The piniple f nsny f ligh eliy is nhing b eqieen f he Nen s fis l, he inei l. Nen s fis l, he inei l, is ine in Glile s nsfin hen he piniple f nsny f ligh eliy is pplie in Glile s nsfin piing he eqin f bles n f he Unling Reliiy h esn he he hee efes esibe. The ie n eliy eqins f bles n n be ien s: s.7.5 s s.8. s The isne beeen he efeenils is eql he p f eliy by ie his y:.9 I esn epen n he ppgin ngle f he y f ligh, being elsiely fnin f eliy n ie, h is, he ppgin ngle f he y f ligh, nly les beeen he ineil efeenil he ppin beeen ie n eliy, eeping he isne nsn in eh en, ny ppgin ngle. The eqins be in fnin f e ien s:, e, e.9 f,,.7 g,.5 f,,.8 g,. Then e he h he isne is fnin f ibles, he ie fnin f hee ibles n he eliy fnin f ibles. he efiniin f en. n enegy.6 e he: p 6. The elee he pe f spplies: p 6. leing he pe f he enegy fl e he: 65/8

66 Whee pplying 6. e he: p p.8 hee e nle h if he ss in esing psiin f pile is nll ze he pile enegy is eql p. 6. Th pplie in 6. spplies: p p 6. p hee e nle h he een f pile ih nll ss in esing psiin lys be he eliy f ligh. ze ill pplying in p he elins yh n y e he: yp p h yh n in he se y p h 6.5 qin h eles he en f pile ih nll ss in esing psiin ih is n y lengh. leing he pe f he fl f en nsfin (.9) e he: p p p p p Whee pplying p n p ps p e fin: p p p p p p p p p K 6.6 Whee pplying 6.5 esls in: h h p p K K inee. K K Whee pplying y n y e he: y y K inee y y K. In e he he eqins. n. pplying he piniple f eliiy he e phse. 66/8

67 7 Tnsfin f. enz bsees in elie een, he eqin h epesens he piniple f nsny f ligh spee f n pin is: y z y z 7. In his eqin neling he syei es e he: Nes neln s es siéis bes: 7. Th e n ie s: 7. If in his eqin e efine he ppin fs n s: hee e s he. ply The eqins 7. hee fis gen by lbe insein. When y f ligh es in he plne y z he bsee O e he = ze n = n sh niins pplie he eqin 7. spplies: 7.5 This esl ill ls be spplie by he eqins n f he gp 7. ne he se niins: hse e he: 67/8 7.6 n 7.7 Whee e he pen h.. he gp 7. e he he Tnsfins f. enz: Inees eqins, n :

68 Sgn effe When bh bsees igins e eql he ie is zee ( = = ze) in bh efeenils n ys f ligh e eie f he n igin, ne in he psiie iein (lise ine ) f he is n ih e fn n nhe in he negie iein (ne-lise ine ) f he is n ih e fn. The ppgin niins be pplie he enz eqins spply he bles n bel: Tble qin Clise y () qin Cne-lise y () S f ys Resl Resl Cniin Cniin Tble 7.9 qin Clise y () qin Cne-lise y () S f ys Resl Resl Cniin Cniin We bsee h he bles n e inese ne nhe. When e f he gp f he s eqins f he ys f bles n : /8

69 Whee he bsee O is he isne beeen he fn es n n hee he bsee O is he isne beeen he fn es n. f he In he eqins 7.5 be, e he ispy f spe n ie n he fn es ys f ligh being he se f bh bsees, he s f ys f ligh e ies s be inible beeen he bsees, hih e n epess by: 7.6 This esl h genees n eqin f ispy f spe n ie n be lle s he nsein f spe n ie piniple. The hee hyphesis f ppgin efine s flls ill be pplie in 7.5 n ese pe he nsein f spe n ie piniple gien by 7.6: yphesis : If he spe n ie e ispi n hee is n een ih n piilege f ne bsee nsiee e he he in n epy spe hen he ppgin geey f ys f ligh n be gien by: n 7.7 This hyphesis pplie he eqin f he gp 7.5 plies he spe n ie nsein piniple gien by 7.6. The hyphesis 7.7 pplie he bles n esls in: yphesis : C 7.8 If he spe n ie e ispi b he bsee O is in n bsle esing psiin in n epy spe hen he geey f ppgin f he ys f ligh is gien by: 7.9 Th pplie he ble n esls in: Sing n in 7. e he: C 69/ i This esl esn ply ih he nsein f spe n ie piniple gien by 7.6 n s esls in siin f f ys f ligh, eh bsee, n eh y f ligh ih is espeie inepenen fn e f he hes. yphesis C: If he spe n ie e ispi b he bsee O is in n bsle esing psiin in n epy spe hen he ppgin geey f he ys f ligh is gien:

70 7. Th pplie he bles n esls in: C 7. Sing C n in 7. e he: This esl esn ply ih he nsein f spe n ie piniple ely he se y s hyphesis gien by 7.6 n s i esls in siin f f ys f ligh, eh bsee n eh y f ligh ih is espeie inepenen fn e f he hes. Cnlsin The hyphesis, n C e pleely pible ih he en f ispy f spe n ie s e n nle ih he geey f ppgins. The esl f hyphesis is ny he esl f hyphesis n C espie f he elie een f he bsees n hnging he fn e eliely he fn e bese he fn es he inepenen een ne f he he n f he bsees. The hyphesis pplie in he nsfins f. enz plies ih he nsein f spe n ie piniple gien by 7.6 shing he pibiliy ih he nsfins f. enz ih he hyphesis. The ppliin f hyphesis n C in he nsfins f. enz spplies he spe n ie efins gien by 7. n 7.6 bese he nsfins f. enz e n pible ih he hyphesis n C. s bin he Sgn effe e s nsie h he bsee O is in n bsle esing psiin, hyphesis C be n h he ph f he ys f ligh be f R : R 7.7 he bsee O he Sgn effe is gien by he ie iffeene beeen he lise y f ligh n he ne-l y f ligh h n be bine sing 7. (C-), 7.7 n 7.: R 7.8 R 7/8

71 9 The Sgn ffe (ninin) The en he igins e he se he ie is zee ( = = ze) bh sies f he efeenil n he ys f ligh e eie f he n igin, ne in he psiie y (lise ine ) f he is n ih e fn n he he ne in he negie y (ne lise ine ) f he is n ih e fn. The pjee y f ligh in he psiie y (lise ine ) f he is n is eqine by n h pplie he Tble I spplies: (.8) 9. K (.7) K K (.5) hse e e h he isne beeen he bsees is gien by: K (.) Whee e he: KK 9. The y f ligh pje in he negie y (ne lise ine ) f he is n is eqine by n : h pplie he Tble I gies: (.8) 9.5 K (.7) K K (.5) K (.) 9.6 hse e e h he isne beeen he bsees is gien by: 9.7 Whee e he: KK 9.8 We s bsee h fis hee is n elinship beeen he eqins ih he eqins Wih he ppgin niins esibe e f he flling Tbles n : Tble qin Cniin Clise y f Cne lise y f qin ligh () ligh () Resl Resl Cniin K. K. S f he ys f ligh 7/8

72 K K K K K.7 K K K.7 Tble qin Cniin Clise y f Cne lise y f qin S f he ys f ligh ligh () ligh () Resl Resl Cniin K. K K K K K K.8 K K K..8 We bsee h f he ys f ligh ih he se iein he Tbles n e inese f eh he. ing he eqins gp f he s f he ys f ligh f he Tbles n : K K K K 9.9 Whee f he bsee O is he isne beeen he e fns n n hee f he bsee O is he isne beeen he e fns n. f he In he eqins be 9.9 e he ispy f he spe n ie n he e fns ys f ligh being he se f bh bsees, he s f he ys f ligh n f ies s be inible beeen he bsees, hih is epesse by: 9. This esl h eqins he ispy f spe n ie n be lle s he spe n ie nsein piniple. The hee hyphesis f ppgins efine ne ill be pplie in 9.9 n ese pe he pline f he nsein f spe n ie piniple gien by 9.. Wih hese hypheses e ee bn beeen he eqins ih he eqins yphesis : If he spe n ie e ispi n hee is een ih ny piilege f ny bsee e eh he in he epy spe hen he ppgin geey f he ys f ligh is eqine by: K K K K 9. Wih hse e e h he isne beeen he bsees is gien by: 9. Resls h pplie in he eqins f he gp 9.9 plies ih he nsein f spe n ie piniple gien by 9., shing h he pple effe in he lise n ne lise ys f ligh e pense in he efeenils. 7/8

73 yphesis : If he spe n ie e ispi b he bsee O is in n bsle esing psiin in he epy spe hen he ppgin geey f he ys f ligh is eqine by: C 9. Wih hse e e h he isne beeen he bsees is gien by: 9. Resls h pplie in he eqins f he gp 9.9 plies ih he nsein f spe n ie piniple gien by 9., shing h he pple effe in he lise n ne lise ys f ligh e pense in he efeenils. yphesis C: If he spe n ie e ispi b he bsee O is in n bsle esing psiin in he epy spe hen he ppgin geey f he ys f ligh is eqine by: C 9.5 Wih hse e e h he isne beeen he bsees is gien by: 9.6 Resls h pplie in he eqins f he gp 9.9 plies ih he nsein f spe n ie piniple gien by 9., shing h he pple effe in he lise n ne lise ys f ligh e pense in he efeenils. In e bin he Sgn effe e nsie h he bsee O is in n bsle esing psiin, hyphesis C be n h he ys f ligh se s be f R : R 9.7 pplying he hyphesis C in 9. n 9.5 e he: K 9.8 K 9.9 he bsee O he Sgn effe is gien by he ie iffeene beeen se f he lise y f ligh n he ne l y f h n be bine ing ( ) n pplying 9.7 ing: R 9. 7/8

74 The eqin is ely he esl bine f he geey nlysis f he ppgin f he lise n ne lise ys f ligh in ifeene shing he heene f he hyphesis pe by he Unling Reliiy. In 9. pplying 9. n 9.6 e he he finl esl e n : R R R 9. The lssi fl f he Sgn effe is gien s: R 9. he ppgin geey e he: 9. The lssi ies l be gien by: R 9. R 9.5 R 9.6 pplying 9., 9.5 n 9.6 in 9. e he: R R 9.7 R R 9.8 R R 9.9 The esls 9.7, 9.8 n 9.9 e pleely iffeen f The Mihelsn & Mley epeiene The iinl nlysis h spplies he slin f he nll esl f his epeiene nsies eie in esing psiin he efeenil f he bsee O h eis ys f ligh, ne hiznl in he iein (lise ine ) n nhe eil in he iein y. The hiznl y f ligh (lise ine ) ns nil i ple in = his pin he y f ligh efles (ne lise ine ) n ens he igin f he efeenil hee = ze. The eil y f ligh ns nil i ple in y = efles n ens he igin f he efeenil hee y = ze. In he iinl nlysis ing he spee f ligh nsny piniple f he bsee O he ys f ligh is gien by: 8. he bsee O he s f ies f he f bh ys f ligh lng he is is: 7/8

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