Unling Reliiy h: lfe is Mei Gi -il: li@sj.sp.g.b 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 ; /75
lng ih he eqins f nsfins beeen efeenes f he UR, e ge he inine f shpe Mell s eqins, sh s: ρ i ; i. i. R. εο R μο. j εο.μο. ; R εο.μο.. We ls ge he inine f shpe he eqin f e n eqin f niniy ne iffeenil shpe: 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 n ehnis ein f in Shöinges eqins /75
h: lfe is Mei Gi. e-il: li@sj.sp.g.b Unling Reliiy Tnsl: Rlf Ms Venâni e-il: se@e..b 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 /75
..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 eing n nsns, he ezns iffeenil f he se s....7 n /75 in.5 s ls be nsn bese f his he s be eql ze f hee e nle, h is ely
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..8..9 pplying.8 in.9 e he, R R R.. 5/75
6/75 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
7/75 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,.,.,...
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 8/75
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 h pplie in R. π π,,.,.9,.,.5, n.6 e he λ λ λ λ, K. 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 y y,. h is ely he lssi fl f he lngiinl pple effe. 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.. 9/75
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.6.... 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..8..9 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 /75
Sphee Obsee O Obsee O efe s s ze, ys s, Cllisin s s, ys fe s s ze, ys Cllisin s s, ys y s s ze, y s s, y s 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.., /75
. 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 /75
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..5.5.6.7 p.6.7.8 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 -negy (f. hee ) (f.7 ih n ) (f. ih n ) /75
-eqeny 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 5 Tnsfins f he es n p 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.. 5., 5. h is n inin beeen he bsees in he Unling.Reliiy. /75
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 5.5. 5... 5. 6 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 ρ ρ ρ n ρ ρ ρ J J ρ 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 6.7 6.8 ρ ρ K 6.9 ρ ρ Κ 6. he syse fe by 6. n 6. e h 6.9 n 6.. 7 Tnsfin f he elei fiels, n gnei fiels, pplying he fes f enz q n q q q q. K K n q q K. hee e ge he inine f flling pnens f eh is. q. 5. 5.5 6.. 6. in 5. n 5. e he, h siplifie bee. K n f beeen he bsees s nseqene f 5. n he 5/75
y z z y yy zz yz zy K y z z y z z K z y y z y y K yzzy y y z z y z z y K y z z y z z K z y y z y y K 7. 7.. 7.. 7. 7.. 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 7. 7. 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 7.5 7.6 y y z 7.5. y y z z z y 7.5. z z y y y K 7.7 y y K 7.8 7.7. K z z K z z y z 7.9 y z z y 7.9. z y 7.. 7.6. 7.6. 7.8. 7. 7.. 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 7.. 7. 7. y y 7.. y y 7.. z z 7.. z z 7.. 6/75
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, 7.5. 7.6 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. 7/75
8 Tnsfins f he iffeenil pes Tble 9, iffeenil pes y y z z K K 8. 8.. 8.. 8. y y z z K K 8. 8.. 8.. 89. 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 8/75
9/75 ze z K y K K 6 6 6 6 6 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 6 6 6 6 6 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
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 z y z y z y z y z y y z z y z y Jz y z 8. y y /75 z z 8.5 8.7 8.9 8. 8. y z y z y z y z y y z z y z y J z y z 8. 8.6 8.8 8. 8. 8.
z y J y z z Jy z y 8.5 8.7 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 8.6 8.8 8. 8. 8. 8.6 8.8 8. 8. 8. We ense he inine f he f Gss in he iffeenil f h f he bsee O is y z 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 8. h eeing esls z y y z y z y z ρk /75
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 z y z h eeing esls in y y hee he fis bln is z z z z J z K y y z y z z z y y z ρ K, e ge y y y y ρ K 8. 8.5 n bese f his eqls ze, he sen bln is eql 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 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. /75
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: 9. 9. 9. 9. 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. /75
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 : /75
. 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 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. 5/75
.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: 6/75
.. 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: 7/75
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: 8/75
.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: 9/75
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. /75
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. /75
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 /75
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 /75 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 he bsee O he se sphees e ne S n S n he he eliies,, yi zi ze befe he llisin lle ing he ble s flls:
The eliy f he sphee S is eqls :. 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: /75
5/75 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:.
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 y z.6.6b z.6 e s nsie h in elin he bsee O n bje es ih eliy: 5,5. / s,5. 6/75
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. 7/75
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 8/75 5.
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, 5.8 5.9 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 5. 5. 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. 9/75
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 l The pil eiies 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 Then e he i j i j j i i j j 5.7 The pil eiies ele ing nsiee nsn n eql : l f he l iffeenil f he ine pnens h l l l, hee in he nsfin i l he il K is /75
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 5. 5. 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 esibe s: l l e /75
/75 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:
/75 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:
Tble 9, iffeenil pes ih he eqins 8. n 8. siplifie: K ze 8. 8.. 8.. K 8. 8.5 ze 8. 8.. 8.. 8. 8.5 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 /75
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: 5/75
6/75 / / / / 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
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 7/75
8/75 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
9/75 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
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: K K ze 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 p p p / K / p / p p p p p / K / 5.5 5.5 The ble 6 in i f bees: J / J J J J J 5.55 K J / J J J J J K 5.56 Inine f he Cniniy qin The niniy eqin he bsee O is eql : J ρ J J J ρ. J J ze J 5.57 5/75
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 5.6 5.6 5/75
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 5.69 5/75
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 5/75
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: 5/75
Tble 7 y y K K K 7. 7.. y y z z K 7.. z z K 7.. 7.5 7.6 y y z y y z 7.5. 7.6. z z y z z y 7.5. 7.6. y z y z 7.9 7. 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: K 7. 7.. 7.. 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 55/75
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 y K z K K 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 z 56/75
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 y z J z 8. 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 y z J z 8.6 Whee pplying he bles 6, 7 n 9 e he: z y y z J y z K K / Ming he peins e he: z y y z J y z y z / 57/75
58/75 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 :
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 y z 8. 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 y z z 8. Whee pplying he bles 7 n 9 e he: 59/75
6/75 / 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
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. 5.85 Ming he p n.r e he y z si sj s. i yj z s sy sz n.r s sy s z n.r s sy s z h pplie he phse spplies ne phse 5.86 5.87 ih he se ening f he peis phse. Repling phse in he f n.r s sy s z 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: f s sy sz. f 5.89 6/75
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. 5.9 5.9 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 5.87 5.88. eiing he pnen in elin n e he: f f f f f f f 5.98 5.99 h pplie in 8.5 spplies / ze f / / f f f / ze ze ensing h i is he phse h s ply ih 8.5. 6/75 ze / ze 5.
s / ze / hen plies ih 8.5. ze / 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 / i e e Whee pplying ze e he: i 6/75
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. 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. 6/75
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: Whee pplying 6. e he: 65/75
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. 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. 66/75
67/75 Th e n ie s: 7. If in his eqin e efine he ppin fs n s: 7. hee e s he. ply 7.. 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: 7.6 hse e he: n 7.7 Whee e he pen h.. he gp 7. e he he Tnsfins f. enz: 7.8 7.9 7. 7. Inees eqins, n : 7. 7.
7. 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 7.8 7.8 7.9 Tble 7.9 qin Clise y () qin Cne-lise y () S f ys Resl Resl Cniin Cniin 7. 7. 7. 7. 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 : 7.5 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. 68/75
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 7. 7. 7. 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: 7. Th pplie he bles n esls in: C 7. 69/75
Sing C n in 7. e he: 7.5 7.6 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 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) K (.) 9. 7/75
hse e e h he isne beeen he bsees is gien by: 9. 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 9. 9. ih he eqins 9.5 9.8. Wih he ppgin niins esibe e f he flling Tbles n : 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.7 K K K. Tble.7 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 7/75
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 9. 9. ih he eqins 9.5 9.8. 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. 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. 7/75
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 (9.8 9.9) n pplying 9.7 ing: R 9. 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. 7/75
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 9.. 8 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: 8. In he iinl nlysis f he bsee O he s f ies f he f bh ys f ligh lng he y is is: y s e he y 8. hee is n inefeene finge n i is pplie he nll esl f he Mihelsn & Mley epeiene. In his iinl nlysis he ienil f he lise n ne lise ys f ligh in he eqin 8. h igines he nll esl f he Mihelsn & Mley epeiene nis he Sgn effe h is ely he ie iffeene eising beeen he f he lise n ne lise ys f ligh. se n he Unling Reliiy e e eepe nlysis f he Mihelsn & Mley epeiene bining esl h plies pleely ih he Sgn effe. Obseing h he eqin 8. espns he hyphesis C f he pgph 9. 7/75
pplying 8. in 9.9 e he: K K K K K K K K K 8. e he: K 8. 8.5 Whee pplying 9.6 e he: 8.6 In 8. e he: Whee pplying 9.5 e he: 8.7 8.8 The eqins 8.6 n 8.8 ense h he pple effe in he lise n ne lise ys f ligh pense iself in he efeenil f he bsee O esling in: y 8.9 ese f his, ing he Unling Reliiy in he Mihelsn & Mley epeiene e n pei h he lise y f ligh hs iffeen f he ne lise y f ligh ing he fl 8.8 bining ls he nll esl f he epeiene n hing hen ih he Sgn effe. This sppsiin nn be e bse n he insein s Speil Reliiy bese ing 7.6 e he: 8. 75/75
76/75 9 Regessin f he peihelin f Mey f 7, e s igine he Sn le in he fs f n ellipse h inies ih he igin f syse f ines (,y,z) ih n een in elin enine fie ss n h he plne Mey is in een gene by he fe f giinl in ih he Sn esibing n ellipi bi in he pln (,y) ing he ls f Keple n he fl f he Nens giinl in l:,8.,98. 67. 6, 9. The sb ine "" iniing ss in elie es he bsee. T esibe he een e ill se he nn fls: 9. 9.. 9. 9.5 The fl f he eliiy fe is gien by: / 9.6 In his he fis e espns he iin f he ss ih he spee n he sen s e ill see le in 9. espns he iin f he enegy ih he ie. Wih his n he peis fls e bin: / 9.7 / / 9.8
77/75 In his e he he nsese n il pnen gien by: / / 9.9 / / 9. s he giinl fe is enl e shl he nll he ese pnen ze s e he: ze / / 9. hee e he: 9. he il pnen e he: / / 9. Th pplying 9. e he: / / 9. Th siplifying esls in: 9.5 This eqle Nens giinl fe esls in he eliisi giinl fe: 9.6
s he giinl fe is enl i shl ssis he hey f nsein f he enegy () h is ien s: = nsn. 9.7 p Whee he inei enegy ( ) is gien by: 9.8 n he penil enegy ( p ) giinl by: p 9.9 Resling in: s he l enegy () i is nsn e shl he: Cnsn. 9. p ze. 9. Then e he: 9. p 9. Resling in: p ze ze 9. This pplie in he eliisi fe 9.6 n eqle he giinl fe 9. esls in: 9.5 In his sbsiing he peis ibles e ge: 78/75
9.6 his e bin he il pnen eqls : 9.7 Th esily bees he eliisi giinl fe 9.6. 9.6 e bin he ese pnen eqls : ze 9.8 his ls ne e he: 9.9 s he giinl fe is enl i shl ls ssis he hey f nsein f he ngl en h is ien s: p p nsn. 9. 9. = nsn. 9. Resling in h is nsn. In 9. e h ze ze ze bese he een is in he plne (,y). 9. 79/75
eiing e fin: h e he: ze 9. 9.5 qling 9. igining f he hey f he enl fe ih 9.9 igining f he hey f nsein f he enegy n 9.5 igining f he hey f nsein f he ngl en e he: 9.6 he ls eqliy e bin 9. n f he f he ile e bin 9.6. slin f he iffeenil eqins e ill se he se eh se in he Nens hey. e s sse 9.7 The iffeenil l f his is 9.8 hee e he e 9.9 he le f he ngl en e he hee e he Whee pplying 9.9 e he 9. 9. 9. Th eie spplies 9. 8/75
8/75 Whee pplying 9. n eiing e he: 9. In his ih 9.6 he il eie is bine his y: / 9.5 / 9.6 Th pplie in 9. spplies: 9.7 Siplifie esls: 9.8 e s fin he sen eie f he ngle eiing 9.: 9.9 In his pplying 9. n 9.5 n siplifying e he: 9.5 pplying in 9. he eqins 9. n 9. n siplifying e he: 9.5 The eqin f he eliisi giinl fe 9.6 eele is: 9.5 In his pplying he fls be e he:
8/75 8 8 8 8 8
8/75 8 8 8 6 8 8 6 8 8 6 8 In his e ill nsie nsn he Nens ngl en in he f: 9.5 Th i is elly he nn heeil ngl en. ze 9.5 Whee e he: 9.55 9.56
8/75 The eqin 9.5 hs s slin: s s 9.57 Whee e nsie ze. I is enine in 9.57. 9.58 The eqin 9.58 is fnin nly f ensing he ininsi nin beeen he iin f he ss ih he iin f he enegy in he ie, bese bh s ley esibe, piipe in he eliisi fe 9.6 in his elies he essenil iffeene beeen he ss n he elei hge h is inible n iniisible in he elegnei hey. 9.57 e bin he y f nil: s s 9.59 Whee is he eeniiy n he iey isne f he fs. eiing 9.57 e he sen 9.6 Th eie esls in s 9.6 pplying in 9.5 he ibles e he: ze. ze s sen s s s 9.6 ze s s s s s ze s s s s s s ze s s 9.6 In his pplying in he fis penhesis e he: ze
In 9.6 pplying in he sen penhesis e he: ze The es f he eqin 9.6 is heefe: ze 9.6 The f he ellipi bi f he plne Mey is []: eniiy f he bi, 6. ge sei-is = = 5,79.. Slle sei-is b 5, 79.,6 56.658.6.5,8. 5, 79.,6 55..955.6,. 5, 79.,6 69..56.65,,6 The bil pei f he h (PT) n Mey (PM) n he Sn in sens e: PT, 6. PM 7, 6. 7 6 s. s. The nbe f ns h Mey ( ) es n he Sn (M ) in ne eny is, heefe: 7, 6. N 5, 79. 9.65 6 7, 6. Theeil ngl en f Mey: 6,67.,98. 5, 79.,6 7,977. 6,67.,98. 8,. 7,. 8. 9.66,65.. 9.67 6,67.,98. 7,.,5. 9.68,6. 8,... 9.69 pplying he nei ih seel eil nbes he es f he eqin 9.6 e he: 8 8,65.,...,65. 69..56.65, 55..955.6,,5. 8,976. 9.7 Resl h e n nsie nll. 85/75
86/75 We ill bin he eliisi ngl en f he es f he eqin 9.6 in his pplying he ibles e he: ze 9.7 ze ze ze ze 9.7 7,978. 9.7 This ls eqin hs he elsie ppey f eling he spee he enine eliisi ngl en h is slle hn he heeil ngl en 9.66. The iin f he eliisi ngl en in elin he heeil ngl en is ey sll n gien by: 7.5.59,,8. 7,977. 7,977. 7,978. 8. 9.7 Th enses he y f he piniple f nsny f he spee f he ligh.
In eliy, he eqin 9.6 pies sel eessin peihelin f Mey, hih is gien by in 5, 79 5, 79 5,...,6.. 9.75 Cneing f he sen e he:,6. 5.8,..6, 7, ". 9.76 This eessin, is n epee in Nenin hey is e eliisi iin f ss n enegy n is she in l bsee peessin f 5599. " 87/75
9 ne f Mey s peihelin f.79 If e ie he eqin f he giinl eliiy enegy R eing he es f he inei enegy, he penil enegy p n he esing enegy: R p p. 9.77 eing he nseie he giinl fe is enegy is nsn. ssing hen h in 9.77 hen he is ens infinie, he spee n penil enegy ens ze, esling hen: R p 9.78 Wiing he eqin he Nen s giin enegy N hing he espnen Nen s es he 9.77: N 9.79 Whee he ineil enegy. is he inei enegy, he penil enegy n he esing enegy bee sying his 9.79 e he: 9.8 eiing 9.79 e he: N ze ze 9.8 88/75
89/75 Ming he eliiy enegy 9.78 eql he Nen s enegy 9.79 e he: p N R 9.8 p 9.8 In h enining he eliiy penil ( ) s: p 9.8 We he: 9.85 In his ne epling he ppiin: 9.86 We he: Th siplifie esls in he Nen s penil: 9.87 Repling 9.8 n he eliiy penil 9.85 in he eliiy enegy 9.78: R 9.88 We he he Nen s enegy 9.79: N
9/75 eiing he eliiy penil 9.85 e he he eliiy giinl elein ls ely s in he Nen s hey: Whee e he: ze N. ese he e be eie is he Nen s enegy iie by h is N h is nsn, esling hen in: In his ne pplying 9.8 e he: 9.89 The e elein is gien by 9.5:
The eliiy giinl elein ls 9.89 is eql he pnen f he e is () hs e he: 9.9 eing nll he nsesl elein e he: ze 9.9 ze Th is eql he eiie f he nsn ngl en ze 9.9 9.9 Reiing se eqins ley esibe e he: n 9.9 9.9 e he: In his ne e 9.9 he spee f 9.8 n he ngl en e he: 9/75
9/75 G ze ze ze ze 9.95 Whee e he: 9.96 The slin he iffeenil eqin 9.95 is: s s. 9.97 Whee e nsie ze Then he is is gien by: s s 9.98 Whee is he eeniiy n he fs isne he iey. eiing 9.97 e he sen n s 9.99 pplying he eiies in 9.95 e he: ze ze s s s s s ze s s s s s s ze s s s s s s s
9/75 ze s s s ze s s s ze s s s ze s s 9. The effiien f he sqe -sine n be nsiee nll bese n is ey lge nbe: ze s 9. Resling f he eqin 9.: ze s 9. e he niiy f he eqin 9. e s he he nly slin h es i nll silnesly he penhesis n he es f he eqin, h is, e s he niqe slin f bh he flling eqins: ze n ze 9. These eqins n be ien s: b 9. 9.5 In hese nes he n e s he single slin hen e he: b 9.6 Wih 9.96 n he heeil en e he: 9.7
I is pplie in 9.5 n 9.6 esling in: b 9.8 9.9 9.8 e he he ise e in 9.5: ze,8. ze 55..955.6, 9. 9. 9.9 e he : 9. I is pplie in 9. esling in 9.: ze 9. e he: 66,67.,98. 8 55..955.6,. 6,999.999.9.599 9. Th espns he ne f Mey s peihelin in ne eny f: 5,79..96.,.5, 79 Clle in his y:.,79 9. In ne ignei n e he 66 6.96., " sens. The ngle in sens n by he plne in ne ignei n is gien by:.96.,.96.,. If, e he egessin..96.,. If, e he n ne..96.,. 9/75
The ngl iin in sens in ne n is gien by:.96.,.96.,.96., If If ze e he egessin. ze e he n ne. In ne eny e he 5,79 ns h spply l ngl iin f: 5,79..96.,.5, 79.,79. If ze e he egessin. If ze e he n ne. Inei Igine in n infinie niese lly epy, pin O hih is he beginning f he ines f he bsee O. In he ses f he bsee O being es in nif in he l f inei eqies h he spheil elegnei es ih spee isse by se le pin O is lys bsee by O, egless f ie, ih spheil spee n heefe he nif in n es e inisingishble f eh he ein li in bh ses he l f inei. T he bsee O he eqins f elegnei hey esibe he spe js lie spheil e. The ige f n bje le in O ill lys be enee n he bje iself n be f ligh eie f O ill lys ein sigh n pepenil he spheil es. Igine nhe pin O h ill be he beginning f he ines f he bsee hih hs he se ppeies s esibe f he ineil bsee O. Obisly iginy pins ih ny f f inein beeen he ein iniilly n gehe pefely eeing he l f inei een hgh hee is nif in beeen he nly eeble e he pesene f bsees h ill be nsiee iniilly in es, seing in in he he efeenil. The ininsi ppeies f hese bsees e esibe by he eqins f eliisi nsfins. Ne: he infinie niese is ne in hih ny pin n be nsiee he enl pin f his niese. ( eleni nslin) Inei (lifiins) Igine in lly epy infinie niese single pin O. e he niqeness ppeies f O is f ligh eie f O s ppge ih eliy. If his y ppges in sigh line, hen O is efine s he igin f n ineil fe bese i is eihe es in nif eiline in. ee, in he hyphesis f ppgin f he ligh y being e he een f O s be inepee s he igin f n elee fe. Theefe he ppgin f y f ligh is sffiien ense hehe O is he igin f n ineil fe elee fe. N igine if in he niese esibe be f he ineil efeene fe O hee is nhe ineil fe O h es n he ny in f physil inein ih O. In he bsene f ny inein beeen O n O he niqeness ppeies e inilble f bh pins n ys f ligh eie f O n O he he se eliy. I is ipssible f he eliy f ligh eie f O be iffeen f he eliy f ligh eie f O bese eh efeene eiss s if he he i n eis. eing O n O he igin f ineil fes he ppgin f ligh ys s in sigh line ih eliy n he elins beeen ies n f eh fe e gien by ble I. 95/75
ne f Mey s peihelin f.79 lle ih he Unling Reliiy ssing (.) ze ze. (.7) (.)....5 (.) (.).6.7.8.9.. 96/75
97/75...6..5
98/75 9.6.6.7..8....9... ns n. ns R n ns R n. R R.
99/75 R R..5.6.6 = nsn.7.....8.9 ze...
/75...5 ze R.6 ze.7 s sen s.8 ze s s s s s.9 ze s s s s s s ze s s s s s s s
/75 ze s s 6 s s s s ze s s s 6 s s s ze s s 6 ze s 6 s ze s s. ze s. ze s. ze ze s ze ze s ze ze. b. R ze b ze b.5.6
/75 b 6.7 The egessin is fnin f psiie enegy h gens he een. R 6 Regessin.8 ze b 6.9 ze ze. R ze 6 ze ze 6 ze ze This egessin is n gene by he psiie enegy
/75.6.5.5.5.5
/75...5...55.56 ns n.57 ns R n ns R n.58 R R.59 R.6 R.6
5/75.6.6.6.6.56.65.66 ze.67.68.69
6/75.7.7 ze R.7 ze.7 s sen s.8 ze s s s s s.7 ze s s s s s s ze s s s s s s s
7/75 ze s s 6 s s s s ze s s s 6 s s s ze s s 6 ze s 6 s ze s s.75 ze s.76 ze s.77 ze ze s ze ze s ze ze.78 b.79 R ze b ze b.8.8
8/75 b 6.8 The ne is fnin f negie enegy h gens he een R 6 6 ne.8 ze b 6.8 R ze ze.78 ze ze ze 6 ze.85.86 ze 6.87 ze ze.88 This ne is n gene by negie enegy ze 6.85 ze 6.89 ze 6 ze 6 6.9 ze s.77 ze s ze s.
9/75 ze s 6 ze s 6 ze s 6.9 ze s 6 ze s 6 ze s 6 9 ze s 9 R ze s 9 ze s 9 ze s.9 ze s 6.9 6 ze s 6 6 6 ze s 6 6 6 ze s 6 8 6 R
/75 ze s 8 6 R ze s 8 6 ze s 8 ze s 6 ze s 6 ze s.9 ze s 6.9 R ze s 6 ze s ze s.9 6 s s s.95
/75 negy Nenin ( N ) N N N ze N ze N ze N ze N ze N y N ze y s sen s ze y sen s s ze y s s s s
/75 ze y s s s s ze y s s s s ze y s s s s ze y s s ze s ze y s ze ze y y N ze ze y ze y y N N N ze N N N
Spil efin This is he spil efin. The lengh es in he efeene fe f he bsee O is gee hn he lengh h is ing ih eliy elie n efeene fe he bsee O. N pe he bsee O he isne beeen O O : Ths e bin he eliy :. N pe he bsee O he isne beeen O O : Ths e bin he eliy :. The spee is he se bh bsees s e he: Whee pplying he elin e bin:. Whee he isne n ies inesely ih he isnes n. /75
/75 In genel, e bin (.,.): ze ze
Vibles ih line Spe n Tie en,,,y, e...they e se in. Geey f spe n ie in he pln y y. y f y s. s f y s. îyĵî sĵ îyĵî sĵ y. y y î ĵ îs ĵîĵ î yĵ î yĵ y î ĵ y s s sen y s y g s y y s s î ze s. ĵ î yĵ îyĵ î sĵ î sĵ y s s s s s s s s s 5/75
6/75 s heeil e y g y g s s y y s y s s s s s s s s s s s s s s s s s.56. s. s s
nsn Viinl Piniple. nsn p gngen. p Wh is he iniil enegy f he pile f ss. p çãs,,,,ze Viinl Piniple This is he eliy pnen in is. S Viin f he in lng he X is. iling he ible in he nge e he seen his hen ze n hee ze e ill he he niins: ze ze ze ze ze ze Then e he ne fnin I G,,,, ze ze 7/75 n hee:,,,,,,,,
8/75 S e he,, I h pies eie: ze,,,, I ze I ze I ze ze I ze ze ze I ze ze This is he X is pnen ze ze z y z y This is he X is pnen z y
9/75 z y z y z y î î X is ĵ y y ĵ y Y is z z z Z is z yĵ î z ĵ y î z z ĵ y y î z z ĵ y y î
/75 z ĵ y yĵ î î z yĵ î z yĵ î z yĵ î z yĵ î z yĵ î =.6 =.9 Viinl Piniple Cninin ns n e. nsn e nsn e ns n e T T p p p p. p p p p p p p R T p T T T p T p T T p T
/75 p T p T p p R p p p p T p T p z yĵ î z ĵ y î.8 z ĵ y î z ĵ y î T p T p z yĵ î z ĵ y î.7 y y z z y y z z ze ze p p T ze p ze p p p ze p ze ze z y z y..
/75 ze z ĵ y î z ĵ y î z ĵ y î =.9 5 gihi spil ze e.7 s sen s.8 e e e e ze e e e e e ze e e e e ze e e ze e e ze ze 6 6.. e ze 6 6 6 6
/75 ze 6 6 6 6 6 6 ze 6 6 6 6 6 6 ze 6 6 6 8 6 ze 6 6 6 ze 6 ze 6 5 gihi Spil (Cninin).7 R ze ze ze
/75 ze ze ze s sen s.8 ze s s s s s ze s s s s s s s ze s s s s s s ze s s s s s s ze s s 6 s s s s ze s s 6 s s s s ze s s s s s s
5/75 ze s s s s s s ze s s R 6 ze ) ( ) ( s ) ( ) ( s ze s s ze s s ze s s ze s s 6 ze s 6 6 s 6 ze s 6 6 s 6 ze s 6 s 6 ze s 6 s 6
6/75 ze s 6 s 6 ze s 6 s 6.6.6. 6 6 s 6 6 6 s 6 6 s 6 6 6 s 6 6 6 s 6 6 6 ze 6 8,55.,997958.,989. 6,67. 55..955.6 6 6 6 6 6 s 8 6 6 6 6 6 s 8 6 8 6 s ze s
7/75 ze s 6 ze M ) ( ze ze s s ze M ze G 6 6 6 ze s s ze s ze s ze s s ze M The pesene f in he fl s ) (, lls i ls esibe spil. 5 gihi Spil Cninin II ze s s 6 ze M ) ( ze
8/75 ze s 6 6 s 6 ze 6 s 6 6 s 6 ze 6 s 6 6 s 6 ze 6 s s 6 6 6 6 s 6 s 6 s 6 s s s s s
s s s s s ze s ze ( ) M ze ze 6 6 6 6 57.99.7.,,5659 55.6.69.568, 6,678..,989.,997958.8.77,89.55.,999.999.9. 6,999.999.9. 6,76.789..5 -.96.,..96., ne Regessin.96., ze ne ze Regessin 9/75
/75.59.89.5",.96., 6.59.876.997",.96., 6 5,.6.9 87,969 65,56.6. PM PT. N,99.98..7" 5,.6.9.59.89.5, N.,99.977.6." 5,.6.9.59.876.997, N. y efiniin ze 6 ze M ) ( ze ze M s ze s s ze s Se If s s negy Nenin ( N ) ze y s s ze s s ze y ze y ze y
yze yze yze N yze y yze yze N N ze ze N ze N N /75
6 neen f he Peiéli f Mey f,99 " Sppsing (.) ze ze. (.7) (.)....5 (.) (.).6.7.8.9.. /75
/75...6.5.5
/75.5.5.5.55.56.57.58.... e ns n e ns R n e ns R n
5/75.59.6.6.6.6.6.6.56.65.66.67 R R R R ze
6/75.68.69.7.7 5 gihi Spil nine.7 R ze
ze ze ze ze ze s sen s.8 The fis hyphesis bin pil slin f he iffeenil eqin is sse he infinie is, hs bining: s zes s s ze ze R ze ze ze ze ze ze ze This esl shs h in infiniy he inflene f he enl ss is ze 7/75 M ze. The sen hyphesis bin nhe pil slin f he iffeenil eqin is bine by bseing h he ngle f he eqin s inies he iein f he infinie is hee he inflene f he enl ss is ze M ze n heefe he iein f he ene f ss is gien by he ngle h eple in he eqin s esls in he ne eqin s h inies iein ppsie he iein f he infinie is hih is he iein f he ene f ss. s s s s s
8/75 s s s R ze ze ze ze ze 6 ze 6 ze 6 ze 6 6 6 pplying he esls f he sen hyphesis in he iffeenil eqin: ze s sen s.8 ze s s s s s
9/75 ze s s s s s s s ze s s s s s s ze s s s s s s ze s s 6 s s s s ze s s 6 s s s s ze s s s s s s ze s s s s s s ze s s R ze ) ( ) ( s ) ( ) ( s ze s s
/75 ze s s ze s s ze s s 6 ze M ) ( ze ze s 6 6 s 6 ze 6 s 6 6 s 6 ze 6 s 6 6 s 6 ze 6 s s 6 6 6 6 s 6 s 6 s 6 s
s s s s s s s s Whee pplying he esl f he sen hyphesis s s : Th i is n ieniy ensing h he esl f he sen hyphesis is e. 6 6 6 57.99.7.,,5659 55.6.69.568, 6,678..,989.,997958.8.77,89.55. /75
/75,999.999.9. 6,999.999.9. 6,76.789..5 -.96.,.96.,. ne Reess.96., ze ne ze Reess.59.89.5",.96., 6.59.876.997",.96., 6 5,.6.9 87,969 65,56.6. PM PT. N,99.98..7" 5,.6.9.59.89.5, N.,99.977.6." 5,.6.9.59.876.997, N. Nenin negy N N N N ze N
/75 ze y s s s s ze y s s ze N ze N ze N ze N y N ze y s sen s ze y sen s s ze y s s s s ze y s s s s ze y s s s s
/75 Nenin negy N ze y s s ze s s ze y ze y ze y ze y ze y ze y ze y N y ze y ze y ze N ze N ze N N N
7 neen f Peihelin f Mey f.99 "n Cniins" e s s f he eqin epessing he eqilibi f fes:.65 On he igh sie e he he giinl fe efine by Nen, n he lef sie e he he physil esipin f e = f he Unling Reliiy. The physil ppeies f eqin.65 eqie is liiy hen is is ies f is gee hn ze n infinie is, s he is ies f ze <, n s e he isin bny niins. The fis bny niin is hen he is is infinie = n he giinl fe is ze, hih ens h he pile is es ih = ze n = ze n he sen bny niin is hen he is is gee hih is ze n slle hn infiniy ze < < hih ens h he pile is in in e he inflene f giinl fe.65 ih ze n ze. In 6 flling he llins is sbsie in.65, he eqliy,.6,.69 n R, e =. fe hese sbsiins e bin he iffeenil eqin: ze 7. This eqin hs be li f he se bny niins s eqin.65, h is, i hs be li f is gee hn ze ( > ze) n infinie is (ze < ). Y slin is gien by: s 7. Whih shl e he n niins ley esibe. pplying slin 7. in iffeenil eqin 7. e he: ze s sen s.8 s s s ze s s s s ze 5/75 s s s s s
6/75 ze s s s s s s ze s s s s s s ze s s 6 s s s s ze s s 6 s s s s ze s s s s s s ze s s s s s s ze s s R ze ) ( ) ( s ) ( ) ( s ze s s ze s s
s s s s ze ze 7. This eqin s he slin f he se n niins f.65. Slin f 7. f he fis bny niin hih is hen he is is infinie giinl fe is ze hih ens h he pile is es n e he = ze n = ze. =, n he pplying in 7. e ge: ( ) ( ) + + ( ) + = ze. ( ) + = ze ε = ( ) 7. qin 7. is ely eql he esl f eqin 7. hen he is is infinie =, = ze n =, s shn in 7.5: = = ( ) [ + εs( )] = [ + εs( )] = + = ze 7.5 Theefe in 7. e he n e esl h esibes h in infiniy he eeniiy ε is ele he ngle f he infinie is f he pile, being ε hih ens h he in f infiniy ill be pbli ih ε = hypebli ih ε >. Ne h by efiniin ε > ze. Slin f 7. f he sen bny niin hih is hen he is is gee hn ze n less hn infiniy ze < < hih ens h he pile is in in e he inflene f giinl fe ih ze n ze. pplying = in 7. e he: s s ze 7. s s 6 6 6 s 6 6 6 s ze s 6 6 6 s s 6 s 6 ze 6 ze 6 ze 7/75
8/75 6 6 6 s 6 s 6 s 6 s s s s s s s s s
( ) + = ze ε = ( ) In he hey f ni f hypeble e he ε =, eqing 7.6 e he ε = =. s( )) hih is he e fl, f he gee hlf is f hypebl. = ( ) 7.6 This esls Theefe in 7.6 e he n e esl h esibes h in he se f ze < < he eeniiy ε is ele he ngle f he pile, being ε hih ens h he in ill be pbli ih ε = hypebli ih ε >. Ne h by efiniin ε > ze 8 Siplifie Peielli ne Peihelin Regessin > Igine h he sn n Mey e piles, ih he Sn being he igin f ine syse n Mey lying pin n he y plne. The e is = nneing he igin pin ill esibe Meys in in he y plne. In he esipin f he een f he plne Mey he bsee O espns he ibles ih line f he bsee O s ih line being se single is = n single ine syse f bh bsees. Tie is fnin f ie h is = () n ie is fnin f ie h is = ( ). = + =. + =. = =. > > =.5 = = +. =. +. = 8. The is n be nsiee fnin f ie = () ie = ( ) = [ ()] i n be nsiee fnin f ie = ( ) ie = () = [( )]. = ( ) = [ ()] = () = [( )] 8. = = + = = = = = = = = = + = = = = = = () = = = () = + = + h spees n eleins e psiie. 8. + 8. + 8.5 = = 9/75
= = = + 8.6 =. =. =. 8.7 =. = +. =. = = = =. = = = 8.8 = + = = + nsne = + = 8.9 = = = + = + 8. In his fis in eliisi inei enegy is gee hn ineil enegy Meys peihelin eee. The plne sees heie e he een. >.This ses =. = =. = =. =. = = =. = =. = =. =. = = 8. = + + = = + = ze = = + = ze = = = = = = = + = + = = = 5/75
= + = + + + + + ze + + = 8. + + + = ze + + + = ze + + + = ze + + + = ze 8. = = [ + εs( )] = ( ) = ( ) ( ) + [ + εs( )] + ( ) ( ) + + ( ) ( ) ( ) ( ) + ( ) ε = = = + ( ) ( ) + [ + εs( )] + ( ) + + ( ) + + ( ) + [ + εs( )] = ze = ze + ( ) + + + ( ) + + = ze + ( ) = ze = ze ( ) ( ) + + ( ) + = ze = 8. ( ) + + ( ) + = ze + 6 ε ε s ( ) + 6 + ε + ε + ε ε + 6 s( ) + ε ε + 6 ε + = ze ε ( ) + + ( ) 6 ( ) ( ) = + + ( ) ( ) + + ±.. = ze + + ( ) + + = ze = ze ( ) = ±. ( ) = ± 5/75
( ) = ± ( ) = = ( ) = ε ( ) = ze 8.5 hypeble eeniiy (ε) is efine s ε = ( ) hee ( ) is he ngle f he sype. ne f he Peielli < = + = > + = = = > = = +. =. +. = = = + = = + = = = = = ( ) = = = ( ) = + + = + + = = = = = + =. =. =. =. = + 8.6. =. = = = =. = = = 8.7 = = = + nsne = = 8.8 = = = = 8.9 In his sen in eliisi inei enegy is slle hn ineil enegy ne f Meys peihelin. The plne elly is lighe e een. 5/75 <.This ses he
=. = =. = = = = =. =. =. =. =. =. = = 8. = + + = = + = ze = = + = ze = = = = = = = + = + = = = = = + ze + = 8. + = ze + = ze + = ze + = ze 8. = = [ + εs( )] = ( ) = ( ) s( ) + ε [ + εs( )] s( ) ε [ + εs( )] ε [ + εs( )] = ze 5/75
( ) + + ( ) ( ) + + ( ) + + ( ) ( ) ( ) ( ) + + ( ) + + ( ) ( ) + + ε = = = ( ) ( ) + + + ( ) + ( ) ( ) + ( ) + ( ) + ( ) = ze ( ) + + ( ) + + = ze = ze = ze + ( ) = ze ( ) = ze ( ) ( ) + + + ( ) + = ze = 8. ( ) + ze ( ) + + + ( ) + + + ( ) 6 ( ) ( ) = + + ( ) + ( ) + ±.. = ze + ( ) + = ze + + + ( ) + = ze = ze ( ) + = ( ) = ±. ( ) = ± ( ) = ± ( ) = = ( ) = ε ( ) = ze 8. hypeble eeniiy (ε) is efine s ε = ( ) hee ( ) is he ngle f he sype. 5/75
The eens f he ellipses ill fs (lef) n he igin f he fe. ll ellipses e esibe by he eqin = () = = =. () ().() In hese he ngle e is (), inies he psiin f he plne Mey in ll ellipses, he een f Mey in he ellipses is nelise, ih he le f being he se f peihelin neen ein. The fis ellipse in ble epesens egessin f he peihelin, hee e he =.. The sen e ellipse epesens he neen f he peihelin, in his e he =.9. In his ellipse he peihelin n phelin ne in he ignei sense, h is, nelise hih is he se iein s he plnes een in he ellipse. The fifh ellipse in geen epesens siny ellipse =. 55/75