Electromagnetic theory and transformations between reference frames: a new proposal

Transcription

1 leoagne heo an ansfoaons beween efeene faes: a new oosal Clauo P. Panana al: lauo.anana@gal.o bsa Ths ae ooses a Gallean-naan heo of eleoagnes, alable a fs oe n /, esbng boh nsananeous an oagae neaons. To een hs heo o hghe oes, efnons of unesal e an oe e ae noue. New ansfoaons beween efeene faes ae suggese. bbeaons: TR: eal Theo of Rela GTR: Geneal Theo of Rela R/s: neal Refeene ae/s GT: Gallean Tansfoaons LT: Loen Tansfoaons nouon The eal Theo of Rela (TR) las a e ole n he hsal eson of he wol. les ha sae an e anno be onee as seaae suues, bu us be aen ogehe as a sngle suue: he sae-e onnuu. s ons he so-alle sae-e eens ae he naual eleens of eal, nael he funaenal eleens aoun whh a onenen eson of he wol an be foulae. ong o TR, hsal laws ae esbe n he sae wa n all neal efeene faes (Rs), an heefoe eah R ossesses eual gn. The onl nea eleens elean o a henoenologal eson ae osons an eloes (of ales o fel subanes) elae o an abal hosen neal obsee. The one of absolue oon s elue fo he hsal eson of eal. ong o TR, he Loen ansfoaons (LT) ae he elaonshs beween easueens, n ffeen Rs, of saal sanes o e neals beween eens. The LT l, hangng he R, naane of he so-alle sae-e neal beween a gene a of eens. The naan neal beween eens an be geoeall neee as a sane beween ons n a fou-ensonal seuo-ulean sae, he so-alle Mnows sae-e. Unle sae-e neals, saal sanes an e neals lose he oe of naane. ne s oelle o abanon he one of absolue sulane of saall seaae eens. n hs one, o eess hsal heoes, he os aoae aheaal enes ae fouensos. Unsusngl, Mawell s eleoagnes an be eesse b hs foals. TR has efe he LT as he ansfoaon laws beween Rs an hose ansfoaons leae he fo of Mawell s euaons unhange ( s sa ha Mawell s euaons ae oaan une LT). The aoon of he TR an heefoe of he LT sas fo he followng obseaon: f we assue he oneoa al of Gallean ansfoaons (GT) an of Mawell s euaons no a lege efeene fae (he osulae eu fo he oagaon of lgh, alle

2 ehe ), hs oues onlusons ha ona he esuls of he Mhelson-Mole nefeoee eeens. Howee, he esoluon of he noabl beween eeens an heo an be heoeall usue (an hsoall has been) a ffeen aoahes. The an be lassfe no he followng oons:. e Mawell s euaons as he oe eson of eleoagne henoena, o a leas as a suffenl oe eson o aue he ue se of sae-e, sse b he euaons of lassal ehans. Ths les aong LT n lae of GT as he oe laws of ansfoaon beween Rs.. Leae boh GT an Mawell s euaons unhange, bu of he oel of he oagaon eu. 3. Manan he GT an hange he eson of eleoagne henoena. 4. Change boh he ansfoaon laws beween Rs an he eleoagne heo. The TR s he full eeloen of he fs oon. Colee o aal ehe aggng heoes ae eales of he seon oon. n eale of a heo ha usues he h ossbl was foulae b T.. Phs. He oose an eleoagne heo ha was foall eual o He s ognal heo, bu eneee n he eanng of he es. l aheas an a new sbol neeaon oue a new heo whh shall efe o as He-Phs eleoagnes slang naane une GT. Phs oose hs heo as a al aoaon o he fs oe n /. The eenson of he heo, whh was esbe b Phs as Neo-Hean an lae o oeoe he laons of he fs oe, eesens a anson o he fouh oon. n fa, he eene heo noles he nouon of a new e of e (he oe e) an a onseuen hange of he GT. The esul s a sl elas heo, sne les no lege efeene fae an he nea uanes ae eesse elael o a gene neal obsee. Howee, he Neo-Hean eson of he heo oues aaoal eons, as wll show lae n hs ae. The h oon was also usue b. elle hough a heo alle Wea Rela. n elle s heo he esene of an absolue efeene fae s assue. ssung he al of Mawell s eleoagnes n hs efee efeene fae, he see of lgh woul onl be soo n ese o. n Rs ffeen fo (whh heefoe hae absolue sees), he see of lgh woul be ansoo. Howee, fo an lose ah, he aeage see of lgh shoul ean onsan n all Rs. n elle s Wea Rela, he LT ae elae b so-alle neal Tansfoaons. The ansfoaons of he eleoagne fels n aoane wh neal Tansfoaons ae esbe n a ae b G.D. Pun 3 an n anohe ae b. uonaua 4. The heo s alle wea beause, unle he TR, eues he esene of an absolue efeene fae, bu anans he nae of ela beause he le sae of absolue oon s loall uneasuable.

3 s a n of ela n whh he slowng own of ong los has an aseal eson. n onas o he eons ae b he TR, wo ffeen neal obsees woul agee on whh of he enal los s beang a slowe ae. uall ase s he lengh onaon. Wea Rela, een f oes ffeen aes fo los a es n ffeen Rs, esoes he absolue sulane of saall seaae eens. n hs heo he absolue efeene fae an be neee as a Loen-e ehe ha jusfes he effes (shoenng of he boes lenghs an slowng of he los as absolue effes) oue b oon. n hs boo Wea Rela an ohe ublaons 5,6, ano elle s aulal effee a showng he weanesses of he TRR neeaon aabl n ffeen aeas (he agna effe, non-neal sses o sella abeaon esons, fo eale). ong he objeons ha elle oses o he TR, he os hlosohal one an be suae as follows. The LT, whh onsue he elaonal suue beween ffeen Rs, ose a fo of onologal onfuson (sne he ean o he funaenal aegoes of beng), gng he sae saus of eal o he as, esen an fuue. ong o he LT, ffeen neal obsees u sae-e n ffeen onsan e sles, eanng ha eah of he obsees abues ffeen olleons of eens o he esen. Ths eans ha eens lae n fuue (nael eens no e anfese n o-ong R esen) a belong o he as of a ffeen neal obsee. The eual gn of all neal obsees also eues he eual gn of all hese ossble ffeen nows. Theefoe, wha all fuue, n s hang o be he as of anohe esen wh eual ghs of eal, s also efe. Ths les an absolue eens. uh a eson s heefoe hlosohall (hough no aheaall) eonlable wh uanu ehans, whh as an nheenl obabls eson of he fuue. f one aes he eson of sae an e oe b LT as oe, one us also ae ha unean (whh s neessa o a uanu eson) s eel a n of age o lluson. These onseaons oae he seah fo heoeal alenaes ha ae aable of beang he se of he LT an enoung he absolue sulane one. elle s heo s neesng n hs sense, an woul be een oe neesng o es eeenall hough easueens of he see of lgh on one-wa ahs n ffeen eons. Measueens of he flgh e of an eleoagne ulse oul be ae ou b eans of san los, no snhone a a sane a he nsen eho, bu snhone n ongu onons an ansoe n uassa onons (low see) o he fnal osons. Clos snhone n hs wa ae ossble o eeal an ffeenes n he flgh e of eleoagne ulses whh oagae n oose eons along he sagh lne jonng wo los. Howee, sne elle s heo s onl alenae o TR, s onl alable o he hsal esons ae b neal obsees. neal obsees us be unesoo n he naowe sense of obsees ong a a onsan see n egons fa fo sgnfan asses; feefallng obsees n egons wh a ga gaen ae heefoe elue. Conseuenl hese ess shoul eall be onue fa awa fo gaaonal soues. 3

4 aong neal Tansfoaons nsea of LT, Wea Rela s no oable wh ha genealaon of TR, he Geneal Theo of Rela (GTR). Thus, f one egas he eenene of he beang of he los on he gaaonal oenal as an eal uh, n absene of an alenae gaaonal heo o GTR ha s able o jusf hs eenene, s no obous how o oe he neal Tansfoaons o ae he alable n eene egons whee a gaaonal fel s esen. lhough agee wh elle s s of TR an onse hs heo lausble, belee ha a eson of he lassal eleoagne eson as nae b Phs n hs boo an sugges fuhe neesng alenaes. Phs s oose ofaon of he lassal eleoagne heo s esuase n he neess o use he oal e eae n oe o full eesen he eeenal aaa esuls egang nuon. Hs heo s naan (no oaan) n ese o GT. wll show ha hs heo us be ofe, sne es esuls ha sagee wh eeene. n fa, usng Phs s heo o alulae foes beween saona uen eleens, one fns esuls ha ae noable wh eal eene. also belee ha hs euaons shoul be enee oable wh he eeenal esuls, ouene n: L. Kholes,.. Msseh, R. no-ruea: Measueen of oagaon elo of boun eleoagne fels n nea one - URNL PPLD PHYC, 359 (7).. Calaea, R. e ango, G. nohao, P. Pae, M. Polo, G. Pella: Measung Poagaon ee of Coulob els ax:.93 [g-] 3 No. oh hese eeenal aoahes show eene of nsananeous neaons. nsananeous neaons ae no eesenable whn Mawell s eleoagnes, sne he lae us obe he sae-e sees esbe b LT. nsananeous neaons l ha he nle of eneg onseaon aes a non-loal fo. nsea, as one ou b enan 7, he onseaon nle ose on eneg b he TR has a sl loal naue. n sho: eneg onseaon n loal fo fsl eans ha eneg s a uan laeable n sae, esbable b a ens funon. also eans ha, f he eneg n a egon hanges, hs a onl ou hough a flow of he sae eneg ossng he bounaes of ha egon. non-loal nle of onseaon aong o whh a ean hsal uan eeases n a lae an sulaneousl neases n anohe lae san fo he foe, so ha he su eans onsan a ee nsan s n onaon wh he TR sne he sulane of saall seaae eens s no shae b ffeen neal obsees. The aeaane-saeaane of eneg a san ons, ealuae as oung sulaneousl b one neal obsee, woul be ealuae b anohe neal obsee as he saeaane of eneg a a ean nsan an he aeaane of he sae aoun of eneg n anohe nsan. Thus hee woul be a e neal wh a shoage o eess of eneg. nsananeous neaons ae, nsea, eesenable n an eleoagnes wh oees of naane o oaane une GT. 4

5 No hang obles wh he oneoane of san eens, hs eleoagnes an be foulae o a nsananeous neaons, whou fallng no onaon. Pu anohe wa, s eessble n a anne ha s onssen wh hese een eeenal obseaons, beause he aous eleal uanes ae no oblge o be oonens of fou eos o fou ensos. Peses Loen Tansfoaons (oe eense noes n en - ) Gen wo Rs: an, le (, ) an (, ) be he saal an eoal oonaes of he sae een n he wo efeene faes. Le be he elo of wh ese o. esse n eo fo, he Loen Tansfoaons ae: ( ) whee: Gallean Tansfoaons an oeaos Gen wo Rs: an, n eo fo, he Gallean ansfoaons ae: o hese ansfoaons, elaons beween oeaos ae ee (eals n en - ): The oeao s heefoe naan, whle he oeao s no. Mawell s euaons n auu (aong o nenaonal se of Uns) () () (3) (4) These fou fel euaons, wh aoae nal an bouna onons, eene he ele fel (,,, ) an he agne fel (,,, ) a a gene on (,, ) n sae a a gene nsan. The euaons ae no naan une GT ue o he non-naane of he oeao whh aeas n euaons (3) an (4). 5

6 Phs ss of Mawell s eleoagnes Une-aaeeaon Mawell s euaons la of eo n onseng oons of hages. n an obsee s neal efeene fae, he oeens of he soue hages ae esbe b he uen ens fel. The oeens of he fel eeo (o absobe), whh s as oneuall essenal as he soue, s absen fo he eson. Mawell s euaons ae une-aaeee wh ese o he eeo s sae of oon beause he o no onelae. The onneon beween fels an eeo s noue hough he efnon of foe: he Loen foe. s easonable o onse a hange of Mawell s euaons, noung a ole fo he sae of oon of he hage eeo-absobe. aaa s obseaons an he use of he e eae oeao lhough base on aaa s obseaons, Mawell s euaon (3) aeas lang as a anslaon of hose sae obseaons. The esuls of aaa s eeens an be suae n he negal fo: L Φ l n whee Φ s he -fel flu hough a sufae boune b a lose onue u L. n aaa s eeens, aaons of he agne flu assng hough a lose eleal u wee eale no onl b ang on he fel soue, bu also hough a hange of he u s shae. eause he ah of he lne negal a be e-aan L L he use of he oal eae n lae of he aal eae s anao. These onseaons also oel he use of he oal eae oeao n he ffeenal foulaon. uaon (3) shoul be elae b: 6

7 The oal eae The aboe suae onseaons sugges an aoah sla o he Lagangan one, use n flu nas, n whh a flu s esbe b he oeen of s aous as. The nual flu ales ae nuall labele an followe n he oons. The alenae aoah, alle ulean, esbes he flu hough funons of oson an e,.e. hough sala o eo fels (see, essue, ens, e.). n he ulean eson, he oson ha aeas as an aguen of a funon s he oson of a geoeal on, egaless of he esene of a sef flu ale. n hs one s naual o efne he aal e eae, enoe b he sbol, as he l of he ao beween he aaon of a uan n a fe on of sae an he e neal of hs aaon, when hs neal ens owas eo. n he Lagangan eson s ahe useful o efne he oal e eae, enoe b he sbol, as he l of he ao beween he aaon of a uan o follow he oon of a ale an he e neal of hs aaon, when hs neal ens owas eo. f,, ealuae on ha ale s ae he oonaes of a ong ale, he e eae of he oe f f f (,,, ) f f f f f f f f n eo noaon: f f ( ) f n he eleoagne faewo he use of he fel eeo. oeao assgns a ole o he oon of he Gen an neal efeene fae, o whh he oonaes of eah elean en ae efee, he use of he oeao s nene o ean ha he eoal aaons of uanes ae no hose seen fo he fe on nsananeousl oue b he eeo, bu ae hose seen n he sae on b he eeo n oon. Moe geneall, he use of he oal e eae oeao n ffeenal euaons of fels eans ha eoal aaons of aables ae no easue on fe ons n, bu on ons n oon, whh shae he sae nsananeous see of he eeo. o eoal aaons of all uanes ae easue n he non-neal (bu no oang) efeene fae,, ong wh see n ese o. elae o Conseng a on ale fel eeo n aba oon wh elo ( ) an abal hosen neal obsee, fo he han ule follows ha: 7

8 8 naane of une GT lng he Gallean law of ooson of eloes o he eeo oon: s he see of he eeo ealuae n. s he see of he eeo ealuae n. s he see of ealuae n. Hang alea esablshe: follows ha: whh efes he fs-oe naane of. s neessal onsan, gen he hohess of neal. s no neessal onsan, as an abue of he non-neal oon of a Lagangan ale.

9 He-Phs leoagnes Refeng o a gene R, n e sae he euaons fo he eleoagne fel al o fs oe of ae: The ae oleene b a efnon of foe: (H) (H) (3H) (4H) (5H) n onas o Mawell s euaons, uen ens seen b a eeo n oon, wh elo elaes. lso, he ens s neee as he n. s elae o he Mawellan uen ens, he uen ens seen b a eeo fe a ha on n, b he followng eesson: (6H) s an eualen uen ue o he eeo oon a elo. eeo saona n eesens he seal ase:. Gen ha:, fo, oal an aal eaes beoe eual. n hs ase, he fel euaons beoe nsngushable fo Mawell s. uheoe, he foe eessons one n he wo heoes. Theefoe, n he ase of a non-aeleae eeo, suosng eual esons of he soues ( an ) n he R n whh he fel eeo s a es ( ), all he eons of Mawell s heo ae eoue b he He-Phs heo. Ths oes no ean, howee, as Phs sae, ha he new heo onsues a oeng heo of Mawell s eleoagnes, sne he eson of he soues ffes, n geneal, n he wo heoes. 9

10 n Mawell s heo, an enses ae ansfoe when he efeene sse hanges, wheeas n he He-Phs heo an enses an ae naan (as wll be shown lae). n he ase of a eeo ong n, a oason beween he wo heoes anno be aolshe b onl onseng he fel euaons, bu b also onseng he ffeenes n he efnons of he eleona foe aoe n he wo heoes. n Mawell s heo he Loen foe s assue: n he He-Phs heo he foe law s: Mawell He s heefoe osulae ha he ele fel eo s suffen o ealuae he foe ang on a hage. The agne fel las an nsensable ole n eenng he ele fel n na suaons, bu oes no aea ell n he law of foe. Mawell Gallean soue ansfoaon euaons Conse wo gene Rs: an. Le n oon be a onsan elo wh ese o. The hage ens ansfoaon euaon s: (, ) (, ) Ths assuon an be unesoo b onseng ha: an efe o he sae on P n sae ewe n an, eseel. fo he assuon of GT al. (, ) us be nene as he ao beween he aoun of ne hage (he algeba su of a fne nube of on-le hages whose onseaon s osulae) onane n a gen nfnesal olue, enee n P, saona n, an he olue self. (, ) us be nene as he ao beween he aoun of ne hage onane n a gen nfnesal olue, wh he sae lnea se of, enee n P, saona n, nael n oon a see n, an he olue self. s nsananeousl onen wh beause of he naane of he lenghs le b GT. The onene of efeene olues n ffeen Rs les he sae aoun of ne hage an heefoe he sae ealuaon fo he ens.

11 To ealuae he ansfoaon of he uen ens, s aoae o begn wh he efnons of an n a gen R. Conse a le egon of sae whee hee s a ose ele hage sbuon wh ens, an a negae eleal hage sbuon wh ens,. s heefoe a ose eal nube, whle s a negae eal nube. The ne hage ens wll be: The olue eleen enfe b, a e, wll hae a oal ele hage: [ ],,, n, f all ose hages onane n shae he elo, whle all negae hages shae he elo, he Mawellan uen ens s efne as: n, he Hean uen ens s efne as: n he R (ong a see wh ese o ), aong he GT, he Mawellan uen ens beoes: ne he eesson of onans he see of wh ese o, shows he non-naane une Gallean ansfoaons (unless he soue s neual). n onas, he Hean uen ens eans unhange (naan) assng fo o :

12 The Hean uen ens, esbe as easue b Phs (alhough s no lea how s easuabl eaes fo he Mawellan one), s no naan beause s easue bu beause s suabl efne. n fa, s efnon onl aes elae eloes elean beween soue hages an eeo. an be eesse as: el el o: el el o: el el el s he elo of ose hages onane n wh ese o he eeo el s he elo of negae hages onane n wh ese o he eeo el s he elo of ose hages n wh ese o negae hages n el s he elo of negae hages n wh ese o ose hages n n an ase, s onl eessble hough he use of he elae sees beween hages. n sne he elae sees une he GT ae he sae n an neal efeene fae, he naane of follows. Gallean ansfoaons of fels ne he oeaos an aeang n he Hean fel euaons ae Galleo-naan, an sne an ae uanes on whh all obsees agee, he fel euaons ae Galleonaan oo. The fel ansfoaon laws, n assng fo o, ae heefoe:

13 Connu euaon Usng he oal eae, one also aes a a Gallean naan onnu euaon: (7H) Ths euaon (le he oesonng Mawellan one) s ll onane n he fle euaons, as an be efe b ffeenang he euaon (H) wh ese o e: ( ) alulang he egene of euaon (4H): ( ) noung he euaon ( ) no ( ): fo whh he euaon (7H) follows. Wae euaons : Tang he oo of (4H) an usng he eo en ( ) ( ) ( ) ( ) onseng (H) an (3H) follows ha: (8H) whh eesses a loal onsan obee b he fel, esbe n, a an on n sae. The ons ae enfe n, bu he eoal aaons of he fel a hese ons ae ealuae as he oe wh nsananeous elo wh ese o. n a sla wa, sang fo (3H), s ossble o oban an euaon fo he fel: (9H) 3

14 (8H) an (9H) ae oagaon euaons wh loal fong es (funons of hage ens an uen ens). The eess loal onsans, bu also een n a non-loal wa on he see of he eeo hough he neeaon of he oal e eae an he efnon of. Coason beween Mawell s heo an he He-Phs heo Le be he obsee s R, n ese o whh he soue osons an he fel ons ae esbe. n he ase of a saona eeo n, f he esons of he soues one, he wo heoes woul ae he sae eons abou he ele fel an he foe ang on he eeo. The eee agne fel woul also hae he sae alue, bu wh ffeen oeaonal eanng. ong o boh heoes, he agne fel oes no oue effes on a saona eeo. Howee, whle n Mawell s heo he agne fel alulae a he oson of he eeo oul be use o ealuae he foe eeene b anohe eeo wh he sae nsananeous oson bu no null see, n He s heo hs s no ue, beause he fel eens on he eeo s oon. The ffeen eson of he soues n he wo heoes s ue o he fa ha, aong o Mawell, hage an uen enses een on he aoe R, whle aong o He-Phs he enses ae naan. The ueson an be bee unesoo usng an eale. Conse a neual agneosa soue onssng of an ele u n whh a onsan uen flows. ssue ha he u s eale b eans of an eal onuo a es n he neal sse of he laboao L. n L he u enfes a egon of sae haaee b an. ong o Mawell, a eeo ong a a ean see n he laboao fae unegoes a foe oall jusfe b he agne oonen of he Loen foe, he ele fel beng eo. nsea, n -,.e. he R nsananeousl o-ong wh he eeo, he foe us hae a uel ele jusfaon an hus he soue anno show loal neual. n - he u enfes a (ong) egon of sae haaee b. s s well nown, ffeen ealuaons of ae b ffeen neal obsees ae ossble, assung he onseaon of hage, b eans of he non-naane of lenghs oe b LT. ong o He-Phs s heo, f he soue s neual n L hen s also neual n an ohe R. Theefoe, he eson of he fel soues ae b an obsee o-ong wh he eeo s ffeen n he wo heoes. n he ase of a ong eeo n, he oason beween Hean an Mawellan eons s oe ole. ne us onse he ffeenes of he oeaos (aal an oal e eae) as well as ffeenes n he efnons of foe. ong o Mawell, n a gene R, a ee nsan an n ee on of he sae, he agne fel us sasf he euaon: 4

15 5 M M s he Mawellan uen ens n. The fel s esbe as neenen fo he ong hage nene o ee. ong o He-Phs, he agne fel us sasf he euaon: The saal an eoal oonaes ha ae he aguens of he eo funon ae he oonaes of. Howee, he aaons of ae alue b an obsee o-ong wh he eeo,.e. as f he ealuaon ons wee ong wh elo n. s he Hean uen ens n o, b eason of s naane, n an ohe R. The eous wo ffeenal euaons also oue ffeen soluons n he ase of a neual soue (when ), as beoes een eanng he oal eae n (eals n en - C): [ ] lal, an be sa ha, aong o Mawell, n he ele fel shoul sasf: M M ong o He-Phs: s he hage ens fel n. ong o Mawell esbes he hage ens elusel n. ong o He-Phs esbes he hage ens boh n an n an ohe R. Ths Hean wae euaon an also be eesse n n he fo (eals n en - C): [ ] Whh hghlghs he ole of he eeo s nsananeous elo on he wae euaon of.

16 n he He-Phs heo, fo he alulaon of fels, s onenen o use he naane oe an hoose -,.e. he R nsananeousl o-oes wh he eeo, as a efeene fae. The aables ealuae n - wll be ae b an ases as sues. The elaons beween - an ae: * * ( ) * * * * n -, he euaons (8H) an (9H) beoe: * * (H) * * * (H) The ae foall enal o he Mawellan ones n -, bu ffeng n he eson of soues. The onl oe a onssen eson a eah nsan n he ase of he eeo s unfo oon (beause n he ase of an aeleae eeo - hanges onnuousl) an l he eae soluons: * (, ) * (, ) * * * (, ( R ) ) * * ( ) 4π 4π * * * (, ( ) ) Whee he eae e R * R * ( ) R R 4π * * * (, ( R ) ) * * ( ) R s ll efne b he euaon: * * ( ) R. The eubaons of he fels ae seen oagang wh see n -. Tha suh saal an eoal fel sbuons ae seen s obousl eahoal an shoul be unesoo as: nfee n oe o jusf he obsee foes, n a anne onssen wh he heo. The oall nfee naue of he fels s aulal een n hs heo. The fels, whose eubaons oagae aong o he wae euaons (H) an (H), ae no esable, een oneuall. an nsan an on n sae no belongng o he wol lne of he eeo, he ele fel a be nene as he foe e un hage eee b a ual eeo shang he nsananeous see of he eal eeo an lae n ha gen on a ha nsan. Howee, whee a eal hage s no esen o oeae as a eeo, hee s no ossbl of noung a hage lae n oe o es a eubaon nene fo anohe eeo. R 6

17 The esene of he eeo us be oneuall onelae a o n he eson of he eleoagne sse n ueson, sne s sae of oon nfluenes he oagaon. lng he GT, one an eue ha, n, he fel eubaons ae seen oagang wh see u K, whee u K s he un eo n he oagaon eon. Ths eans ha, gen a soue onane n a le an oonless saal egon n, he eubaons of he fels oagang fo he soue ael a sees hghe (o lowe) han f assoae wh a eeo ong awa fo (o aoahng) he egon. Ths wae aggng ehans, whh aes he see of fel eubaons eual o wh ese o he eeo, s oable wh he null esuls of he Mhelson-Mole nefeoee eeens. esee of he soue-eeo sane, he aae eleoagne fel, behaes as f new s esnaon, hangng as a funon of he eeo-absobe oon. e hage s onnee wh eah ohe egaless of he sane (sne all he hages us shae nfoaon abou he uual sae of oon). Theefoe, he oal e eae as use n He-Phs s heo noues a ln beween fels an eeos of a non-loal naue, a n of enangleen. shoul be noe ha, ese he non-loal onsan ha as on he oagaon see, he eleoagnes foulae b eans of he euaons (H), (H), (3H) an (4H) onl les oagae soluons,.e. elae. 7

18 oluons of he wae euaon Conse a hoogeneous wae euaon (of, bu he oeue also ales o ): an a lneal olae onohoa lane-wae soluon: ( ω ) sn φ o ne a onsan hase an be se o eo b a suable hoe of he eo alue fo e, fo he hase one a onse he slfe eesson: ω ω whh oesons o a hase elo: h ω. noung he soluon n he wae euaon (eals n en - D), onseng onl nonaeleae eeos, one obans: [ ω ( )] follows ha: n so: [ ω ( )] ω ± (eo) n R, he wae hase oagaon see of a lane wae hooe on a gen non-aeleae eeo, wh see n, s: h ω ± (eo) Conseng a ffeen R, ong wh see n, he naane of fels les: so: whh eans:, ( ), ω ω Ths esbes a onsan hase alue of he wae n he wo Rs. 8

19 lng he GT:, he eous elaon beoes: ω ( ) ω so: ( ) ( ω ω ) ne an ae aba an neenen aables, he al of eual les ha he oeffens ha ull he ae eo. follows ha: ω ω (eo3) The fs esul shows ha, n he eson of a lane wae assoae wh ha gen eeo, he wae oagaon eon oes no hange n he anson fo o. The seon esul esbes a aaon of he feuen asbe o he lane wae n he hange of R. Ths feuen aaon s no a easue uan, sne he easueens onl ean o he eeo. ssung ha ones wh -,.e. he R nsananeousl o-ong wh he neal eeo ( ), follows ha: ω ω (eo4) Now he seon esul bns he angula feuen easue b he eeo wh he angula feuen of he wae esbe n, n whh he eeo has elo. alng he GT o he R aaon, he fgue shows how he eon of an gen on of he wae fon hanges, whle oes no hange he eon of he fon as a whole. Phs nees he esul as f he wee esbng he abeaon of lgh, whh, heefoe, woul be eo n he fs oe eson of he heo. The onluson s wong. The esul onl sas ha he wae eo of a lane wae assoae wh a gen eeo s naan when we hange he R n whh he wae s esbe. To ealuae he He-Phs heo abou lgh abeaon, he hange of he R n whh he oon of he neal eeo s esbe s no a elean ueson. 9

20 nsea, hang hosen he soue, s oan o unesan f an how he wae eo aes when he see of a eeo n he sae R hanges. lenael, we us unesan how he wae eos of eleoagne waes, geneae b he sae soue bu assoae wh wo eeos wh ffeen sees n he sae R, ae lne. Plane waes assoae wh ffeen nsananeousl onen eeos bu wh ffeen sees ae no neessal subje o he onsan of eual fo he hases n a gene on of he sae a a gen nsan. The sae an be sa fo he waes assoae wh he sae eeo, hang ffeen unfo sees a ffeen es. Theefoe, he eousl use oeue anno be followe o nfe he naane of o he hange of he eeo o s sae of oon. o hs uose s, nsea, useful o onse he ole of he nal onons fo eenng he oenaon of a lana aellng wae soluon. a fo he soue, he wae euaon s: [ ( ) ] ( ) ( )( ) an be ewen n he fo: ( a ) ( ) ( ) Conse a onohoa soue of angula feuen ω soue, lae a a gea sane fo he eeo, so ha a lane soluon onsues an auae loal aoaon of he sheal sufae of he fon a he eeo s oson. s a slfe oel of a sella soue. Le be he R n whh he soue s a es. Le an be wo ffeen eeos:, saona n, an, wh onsan elo n. uose hee s an nsan, efne as, n whh he wo eeos an ae onen, a a (lage) sane L fo he soue. Le he -as be oene as he sagh lne jonng he soue an. Le he ogn of he aes be lae no he soue. Le be he R n whh he eeo s saona. Le he ogns of he aes of he wo Rs be onen a.

21 Wae euaon assoae o eeo, n : n L u Ths euaon as, n, a soluon wh a oagaon see eual o, of fo: Wh: ( ) sn ω ngula feuen Waelengh ω n ω ω λ soue π h n h n n f n ω n π ω soue Waenube ω h n n ω soue The oagaon eon s onen wh he as: u u

22 Wae euaon assoae o eeo, wh ong awa fo he soue n he eon of he ose -as ( // u ) : n : ( ) ( ) n : n ( ) u - n u The seon euaon as, n, a soluon wh a oagaon see eual o, of fo: ( ) sn ω Due o he se of he sse, he oagaon eon s onen wh he -as: u Ths eans ha he fs euaon (esbng he sae en n ) as a soluon, wh oagaon elo n, ealuae b (eo), eual o u. Ths soluon an be eesse as: sn ( ω ) uang he hases of he wo esons an noung he oonae ansfoaons, he eessons (eo4) ae obane. Theefoe: u u.

23 n : ω n ω ω soue λ n f h n n π h n π ω n ( u ) π ( ) ω soue ω soue ω ω n n soue n (eo5) h n u ω n, onseng (eo4): ω n n ω ω ω n ω ω n u u ω ωsoue soue ωsoue Theefoe: λ λ n h n π h n π f n ω n f ωsoue soue ω ω soue (eo6) λ (eo7) f soue (eo6) an (eo7) esbe he Dole effe, n es of angula feuen an waelengh, assoae wh a eeo ong awa fo he soue, aong o he fs oe heo. n ase of aoahng he soue along he -as: ω ω soue (eo8) λ (eo9) f soue 3

24 Wae euaon assoae wh eeo, wh ong n ese o he soue n he eon of he ose -as ( u ) : n : ( ) ( ) n : The seon euaon as, n, a soluon wh a oagaon see eual o, of fo: ( ) sn ω o he fs euaon (esbng he sae en n ) as a soluon wh oagaon elo n eual o u : ( ) sn ω o he eual of he hases n he wo esons, one sll obans:. Howee, n hs ase: u u. n fa, he oagaon eon us be onssen wh he eson of he soue n he R n whh he soluon s eesse. n, whee s saona, he soue s esbe n oeen wh elo. Le o be he oagaon e,.e. he e aen b a wae fon o aese he sae beween he soue an he eeo. follows ha he sane aese b he wae fon (he hoenuse of he gh angle wh ahes D an L n he fgue below) s: o D has a lengh eual o:. o o he followng us al: sn( θ ) nall, follows ha: ( θ ). o sn (eo) The eesson (eo) esbes abeaon aong o he fs oe heo fo a eeo ong wh ese o he soue n a eon eenula o he sagh lne jonng soue an eeo. o 4

25 oue Poson a - o n - D θ u oue Poson a L Theefoe: n : ω n ω ω λ n π ω h n soue n π ( u ) π ( sn( θ )) ω soue ω soue π ω soue ω ω n ωsoue (eo) u n n h n 5

26 n, onseng (eo4): ω n n ω ω Theefoe: ω soue ω ω n u ωsoue ω soue ω ωsoue (eo) aen oue Poson a n D θ () u Real oue Poson a ( - o ) L 6

27 Hean oenals eo egene eo fel an be eesse as he ul of anohe eo fel. Theefoe: (H) eo oenal s eene u o he gaen of an aba sala fel; on he bass of (H) he eo fels an ae eualen. noung (H) no euaon (3H): ( ) f he euabl beween he oal e eae an ul oeao s no een, see he oof n en -. so s onseae an heefoe a be eesse as a gaen of a sala oenal. (3H) Gauge ansfoaons The onsan (H) beween an leaes he egene of unefne. Ths les he ossbl o a he gaen of an sala funon, he so-alle gauge, o whou aleng he agne fel. alng a aaon n he gauge of : ^ noung (4H) no (3H): ^ ^ ( ) ^ (4H) 7

28 Hene: ^ ^ angng he es an subsung ^ ^ ^ wh s eesson, follows ha: ^ ^ o he oason of he eous eesson wh (3H), follows ha: ^ ^ (5H) whh shows how he sala oenal s affee b he hoe of he eo oenal s gauge. Gauge naane s nown ha Mawell s eleoagnes s gauge-naan. Ths oe s ue o he suue of Mawell s euaons an he Loen foe, whh elue he nfluene of he egene of n an easuable hsal anfesaon. The gauge naane also ales o eleoagnes esbe b he He-Phs euaons. To ef hs asseon s suffen o seah he eesson of he foe (o of he ele fel) ang on a hage-eeo aong o He-Phs s euaons, eesse n es of oenal, an al a aaon of he gauge. ( ) (6H) ne: ( a b ) ( a ) b ( b ) a a ( b ) b ( a) b lang: a b ( ) ( ) ( ) follows ha: ( ) ( ) (7H) ne: 8

29 9 follows ha: (8H) (9H) (H) Loen (H) o he foe on he eeo, He-Phs s euaons l an eesson eual o ha of Loen, oee wh an aonal e. lng a gauge ansfoaon o oenals,.e. noung (4H) an (5H) no (H): ^ ^ ^ ^ ^ [ ] [ ] ^ [ ] [ ] [ ] ^ ^ ^ ^ Usng he eo en: a b b a a b b a b a seng: a b follows ha: eause he ul of a gaen s null an behaes le a onsan wh ese o an saal ffeenal oeao, we an sa: Theefoe: ^ whh eans ^. o gauge naane ales.

30 Hean euaons eesse hough oenals noung (H) no (H): noung (H) an (3H) no (4H): Usng he eo en: ( ) ( ) ( ) ( ) Thus he new sse of ffeenal euaons, whh elaes (H), (H), (3H) an (4H), s: (H) whh us be olee b he foe law eesse n es of oenal,.e. (8H). The eoulng of he euaons of sse (H) an be obane b usng he oees of gauge naane. Loen-le gauge: L L ( ) Wh suh a gauge, sse (H) aes he fo: wh: L L L L L L (3H) - Loen 3

31 The euaons hae oagae soluons fo boh oenals. shoul be noe ha L s no unuel efne. The gauge ( ) as on he egene bu oes no ae unue. f L an L ae a a of oenals sasfng ( ), hen hs sae onon ( ) s also sasfe b L an L, efne b λ whee λ s he soluon of λ λ L L λ L L, The oenals, as well as he fels, ae abue o ons n sae a es n he R, bu ealuae n ons ha ae ong wh he eeo s elo. n he R nsananeousl o-ong wh he eeo, he oenals alulae usng he gauge ( ) ae esbe as oagang a see. nsea, he oenals oagaon see s esbe as ffeen fo n ohe Rs. Usng a ffeen gauge, fo eale: Coulob-le gauge: C ( ) sse (H) aes he fo: C C C C (4H) - Coulob wh: C C The sse s fs euaon has an nsananeous Coulob oenal as a soluon. The seon euaon oes oagae soluons. The fong e has a loal oonen, whh s ooonal o he uen ens, an a nonloal oonen, whh s ooonal o he gaen of he eoal eae of he nsananeous sala oenal. s lea ha, jus le fo Mawell s eleoagnes, gauge naane allows he use of ffeen as of oenals, wh ffeen oagaon sees, obanng he sae hsal eons. 3

32 Calulaon of foe beween sea uen eleens, aong o he He-Phs heo Conse a agneosa suaon wh wo saona us n whh onnuous uens flow. aluae he foes ang beween he us eleens. Le he lose u Γ, aese b a onsan uen wh agnue, be he soue. Le Γ be he seon lose u, aese b a onsan uen. Le he onuo of he us be hn bu wh neglgble eleal essane. Le L be he R of he laboao n whh he onuos ae fe. Le l be an nfnesal oon of Γ, lae n oson an oene as he ose eon of he uen. s a eeo, hoose he hage eleen an assoae wh an nfnesal lne eleen he uen. Le be he oson of l. The ln beween hese uanes s obousl: (onuon eleons), ong a a onsan see l of Γ, oene n he ose eon of l. Teang he we seons as nfnesal, he us ae esbe b lnes, one-ensonal enes ese n R 3. Le: l s s s l u u u foe eee on foe eee on l b l b l l (ee en - fo a noe on alenae hsoall oose eessons of he foe beween uen eleens). n L, n whh he onuos ae saona, he wes ae esbe b hage enses: he ose, eesenng fe hages an he negae,, eesenng onuon eleons n oon. ong he hohess of neual fo onuos les: ( ) ; ( ). eause GT esee lengh naane, hs neual eans ue n all efeene faes. uheoe, ue o he neual, Mawellan an Hean uen enses assoae wh he soues ae eual: 3

33 Conse he oenal euaons wh he onsan ose b he Loen-le gauge, sse (3H). ong he hohess of neual of he soue, he sse s eue o he sngle euaon (he subs L s oe fo be): n -,.e. he R nsananeousl o-ong wh he eeo, he euaon aes he fo: * ( ) n -, he onuos ae no saona, bu ong wh elo The eo oenal oue b he sngle uen eleen s: * l (, ) * * 4 π R. l, n he eeo oson, Une he onon: R * * ( ) R Theefoe negleng he oagaon ela: * l, * 4 π * Reebeng ha he foe on a hage, n an R, an be eesse as: ha, n -, beoes: * Conseng he neual of he soue an he onsan beween he aal e eaes eesse n L an -, he foe eee on he uen l b he eleen l wll be: o he assue saona of n L ( ), follows ha: L o: 33 l l * L

34 34 l * 4 4 l l l l π π Usng he en C C C : l l l π 4 l l l 4 π Maheaal noe: 3 u l u l l 3 (Poof n en - G). Usng he eous esuls: l u u l l 4 π { } l u u l l 4π ne: l l (he ong hage s negae, heefoe s oose o he ose eon of he uen, whh s he eon of l ), follows ha: [ ] l u l u l l 4π [ ] u l l u l l 4π The seon of he wo es ones wh he eesson of Gassann s foe, he foe eee b Mawell s eleoagnes. G u l l 4 π

35 Usng he en: a ( b ) ( a ) b ( a b) s ossble o eess l, so: ( l u ) ( l u ) l ( l l ) u n anohe fo: 4π [ ( l l ) u ( l u ) l ( l l ) u ] 4π [( l u ) l ] o he foe s oene le l. eesng he oles of soue an eeo, he followng eesson s obane: 4π [( l u ) l ] These eessons ae no oable wh eal. Conseng, fo eale, he ase of lnea aallel onuos lae a sane fo eah ohe, he negaon of he eleenal foes along he wes ah aong o Mawell s heo (bu also aong o he ee s ognal law) les a foe e un lengh, aae n he ase of uens n he sae eon, wh alue: L L G π eeang he sae negaon, usng he eesson of he foe aong o He-Phs s heo, sne he eleenal foes le on he sae lne of he wes, no aaon s ee. n s esen wong, he heo s heefoe o be abanone ue o noabl wh eeenal obseaons. 35

36 oe onseaons on ee falue an an nal oosal fo aenen The eous eessons of he foes beween uen eleens show ha he elaon beween foe an agne fel, eae onl ll b he use of he oal e eae n he fel euaons, fals o eesen he foe oonen ee eenula o he oon of he eeo. s no ossble o ajus he law of foe b sl ang a e, hus aong Loen foe, ue o he naan naue of he fels. ne he fels an ae naan, he foe s also naan on aoun of he naane of, whle he foe s no, ue o he eenene of on he R. n oes no see ossble o onee a efnon of foe wh an naan an el ole fo ha s oable wh obseaons. oes no een see ossble o of he heo o ae oaan wh ese o he hange beween Rs, sne oaan fels eue oaan soue es. uheoe, s ossble o hae a oaan hage ens f one assues he al of he GT an onseaon of hage. oose an nal aenen o he heo, whh esees s naan haae bu hanges he eson of he soues wh ese o he eeo s oon. ne he effe of hs oon us be naan, he eeo s sae of oon anno be esbe usng, whh s he see ealuae b an neal obsee, ue o he abaness of he obsee. o he elae eloes beween eeo an soues us be onsee. Theefoe, assue ha he egene of he ele fel (fo he eeo s on of ew ) oes no een solel on he soue hage ens (unesoo as he nube of hages n he un olue), bu on a sala funon, an eualen ens, funon of, whh s eue o fo. ( ) shoul be noe ha he uen ens n he Hean heo alea las a sla ole. n fa, n he geneal ase of a non-neual soue, noues an effe of he elae elo of he eeo an soue on he ul of he agne fel oue b he soue. To laf hs oosal, onse a le egon of sae wh he esene of a ose hage wh ens (,) an a negae hage wh ens (,). The ne hage ens s: The olue eleen, loae n, a he nsan, has a oal ele hage: (, ) [ (, ) (, ) ] f, n, all ose hages onane n shae he sae aeage elo, whle all he negae hages shae he sae aeage elo, s ossble o efne he followng uen enses: ( ) (5) (6) 36

37 37 The Hean uen ens n s hen eessble as: (7) nee: whh ones wh he ognal efnon. has eousl been shown ha, b aong GT, he Hean uen ens s naan o hanges of R. has also been shown ha s naane oesons o he ossbl of beng eesse b eans of he onl elae sees beween hages: el el el el el el o efne: (8) (9) ˆ (3) The sala fels, an, ae naan eanng he hae alues on whh all neal obsees agee beause he een on he onl elae sees. follows ha:

38 38 lenael: ne, ha eans:, follows ha: (3) : (3) Refeng o a gene R, n e sae he euaons fo he eleoagne fel, al o fs oe of, beoe: (H) (H) (3H) (4H) The foe law: (5H) shoul be noe ha hs foulaon oes fo he esene of foes eee b uens on fe ele hages.

39 Calulaon of foe beween sea uen eleens, aong o he ofe heo n L, he sse s esbe hus: oue: he u Γ, n whh flows. The eleen l of Γ onans he negae hage (onuon eleons ong a elo n L ) an he ose hage (osel hage eal ons, oonless n L ): l Deeo : he negae hage assoae wh he nfnesal oon (onuon eleons ong a elo n L ), l of he u Γ. l Deeo : he ose hage (eal ons, oonless n L ), assoae wh l. l n a gene R, he oenal euaons wh he Loen-le gauge beoe: Refeng o -,.e. he R nsananeousl o-ong wh he eeo, hs sse beoes: * * ( ) ( ) * * ( ) ( ) Two ffeen Rs wll heefoe be use, eah o-ong wh he esee eeo. Gen he naane oees, an an be alulae n an R, heefoe also n L : 39

40 4 o eeo, n L :,, [ ] o, n - : * * Theefoe, he nfnesal onbuon o sala oenal eee b eeo, lae n oson * an geneae b he nfnesal soue lae n oson *, s: R R * * * * * 8 4, π π Conseng ha l l e, follows ha: R R l l e * * * * * 8 8, π π Negleng he oagaon ela: l * * * 8, π The nfnesal onbuon o eo oenal s: l l R R * * * * * * * 4 4 4, π π π

41 4 o eeo, n L :,, Then, n -, whh s onen wh L, one an foulae he followng eessons: L L Theefoe, he nfnesal onbuon o sala oenal eee b eeo, s: R R π π 8 4, R R l l e π π 8 8, l π 8, The eo oenal eee b eeo s elean fo foe alulaon, gen ha s oal e eae s eanl eo, beause of he e neenene of he uen ha nullfes he aal eae e n L an he sllness of he eeo ha nullfes he effe of oeao.

42 4 The foe on eeo s: * l l L [ ] l l l l π π 4 8 l l l 4 π l l l l 4 π ( a ) l l l l l 4 π l l l 4 π l u u l l 4 π l u u l l 4π l u l u l l 4π ne he ong hage s negae, s oene n ooson o he ose eon of he uen. o: l l u l l u l l 4π (33) ( a ) Usng he eo en: C C C

43 43 The foe on eeo, eebeng l, s: l l L l l l l 4 8 π π 4 4 u l l l l π π o: u l l 4π (34) The oal foe ang on he u oon l, hen, s: u l l u l l u l l 4 π u l l 4π (35) esson (35) ones wh he eesson of Gassann s foe: G Mawell s heo an he ofe eson of He-Phs s heo e he sae foes beween us n saona onons. nsea, he wo heoes woul ae ffeen eons f one elaes he neuale uen of u Γ wh an eleon bea n auu. f, howee, one onses he ase of a sngle on hage eeo Q, oonless n L, he new heo es ha he eleen of he uen l shoul ee a foe on eual o: Q Q Q L u l Q l Q 8 8 π π

44 seon oosal fo aenen s alea enone n he nouo a of hs wo, eeenal esuls sugges he nouon of soe n of nsananeous neaon no he eson of he eleoagne henoena. The esene of nsananeous neaons aeas oneuall oable wh he non-loal noue b usng he oal e eae. s neessa o fuhe of he heo esbe b he euaons of fels (H), (H), (3H) an (4H), b he efnon of foe (5H) an b he esons of soues (5), (6), (7), (8), (9) an (3) whh onl oes soluons wh fne oagaon see. The new euaons ae: (a) (a) (a3) (a4) (a5) (a6) (a7) (a8) wh: (a9) (a) (a) ( ) (a) ( ) (a3) ( ) (a4) ( ) (a5) (a6) 44

45 Connu euaon Dffeenang he su of (a3) an (a5) wh ese o e: ( ) [ ( )] Calulang he egene of he su of (a8) an (a): ( ) ( ( ) so: ( ) ( ) noung ( ) no ( ): so: (a7) Posson s euaons an wae euaons (a8) nsananeous (a9) oagae (a) nsananeous (a) oagae Poof n en - H. an an be alle he nsananeous oonens of he ele an agne fels. an an be alle he nue oonens of he ele an agne fels. 45

46 46 bsene of soues Conseng he fels nsananeous oonens, he onon of absene of soues s ue n ee on of he sae n whh he hage ens an uen ens ae eual o eo. n hose egons: nsea, onseng he nue oonens of he fels, n ee on n whh he hage ens an uen ens ae eual o eo, he wae euaons eue o: The fong es ae seon oe oal e eaes of he nsananeous fels an he ae onnuous funons whh een beon he egons oue b he hages. o he wae euaons o no beoe hoogeneous eael ouse hese egons, bu onl a sanes ha ae lage enough fo he hages o ene he onbuons of hese onnuous funons neglgble. Thus, he aaon soues woul no be on-le n he sae wa as he eleena hages whh onsue he soues, bu woul be eene aoun he hages an esbe b onnuous sbuons, eeasng aong o. a gea sane fo he soues, he euaons ae aoae b he hoogeneous fos: whh esbe he fa fel oagaon. The nsananeous euaons (a8) an (a) ean unalee n an R. The oagaon euaons (a9) an (a), fo non-aeleae eeos, n -, ae he fo: Wh ese o he wae euaon a a gea sane fo he soue, he oeues followe usng he eous eson of he heo ae sll alable. Theefoe, egang abeaon an Dole effes, he elaons (eo6), (eo7), (eo8), (eo9), (eo), (eo) an (eo) ae sll al.

47 nouon of oenals eans ha s onseae, so an be eesse as: (a) ( ) an l ha an an be eesse b eans of oenal eos: (a3) (a4) o, allng: ( ) (a5) noung (a5) no (a6): ( ) eans ha s onseae, so an be eesse as:. o: (a6) uaons (a3) an (a4) leae he egene of an nefne. Ths les he ossbl of ang gaens of an sala funon (naeable gauges) o oenals an, whou aleng he oesonng agne fels an. ffe of a hange n he gauge of ^ alng a gauge aaon: ^ o: an noung hs eesson no euaon (a6), follows ha: ^ ( ) ^ ^ 47

48 ^ ^ ^ o: ^ o he oason of he eous eesson wh (a6), follows ha: ^ ^ ^ ^ Conseuenl: ^ whh shows he effe of he hoe of he eo oenals gauge on he sala oenal. Gauge naane Gauge naane ales. To ef hs asseon, s suffen o eess he ele fel ang on a eeo aong o he new euaons eesse n es of oenal, an al a hange of he gauge (eals n en - ). uaons eesse hough oenals noung (a) no (a3): ( ) so noung (a6) no (a5): so noung (a3) no (a8): ( ) ( ) ( ) noung (a4) no (a): ( ) so 48

49 49 n hs wa one aes a a sse of fou euaons: (a7) The sse an be ewen n a sle fo b aong he followng hoes of gauge: () () n suh a ase he sse beoes: (a8) whh us be olee b he foe law eesse n es of oenals: (a9) o: [ ] [ ] (a3)

50 5 oluons of sse (a8) o a non-aeleae eeo, n -, sse (a8) beoes: (a3) The nsananeous euaons, whh ean he sae fo n an R, hae he nsananeous soluons:, 4, e a sae all π (a3), 4, e a sae all π (a33) The wae euaons wh non-loal fong es hae soluons wh oagaon see eual o. The fong es ae no onenae on hages bu sbue n he sae aoun he. The ae onnuous funons, eeasng aong o. The soluons ae: [ ], 4, 4, R R R π π (a34) [ ], 4, 4, R R R π π (a35) R s ll efne b: R R

51 oe aonal onseaons on he new heo The elas onsan whh eues he see l fo he oagaon of eneg an nfoaon s onssen wh he ea of loang he eneg n he fel. s assue ha he eleoagne eneg has he oe of beng lae wh a ean ens n he sae beween he hages. u s nene as eneg e un olue; aea; s nene as eneg ansfee e un e e un oss-seonal g s nene as oenu e un olue. Ths neeaon shows s usefulness n ffeen suaons, bu s less han sasfao beause s eonlable wh a on-le eson of he ele hages. n fa, fo he aboe efnon of eleoagne eneg ens, he eneg assoae wh he fel of a on hage has an nfne alue. Conseng he hage nas, he nee o elue he (nfne) onbuon of suh eneg fo gaaonal o neal oees (.e. he ass) of he hage shows he ls of onssen n he eeng assuons. n he new heo, assung he esene of nsananeous neaons (aon a a sane) an he fels eenene on he eeo s sae of oon, he ea of saall laeable eleoagne eneg sees o loose fuhe ebl. sees oe naual o nee he eneg as a elaonal oe assoae wh he whole sse of ele hages. nl he henoena of absoon an esson woul hae a loaon n sae. 5

52 Possble eensons of he heo Hghe-oe heo aong o Phs Phs oose o een he alabl of hs heo beon he l of he fs oe (allng hs eson Neo-Hean), assung he naane of he followng ffeenals: : wh: : δ n he fs ase, he ffeenals an efe o oonae neens assoae wh as of suesse eens, belongng o he ajeo of a sngle ale. s he e neal beween suh eens, easue b a lo a es n he hosen R, alle. s heefoe a ffeenal of R oonae e. s he e neal beween suh eens, easue b a lo a es wh ese o he ale. s he saal sane beween suh eens, ealuae n. The naane of us be nene n he sense ha all obsees n whaee sae of oon wll agee on s nueal alue. The alue of oe e a be (eall) ea on a lo o-ong wh he ale. enoes a seaaon beween eens on he sae wollne. n he seon ase, δ naes a saal seaaon beween ons belongng o an eene suue (le a sana ee), whh eans a sane easue hough g boes a es n an aba R. δ enoes a seaaon beween eens on ffeen wollnes a he sae oonae e. o he efnon of (onseng he ose oo onl): The oe e use n eleoagne heo s he oe e of he eeo:. The new fel euaons ae obane b subsung he non-naan e wh he assue naan e. 5

53 Poblea naue of suh hoes The aoon of a oe e n aoane wh Phs s assuons leas o aaoal eons n ese o he assuon ha absolue sulane ess beween saall seaae eens. To laf hs saeen, onse wo gene neal eeos an,.e. wo lassal ales wh a unfo sae of oon wh ese o an R. Le an be he Rs eseel o-ong wh an. Theefoe, seang of he eeos oe es s eualen o seang of he oonae e of an. To slf he suaon, assue ha hee s an neseon of he wollnes of he eeos an. Ths eans ha he ajeo of neses he ajeo of, an ha he osons of he wo ales one n he neseon. Le us use hs een of onene o ae a unoal snhonaon of he los ae b an. Conssen wh Phs s efnon of oe e, one an sa he followng. n, he nsan ) of s oonae e us oeson o alue ), ea on he lo ae b, an o alue ) ea on he lo ae b, wh: ) ) n n n ) ) n n > < n, he nsan ) of s oonae e us oeson o alue ), ea on he lo ae b *, an o alue ea on he lo ae b, wh: ) * n ( n ) n * ) ) n n > < <. We a heefoe onlue ha, n he een of he lo ae b ang he e ), he wo neal obsees anno agee on he e ae b he lo ae b. ne s a onoonall neasng funon of he oule of he eeo-neal obsee s elae elo, he oe e of an eeo n oon wh ese o he aoe R shoul flow slowe han oonae e. n us be ue n ee R, gen he abaness of s hoe. The aoon of hese onsans ules ou he ossbl of a eson of sae an e whh as absolue sulane beween san eens. 53

54 eahng fo alenaes The sles alenae aoah, whh ees he nae of naane of he wo ffeenals δ an, s he followng. osulaes he esene of a lege efeene fae, n ese o whh he e flow s he au. Nael he beang of a lo a es n woul hae he au feuen. The os naual anae fo suh a lege R woul be one n whh he ole oonen of he os bagoun aaon anshes. Le be he oonae e of. Gen a gene ale, enfe n b he eo he sense ha all obsees wll agee on s alue) of:, one an assue he naane (n nene as he ale s oe e ffeenal. follows ha: s he ale elo n (s absolue elo). Usng he onene of he oe e of a unfo ong ale an he oonae e of he o-ong R, new ansfoaon ules beween R an be obane. Conse a gene R. Le (,) an, be he saal an eoal oonaes of he sae een n an. Le be he elo of n. Le us hoose he oonae sses so ha he ogns of he aes n he wo Rs one a he hosen eo e. The ansfoaons an be eesse eoall b: The nese ansfoaons ae: (T) (T) wh: 54

55 55 The ogn of he -aes s enfe b he eos:, n, n Theefoe, n s esbe as ong wh elo:. The ogn of he -aes s enfe b he eos:, n :, n Theefoe, n s esbe as ong wh elo: The see of ealuae b s geae han he see of ealuae b. bsolue sulane s eoee wh hese ansfoaons, bu he an-se oe of he elae elo s los. (T) an (T) allow lnng he esons of he oon of a gene ale n an n. o o (T3) o o o (T4) nall, onseng a h R, wh he absolue elo, suosng he onene of he ogns of all hee Rs a e eo, follows: lnang he aables wh subs : (T5) (T6) P

56 56 Usng (T5) an (T6) s eas o eess he elae oon of he ogns of he aes of an : (T5) an (T6) allow lnng he esons of he oon of a gene ale n an n. (T7) (T8) uh a foulaon, alhough onssen an enowe wh a leas one neesng feaue, has he heoeal saanage of easng he nle of ela an he aal efe of sageeng wh eal eene. The neesng feaue onens he e-ealuaon of he eneg ole. n fa, he eesson of he eeo s oe e an be ewen n he fo: T T Ths foulaon osulaes a e ln beween ne eneg an he oe e of he ale. long wh he usual oees of nesubl, hs foulaon also abues eneg wh he haae of hang an absolue alue. eoes eneg s ole of funaenal subsane. Howee, he eenson of he oe e s eenene on ohe fos of eneg, suh as he gaaonal oenal eneg, oes no see oable wh eeenal esuls (fo eale wh aa sule fo ao los use n GP saelles). has been eeenall esablshe ha los lae n fe osons n a gaaonal fel wh sheal se show a eenene on Newonan gaaonal oenal (wh eo oenal loae o nfn) n soun ageeen wh he elaon: Φ Φ whee: G M Φ s he oe e ffeenal of a lo lae n a gene fe oson wh ese o he gaaonal fel soue. s he oe e ffeenal of a lo a es elae o he gaaonal fel soue an osone a saal nfn. M s he ass of he soue.

57 Usng he fs eesson fo he ffeenal o esbe he behao of a lo ha s oonless elae o he soue of he gaaonal fel an lae a nfn, we an sa: s he absolue elo of he gaaonal soue. Theefoe, he behao of a lo whh s oonless elae o he soue, bu loae a a gene sane fo, woul be esbe b: Φ Φ To esbe he behao of a lo ong elae o he soue, s lausble o osulae: Φ T Φ T G M Le us use hs eesson o ealuae he e of a lo ae b a GP saelle. Conse a slfe sse onssng of he un, ah an a saelle. The obs ae onsee ula. The absolue elo of he saelle (eeo) s: un ah _ un a _ ah un absolue elo of he un. ah _ un elae elo he ah-un n. elae elo aelle-ah n. a _ ah un 37 se, on he bass of he ole oonen of he os bagoun aaon. ah _ un a _ ah se se o: un ah _ un a un ah _ un ah _ un a _ ah un a _ ah The e e un a _ ah shoul oue a hange n he hh of he saelle lo wh 8 alue n he oe of un a _ ah.6. The au alue ous when he lane of he saelle s ob s aallel o he elo eo of he un. uh a aaon woul hae a eo of half a seeal a. Ths eon leal onfls wh he aalable aa, ang neessa o eje he eous aoah. 57

58 ehao of he los ae b he GP saelles, aong o GTR Relas esons ale o he GP sse (see fo eale efeene 8 ) aea o agee wh he eeenal aa, alhough he fal o jusf he eleane of he sola an luna gaaonal oenal n hese esons. These esons begn b alng he hwashl e n he so-alle ah Cenee neal fae (C), a efeene fae fe o he ah s ene of ga bu no oang. The hwashl e s an ea soluon of nsen s euaons, al n he ase of a sngle sa an sheall se gaaonal soue. n hs e, he eesson of he lne eleen s: s G M ( ϑ) θ sn G M o wea fels, eanng he oeffen (as a funon of he aable M ) no he Talo sees, he aboe foula an be aoae b he lneae hwashl e: G M G M s θ sn Φ Φ ( θ sn ( ϑ) ) ( ϑ) The esuose sheal se auoaall elues an ole fo soues ohe han he soue lae n he ogn of he oonaes. The eesson of he oe e ffeenal, alable o a saelle obng he ah, obane fo hs e s: Φ ( ) s he ffeenal of a oonae e, an s enal o he oe e of a lo n a fe oson wh ese o he enal soue an loae a nfn. The oonae e ffes fo he GP oonae e b a sale fao, whh sees o ansfo no he oe e easue b a lo gl oule o he efeene geo. n he ase of a ula ob, he eesson ( ) les wo oee es of oose sgn fo he e nae b he obng lo, oae o he e nae b a lo a es n he C efeene fae a goun leel. The fs e eans ha e beas oe ul ue o he fa ha he oule of gaaonal oenal a obal alue has a lowe alue oae o goun leel. The seon e onsues a slowown n he bea of e ue o he obal elo. n he sef ase of a GP saelle ob (obal aus 66 ), hee s an oeng nfluene of he fs e ang an nease n see (abou 38 s/a). f one assues a Kelean ob fo he saelle, eesson ( ) also allows a oe eon of he eo aaons n he oe e, assoae wh he obal een. 58

59 ssung ha eesson ( ) s eenable o he ase n whh he oenal s no onl assoae wh he enal soue, bu also nlues a onbuon fo a eoe soue (suh as he un), he sola oenal shoul anfes easuable effes on he hh of he lo. Ths eans ha f we assue he al of he eesson Φ ah Φ un ( ) he effes ause b hanges n he sola oenal Φ un aoun he saelle s obal ah as a esul of aaons n he saelle-un sane shoul be easuable fo los obng n Kelean oon. n he ona one oul eje ( ), sne ( ) s obane fo ( ) b eenng s al o he oal oenal an oeaon ha neessal esos he assue sheal se. n hs ase, howee, s unlea how o uanf he effes of he un s an he Moon s oenals. n he elas leaue onenng he GP sse, hee s no sasfao jusfaon fo he aaen eleane of he sola oenal on he obng los. Conse fo eale efeene 9, whh las o elan he henoenon b nong he nle of eualene an he fee-fallng sae of he ah an s sse of saelles whn he un s gaaonal fel. f suh onseaons wee oe, he oul be ale o eoneousl eue he eleane of he eesal oenal on a GP saelle lo, sne he saelle s fee-fallng aoun he ah. ssung he al of ( ), woul be ease o jusf he non-obseaon of saelles oe e eenene on he ang sane fo he un f we abanon he hohess of sl Kelean obs. Changes n a saelle s elo, ue o he esene of he un s gaaonal fel, shoul oue effes on he lo ha oensae fo hose oue b he aaon of he sane fo he un. n oe geneal ones, wh ulle gaaonal soues n elae oon, he ossbl o eess he oe e of a gene sae obe whh s no onsane o follow a lose ob aoun a onan soue oes no see o be a sole oble aong o he GTR. 59