Lorentz-invariant gravitation theory.

Transcription

1 Alexande G. Kyiakos oenz-invaian aviaion heoy. 4 с 4 A A с d d a M d d d d ds s s s sin 4 sin 6

2 Annoaion The oden heoy of aviy, whih is alled Geneal Theoy of Relaiviy (GTR o GR), was veified wih suffiien auay and adoped as he basis fo sudyin aviaional phenoena in oden physis. GR is he eoei heoy of aviaion, in whih he ei of Rieannian spae-ie plays he ole of elaivisi aviaional poenial. Theefoe i has eain feaues ha ake i ipossible o onne i wih ohes physis heoies in whih eoey plays only a suppoin ole. Anohe foal feaue of eneal elaiviy is ha he sudy and he use of is aheaial appaaus equie uh oe ie han he sudy of any of he banhes of oden physis. This book is an aep o build a non-eoeial vesion of he heoy of aviaion, whih is in he faewok of he oden oenz-invaian field heoy and would no ause diffiulies when eahin sudens. A haaeisi feaue of he poposed heoy is ha i is buil on he basis of he quanu field heoy. Table of onens Fo auho.. oaions Chape. aeen of poble.4 Chape. Oiin of he aviaion field soue... 6 Chape. The axioais of IGT and is onsequenes. 9 Chape 4. Conneion of eleoanei heoy and aviaion... Chape 5. Eleoanei base of elaivisi ehanis... 7 Chape 6. The equaion of oion in IGT.. 6 Chape 7. Geoey and physi of IGT and GTR.. Chape 8. The equivalene piniple and ei enso of IGT.. 6 Chape 9. oluion of he Keple poble in he faewok IGT. 44 Chape. The soluion of non-osoloial pobles in faewok of IGT 48 Chape. Cosoloial soluions in faewok of IGT.. 57 Chape. Quanizaion of aviaion heoy.. 58 Refeenes. 6 Fo auho The oenz-invaian heoy of aviaion (IGT) is he ondiional nae of he poposed heoy of aviy, sine oenz-invaiane is a vey ipoan, alhouh no he only feaue of his heoy. oe ha ou appoah was used in he pas in elaion o he aviaional heoies ha have soe siilaiies wih ou heoy. Theefoe he esuls obained by well-known sieniss ae widely ied in he book. Howeve, fo posin he poble and fo soe of he basi eleens of he heoy whih ae obained by he auho of he book, he only peson esponsible is he auho.

3 OTATIO. (In alos all insanes, eanins will be lea fo he onex. The followin is a lis of he usual eanins of soe fequenly used sybols and onvenions). Maheaial sins,,,... - Geek indies ane ove,,, i, j, k - ain indies ane ove,, ˆ ˆ, - Dia aies A A - -diensional veo - 4-diensional veo A - Tenso oponens - Covaian deivaive opeao Physial values u - veloiy - aplaian - dalebeian opeao a - 4-aeleaion du d p - 4-oenu T - ess-eney enso F - Eleoanei field enso Abbeviaions: IGT - oenz-invaian aviaion heoy; EM - eleoanei; EMTM - eleoanei heoy of ae; EMTG - eleoanei heoy of aviaion; M - andad Model; QFT quanu field heoy Indexes e - eleial - anei, e - eleoanei, - ei enso of uvilinea spae-ie GR - ei enso of GR spae-ie R - Rieann enso R - Rii enso R R - Rii sala R G - Einsein enso - Minkowski ei h - Mei peubaions - oenz ansfoaion aix j - Cuen densiy J - Anula oenu enso - ewons onsan of aviaion - oenz fao (-fao) - ass of paile M - ass of he sa (un) M, - anula oenu QFT - nonlinea quanu field heoy; QED - quanu eleodynais. EHE - Einsein-Hilbe equaion HJE Hailon-Jaobi equaion GTR o GR - Geneal Theoy of Relaiviy -ansfoaion - oenz ansfoaion -invaian - oenz-invaian - aviaional e - avio-elei, - avio-anei - ewonian

4 4 4 Chape. aeen of poble.. The plae of aviaion heoy in a nube of ohe physial heoies The fis lassial ehanis heoy was eaed by ewon. Two ypes of laws of ehanis exis: laws of oion of aeial bodies unde he aion of foes, and laws ha define hese foes (whih ae ofen alled equaions of soues). In faes of ewons heoy, his seond law is he piay law of oion, while ewon s aviaional law defines he foe of aviy. I should also be enioned ha duin he fuhe developen of ehanis, nueous aheaial foulaions of he oiinal laws of ewonian ehanis wee found, whih ae physially alos equivalen, inludin he ones ha use eney haaeisis of he oion of bodies, ahe han foe... Relaivisi heoies As was evealed lae, ewonian ehanis is valid fo speeds, well below he speed of lih k/se. Mehanis, whih is valid fo speeds fo zeo o he speed of lih was ondiionally naed elaivisi ehanis (deailed oveview of he heoy see (Pauli, 98)). Unde he ondiion, ewons laws have vey hih auay. The definiion "elaivisi" is equivalen o he equieen "o be invaian unde oenz ansfoaions". Theefoe we will use he definiion of "elaivisi" equally wih he definiion of "oenz-invaian (biefly "-invaian"). In elaivisi ehanis, hee ae also seveal fos of equaions of oion and equaions of soues. As he elaivisi law of oion (inludin he heoy of aviy) he elaivisi Hailon-Jaobi equaion is ofen used... How non-elaivisi ehanis is elaed o he elaivisi ehanis on-elaivisi heoies ive oe pediions a speeds uh less han he speed of lih. The elaivisi heoies ive exa values in he enie ane of speed fo unil he speed of lih k/se. The inauay of non-elaivisi heoies opaed o he elaivisi an be aibued o he oenz fao, a fao of he oenz ansfoaion (see in efeene book he diaa of oenz-fao as a funion of speed). As seen fo he aph, fao is no vey diffeen fo he uni, up unil he veloiy of he paile eahes he / of he veloiy of lih (i.e., abou k/). The axiu speeds of he planes and he assive bodies on he Eah and in he sola syse ae: pojeile -.5 k/s, he oke - - k/s, eeoies k/s, he Eah aound he un - k/s, he un in he dieion o he alai ene - k/s, ou alaxy - up o 4 k/s. Hihe speed is ahieved only by eleenay pailes in osi spae o in aeleaos, bu hey do no play any ole in he heoy of aviy. Thus, he value in eal pobles of ehanis is vey sall and he oenz fao is no vey diffeen fo uni. This eans ha ewonian ehanis is valid in paial appliaions wih ea auay. This was aleady undesood by one of he foundes of he oenz-invaian physis A. Poinae, who had waned (Poinaé, 98): I ied in a few wods o ive he fulles possible undesandin of new ideas and explain how hey wee bon... In onlusion, if I ay, I expess a wish. uppose ha in a few yeas, his new heoy will be esed and oe ou vioious fo his es. Then, ou shool eduaion is in seious dane: soe eahes will undoubedly wan o find a plae o new heoies... And hen [he sudens] will no asp he usual ehanis. Is i ih o wan sudens ha i ives only appoxiae esuls? Yes! Bu lae! When hey will be peeaed by i, so o speak, o he bone, when hey will be ausoed o hink only wih

5 5 5 is help, when hee will no be a isk ha hey foe how o do his, hen we an show he is bodes. They will have o live wih he odinay ehanis, he only ehani ha hey will apply. Whaeve he suess of auoobilis would be, ou ahines will neve eah hose speeds whee odinay ehanis is no valid. Ohe ehanis is a luxuy, bu one an hink abou a luxuy only when i is unable o ause ha o he neessay... The eneal heoy of elaiviy The oden heoy of aviaion, alled he eneal heoy of elaiviy (GTR o GR), efes o lassial ehanis. As he equaion of soue is onsideed o be he Einsein-Hilbe equaion (EHE) of eneal heoy of elaiviy (GTR) o (GR), whih was found by hese eseahes alos independenly and alos siulaneously (Pauli, 98; Vizin, 98). As he basis fo heoy buildin, Hilbe used a vaiaional piniple. The appoah of Einsein was heuisi, eanain fo he expeienal fa of equaliy of aviaional and ineial asses (noe ha his equivalene is also valid in nonelaivisi heoies). A vey diffiul quesion, is whehe he GTR and is equaion ae elaivisi in es of he oenz invaiane. ily speakin, i is no (Kaanaev,, p. 74). Einsein assued ha he eneal ovaiane of he equaions of eneal elaiviy inludes speial elaiviy. As is known, EHE is vey diffeen fo ohe equaions of ehanis, sine i is based on he Rieann eoey in eneal syse of oodinaes. Besides, GR has seveal disadvanaes, whih have no been oveoe o dae (Fok, 964; Rashevskyi, 967); ounov, ). These disadvanaes have been fo any yeas he ause of seahin he new heoy of aviy. The -invaian heoy of aviaion is eaded as one of he basi, beause i ould opleely eliinae he disadvanaes of he GR. e us enueae basi disadvanaes. ) In 98, hodine (hoedine, 98) fis showed ha by he appopiae hoie of oodinae syse all oponens of pseudo-enso of he eney-oenu, whih in he faewok of GR is he soue of he aviaional field, an be uned ino zeo. This was onfied by D. Hilbe and ohe sieniss (Baue, 98; Fok, 964; ounov, (; Pauli, 958;) (Fo oe infoaion abou his issue, see hape ). ) GTR has no onneion wih quanu field heoy (i.e., wih he heoy of eleenay pailes - he salles pailes of ae, apable o podue he aviaional field). oe poinen sieniss even aue ha aviy is soe independen obje of naue, whih has no onneion wih he es of physis. 4.. The sienifi oals The naue of ie, spae and ealiy ae o lae exen dependen on ou inepeaion of - peial (RT) and Geneal Theoy of Relaiviy (GTR). In TR essenially wo disin inepeaions exis; he eoeial inepeaion by Einsein based on he Piniple of Relaiviy and he Invaiane of he veloiy of lih and, he physial oenz-poinae inepeaion wih undepinnin by od onaions, lok slowin and lih synhonizaion, see e.. (Boh, 965; Bell, 987). I an be quesioned whehe he oenz-poinae-inepeaion of TR an be oninued ino GTR (Boekae, 5). I an be said ha he pupose of eaion of oenz-invaian heoy of aviaion (IGT) is o show ha he oenz-poinae-inepeaion of TR an be oninued ino aviaion heoy. uh a heoy ould allow o oveoe all he shooins of eneal elaiviy. ine he Hilbe-Einsein equaions ive poven esuls, obviously, we have o show ha suh a IGT ives equivalen esuls. Ou addiional oal will be o explain he feaues of eneal elaiviy wihin he faewok of noneoei physis.

6 6 6 Fo he puiy of he heoeial onlusions of IGT we will no use anywhee of ideas of GTR o of siila ei heoy as he basis of ou heoy (his does no apply o hose ases, in whih we will opae he esuls of hese heoies). In he book we shall use he CG syse of unis, in paiula, he syse of unis of Gauss, sine hee all unis ae a unified syse of ehanial unis. Chape. Oiin of he aviaion field soue.. The soue of aviaion in he heoies of aviaion and onsevaion laws.. The soue of aviy in eneal elaiviy Iniially Einsein assued ha he soue of aviy in he Hilbe-Einsein equaions is syei eney-oenu enso T of he oenz-invaian ehanis saisfyin he law of eney-oenu onsevaion: k T x ik k, (.) whih oesponds o en ineals of oion of oenz-invaian ehanis. As he enealizaion of T in GR should be he eneal ovaian deivaive and insead (.) we have: T T T x, (.) Bu, i appeas ha (andau and ifshiz, 97) in his fo, howeve, his equaion does no eneally expess any onsevaion law whaeve. As a way ou of his siuaion Einseins foulaion of eney-oenu onsevaion laws in he fo of a diveene involved he inoduion of a pseudo-enso quaniy ik whih is no a ue enso (alhouh ovaian unde linea ansfoaions). To deeine he onseved oal fou-oenu fo a aviaional field plus he ae loaed in i, Einsein hoose a syse of oodinaes of suh fo ha a soe paiula poin in spae-ie all he fis deivaives of he ik vanish. Then we an ene he value ik by he followin expession: ikl h, (.) x ik ik T l 4 ikl ik l il k whee h 6 x ik ik Fo he definiion (96.4) i follows ha fo he su T he equaion ik ik T, (.4) k x is idenially saisfied. This eans ha hee is a onsevaion law fo he quaniies P i ik ik T d k., (.5) ik In he absene of a aviaional field, in alilean oodinaes,, and he ineal oes ove ino ino he fou-oenu of he ae. Theefoe he quaniy (.5) us be idenified wih he oal fou-oenu of ae plus aviaional field. Bu i is obvious ha his esul depends on he hoie of oodinaes and is abiuous.

7 7 7 Unfounaely, hee is sill no eneally aeped definiion of eney and oenu in GR. Aeps aied a findin a quaniy fo desibin disibuion of eney-oenu due o ae, non-aviaional and aviaional fields only esuled in vaious eney-oenu oplexes, whih ae non-ensoial unde eneal oodinae ansfoaions... The soue of aviy in IGT and onsevaion laws In he oenz-invaian ehanis, in eneal, he values ha ake up he eney-oenu enso (see above), ae used in he heoy, wihou bein eoded in he fo of he enso (Fok, 964) (i is noewohy ha W. Fok alled his enso he ass enso (Fok, 964, )). oe, ha afe bein divided by he squae of he speed of lih, hese values ae idenial o he ass and ass flow (in eneal ase, densiies of ass and ass flow). Theefoe followin o V. Fok (Fok, 964, 54), in foulain Einseins heoy we shall likewise sa fo he assupion ha he ass disibuion is insula. This assupion akes i possible o ipose definie liiin ondiions a infiniy as fo ewonian heoy, and so akes he aheaial poble a deeined one. Theoeially, ohe assupions ae also adissible. (As ass disibuion of insula haae V. Fok desibes he ase ha all he asses of he syse sudied ae onenaed wihin soe finie volue whih is sepaaed by vey ea disanes fo all ohe asses no foin pa of he syse. When hese ohe asses ae suffiienly fa away One an nele hei influene on he iven syse of asses, whih hen ay be eaed as isolaed. ) The foeoin allows us in faewok of ou heoy o all, fo he sake of beviy, he soue of aviy - "ass/eney" o siply ass (eanin by his e any eleen of he eneyoenu enso of iven ask). Mass as a soue of aviaion is alled aviaional ass o aviaional hae. Cuenly, he oiin of he aviaional ass is unknown. Bu we know ha i is equal wih ea peision o ineial ass, whih appeas in he laws of oion in ehanis. Thus, if we find ou he oiin of ineial ass/eney, we an onlude ha aviaional ass/eney and aviaion field have he sae oiin. The quesion now is wha do we know abou he oiin of ineial ass, paiulaly, of he eleenay pailes as iniial soue of aviaion?.. The ass heoies (lassial and oden views) To sae he exisin views on he onsideed issues, we will use he woks of onepoay sieniss (Feynan e al, 964; Qui, 7; Dawson, 999; e):.. Classial views Mass eained an essene - pa of he naue of hins - fo oe han wo enuies, unil J.J. Thoson (88), Abaha (9) and oenz (94) souh o inepe he eleon ass as eleoanei self-eney, ( Qui, 7). Theoy, eaed by J.J. Thoson and H. oenz (88-96), lies eniely in he field of lassial eleoanei heoy. Aodin o his heoy, he ineial ass has eleoanei oiin. The eleoanei oiin of he ass of all eleenay pailes, as well as he weakness of he aviaional field opaed o he eleoanei field, allowed o O.F. Mossoi (Mossoi, 96) o assue ha he aviaional field is a esidual eleoanei field Wilhei Webe (84-9) of Goinen and Fiedih Zollne (84-8) of eipzi developed his onepion ino he idea ha all pondeable oleules ae assoiaions of posiively and neaively haed eleial opusles, wih he ondiion ha he foe of aaion beween opusles of unlike sin is soewha eae han he foe of epulsion beween opusles of like sin. If he foe beween wo elei unis of like hae a a eain disane is a dynes, and he foe beween a posiive and a neaive uni hae a he sae disane is y

8 8 8 dynes, hen, akin aoun of he fa ha a neual ao onains as uh posiive as neaive elei hae, i was found ha need only be a quaniy of he ode -5 in ode o aoun fo aviaion as due o he diffeene beween and (Whiake, 95). A he eein of he Aseda Aadey of ienes on Mah 9, oenz ouniaed a pape eniled Consideaions on Gaviaions on Gaviaion, in whih he eviewed he poble as i appeaed a ha ie (Whiakke, 95). Unfounaely, aeps o apply his heoy o quanu heoy has no been undeaken. Howeve, unil now hee was no evidene of ha he ineial ass is no fully eleoanei (Feynan e al, 964): We only wish o ephasize hee he followin poins: ) he eleoanei heoy pedis he exisene of an eleoanei ass, bu i also falls on is fae in doin so, beause i does no podue a onsisen heoy and he sae is ue wih he quanu odifiaions; ) hee is expeienal evidene fo he exisene of eleoanei ass; and ) all hese asses ae ouhly he sae as he ass of an eleon. o we oe bak aain o he oiinal idea of oenz - ay be all he ass of an eleon is puely eleoanei, aybe he whole.5 MeV is due o eleodynais. Is i o isn i? We haven o a heoy, so we anno say. As we will be onvined lae, he esuls of oden heoy of eleenay pailes do no onadi o he oiinal idea of oenz ha all he ass of an eleon ay be puely eleoanei... Moden views The oden ass heoy is he, so-alled, His ehanis of he andad Model heoy (M) (Qui, 7; Dawson, 999; e). Ou oden onepion of ass has is oos in known Einseins onlusion: "The ass of a body is a easue of is eney onen. Aon he viues of idenifyin ass as, whee desinaes he bodys es eney, is ha ass, so undesood, is a oenz-invaian quaniy, iven in any fae as p. Bu no only is Einseins a peise definiion of ass, i invies us o onside he oiins of ass by oin o es wih a bodys es eney. We undesand he ass of an ao o oleule in es of he asses of he aoi nulei, he ass of he eleon, and sall oeions fo bindin eney ha ae iven by quanu eleodynais. uleon ass is an eniely diffeen soy, he vey exepla of. Quanu hoodynais (QCD), he aue heoy of he son ineaions, eahes ha he doinan onibuion o he nuleon ass is no he asses of he quaks ha ake up he nuleon, bu he eney soed up in onfinin he quaks in a iny volue. The asses and of he up and down quaks ae only a few MeV eah. The quaks onibue no oe han % o he 99MeV ass of an isosala nuleon (aveain poon and neuon popeies). Hadons suh as he poon and neuon hus epesen ae of a novel kind. In onas o aosopi ae and beyond wha we obseve in aos, oleules and nulei, he ass of a nuleon is no equal o he su of is onsiuen asses - quaks; i is, basially, a onfineen eney of luons! (Qui, 7). The His ehanis, unde eain assupions, allows us o desibe he eneaion of asses of fundaenal eleenay pailes: ineediae bosons, lepons and quaks. Bu as i is enioned above (Qui, 7), oe han 98% of he visible ass in he Univese is oposed by he non-fundaenal (oposie) pailes: poons, neuons and ohe hadons. Thus, he His ehanis an no be used in he aviaion heoy. u d

9 9 9.. Eleoanei oiin of eleenay pailes and hei ineaions ain wih quanizaion of Maxwells heoy of eleoaneis, physiiss have ade eendous poess in undesandin he basi foes and pailes onsiuin he physial wold. Moden quanu heoies of eleenay paile, suh as he andad odel, ae quanu Yan-Mills heoies. In a quanu field heoy he quana of he fields ae inepeed as pailes. In a Yan-Mills heoy hese fields have an inenal syey: hey ae appea by a spae-ie dependan non-abelian oup ansfoaions. These ansfoaions ae known as loal aue ansfoaions and Yan-Mills heoies ae also known as non-abelian aue heoies. If we will poeed o he aue heoies, we will see ha Maxwells equaions ae a speial ase of he Yan-Mills equaions, whih desibe no only eleoaneis bu also he son and weak nulea foes. Maxwell s equaions an be eaded as a lassial Yan-Mills heoy wih aue oup U(). Quanu eleodynais is an Abelian aue heoy wih he syey oup U() and has one aue field, he eleoanei fou-poenial, wih he phoon bein he aue boson. The andad Model is a non-abelian aue heoy wih he syey oup U() U() U() and has a oal of welve aue bosons: he phoon, hee weak bosons and eih luons. Fo us i is ipoan o ephasize ha he Yan-Mills heoy is a enealizaion of Maxwells heoy (Ryde, 985). We have he wokin enoalizable heoy of son, eleoanei and weak ineaions... This is of ouse he Yan-Mills heoy Essenially, all ha we anaed o do is jus o enealize quanu eleodynais (QED). QED was invened aound 99 and sine hen has neve haned... ow QED is enealized and inludes son and weak ineaions alon wih eleoanei, quaks and neuinos, alon wih eleons (Gell-Mann, 985). As we know, hese heoies ove all ypes of eleenay pailes: assless phoons and assive lepons, bosons and hadons. Theefoe, i an be aued ha he ass of eleenay pailes and hene of he whole ae has eleoanei oiin. This answes he Feynan quesion in he above passae. Fo his follows ha aviaional ass/eney and aviaional field also have an eleoanei oiin. Obviously, hen he heoy of aviy should be soe vaian of he nonlinea heoy of he eleoanei field. We will pesen an aep o build suh a heoy in he followin hapes. oe fis, ha we won be o deive he equaions fo a aanian (i.e., fo leas-aion) foulaion. A full exposiion of hese ideas would add oo uh exa lenh o he book. eond, eveyhin we will do is lassial. To e o he sandad odel o he ohe quanu field heoies, we need o quanize he heoy. Chape. The axioais of IGT and is onsequenes.. ea of eleoaneis In he pevious hape of ITG, we pesened evidene of he eleoanei oiin of ineial ass. Feynan noed (see above), ha his saeen does no onadi he expeienal daa. On his basis, we sae hee he followin lea, whih will seve as a foundaion fo buildin IGT (le us all i ondiionally "ea of eleoaneis"). ea of eleoaneis: The eleoanei field is he basis fo he oiin of ae.

10 Fo hee follow a nube of onlusions ha ae ipoan fo he heoy of aviaion. ) The equivalene of aviaional and ineial asses leads o he onlusion ha aviy has an eleoanei oiin. This onlusion is of fundaenal ipoane fo he onsuion of he oenz-invaian heoy of aviaion. ) The oenz-invaiane of he laws of eleoaneis, deeines oenz-invaiane of he laws of aviy. ) Eleenay pailes ae he piay aies of ae and is haaeisis. Hene, he equaion of aviaion should follow fo he equaions of eleenay pailes. 4) Mae is involved in he eaion of he aviaional field as is soue, wihou quanizaion of his soue. Thus, he aviaional field an be eaded as a lassial field, whih does no equie quanizaion. The assued oiin of his equaion fo quanu equaions of eleenay pailes, is no a liiaion hee, beause a ansiion exiss fo quanu o lassial equaions. 5) In he eleenay pailes heoy, ineial ass is assoiaed wih eney and oenu of paile by he equaion: 4 p, whee is he es ass (invaian quaniy). Fo his follows, wha in eneal is he equivalene of ass and eney-oenu p, Aodin o he above enioned ause we an onside ass, eney and oenu as he aviaion soues. 6) ine in eneal ase, he oiinal equaions of ioos ae nonlinea, we should assue ha he aviaional equaions ae non-linea. Based on foulaed above ea of eleoaneis, we an hoose he followin axios fo IGT, whih do no onadi o he expeienal daa... Axioais of IGT As he fis and seond posulaes we will ake he expeienal fas:. Posulae of soue: he soue of he aviaional field is ae in he fo of an island ae o a field ass/eney.. Posulae of he asses equivalene: he aviaional hae (ass) is popoional o he ineial ass/eney.. Posulae of Mossoi -oenz: Posulae of Mossoi-oenz: he aviaional field is a esidual eleoanei field, whih is eained as a esul of inoplee opensaion of elei and anei fields of diffeen polaiy. (oe: we do no assoiae his axio wih he Mossoi odel whih explains how his esidue is foed, bu have in ind he eneal idea ha he aviaional field is a sall pa of he eleoanei field, whih as aaively). 4. The loaliy posulae: aviaional field is loally oenz-invaian, ha is oenzinvaian on any infiniely sall ie ineval and on any infiniely sall disane. (oe: sine he EM field is iself oenz- invaian, his axio an be seen as a onsequene of he axio of Mossoi-oenz. Bu lassial ehanis is lobally oenz- invaian. Wih

11 he inoduion of posulae 4 we aually ephasize ha aviaion, in he eneal ase, is no lobally oenz-invaian). Fo hese axios he nex onsequenes follow, poof of whih ay seve as a onfiaion of he axios. Coollay : sine he aviaional field is esidual, i is uh weake han he eleoanei field, bu in he ase of a neual ae (in he eleoanei sense), he aviaional field is deisive. Coollay : he aviaional onsan is deeined as a poion of full eleoanei ineaion. Coollay : as in he heoy of eleoaneis he ineaion is desibed by he oenz foe, he sae (o is odifiaion) desibes he heoy of aviaion. Coollay 4: he equaions of assive eleenay pailes an be eaded as he soue equaions of he aviaional field. Coollay 5: all he feaues of oion of ae in he aviaional field oe fo he eleoanei heoy, in paiula, fo he effes assoiaed wih he oenz ansfoaions. Coollay 6: all he haaeisis of aviaional field (is eney, oenu, anula oenu, e) have an eleoanei oiin and obey he laws of eleoaneis. Chape 4. The onneion of eleoanei heoy and aviaion Hee, we will show ha he adoped by us he Mossoi-oenz posulae does no onadi he exisin esuls of physis, inludin eneal elaiviy... Tansiion fo EM field heoy o aviaional field heoy In he woks of oenz (see, e.., (oenz, 9)) i was shown in suffiien deail ha in he heoy of eleoanei fields he esidual eleoanei field an aually be desibed. Bu fo he speifi pupose of is inoduion i is easie and oe onvenien o use he ehods of siilaiy heoy and diensional analysis (edov, 99). We will opae he expessions of EM heoy wih he paallel expessions of aviaional heoy and sele he oespondenes beween he. Fo he onol of he onlusions we use diensional analysis. The ain haaeisi of he soue field in he one and in he ohe heoy is he expession of he ineaion foe o he oespondin ineaion eney beween he wo bodies... Gaviy eleosai (e-) field. The ansiion fo he Coulob s field o he ewon s field If we assue ha aviy is eneaed by elei field, bu quaniaively, by vey sall pa of i (see Appendix A), hen ewon s aviaion law: M F, (.) should ake he fo of Coulobs law: q Q F C k, (.) whee and q ae he ass and elei hae of he paile, M and Q ae he ass and elei hae of he soue, is ewons aviaional onsan, and he oeffiien k in

12 Gauss s unis is k. In his ase, he definiions of aviaional field senhs of ewon and F M Coulob elei field have he fo E FC Q and E k, q espeively. We inodue he aviaional hae q, oespondin o ass (Ivanenko and okolov, 949), by eans of he elaion: q q, (.) In his ase, ewons law an be ewien in he fo of Coulobs law: q Q M F F, (.4) whee Q M is he aviaional hae of soue, oespondin o he ass M of he soue. Fo he opaison of equaions (.) and (.4) i follows ha he diensions of he eleoanei and aviaional haes oinide. A he sae ie, a aviaional hae (.) has eleoanei oiin, and, hene, he oespondin ass is he ineial ass. On he ohe hand, he law (.4) opises he aviaional asses. This iplies he equivalene of ineial and aviaional asses. We inodue he -field senh wihin faewok of EMGT as: E E, (.5) whee he ension of he Coulob field is equal o: E Q. ubsiuin he values of aviaion heoy hee, we e: Q M E E, (.6) whee E is he senh of he ewon aviaional field. e us inodue he sala aviaional poenial wihin he faewok of EMTG as: whee he poenial of he Coulob field is: hee, we e:, (.7) Q. ubsiuin he values of aviaion heoy Q M, (.8) whee is he poenial of he ewon aviaional field. The Poisson equaion fo he -field an seve as es fo (.7). Indeed, fo he EM field he Poisson equaion an be wien as: 4, (.9) e dq whee e is he elei hae densiy, d is he volue eleen. We inodue he d densiy of aviaional hae siilaly o he elei densiy:

13 dq e, (.) d d whee is ass densiy. Then, eplain he poenial and he hae densiy in (.9) d aodin o (.7) and (.), we obain he Poisson equaion fo he aviaional poenial: 4 4, (.) whih oesponds o he Poisson equaion fo he ewon aviaional field... Gavi-anei field (-field) In his ase, by analoy wih eleodynais, he exisene of he vaiable e-field and assoiaed wih he alenain o die -fields is assued. The exisene of a siila field is onfied by eneal elaiviy and expeiens. Unfounaely, sine he ewon heoy does no onain an analo of anei field, he veifiaion of exisene of he -anei field wihin faewok of EMGT, an pesenly be done only by diensional analysis. eious onfiaion should be obained by he soluion of he oespondin equaions of aviaion, whih will ive equivalen esuls o he eneal heoy of elaiviy. As is known, he anei field is eneaed by he oion of elei haes o oveen of an elei field. In his ase, we need o obain an expession fo he anei field, siila o Coulobs law fo he elei field. This is he Bio-ava aplae law. Fo sipliiy, we will onside he speial ase of unifo oion of a soue hae Q, whih eae a uen I (uen fo oion of hae q will be denoed by i ). In eal asks, of ouse, haes and asses ae divided ino poin (diffeenial) values, and field alulaed by ineain ove a se of poin haes. Manei veo H ha ous when he hae Q oves a a speed in iui eleen dl, will be: Q Idl H, (.) Usin he aviaional hae densiy aodin o (.), siilaly o he elei uen i dq d and he uen densiy j, we will define espeive -uen (o uen of ass) as: dq i i nd nd, (.) d and he densiy of -uen of ass, as: j j i d n, (.4) n whee is he veloiy of he hae in a onduo wih a oss seion d, and n he pojeion of he veloiy on he noal o d. If he e-hae q oves lose o he e-uen (o peanen ane field H ), his uen (o field H ) as on he hae via he anei pa of he oenz foe F q Q i I H dl dl F : q, (.5) e us inodue he senh of -field wihin faewok of EMGT as: H H, (.6)

14 4 4 whee he anei field H is iven by (.). ubsiuin he oespondin physial quaniies aodin o (.) and (.6) in (.), we will obain he avi-anei (-) veo ha aises when he hae Q M oves a a speed in an eleen of a iui dl : Q M dl H, (.7) o H dl dl d I n, (.8) whee is he disane beween he es paile and he ovin haed soue o eleen of uen I, whih eneae he -veo. Usin (.5), fo he avio-anei oenz foe we obain: qq M F, (.9) ine he anei field H in he eleodynais an be expessed via a veo poenial A by he expession: H oa, (.) i is useful o define he ansiion fo he EM veo poenial A o he aviaional A. We assue ha: A A, (.) Then, usin (.6), we an ewie (.) in he fo: H o, (.) A Expession (.) also saisfies he full EM expession fo he elei senh veo: A E ad, (.) Usin (.5), (.7) and (.), we obain fo -field: A E ad, (. ) Thus, we have shown ha he basi EM quaniies and equaions an be assoiaed wih siila quaniies and equaions fo he -field. As an illusaion of he oeness of he elaionship of eleoanei and avioeleoanei quaniies, we pu in Appendix A o his hape he able of diensions of physial quaniies, onsideed above.

15 5 5.. GTR, EMG and EMGT A lo of he soluions of eneal elaiviy ae obained in linea appoxiaion, usin he ehod of peubaion. I was found ha he esuls of his linea heoy ay be pesened in he fo of Maxwells equaions. uh a epesenaion has been alled avioeleсoaneis, o, biefly, GEM... Gavio-eleoaneis (EMG) In eneal elaiviy (GR) (Oveduin, 8), spae and ie ae inexiably bound oehe. In speial ases, howeve, i beoes feasible o pefo a "+ spli" and deopose he ei of fou-diensional spaeie ino a sala ie-ie oponen, a veo ie-spae oponen and a enso spae-spae oponen. When aviaional fields ae weak and veloiies ae low opaed o, hen his deoposiion akes on a paiulaly opellin physial inepeaion: if we all he sala oponen a "avio-elei (e-) poenial" and he veo one a "avio-anei (-) poenial", hen hese quaniies ae found o obey alos exaly he sae laws as hei ounepas in odinay eleoaneis. In ohe wods, one an onsu a "avio-elei field" E e and a "avio-anei field H, and hese fields ae obeyed equaions ha ae idenial o Maxwells equaions and he oenz foe law of odinay eleodynais. Fo syey onsideaions we an infe ha he eahs avio-elei field us be adial, and is avio-anei one dipola, as shown in he diaas. and.. below: Fi... Radial aviaion field lines of Eah Fi... Dipole aviaion field lines of Eah These fas allow one o deive he ain pediions of eneal elaiviy, siply by eplain he elei and anei fields of odinay eleodynais E and H by E e and H espeively. The aheaial aspe of GEM heoy is desibed in any papes (see, fo exaple, (Fowad, 96; Wald, 984; Ruieo and Taalia, ; Gøn and Hevik, 7; Mashhoon, 8; Foese, ; )) To avoid isundesandin, i should be noed ha he eleoanei heoy of aviaion (EMGT) and avioeleoaneis (GEM) - ae no he sae (Mashhoon, 8). GEM is an auxiliay epesenaion of GR, whih allows o physially iaine of esuls of he ei heoy. In onas, EMGT is an independen heoy of aviaion, whih aose on he basis of he hypohesis Mossoi and hen was developed by nube of sieniss, inludin O. Heaviside, H.oenz and ohes (Heaviside, 9; oenz, 9; Webse, 9; Wilson, 9; e).

16 6 6 Appendixes: A. Relaionship beween elei and aviaional haes I is easy o show ha he aviaional field is a sall faion of he eleoanei field. To his oesponds he fa ha he aviaional hae of he eleon is less han is elei hae e, whee q (hee q CGEq = / s / s - ), e e,9 7 e 4,8 uni. GEq is eleon hae ( uni is eleon ass, 8 6,67 / se is he aviaional onsan. I is easy o see ha he diension of he aviaional hae of he eleon oinides wih he diension of elei hae and is aniude in ies less. Indeed, e. e Fo a poon (he only sable heavy paile), his value is of he ode e p 8 heavies known eleenay pailes ae he hihly unsable bosons W ± (ass 8 GeV). This is 6 abou ies oe han he ass of he poon, ivin a aio of no less han.. The A. Diensions of eleoanei and avi-eleoanei quaniies Fo he veifiaion of he oeness of oelaions in he ansiion fo he EM physial quaniies o he aviaion quaniies, he aodane of hei diensions plays an ipoan ole. The woded below lis onfis ha eleodynais an be onsideed as he basis of ehanis. Eleoanei heoy e-hae [ q ] = / / s e-hae densiy [ e ] = / / s e-uen [ i ] = / / s e-uen densiy [ j ] = / -/ s Coulob foe [ F C ] = s enh of e-fields [ E ] = / / s enh of -field [ H ] = / / s ala poenial [ ] = / / s Veo poenial [ A ] = / / s Field eney [ e ] = s = [ q ] Field eney densiy [ ] = s = [ E ] = [ H ] Gaviaion heoy of ewon ewon s foe [ F ] = s ewon s aviaional onsan [ ] = s ( [ ] = / / s ) Field senh [ E ] = /s (aeleaion) ala poenial [ ] = /s (/s) (veloiy squae) ala poenial [ ] = [ ] = /s Eleoanei aviaion heoy (EMGT) -hae [ q ] = / / s = [ q e ] = [ ] -hae densiy [ ] = / / s -uen -uen densiy [ i ] = / / s [ j ] = / -/ s

17 7 7 -foe [ F ] = s = [ F ] enh of e-fields [ E ] = /s = [ E ] = [ E enh of -field [ H ] = /s = [ H ala poenial of -fields [ ] = /s = [ Veo poenial of -fields [ A ] = /s = [ A -field eney ] ] ] [ ] = /s = [ ] ] Chape 5. Eleoanei base of elaivisi ehanis.. Geneal piniples of eleoanei heoy of ae Unde he ovin asses (aviaional haes) we will undesand he wo ineain bodies, one of whih we all he soue of he aviaional field, and he ohe - he es paile. Ou appoah o he heoy of aviaion is based on a oden vesion of he eleoanei heoy of ae (EMTM) (oenz, 96; Rihadson, 94; Beke, 9). In faewok of EMTM he ass is of eleoanei (EM) oiin. Theefoe in faewok of ou axioais, he ain esuls of EM heoy ae equivalen o esuls of he heoy of aviaion. The Maxwell EM heoy was he fis heoy, whose popeies wee found o depend on he speed of he hae (in his ase, elei). This heoy is alled he oenz-invaian (-invaian) o elaivisi heoy. A speeds of up o one-enh of he speed of lih, hese paaees ae hadly diffeen fo he paaees of sai objes. This suess ha he basis of ehanis is sill he lassial ewonian ehanis, and elaivisi ehanis is ewons ehanis plus ino aendens heeo. Moeove, in he faewok of EMTM i is easy o show ha he alulaion of oeions o he non--invaian heoy is deeined by he non--invaian heoy. The aendens ae alulaed on he basis of non--invaian laws ha ake ino aoun he hanes in he paaees a hih speeds. The alulaion poedue is equivalen o he ehod of alulaion whih is based on peubaion heoy, when he zeo appoxiaion is he non-elaivisi heoy. Below we show his, based on he known esuls pesened in exbooks... The Maxwell-oenz equaions.. The Maxwell-oenz equaions wien in es of field senhs The eneal equaions of he eleoanei heoy of ae (EMTM) ae foulaed on he basis of Maxwells equaions, akin ino aoun he oenz hypohesis. Unde his hypohesis, all eleenay pailes (and, onsequenly, aos, oleules and bodies) ae oposed of an eleoanei field, whih is in a onenaed ("ondensed" aodin o Einsein) sae. ine aon hese pailes ae he fee EM fields (phoons), hey ae also inluded in his lis. A he sae ie, haes and uens ae also deeined by he eleoanei fields. Consequenly, hee is only one kind of veos, desibin he field, naely he eleoanei (EM) field senhs in vauo E and H o equivalen quaniies. The self-onsisen Maxwell-oenz iosopi equaions ae he independen fundaenal field equaions. The Maxwell-oenz equaions ae followin fou diffeenial (o, equivalen, ineal) equaions fo any eleoanei ediu (Jakson, 965; Tonnela, 966): E 4 ob j, (.)

18 8 8 B oe, (.) die 4, (.) di B, (.4) whee E, H, D, B ae elei field veo, anei field veo, elei induion veo, anei induion veo, oespondinly; in vauu D E and B H ; A E ad, B oa, whee, A ae sala and veo poenials, oespondinly; is he hae densiy; j is he uen densiy: is he speed of lih. The diffeene beween hese equaions and Maxwells equaions is ha E and H, as well as all ohe quaniies needed o desibe a ae, efe o an abiaily sall volue of spae. In his ase he equaions (.-.4) ae alled he Maxwell-oenz (M) equaions. The Maxwell s aosopi quaniies E and H an be dedued fo he iosopi quaniies E and H only by aveain ove spae and ie. This aveain and deduion of he aual Maxwell equaions fo (.-.4) is onsideed in any ouses on eleoaneis (Beke, 9). In he equaions (.-.4) nohin is said abou how he veloiy of he haes hanes ove ie. Fo his pupose he oenz law is used. Aodin o oenz he densiy of foe has he fo: f E B, (.5) hene fd is a foe, ain on he volue d... The Maxwell-oenz equaions wien in es of field poenials Fo he analysis of he field equaions (.-.4), i is advisable o o fo fields heselves o he eleoanei field poenials (Beke, 9). This is done as follows: fis of all, we saisfy he equaion di H (.4) by subsiuin: H oa, (.6) whee veo A is naed he veo poenial. Then fo he equaion of (.) i follows ha E A should be zeo. Theefoe, we deand ha he value oe A is equal o he adien of a sala : A E ad, (.7) whee veo is naed he sala poenial. The veo field is uniquely deeined by diveene and voiiy of his field. Unil now, we deeined only oa. ow we an in addiion feely dispose by he diveene of he veo A. We will use his in ode o pu dia, (.8) If we now subsiue (.6) and (.7) in he wo eainin equaions (.-.4), hen wih he help of (.8), we obain wo equaions fo he poenials:

19 9 9 A A A A 4, x y z, (.9) 4, x y z.. The oenz ansfoaion as ansiion fo es o oion Fo hei ineaion we use he well-known fa ha he field a ie a in any poin is equal o he field a ie d a he poin, shifed bak o he seen d. This eans ha fo all he quaniies, haaeizin he field, we will aain have he elaion: ad x, y, z is a funion of he field. whee Fo exaple, fo hane in ie of he elei veo E we will obain: E ad e Thus, if he veloiy is paallel o he posiive x -axis, in ou equaions fo he poenials (.9), seond ie deivaives ae eplaed by deivaives wih espe o he oodinae x aodin o he foula x Theefoe, fo he poenials A and we e he equaion: A A A 4, x y z, (.) 4, x y z oe ha he equaions fo he oponens of he veo poenial diffe fo he equaion fo he sala poenial only by onsan fao. Theefoe, if we esolve he equaion fo, hen a soluion fo he veo poenial follows diely fo i: A, (.) Fo his we obain wo equaions: oa o ad A ad If we inodue he o he definiions (.6) and (.7), we e: E ad ad and H E, (.) I uns ou ha he elaion (.) beween he veo and sala poenials leads o known dependene H E beween H and E. Theefoe, o solve ou poble, we an onfine ouselves o ineain he equaion (.) fo. oe ha his equaion diffes fo he equaion fo he odinay eleosai poenial only by

20 onsan oeffiiens a x. o ehnially we an edue ou poble o a siple eleosai poble, if insead of oodinaes x, y, z, we inodue he new oodinaes x, y, z, usin he ansfoaion: x x, y y, z z,, (.) whee fo beviy we pu. Due o his hane, he funions (х, y, z, ) and (х, y, z, ) pass o funions and fo x, y, z,, so ha we have he ideniies: x, y, z, x, y, z,, x, y, z, x, y, z,, Theefoe, ou equaion fo he poenial in he pied oodinaes is (.4) 4, (.5) x y z As suh, his equaion is opleely idenial o he equaion ha deeines he poenial of he fixed hae syse. Theefoe, is ineaion an podued aodin o he well-known heoy of he eleosai poenial. We e:,,, d d d x, y, z, x y z If we aain un o he unpied oodinaes wih he help of (.) and (.9), we will obain he soluion of equaion (.) fo he sala poenial in he fo,,, ddd x y z x, y, z,, (.6) ow le us find a paiula soluion fo he ie when he eleon is in he beinnin of he oodinae syse, and esi ouselves o he ase of he poin eleon, i.e., assue ha he hae densiy is diffeen fo zeo only in he iediae viiniy of he oiin of oodinaes. Then he ineaion an be done, and we e he soluion: E x, y, z,, (.7) x y z Fo he puposes of beviy, we inodue fo he expession ha appeas in he denoinao insead of he disane, he desinaion: s y z x, (.8) Then we will be able o pesen he soluion o ou poble in he fo e e, Ax, Ay Az, (.9) s s Usin hese poenials we an alulae he field E and H by he foulas (.6), (.7) o (.), and akin ino aoun ha diffeeniaion by ie is always eplaed by, fo he x eleial senh in veo fo we will e: e E, (.) s Fuhe, he anei senh H E ; hene:

21 H e e, H y z, H z y, (.) s s x.. oenz ansfoaions and hei onsequenes Ou ai (oenz, 94; oenz, 96; Poinaé, 95) us aain be o edue he equaions fo a ovin syse o he fo of he odinay foulae ha hold fo a syse a es. I is found ha he ansfoaions needed fo his pupose ay be lef indeeinae o a eain exen; ou foulae will onain a nueial oeffiien l, of whih we shall povisionally assue only ha i is a funion of he veloiy of anslaion, whose value is equal o uniy fo, and diffes fo by an aoun of he ode of aniude fo sall values of he aio. If х, y, z ae he oodinaes of a poin wih espe o axes fixed in he vauu, o, as we shall say, he absolue oodinaes, and if he anslaion akes plae in he dieion of OX, he oodinaes wih espe o axes ovin wih he syse, and oinidin wih he fixed axes a he insan, will be x x, y y, z z, (.) ow, insead of x, y, z we shall inodue new independen vaiables diffein fo hese elaive oodinaes by eain faos ha ae onsan houhou he syse. Puin we define he new vaiables by he equaions o x, (.) x lx, y ly, z lz (.4) lx, y ly and o hese we inodue as ou fouh independen vaiable l l, z lz, (.5) x l x, (.6) I was Poinaé (Poinaé, 95) who fis inodued ha he eal eanin of he subsiuion (.5), (.6) lies in he elaion x y z l x y z, (.7) ha an easily be veified, and fo whih we ay infe ha we shall have when x x y z, (.8) y z, (.9) This ay be inepeed as follows. e a disubane, whih is podued a he ie = a he poin х =, y =, z = be popaaed in all dieions wih he speed of lih, so ha a he ie i eahes he spheial sufae deeined by(.9). Then, in he syse х, y, z,, his sae disubane ay be said o sa fo he poin х =, y =, z =, a he ie = and o eah he spheial sufae (.8) a he ie. ine he adius of his sphee is, he disubane is popaaed in he syse х, y, z, as i was in he syse х, y, z,, wih he speed. Hene, he veloiy of lih is no aleed by he ansfoaion... The sai fields of Coulob and ewon as fundaenal fields wih espe o fields of ovin soues. The ovin soue of he aviaional field is a aviaional uen, i.e., he oveen of aviaional hae (ass). As we have seen (see above), he ansiion fo he fixed hae and

22 hei fields o he obile hae and fields, and vie vesa, is desibed by oenz ansfoaions. In he heoy of aviy he ansiion fo he non--invaian heoy o he -invaian heoy equies, fis and foeos, o find he -invaian expession fo he sai ewons law of aviy F M, o dig 4, whee G F (o by inoduin poenial houh F ad, in he fo 4 ). Aodin o IGT he ewon law of aviy is a onsequene of he law of he sai ineaion beween haes of Coulob (o of oe eneal asseion - of Gauss heoe). This ives us an oppouniy o onside he aviy poble on he basis of he eleoanei poble ha has been solved. A fis lane, hee lies he onadiion. ai (non--invaian) Coulobs law FC eqq (whee in he CG e ) in he fo die 4 e o e 4 e, is inluded as pa in he M- equaion, whih, in is oaliy, is of ouse, -invaian. The exi fo his onadiion is soewha unexpeed. We will show below ha in he ansiion fo soue шт a saionay efeene fae o he sae soue in ovin fae, new addiional fields ae eneaed, whih oehe wih he sae sai field, ee he equieens of he -invaiane. Fahe we assue ha all he saeens ha we an ake wih espe o EM heoy, ae valid fo he heoy of aviy, akin ino aoun he esablished einoloy (fo exaple, he hae in EM heoy is alled ass in he heoy of aviy, e). (Fahe o onfi ou ideas, we will use he quoes fo he book of E. Puell (Puell, 985))... Gausss law The flux of he elei field E houh any losed sufae, ha is, he ineal E ds ove he sufae, equals 4 ies he oal hae enlosed by he sufae: E ds 4 qi 4 d, (.) i We all he saenlen in he box a law beause i is equivalen o Coulobs law and i ould seve equally well as he basi law of eleosai ineaions, afe hae and field have been defined. Gausss law and Coulobs law ae no wo independen physial laws, bu he sae law expessed in diffeen ways. This suess ha Gausss law, ahe han Coulobs law, offes he naual way o define quaniy of hae fo a ovin haed paile, o fo a olleion of ovin haes. I would be ebaassin if he value of Q E ds so deeined depended on he size 4 and shape of he sufae. Fo a saionay hae i doesn-ha is Gausss law. Bu how do we know ha Gausss law holds when haes ae ovin? We an ake ha as an expeienal fa... Invaiane of hae Thee is onlusive expeienal evidene ha he oal hae in a syse is no haned by he oion of he hae aies. This invaiane of hae lends a speial sinifiane o he fa of hae quanizaion. I is known he fa ha evey eleenay haed paile has a hae equal in aniude o ha of evey ohe suh paile. And his peise equaliy holds no only fo wo pailes a es wih espe o one anohe, bu fo any sae of elaive oion.

23 . Elei field easued in diffeen faes of efeene If hae is o be invaian unde a oenz ansfoaion, he elei field E has o ansfo in a paiula way. Tansfoin E eans answein a quesion like his: if an obseve in a eain ineial fae F easues an elei field E as X vols/, a a iven poin in spae and ie, wha field will be easued a he sae spae-ie poin by an obseve in a diffeen ineial fae F? Fo a eain lass of fields, we an answe his quesion by applyin Gausss law o soe siple syses. Gausss law ells us ha he aniude of E us be E E E Bu his onlusion holds only fo fields ha aise fo haes saionay in F. As we shall see below, if haes in F ae ovin, he pediion of he elei field in F involves knowlede of wo fields in F, he elei and he anei..4 Foe on a ovin hae A soe plae and ie in he lab fae we obseve a paile ayin hae q whih is ovin, a ha insan, wih veloiy houh he eleosai field. Wha foe appeas o a on q? Foe eans ae of hane of oenu, so we ae eally askin, Wha is he ae of hane of oenu of he paile, dp d, a his plae and ie, as easued in ou lab fae of efeene? Tha is all we ean by he foe on a ovin paile..5 Ineaion beween a ovin hae and ohe ovin haes We know ha hee an be a veloiy-dependen foe on a ovin hae. Tha foe is assoiaed wih a anei field, he soues of whih ae elei uens, ha is, ohe haes in oion..5. Maneis as a onsequene of oenz s lenh onaion Model a uen-ayin wie (hoede, 999) as a line of neaive haes ( q ) a es and a line of posiive haes ( q ) ovin o he ih a speed x, whee x is uni veo of x -axis,. The aveae linea sepaaion beween haes is l. Conside a es hae Q ovin paallel o he wie, a he sae speed (fo sipliiy). In he fae of he es hae i is a es and so ae he (+)-haes in he wie, bu he haes ae ovin o he lef. Aodin o elaiviy, he disane beween he ( )-haes is lenh-onaed o he disane beween he (+)-haes is un-lenh-onaed o l l, while l l. Theefoe he wie aies a ne neaive hae and exes an aaive eleosai foe on he es hae. Bak in he lab fae, we all his a anei foe. ab Fae: Tes Chae Fae: es hae (a es) To alulae he senh of he foe, fis we find he linea hae densiy of he wie in he es hae fae (assuin fo sipliiy):

24 4 4 q l q l q l q q In a ypial household wie ~ l l, (.), so he oenz fao diffes fo by only abou one 6 pa in. This iny aoun of lenh onaion is sill obsevable, beause he oal hae of all he ovin eleons is enouh o exe enoous eleosai foes. The sae deivaion an be adaped o oe opliaed ases whee he es hae has an abiay veloiy, in eihe dieion. To undesand he ase whee he es hae is ovin owad o away fo he wie, you need o diess o show how he elei field of a poin hae in oion is weake in fon of and behind he hae bu sone in he ansvese dieions. (This an be deived usin lenh onaion and soe siple edanken expeiens.) Fo ou pesen vanae poin (Puell, 985), he anei ineaion of elei uens an be eonized as an ineviable oollay o Coulobs law. If he posulaes of elaiviy ae valid, if elei hae is invaian, and if Coulobs law holds, hen, as we shall now show, he effes we oonly all "anei" ae bound o ou. They will eee as soon as we exaine he elei ineaion beween a ovin hae and ohe ovin haes. Two hae disibuions expeiene oenz onaion of vaious values - his is he soluion of he poble. A oe eneal and deailed analysis of he poble is desibed, fo exaple, in he book e us use he esuls of book (Puell, 985) o e he aheaial expession of he aisin foe and anei field (fo beviy we use he noaion inodued ealie, ) In eneal ase he oal linea densiy of hae in he wie in he es hae fae,, an be alulaed:, (. ) (he eanin of he unknown vaiables in (. ) is explained below The wie is posiively haed. The use of Gausss law (applied o he ylinde whih suounds he line) uaanees he exisene of a adial elei field E iven by he foula fo he field of any infinie line hae: 4 E Hene, he es hae q will expeiene a foe, whih is dieed inwadly adially, (.) q 4q F qe, (.4) ow les eun o he lab fae. Wha is he aniude of he foe on he hae q as easued hee? If is value is qe in he es fae of he es hae, obseves in he lab fae will epo a foe salle by he fao. ine, he foe on ou ovin es hae, easued in he lab fae, is: F F 4q, (.5) ow is jus he oal uen I in he wie, in he lab fae, fo i is he aoun of hae flowin pas a iven poin pe seond. Well all uen posiive if i is equivalen o posiive hae flowin in he posiive x dieion. Ou uen in his exaple is neaive. Ou esul an be wien his way:

25 5 5 qi F, (.6) We have found ha in he lab fae he ovin es hae expeienes a foe in he y dieion whih is popoional o he uen in he wie, and o he veloiy of he es hae in he x dieion. If we had o analyze evey syse of ovin haes by ansfoin bak and foh aon vaious oodinae syses, ou ask would ow boh edious and onfusin. Thee is a bee way. The oveall effe of one uen on anohe, o of a uen on a ovin hae, an be desibed opleely and onisely by inoduin a new field, he anei field..6 Inoduion of he anei field Thus, a hae whih is ovin paallel o a uen of ohe haes expeienes a foe pependiula o is own veloiy. We an see i happenin in he defleion of he eleon bea. e us sae i aain oe aefully. A soe insan a paile of hae q passes he poin ( x, y, z) in ou fae, ovin wih veloiy. A ha oen he foe on he paile (is ae of hane of oenu) is F. The elei field a ha ie and plae is known o be E. Then he anei field a ha ie and plae is defined as he veo B whih saisfies he veo equaion q F qe B, (.7) Wha kind of veo should be B, in ode o ake he equaion (.6) opaible wih he equaion (.7). Fo fields ha vay in ie and spae equaion (.7) is o be undesood as a loal elaion aon he insananeous values of F, E, and B. Of ouse, all fou of hese quaniies us be easued in he sae ineial fae. In he ase of ou "es hae" in he lab fae, he elei field E was zeo. Wih he hae q ovin in he posiive x dieion, x, we found ha he foe on i was in he neaive y dieion, wih aniude q I qi F y, (.8) In his ase he anei field us be I B z, (.9) fo hen equaion (.7) beoes q q I qi F B x z y, (.) in aeeen wih equaion (.8)..7 Veo poenial We found ha he sala poenial funion x, y, z eleosai field of a hae disibuion. If hee is soe hae disibuion x, y, z poenial a any poin x, y, z is iven by he volue ineal x, y, z x, y, z d, (.) ave us a siple way o alulae he, he

26 6 6 The ineaion is exended ove he whole hae disibuion, and is he aniude of he disane fo x, y, z o x, y, z. The elei field E is obained as he neaive of he adien of p: E ad, (.) The sae ik won wok hee, beause of he essenially diffeen haae of B. The ul of B is no neessaily zeo, so B an, in eneal, be he adien of a sala poenial. Howeve, we know anohe kind of veo deivaive, he ul. I uns ou ha we an usefully epesen B, no as he adien of a sala funion bu as he ul of a veo funion, like his: B oa, (.) By obvious analoy, we all A he veo poenial. I is no obvious, a his poin, why his ai is helpful. Tha will have o eee as we poeed. I is enouain ha equaion (.4) ( di H ) is auoaially saisfied, sine di oa, fo any A. In view of equaion (.), he elaion beween J and A is 4J ooa. (.4) Equaion (.4), bein a veo equaion, is eally hee equaions. We shall wok ou one of he, say he x-oponen equaion. Aon he vaious funions whih ih saisfy ou equieen (.), le us onside as andidaes only hose whih also have zeo diveene di A. Then, afe a seies of ansfoaions we e fo (.4): A x x A y x A z x 4J x, (.5) Thus, we shown ha he alulaion of -invaian aendens o he non--invaian heoy is deeined by he non--invaian heoy. Chape 6. The equaion of oion in IGT.. Equaion of assive boson e us use he eleoanei epesenaion of h Dia equaion (see in deails (Kyiakos, ; 4; 9)). Moe ofen he Dia equaion is desibed in he bispino fo. Enein he funion:, (.) 4 alled bispino, he Dia equaions an be wien in one equaion. Thee ae wo bispino Dia equaion fos: ˆ ˆ ˆ ˆ ˆ p, (.) o e ˆ ˆ ˆ pˆ ˆ o e, (.) whih oespond o he wo sins of he elaivisi expession of he eney of he eleon: