Lorentz-Invariant Gravitation Theory
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1 Pepaeie Jounal Apil 6 Volue 7 Iue 6 pp 963 Kyiako, A G, LoenzInvaian Gaviaion Theoy 6 LoenzInvaian Gaviaion Theoy Chape oluion of he Keple pole in he faewok of LIGT Inoduion In peen hape, aed on eul of peviou hape 9, we onide he oluion of he Keple pole, ie, he oluion of he pole of oion of wo odie in a enally yei gaviaional field of a aionay oue I i hown ha hi oluion oinide wih ha oained in GR A he oion euaion of LITG we ue he HailonJaoi euaion (Chape 8) Aoding o Chape 7, he euaion of oion of HailonJaoi ha a oneoone onneion wih he uae of he ineval (uae of a eleen of aeoy) in faewok of LITG Theefoe, a we will how elow, i i no neeaily o find an appopiae ineval o wie he oeponding HailonJaoi euaion fo paile oion in gaviaion field Effe of Loenz anfoaion A oneuene of he peviouly adoped aioai (hap 3) of Loenzinvaian gaviaion heoy (LIGT) i he aeion ha all feaue of he oion of ae in he gaviaional field owed hei oigin o effe aoiaed wih he Loenz anfoaion Thi ean ha he elaoaion of he euaion of Newon gaviaion u follow fo onideing of hee effe A i well known (Beke, 3), hee ueion an e onideed wihou peial elaiviy heoy, uing only he Mawell euaion Effe, ha owe hei eiene o he Loenz anfoaion ae diued in any eook devoed o he EM heoy o RT (Beke, 3; Pauli, 98; e al) We will no dwell on hei wihdawal, and we will only iefly enion oe of he Fo he Loenz anfoaion follow he veloiy anfoaion, howing ha no ody an oveoe he peed of ligh Fo he Loenz peed anfoaion follow he ie dilaion and lengh onaion in a oving fae of efeene, a well a he anfoaion of enegy and oenu The ue of invaiane popeie of he wave phae wih epe o he Loenz anfoaion, allow o oain he elaivii foula of Dopple effe, aeaion, efleion fo a oving io, Wien diplaeen law, e The aniion fo Newonian ehani o he Loenzinvaian ehani Le u y (Беккер, 3) o ale he Newonian euaion o ha hey aify he Loenz anfoaion We egin y onideing he oion of a paile in a given foe field (eg, eleoagnei o gaviaional) Newonian euaion of oion ead a follow: du F L, () d whee F L i, eg, he Loenz foe : F L E u H, () IN: 5383 Pepaeie Jounal Pulihed y QuanuDea, In wwwpepaeieo
2 Pepaeie Jounal Apil 6 Volue 7 Iue 6 pp 963 Kyiako, A G, LoenzInvaian Gaviaion Theoy 7 Now we will y o give hi euaion he Loenzinvaian fo Oviouly, he Loenzinvaian veion of he euaion () inead of he laial ie u onain he pope ie : du F L, () d In ode o find hi veion of he euaion, we eplae in () i pope ie in line wih he aio fo he Loenz ie dilaion d d on d : d u E u H, (3) d A i known, he euaion (3) i he Loenzinvaian euaion of oion of a haged paile in an EM field Below we will onienly apply hi ehod o oain he elaivii euaion of gaviaion in he fo of HailonJaoi euaion oluion of he Keple pole in he faewok of LIGT Two of he o ipoan effe fo he poin of view of ehani ha aie due o he Loenz anfoaion, ae he Loenzian ie dilaion and onaion of lengh: d d, whee, a hown peviouly, d d, (), and i he hwazhild adiu The fee paile oion i deied y he HailonJaoi euaion Landau and Lifhiz, 97): ( Ñ), () In a pheial oodinae ye (aking ino aoun oh elaivii effe) i ake he fo:, (3) in whee and ae eaued in a fied oodinae ye aoiaed wih a aionay pheial a M We will a wih he aoun of he fi effe The euaion of oion of a paile in a gaviaional field, aking ino aoun he elaivii effe of ie dilaion Taking ino aoun ha he oion of a paile aound he oue ou in he plane, we define hi plane y ondiion p In hi ae, he euaion (3) ake he fo:, () IN: 5383 Pepaeie Jounal Pulihed y QuanuDea, In wwwpepaeieo
3 Pepaeie Jounal Apil 6 Volue 7 Iue 6 pp 963 Kyiako, A G, LoenzInvaian Gaviaion Theoy IN: 5383 Pepaeie Jounal Pulihed y QuanuDea, In wwwpepaeieo 8 Taking ino aoun only he anfoaion of ie d d (ee ()), euaion () an e ewien a follow:, (5) uiuing, we oain:, (6) Le u iplify hi euaion, aking ino aoun he epanion ( ) n fo << ine fo he aual ize of he plane and un and he diane eween he, value <<, we an e liied y fi wo e of he epanion A he ae and he euaion () ake he fo:, (7) We will how ha Linvaian ie dilaion lead o he appeaane of Newon gaviaional field Newon appoiaion Le u peen hi euaion o he nonelaivii ind, uing he anfoaion : uiuing hi in (7), we find û ë é Epanding he ake, we oain: Dividing hi euaion y, we find:, (8)
4 Pepaeie Jounal Apil 6 Volue 7 Iue 6 pp 963 Kyiako, A G, LoenzInvaian Gaviaion Theoy IN: 5383 Pepaeie Jounal Pulihed y QuanuDea, In wwwpepaeieo 9 Taking ino aoun ha gm, we oain U M N g, whee U i he enegy of he gaviaional field in he Newonian heoy In he nonelaivii ae we pu Fuheoe, fo eal diane of he ody oveen aound oue wih hwazhild adiu, we have << and <<, and hen we an ignoe he e В пределе при уравнение (8) переходит в известное классическое уравнение ГамильтонаЯкоби для гравитационного поля Ньютона: In he lii a, euaion (8) goe ove ino he laial HailonJaoi euaion fo Newon gaviaion field U, (9) A i known, he oluion of hi pole lead o a loed ellipial (no peeing) aellie oi aound he pheial enal ody Fo hi i follow ha he inluion only of Loenz ie dilaion ino he fee HailonJaoi euaion lead o he Keple pole in nonelaivii heoy of gaviaion Noe alo ha euaion (9) i a oneuene of he Linvaian HJE wih he Newon poenial field: in U, () Thu, he euaion (6), (9) and () ae euivalen fo poin of view of hei eul The euaion of oion of a paile in a gaviaional field wih he Loenz ie dilaion and lengh onaion Now in ode o ake ino aoun he lengh onaion effe along wih he effe of ie dilaion, we will ue he HailonJaoi euaion (3) in fo: in û ë é, () uiuing in () no only d d, u alo d d, we oain: ( ) in û ë é, () Taking ino aoun ha in ou heoy, we oain fo () he wellknown HailonJaoi euaion fo geneal elaiviy in he ae of he hwazhilddoe ei (hwazhild, 96; Doe, 97):
5 Pepaeie Jounal Apil 6 Volue 7 Iue 6 pp 963 Kyiako, A G, LoenzInvaian Gaviaion Theoy é, (3) in ë û A i well known (Landau and Lifhiz, 97), he oluion of hi euaion ae hee wellknown effe of geneal elaiviy, well onfied y epeien: he peeion of Meuy oi, he uvaue of he aeoy of a ay of ligh in he gaviaional field of a enally yei oue and he gaviaional feueny hif of EM wave A we noed, in he Keple pole oluion, aed on hi euaion, hee i an addiional e in he enegy, whih i iing in Newon heoy: g M M M M U( ) N g N, () 3 whih i eponile fo he peeion of he oi of a ody, oaing aound a pheially yei aionay ene Fo he aove analyi i follow ha he appeaane of hi e i povided y Loenz effe of he lengh onaion We found aove ha he e onaining he Loenz ie dilaion effe in he laial appoiaion lead o he euaion of Newon gaviaion wih Newon gaviaional enegy Fo hi i follow ha he peeion of he oi enue he inoduion of an addiional e 3 Gaviaional defleion of ligh ay aeoy The pah of a ligh ay (Landau and Lifhiz, 97, p 3839) in a enally yei gaviaional field i deeined y he eikonal euaion whih diffe fo HailonJaoi euaion only in having, a he ae ie, in plae of he enegy e of he paile we u wie he feueny of he ligh w l Y p g ik Y Y i k, (5) The oluion how ha unde he influene of he field of aaion he ligh ay i en: i aeoy i a uve, whih i onave owad he ene (he ay i aaed owad he ene), o ha he angle eween i wo aypoe diffe fo p y g N M d, (6) In ohe wod, he ay of ligh, paing a a diane fo he ene of he field, i defleed hough an angle d Gaviaional ie dilaion and ed hif of he feueny Wihin he faewok of geneal elaiviy, hee gaviaional effe ae ook ino onideaion on he ai of he hwazhilddoe ei In he faewok of LIGT, hi oluion i aed on he aoun of effe euling fo he Loenz anfoaion, and ha no elaion o he ei Nevehele, he indiaed effe ae eaily olved hee IN: 5383 Pepaeie Jounal Pulihed y QuanuDea, In wwwpepaeieo
6 Pepaeie Jounal Apil 6 Volue 7 Iue 6 pp 963 Kyiako, A G, LoenzInvaian Gaviaion Theoy We ae ale o pove a geneal aeen egading he influene of a gaviaional field on lok (Pauli, 98) Le u ake a efeene ye K whih oae elaive o he Galilean ye K wih angula veloiy w A lok a e in K will hen e lowed down he oe, he fahe away fo he ai of oaion he lok i iuaed, eaue of he anvee Dopple effe Thi an e een iediaely y onideing he poe a oeved in ye K The ie dilaaion i given y, (7) u w The oeve oaing wih K will no inepe hi hoening of he ie a a anvee Dopple effe, ine afe all he lok i a e elaive o hi Bu in K a gaviaional field (field of he enifugal foe) ei wih poenial w Thu he oeve hi K will oe o he onluion ha he lok will e lowed down he oe, he alle he gaviaional poenial a he paiula po In paiula, aking ino aoun ha u, he ie dilaaion D i given, o a fi appoiaion, y D ;, (8) Einein 93 applied an analogou aguen o he ae of unifoly aeleaed ye We hu ee ha he anvee Dopple effe and he ie dilaaion podued y gaviaion appea a wo diffeen ode of epeing he ae fa, naely ha a lok will alway indiae he pope ie ò d i Relaion (8) ha an ipoan oneuene whih an e heked y epeien The anpo of lok an alo e effeed y ean of a ligh ay, if one egad he viaion poe of ligh a a lok If, heefoe, a peal line podued in he un i oeved on he eah, i feueny will, aoding o (8), e hifed owad he ed opaed wih he oeponding eeial feueny The aoun of hi hif will e whee Dn E, (9) n E i he value of he gaviaional poenial on he eah, n The nueial alulaion give k/e Einein (Einein, 9) ye D, n 6 ha on he ufae of he un, oeponding o a Dopple effe of,63 applied an analogou aguen o he ae of unifoly aeleaed Le u aue (ivukhin, 5) ha he lok A elaively o he ye i oving wih onan aeleaion a We will oun he ie fo he oen when he veloiy wa zeo Then u a, whee i he diane ha he lok A oveed duing he ie Theefoe: IN: 5383 Pepaeie Jounal Pulihed y QuanuDea, In wwwpepaeieo
7 Pepaeie Jounal Apil 6 Volue 7 Iue 6 pp 963 Kyiako, A G, LoenzInvaian Gaviaion Theoy d d a, () Now le u inodue an aeleaed efeene fae, whih ove ogehe wih he lok A In hi ye he lok A i ioile, u hee ae ineial foe If all he phenoena will e deied, aking a a efeene fae, hen a he aue of ie dilaion he ineial foe hould e onideed The ineial foe pe uni a of he oving ody i a Bu, aoding o he piniple of euivalene, he ineial foe ae indiinguihale fo he gaviaional field, he ineniy of whih in ou ae i g a Then he gaviaional poenial i g and he foula () eoe: o» d ( ) d d, () d d d, () A zeo gaviaional poenial, he poenial of poin i onideed, a whih he oving and aionay lok un eually fa Theefoe, in foula () and (), he ie ineval d an e ouned no y he lok of he ineial ye, u y he lok ha i in e in ye, whih i loaed a he poin B wih zeo poenial In geneal, we an e he iniiaion of oun of gaviaional poenial a any poin, if he foula () ha he fo: d A d d A B B A, (3) whee he ie ineval d A and d B ae ouned y wo lok, whih ae in e in an aeleaed efeene fae a poin A and B wih gaviaional poenial A and B Conluion Thu, we an ay ha, in he ae of enally yei gaviaional field, wihin he faewok of LIGT we ge he ae eul a in he faewok of geneal elaiviy I i noewohy ha in ode o oain hee eul ino aduen in euaion of oion ae euied, whih ae enued y wo effe following fo he Loenz anfoaion IN: 5383 Pepaeie Jounal Pulihed y QuanuDea, In wwwpepaeieo
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