DERIVATION OF LORENTZ TRANSFORMATION EQUATIONS AND THE EXACT EQUATION OF PLANETARY MOTION FROM MAXWELL AND NEWTON Sankar Hajra

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1 PHYSICAL INTERPRETATION OF RELATIVITY THEORY CONFERENCE, LONDON, 4 DERIVATION OF LORENTZ TRANSFORMATION EQUATIONS AND THE EXACT EQUATION OF PLANETARY MOTION FROM MAXWELL AND NEWTON Sanka Haja CALCUTTA PHILOSOPHICAL FORUM Sal Lake, AC-54, Seco-1, Fla No.A1, Kolkaa ankahaja@yahoo.com Abac: The cala and veco poenial of a yem of chage and cuen aionay in fee pace ae govened by Poion equaion wheea he imila poenial of a yem of chage and cuen eadily moving in fee pace ae govened by D Alembe equaion. Heaviide (1888, 1889) and Thomon (1889) fi coecly calculaed he cala and veco poenial of a eadily moving poin chage by anfoming he D Alembe equaion of he poenial fo he eadily moving chage ino Poion fom fo a aic chage by elongaing a coodinae ai lying along he diecion of he anlaion of he chage. They hu developed a way o olve dynamic poblem like aic poblem uing an auiliay equaion in he fom of Poion poenial equaion. We have ued hi ingeniou mahemaical eamen o deive he field of a eadily moving poin chage, i elecomagneic momenum, i longiudinal and anvee elecomagneic mae and many ueful elecodynamic equaion including Loenz Tanfomaion Equaion fom Mawell field equaion alone. In he la pa of he pape, aing fom he longiudinal elecomagneic ma of a poin chage eadily moving in fee pace, we have peened a deducion of he eac equaion of planeay moion following Mawell and Newon conideing ha a plane conain ma and chage and auming ha in he gaviaional field a poin chage ac a a ma poin, ma of he ma poin being popoional o he longiudinal elecomagneic ma of he chage. Thi udy implie ha elecomagneic field, chage and elecomagneic enegy ae eal phyical eniie. Ju like all ohe maeial bodie hey ae no only ubjec o gaviaion, hey have he ame acceleaion a ha of maeial bodie in he ame gaviaing field and hi imple conideaion i equivalen o pecial and geneal elaiviy. 1. Inoducion All elecodynamic equaion, he upiing fac ha all elecodynamic phenomena obeved on he uface of ou Eah ae independen of he moion of he Eah, advance of he peihelion of Mecuy, gaviaional ed hif of aal ay and bending of ligh ay gazing he uface of he un ae eplained by iniuional phyici fom he conideaion of pecial and geneal elaiviy which ue many abud meaphyical pinciple. In hi pape we hall imply deive all hoe equaion fom Mawell and Newon and eablih ha all elecodynamic eniie ae ubjec o gaviaion.. Heaviide-Thomon Auiliay Equaion The cala poenial ( ) of a yem of chage moving eadily in fee pace wih a velociy u (ay S yem) i govened by D Alembe equaion: (1) 1 whee i he chage deniy of he yem, i he pemiiviy and μ i he pemeabiliy of fee pace uch ha c 1, and,y,z ae he Caeian co-odinae inoduced in he fee pace. Aume u in he OX diecion. Then, i a funcion of wo independen vaiable, and. Fom he definiion of wo independen vaiable we have, d d d ud d ( d being equal o ud ). Moeove, in uch a iuaion, he poenial a he poin (, y, z ) a he inan and he poenial a he poin ud, y, za he inan d in fee pace will be he ame; i.e.,, y, z, ud, y, z, d, y, z, d Fom he above wo equaion we have ud d u ()

2 PHYSICAL INTERPRETATION OF RELATIVITY THEORY CONFERENCE, LONDON, 4 u () Le u now fi a co-odinae yem, y, z o he yem of chage moving eadily in fee pace wih a velociy u in he OX diecion a defined in he equaion (1) and aume ha a =, boh coincide. Le u now imagine an elongaion of he moving yem owad i diecion of moion by he faco whee 1 k and k 1 u c. Then he volume chage deniy of hi imaginaily elongaed yem hould be, k and we have,, y y, z z (4) The yem imaginaily elongaed by he equaion (4) owad OX ai which i he diecion of moion of he yem of chage may be called he auiliay yem S compiing of, y, z. The cala poenial of hi Auiliay Syem hould be (5). Now ubiuing, y y, z z a in he equ. (4). and uing he equ.(), he equaion (1) could be anfomed o (6). Compaing equaion (5) & (6), we have, k (7) fom which we have E E E E, E. (8), y y z E z Whence we can ge he induced magneic field fom he elaion * B ue c. (9). Theefoe, he elecic and magneic field of a eadily moving poin chage a ha been noed by Heaviide [1,] hould be eacly he ame a hoe of a conducing imilaly chaged and imilaly moving ellipoid (Heaviide Ellipoid) having i ae wih he aio k:1:1, k being in he diecion of movemen of he chage. Fo, in he Auiliay Syem, boh have he ame eenal field a hoe of a aionay imilaly chaged phee. The above device of he Auiliay Syem compiing of he equaion (4) wa ued by Heaviide (1888,1889), Thomon (1889) and hei conempoaie o udy he elecodynamic of moving yem of chage. Theefoe, we may call he equaion (4) he Heaviide-Thomon Auiliay Equaion. Field of a Seadily Moving Poin Chage Following Heaviide, he elecic field and he induced magneic field of a poin chage eadily moving wih a velociy could be calculaed a a poin ( ) in fee pace (conideing a a paicula inan, he poin chage a oigin and OX a he diecion of moion of he chage) in he following way: whee, and ae he componen of auiliay elecic field in he Heaviide-Thomon auiliay yem a ( ) Now, he diance of he poin fom he oigin in hi yem, whee ( ) of he yem i he coeponding poin ( ) of he yem. The Coodinae of he coeponding poin i ( ) in Caeian Syem and ( ) in pheical pola coodinae whee i he diance fom he oigin, i he angle down fom he ai and i he angle aound fom he ai. Theefoe,, coodinae] of ] Now,,, fom which and. Theefoe: [uing Eq. (4)] Thu, independen of, we have: [in pola [a eul independen [uing Eq. (8)]

3 PHYSICAL INTERPRETATION OF RELATIVITY THEORY CONFERENCE, LONDON, 4 The auiliay i dieced along. Theefoe, he eal field i dieced along. Now, fom, we have, fom which he induced magneic field i 4. A Coollay I can eaily be hown ha Heaviide field obey Mawell equaion ju like Coulomb field do. Theefoe, if a aionay dipole adiae in fee pace, i will alo adiae while moving in fee pace a conan anlaional velociy. 5. Elecomagneic Momenum of a Seadily Moving Poin Chage Auming ha he poin chage i pheical having he adiu R, he elecomagneic momenum of a Heaviide ellipoid (wih he ae : : ) while moving ecilinealy wih a velociy in he OX diecion in fee pace can be calculaed a: * P D B * D B d : y Z Z Y u E E kd Y Z c u EY EZ d [cf. eq.(8&9)] c whee d i he volume elemen in he moving yem (S) and d i he coeponding volume in he yem, D i he elecic diplacemen veco in (S). Hee, in he yem he Heaviide ellipoid i eacly a phee. Theefoe, when evaluaed beween and, E d Q 4 R and each inegal in he peviou equaion i equal o. Q 1 R Theefoe, Q u 6 c Rk mu (1) P which i he elecomagneic momenum of a poin chage moving ecilinealy in fee pace wih velociy, whee and ae he elecomagneic mae of he poin chage moving wih he velociie nea zeo and epecively in fee pace uch ha Q and 6 c R m (11) m k m (1). Theefoe, Longiudinal Elecomagneic ma of a poin chage = P u m (1) and Tanvee Elecomagneic ma = P u m (14). All elecodynamic equaion imila o hoe of pecial elaiviy could eadily be deduced fom he equaion (1) [] 6. Auiliay Tanfomaion Equaion of Loenz Thi Secion deive Loenz auiliay equaion fom hoe of Heaviide. Following Heaviide and Thomon, Loenz engaged himelf in developing ome new auiliay anfomaion equaion. He liked o olve opical poblem of moving bodie a well a elecodynamic poblem of chage moving wih low and high velociie hough hoe anfomaion equaion. Thee anfomaion equaion hould educe he equaion of moving yem o he fom of odinay fomula ha hold fo yem a e. The poblem Heaviide and Thomon addeed wa o model he poenial fo chage having a conan anlaional velociy in fee pace. They olved hi poblem by anfoming he D Alembe equaion in an invaian fom wih Poion in he auiliay yem. Loenz poblem wa o model he adiaion fom moving bodie. He olved hi poblem by anfoming Mawell equaion fo he moving adiaing body o he ame fom fo he auiliay yem, which coelae beween he aic and dynamic adiaion ae. By din of he Coollay aed in he ecion 4, we have, E i.e., (15) whee i he elecic field of a adiaing dipole moving wih a conan anlaional velociy in fee pace. Fom hi we ge (15a) To olve adiaion poblem in a way analogou o ha hown in Secion, we ae o keep he Mawell equaion in he ame fom in he yem; i.e., i i now equied ha E

4 PHYSICAL INTERPRETATION OF RELATIVITY THEORY CONFERENCE, LONDON, 4 i.e., (16) whee i he auiliay elecic field of Heaviide and Thomon in he yem. Tha i, whee, y, z i o be choen appopiaely. (16a) ae defined in he ecion and Subacing Eq. (15a) fom Eq. (16a) and uing he Heaviide-Thomon auiliay equaion (17) [if elecomagneic acion i conideed afe he ime of he inan when he coodinae aached o he yem coincided wih he coodinae aached o fee pace, Eq. (4) hould be changed o he equaion (17) ] o eplace he pimed vaiable, we ge o (18) he famou auiliay ime equaion of Loenz. An ineeing fac abou he equaion i ha he invee Loenz Tanfomaion equaion (which could be deduced fom Loenz Tanfomaion equaion) have he ame fom a he Loenz Tanfomaion equaion hemelve; i.e., if hen (19 a-d) ( a-d) Theefoe, Loenz anfomaion equaion can be deduced in evee fom he dyad of Eq. (15a) & (16a) and (19a-c) &(a-c). (Thee anfomaion equaion wee lae obeved and ued by A. Einein in hi heoy). I hould be menioned hee ha all he quaniie,, ec., ae auiliay, o Eq. (16) i auiliay. Thu, fom he andpoin of elecodynamic, Loenz Tanfomaion Equaion ae acical, ju like he acical equaion of Heaviide and Thomon. Thee auiliay equaion ae vey ueful o olve coecly he adiaion poblem aociaed wih eadily moving elecomagneic bodie. The auiliay equaion of Heaviide and Thomon (17) ae geneal, bu Loenz auiliay ime Eq. (18) i no geneal. I i only applicable in adiaion poblem of moving yem and in ome pecial cae. To conide Loenz auiliay ime equaion a geneal i conay o he pinciple of elecodynamic, and o conide all fou Loenz anfomaion Eq. (17) & (18) a eal i an ove implificaion of elecodynamic and of naue. 7. Naue of Elecic and Magneic Field a he Viciniy of he Eah Suface All elecodynamic phenomena on he uface of he moving eah ae een o be independen of he movemen of he eah, which implie ha he eah caie elecic and magneic field along wih i uounding ju like i caie all ohe phyical objec wih i. Thi imple conideaion along wih Heaviide elecodynamic a aed above may eve a an equivalen alenaive o pecial elaiviy in he domain of elecodynamic wihin he Newonian famewok [,]. We may now eend hi conideaion o he cae of chage and elecomagneic enegy oo and each o ome ineeing eul a given below. 8. Ineacion of Chage wih Gaviaing Field Le a gaviaing body wih he maeial ma M be a e a he oigin and he poin chage Q be moving wih a velociy u aco a gaviaing field. Now aume ha chage ae ubjec o gaviaion and he longiudinal elecomagneic ma of he chage 'q' i popoional o i ma. Thu, in a gaviaing field, a poin chage will behave ju like a ma poin. Theefoe, he equaion of moion of a poin chage Q moving aco a gaviaing field could be wien a follow [when he moving poin chage i a, wih pola co-odinae, pole being he ouce of he field]: F G m, GM and eplacing F G by m we have, 4

5 PHYSICAL INTERPRETATION OF RELATIVITY THEORY CONFERENCE, LONDON, 4 GM (1) (which implie ha poin chage have he ame acceleaion a ha of maeial bodie in he ame gaviaional field) and 1 d F G m () d whee G i he gaviaional conan, F and G G F ae he adial componen and he anvee componen of he gaviaing foce acing on he chage, m i he ma of he chage a well a he longiudinal elecomagneic ma of he chage when he aio of popoionaliy of boh he quaniie i uniy. The quaniy of chage, lage o mall i being aumed concenaed in a vey mall pheical volume wih a vey negligible mall adiu δr. Fom () we have, Mechanical Angula Momenum of he chage, A m = Conan () whence we ge, = H, a conan. (4) Now, we can eplace equaion (1) a unde: Le U 1 (5) Fom () we have HU k (6), du Hk (7) d 6 d U H U k (8). d Subiuing in (1), we have, d U GM 6 GM U d H u 1 H c [when u i vey mall in compaion o c and noing ha fo cicula moion u ] GM GM (9) H c 9. Eac Equaion of Planeay Moion Suppoe ha a plane of ma m conain Q amoun of poiive and negaive chage in oal (ignoing he ign of he chage) and fo imple calculaion le u aume ha he oal chage ae concenaed a he cene of he plane. In hi p iuaion, if we aume, a befoe, ha he longiudinal elecomagneic ma of a poin chage i popoional o i ma, he equaion of planeay moion will be a unde: Radial Foce= F m m G p GM o, m m p m m p O, GM () whee m i he longiudinal elecomagneic ma a well a he ma of he chage aociaed wih he plane, popoionaliy conan being aken a uniy. M i he ma of he un and G i he gaviaional conan. Co-adial Foce= 1 d FG m m p d (1) fom which mechanical angula momenum of he plane, A mp m. In cae when u i vey mall wih epec o c and m i much lage han u mp c, A= m 1 u c m p fom which we have, H Conan (). Fom equ. () & () we may deduce d U GM GM U () d H c 1. Gaviaional Red Shif Suppoe ha a ay i coming fom he uface of a a of adiu R and of ma M o he uface of he eah, diance away fom he a. A pe ou peviou dicuion, we aume ha elecomagneic enegy ha he ame acceleaion a ha of maeial bodie a well a poin chage in he ame gaviaional field. Theefoe, he velociy of he ay a he uface of he eah 5

6 PHYSICAL INTERPRETATION OF RELATIVITY THEORY CONFERENCE, LONDON, 4 GM d1 V c c R 1 ( 1 being he diance of an abiay poin on he ay fom he a) GM c 1 (4) Rc fom which we have, GM 1 (5) Rc i he adian fequency of ligh a he uface of he a and i he adian fequency of he ame ligh ay a he uface of he eah, which i aid o have been veified by many efined epeimen. 1. Bending of Ligh Ray Gazing he Suface of he Sun The above udy empowe u o apply equaion (1) &() o he cae of popagaion of elecomagneic adiaion gazing he uface of he un having gaviaional ma M and adiu R. Theefoe, equaion of moion of a ligh beam paing hough a medium (uch ha he velociy of he ligh beam u i much malle han c in he medium), medium being fied wih he un uface will ake he fom: d U GM U GM d H c R Now fo a mall volume of adiaion enegy, he mechanical angula momenum A i finie. Theefoe, H fo hi mall volume of adiaion enegy will be infinie, a m fo hi mall volume of adiaion enegy =. In uch a iuaion, he above equaion fo he popagaion of a mall volume of adiaion enegy hough he un adjacen gaviaional field inide a medium wheein u c hould be d U d U GM. (6) c R which i widely believed o have confimed by billion dolla epeimen. Elecomagneic enegy i neual. Sill we ee ha he equ. (9) i fully applicable o he cae of elecomagneic enegy a good a o he cae of poin chage. Theefoe, we ae conained o peculae ha elecomagneic enegy, oo, may have poin eience coniing of equal amoun of oppoie poin chage.[] 1. Concluion Elecomagneician have amply demonaed ha elecomagneic field, chage and elecomagneic enegy ae all eal phyical eniie. We have hown in hi pape ha all hoe eniie ae no only ubjec o gaviaion ju like all ohe phyical objec, hey have he ame acceleaion a ha of phyical bodie in he ame gaviaing field, and hi imple conideaion i equivalen o pecial and geneal elaiviy. To wie hi pape, I have been much influenced by he wok of D.B.Ghoh [] and of Pof. K.C. Ka [4]. [1] (i) O. Heaviide, Elecical Pape, ii (189), 514.(ii) J.J. Thomon, The Phil. Mag., pp (1989). [] Sanka Haja and Debabaa Ghoh, Phyical Inepeaion of Relaiviy Theoy, edied by D.M.C.Duffy,, [] Thi communicaion i he gi of hee pape cheduled o be publihed in he Galilean Elecodynamic. [4] Pof. K.C. Ka, New Appoach To The Theoy of Relaiviy (Iniue of Theoeical Phyic, Bignan Kui, 41 Mohan Bagan Lane, Calcua- 4, 197),

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