Scholars Research Library. Archives of Applied Science Research, 2014, 6 (5):36-41 (

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Available online a wwwsholaseseahlibayom Ahives o Applied Siene Reseah 4 6 5:6-4 hp://sholaseseahlibayom/ahivehml ISSN 975-58X CODEN USA AASRC9 Einsein s equaions o moion o es pailes exeio o spheial

Pue and Applied Physis Fedeal Univesiy Wukai Wukai Taaba Sae-Nieia Depamen o Mahemais Nasaawa Sae Univesiy Kei Nasaawa Sae-Nieia ABSTRACT In his aile he mei enso exeio o hypoheial spheial disibuions

ime on a lok movin in his aviaional ield Fo pue pola moion es pailes move wih veloiy ha has an invese dependene on he adial disane The esuls show ha he inoduion o in his ield does no ale he invese

iven by µν T µν whee Gaviaional onsan G µν Einsein s enso T µν sess eney enso and is he speed o lih [] Sine hen he seah has been on o he developmen o hei soluions in he spae-ime o all disibuions o

onsidein spheial bodies suh as he sun and he sas In Shwazhild s mei he enso ield vaies wih adial disane only I is he mei enso exeio o an ideal sai spheially symmei body siuaed in empy spae []; [];

eion []; [4] Sine he eah is no peely spheial [9] is ield anno be a union o only he adial disane as assumed by Shwazshild This pape inodues an asophysial disibuion o mass wihin he eion o spheial

1 Available online a wwwsholaseseahlibayom Ahives o Applied Siene Reseah 4 6 5:6-4 hp://sholaseseahlibayom/ahivehml ISSN X CODEN USA AASRC9 Einsein s equaions o moion o es pailes exeio o spheial disibuions o mass whose enso ield vaies wih ime adial disane and pola anle Lumbi Luas Williams Ewa Ibahim Inusa and Tsaku Nuhu Depamen o Physis Nasaawa Sae Univesiy Kei Nasaawa Sae-Nieia Depamen o Pue and Applied Physis Fedeal Univesiy Wukai Wukai Taaba Sae-Nieia Depamen o Mahemais Nasaawa Sae Univesiy Kei Nasaawa Sae-Nieia ABSTRACT In his aile he mei enso exeio o hypoheial spheial disibuions o mass whose enso ield vaies wih ime adial disane and Pola anle is exended o deive equaions o moion o es pailes in he aviaional ield The ime equaion is used o deive he expession o he vaiaion o he ime on a lok movin in his aviaional ield Fo pue pola moion es pailes move wih veloiy ha has an invese dependene on he adial disane The esuls show ha he inoduion o in his ield does no ale he invese dependene o veloiy on he adial disane Keywods: Tenso Field Tes Pailes Gaviaional Field Time Radial Disane Pola Anle INTRODUCTION 8πG G 4 In he yea 95 Einsein published his aviaional ield equaion iven by µν T µν whee Gaviaional onsan G µν Einsein s enso T µν sess eney enso and is he speed o lih [] Sine hen he seah has been on o he developmen o hei soluions in he spae-ime o all disibuions o mass in naue and hei appliaions o he moion o es pailes o non zeo es masses and phoon The exa soluion o his ield equaion was is onsued in sai and pue adial spheial pola oodinaes by Shwazhild in 96 by onsidein spheial bodies suh as he sun and he sas In Shwazhild s mei he enso ield vaies wih adial disane only I is he mei enso exeio o an ideal sai spheially symmei body siuaed in empy spae []; []; [5] Shwaz hild s mei is he mahemaially mos simple and asophysially mos saisaoy soluion o Einsein s eomeial aviaional ield equaion in he spae exeio o a sai homoeneous disibuion o mass wihin a spheial eion []; [4] Sine he eah is no peely spheial [9] is ield anno be a union o only he adial disane as assumed by Shwazshild This pape inodues an asophysial disibuion o mass wihin he eion o spheial eomey whose enso ield vaies wih ime adial disane and pola anle only The equaion o moion o pailes o non zeo es masses ae deived o his aviaional ield This eseah will help in sudyin asophysial spheial disibuion 6

2 Lumbi Luas Williams e al Ah Appl Si Res 4 6 5:6-4 o mass whose enso ield vaies wih ime adial disane and pola anle An example o suh disibuion is he homoeneous disibuion o mass wih spheial eion whih is oain wih uniom speed abou a ixed diamee [8]; [9]; [] Theoeial Analysis Consuion o mei enso and oeiiens o aine onneion Shwazhilds mei is he soluion o Einsein s ield equaions exeio o a sai homoenous spheial body [4]; [];[4] iven by sin 4 ; ohewise µν 5 whee is he speed o lih in vauum is an abiay union deemined by he disibuion; i is a union o he adial oodinae only sine he disibuion and hene is exeio aviaional ield possess spheial symmey Fom he ondiion ha his mei omponen should edue o he ield o a poin mass loaed a he oiin [4] and onains Newon s equaion o moion in he aviaional ield o he sai homoenous spheial body i ollows ha in his ield as is he Newonian aviaional sala poenial in he exeio eion o he body deined GM > R 6 whee G is he univesal aviaional onsan M is he mass o he spheial body and R is he adius o he spheial body Le us onside an asophysial mass disibuion wihin spheial eomey in whih he enso ield vaies wih ime adial disane and pola anle The ovaian mei enso o his disibuion o mass o pessue is iven as [];[] whee sin ;ohewise µν is an abiay union deemined by he mass o pessue and possesses all he symmeies o he lae In appoximae aviaional ields i is equal o Newon s aviaional sala poenial exeio o he spheial mass disibuion To obain he oespondin ona vaian mei enso o his aviaional ield we impose he Quoien Theoem [6] o he enso analysis o obain he omponens o he ona vaian enso as 7

3 Lumbi Luas Williams e al Ah Appl Si Res 4 6 5:6-4 4 sin 5 µν ;ohewise The oeiiens o aine onneion deined by he mei enso o spae-ime ae deemined usin he enso equaion { } 7 δ δν µλ µλ v νλ µ µ v λ s They ae ound o be iven expliily in ems o as whee he omma denoes paial dieeniaion w Equaion 8 o an be wien expliily in ems o as 6 8

4 Lumbi Luas Williams e al Ah Appl Si Res 4 6 5: sin 4 4 sin os ;ohewise α β 48 Thus his aviaional ield has 5 non-zeo oeiiens o aine onneion; unlike he Shwazhild s ield whih has 9 aine onneions Thus we expe he aviaional ield in his aile o have some peuliaiies no ound in Shwazhild s ield Moion o es pailes exeio o spheial bodies whose enso ield depends on ime adial disane and pola anle A es mass is one whih is so small ha he aviaional ield podued by i is so neliible ha i doesn have any ee on he spae mei A es mass is a oninuous body whih is appoximaed by is eomeial ene; i has nohin in ommon wih a poin mass whose densiy should obviously be ininie [7] The eneal elaivisi equaions o moion o es pailes in a aviaional ield ae iven by: d x dτ µ µ νλ ν λ dx dx dτ dτ 49 9

5 Lumbi Luas Williams e al Ah Appl Si Res 4 6 5:6-4 4 whee τ is he pope ime We used equaion 49 o onsu ime adial and pola equaions o moion o pailes o non-zeo es mass in he aviaional ield unde onsideaion Sein µ ino equaion 49 and subsiuin equaions o 6 ives he ime equaion o moion as ` 5 5 ln ln τ 5 Ineain equaion 5 ives 5 Equaion 5 is he expession o he vaiaion on he ime on a lok wih his aviaional ield Similaly sein and µ ino equaion 49 ives he adial pola and azimuhal equaions o moion as and 56 espeively sin φ 54 os sin φ 55 φ φ 56 Fo pue adial moion φ and hene he adial equaion 55 beomes Equaion 57 is he pue adial equaion o moion o pailes o nonzeo es masses in his ield Conside he pola moion o es pailes ha has adial dependene in his ase equaion 55 edues o 58 o 59

6 Lumbi Luas Williams e al Ah Appl Si Res 4 6 5:6-4 Ineain equaion 58 ives he insananeous pola veloiy as A φ 6 whee is he onsan o ineaion This moion has an invese squae dependene on he adial disane Resul in his ield is simila o he esul obained in ou ealie publiaion when [] We an heeoe onlude ha he inoduion o o his ield does no hane he invese squae dependene on he adial disane CONCLUSION The ime adial pola and Azimuhal equaions o moion o es pailes exeio o asophysial eal spheial disibuion o mass wee ound o be equaions and 57 espeively The soluion o he ime equaion o moion ives he vaiaion o he ime on a lok wih he aviaional ield Thus he expession o aviaional ime dilaion in his aviaional ield has been obained as equaion 5 The adial equaion o moion iven by equaion 55 an be used o obain he insananeous speed o a paile o nonzeo es mass in his ield The oeiiens o aine onneion obained an be used o onsu he Riemann-Chisoel Rii and Einsein s enso o his ield and hene he Einsein s ield equaions o his aviaional ield an be obained he Einsein s equaion onain only a sinle unknown values o This is opened up o uhe eseah REFERENCES and hus an be solved ompleely o obain explii [] JL AndesonPiniples o Relaiviy Physis Aademi Pess New Yok 967Pp5- [] G Aken Mahemaial Mehods o Physiiss 5 h ed Aademi Pess New Yok995 Pp [] PG Bemann Inoduion o he Theoy o Relaiviy Penie Hall India987Pp- [4] EN Chiu; SXK Howusu Jounal o NAMP [5] EN Chiu; SXK Howusu;LW Lumbi Poess in Physis9 8 - [6] R Dmii; B Laissa Poess in Physis [7] P Dunsby An Inoduion o Tenso and Relaiviy Shiva Cape Town 5- [8] A Einsein The Foundaion o he Geneal Relaiviy Annalen de Physik 96 [9] SXKHowusu Disouse on eneal elaiviy Jos Univesiy Pess Ld 98-5 [] SXK Howusu Naual Philosophy Soiey Jos7-7 []WL Lumbi;MS Liman;RN Jaau;I Uma Ineaed Jounal o Sieneand Enineein 9-5 [] K Shwazshild Ube das Gaviaionseld eines Massen-punkes nah de Einseinshen Theoie Sizunsbeihe de Konilih Peussishen Akademie de Wissenshaen published in Enlish as: Shwazshild K Abaham Zelmanov Jounal 8-9 [] S Weinbe Gaviaion and Cosmoloy J Willey and sons New Yok 97-7 [4] Wikipedia The Fee Enylopaedia Geneal elaiviy8 hp://wwwenikipediao 4

Physics 442. Electro-Magneto-Dynamics. M. Berrondo. Physics BYU

Physics 442. Electro-Magneto-Dynamics. M. Berrondo. Physics BYU Physis 44 Eleo-Magneo-Dynamis M. Beondo Physis BYU Paaveos Φ= V + Α Φ= V Α = = + J = + ρ J J ρ = J S = u + em S S = u em S Physis BYU Poenials Genealize E = V o he ime dependen E & B ase Podu of paaveos: