Consequences of Using the Four-Vector Field of Velocity in Gravitation and Gravitomagnetism 1

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1 Consequenes of Usin he Four-eor Field of eloi in Graviaion and Graviomaneism 1 Mirosław J. Kubiak Zespół Skół Tehninh, Grudiąd, Poland In his paper are disussed he phsial onsequenes of usin he fourveor field of veloi ( ) µ in raviaion and raviomaneism. PACS number: 4.4.Nr kewords: raviaion, raviomaneism, four-veor field of veloi I. INTRODUCTION The analo beween raviaion (G) and eleromaneism has a lon hisor [1]. The onjeure ha mass urrens should enerae a field alled, b analo wih eleromaneism, he raviomanei field, oes bak o he beinnins of eneral relaivi. Indeed, aordin o eneral relaivi, movin or roain maer should produe a onribuion o he raviaional field ha is he analoue of he manei field of a movin hare or manei dipole []. The erm raviomaneism (GM) ommonl indiaes he olleion of hose raviaional phenomena reardin orbiin es pariles, preession of rosopes, movin loks and aoms and propaain eleromanei waves whih, in he framework of he General Theor of Relaivi (GTR), arise from non-sai disribuions of maer and ener. In he weak-field and slow moion approimaion, he Einsein field equaions of GTR, whih is a hihl non-linear oren-ovarian ensor heor of G, e linearied, hus lookin like he Mawellian equaions of eleromaneism [3]. II. THE FOUR-ECTOR FIED OF EOCITY Wh does he salar poenial of a G field ϕ(r, ) have he dimension of he square of he veloi [m /s ]? Wh does he veorial poenial of he GM field A (r, ) have he dimension of he veloi [m/s]? Are hese imporan quesions? Or onl a onsequene of our perepion of reali? We will r o answer for hese quesions. e's replae he salar poenial of a G field, ϕ(r, ), b (r, ), where (r, ) - ϕ(r, ). e s name he (r, ) as he salar field of he square of he veloi. e's replae he veorial poenial of he GM field A (r, ) b he m (r, ), where m (r, ) A (r, ). e s name he A (r, ) as he veorial field of he veloi. e's replae of he G and GM four-poenial A µ (ϕ/, A ) b he four-veor field of he veloi ( ) µ, whih we will define in he form 1 Paper has been aeped for publiaion in he Journal of eorial Relaivi, Sepember 11 issue, (JR 6 (11) 3 1-8). Now all publiaions of he journal were hun on he undefined ime beause of he breakdown of he server. 1

2 def µ, m (1) where: speed of propaaion of field (equal o, b GTR, he speed of lih ). The ( ) µ has dimension [m/s], from here he name - he four-veor field of he veloi. III. THE AGRANGIAN The enire ssem of bodies and fields onsiss of a mehanial par, an ineraion par and a field par. We herefore assume ha he oal arane densi o for his ssem an be epressed as o meh in field + () + where: is he mehanial arane densi, meh ρv (3) in ρv + ρ (4) m is he ineraion arane densi for he bod inerain wih he ( ) µ field, and field ( F ) ( F ) 16πG (5) is he field arane densi. Beause field ener differene epressed in he ensor field of he veloi (see Appendi A I), i.e. he differene beween he G and GM field ener densiies, has he form 16 and equaion (5) beomes 1 ( F ) ( F ) + π G 8πG m field 1 + m (5a) 8πG where: v veloi of he bod, is he inensi of he G field or he field of he aeleraion, m is he inensi of GM field or field of he roaion, ρ mass densi, G raviaional onsan. I. THE AGRANGE S EQUATION OF MOTION The equaion of moion for he bod movin in he ( ) µ field an be alulaed from he arane s equaion d d v r

3 and for he laranian meh + in we e d d ( mv ) m + mv m (6) where: m ( m /), rad ( ) - (/)( m ), mv m is he Coriolis fore, m mass of he bod.. THE AGRANGE S FIED EQUATIONS Field equaions for and for he m have a form (see Apendi A II ) m (7a) ρ (7b) 1 m ρv + (7) (7d) These equaions are similar o he field equaions in oren-invarian heor of raviaion in he weak raviaional field aordin o he Einsein field equaions for GTR [4]. I. THE WAE EQUATIONS If we appl he url operaor ( ) o boh sides of he equaions (7a) and (7b), hen we obain ( ( ) 1 ( ) ( ) m m ) ( ρv) Furher alulaions ive he wave equaions for he veors and m in he form 1 ( ρv) ρ (8a) 1 m π m G ( ρv) (8b) The wave equaions for he veor field of m and salar field of ( ), have he forms 3

4 1 ρv m m (8) 1 ρ (8d) if oren aue ondiion for he m and ( ) is fulfilled, hen ( ) 1 (8e) m In his sense, he followin wave equaions: (8a) and (8b) are he raviaional and GM analoous o wave equaions for eleromaneism. (8) and (8d) desribes how he veorial waves of he m and he salar waves of he ( ) propaae hrouh he spae. II. PHYSICA INTERPRETATION OF ( ) e s onsider equaion (8d). For he saionar field his equaion beomes he Poisson s field equaion ρ (9) In pariular, solvin equaion (9) for he spherial smmer we obain well-known equaion GM (r) (9a) r where: M is he mass of he sar, r disane from he sar. If we subsiue he mass of he Sun and he averae radius of he orbi for eah plane ino he equaion (9a), hen we obain averae of he (r) for planes in he Solar Ssem (see he Table 1). Table 1. Calulaed (wihin he model) averae of he (r) and observed averae veloi v(r) for he planes in he Solar Ssem. Averae radius of he orbi [AU * ] Calulaed averae of he (r) [kms -1 ] Observed averae orbial veloi v(r) [kms -1 ] Merur enus Earh Mars Jupier Saurn Uranus Nepune Pluo * 1AU km 4

5 In our model he G is he salar he field of he square of he veloi. The sar wih he ρ densi eneraes he salar field of he square of he veloi equaion (8d). In pariular (equaion (9a)), he orbial veloi of he planes v(r) (r). III. CONCUSION Simple replaemen of he four-poenial A µ (ϕ/, A ) b he four-veor field of he veloi ( ) µ (-( ) /, m ) ives a new perepion for raviaion and he raviomaneism. In our model he G is he salar field of he square of he veloi and he raviomaneism is he veorial field of he veloi. REFERENCES 1. B. Mashhoon, Gravioeleromaneism: A Brief Review, arxiv:r-q/3113v1, 8... B. Beerra, A. Barros, C. Romero, On some aspes of raviomaneism in salar-ensor heories of ravi, Brailian Journal of Phsis, vol. 35, No. 4B, Iorio, C. Corda, Graviomaneism and raviaional waves, arxiv: v, S. G. Fedosin. Eleromanei and Graviaional Piures of he World, Apeiron, ol. 14, No. 4, 7, hp://redshif.vif.om/journalfiles/14no4pdf/14n4fed.pdf. 5. B. Thidé, Eleromanei Field Theor, nd ediion, hp:// Jul 11. APPENDIX A I. The Field of he eloi Tensor In Seion II we defined he four-veor field of he veloi in he onravarian form def µ, m (A I. 1) Now we define he field of he veloi ensor in he onravarian form ν µ def (A I. ) ( F ) Mari represenaion of he field of he veloi onravarian ensor has form µ ν 5

6 6 ( ) F (A I. 3) Mari represenaion of he field of he veloi ovarian ensor has form ( ) F (A I. 4) A II. The Field Equaions The field equaions we an alulae from he Euler-arane equaions of moion for he field [5], whih were adoped for our onsideraion ( ) i m i m k i m k k + / / 3 1 i 1,, 3 (A II.1) 3 1 k k k + / / (A II.) For he arane funion o (see equaion ()) alulaions ives field equaions (7a), (7b), (7) and (7d).