GAVITOELECTOMAGNETIM AND NEWTON LAW O UNIVEAL GAVITATION Antoine Ake etied Pofesso, Depatent Industial ienes Uniesity Collee Kaho int-lieen Gent - Beliu ant.ake@skynet.be Abstat Takin into aount the kineatis of the aitatin objets, aitoeletoanetis (GEM) is a onsistent lassial field theoy about the aitational phenoena in whih ontext the piniple of elatiity and the piniple of equialene ae alid. In this atile it is shown that, in the faewok of GEM, Newton s law of uniesal aitation pefetly an be dedued and that it an be extended to the ase of oin point asses. Keywods: aito-eletoanetis, aitation. INTODUCTION TO GAVITOELECTOMAGNETIM In aito-eletoanetis [],[],[3] (GEM) we think about the foe atin between two patiles and in tes of aitational fields :. Patile sets up a aitational field in the spae aound itself;. That field ats on patile, this shows up in the foe that expeienes. Genealized: in GEM the aitational field plays an inteediay ole in the inteation between asses.. The aitational field is set up by a ien distibution of - whethe o not oin - patiles and it is - just as the eletoaneti field - desibed as a obination of two theediensional, tie-dependent, intetwined eto fields: the -field E and the indution B. These eto fields ae, elatie to an inetial efeene fae O, funtions of the spae and tie oodinates. Just like the eletoaneti field ( E, B ), the aitational field ( E, B ) is atheatially desibed by a set of fou patial diffeential equations, the GEM-equations (o the Maxwell-Heaiside equations ) that desibe how E and B ay in spae due to thei soues - the asses and the ass flows - and how they ae intetwined. At a point P of a aitational field - whee ρ G is the ass density and J G is the density of the ass flow - E and B ust obey the followin equations [4] : also alled: aito-aneti field
. di E =. di = B 3. ote ρ G η B = t E 4. otb = ν. J G t And: η. ν = with η = 4. π.g Neithe these equations no thei solutions indiate an existene of ausal links between the -field E and the -indution B. Theefoe, we ust onlude that a aitational field is a dual entity hain a field oponent and an indution oponent siultaneously eated by thei oon soues: tie-aiable asses and ass flows.. The aitational inteations an be explained [4] as the effet of the tendeny of a ateial objet to aeleate in ode to beoe blind fo the aitational fields eneated by othe objets. In the ontext of GEM, the ation of the aitational field on a patile is desibed by the foe law of GEM, a law analo to Loentz foe law: A patile, that is oin - elatie to an inetial efeene fae O - with eloity in a aitational field ( E, B ), will aeleate elatie to its pope inetial efeene fae O with an aount a ': a' = E + ( B We an intepet this by sayin that the aitational field exets an ation on a patile in that field. We all that ation the aitational foe G. It is defined as: ) G =. a' =. [ E + ( B )] whee is the est ass of the patile. o this foe law it follows [4] that the effet of the aitational foe on the state of oeent of the patile an be expessed by: dp = dt G a efeene fae oin elatie to O with eloity
3 p =. is the linea oentu of the patile elatie to the inetial efeene fae O. It is the podut of its elatiisti ass = with its eloity in O. The linea oentu of a oin patile is a easue fo its inetia, fo its ability to pesist in its dynai state. GEM an be onsideed as an upade of Newtonian aity. Unlike Newton s law of uniesal aitation, GEM takes the effet of the kineatis of the aitatin objets into aount and it is based on the ideas of otion deeloped in the ontext of speial elatiity. Beause the equations of GEM ae analoue to Maxwell's equations and beause Maxwell's equations ae inaiant unde a Loentz tansfoation, the equations of GEM ae inaiant unde a Loentz tansfoation so that the Piniple of elatiity is alid in the ontext of GEM. o the postulate of the aitational ation, it follows that he sae is tue fo the Piniple of Equialene. It is ipotant to state that, in the ontext of GEM, the field is onsideed as a substantial eleent of natue and not as a puely atheatial onstution. We an the substane of the aitational field identify as -infoation [4]. The statin point of GEM diffes fundaentally fo the statin point of GT beause, in the desiption by GEM of the aitational phenoena and laws, spae and tie don t play an atie ole. It ae eleents of the desiption of natue that do not patiipate in the physial poesses. It has been shown [7],[8] that etain onete peditions ade on the basis of the aitoeletoaneti desiption of aity ae pefetly in line with the esults of osoloial obseations. NEWTON UNIVEAL LAW O GAVITATION The phenoenon of the aitational inteation between two patiles at est is desibed by Newton s uniesal law of aitation [5] : The foe between any two patiles hain asses and sepaated by a distane is an attation atin alon the line joinin the patiles and has the anitude.. = G. =. whee G = is a uniesal onstant hain the sae alue fo all pais of patiles. We will show that this law pefetly an be dedued in the faewok of GEM.
4 P e e P i In fi we onside two patiles with est asses and anhoed at the points P and P of an inetial efeene fae.. eates and aintains a aitational field that at P is defined by the -field: E =. e. 4. π. η. Indeed. The fist GEM-equation - that an be intepeted as the atheatial expession of the onseation of -infoation [4] is equialent to the stateent that the flux of the aitational field thouh an abitay losed sufae is deteined by the enlosed ass in aodin to the law: in E. d = () η Let us apply this equation to an hypothetial sphee with adius enteed on P. - Beause of the syety, E is at eey point of that sphee pependiula to its sufae and has the sae anitude. o, at an abitay point P of the sphee, E an be expessed as E = E whee e and E espetiely ae the unit eto and the oponent (with onstant anitude) of E in the dietion of P P. uthe, at eey point of the sufae of the sphee:. e d = d. e. With this infoation we alulate E. d: E. d E. e. d. e = E. d = E. d = E.4π = () - The enlosed ass is, so
5 Takin into aount () and (3), () beoes: E in = (3).4π η = (4) We onlude: at a point P at a distane fo P the aitational field is pointin to P and deteined by: E = E. e =. e In patiula at the point P: E =. e 4. π. η.. If was fee, aodin to the postulate of the aitational inteation it would aeleate with an aount a : a = o the aitational field of exets a aitational foe on : E.. e = a =. E =. 4. π. η. In a siila anne we find. : =. e = 4. π. η. 3 THE INTEACTION BETWEEN TWO MOVING POINT MAE Z=Z X X O Y O Y i
6 Two patiles with est asses and (fi ) ae anhoed in the inetial fae O that is oin elatie to the inetial fae O with onstant eloity =. ez. The distane between the patiles is. elatie to O the patiles ae at est. Aodin to Newton s law of uniesal aitation, they exet on eah othe equal but opposite foes: ' = ' = '. = G... =. 4. π. η elatie to O both patiles ae oin with onstant speed in the dietion of the Z-axis. o the tansfoation equations between an inetial fae O and anothe inetial fae O, in whih a patile expeienin a foe is instantaneously at est, we an iediately dedue the foe that the point asses exet on eah othe in O [6] is: = = = '. ( ) = '. β We will show that also this fo of Newton s law of uniesal aitation pefetly an be dedued in the faewok of GEM.. Z=Z P=O θ P e O Y X i 3 At a point P whose position is deteined by the tie dependent position eto (fi 3) - the aitational field ( E B, ) of a patile with est ass that is oin with onstant eloity =. ez alon the Z-axis of the inetial efeene fae O (fi 3) is deteined by [4] : w E B β =.. =.. e 3 3 3 =. ( β.sin θ ) 3. β 3 ( β.sin θ).( ) β ( β.sin θ )
7 with β =, the diensionless speed of. One an eify that these expessions satisfy the laws of GEM.. In the inetial fae O of fi, the patiles and ae oin in the dietion of the Z- axis with speed. oes thouh the aitational field eneated by, and oes thouh that eneated by. Aodin the aboe foulas, the anitude of the GEM field eated and aintained by at the position of is deteined by: E =. and B =.. β β And aodin to the foe law.[ E + ( B )] exeted by the aitational field (, E B is: G =,, the anitude of the foe ) on - this is the attation foe of on - =.( E. B ) ' Afte substitution: =.. β =. β In the sae way we find: ' =.. β =. β We onlude that the oin asses attat eah othe with a foe: = = = '. β This esult pefetly aees with that based on..t. 4 CONCLUION o the aboe we an onlude that it s easy to atheatially dedue all aspets of Newton s law fo the laws of GEM. The sin of the esult indiates that the aitational inteation between asses is always attatie. Note that E is pointin to the atual position of the patile and not to its liht-speed delayed position.
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