Lorentz-invariant theory of gravitation

Transcription

1 oentz-invaiant theoy of gavitation (summay) Alexande G. Kyiakos * Annotation This atile is a summay of the non-geometial oentz-invaiant theoy of gavitation (IGT) (efeenes and itations hee allow to familiaize oneself with known esults fom the theoy of gavitation in moe detail). In the famewok of the poposed theoy the physial meaning of the meti tenso and squae of inteval in pseudo-eulidian spae was laified, all the exat solutions of GR wee obtained, the violation on the law of onsevation of enegymomentum was eliminated, as well as othe diffiulties have been oveome. A haateisti featue of the poposed theoy is that it is built on the basis of the quantum field theoy. Abbeviations: IGT - oentz-invaiant gavitation theoy; EM - eletomagneti; EMTG - eletomagneti theoy of gavitation; QFT - quantum field theoy M - tandad Model; QED quantum eletodynamis. GTR o GR Geneal Theoy of Relativity -tansfomation oentz tansfomation -invaiant oentz-invaiant.. Intodution. tatement of the poblem The moden theoy of gavity, whih is alled Geneal Theoy of Relativity (GTR o GR), was veified with suffiient auay and adopted as the basis fo studying of gavitational phenomena in moden physis. Howeve, GR has etain disadvantages (see, e.g., (ogunov and oskutov, 987; Kogdahl, 7)): ) in the geneal ase, the enegy and momentum onsevation is violated; ) it an not be quantized; ) it is based on a geometi basis, unelated to othe existing physial theoies. These defiienies have esulted in the fat that attempts to impove this theoy ae being made. Howeve, as the analysis shows (fo example, see (Feynman, Moínigo and Wagne, )) - thee is no doubt that the GRT ontains elements that will be pesent in any othe theoy of gavitation. It is known that geneal elativity is a lassial theoy. At the same time, the base of the moden appoah to the study of natue is quantum field theoy (QFT); and lassial physis is onsideed to deive fom the quantum theoy. o the question aises if thee is a possibility to impove the gavitation theoy on the basis of QFT? In othe wods, an we get the tested esults and eliminate the diffiulties of GRT, if we put the quantum field theoy in the foundation of the gavitation theoy? Of ouse, fist of all, we should make sue that suh an appoah, has a pospet. What ae the peequisites that allow us to put QFT in the basis of the gavitation theoy? What hallenges need to be oveome fo the onstution of suh a theoy? * aint-petesbug tate Institute of Tehnology, t.petesbug, Russia Pesent addess: Athens, Geee, a.g.kyiakathotmail.om

2 .. Bakgound of the existene of a oentz-invaiant theoy of gavitation based on QFT ) Moden QFT (e.g., in the fom of the tandad Model - M) desibes all the elementay patiles and thei inteations. Moeove, it was found that all the obsevable matte in the univese is omposed by these elementay patiles and the enegy of thei inteation. Thus the elementay patiles and the enegy of thei inteations ae the pimay embodiment of the matte. Gavitation is one of the fundamental popeties of matte. Theefoe elementay patiles ae, so to speak, the pimay "aie" of gavitation. Theefoe, we an assume that thee must be a onnetion between the theoy of elementay patiles and gavitation theoy. ) QFT is oentz-invaiant (-invaiant) theoy. This, in geneal, means that it is invaiant with espet to the oentz goup. Aoding to oethe s theoem, this invaiane povides all the neessay onsevation laws in mehanis. ) Geneal elativity, as a lassial theoy, efes to the maowold, while QFT is the theoy of the miowold. Can the fist theoy be edued to the seond? It is known that the lassial mehanis is a onsequene of quantum mehanis. o ewton's equation of motion an be obtained fom the hodinge equation, and the equation of motion fo a haged patile an be obtained by squaing of the Dia equation. Theefoe, it is possible that by using one of the known aveaging tehniques it is possible to obtain the esults of geneal elativity fom QFT. In addition in this ase thee is hope that gavitation theoy an be assoiated with quantum theoy. 4) Fom expeiene it follows that enegy, momentum, mass, and othe mehanial quantities ae the same in both theoies - geneal elativity and QFT. 5) To pass on to the atual onstution of the theoy, it is equied to ente the physial haateistis of gavitation theoy. One of the key hee is, appaently, the soue of gavitation. The soue of gavity in geneal elativity is the pseudo-tenso of enegy-momentum, whih is the sum of the oentz-invaiant enegy-momentum tenso and some pseudo-tenso (you an ead moe about it below). This pseudo-tenso is not a oentz-invaiant tenso, and, in geneal does not desibe the onsevation laws of physial quantities in GRT. The enegy-momentum tenso of a lassial field theoy ombines the densities and flux densities of enegy and momentum of the fields into one single objet. Howeve, the poblem of giving a onise definition of this objet able to povide the physially oet answe unde all iumstanes, fo an abitay agangian field theoy on an abitay spae-time bakgound, has puzzled physiists fo deades. (Foge and Röme, ). 6) The theoy of gavitation, as we know, efes to the gavitational mass / enegy (gavitational hage). A mass / enegy of the elementay patiles is onsideed to be inetial. The onnetion of the theoy of gavity with QFT hee ensues expeimental fat of equality between gavitational mass/enegy and inetial mass/enegy. Thus, if we will undestand the natue of the inetial mass of elementay patiles, we will also define the natue of the gavitational mass. We will also analyze the diffiulties of the oentz-invaiant theoy when they aise. We will also show how to oveome them. Fist of all, we will disuss in moe detail the diffeenes of the gavitation soues between GR and -invaiant theoy of gavitation based on QFT. The impotant physial onsequenes of the theoy ae assoiated with these diffeenes; in patiula, the ompliane with the enegy and momentum onsevation law.

3 .. The soue of gavitation.. The soue of gavitation in GRT Initially Einstein assumed that the soue of gavity in the Hilbet-Einstein equations is symmeti enegy-momentum tenso T of the oentz-invaiant mehanis satisfying the law of enegy-momentum onsevation: T x, (.) whih oesponds to ten integals of motion of oentz-invaiant mehanis (Fok, 964). The wod "elativity" in the title of geneal elativity suggests that geneal elativity is a elativisti theoy. Einstein assumed that geneal ovaiane of equations eflets some geneal elativity, whih inludes the speial elativity. But the question of whethe the GR equation is elativisti in the sense of oentz-invaiane, is not tivial. It is known that geneal elativity is onsideed as elativisti theoy, but it is not a -invaiant theoy (Katanaev,, pp. 74): «oentz meti satisfies the Einstein s vauum equations. [But] "in GTR is postulated that spae-time meti is not a oentz meti, and is found as a solution of Einstein's equations. Thus, the spae-time is a pseudo-riemanian manifold with meti of a speial type that satisfies the Einstein equations." Fom this it follows that geneal elativity is a geneal ovaiant theoy, but, stitly speaking, it is not a oentz-invaiant theoy (i.e., a theoy within the famewok of RT). It is easy to see that this leads to diffiulties with the law of onsevation of enegy in geneal elativity. As we know, a genealization, that is, the tansition fom the oenz invaiane to the geneal invaiane, is the tansition to abstation of highe level. uh a tansition an be ahieved only by the intodution of new postulates. In patiula, a few ules ae fomulated fo obtaining the geneal-ovaiant expessions fom oentz-invaiant expessions. Often this genealization involves only the eplaement of patial diffeentiation by ovaiant diffeentiation ("omma-goes-to-semiolon ule"); fo example the genealization of the equations T of motion is fom T, T ; ; this latte [tenso], with semiolons, inludes the effets x of gavity. (ightman, Pess et al., 979). As the genealization of T in GR should be the geneal ovaiant deivative, instead (.) we will have (andau and ifshits, 97; Fok, 964): gt T T g x, (.) T ;, and But, it appeas that (andau and ifshitz, 97; 96 The enegy-momentum pseudotenso ) in this fom, howeve, this equation does not geneally expess any onsevation law whateve. As a way out of this situation, Einstein, and othes poposed to intodue an additional tem pseudotenso. It is alled like that beause it is not a tue tenso. Due to t, in some oodinate systems a kind of onsevation law of enegy-momentum an atifiially be fomulated. Violations of this law aise in othe oodinate systems. On this basis, Einstein even poposed to onside that the violation of the law of enegymomentum onsevation is a peuliaity of the gavitational field. In patiula, at least inside the bodes of ou galaxy, ewton's theoy of gavity povides at least 99% of auay in the alulation of gavitational poblems in ompaison with geneal elativity. The law of onsevation of enegy and momentum is fully espeted hee. Theefoe, it is easie to assume that geneal elativity ontains a mistake, than to doubt the esults of ewtonian mehanis.

4 4 Enegy-momentum is an impotant onseved quantity whose definition has been a fous of many investigations in geneal elativity (GR). Unfotunately, thee is still no geneally aepted definition of enegy and momentum in geneal elativity. Attempts aimed at finding a quantity fo desibing distibution of enegy-momentum due to matte, non-gavitational and gavitational fields only esulted in vaious enegy-momentum omplexes (whih ae nontensoial unde geneal oodinate tansfomations) whose physial meaning have been questioned. Moeove (Foge and Röme, ; page 6) the expession in equation. (.) above does not epesent a physial enegy-momentum tenso fo the gavitational field: as is well known, suh an objet does not exist. What we have in the oentz-invaiant (-invaiant) mehanis?.. The soue of gavity in oentz-invaiant mehanis ine QFT is a oentz-invaiant (-invaiant) theoy, the gavitation theoy, built on its basis, will be the -invaiant gavitation theoy (IGT). What an we say about the soue of gavity in suh a theoy? In geneal in the oentz-invaiant mehanis, the elements that make up the enegymomentum tenso T (see above), ae used in theoy. But in patie they ae aely used in the fom of tenso (Fok, 964, 7-9). In fat, the enegy-momentum tenso, whih is inluded in the fomulation of the equations of geneal elativity, is also not used hee as a tenso. The tensoial equation of GR is a shot notation of equations. Eah of these equations ontains only one tem of the enegymomentum tenso. ote that afte its division by the squae of the speed of light, the tenso omponents ae idential to the mass density and the densities of mass flow. Pehaps, in this egad, Fok alled this tenso - the tenso of mass (Fok, 964, ) Theefoe, following to V. Fok (Fok, 964, 54), in fomulating Einstein's theoy we shall likewise stat fom the assumption that the mass distibution is insula. This assumption makes it possible to impose definite limiting onditions at infinity as fo ewtonian theoy, and so makes the mathematial poblem a detemined one. Theoetially, othe assumptions ae also admissible. (As mass distibution of insula haate V. Fok desibes the ase that all the masses of the system studied ae onentated within some finite volume whih is sepaated by vey geat distanes fom all othe masses not foming pat of the system. When these othe masses ae suffiiently fa away One an neglet thei influene on the given system of masses, whih then may be teated as isolated. ) The abovementioned allows us, in famewok of the oentz-invaiant poblem, to all the soue of gavitation, the mass/enegy, o simply mass, implying by this tem all tems of the enegy-momentum tenso of the speifi task. ext, we will analyze the question of what we know about the oigin of the inetial mass/ enegy of elementay patiles as the pimay soues of gavity. 4.. The mass / enegy theoies. Classial and ontempoay point of view Mass emained an essene - pat of the natue of things - fo moe than two entuies, until J.J. Thomson (88), Abaham (9) and oentz (94) sought to intepet the eleton mass as eletomagneti self-enegy, ( Quigg, 7). Theoy, eated by J.J. Thomson and H. oentz (88-96), lies entiely in the field of lassial eletomagneti theoy. Aoding to this theoy, the inetial mass has eletomagneti oigin.

5 5 Unfotunately, attempts to apply this theoy to quantum theoy has not been undetaken. Howeve, until now thee was no evidene of that the inetial mass is not fully eletomagneti (Feynman et al, 964): We only wish to emphasize hee the following points: ) the eletomagneti theoy pedits the existene of an eletomagneti mass, but it also falls on its fae in doing so, beause it does not podue a onsistent theoy and the same is tue with the quantum modifiations; ) thee is expeimental evidene fo the existene of eletomagneti mass; and ) all these masses ae oughly the same as the mass of an eleton. o we ome bak again to the oiginal idea of oentz - may be all the mass of an eleton is puely eletomagneti, maybe the whole.5 MeV is due to eletodynamis. Is it o isn t it? We haven t got a theoy, so we annot say. The moden mass theoy is the, so-alled, Higgs mehanism of the tandad Model theoy (M) (Quigg, 7; Dawson, 999; et). The Higgs mehanism, unde etain assumptions, allows us to desibe the geneation of masses of fundamental elementay patiles: intemediate bosons, leptons and quaks. But as it is mentioned above (Quigg, 7), moe than 98% of the visible mass in the Univese is omposed by the non-fundamental (omposite) patiles: potons, neutons and othe hadons. Thus, the Higgs mehanism an not be used in the gavitation theoy. 4. The oigin of mass/enegy in QFT tating with quantization of Maxwell's theoy of eletomagnetism whih led to the onstution of the QED, physiists have made temendous pogess in undestanding the basi foes and patiles onstituting the physial wold. Moden patile theoies, suh as the tandad model, ae quantum Yang-Mills theoies (Houghton, 5; ielsen, 7). In a quantum field theoy the quanta of the fields ae intepeted as patiles. In a Yang-Mills theoy these fields have an intenal symmety: they ae ated on by a spae-time dependant non-abelian goup tansfomations. These tansfomations ae known as loal gauge tansfomations and Yang-Mills theoies ae also known as non-abelian gauge theoies. Maxwell s equations an be egaded as a Yang-Mills theoy with gauge goup U(). We have the woking enomalizable theoy of stong, eletomagneti and weak inteations... This is of ouse the Yang-Mills theoy Essentially, all that we managed to do is just to genealize quantum eletodynamis (QED). QED was invented aound 99 and sine then has neve hanged... ow QED is genealized and inludes stong and weak inteations along with eletomagneti, quaks and neutinos, along with eletons (Gell-Mann, 985). As we know, these theoies ove all types of elementay patiles and thei inteations. They make up the matte of the univese: massless photons and massive leptons, bosons and hadons. Fo us it is impotant to emphasize that the Yang-Mills equations ae nonlinea genealization of Maxwell's theoy and an be epesented in both lassial, and quantum fom (see, fo example, in details, (Ryde, 985)). Theefoe it an be agued that the mass/enegy of elementay patiles, and thus the whole matte, has eletomagneti oigin. This answes the question of Feynman in the passage above. This also implies that the gavitational mass/enegy and gavitational field also have eletomagneti oigin. Obviously, it follows that the theoy of gavitation, in the geneal ase, must be a vaiant of the nonlinea theoy of eletomagneti field. It is undestood that the new theoy must be - invaiant, sine it is based on the eletomagneti theoy. Can the -invaiant theoy give the same esults as geneal elativity? we will ty to answe this question below. But fist, let us note that the base (not the poof!) to assume that the eletomagneti theoy of gavity is possible, aleady exists in the GR. It will be useful to biefly dwell on this.

6 6 4.. Connetion between GTR and EM theoy Whih onsequene of geneal elativity onfims that the eletomagneti gavitation theoy (EMGT) is possible? Fist of all, let us undeline again that this is not the poof of the existene of EMGT. evetheless, it is a efeene to the possibility of its existene. We ae talking hee about the known solution of the lineaized equations of geneal elativity. Even Einstein himself pointed out some paallels between this appoximate solution and the EM fomulas (Moelle, 95). Aound the 6s these paallels dew moe attention (Fowad, 96; Ruggieo and Tataglia, ; et.). To date is shown an almost omplete ageement of fomulas of lineaized equations of geneal elativity and eletomagneti theoy (the diffeene in some numeial oeffiients, is not to take into aount, a quite undestandable fat in tems of featues of gavity and QFT). In the linea appoximation, the left side of the equation of geneal elativity Rii s uvatue tenso is equal to the D'Alembet opeato (see analysis (Fok, 964, 68 and Annex B)). This opeato ats on the meti tenso with a fist ode of appoximation. Geneal elativity equation, taking into aount the oentz gauge, beomes a d'alembet wave equation with espet to the gavitational 4-potential. This equation up to numeial oeffiients oinides with the EM field equations: fou d'alembet equation: one equation fo sala and equations fo the omponents of veto potential. Moeove, in this ase, the fat that these ae appoximations, athe than exat solutions, is not a eason to ejet the onnetion of GR theoy with EM theoy. Point is that these appoximate solutions wee the fist solutions of Einstein, whih onfimed expeimentally the existene of elativisti effets. Howeve, we emphasize again that this does not pove the existene of EMGT. The eletomagneti epesentation of these solutions is only thei intepetation. We must, independent obtain the exat solutions of geneal elativity exlusively on the basis of eletomagneti theoy and explain the physial meaning of the geometi appaatus of GTR. 5.. Eletomagneti gavitation theoy (EMGT): fomulation of the poblem o, we will ty to onstut a theoy of gavity based on EM theoies of Yang-Mills. And, pobably, it is bette fo the beginning to hoose the simplest of them - the Maxwell-oentz theoy. It is lea that to build the EM gavitation theoy (EMGT) - does not mean to take the EM theoy and use it as a theoy of gavity. Thee ae impotant diffeenes between these theoies. The main hallenges to oveome ae the diffeenes between the eletomagneti and gavitational field: ) the weakness of the gavitational in ompaison with the eletomagneti field, ) its neutality, and ) the absene of epulsion. In addition, thee ae some seeming diffiulties. Equations of eletomagneti fields (inluding a genealization of Yang-Mills) do not ontain mass. This seems to make them unsuitable even fo desibing the ewtonian s theoy of gavity. But let us not foget that these equations ontain the intensities of EM fields in the foe pesentation. The squae and veto podut of these fields ae popotional to the densities of enegy and momentum, espetively. And in the enegy epesentation, they ae dietly haateized by enegy and momentum pe unit of gavitational hage. In ase of self-ating of EM fields, these equations an tigge a mass aoding to the piniple of mass-enegy equivalene. elf-ation is desibed by nonlinea equations, whih is typial fo the Yang-Mills equations. et us eall, that Higgs mehanism geneates the masses due to the self-ating of fields. But thee ae also othe mehanisms of self-ating of the fields. Anothe featue of the EM equations is that they desibe only the veto bosons (e.g., photons). But this does not ontadit with the theoy of gavity beause, as was eognized in geneal elativity, the quanta of gavitational field - gavitons - ae also bosons.

7 7 5.. The hypothesis of esiduality The idea of an eletomagneti oigin of gavitation appeaed sine fomulation of the eletomagneti theoy of matte. The fist hypothesis of explaining the gavity on the basis of eletomagneti theoy was put fowad in 86 by O.Mossotti As shown by futhe analysis, its justifiation was not satisfatoy. The eletomagneti oigin of the mass of all elementay patiles, as well as the weakness of the gavitational field ompaed to the eletomagneti field, allowed to O.F. Mossotti (Mossotti, 96) to assume that the gavitational field is a esidual eletomagneti field. Wilheim Webe of Gottingen and Fiedih Zollne of eipzig developed this oneption into the idea that all pondeable moleules ae assoiations of positively and negatively haged eletial opusles, with the ondition that the foe of attation between opusles of unlike sign is somewhat geate than the foe of epulsion between opusles of like sign. If the foe between two eleti units of like hage at a etain distane is a dynes, and the foe between a positive and a negative unit hage at the same distane is y dynes, then, taking aount of the fat that a neutal atom ontains as muh positive as negative eleti hage, it was found that need only be a quantity of the ode -5 in ode to aount fo gavitation as due to the diffeene between and (Whittakke, 95). At the meeting of the Amstedam Aademy of ienes on Mah 9, oentz ommuniated a pape entitled Consideations on Gavitations on Gavitation, in whih he eviewed the poblem as it appeaed at that time (Whittake, 95). In othe wods, aoding to Mossotti, Webe and Zolneu (Zoellne, We be, and Mossotti, 88) gavitational field - is a esidual eletomagneti field. ate, sientists ome to the onlusion that patiles ae the field, and theefoe, it is neessay to efomulate the hypothesis at the level of eletomagneti fields. As shown by H. oentz (and then othes), it an be done without enteing into onflit with the expeimental fats. At the meeting of the Amstedam Aademy of ienes on Mah 9, oentz ommuniated a pape entitled Consideations on Gavitations on Gavitation, in whih he eviewed the poblem as it appeaed at that time (Whittakke, 95). In the seond half of this atile oenz examined this onept at the field level and eeived enouaging esults. But the final theoy had not been eeived yet. And afte the ouene of the geneal elativity, the inteest in the eletomagneti gavitation theoy was lost (moe infomation an be found in the book: Vizgin, V.P. (Vizgin, 98), hapte "Eletomagneti theoy of gavitation"; whith suffiient bibliogaphy. We will not ty to finish this solution in famewok of QFT, sine the solution of this poblem equies seious analysis and a lot of time. At the same time, this solution is not impotant fo the onvesion of the eletomagneti theoy to the gavitation theoy. We will poeed diffeently: we will onside this idea as a postulate (alled the Mossotti-oentz postulate). Postulate of Mossotti-oentz: the gavitational field is a esidual eletomagneti field, whih emained as a esult of inomplete ompensation of eleti and magneti fields of diffeent polaity. It is easy to hek, if this postulate ontadits the ondition of neessity: an all eletomagneti quantities be ewitten in suh a way that they give the oet fomula of the gavitation theoy (fo example, the fomula fo ewton's foe, field, enegy, et.); We must also hek the dimensions of all obtained haateistis of the theoy of gavitation (gavitational hage, the field intensity, potentials of the field, et.).

8 8 In pat, this has been done befoe by many sientists. A detailed veifiation within ou theoy shows that no ontadition aises. Below we show these tansfomations on an impotant example. 5. ewton's law of gavitation, as a esult of EM theoy If we assume that gavity is geneated by eleti field, but quantitatively, by vey small pat of it, then ewton s gavitation law: m M F, (5.) should take the fom of Coulomb's law: q Q F C k, (5.) whee m and q ae the mass and eleti hage of the patile, M and Q ae the mass and eleti hage of the soue. In Gauss s units =6,67 8 см³/(г с²). is ewton's gavitational onstant, and the oeffiient k is k. We intodue the gavitational hages q g and Q g, oesponding to mass m and M (Ivanenko and okolov, 949), by means of the elations: q q m, Q Q M (5.) g g In this ase, ewton's law an be ewitten in the fom of Coulomb's law: qg Qg m M Fg F, (5.4) imilaly, we an tansfom all othe EM quantities in the gavitational quantities. If we talk speifially about the weakness o stength of the field, the intensity of the inteation is usually haateized by some bond onstant. In the EM theoy as suh a onstant an be onsideed k in Coulomb aw fo vauum. In the CGE system units, the hage is seleted so that k The gavitational onstant in CG is appoximately = m /(g s²). It is haateized by ( ) / in the Mossotti et al. appoah. 5.. The elativization of the ewton law Above we have eeived a non-elativisti equation of ewton's gavity. But the oentz invaiane of EM theoy pomises the existene of a -invaiant vesion of this equation. The motion of haged patiles in the non-elativisti ase is pefomed unde the ation of the non-elativisti Coulomb foe. This allowed us to obtain the foe of ewton. Both of these foes oespond to the stationay soue. But how an we give them a elativisti fom oesponding to motion of the soue? Obviously, we need to stat with the Coulomb foe. It tuns out that Coulomb foe takes the -invaiant fom, if we add the oentz magneti foe. In this ase, the total foe ating on the hage, onsists of two tems, and alled a (omplete) oentz foe. It is lea that a simila tem needs to be added to the foe of ewton, to take a elativisti fom. What is needed to be done fo this? The most signifiant hee is that in both ases, it is not neessay to add this tem atifiially, e.g., by a postulate. A stiking unifiation of eletomagneti theoy was published in 9 by eigh Page. It had been ealized long befoe by Piestley that fom the expeimental fat that thee is no eleti foe in the spae inside a haged losed hollow onduto, it is possible to dedue the law of the invese squae between eleti hages, and so the whole siene of eletostatis. It was now shown by Page that if a knowledge of the elativity theoy of Poinae and oentz is assumed, the effet of eleti hages in motion an be dedued fom a knowledge of thei behavio when

9 9 at est, and thus the existene of magneti foe may be infeed fom eletostatis: magneti foe is in fat meely a name intodued in ode to desibe those tems in the pondeomotive foe on an eleton whih depend on its veloity. In this way Page showed that Ampee s law fo the foe between uent-elements, Faaday s law of the indution of uents and the whole of the Maxwellian eletomagneti theoy, an be deived fom. (Whittakke, 95) It tuns out that in the eletomagneti theoy, the additional tem to the Coulomb foe aises automatially due to the oentz tansfomations. As is known, the magneti field ous when the hage moves. Duing this the Coulomb field emains unhanged. But if we apply the oentz tansfomation to eleti hage then automatially due to spatial ompession of hages, a magneti field aises, as well as an additional tem of the magneti foe (see. Details (Puell, 975)). It is not diffiult to tansfe this esult to gavitation theoy. As is known, GTRatually ontains a field whih we an name gavito-magnetial. But in ou appoah, it is a onsequene of -invaiane, not of geneal ovaiane and the Riemann spaetime. The equations of EM field an be witten though the field stengths, but an also be witten though the potentials - sala and veto. These equations, taking into aount the oenz gauge ondition A, ae inhomogeneous equations of d'alembet in the 4-dimensional с t spae-time, whih ae fully equivalent to Maxwell's equations: A 4 A, (5.4) с t whee A is 4-potential. The equations of ewton's theoy of gavity an also be witten in these two foms. And the enegy fom of equation in the stationay state oesponds to the Poisson equation, whih is the stationay limit of the GR equation. Obviously, afte the tansition to the elativisti fom of ewton's law, we will obtain an equation whih is mathematially idential to the EM field equation and also ontain the veto potential. As we know, the geat ahievement of geneal elativity is the pesene in its equation of the veto potential. 6.. Geomety and physis of geneal elativity and the -invaiant theoy of gavitation 6.. Fomulation of the poblem in geneal elativity The patial side of the Einstein-Hilbet theoy is following: "All the peditions of geneal elativity follow fom: ) the solution of the field equations: R Rg T, (6.) 8 whee, 4 R x x is the Rii uvatue tenso, ae the Chistoffel symbols, R is the sala uvatue, is the speed of light in vauum, T is the stess enegy tenso, and g is the meti tenso of Riemannian spae; ) the law of motion in fom of geodesi line equation o the Hamilton-Jaobi equation fo a massive body: g x x m, (6.)

10 ) the postulat that in the Riemann geomety the meti tenso gavitational field - GR g. g is a funtion of the 4) the equiement that the equations of geneal elativity in the non-elativisti ase is edued to the equation of ewton's gavity (in the fom of Poisson's equation) The equation (6.) allows to detemine g. The non-elativisti limit allows to intodue in this solution the ewton gavitational potential and so allows fom g to obtain g GR. Putting this value in (6.) we obtain the solution of given poblem of the body motion in the gavitation field of a soue. ine the meti tenso is ontained in the squae of inteval of Riemannian spae: ds g, (6.) it is often said that the pupose of solution of equation (6.) is to find the inteval (6.). It is often said that inteval in TR is a genealization of inteval of Eulidean geomety on pseudo-eulidean geomety. In tun, the inteval in geneal elativity is a genealization of inteval of pseudo-eulidean geomety on pseudo-riemannian geomety. But it is easy to make sue, that the intodution of inteval in TR and GTR is a postulates athe than a logial onlusion. Indeed, the intevals in TR and GTR ae a genealization of inteval of Eulidean geomety, but the eason fo the intodution of these new intevals is not geomety, but physis. Obviously, the patial pat of the GRT ideas must be enteed into any new theoy of gavitation. It follows that within the famewok of ou fomulation of the poblem (i.e., within the QFT and namely, EM theoy) it is desiable to find out the funtional meaning of the meti tenso and the inteval. ext it is also neessay to obtain the equation of motion, to find out the onnetion of the meti tenso with the field of gavity. et us ty to implement this pogam. 6.. Deivation of the pseudo-eulidean inteval fom QFT The vetos of the oentz-invaiant (i.e., elativisti) theoies neessaily depend on fou oodinate: one time oodinate and thee spae oodinates. Does these theoies ontain the equations, whih have a sum of tems, eah of whih is assoiated with one of the fou oodinates, like as in the squae of the inteval? As we know, in the fist time suh equations in lassial eletodynamis appea, and then in quantum field theoy. The wave equations of these theoies inlude a sum of tems, eah of whih is assoiated with one of the vaiables t, x, y, z. It would be logial, to seek the ause and the meaning of the appeaane of 4-inteval in them, instead of intoduing them atifiially, as did Minkowski. The well-known elation between the enegy, momentum and mass of elementay patiles follows fom the wave equation of the patiles: o in the Catesian oodinate system: 4 p m, (6.4) p x p y p ine p i m i m i dt ewitten as: z m, a m, (whee, (6.4') and ae the oentz fato and anti-fato, espetively), this elation an be dt dy dz, (6.5) dt Multiplying it by, we get:

11 dt dt dy dz, (6.6) Taking into aount dt dt ds witten as squae of a 4-inteval: ds dt dy dz, the expession (.7) an be, (6.7) In the ase of genealized uvilinea othogonal oodinate, this inteval takes the fom ds g : whee g is the meti tenso that is not assoiated with gavity (setion 6.4 is dediated to laifying its physial meaning). Obviously, if we go in the opposite dietion, we an obtain the equation (6.4 ) fom the squae of the inteval (6.7). This implies, fistly, that these equations - (6.4) and (6.7) - losely bind the massive elementay patiles physis and geomety. Fom this it follows that (6.7) is not a meti of pseudo-eulidean geomety, but it is a meti of Eulidean geomety that desibes the oentz-invaiant field equations. The only hange in the geomety, whih we an obseve in this ase is the tansition fom etilinea to uvilinea geomety. 6.. The deivation of motion equation in famewok of QFT In addition, anothe link between the inteval (6.4) and the physial equation is deteted. As we have shown in haptes 6, using the hödinge definition of ation ( p x ), fom the equation (6.4 ) it is easy obtain oentz-invaiant Hamilton-Jaobi equation in geneal view. Fo this it is enough to wite the equation (6.4 ) in a fom, suitable fo any of the Eulidean oodinate system: g p p m, (6.8) whee, we eall, g is the meti tenso of geometial spae, but not of the gavitational spae-time of geneal elativity (in othe wods, in this ase the tenso g does not inlude the physial haateistis of the field). In this ase the Hamilton-Jaobi equation of fee patiles obtains the fom: g x x m, (6.9) Thus, we onlude that the thee equations (6.4), (6.7) and (6.9) ae losely bonded to eah othe and, in fat, follow fom one diffeential equation. Fom this follows that the inteval (6.7) within a elativisti physis is the physial law, and not a geometi elation. Below we will onside the physial meaning of the meti tenso in the famewok of QFT 6.4. The physial sense of the meti tenso Reall the tansition fom Catesian s system of oodinates to the genealized oodinate system (Kon and Kon, 968). et us intodue a new set of oodinates q, q, q, so that among x, y, z and q, q, q thee ae some elations: q, q,. q, y yq, q,. q, z zq, q q x x, (6.), The diffeentials ae then x x x dq dq dq, (6.) q q q and the same fo dy and dz. In Catesian oodinates the measue of distane, o meti, in a given oodinate system is the a length ds, whih is defined by

12 ds dy dz, 6.) In geneal, taking into aount (6.), fom (6.) we obtain ds gdq gdqdq... gijdqidq j, (6.) whee whee g ij is the meti tenso. Thus in othogonal system we an wite h i s ae: ds h dq h dq h dq ij, (6.4) x y z h i, (6.5) qi qi qi ae alled ame oeffiients o sale fatos, and ae fo Catesian oodinates. Thus, the meti tenso, eoded in oodinates q i, is a diagonal matix whose diagonal ontains the squaes of ame oeffiients: Fo example, in the ase of spheial oodinates, the bond of spheial oodinates with Catesian is given by: x sin os, y sin sin, z os, (6.6) The ame oeffiients in this ase ae equal to: diffeential of a (inteval) is: ds h, h, h sin, and the squae of the d d sin d, (6.7) ine the meti tenso is detemined by means of ame oeffiients, let us eall the geometi meaning of the latte: the ame oeffiients show how many units of length ae ontained in the unit of length of segment of oodinates of the given point, and used to tansfom vetos when tansition fom one system to anothe takes plae. This means that the meti tenso in Eulidean geomety defines esaling of thee oodinates,,, and in the pseudo-eulidean o pseudo-riemannian geomety it detemines esaling of fou oodinates t,,,. Thus, the elements of the meti tenso g allow the hanging of the pojetions of body tajetoy segment on the oodinate axes duing the tansition fom the Catesian oodinate system to anothe. In the Catesian system, all elements equal to one. In othe systems, takes plae the inommensuability of uve lines elative to the staight, simila to the inommensuability of the diamete of iumfeene in elation to its length. Theefoe, g elements ae appeaed othes than. This analysis aises a question, the answe to whih, in patie, detemines the alulation of the meti tenso in famewok of ITG: what hanges of the sales of the oodinates follow fom -invaiant tansfomations? Fist of all, we ae talking about hanging the sales of t and oodinates. As it is known, the sale hanges of t and in the -invaiant mehanis, ae aused by the effets of time dilation and length ontation of oentz-fitzgeald. Fom the peeding analysis follows that in aodane with the laws of natue the inteval of 4 spae-time an be obtained only fo the pseudo-eulidean spae-time as an embodiment of the physial law of motion of elementay patiles. ine theeis no othe motion law fo massive patiles, we an assume that the meti tenso of GTR has the same physial meaning.

13 Thus, we onlude that the equations of Hilbet-Einstein and Hamilton-Jaobi ontain g as a fato that takes into aount the hange in the sales of time and distane, due to -invaiant effets aising fom the motion of bodies. Indeed, it an be shown that the squaes of the intevals, oesponding to the exat solutions of the Hilbet-Einstein equations, ae defined by -invaiant effets The squae of inteval of -invaiant gavitation theoy in the ase of a hange of sales of the spae oodinates and time We have shown that MT elements ae detemined by the ame oeffiients. The linea a element in the -dimensional mehanis is expessed though ame's sale fatos in the fom of linea elements: ds hi i h h h, (6.8) i whee x i x, x, x, i,,.. In a Catesian oodinate system x i x, y, z, and all the ame oeffiients equal to one. In the -invaiant mehanis it is impossible to ente the line element of the a sine the physial equation, fom whih follows the magnitude of the a, onnets the squaes of the enegy, momentum and mass, and not the fist degees of these values. The exat expession is obtained in the fom of the squae of length of a element, whih is often efeed to simply as an inteval. In the 4-geomety it is of the fom: ds h h h h h o, taking into aount that, (6.9) h h, we eeive fom (6.9) the fom: ds, (6.9 ) whee is meti tenso in IGT. ow let us onside whih view takes the squae of the inteval in the onete patiula ase of the hange of sales of the spae and time oodinate Time dilation and length ontation as a hange of the sales of time and spae oodinates Using the definition of the meti tenso in ITG given above, let us alulate it in the simplest ase. Conside (Pauli, 958) oentz tansfomation in the tansition fom the oodinate system K to K', whih is uently moving at a speed along the axis x. In this ase only the oodinate x and time t undego tansfomations. The oentz effets of length ontation and time dilation ae the simplest onsequenes of the oentz tansfomation fomulae, and thus also of the two basi assumptions of RT. t' x' x' t ' x, y y', z z', t, (6.) The tansfomation whih is the invese of (6.) an be obtained by eplaing by : t' x' x' t ' x, y y', z z', t, (6.а) Take a od lying along the x-axis, at est in efeene system K. The position oodinates of its ends, x and x ae thus independent of t and x x l is the est length of the od. On

14 4 the othe hand, we might detemine the length of the od in system K' in the following way. We find x and x as funtions of t. Then the distane between the two points whih oinide simultaneously with the end points of the od in system K will be alled the length l of the od in the moving system: x t x t l ine these positions ae not taken up simultaneously in system K, it annot be expeted that l equals l. In fat, it follows fom (6.) : and theefoe x. x t ; l t' x x t t' l fo infinitesimal time intevals of length has fom Fom hee the saling fato of the oentz tansfomation of oodinates (denote it as k x ) will be equal to: k x, (6.) The oesponding element xx of the meti tenso of the oentz tansfomation will be: xx ' ', (6.) The od is theefoe ontated in the atio :, as was aleady assumed by oentz. It theefoe follows that the oentz ontation is not a popety of a single measuing od taken by itself, but is a eipoal elation between two suh ods moving elatively to eah othe, and this elation is in piniple obsevable. Analogously, the time sale is hanged by the motion. et us again onside a lok whih is at est in K. The time t whih it indiates in x is its pope time, and we an put its oodinate x' equal to zeo. It then follows fom (6.a) that t, whih fo dt infinitesimal time intevals dt give: dt. Fom hee the saling fato of the oentz tansfomation of time (denote it as k t ) will be equal to: dt kt, (6.) dt The oesponding element xx of the meti tenso of the oentz tansfomation will be: dt' dt' tt, (6.4) dt dt Measued in the time sale of K, theefoe, a lok moving with veloity will lag behind one at est in K in the atio :. While this onsequene' of the oentz tansfomation was aleady impliitly ontained in oentz's and Poinae's esults, it eeived its fist lea statement only by Einstein. Then, in famewok of ITG the squae inteval will be as follows:

15 5 ds,(6.5) whee is the geometi meti tenso in IGT (tenso of pseudo-eulidian spae); is the physial meti tenso in IGT. Using the values tt и xx ( ) aoding to (6.) and (6.4), we obtain in the Catesian system of oodinates: o ds dt dy dz, (6.6) ds dt ( ) dy dz, (6.6 ) 6.7. Connetion of the meti tenso of GR with the gavitational field Fistly, let us efe to GR fo guidane, sine a simila poblem aises also in the solution of the equations of geneal elativity (see. (andau and ifshitz, 97,. «The entally symmeti gavitational field"). et us skip the details, ontained in this book and fous on the question of ou inteest. et us stat fom the point when the squae of the inteval is speified. Hee, in the squae of the inteval, the designation h oesponds to MT element xx, and l oesponds to the element tt, and it is assumed that h e, as well l e, whee and ae some funtions of and t. Afte substituting these exponents into the vauum equation of geneal elativity R and its integation, we get that e e onst. At that, the onstant onst emains unknown (i.e., it is not defined by the solution of the equation of geneal elativity). Aoding to hwazshild solution, this onstant an be easily expessed though a mass body, equiing that at lage distanes, whee the field is weak, the law of ewton was oppeating. Fom the esults of the peliminay analysis (see (andau and ifshitz, 97), 87 «Motion of a patile in a gavitational field" and 8 «Piniple of least ation."), g g, tt whee is the ewtonian potential. In ompliane with this, we assume that onst. Howeve, by taing the sequene of this analysis, we an easily onfim that the "deivation" of expession g is not onneted with geneal elativity. This means that the deivation of hwazshild s solution is also found by tial and eo method. Howeve, this solution is poved to be oet, and shows us that the non-elativisti theoy of gavitation of ewton is the basis of elativisti theoy of gavitation fo the hwazshild poblem. In othe wods, the ewton solution is the zeo appoximation of the poblem, and the elativisti theoy should only add mino hanges to this esult in aodane with petubation theoy Global and loal oentz tansfomation It is inteesting that the equiement to elate the veloity of the body with the gavitational field in famewok of IGT also follows fom the fat of the equivalene of gavitational and inetial mass. As we know, the usual oentz tansfomations in lassial elativisti mehanis ae global; that is they take plae fo all spatial and tempoal spae-time points. Based on the fato of the equivalene of gavitational and inetial mass, Einstein and othes showed, that oentz's tansfomation of gavitation theoy must be aied out loally, that is,

16 6 independently at eah point of spae-time. Thus, the question aises, what ae these loal - tansfomation? et us use the analogy with phase invaiane in quantum field theoy, bette known as "gauge invaiane". The phase invaiane is the invaiane of wave equations egading the intenal otation of the patile field. As it is known, in quantum field theoy exist both global, and loal phase (gauge) invaiane. Moeove, all suesses in QFT afte the eation of QED ae linked to the tansition fom the fist to the seond. Global invaiane is disoveed though the addition of a onstant omplex value i to the phase of the wave funtion of the patile, whee is independent of plae and time: onst. Physial quantities, alulated aoding to the new wave funtions ae the same as they wee befoe the intodution of this quantity (this fat means invaiane). ate, loal invaiane was intodued to QFT, i.e., the invaiane that is tue only at evey point of spae and time, but not fo the entie spae-time as a whole. It is haateized by a vaiable alpha: ( x, y, z, t). In the -invaiant theoy we ae talking about the oentz tansfomations at the inetial motion. This is a global tansfomation: in elation to the spatial oodinates x x ( ) and similaly fo the time t t' ( ), whee is the patile speed onstant. Aoding to Einstein's equivalene piniple, the gavitational field is equivalent to the non-inetial motion of the patile (movement with a vaiable speed). Theefoe, we must intodue the oentz tansfomations fo non-inetial motion, by taking into aount the loal inteation of the field with the patile. The obvious way to do this is desibed futhe. We intodue loal tansfomation though tansition to infinitely small distanes and times: ' ( ) and dt dt' ( ), in whih ( x, y, z, t) is the vaiable speed of patile motion. This tansfomation does not violate any laws of physis, but makes them loal in spae and time. Thei oetness is veified by esults, whih ae onfimed by expeiment. In this ase, thei use allows us to alulate the elements of the MT, idential to those that we have fom the solutions of geneal elativity (this is the topi disussed in the following setions). ote that aoding to Poinae and ommefeld, oentz tansfomations desibe invaiane in elation to otations in 4-dimensional spae. This bings them lose to phase tansfomations. In addition, they an be eoded though hypeboli (i.e., exponential) funtions. The question of how fa these analogies go, equies a sepaate analysis The onnetion of motion veloity of the body in the gavitation field with haateistis of the gavitation field in the famewok of IGT In etion 6.6 we noted that fo the bond of the meti tenso with gavitational field, the speed of a body in the oentz tansfomation must depend on the gavitational field. In the pevious setion we ame to the onlusion that the loal -invaiane equies this ate to be vaiable. We show that both these equiements an be met within IGT. et us onside the equivalene piniple of gavitational and inetial masses. Begin with the ewtonian law of motion of a patile with inetial mass m in in a gavitational field of soue with a mass M : d mg M min, (6.7) dt whee m is gavitational mass. ine m m m, then dividing (.) by m we obtain in the g in g ase of gavitation the movement equation of the fom: d M, (6.7 ) dt whee aeleation is on the left and the ewton foe pe unit mass is on the ight. It is easy to see that this equation is the mathematial expession of Einstein's piniple of equivalene of gavitational and inetial foes of Einstein. As we know, on the basis of this '

17 7 piniple, Einstein onluded that spae should be heteogeneous and gavity must be desibed as a uved spae-time of Riemann. The question aises whethe it is possible to give anothe explanation to this piniple. It appeas, that based on the same mathematis, we an atually find this onnetion. As is known, the equation (6.7') an be epesented in the enegy fom. Fo this let us ewite the ewton's motion law in the fom: M d dt, (6.8) Multiplying the left and ight hand side of equation (6.8) on the speed, and taking into aount that d dt d d, we have fom (6.8) afte integation: and M onst, (6.9) whee m is the kineti enegy of the moving patile pe unit mass, and M m is the potential enegy of a patile pe unit mass at a given point of the pot gavitational field. Thus, taking into aount the postulate of equivalene and the expession fo the potential of the gavitational field M, we obtain fom (.), the elationship between the veloity of the patile and potential of the gavitational field at the position of the patile: onst, (6.) If at the initial moment a patile was at est, and the motion is only aied out via the potential enegy outlay, then duing the whole peiod of motion onst =. Fo example, this ous when the efeene fame, that is elated to the obseve, falls feely to the ente of gavity soue along the adius (adial infall) fom infinity, whee it had a zeo veloity. In this ase, we have: M, (6.) Thus, as a mathematial onsequene of ewton's theoy of gavity, we have eeived anothe intepetation of the fat of the equality of inetial and gavitational mass. Following the example of Einstein's equivalene piniple, it an be expessed as follows: the potential of the gavitational field is equivalent to the squae of the veloity of the motion of patiles in this field. In addition, (see hapte 4) the eletomagneti basis of gavitational equations allows one to wite the veto potential of the gavitational field though the sala potential A ( ). Expession (6.) was obtained on the basis of non-elativisti enegy onsevation law (6.9). Obviously, to obtain the post-ewtonian oetions to (6.), it is neessay to tansition to the elativisti law of onsevation (see moe details in the full vesion of the theoy). 6.. Deivation of the hwazshild meti It is easy to see that by substituting (6.) in the -invaiant squae of the inteval (6.6 ), obtained in paagaph 6.6, we obtain the inteval of hwazshild-doste: d ds dt sin d d, (6.) Thus, we have, indeed, eeived the fist tested and most impotant esult of GTR, only in famewok of the -invaiant gavitation theoy.

18 8 7.. The Keple poblem The Keple poblem is the poblem of motion of a body of litle mass in a entally symmeti gavitational field of a stationay soue of geat mass. 7.. Diet solution of the Keple poblem in the famewok of IGT As the motion equation of ITG we use the Hamilton-Jaobi equation. As we have shown above, the equation of motion of Hamilton-Jaobi has a one-to-one onnetion with the squae of the inteval (squae of a element of tajetoy) in famewok of ITG. Theefoe, as we will show below, it is not neessaily to find an appopiate inteval to wite the oesponding Hamilton-Jaobi equation fo patile motion in gavitation field. Aoding to ou esults all featues of the motion of matte in the gavitational field owed thei oigin to effets assoiated with the oentz tansfomations. Two of the most impotant effets fom the point of view of mehanis that aise due to the oentz tansfomations, ae the oentzian time dilation and ontation of lengths: ~ dt t d, ~ d d, (7.) whee, as shown peviously,, and is the hwazshild adius. 7.. The equation of motion of a patile in a gavitational field with the oentz time dilation and length ontation We will use the Hamilton-Jaobi equation (7.) in fom: sin ~ ~ m t, (7.) ubstituting in (7.) ~ dt t d and ~ d d, we obtain: 4 sin m t, (7.) Taking into aount that in ou theoy s, we obtain fom (7.) the well-known Hamilton-Jaobi equation fo geneal elativity in the ase of the hwazshild-doste meti (hwazshild, 96; Doste, 97): 4 sin m t s s, (7.) As is known (andau and ifshitz, 97) the tem ' t s (whih ontains the oentz time dilation effet) in the lassial appoximation leads to the equation of motion with ewton's gavitational enegy. Fom this it follows that the peession of the obit ensue the intodution of an additional tem s. As is known, the Keple poblem solution, based on this equation, gives an additional tem in the enegy, whih is missing in ewton's theoy: