Geometry and Physic in Gravitation Theory

Transcription

1 48 Artle Geometr and Phs n Gravtaton Theor Aleander G. Krakos * Abstrat Ths artle s devoted to analss of the relaton of geometral and phsal quanttes n the Newtonan theor of gravtaton, general relatvt and orent-nvarant gravtaton theor (IGT), and also to larfaton of the phsal meanng of the metr tensor and the spae-tme nterval n the Euldean, pseudo-euldean and pseudo-remannan spaes. The suesson of the use of geometr onepts n these three theores s shown. It s shown that the math epresson of nterval s mutuall unquel assoated wth phsal equatons of elementar partles and IGT. It s also shown that n IGT the metr tensor has the phsal meanng of the sale fator, defned b means of the orent-nvarant transformatons. Evdene are gven of that the metr tensor n general relatvt should have the same meanng. Ke wor: Euldean spae and tme; pseudo-euldean spae-tme; pseudo-remanan spae-tme. Introduton: Geometr and Phs n the GR Equaton The Ensten-Hlbert feld equatons ma be wrtten n the form: R Rg T, (.) The pratal sde of the Ensten-Hlbert theor (Tonnelat, 965/966) s followng: All the predtons of general relatvt follow from the feld equatons: G g, g, g T m, u g, (.) 8G where G R Rg,, 4 R s the R urvature tensor, are the Chrstoffel smbols, R s the salar urvature, N s Newton's gravtatonal onstant, s the speed of lght n vauum and T s the stress energ tensor. and g s the metr tensor of Remannan spae, and ) The law of moton (geodes equaton) for a massless bod (photon): 0, (.) or the Hamlton-Jaob equaton for a massve bod (andau and fsht, 97): * Correspondene: Aleander G. Krakos, Sant-Petersburg State Insttute of Tehnolog, St.-Petersburg, Russa. Present address: Athens, Greee, E-mal: a.g.krak@hotmal.om ISSN: 5-80 Prespaetme Journal Publshed b QuantumDream, In.

2 48 g k S S k m The equaton (..) allows to determne 0, (.4) g and to put ths value n (.). Sne the metr tensor s ontaned n the square of nterval of Remannan spae: g d d, (.5) t s often sad that the purpose of soluton of equaton (.) s to fnd the nterval (.5). Dependng on the tpe of energ-momentum tensor the solutons of (.) an be dvded nto several tpes. The most mportant of them are the vauum solutons, sne t was possble to verf some of them epermentall. Suh solutons an be obtaned from the equaton (.), f the energmomentum tensor vanshes: T 0. These solutons desrbe the empt spae-tme around a massve ompat soure of the gravtatonal feld, down to ts surfae or sngulartes. These nlude the Shwarshld metr, the ense- Thrrng, Kerr, Ressner - Nortrom, Kerr - Newman and others metrs. In general relatvt, a vauum solutons are a orentan manfold,.e., the relate to asmptotall flat spae-tme. A orentan manfold s an mportant speal ase of a pseudo-remannan manfold n whh metr s alled orentan metr or the pseudo-euldan metr of speal relatvt. In the vauum equatons of general relatvt onl the left sde s used - purel geometr part of ths equaton. At the same tme, the larfaton of the phsal meanng of sgnfant elements of the metr tensor (MT) takes plae on the bass of a omparson wth Newton's theor of gravtaton. Htherto, the queston, wh a purel geometral funtons produe phsal results, has not larfed. In other wor, we do not know, how the MT s assoated wth phss. The bass for the ntroduton and use of MT s the nterval (often the are dental). Then the queston an be reformulated n a dfferent wa: how nterval and MT n ths omposton relates to phss? It s often sad that nterval n SRT s a generalaton of nterval of Euldean geometr on pseudo- Euldean geometr. In turn, the nterval n general relatvt s a generalaton of nterval of pseudo- Euldean geometr on pseudo-remannan geometr. But t s eas to make sure, that the ntroduton of nterval n SRT and GTR s a postulates rather than a logal onluson. The ntervals n SRT and GRT are a generalaton of nterval of Euldean geometr. And the reason for the ntroduton of these new ntervals s not geometr, but phss. Then what was postulated and on what bass dd t take plae n eah of these ases? ISSN: 5-80 Prespaetme Journal Publshed b QuantumDream, In.

3 484. Geometr and Phss of Newtonan Mehans (Euldean Spae) et us begn from the relaton of the Euldean nterval wth phss. To do ths, we need to reall the meanng of the geometral pont and the materal pont, as well as of the geometr and phsal traetor, as the lne of moton of a materal pont. The lne n geometr s an ndependent geometr obet, almost not related to phss. Not strtl speakng, the lne s a ontnuum (ontnuous sequene) of dots, for eah adaent par of whh the same relatonshp s set. If ths relatonshp an alwas be redued to a onstant number, a lne s alled Euldean; f ths relatonshp s a funton of the poston on the lne, the lne s alled Remannan. The lne n geometr s defned (desrbed) b spefng oordnates,.e., some of the numbered lnes, whh are spefed b the loaton of the materal ponts (obets) of the real world. In phss, the lne s a ontnuum of ponts, whh a materal pont passes suessvel whle movng b nerta or under the nfluene of fores. And ths lne s determned b the law of moton of a materal pont wth respet to the others, outsder materal ponts whh allows to establsh a base oordnate sstem of lnes. Namel here, geometr omes n ontat wth phss. The nterval n Euldean geometr s a generaled desrpton of Pthagoras theorem for an nfntesmal segment of lne (ar): square of the length of an lne segment s equal to the sum of the squares of the proetons of the segment on the three oordnate lnes. The obetves of the geometr, whh requres the use of ths law, has no onneton wth phss. But for the traetor of a materal pont the theorem of Pthagoras s some ondton - restrtve law, whh must take plae n an problem of the moton of materal bod. Condtonall speakng, the law of Pthagoras must be ontaned n the law of moton. Obvousl, ths one-to-one relatonshp should allow to restore the movement law b means of the known nterval. Appromatel n ths manner the problem s set on the theor of gravtaton of Hlbert and Ensten. Atuall, the nterval at an pont of the traetor of moton of a pont must be mutuall unquel assoated wth the soluton of the dnam (phsal) problem. Otherwse the deson wll be wrong,.e., the traetor wll not be one that s dtated b the law of moton. But ths bond an not be assoated wth a oordnate sstem, sne the latter s not related to the phsal problem, and t an be hosen n man was. Ths bond must our n an oordnate sstem, n whh the law of Pthagoras ats. In ths ase, the ntroduton and the hoe of the oordnate sstem s a agreement, requred for a quanttatve alulaton of the phsal problem. et us demonstrate the orretness of our onluson n the framework of non-relatvst and then relatvst (.e., the orent-nvarant) mehans... Cartesan sstem of oordnates Subet of mehans (see (Webster, 9) ) s stud of moton n spae and tme of the matter partle or sstem of partles, as sold bod, under the aton of fores. ISSN: 5-80 Prespaetme Journal Publshed b QuantumDream, In.

4 485 Sne the moton desrpton of a materal pont nvolves four varables alled b arange geometr of four dmensons.,,, t, knemats was Suppose that we have a sstem of n materal ponts. If the are free to move, a sngle partle requres oordnates,,, and a sstem of partles requre n oordnates:,,,,,,, n, n, n If an partle at,, s plaed b a small amount, t has the oordnates d, d, d If a number of partles are plaed, we must take the sums lke the above for all the partles. The nfntesmal tane between two ponts d d d, (.) s a salar, whereas the geometral dfferene n poston of the two ponts s known onl when we spef not merel the length, but also the dreton of the lne onng them. Ths s usuall done b gvng ts length s and the osnes of the angles,, made b the lne wth the three retangular aes, os,os, os, whh n vrtue of the relaton leaves three ndependent data. os os os, (.) We ma otherwse make the spefaton b gvng the three proetons of the lne upon the oordnate aes: sos d, sos d, sos d, (.) Squarng and addng we have n vrtue of relaton (.):, (.4) The quanttes d, d, d are the dreton osnes of the tangent to the ar. The vetor denned b the produt of the salar quantt mass b the vetor quantt aeleraton (vetor quantt), whose omponents are d d d F m, F m, F m, (.5) dt dt dt s alled the fore atng upon the bod, and s the appled fore of the Newton seond law. The seond and thrd laws taken together aordngl gve us a omplete defnton and mode of measurement of fore. It s ustomar to haratere the produt of the mass b the vetor velot as the momentum of the bod, a vetor whose omponents are ISSN: 5-80 Prespaetme Journal Publshed b QuantumDream, In.

5 486 d d d p m m, p m m, p m m, (.6) dt dt dt Ths s the momentum whose rate of hange measures the fore, so that equatons (.5) ma be wrtten dp F dp dp, F, F, (.7) dt dt dt These equatons are a generalaton of equaton (.5), sne the ma be appled n the ase when mass m hanges, for eample, n the ase the engne of the roket s runnng. d d d T m m, (.8) dt dt dt the half-sum of the produts of the mass of partle b the square of ts velot, s alled the knet energ of the partle T. If we have a sstem of n materal ponts then: n n d d d T m m, (.9) dt dt dt The knet energ ma be wrtten, bearng n mnd the defnton of momentum, as: T m It s easl to see: whene d T dt p p p m p p p p m d d d d m F F F, dt dt dt dt dt F d F d F d s the work done upon the partle at reloaton t on nfntesmal tane. The equaton, (.0) T T F d F d F d, (.) t t 0 t t 0 s alled the equaton of energ, and states that the gan of knet energ s equal to the work done b the fores durng the moton. ISSN: 5-80 Prespaetme Journal Publshed b QuantumDream, In.

6 487 In the ase that the fores atng on the partles depend onl on the postons of the partles, and that the omponents ma be represented b the partal dervatves of a sngle funton of the U,, so that oordnates F the equaton of energ then s du du du, F, F, (.) d d d T T U U, (.) t t0 t t0 The funton U s alled the fore -funton, and ts negatve W= U s alled the potental energ of the sstem. Insertng W n (.) we have T W T W, (.4) t t t0 t0 the prnple of onservaton of energ. Suppose that the partle nstead of beng free s onstraned to le on a gven surfae. The path desrbed must then be an ar of a shortest or geodes lne of the surfae. The alulus of varatons enables us to fnd the dfferental equatons of suh a lne. The prnple of least aton sas that n the natural or unonstraned moton t wll go from P to Q along the shortest path, that s, an ar of a great rle... Generaled sstem of oordnates As was shown b Beltram (Beltram, 869), and worked out n detal b Hert, that the propertes of agnterval's equatons have to do wth a quadrat form, of eatl the sort that represents the ar of a urve n geometr. For nstane f a partle s onstraned to move on the surfae of a sphere of radus r, we ma spef ts poston b gvng ts longtude and olattude. These are two ndependent varables. The potental energ dependng onl on poston wll be epressed n terms of p and #. The knet energ wll depend upon the epresson for the length of the ar of the path n terms of and : r d sn d Dvdng b dt and wrtng d dt, d dt, we have T mr d sn d, (.5) The parameters and are oordnates of the pont, sne when the are known the poston of the pont s full spefed. Ther tme -dervatves and beng tme-rates of hange of oordnates ma be termed velotes, and when the together wth and are known, the velot of the partle ma be alulated. The knet energ n ths ase nvolves both the oordnates and and the velotes and. Inasmuh as the partle n an gven poston ma have an gven ISSN: 5-80 Prespaetme Journal Publshed b QuantumDream, In.

7 488 velot, the varables,,, are to be onsdered n ths sense as ndependent, although n an gven atual moton the wll all be funtons of a sngle varablet. The form of the funton T s worth of attenton. It s a homogeneous quadrat funton of the velotes and, the oeffents of ther squares beng funtons of the oordnates and, the produt term n and beng absent n ths ase. We ma prove that f a pont moves on an surfae the knet energ s alwas of ths form. We ma prove that f a pont moves on an surfae the knet energ s alwas of ths form. In the geometr of surfaes t s onvenent to epress the oordnates of a pont n terms of two parameters q and q. Suppose f q q, f q, q, f q,,, q from these three equatons we an elmnate the two parameters q, q, obtanng a sngle equaton between,,, the equaton of the surfae. The parameters q and q ma be alled the oordnates of a pont We ma obtan the length of the nfntesmal ar of an urve n terms of q and q. We have d q dq q Squarng and addng, where dq, d dq dq, d dq dq, (.6) q q q q d d d Edq Fdq q Gdq, (.7) E q q F q q q q q q q G q q q Thus the square of the length of an nfntesmal ar s a homogeneous quadrat funton of the dfferentals of the oordnates q and q, the oeffents E, F, G beng funtons of the oordnates q, q themselves., (.8) If the oordnate lnes ut eah other everwhere at rght angles we shall have Edq Gdq, (.9) The oordnates q, q are then sad to be orthogonal urvlnear oordnates. In general we have the equatons of hange of oordnates, ISSN: 5-80 Prespaetme Journal Publshed b QuantumDream, In.

8 489 r sn os, r sn sn, r os, (.0) from whh r os os, r os sn, r sn, r sn sn, r sn os, 0 and E r, F 0, G r sn Emplong the epresson (.7) for the length of the ar, dvdng b dq q dt we fnd for the knet energ, dq dt, q dt and wrtng T m Edq Fdq q Gdq, (.) Ths s a tpal eample of the emploment of the generalsed oordnates ntrodued b agrange nterval, q and q beng the oordnates, q, q, the velotes orrespondng, and T beng a homogeneous quadrat funton or quadrat form n the velotes q[, q, the oeffents of the squares and produts of the velotes beng funtons of the oordnates alone. We shall show that ths s a haraterst propert of the knet energ for an sstem dependng upon an number of varables. In the ase of a sngle free partle we ma epress the oordnates х, у, n terms of three parameters q, q, q, and we shall then have as n (.6) and (.7) where E dq E dq E dq E q q E q q E dq dq, (.) E, (.) q q q q q q (here,,, ) Proeedng now to the general ase of an number of partles, whether onstraned or not, let us epress all the oordnates as funtons of m ndependent parameters, q, q,... qm, the generaled oordnates of the sstem, k r q q,... q, q, q,... q, q, q q,..., (.4) where, m k r m k r k,,,..., n Dfferentatng, squarng and addng, we obtan k E k k k k k dq E dq... Emn dqm E dqdq E dqdq..., (.5) m ISSN: 5-80 Prespaetme Journal Publshed b QuantumDream, In.

9 490 where E k k k k k k k (.6) q q q q q q Thus the square of eah nfntesmal ar s a quadrat form n the dfferentals of all the oordnates q. Dvdng b dt, multplng b m and takng the sum for all the partles, we obtan k k k k k k T m E dq E dq... E dq E dq dq mn m E dq k dq..., (.7) Thus the knet energ possesses the haraterst propert mentoned above of beng a quadrat form n the generaled velotes q, the oeffents E beng funtons of onl the generaled oordnates q. The must satsf the ondtons neessar, n order that for all assgnable values of the q 's T shall be postve. It s sometmes onvenent to emplo the language of multdmensonal geometr. Ths sgnfes nothng more than that when we speak of a pont as beng n n dmensonal spae we mean that t requres n parameters to determne ts poston. Inasmuh as n moton along a urve, that s n a spae of one dmenson we have for the length of ar dq dq on a surfae, that s n a spae of two dmensons, and n spae of three dmensons Edq Fdq q Gdq, F dq q so b analog, n spae of n dmensons, n n F dq q, (.8) That s to sa a quadrat form n n dfferentals ma be nterpreted as the square of an ar n n dmensonal spae. Thus we ma assmlate our sstem dependng upon m oordnates to a sngle pont movng n spae of n dmensons. To eah possble poston of ths pont orrespon a possble onfguraton of our sstem. No matter what be taken as the mass of the pont, n, ts knet energ, T m dt s equal to the knet energ of our sstem, the oeffents n the quadrat form for and Т beng proportonal. ISSN: 5-80 Prespaetme Journal Publshed b QuantumDream, In.

10 49 The advantage of ths mode of speakng (for t s no more) ma easl be seen from the man analoges that arse, onnetng the dnamal theor of least aton wth the purel geometral theor of geodes lnes. Ths method s adopted b Hert n hs book (Hert, 894). The deas nvolved were frst set forth b Beltram. (Beltram, 869 ). Analoges that arse, onnetng the dnamal theor of least aton wth the purel geometral theor of geodes lnes были развиты далее благодаря prnple of varng aton of Hamlton Hamlton showed that the funton S, whh s named aton S t t 0 dt, (.9) where T W s agrange funton, satsfes a ertan partal dfferental equaton, a soluton of whh beng obtaned, the whole problem s solved: where S S S H t, q,... q,,..., m t q qm H T W s the Hamlton funton 0, (.0) The equaton s of the frst order sne onl frst dervatves of S appear, and, from the wa n whh T ontans the momenta, s of the seond degree n the dervatves. Sne S appears onl through ts dervatves an arbtrar onstant ma be added to t. Hamlton's equaton (.0) assumes a somewhat smpler form when the fore-funton and onsequentl H are ndependent of the tme that s when the sstem s onservatve. We ma then advantageousl replae the prnpal funton S b another funton alled b Hamlton the haraterst funton, whh represents the aton A. Makng use of the equaton of energ, T W h, to elmnate W, we have t 0 0 A Tdt S h t t, (.) t and the above partal dfferental equaton (.0) beomes merel A A H q qm h q q,...,,..., (.) m In these sstem a new varatonal prnple wll work; ths prnple was obtaned n 87 b Jaob (Enlpeda of matemats, 0). The knet energ of a sstem ma be epressed n generaled oordnates q as follows: n T aq q, (.), The metr of the oordnate spae s gven b the formula ISSN: 5-80 Prespaetme Journal Publshed b QuantumDream, In.

11 49 n a qq, (.4) The ntal and fnal postons r 0 and r of the sstem n some atual moton are also gven. Jaob's prnple of statonar aton: f the ntal and fnal postons of a holonom onservatve sstem are gven, then the followng equaton s vald for the atual moton: r r 0 h W 0, (.5) as ompared to all other nfntel near motons between dental ntal and fnal postons and for the same onstant value of the energ h as n the atual moton. Jaob's prnple redues the stud of the moton of a holonom onservatve sstem to the geometr problem of fndng the etremals of the varatonal problem (.5) n a Remannan spae wth the metr (.4) whh represents the real traetores of the sstem. Jaob's prnple reveals the lose onneton between the motons of a holonom onservatve sstem and the geometr of Remannan spaes. If the moton of the sstem takes plae n the absene of appled fores,.e., U 0, the sstem moves along a geodes lne of the oordnate spae q,...q n at a onstant rate. Ths fat s a generalaton of Galle's law of nerta. If U 0, determnng the moton of a holonom onservatve sstem s also redued to the task of determnng the geodess n a Remannan spae wth the metr n U h bqq (.6), In the ase of a sngle materal pont, when the lne element s the element of three-dmensonal Euldean spae, Jaob's prnple s the mehanal analogue of Fermat's prnple n opts. These results prove that n a Remannan form we an wrte all lassal potental fel, not ust gravt feld.. Geometr and Phss of Theor of Elementar Partles (Pseudo- Euldean or orent-nvarant Spae) et us now onsder the onneton of nterval wth phss n the ase of the pseudo-euldean geometr. A stud of the lterature shows that the pseudo-euldean oordnates and nterval of the fourdmensonal spae-tme are ntrodued nto phss b analog wth the nterval of Euldean geometr (andau and fsht, 97): It s frequentl useful for reasons of presentaton to use a fttous four-dmensonal spae, on the aes of whh are marked three spae oordnates and the tme. ISSN: 5-80 Prespaetme Journal Publshed b QuantumDream, In.

12 49.. Interval and square of 4-tane dfferental In Cartesan oordnate sstem of the Euldean geometr an nterval s the tane s between two ponts on a straght lne n spae, whh s alulated aordng to the Pthagorean theorem. Sne n phss traetores are often urved lnes, the Pthagorean theorem n ths ase s vald onl for the nfntelsemal tanes. Therefore, an nterval s defned here aordng to (.) as the square root of the square of the tane dfferental n Euldean spae. In the pseudo-euldean geometr an nterval s defned as the square root of the square of the 4- tane dfferental and s gven b the sum (takng nto aount the summaton of Ensten) d d, where 0,,, d0 dt. The square of the nterval looks lke: dt dr ( dt) ( d) ( d) ( d) Note that urrentl the magnar tme oordnate s rarel used (although t s b no means a mstake and has ertan advantages), and s wrtten as: dt dr ( dt) ( d) ( d) ( d), (.) d d, (.') where,,, 4, and d4 dt. In addton, the squares of dfferentals are often wrtten wthout parentheses:, d, nstead of ( ), ( d), et.. Thus, the use of haratersts of the -dmensonal spae n the ase of 4-dmensonal spae tme s a postulate,.e., some hosen mathematal epresson, whh s neessar for the onstruton of speal relatvt b Mnkowsk. It also follows from the fat that n nature the length of the ar n the 4- spae-tme s not measurable. Therefore the queston of the phsal meanng of the 4-nterval arses. et's tr to answer t... Dervaton of pseudo-euldean nterval from the phsal equatons The vetors of the orent-nvarant (.e., relatvst) theores neessarl depend on the 4- oordnate: one tme oordnate and three spae oordnates. In other wor, these equatons are "workng" n a 4-dmensonal spae-tme. Does ths theor ontan the equatons, whh have a sum of terms, eah of whh s assoated wth one of the four oordnates, lke the square of the nterval? As we know, n the frst tme suh equatons n lassal eletrodnams appear, and then n quantum feld theor. The wave equatons of these theores nlude a sum of terms, eah of whh s assoated wth one of the varables t,,,. It would be logal, to seek the ause and the meanng of the appearane of 4-nterval n them, nstead of ntrodung them artfall, as dd Mnkowsk. Reall that our stud of the gravtatonal feld s based on an nhomogeneous wave equaton of the so-alled "massve photon" (whh n mathematal notaton s smlar to the Klen-Gordon ISSN: 5-80 Prespaetme Journal Publshed b QuantumDream, In.

13 494 equaton). It s an equaton for the two vetors of the eletr and magnet fel that gve ths photon a mass. From ths equaton follows the well-known equaton of onservaton of energ and momentum for massve partles, whh s eas to obtan also from the defntons of 4-vetors of momentum and energ (see above). (andau and fsht, 97) From (.) we an easl obtan: ( dr dt dr ( dt) ( dt) dt) At the same tme nterval s assoated wth proper tme, (.) d b relaton: dt d, (.) For a free materal pont the onept of the 4-momentum s ntrodued: where p 0 From ths: p or p p, mu 0 p, (.4) m m m, p, ; u s the 4-velot. p m, (.5) where the energ and momentum s rewrtten for onvenene as follows: p m m d dt (where and fator and antfator, respetvel). Hene, n the Cartesan oordnate sstem: p p p p m Sne m dt d d d dt p m, 0, (.5 ) m m d dt, a, ths relaton an be rewrtten as:, (.6) Multplng t b, we get: dt dt d d d, (.7) Sne (see above (.)) we got are the orent dt dt, the epresson (.7) an be wrtten as square of a 4-nterval: dt d d d, (.') In general ase of use n Euldean spae of an other, than the Cartesan, oordnate sstem for reordng of the relaton (.5 ), partularl, the orthogonal urvlnear oordnates, ths nterval takes the form: g dd, (.8) ISSN: 5-80 Prespaetme Journal Publshed b QuantumDream, In.

14 495 where g s a so-alled metr tensor, whose elements take nto aount the hanges n the proetons of the segments of the traetor of the bod on the oordnate aes, at the transton from the Cartesan oordnate sstem to an other. In a Cartesan sstem, all elements g are equal to untes. Obvousl, f we go n the opposte dreton, we an obtan the equaton (.5 ) from the square of the nterval. Ths mples, frstl, that these equatons - (.) and (.5 ) - losel bnd the massve elementar partles phss and geometr. Seondl, the equaton of "massve photon" s derved from Mawell's equatons of a massless photon as a result of hs self-nteraton of fel (Krakos, 04a). Ths non-lneart of a self-atng fel of the massve photon does not mean transton from Euldean to some new geometr. From ths t follows that (.) s not a metr of pseudo-euldean geometr, but t s a metr of Euldean geometr that desrbes the orent-nvarant feld equatons. The onl hange n the geometr, whh we an observe n ths ase s the transton from retlnear to urvlnear geometr. In addton, another lnk between the nterval (.) and the phsal equaton s deteted. As we have shown n a prevous artle (Krakos, 00; 04b), usng the Shroednger defnton of aton p S ), from the equaton (.5 ) t s eas obtan orent-nvarant Hamlton-Jaob ( equaton n general vew. For ths t s enough to wrte the equaton (.5 ) n a form, sutable for an of the Euldean oordnate sstem: g p p m, (.9) where, we reall, g s the metr tensor of geometral spae, but not of the gravtatonal spaetme of general relatvt (n other wor, n ths ase the tensor g does not nlude the phsal haratersts of the feld). In ths ase the Hamlton-Jaob equaton of free partles obtans the form: g S S m 0, (.0) Reall that the phsal feld (e.g., eletromagnet feld) s nluded n Hamlton-Jaob equaton n the followng wa: g S p e S p e m 0, (.) Thus, we onlude that the three equatons: (.) (.5 ) and (.0) are bonded to eah other one-toone and, n fat, are equvalent. From ths follows that the nterval (.) wthn a relatvst phss s the phsal law, and not a geometr relaton. Net, we onsder how the 4-nterval s ntrodued n the transton from Euldean geometr to Remann geometr. ISSN: 5-80 Prespaetme Journal Publshed b QuantumDream, In.

15 Geometr and Phss n the Pseudo-Remannan Spae In general relatvt an nterval smlar to (.8) s ntrodued (postulated), where g takes nto aount the peulartes of the Remann geometr. But the most mportant thng here s other: n general relatvt, t s postulated that, due to the transton to the Remann geometr, the metr tensor s a funton of the gravtatonal feld. Whether ths s proved b eperment, we do not know beause all the epermental onfrmaton of general relatvt are obtaned for problems n the pseudo-euldean metr. Another fat also rases the queston about the sgnfane of Remann geometr n phss. As we know, all theores of phss, eept the GTR, are bult n a Euldean spae, although mathematall, relatvst theores an be onstruted n the pseudo-euldean spae. But there s no suh theor, whh nee the ntroduton of the Remann geometr. et us wrte the nterval GRT as follows: where the metr tensor GR g d d, (4.) GR g ontans the haratersts of the gravtatonal feld. In addton, nstead of the equaton for the eternal feld (.) n general relatvt the equaton of GR eternal feld of tpe (.0) s taken, but wth the approprate metr tensor g : S S m 0, (4.) g GR The queston s, wh s there suh a dfferene between (.0) and (4.), as well as between (.8) and (4.), and wh s the eternal feld n GTR nserted through the metr tensor? To answer ths queston, we wll tr to fnd out the phsal sense of the metr tensor. et us turn frst to Euldean geometr. 4.. The phsal sense of the metr tensor of urvlnear oordnates sstem of the Euldean geometr Reall the generaled oordnate sstem and partularl, urvlnear oordnates. (Korn and Korn, 968) et us ntrodue a new set of oordnates q, q, q, so that among,, and q, q, q there are some relatons q, q,. q, q, q,. q, q, q q, (4.) The dfferentals are then, d dq dq dq, (4.4) q q q ISSN: 5-80 Prespaetme Journal Publshed b QuantumDream, In.

16 497 and the same for d and d. In Cartesan oordnates the measure of tane, or metr, n a gven oordnate sstem s the ar length, whh s defned b d d d, (4.5) In general, takng nto aount (4.4), from (4.5) we obtan gdq gdqdq... gdqdq, (4.6) where g s the metr tensor. Thus n orthogonal sstem we an wrte where the H s где H dq H dq H dq, (4.7) H H, (4.8) q q q are alled ame oeffents or sale fators, and are for Cartesan oordnates. Thus, the Remann metr tensor, reorded n oordnates ontans the squares of ame oeffents: q, s a dagonal matr whose dagonal For eample, n the ase of spheral oordnates, the bond of spheral oordnates wth Cartesan s gven b (.0). The ame oeffents n ths ase are equal to: dfferental of ar (nterval) s: dr r d r sn d H r, H r, H r sn, and the square of the Sne the metr tensor s determned b means of ame oeffents, let us reall the geometr meanng of the latter: the ame oeffents show how man unts of length are ontaned n the unt of length of oordnates of the gven pont, and used to transform vetors when transton from one sstem to another takes plae. Ths means that the metr tensor n Euldean geometr defnes resalng of three oordnates r,,, and n the pseudo-euldean or pseudo-remannan geometr t determnes resalng of four oordnates t, r,,. As we have seen from the soluton of the Kepler problem wthn IGT (Krakos, 04), the relatvst orretons wthn IGT orrespond to hanges of sales t and r, aused b the orentnvarant effets (tme dlaton and orent-ftgerald length ontraton). In the net artle, we wll show that the same thng ours n problems of a movng soure. ISSN: 5-80 Prespaetme Journal Publshed b QuantumDream, In.

17 498 Thus, we onlude that relatonshps (4.) and (4.) have metr tensor GR g as a fator that takes nto aount the hange of sales of tme and tane due to relatvst effets assoated wth moton of bodes. 5. Consequenes From the foregong analss follows that b regular wa the nterval of a 4-spae-tme an be obtaned onl for the pseudo-euldean spae, as a varant of the phsal law of moton of elementar partles. Sne there s no other law of moton for massve partles, we an assume that the hpothess of Ensten that the gravtatonal feld s reated b the urvature of spae-tme, whh requres a transton to a pseudo-remannan geometr, nee onsderable adustment. Followng the theor of mass generaton (Krakos, 04a), we have to onlude that the gravtatonal feld arses from the self-aton of massless fel. It s ndeed aompaned b the transformaton of the lnear movement of fel n urvlnear moton (mathematall, ths s the transton from lnear equatons to nonlnear equatons). But ths has nothng to do wth the Remann geometr. It an be assumed that the use of the Remann geometr n GR s possble for the reason that math phss n the ase of the Remann geometr s ver lose to the math phss usng generaled oordnates of Euldean geometr. Formall, the oordnates of the Remann geometr, an be onsdered as generaled oordnates of the set of n materal ponts, or as one pont n the n - dmensonal spae. Ths s evdened b the form of squared length of ar (traetor) element usng generaled oordnates (.5) wth the values of the Gauss oeffents (.6). In the transton to the Remann geometr the Gaussan quadrat form oeffents E are replaed b elements of the metr tensor g. In ths sense, the Remann geometr should not be opposed to Euldean geometr. From a formal pont of vew (Bogorok, 97) the Remann spae an be determned, lke the Euldean multdmensonal spae, as a feld of the metr tensor n the n-dmensonal ontnuum n whh the tane between the nfntel near ponts s usng quadrat forms g, and the angle between two lnear elements - at os g lass of spaes and nludes Euldean geometr as a smple speal ase. s. Remann geometr overs a wde Moreover, t s possble to hoose the oordnates n Euldean spae n an wa, that all g and ther frst dervatves wth respet to oordnates n Remannan and Euldean metrs were the same values n all ponts of the lne. In ths ase, the Euldean metr s n ontat along a gven urve wth the Remann metr. In an nfntel thn tube ontanng the urve, Euldean spae s Remann spae up to the seond order. Ths s alled the ontguous Euldean spae. ISSN: 5-80 Prespaetme Journal Publshed b QuantumDream, In.

18 499 Another reason to beleve that n the theor of gravt t s suffent to use the pseudo-euldean metr s the possblt to present the Hlbert-Ensten equatons n the form of generaled d'alembert equaton (Fok,.964): In the prevous seton we saw that, at least f 0 (.e., n harmon oordnates) Ensten s equatons are of the tpe of the wave equaton, beause ther man terms nvolve the d Alamber operator. A smlar result an be obtaned based on the nonlnear theor of elementar partles. Sne the equaton (.9) s a formal onsequene of squarng of the Dra eletron equaton, one mght thnk that there s a onneton between the Dra matres, and tensor metr spae. Indeed, suh a onneton ests. And t gves the possblt to reeve the general ovarant form of the squared Dra eletron equaton n a gravtatonal feld (for detals, see. (Shroednger, 9; Krakos, 0). The onneton between the Dra matres and metr tensor s defned b relatons g S, from whh follows g S. and where s the Dra matres The seekng equaton s the d'alembert equaton: g where R s an nvarant urvature Remann tensor R 4 gg f S m, (4.9) 8 R g g R, R, S S, and, R s a smmetr In the frst term of equaton (4.9) s eas to fnd a regular operator of the Klen seond order equaton n the Remann geometr. In the thrd term on the left s reogned well-known term assoated wth the spn magnet and eletr moments of the eletron (tensor S ). The seond term provoked partular nterest of Shrödnger: To me, the seond term seems to be of onsderable theoretal nterest. To be sure, t s muh too small b man powers of ten n order to replae, sa, the term on the r.h.s. For m s the reproal Compton length, about 0 m. Yet t appears mportant that n the generalsed theor a term s enountered at all whh s equvalent to the engmat mass term. Nonlnear theor of elementar partles an eplan the phsal meanng of ths term: t defne the harge and mass of elementar partle. Referenes Beltram, E. (869). Sulla teora generale de parametr dfferenal (Memore del' Aadema delle Sene der lsttuto d Bologna, Sere, t. VIII, p. 549; 869.) Bogorok, A.F. (97). Unversal gravtaton. (n Russan) Kev, "Naukova Dumka". Buhhol, N. (97) Bas Course of Theoretal Mehans, vol. (n Russan). Mosow, Nauka. pp ISSN: 5-80 Prespaetme Journal Publshed b QuantumDream, In.

19 500 Enlpeda of matemats. (0). Varatonal prnples of lassal mehans. Fok, V. (964). The theor of spae, tme and gravtaton. Pergamon Press, Oford. Hert, H. (894). De Prnpen der Mehank n neuem Zusammenhange dargestellt. epg, J.A. Barth, 894 Korn, G. A. and Korn, T. M. (968). Mathematal Handbook for Sentsts and Engneers. New York/San Franso/Toronto/ondon/Sdne 968. MGraw-Hll Book Compan. Krakos, A.G. (00) Nonlnear Theor of Elementar Partles: V. The Eletron & Postron Equatons (near Approah). Prespaetme Journal, Vol, No 0 Krakos, A.G. (0). On orent-nvarant Theor of Gravtaton Part : Revew. Prespaetme Journal Vol, No 6 (0) Krakos, A.G. (04a) Orgn of Gravtaton Mass n orent-nvarant Gravtaton Theor (Part I) and (Part II). Prespaetme Journal, Vol 5, No 5 Krakos, A.G. (04b). The mathematal apparatus of orent-nvarant gravtaton theor Prespaetme Journal, Vol 5, No 8: --- b Krakos, A.G. (04). Soluton of the Kepler Problem n the Framework of orent-invarant Gravtaton Theor. Prespaetme Journal, Vol. 5, Issue 4 pp andau.d. and fsht E.M. (97). The Classal Theor of Fel. (Vol. of a ourse of theoretal phss). Pergamon Press. Shroednger, E. (9). Drashes Elektron m Shwerefeld I. Stungsber. Preuss. Akad. Wss. Phs. Math. Kl. 05. Tonnelat, Mare-Antonette. (965/966). es verfatons epermentales de la relatvte generale. Rend. Semn. mt Unv. e Polten. Torno, 965/966, 5, 5-5. Webster, A.G. (9). The dnams of partles and of rgd, elast, and flud bodes. epg, B.G. Teubner. ISSN: 5-80 Prespaetme Journal Publshed b QuantumDream, In.