THE MANNHEIM-KAZANAS SOLUTION, THE CONFORMAL

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1 THE MANNHEIM-KAZANAS SOLUTION, THE CONFOMAL GEOMETODYNAMICS AND THE DAK MATTE M.V. Gobatnko, S.Yu. Sdov ussian Fdal Nucla Cnt - All-ussia sach Institut of Expimntal Physics, Saov, Nizhny Novgood gion Abstact Within th famwok of th Einstin's standad quations of th gnal thoy of lativity, flat galactic otational cuvs of galaxis cannot b xplaind without hypothsis attacting th dak matt, th paticls of which had not yt bn idntifid. Th vacuum cntalsymmtic solution of th quations of confomal gavitation is wll known as mtics of Mannhim-Kazanas, on th basis of which ths cuvs civ puly gomtical xplanation. W show in ou wok that th mtics of Mannhim-Kazanas is th solution of not only Bach quations civd fom confomal-invaiant Wyl Lagangian, but also th solution of quations of th confomal gomtodynamics at a nonzo vcto of Wyl. In this connction th hypothsis is fomulatd that th spac on galactic scals can b dscibd not only by imannian gomty, but with gomty of Wyl. Kywods: confomal xtnsions of th gnal thoy of lativity, galactic otational cuvs. INTODUCTION It is wll known that th mysty of th dak matt (DM) is still not solvd. Its xistnc is confimd by numous obsvation data that in paticula f to th flat galactic otational cuvs. Th situation sms as if insid visibl (light) matt dak matt is spad in th volum of galaxis, th psnc of which vals itslf in not only additional gavity foc that affct th visibl matt. otational cuvs of th otation of stas, hydogn and oth typs of light matt aound th cnt of th spial galaxy stating with th adius of about. has a flat viw, as thy say, i.. th obiting vlocitis in a wid ang of adii do not dcas with th distanc. H, is th thicknss of th optical disc of th galaxy. Th otational cuv fo galaxy NGC733 is

2 shown in Figu adoptd fom []. Oth galaxis hav simila otational cuvs (s []-[5], fo xampl). It is wll-known []-[6] that th otational cuvs a dscibd wll by th mpiical dpndncy of th otational vlocity of stas v on th adius that is wittn as follows an v bn c, () wh is th distanc to th cnt of th galaxy, N is th numb of stas in th galaxy, and a, b, c a phnomnological paamts. This fomula bsids th fist Nwtonian tm has oth tms that a popotional to th adius. Ths tms do not com fom th gnal thoy of lativity (GT) by Einstin, but thy giv th flat shap of otational cuvs. Figu - Th obiting vlocitis of th visibl matt (in km/s) aound th cnt of galaxy NGC733 as a function of atio /, wh is th thicknss of th optical disk, is th distanc fom th cnt of th galaxy. Dashd lin shows th contibution of th Nwtonian componnt. Dot-dash lin is th contibution of th lina potntial. Solid lin is th accumulation cuv. Th data a fom []. To xplain th psnc of flat sctions, th w usd two kinds of hypothsis. Th most popula is th hypothsis basd on th psnc of dak matt (DM) assumption. Within this thoy thy a looking fo th DM cais. Th a many thois considing th natu of th DM cais [6]; howv, non of thm has not achd th xpimntal confimation yt. This fact maks us look fo som altnativ ways to xplain flat otational cuvs. Th oth altnativ sach aa is in fact th gomtic on, i.. th on whn th obsvd ffcts (flat otational cuvs in this cas) a th sult of th solution of th quations of som sot modifid gavitation thoy. Th pogss in th diction of th gomtical xplanation of th otational cuvs is don in th wok by Mannhim-Kazanas [7], wh an xact solution of th confomal gavitation quation is obtaind using Wyl Lagangian. It appad that spac mtic in this solution has a tm that is popotional to th adius, which is capabl to xplain flat galactic otational cuvs in puly gomtic way, without th hypothsis of th dak matt xistnc. Th dynamic quations of spac h should b wittn not in th fom of standad GT quations, but in a mo gnal fom as gavitation quations gnalizd with th

3 quimnts of th confomal invaianc. On of ths ways of such gnalization is to us of th Ladangian quadatic by Wyl tnso in imann spac. W call th sultd quations Bach quations in complianc with [8]. That was th xact solution of Bach quations in vacuum that was obtaind in [7]. Quadatic vaiant of confomal gnalization of GT quations is not th only possibl on. In [9], [] a mo simpl vaiant of such gnalization was offd in th Wyl spac. Th govning quations in [] w calld th quations of confomal gomtodynamics (CG). Thy had th viw of standad GT quations, but with a spcific tnso of ngy-impuls constuctd fom a nw gomtic objct A, which w call a Wyl vcto. In cas whn Wyl vcto is bought to th gadint fom som scala fild, CG quations coincid with Bans-Dick quations [] whn paamt 3/. If w add th ngy-impuls tnso of th matt to ths quations, ths quations will comply with th quations poposd in [] as a scalinvaiant gavitation thoy. It is intsting to consid th issu if th Mannhim-Kazanas solution fom [7] is th solution of CG quations of [9], and [], that, as it was said bfo, is also confomal-invaiant. Claification of this issu is und considation of ou wok h. W will show that th answ to this qustion is positiv. Th spcific fatu of th sult obtaind lis in th fact that it boadns th possibilitis fo intptation of th Mannhim-Kazanas solution. Actually, in cas of CG th ffctiv ngyimpuls tnso is constuctd on th basis of th Wyl vcto, i.. fom th gomtic chaactistics of th spac itslf. Such ngy-impuls tnso can b analyzd th sam way as it is don in th GT. Th wok povids th analysis that bings us to th conclusion about th abnomal chaact of thmodynamic poptis of th poducd gomtodynamic mdium. On of th poptis that confim th abnomal chaact of th ffctiv ngy-impuls tnso is th ngativ chaact of th ngy dnsity at low nough valus of th adial vaiabl. In th nd th sults of th pfomd analysis a discussd. 3 MANNHEIM-KAZANAS SOLUTION In th basis of quadatic confomal-invaiant gnalization of th gnal thoy of lativity th is Wyl tnso C, that is dfind though th imann tnso of cuvatu, icci tnso g and contactd icci tnso g in th following way: C g g g g g g g g. () 6 It is known fom [7], [8] that if a scala is usd L C C (3) as a Lagangian, a sult of a standad vaiation pocdu a Bach quations: wh () () B B B, (4)

4 () ; ; B ; ; ; ; ; ; ; g ;, (5) () B g g. (6) Distinctiv fatus of quations (4)-(6) a as follows: () Thy a wittn only in tms of valus, and mtic. () Th quations involv th foth od divativs fom th mtic. (3) Th quations a invaiant with gad to confomal tansfomations of th mtic * x g g g. (7) In Mannhim and Kazanas solution [7] th squa of th intval is wittn in th fom of and function B in th fom of H,,, a som intgation constants. d ds B dt d sin d B 3 B 3 4, (8). (9) Solutions (8), (9) appad to b fuitful fom th point of viw of xplanation of flat galactic otational cuvs and undstanding of th dak matt natu. Th a many publications on this issu,.g., s []-[5]. 3 EXACT SOLUTION FO THE CG EQUATIONS It is known that th static solution of GT quations g g () in cas of cnt-symmtic static (CSS) poblm is Schwazschild solution in th spac of d Sitt, that has th following fom: wh ds dt d d sin d, () 3. () H, is gavitation adius, Const - is lambda-tm. Equations () a a paticula cas of CG quations g A A g A g A ; A ; A ; g (3) and a obtaind fom thm if vcto A is takn qual to zo. If w consid () as a solution of quation (3), thn it should b wittn lik this

5 5 ds dt d d sin d, A 3,, Const. Lt s psum that in cas of CSS poblm vcto A has a adial componnt as th only componnt diffnt fom zo, which in (4) is dnotd as A. Equations (3) a invaiant with gad to confomal convsions g g g * * A A A * In confomal convsions (5) not only mtic in involvd, but vcto Wyl A and lambda-tm as wll. So, if xpssions (4) a tansfomd accoding to (5), thn th tansfomd xpssions will also b th solution of initial quation (3). Lt s us this popty and subjct solution (4) to th confomal tansfomation with confomal facto xp, wh function. dpnds only on th adial vaiabl. Th nw solution will b again cntsymmtic and static and hav th following fom: * ds dt d d sin d, A d d * *,. Thn will go though coodinat tansfomation and plac adial coodinat with coodinat in such a way, that th nw coodinat would b bight. That is lt,,, (4) (5) (6). (7) Th diffntials of th old and nw coodinats, as it coms fom (7), a latd though th atio d d. (8) Aft substitution (7), (8) in (6) w gt * d ds dt d sin d, 3 3 A d d * *,. (9) In (9), tm is to b considd as a function of a nw adial vaiabls, i.. as a function, dtmind fom quations (8). On this stag function choos so that th following atio would b tu stays unctain. Lt s * * gg. () To mt atio (), as it coms fom (9), th following atio should b tu

6 6. () It is intsting to not that atio () is obtaind th sam ispctiv to what fom th initial function was usd. Thn w tak th squa oot fom both pats () and gt a diffntial quation to find function in th fom, () wh. (3) Th solution of quations () is. (4) H, is a constant with th dimnsion of th lngth. Fom th condition of th positiv chaact of xponnt w find that paamt should b qual to, (5) and th valu should b positiv,. (6) As a sult, w obtain. (7) With th account of (7) this sults giv th following lation btwn and :,. (8) As a sult, a confomal facto intoduc a nw symbol Solution (9) will b wittn now lik this H, F has th fom and xpssion fo a nw adial vaiabl a found. Lt s * * * F g. (9) * d ds F dt d sin d, F A,. (3) F ( ). (3) 3

7 7 wh Lt s consid paticula cass. Schwazschild solution. W gt it if in (3), (3) w hav and stting. Kottl solution. W gt it if in (3), (3) stting. Mannhim-Kazanas solution. Squad intval in this solution has th fom ds dt d d sin d, (3) MK MK 3 3. (33) MK If w tak into account idntity of function paamts,, with paamts, in th following way: in (33) and function F in (3) and lat MK 3 3 3,,, (34) thn it will bcom vidnt that (3), (3) is th solution fom th class of Mannhim-Kazanas solutions. Actually: Both solutions lat to th class of cnt-symmtical static solutions, wh bight adial vaiabl is implmntd. -componnt of mtic in both solutions compis th tms with dgs,,,. Equalization of th cofficints at th sam dgs of adial vaiabl sult into fou atios. W gt th lation (34) btwn th paamts fom th atios. Th foth atio is mt automatically. 4 POPETIES OF GEOMETODYNAMICS MEDIUM Fom th point of viw of GT quations solution (3), (3) is th solution with a nonzo ngy-impuls tnso. Componnts of tnso T diffnt fom zo can b found fom GT quations T, (35) wittn fo static cnt-symmtic cas (s [3], [4], fo xampl). Ths quations sult into th following fomulas: T, (36) T, (37) adial vaiabl, which is makd as in Mannhim-Kazanas solution (8), (9), h is makd as.

8 8 T. (38) Engy-impuls tnso componnts a dtmind only though function, as it coms fom (36)-(38). Using fomulas(3), (3) w gt: U P T U, P T T At low valus of adial vaiabl gomtodynamic mdium dscibd with tnso T (39) has th following abnomal fatus in quations of stat: ngativ dnsity of ngy U, not qual btwn thmslvs stss tnso componnts P P, adial pssu is qual to th ngy dnsity with th opposit sign, P U. (39) 5 DISCUSSION Th basic sults of th wok can b summaizd as follows. It is shown that th cnt-symmtic static Mannhim-Kazanas solution is th solution of not only Bach quations (4)-(6), but th solution of quations of confomal gomtodynamics (3). It coms fom CG quations that th scala fild containd in th confomal facto bings a non-zo ngy-impuls tnso with componnts (36)-(38). Analysis of tnso (36)-(38) shows that at small valus of adial vaiabl gomtodynamic mdium fo this CG solution has abnomal quations of stat: ngativ ngy dnsity U, non-qual componnts of stss tnso P P, but th adial pssu is positiv and modulo is qual to th ngy dnsity, P U. In fncs []-[5], [5] thy not that within standad GT it is impossibl to xplain numous obsvation data that f to th flat galactic otational cuvs without attaction of a dak matt hypothsis. Dak matt cais a not found yt, and this maks it impotant to look fo an altnativ xplanation of th obsvd data. Natual xplanation fo ths data is obtaind within Mannhim-Kazanas solution (8), (9). But this solution is not th solution of GT quations in th standad fom. At th sam tim it is both th foth od solution of Bach quations and th scond od solution of CG quations (that has confomal invaianc and is latd to th Wyl spac). Bach quations [6] obtaind fom quadatic action by Wyl tnso a th most known vaiant of confomal invaiant gnalization of th GT quations. CG quations on th basis of Wyl gomty [9], [] a lss usd as an altnativ fo GT (howv not [7], [8]), but thy a th simplst vaiant of alization of confomal invaianc, as th gnalization is don by intoduction of a so calld gaug vcto fild A into GT quations whn psving th od of th obtaind diffntial quation. This

9 pocdu is analogous to th on that is usd in physics whn gaug filds a intoducd. CG quations poducd in this cas allow th analysis of th poptis of th gomtodynamic mdium that dpnd on th compnsating vcto fild, in th tms of som ffctiv ngyimpuls tnso. Substitutions of GT quations with CG quations [] sults to th tansition fom imann spac to spac of Wyl, and this quis considation of th concptual poptis 9 of spac. Th spac stays 4D with signatu. But in spac of Wyl it is ncssay to gist not only th fnc fam, but also th scal of lngth (tim). This pocdu is analogous to th calibation of th vcto-potntial in Maxwll lctodynamics. Th slction of a singl in spac gaug cosponds to th vsal into zo fo vcto A. Mannhim-Kazanas solution taks h th fom of Schwazschild solution. Anoth choic of slctd gaug lngth is possibl, whn th squa of th intval has gnal xpssion (4). This choic unambiguously lads to Mannhim-Kazanas solution (8),(9). Dpndncy of th intptation of th solution of CG quations on th choic of calibation of vcto A in spac of Wyl has a gnal chaact. It is possibl to psum that w can nglct this dpndncy in th ang of lngths fom th ons of chaactistic lmntay paticls to th sizs of sta systms and us a constant gaug, i.. assum A. But stating with galactic distancs th situation changs, which is indicatd by flat galactic otational cuvs. W may mak a conclusion that if not involving th dak matt concpt th ality in galactic scals is btt dscibd not with GT quations in thi standad fom, but with th gavitation quations in confomal invaiant fom. fncs [] Ph.D. Mannhim. Altnativs to Dak Matt and Dak Engy. axiv: 5566v [asto-ph.co] (5). [] Ph.D. Mannhim and J. G. O Bin. Impact of a global quadatic potntial on galactic otation cuvs. Axiv: 7.97v [asto-ph.co] (). [3] Ph.D. Mannhim and J. G. O Bin. Fitting galactic otation cuvs with confomal gavityl. Axiv:.3495v4 [asto-ph.co] (). [4] Ph.D. Mannhim and J. G. O Bin. Fitting dwaf galactic otation cuvs with confomal gavity and a global quadatic potntial. Axiv: 7.59v [asto-ph.co] (). [5] G.M. Gaipova. sach on galactic halo within confomal thoy of gavitation. Thsis. Scintific Libay of Chliabinsk StatUivsity (4). [6] G.VВ. Klapdo-Klingothaus, K. Tsiub. Astophysics of lmntay paticls. M.: Editd by YFN (). [7] Ph.D. Mannhim and D. Kazanas. Exact vacuum solution to confomal Wyl gavity and galactic otation cuvs. // Th Astophysical Jounal (989). [8] H.-J. Schmidt. Non-tivial solutions of th Bach quation xist. Ann. Phys. (Lipz.) (984). (AXiv: g-qc/58). [9] M.V. Gobatnko, A.V. Pushkin. Spac dynamics of lina affin connctdnss and confomal-invaiant xpansion of Einstin quations. // Voposy Atomnoy Nauki I Tkhniki. S.: Thotical and Applid Physics. Issu. pp (984). [] M.V. Gobatnko. Som Consquncs of th Confomally Invaiant Gnalization of Einstin s Equations. // Gnal lativity and Gavitation. 37. No (5).

10 [] C. Bans,.H. Dick. Mach s Pincipl and a lativistic Thoy of Gavitation. // Phys. v., 4, 95 (96). [] V. Canuto, P.J. Adams, S.-H. Hsih, and E. Tsiang. Scal-covaiant thoy of gavitation and astophysical applications. // Phys. v. D. Vol. 6. No. 6. P. 643 (977). [3]. Tolmn. lativity,thmodynamics and cosmology. M.: Nauka (974). [4] J. Sing. Gnal thoy of lativity. M.: IL (963). [5] Massimo Psic, Paolo Salucc and Fulvia St. Th univsal otation cuv of spial galaxis - I. Th dak matt connction. // Mon. Not.. Aston. Soc. 8, 7-47 (996). [6] M.V. Gobatnko, A.V. Pushkin, H.-J. Schmidt. On a lation Btwn th Bach Equation and th Equation of Gomtodynamics. // Gnal lativity and Gavitation. 34, No., 9- (). AXiv: g-qc/65. [7] Scholz, Ehad. Cosmological spactims balancd by a Wyl gomtic scal covaiant scala fild. // Foundations of Physics (9). AXiv:85.557v3. [8] Isalit, Mak; osn, Nathan. Wylian dak matt and cosmology. // Foundations of Physics (993).

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