The Problem of General Relativity for an Orthotropic Solid Sphere

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1 Appled Physcs Reseach; Vol. 5, No. ; 3 ISSN E-ISSN Publshed by Canadan Cente of Scence and Educaton The Poblem of Geneal Relatvty fo an Othotopc Sold Sphee Valey Vaslev & Leond Fedoov Insttute of Poblems of Mechancs, Russan Academy of Scences, Venadskoo, Moscow, Russa Coespondence: Valey Vaslev, Insttute of Poblems of Mechancs, Russan Academy of Scences, Venadskoo, Moscow 956, Russa. Tel: E-mal: vvvas@dol.u Receved: Octobe, Accepted: Novembe 6, Onlne Publshed: Januay 6, 3 do:.5539/ap.v5np5 URL: Abstact The pape s concened wth analyss and soluton of the statc poblem of the Geneal Theoy of Relatvty fo an othotopc sold sphee and the suoundn space. It s shown that, n contast to lqud and sotopc elastc sphecal solds, the snulaty does not appea at the cente of the sphee whose adus s educed to the avtatonal adus. Keywods: eneal elatvty, sphecally symmetc poblem, othotopc sold. Intoducton The sphecally symmetc statc poblem of the Geneal Theoy of Relatvty (GTR) was solved by Schwazschld n 96 just afte the theoy was developed. The soluton fo the empty space (Schwazschld, 96a) suoundn the sphee wth adus R shows that the adal space metc coeffcent becomes snula f the sphee adus appoaches the avtatonal adus. Howeve, the analyss of ths soluton (Syne, 96) allows us to conclude that the suface wth the adus s located nsde the sphee, and the snulaty does not exst n the extenal space. The soluton fo nsde space smulated wth a pefect ncompessble lqud (Schwazschld, 96b) becomes snula at the sphee cente f R appoaches, and a natual queston ases as to whethe ths snulaty s actual (Hawkn & Penose, 97) o t s a fomal esult havn no physcal meann (Fock, 959; Lounov & Mestvshvl, 985). It s also can be supposed that the snulaty appeas n soluton fo the ncompessble lqud only and can dsappea fo a moe ealstc mateal model allown fo stesses and stans nduced by the avtaton. Two man poblems ase n connecton wth ths queston. Fst, fo the sphee mateal whch s moe eneal than a pefect ncompessble lqud, the GTR equatons ae not complete and must be supplemented wth the coespondn mateal consttutve equatons (Feynman, Mono & Wane, 995). Second, only numecal solutons ae possble fo a complete set of equatons (n case such a set s obtaned), but such solutons cannot be used to dentfy snulates because they do not convee not only fo snula poblems, but also f the soluton of the bounday poblem does not exst. Numecal solutons whch do not convee at the sphee cente have been constucted by Vaslev and Fedoov () fo compessble lqud and elastc sold sphees. In the pesent pape, the GTR poblem fo the othotopc sphee s studed. As follows fom the Theoy of Elastcty, snulaty does not appea at the cente of the othotopc sphee unde some condtons mposed on the mateal elastc constants. Thus, the snulaty at the sphee cente can be elmnated, wheeas, n the vcnty of the oute suface, the sphee mateal can be smulated wth ncompessble lqud and the analytcal soluton fo the ncompessble can be used to dentfy the possble snulaty.. Govenn Equatons Consde a sphecal sold wth adus R suounded by the nfnte empty space. In accodance wth the classcal Schwazschld soluton (Syne, 96), the lne element n sphecal coodnates,, s taken as ds d ( d hdt sn d ) () whee () and h() ae the coeffcents of the metc tenso of sem-remannan space nduced by avtaton. Fo the extenal ( R) and ntenal ( R) spaces the follown consevaton equaton must be satsfed (Syne, 96): 5

2 Appled Physcs Reseach Vol. 5, No. ; 3 h 4 (T ) (T T ) (T T 4 ) () h Hee, (...) d(...) / d and T j ae the components of the eney tenso whch fo the statc poblem n sphecal coodnates ae expessed n tems of the adal,, and the ccumfeental,, stesses nduced by the avtaton nsde of the sold sphee wth constant densty,.e., T, 3 3 T T, T 4 4 c (3) Accodn to GTR, Equaton () s dentcally satsfed f the components of the eney tenso n tems of the components of the Ensten tenso G as j T ae expessed j n whch T h h G 3 h h h h T T3 G h h h h T G4 (4) (5) (6) 4 8 / c (7) and s the avtaton constant. Because substtuton of Equatons (4)-(6) dentcally satsfes Equaton (), only thee of fou Equatons () and (4)-(6) ae mutually ndependent. The smplest set of equatons whch s used futhe fo analyss ncludes Equatons (), (4) and (6), wheeas Equaton (5) s satsfed dentcally (Syne, 96). Usn Equatons (3), we can pesent the fnal set of equatons n the follown fom: h ( ) ( c ) (8) h h h (9) c () Consde the soluton of these equatons fo the extenal and ntenal spaces. 3. Extenal Space The extenal space ( R ) s empty, so that and. In ths case, Equaton (8) s satsfed dentcally, wheeas Equatons (9) and () educe to e h ( ), h e e () Subscpt e coesponds to the extenal space. Inteaton of Equatons () yelds e, he C ( C / ) () C / Ths esult coesponds to the well known Schwazschld soluton. Detemne the nteaton constants C and C. At an nfnte dstance fom the sphee,.e., at, Equatons () must educe to the soluton of the classcal avtaton theoy (Landau & Lfshtz, 96 ),.e., to 6

3 Appled Physcs Reseach Vol. 5, No. ; 3 e f / c he (3) whee f m/ s Newtonan avtaton potental n whch m s the sphee mass. Intoduce the so-called avtatonal adus m / c (4) Then, the condtons e( ) e and become h( e ) he yeld C, C and Equatons () e, h e / (5) / Fo the Eucldean space, and e he. 4. Geomety of Intenal Space Fo the sphecal sold, = constant and nteaton of Equaton () esults n the follown expesson: C c 3 3 whee subscpt coesponds to the ntenal space. Because at the sphee cente, we have C3 and Equaton (6) becomes 8 4, c (7) 3 3c 3m n whch Equatons (7) and (4) ae used to tansfom paamete. 5. Bounday Condtons Fo the lne element n Equaton (), the aea of the suface = constant s 4, but the dstance of ths suface fom the sphee cente s not equal to. Fo the ntenal space, we have usn Equaton (7) (6) d (8) d sn ( ) Fo the extenal space, n accodance wth Equaton (5), we et e e ( R) d ( R) R R d / sn R ( ) R( R ) ln ( ) R R( R ) (9) Fo R, Equatons (8) and (9) yeld e whch looks natual because the sphee oute suface s a physcal object. The metc coeffcents of the ntenal and the extenal spaces must be contnuous at e o at R,.e., (R) e(r). Usn Equatons (5) and (7), we have R R () 3c Recall that s specfed by Equaton (4). Matchn Equaton (4) wth Equaton (), we can fnd the sphee mass,.e., 7

4 Appled Physcs Reseach Vol. 5, No. ; m R 3 () Ths s the mass of the sphee n Eucldean space. Howeve, accodn to CTR, the space nsde the sphee s sem-remannan, the metc coeffcent s specfed by Equaton (7) and the sphee mass must be whee R m d snd d R sn () 3 / R (3) To analyze Equaton (), decompose ts ht-hand pat nto the powe sees,.e., m R Ths esult concdes wth Equaton () only fo,.e., only fo Eucldean space. Recall that Equaton () follows fom the bounday condton fo the metc tenso whch must not be volated. Thus, fo the lne elements n Equaton () the sphee mass coesponds to the Eucldean space thouh the metc coeffcents of the space coespond to the Remannan space. Ths allows us to suppose that n GTR these coeffcents descbe some othe physcal popetes of space athe than the space eomety. 6. Intepetaton of Remannan Space As known (Rashevsky, 967), Remannan space wth n R dmensons can be embedded nto Eucldean space wth ne n R(nR )/ dmensons. Fo two-dmensonal Remannan space ( n R ),.e. fo an actual suface, we have n E 3 whch coesponds to the actual thee-dmensonal Eucldean space. Howeve, fo nr 3, we et n E 6,.e. sx-dmensonal Eucldean space whch has no physcal ntepetaton. To demonstate the dea of the poposed ntepetaton of the thee dmensonal Remannan space (Vaslev, 989), consde a two-dmensonal Eucldean space,.e., a dsk nsde the ccle wth adus R (Fue (a)) wth aea R and the metc coeffcents, and a suface of evoluton n a thee-dmensonal Eucldean space (Fue (b)) wth the metc coeffcents (z),. (a) (b) Fue. To the ntepetaton of Remannan space If we move pont A n Fue (b) to the ccle cente O compessn the suface n the adal decton, the aea of the esultn flat dsk becomes equal to R, but ths dsk exsts now n non homoeneous and ansotopc two-dmensonal Eucldean space. Intoducn the space densty n the adal and the ccumfeental dectons as / (z), 8 / (4) we can teat the Remannan eomety as the mathematcal model of non-homoeneous and ansotopc Eucldean space. Note that havn Equatons (9) and () whch lnk the stess and the metc tensos, we can fnd the space denstes as functons of stesses and, concentatn the mateal n the aeas whee the denstes ae hh, ave at some optmal mateal dstbuton coespondn to the actn stesses. Applcaton of ths appoach to stuctual optmzaton s pesented by Vaslev and Fedoov (6).

5 Appled Physcs Reseach Vol. 5, No. ; 3 In accodance wth the foeon ntepetaton of Remannan space, avtaton causes not the space cuvatue whch can be hadly maned, but affects the densty of the Eucldean space. The poposed ntepetaton does not chane the physcal esults obtaned n GTR. Fo example, the cuvatue of the lht beam n the vcnty of a massve sold s nduced n Newton theoy by avtatonal foce, n GTR the beam follows the eodesc lne of the space cuved by the sold, wheeas wthn the poposed ntepetaton, the beam popaates n space lke n a tanspaent meda wth vaable efacton coeffcent. 7. Snula Solutons As follows fom Equatons (5), the metc coeffcent e (o the densty ) of the extenal space can become snula f the adal coodnate eaches the avtatonal adus. Hence, a natual queston ases as to whethe can be equal to n the extenal space. In the Eucldean space (see Secton 6), the mass of the sphee wth an abtay densty () s R m 4 () d Expessn ρ fom Equaton () whch s vald fo any functon R c (), we et 4 4R m d c R whee R (R) e (R). Substtutn ths esult n Equaton (4) fo and usn Equaton (7) fo χ, we ave at the follown expesson fo : R R (5) Applyn Equaton (5) fom whch t follows that e(r) R s hhe than unty, we can conclude fom Equaton (5) that R,.e., that the suface wth adus s located nsde the sphee. Howeve, Equaton (5) fo e s vald outsde the sphee and thus, cannot be snula. Note that Equaton (5) s vald espectve of the sphee stuctue and mateal. As known, the snulaty can appea at the sphee cente. Such snulaty occus fo the pessue n the lqud sphee f R 9/8R (Wenbe, 97) and n elastc sphee f R appoaches (Vaslev & Fedoov, ). Consde the othotopc sphee fo whch no snulaty can exst at the sphee cente unde some condtons mposed on the elastc popetes of the sphee mateal. 8. Classcal Soluton fo an Othotopc Sphee Wthn the famewok of Newton avtaton theoy, the set of equatons descbn the equlbum of a sphecally othotopc sold unde the acton of avtaton foces ncludes the equlbum equaton (Love, 97) consttutve equatons and stan-dsplacement equatons ( ) k,, E E E E 3 4 c k (6) 3 R (7) u, u/ (8) n whch E s the adal elastc modulus and E s the ccumfeental modulus of the mateal stffness, ae Posson s atos such that E E, and u( ) s the adal dsplacement. Elmnatn u fom Equatons (8), we ave at the follown compatblty equaton whch lnks the adal,, and the ccumfeental,, stans: 9

6 Appled Physcs Reseach Vol. 5, No. ; 3 ( ) (9) To solve the poblem, satsfy the equlbum equaton, Equaton (6), ntoducn the stess functon F as F F, k (3) Substtutn the stans fom Equatons (7) n Equaton (9) and expessn the stesses n tems of the stess functon wth the ad of Equatons (3), we obtan the follown ovenn equaton fo the stess functon: 4 k F m F [3( ) ], m E ( ) E( ) (3) The soluton of ths equaton whch satsfes the eulaty condton at the sphee cente s n 4 F() C kb, B 3( ) (3) ( s )( ) On the sphee suface, we have the bounday condton (R). Detemnn the nteaton constant n Equaton (3) fom ths condton, we ave at the follown fnal expessons fo the stesses: whee, n 4m n, kbr kr B n 4 n / R (34) The othotopc popetes of the sphee mateal ae specfed by paametes m and n n Equatons (3) and (3). If the adal modulus E s hhe than the ccumfeental modulus E,.e., the sphee s enfoced n the adal decton, we have m, n and the stesses n Equatons (33) become snula at the sphee cente. If E E,.e., the sphee s enfoced wth concentc shells, we have m, n and the stesses n Equatons (33) become zeo at the sphee cente. Thus, the classcal soluton shows that no snulaty can appea at the cente of an othotopc sphee f E E. 9. GTR Soluton fo an Othotopc Sphee The metc coeffcent of the ntenal space s specfed by Equatons (7) and () whch yeld (35) 3 / R Substtutn ths esult n Equaton (9) and takn nto account Equatons (7), (4) and () fo, and m, we et (33) h 3 3 h R c (36) The fst ovenn equaton of the poblem unde study follows fom the consevaton equaton, Equaton (8), f we substtute h / h fom Equaton (36),.e., c 3 3 ( R ) c c ( ) (37) Compae ths equaton wth Equaton (6) of the classcal avtaton theoy. Fo ths pupose, pefom lneazaton of Equaton (37) nelectn the tems / c n compason wth unty. Then, Equaton (37) educes to If we futhe nelect c 3 ( ) ( R ) n compason wth R 3, we ave at Equaton (6) of the classcal avtaton theoy. (38)

7 Appled Physcs Reseach Vol. 5, No. ; 3 The last tem n Equaton (38) coesponds to the body foce n Equaton (6) and becomes nfntely hh at the sphee suface R f R. Ths popety of the equaton causes possble specfc featues of ts soluton dscussed futhe n Secton. Retun to Equaton (37) whch ncludes two unknown stesses and. In the classcal avtaton theoy (Secton 8), ths equaton s supplemented wth the compatblty equaton, Equaton (9). To constuct the analoous equaton fo GTR, apply the nvaant condton of the Ensten tenso n Equatons (4)-(6) ( Vaslev & Fedoov, 6), assumn that ths tenso dentcally satsfes the consevaton equaton, Equaton (37), not only fo the ntal space, but fo the defomed space as well. Intoduce the metc tenso of the defomed space as ( ) ( ), ( ) Hee, ndex coesponds to the defomed space and the stans and ae assumed to be small n compason wth unty. As shown by Vaslev & Fedoov (6, ), fo the poblem unde study, the Ensten tenso satsfes the consevaton equaton fo both ntal and defomed states f G(, ) G(,), hee G s specfed by Equaton (4). As a esult, we ave at the follown equaton: h h h t h Fo and h ths equaton educes to Equaton (9) of the theoy of elastcty. Substtutn and h / h fom Equatons (35) and (36) n Equaton (39), we et 3 3 R c c ( ) 3 If we substtute the stans expessed n tems of stesses wth the ad of Equatons (7) n Equaton (4), we can obtan the second equaton whch, ben added to Equaton (37), povdes the set of two equatons fo the stesses and. The fnal fom of ths set s (39) (4) s u ( )(3s )(s ) (4) u ( )u ( )s ( k )s ( ) ( k )s ( )u ( 3s )[ u ( )s 6s( ks u ) (4) whee s, u ( ), c c ( ) ( ) E k,,, E and, ae specfed by Equatons (3) and (34). The obtaned equatons must be nteated unde the follown bounday condtons: () ( ) and ( R) whch yeld u( ), s( ) (43) As an example, consde an othotopc sphee wth zeo Posson s atos,.e., take. Then, Equaton (4) s smplfed as u u s ( k )s ( ) ( k )s u (3s )[ u s 6ks ) (44) and ncludes only one mateal paamete k E / E. Numecal (MAPLE-7) solutons of Equatons (4) and (44) unde bounday condtons n Equatons (43) fo. and thee values of paamete

8 Appled Physcs Reseach Vol. 5, No. ; 3 k (.9,.,.5 ) ae pesented n Fue. (a) (b) Fue. Dependences of the nomalzed adal (a) and ccumfeental (b) stesses c on the adal coodnate fo. and vaous values of paamete k Fo compason, the dashed lnes demonstate the classcal soluton n Equatons (33) fo an sotopc sphee wth zeo Posson s ato whch takes the fom 3, 3 (45) As can be seen, fo k.9 the stesses ae snula at the sphee cente, fo k the stesses ae fnte and athe close to the classcal soluton, wheeas fo k.5, the stesses ae zeo at the sphee cente. Fo the sphee whose adus R s close to the avtatonal adus,.e., fo.99, the stesses ae shown n Fue 3 fo k.5 (sold lnes) and fo k (dashed lnes). Note that the classcal soluton n Equatons (45) fo the sphee wth.99 ves the stesses that ae by an ode of mantude lowe than the stesses pesented n Fue 3. Fo, the numecal pocedue does not convee. Fue 3. Dependences of the nomalzed stesses c on the adal coodnate fo k.5 () and fo k ( ) and.99

9 Appled Physcs Reseach Vol. 5, No. ; 3. GTR Soluton fo an Othotopc Sphee Coveed wth Lqud As follows fom Fue 3, at the cente of the othotopc sphee wth k.5 and.99 the stesses ae zeo. On the sphee suface, the adal stess s zeo, wheeas the ccumfeental stess s extemely hh (ecall that c ). Real mateals cannot take such hh stess and we can expect that whle R educes to, the suface laye of the sphee fals unde nceasn stess. To evaluate the condtons unde whch such falue occus, we apply the stess ntensty whch fo the poblem unde study has the follown fom (Jones, 9): (46) The mateal fals when eaches some ultmate value establshed expementally. Note that f, and the falue does not occu. The condton p s satsfed fo the lqud whch does not fal unde any pessue p. Fo a lqud sphee, the pessue can be found fom Equatons (37) whch s smplfed as p (3p )( p ) ( ) (47) and has the follown soluton: p( ) C 3C n whch p p / c and the nteaton constant C can be found fom the bounday condton on the sphee suface,.e., The esultn expesson fo the pessue s (Syne, 96) (48) p( ) (49) p( ) 3 The dea of the poposed soluton s demonstated n Fue 4 whch coesponds to. and k.5. (5) Fue 4. Dependences of the nomalzed stesses and pessue on the adal coodnate stesses n sold sphee pessue n lqud stess ntensty n sold sphee 3

10 Appled Physcs Reseach Vol. 5, No. ; 3 Usn the stesses pesented n Fue, we can plot the stess ntensty n Equaton (46) as a functon of (the dotted lne n Fue 4). Assume that eaches the ultmate value at some adus s ( s.5 n Fue 4). Then, the mateal of the extenal pat of the sphee (s ) can be smulated wth lqud. The pessue n lqud s specfed by Equaton (5) whch allows us to fnd the pessue p( s ) s. The ntenal pat of the sphee ( s ) s elastc and othotopc. The stesses can be found by nteaton of Equatons (4) and (44) unde the follown bounday condtons: u( ) and s( s ) ps. Now etun to the case. Fo the lqud sphee, the pessue specfed by Equaton (5) becomes nfntely hh at the sphee cente f the sphee adus R s educed to 9/8 (Wenbe, 97). Howeve, fo the lamnated sphee unde consdeaton, the cental pat s a sold othotopc sphee wth k.5 (Fue 3) and thee s no snulaty at the sphee cente. Fo a sold othotopc o sotopc sphee, the numecal soluton ves hh ccumfeental stess on the sphee suface R and does not convee f R whch means that ethe the soluton s snula at R o that the soluton does not exst fo R. Unfotunately, the numecal soluton does not allow us to dentfy whch of these two cases takes place. Howeve, fo a sphee wth a lqud suface laye, we have the analytcal soluton, Equaton (48), whch can be used to clea out the stuaton. Takn and n Equaton (48), we et p / 3 whch has no physcal meann, fst, because the pessue cannot be neatve and, second, because the bounday condton n Equaton (49) cannot be satsfed. Thus, the GTR poblem has no soluton fo the sphee wth adus whch s equal to the avtatonal adus. The eason fo ths s assocated wth Equaton (38) n whch the last tem s analoous to the body avtaton foce n Equaton (6) of the classcal avtaton theoy. Ths foce (o the space densty n Equatons (4)) becomes nfntely hh on the sphee suface R f R. It seems evdent that n ths case the bounday condton n Equaton (49) cannot be satsfed, because the nfntely hh foce actn on the suface R cannot be accompaned wth zeo pessue actn on ths suface.. Concluson The foeon esults allow us to ave at the follown conclusons. As follows fom Equaton (5) whch lnks the avtatonal adus and the value of the metc coeffcent on the sphee suface and s vald fo any model of the sphee mateal, the suface coespondn to the avtatonal adus s always located nsde the sphee. Thus, fo the metc coeffcent of the extenal space specfed by Equaton (5), cannot become equal to and the snulaty of the metc coeffcent n the extenal space does not exst espectve of the stuctue and the mateal of the sphecal sold. The condton accodn to whch the GTR soluton fo the extenal space must deeneate to the classcal soluton at a dstance fom the sphee suface actually means that the eomety of the space nsde the sphee s Eucldean, and that Remannan eomety of the nsde space can be teated as a mathematcal model of a non-homoeneous and ansotopc Eucldean space. As shown n Sectons 8 and 9, both classcal and GTR solutons fo the othotopc sphee whose ccumfeental modulus s hhe than the adal modulus, n contast to the solutons fo the ncompessble lqud and the sotopc elastc sold sphees, do not demonstate the snulaty at the sphee cente f the sphee adus educes to the avtatonal adus. It s shown that fo a specally constucted sphee whch conssts of an ntenal othotopc elastc sold coveed wth ncompessble lqud and whose extenal adus s equal to the avtatonal adus, the GTR soluton does not exst. Refeences Feynman, R. P., Mono, F. B., & Wane, W. G. (995). Feynman Lectues on Gavtaton. Addson-Wesley Publsn Company. Fock, V. (959). The Theoy of Space, Tme and Gavtaton. London: Peamon Pess. Hawkn, S. W., & Penose, R. (97). The snulates of avtatonal collapse and cosmoloy. Poc. of the Royal Soc. A, 34, Jones, R. M. (9). Defomaton Theoy of Plastcty. Blacksbu, Vna: Bull Rde Publshn. Landau, L. D., & Lfshtz, E. M. (96). Feld Theoy. Moscow: Nauka (n Russan). Lounov, A. A., & Mestvshvl, M. A. (985). Foundatons of Relatvstc Gavtaton Theoy. Moscow State Unvesty (n Russan). Schwazschld, K. (96a). Ube das Gavtatonsfeld enes Massenpunkes nach de Enstenschen Theoe. 4

11 Appled Physcs Reseach Vol. 5, No. ; 3 Stzunsbechte de Deutschen Akademe de Wssenschaften zu Beln, Klasse fu Mathematk, Physk, und Technk. 89. Schwazchld, K. (96b). Ube das Gavtatonfeld ene Kuel aus ncompessble Flussket nach de Enstenschen Theoe. Stzunsbechte de Deutschen Akademe de Wssenschaften zu Beln, Klasse fu Mathematk, Physk, und Technk. 44. Syne, J. L. (96). Relatvty: the Geneal Theoy. Amstedam: Noth Holland. Vaslev, V. V. (989). Stessed state of solds and some eometcal effects. Mechancs of Solds, 5, Vaslev, V. V., & Fedoov, L.V. (6). Geometc theoy of elastcty and shape optmzaton of solds. Mechancs of Solds,, 6-7. Vaslev, V. V., & Fedoov, L.V. (). On snula solutons n sphecally symmetc ststc poblem of eneal elatvty. Appled Physcs Reseach, 4(), 66-74, Wenbe, S. (97). Gavtaton and Cosmoloy. New Yok, NY: Wlley. 5