The Precise Inner Solutions of Gravity Field Equations of Hollow and Solid Spheres and the Theorem of Singularity

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1 Intnational Jounal of Astonomy and Astophysis,,, 9-6 doi:.6/ijaa..6 Publishd Onlin Sptmb ( Th Pis Inn Solutions of Gavity Fild Equations of Hollow and Solid Sphs and th Thom of Sinulaity Abstat Xiaohu Mi Institut of Innovativ Physis in Fuzhou, Dpatmnt of Physis, Fuzhou Univsity, Fuzhou, China ivd May 6, ; visd Jun 8, ; aptd July, Th pis inn solutions of avity fild quations of hollow and solid sphs a alulatd in this pap. To avoid spa uvatu infinit at th nt of solid sph, w st an intal onstant to b zo ditly at psnt. Howv, aodin to th thoy of diffntial quation, th intal onstant should b dtmind by th known bounday onditions of sphial sufa, in stad of th mti at th sphial nt. By onsidin that fat that th volums of th dimnsional hollow and solid sphs in uvd spa a diffnt fom that in flat spa, th intal onstants a povd to b nonzo. Th sults indiat that no matt what th masss and dnsitis of hollow sph and solid sph a, th xist spa-tim sinulaitis at th nts of hollow sph and solid sphs. Manwhil, th intnsity of pssu at th nt point of solid sph an not b infinit. That is to say, th matial an not ollaps towads th nt of so-alld blak hol. At th nt and its nihboin ion of solid sph, pssu intnsitis bom nativ valus. Th may b a ion fo hollow sph in whih pssu intnsitis may bom nativ valus too. Th ommon hollow and solid sphs in daily liv an not hav suh impntabl haatistis. Th sults only indiat that th sinulaity blak hols pdiatd by nal lativity a ausd by th dsiptiv mthod of uvd spa-tim atually. If blak hols xist ally in th univs, thy an only b th Nwtonian blak hols, not th Einstin s blak hols. Th sults vald in th pap a onsistnt with th Hawkin thom of sinulaity atually. Thy an b onsidd as th patial xampls of th thom. Kywods: Gnal lativity, Inn Solutions of Hollow and Solid Sphs, Blak Hol, Thom of Sinulaity. Intodution W know that th stati solutions of th Einstin s quation of avity fild with sphial symmty a th Shwazshild solutions whih inlud inn and xtnal ons. W onsid a stati and unifom sph with adius and onstant dnsity, inn pssu intnsity p is latd to oodinat but dos not dpnds on tim. By onsidin stati ny momntum tnso of ida fluid, th Shwazshild inn solution is [,]. dt ds d d sin d () H 8 G. Th mti is finit at th nt point of sph. Howv, it should b pointd out that in th poss of solvin th Einstin s quation of avity fild, what w obtain is atually A () It an b povd basd on () that th spa uvatu is infinit at point. In od to avoid th infinity, w lt intal onstant A to b zo ditly in th unt thoy. Howv, aodin to th thoy of diffntial quation, intal onstant should b dtmind by th known bounday onditions on sphial sufa, in stad of th mti at th sphial nt whih is unknown. By onsidin th fat that th volum of sph in uvd spa is diffnt fom that in flat spa, Copyiht Sis.

2 X. C. MEI w an pov A. Thfo, no matt what th mass and dnsity of solid sph a, th uvatu infinity at th nt of sph is invitabl. On th oth hand, aodin to th unt thoy, th inn pssu intnsity of sph is []. p () On th sphial sufa w hav p. To mak pssu intnsity to b finit at th nt of sph, w hav to intodu a onstaint ondition fo sphial adius with 8 9 o () 9 8 H GM is th Shwazshild adius. If 9 8, pssu intnsity will bom infinit. In this as, stabl solution is impossibl and matial would ollaps towads th nt of sph so that sinulaity blak hols appa. Howv, if intal onstant A, pssu intnsity () and onstaint ondition () will b hand. All alulations basd on () and () about hih dnsity lstial bodis in th unt astophysis should b onsidd. Lt s fist stitly alulat th solutions of avity fild quations of hollow and solid sphs, and thn disuss th poblms of sinulaitis blow.. Th Stit Inn Solution of Gavity Fild of Hollow Sph Suppos that th inn adius of hollow sph is and th xtnal adius is, th avity mass is M. Th ion I with and th ion I with a vauum. Th ion I insid two sphial shlls with is omposd of omplt liquid with onstant dnsity and pssu intnsity p. Baus matial is distibutd with sphial symmty, th mti an b wittn as ) ds v ( dt ( ) d d sin d (5) Th solution of th Einstin s quation of avity in th ion I is th wll-known Shwazshild mti with v A A (6) In od to dtmin intal onstant A, w ompa (6) with th Nwtonian thoy und th asymptoti ondition with GM, (7) H M is th stati avity mass of hollow sph in th Nwtonian thoy. By ompain (6) with (7), w obtain A GM. So w hav th sam sult fo hollow sph, GM GM (8) To alulat th mti in th ion I bnath two sphial shlls, th mixin ny momntum tnso of omplt fluid is usd [] T, T T T p, (9) Th Einstin s quation of avity fild is 8 GT () Aodin to th standad podu of alulation in nal lativity, w obtain 8G ( ) ( ) v 8Gp ( ) v v v v 8Gp Th intal of () is ( ) () () () A () In whih 8 G and A is an intal onstant. W will pov A in th nxt stion. () minus () thn multiplid by and onsidin (), w obtain Th intal of (5) is dp v p d p v ( ) (5) B (6) H B is an intal onstant. By onsidin (), () and (6), w obtain v ( ) ( ) v B G (7) Copyiht Sis.

3 X. C. MEI On th oth hand, by takin th diffntial of () with spt to, w t ( ) A (8) Substitutin () and (8) in (7), w t ( ) A v A v B G By onsidin th lation (9) an b wittn as h d d P v ( ) K Th intal of () is P ( ) v v ( ) v d v ( ) (9) d, B K A () () 8 G A B K C v ( ) P( )d P( ) d d H C is an intal onstant. If lt A in () and (), w ah th sult of unt thoy [] B C v ( ) ( ) In whih onstants B B G A h lt, w hav Q () () 8, C C ln P( )d Q ( )d Q ( )d A A A A Th foms of onstants. If () (5) (6), i i A a omplx, but it is unnssay fo us to wit thm out. W hav d Q A ln A 6 ln( ) ln (7) w wit F Q ( )d ( ) A/ ln D lt in (8), w t F Thfo, wit () as K d F A ( 7 ) 5 A6 ( 7) ln In th ion A6/( 7) 6 57 (8) (9) () D is finit at point. W an ( ) F v C B D I, th mti an b wittn as at last F ds C B D dt A d d sin In th vauum ion d () () I of hollow sph avity, th Copyiht Sis.

4 X. C. MEI solution of th Einstin s quation of avity fild is still th Shwazshild solution and, A A Lt s dtmin th intal onstants A, C blow. () A, B. Th Calulations of Intal Constants fo Hollow Sph By onsidin th ontinuity of mti tnsos on th xtnal sphial sufa, aodin to (8), () and (), w hav GM GM 8G A () F GM C BD Similaly, on th intnal sph sufa with, aodin to (), (), (5) and (6), w hav (5) GM 8G A (6) F A C BD (7) Lt s pov A in uvd spa now. If spa is flat, th lation btwn mass M and volum V of hollow sph is M V with G V (8) It should b mphasizd that M is th Nwtonian avity mass. W intodu it by onsidin th asymptoti lation (7) btwn th Einstin s thoy and th Nwtonian thoy of avity. Substitut (8) in (7), w t A. This is just th unt alulatin sult of nal lativity. Howv, (8) an not hold in uvd spa. Baus th is a lnth ontation alon th dition of adius, w should hav dl dl and d d, so dvdld dv. In th uvd spa, th volum should b alulatd by th followin fomula V d d V (9) A Th intal of (9) is diffiult. If th thid itm in adial sin is nltd, w obtain []. d V asin 5 5 Q,, 5 () If w onsid fato A in (9), th intal boms mo omplx. So in uvd spa, w hav: () M M Substitut () in (7), w t GM A () Baus, and an b hosn abitaily, w hav M and A in nal. Thfo, fom (5) and (7), w obtain C B GM GM F F D D F F D D D () GM D GM () Similaly, baus, and a abitay, w hav B and C in nal. In this way, all intal onstants a dtmind. In th ion I of sphial avity, w an wit () as GM, GM (5) Aodin to th Nwtonian thoy, th matial distibutd outsid th sphial avity with sphial symmty dos not afft th avity fild in th avity. But aodin to (5), it will hav som fft fo th avity.. Th Sinulaity of th Inn Mti of Hollow Sph Aodin to (5), th mti and uvatu has sinulaity at th point. This is inhnt sinulaity whih an not b liminatd by th oodinat tansfomation. Th siousnss of poblm is that fo any hollow sph omposd of ommon matial, no matt what a its mass and dnsity, sinulaity always xists at its nt. This dos not a with patial obsvation. It is impossibl atually. On th oth hand, w onsid Copyiht Sis.

5 X. C. MEI () and lt A o (6) A It sms that th is a sinulaity sufa bnath two sphial shlls. W now disuss this poblm. Th al numb solution of (6) is A A A A (7) Howv, if is ally a al numb, th followin lation should b satisfid Lt A in (6) and onsidin (), w hav GM GM A (8) (9) In th wak fild, th itm ontainin ΔM an b nltd. By onsidin (8) and (9), w t 9.6 (5) 7 8G W know that vn fo hih dnsity lstial body just as whit dwaf, th diffn is still vy small whn w do alulation basd on both nal lativity and th Nwtonian thoy. Th matial dnsity of whit dwaf is K m. By usin this valu in (5), w t 7.7 m, simila to th siz of whit dwaf. Fo ommon alaxy, w hav K m. By usin this valu in (5), w hav m, whih is just th siz of alaxis. So (5) an b satisfid fo ommon sphs and (7) boms GM GM (5) By dvlopin (5) into th Taylo sis, if M, w hav. That is to say, th is a sinulaity sufa in th avity. Baus th mti of avity is (), in stad of (), th is no sinulaity sufa in th ion insid th hollow sph. If M, w hav. In this as, th is a sinulaity sufa in th ion. Aodin to (6), th pssu intnsity in th hollow sph is ( ) v p B F C BD B (5) If th is a sufa with adius insid th hollow sph on whih w hav C B D (5) Th pssu intnsity on th sufa will bom infinit. Thfo, if th xists blak hol in hollow sph, th blak hol would b a sphial sufa. Substitut () and (5) in (5), w t D G V D GM F F D G V GM F F (5) Baus th adii and a abitay, fom (5) that w may find a pop so that (68) an b satisfid. Howv, on this sphial sufa omposd of blak hols, spa-tim has no sinulaity. That is to say, th sufa of spa-tim sinulaity dos not ovlap with th sufa on whih matial ollapss. This is inomphnsibl. It should b notd that up to now w hav no any stition on th mass and dnsity of hollow sph. This sult indiats that ommon hollow sphs may b unstabl. Thy may ollaps into th blak hol of sphial sufa! Similaly, baus th intnal and xtnal adii a abitay, lt o, w hav p and p in nal. Baus th hollow sph is plad in vauum without matial outsid and insid its two sufas, this sult is also inomphnsibl. Th sinulaitis of hollow sph a shown in Fiu. It is obvious that th sults an not b tu. Fiu. Th sinulaity of hollow sph. Copyiht Sis.

6 X. C. MEI 5. Th Sinulaitis of Solid Sph s Mti and Blak Hols Aodin to th psnt alulation of nal lativity, th intnal mti of a ommon solid sph has no sinulaity whn th adius of sph is at than th Shwazshild adius. Aodin to th stit alulation in this pap, th situation is ompltly diffnt. Th solid sph is a spial situation of hollow sph whn its intnal adius boms zo. Th intnal mti of solid sph is still dsibd by (), but th onditions of bounday a diffnt. On th sphial sufa, w hav GM A (55) GM F C B D (56) In od to dtmin A in (55), w hav to know th lation btwn M and. In uvd spa, w hav M d d V (57) A By substitutin (57) in (55), w an did A in pinipl. W hav A in nal. If suppos A, w hav d M 5 O, 5 M Baus M, th bounday ondition (55) an not b satisfid. W stimat th manitud of volum s han in uvd spa basd on (58). Lt V V V and omittin hih od itms, w hav (58) V G (59) V 5 5 Fo nuton stas, w hav ~ and ~, so VV.8. If onsidin th univs as a ~ 7 ~ 6 unifom sph, w hav and, so VV.8. Fo so-alld blak hol, w hav G aodin to () and VV 5.7. Fo ommon sphs just as th sum and th ath, V V is a vy small but non-zo quantity. Aft A is dtmind, by substitutin it into (8) and (9), w an dtmin F and D. Howv, w an not yt dtminat C and B only basd on (56). Anoth ondition is ndd. By onsidin th fat that th pssu intnsity on th sufa of sph should b zo with p, fom (6) and () w hav B F C BD (6) Fom (56) and (6), w obtain C B GM GM D F (6) Now, all intal onstants a dtmind. Th intnal mti of solid sph is F ds C B D t d A d d sin d (6) Baus F and C BD in nal, w hav and at point, so th infinit of spa-tim uvatu also appas at th nt of sph. Th pssu intnsity of solid sph is also psntd by (5), but th intal onstants should b psntd by (6). In nal, w hav F and B D, so w hav p C. That is to say, no mat what a th mass and dnsity of sph, th pssu intnsity at th nt of sph an not b infinit. So-alld th sinulaity blak hols in whih matial ollapss towads its nt a impossibl. Manwhil, th pssu intnsity may bom nativ valu at th nt of sph and its naby ion too. On th oth hand, if th is sphial sufa with adius so that w hav C B D (6) th pssu intnsity may bom infinit on th sufa. Substitut (6) in (6), w obtain D D (65) F Thfo, if blak hol xists in solid sph, it an Copyiht Sis.

7 X. C. MEI 5 Fiu. Th sinulaity of solid sph. only tak th fom of a sphial sufa. Suh solid sph is not stabl fo matial will ollaps to its sphial sufa. But aodin to ou ommon xpins th is no sinulaity of spa-tim uvatu on th sphial sufa. Suh sult is also unanny. Th sinulaity of solid sph is shown in Fiu. Noti that w did not impos any stition fo mass and dnsity, and th a so many stan haatistis fo ommon solid sphs. Th sults a ompltly diffnt fom th unt undstandin in nal lativity. Th thoy of sinulaity blak hol in th unt astophysis and osmoloy has to b onsidd. 6. Disussions on th Thom of Sinulaity and th ationality of Cunt Sinulaity Blak Hol Thoy S. W. Hawkin t. povd th thom of sinulaity by mans of th mthod of diffntial omty []. Th thom was basd on th pquisit onditions.. Gnal lativity was tnabl.. Th law of ausality was tnabl.. Th w som points in spa-tim at whih matial dnsitis w non-zo. Th thom laimd that if thss th onditions w satisfid, sinulaity invitably xistd in spa-tim. Hawkin t. onsidd sinulaitis as th binnin and ndin of tim. Th Bi Ban thoy was onsidd as th binnin of tim and th blak hols w add as th ndin of tim. W not that th thom had no stition on matial s mass and dnsity and did not dmand that sinulaitis w mbodid in matial. That is to say, aodin to th thom, sinulaitis may ba in vauum. In od to avoid this mbaassin situation, Pnos poposd th so-alld pinipl of th univs supviso. Th pinipl dlas that th xists th univs supviso who pohibit th appaan of ba sinulaitis in vauum. In oth wod, du to th xistn of th univs supvisos, all sinulaitis will b wappd in th nts of blak hols with at masss and hih dnsitis. Aodin to th solutions of th Einstin s quation of avity, th xist Shwazshild blak hols with sphial symmty and th K blak hols with axial symmty and so on. Th sinulaitis w hiddn in th nts of matial. In this way, thy an not b pivd ditly, and physiists sm tolat thi xistn. Th sults vald in th pap a onsistnt with th Hawkin thom of sinulaity atually. W an onsid thm as th patial xampls of th thom. By onsidin th fat that th volum of hollow and solid sphs in uvd spa a diffnt fom that in flat spa, th stit alulation vals that sinulaitis an not b avoidd at th nts of ommon hollow and solid sphs with small masss and low dnsitis. On th oth hand, baus th pssu intnsity an not b infinit at th nt of sph, matial an not ollaps towads th sphial nt. Also th sult shows that th pssu intnsity may bom nativ valus at th nt and its naby ion. Manwhil, th may b uvd sufas insid th ommon hollow and solid sphs on whih pssu intnsitis an bom infinit so that matial will ollaps to thm. But th spa-tim uvatus a still finit on th sufas. Th sufas of spa-tim sinulaity do not always ovlap with th sufas with infinit pssu intnsitis. All ths haats an not a with ou patial xpins of ommon hollow and solid sphs. Thy a inomphnsibl in physis. Aodin to th unt undstandin, th blak hols xist at th nt of Quasas. Howv, aodin to th obsvations of udolf E. Shild and Dayl J. Lit, th nt of Quasa is a los objt, alld MECO (Massiv Etnally Collapsin Objt) [5]. It is not a sinulaity blak hol, and is suoundd by a ston manti fild. Th obsvation of udolf E. Shild was onsistnt with th alulation and analyss in this pap. That is to say, if th a blak hols in th univsal spa, thy an only b th Nwtonian blak hols, not th Einstin s sinulaity blak hols! Mo ssntially, th tu wold xluds infinits. A ot thoy of physis an not tolat th xistn of infinits, spially sinulaitis in daily lif s hollow and solid sphs omposd of ommon matial. It is wll known that th histoy of physis is on to ovom infinits. Modn physis ows up in th poss to sumount infinits. As vald in this pap, sinulaity in nal lativity is atually ausd by th dsiption mthod of uvd spa-tim. Physiists and osmoloists should tak autious and indulous attitud towad th poblms of sinulaity blak hols. It is not a sintifi attitud to onsid sinulaity blak hols as objtiv xistn without any qustion to thm. W Copyiht Sis.

8 6 X. C. MEI should think in dp, whth o not ou thoy has somthin won. Whn w njoy th bauty and symmty of th Einstin s thoy of avity, mmb that w should not nlt its limitations and possibl mistak. 7. fns [] J.. Oppnhim and H. Snyd, On Continud Gavitational Contation, Physial viw, Vol. 56, No. 5, 99, pp [] Zhan Yonli, Intodu to lativity, Th Publishin Company of Yunnan Popl, Kunmin, 989, p. 88. [] L. B. Fn, X. C. Liu and M. C. Li, Gnal lativity, Jilin Sin Publishin Company, Jilin, 995, p. 9. [] S. W. Hawkin and G. F.. Eills, Th La Sal Stutu of Spa Tim, Cambid Univsity Pss, Nw Yok, 97. [5]. E. Shild, D. J. Lit and S. L. obtson, Blak Hol o Mo: Didd by a thin Luminous in Stutu Dp within Quasa Q , Jounal of Cosmoloy, Vol. 6,, pp.-7. Copyiht Sis.