Singularities in Global Hyperbolic Space-time Manifold

Transcription

1 MPRA Much Persoal RePEc Archve Sgulartes Global Hyperbolc Space-tme Mafold Haradha Mohaja Assstat Professor, Premer Uversty, Chttagog, Bagladesh. 5 March 16 Ole at MPRA Paper No. 8953, posted 9 November 17 5:33 UTC

2 Asa Joural of Appled Scece ad Egeerg, Volume 5, No 1/16 Sgulartes Global Hyperbolc Space-tme Mafold Haradha Kumar Mohaja Assstat Professor, Faculty of Busess Studes, Premer Uversty, Chttagog, Bagladesh E-mal: Abstract If a space-tme s tmelke or ull geodescally complete but caot be embedded a larger space-tme, the we say that t has a sgularty. There are two types of sgulartes the spacetme mafold. Frst oe s called the Bg Bag sgularty. Ths type of sgularty must be terpreted as the catastrophc evet from whch the etre uverse emerged, where all the kow laws of physcs ad mathematcs breakdow such a way that we caot kow what was happeed durg ad before the bg bag sgularty. The secod type s Schwarzschld sgularty, whch s cosdered as the ed state of the gravtatoal collapse of a massve star whch has exhausted ts uclear fuel provdg the pressure gradet agast the wards pull of gravty. Global hyperbolcty s the most mportat codto o causal structure space-tme, whch s volved problems as cosmc cesorshp, predctablty, etc. Here both types of sgulartes global hyperbolc space-tme mafold are dscussed some detals. Keywords: Bg Bag, global hyperbolcty, mafold, FRW model, Schwarzschld soluto, space-tme sgulartes. 1. Itroducto I the Schwarzschld metrc ad the Fredma, Robertso-Walker (FRW) cosmologcal soluto cotaed a space-tme sgularty where the curvature ad desty are fte, ad kow all the physcal laws would break dow there. I the Schwarzschld soluto such as a sgularty s preset at r whch s the fal fate of a massve star (Mohaja 13a), whereas the FRW model t s foud at the epoch t (bg bag), whch s the begg of the uverse, where the scale factor S t also vashes ad all objects are crushed to zero volume due to fte gravtatoal tdal force (Mohaja 13b). Schwarzschld metrc of Este equato s establshed assumg a star solated from all the gravtatg bodes. It s also mportat for the terpretato of black hole. Schwarzschld establshed hs metrc by cosderg asymptotcally flat solutos to Este s equato (Mohaja 13a). Fredma, Robertso-Walker (FRW) model s establshed o the bass of the assumpto that the uverse s homogeeous ad sotropc all epochs. Eve though the uverse s clearly homogeeous at the local scales of stars ad cluster of stars, t s geerally argued that a overall homogeety wll be acheved oly at a large eough scale of about 14 bllo lght years. I the 196s, Stephe W. Hawkg ad Roger Perose dscovered the sgulartes the FRW model (Hawkg ad Ells 1973, Mohaja 13b). Hawkg ad Perose (197) expla that 1

3 Asa Joural of Appled Scece ad Egeerg, Volume 5, No 1/16 sgulartes arse whe a black hole s created due to gravtatoal collapse of massve bodes. A space-tme sgularty whch caot be observed by exteral observers s called a black hole. Posso (4) descrbes that whe the black hole s formulated due to gravtatoal collapse, the space-tme sgulartes must occur. The exstece of space-tme sgulartes follows the form of future or past complete ospacelke geodescs the space-tme. Such a sgularty would arse ether the cosmologcal scearos, where t provdes the org of the uverse or as the ed state of the gravtatoal collapse of a massve star whch has exhausted ts uclear fuel provdg the pressure gradet agast the wards pull of gravty (Mohaja 13c). We cosder a mafold M whch s smooth,.e., M s dfferetable as permtted by M. We assume that M s Hausdorff ad paracompact. Global hyperbolcty s the strogest ad physcally most mportat cocept both geeral ad specal relatvty ad also relatvstc cosmology. Ths oto was troduced by Jea Leray 1953 (Leray 1953) ad developed the golde age of geeral relatvty by A. Avez, B. Carter, Choquet-Bruhat, C. J. S. Clarke, Stephe W. Hawkg, Robert P. Geroch, Roger Perose, H. J. Sefert ad others (Sáchez 1). Each geerator of the boudary of the future has a past ed pot o the set oe has to mpose some global codto o the causal structure. Ths s relevat to Este s theory of geeral relatvty, ad potetally to other metrc gravtatoal theores. I 3, Atoo N. Beral ad Mguel Sáchez showed that ay globally hyperbolc mafold M has a smooth embedded 3- dmesoal Cauchy surface, ad furthermore that ay two Cauchy surfaces for M are dffeomorphc (Beral ad Sáchez 3, 5). Despte may advaces o global hyperbolcty however, some questos whch affected basc approaches to ths cocept, remaed usolved yet. For example, the so-called folk problems of smoothablty, affected the dfferetable ad metrc structure of ay globally hyperbolc spacetme M (Sachs ad Wu 1977). The Geroch, Krohemer, ad Perose (GKP) causal boudary troduced a ew gredet for the causal structure of space-tmes, as well as a ew vewpot for global hyperbolcty (GKP 197). I ths study, we descrbe how the matter felds wth postve eergy desty affect the causalty relatos a space-tme ad cause focusg the famles of o-spacelke trajectores. Here the ma pheomeo s that matter focuses the o-spacelke geodescs of the space-tme to pars of focal pots or cojugate pots due to gravtatoal forces.. Some Related Deftos I ths secto, we gve some deftos whch are related to our study (Mohaja 13e). The deftos wll provde ecessary formato to uderstad the paper perfectly. Mafold: A mafold s essetally a space whch s locally smlar to Eucldea space that t ca be covered by coordate patches but whch eed ot be Eucldea globally. Map : O O s sad to be a class m where O R ad O R C r r satsfed. If we choose a pot (evet) p of coordates 1 x,..., f the followg codtos are x o O ad ts mage p of

4 Asa Joural of Appled Scece ad Egeerg, Volume 5, No 1/16 1 r coordates x,..., x o O the by C map, we mea that the fucto s r-tmes r dfferetal ad cotuous. If a map s C for all r the we deote t by C ; also by C map we mea that the map s cotuous (Hawkg ad Ells 1973, Mohaja 16). A -dmesoal, M has a r C, real dfferetable mafold M s defed as follows: r C atlas U, where U are subsets of M ad correspodg U to ope sets R such that (fgure 1);. U cover M.e., M,. If U U U, the the map 1 : U U U U ope subset of R to a ope subset of s a R. are oe-oe maps of the r C map of a Codto () s very mportat for overlap of two local coordate eghborhoods. Now suppose U ad U overlap ad there s a pot p U U ad a pot r 1 1 U. Now r p, p r q 1 ad those of r be y,..., y U. Now choose a pot q. Let coordates of q be x 1,..., x. At ths stage we obta a coordate trasformato; U U M U U R q U p 1 1 r U Fgure 1: The smooth maps 1 o the -dmesoal Eucldea space R gvg the chage of coordates the overlap rego. y y x,..., x y 1 y x,..., x 3 1 y y x,..., x.

5 Asa Joural of Appled Scece ad Egeerg, Volume 5, No 1/16 The ope sets U, U ad maps 1 ad 1 are all -dmesoal, so that r C mafold M s r-tmes dfferetable ad cotuous,.e., M s a dfferetable mafold (Hawkg ad Ells 1973). Hausdorff Space: A topologcal space M s a Hausdorff space f for a par of dstct pots p, q M there are dsjot ope sets U ad U M such that p U ad q U (Mohaja 16). Paracompact Space: A atlas U, s called locally fte f there s a ope set cotag every p M whch tersects oly a fte umber of the sets U. A mafold M s called a paracompact f for every atlas there s locally fte atlas O, wth each O cotaed V be a tmelke vector, ad the paracompactess of mafold M mples that some U. Let there s a smooth postve defte Rema metrc K defed o M (Josh 1996). Compact Set: A subset A of a topologcal space M s compact f every ope cover of A s reducble to a fte cover (Mohaja 16). k Taget Space: A C -curve M s a map from a terval of R to M (fgure ). A vector 1 whch s taget to a C -curve t t t at a pot t s a operator from the space of all smooth fuctos o M to R ad s deoted by (Josh 1996); f t t f t t Lm s f b t s f t s. t a a t b Fgure : A curve a dfferetal mafold (Mohaja 13d). If x are local coordates a eghborhood of p the, t 4

6 Asa Joural of Appled Scece ad Egeerg, Volume 5, No 1/16 f t dx f. dt x t t. Thus, every taget vector at p M ca be expressed as a lear combato of the coordates dervates, 1,...,. Thus, the vectors x p x p spa the vector space T. The the vector x p space structure s defed by X Y f Xf Yf. The vector space Tp s also called the taget space at the pot p. A metrc s defed as; ds g dx dx (1) where g s a defte metrc the sese that the magtude of o-zero vector could be ether postve, egatve or zero (Mohaja 13d). The ay vector X Tp s called tmelke, ull, spacelke or o-spacelke respectvely f; g X, X, gx, X, gx, X, X, X g. () Oretato: Let B be the set of all ordered bass e for T p, the taget space at pot p. If e ad e j are B, the we have ej a je. If we deote the matrx a j the deta. A -dmesoal mafold M s called oretable f M admts a atlas U, such that wheever x U U j the the Jacba, det x j are local coordates j x U ad U j respectvely. The Möbous strp s a o-oretable mafold. A vector defed at a pot Möbous strp wth a postve oretato comes back wth a reversed oretato the egatve drecto whe t traverses alog the strp to come back to the same pot (Mohaja 15). J, where x ad Space-tme Mafold: Geeral Relatvty models the physcal uverse as a 4-dmesoal Hausdorff dfferetable space-tme mafold M wth a Loretza metrc g of sgature,,, whch s topologcally coected, paracompact ad space-tme oretable. These propertes are sutable whe we cosder for local physcs. As soo as we vestgate global features the we face varous pathologcal dffcultes such as the volato of tme oretato, possble o-hausdorff or o-papacompactess, dscoected compoets of space-tme, etc. Such pathologes are to be ruled out by meas of reasoable topologcal assumptos oly (Mohaja 13d). However, we lke to esure that the space-tme s causally well-behaved. We wll cosder the space-tme Mafold M, g, whch has o boudary. By the word boudary we mea the edge of the uverse whch s ot detected by ay astroomcal observatos. It s 3 commo to have mafolds wthout boudary; for example, for two-spheres S R o pot C 5

7 Asa Joural of Appled Scece ad Egeerg, Volume 5, No 1/16 S s a boudary pot the duced topology o the same mpled by the atural topology o 3 R (Mohaja 13d). All the eghborhoods of ay p S wll be cotaed wth S ths duced topology. We shall assume M to be coected.e., oe caot have M X Y, where X ad Y are two ope sets such that X Y. Ths s because dscoected compoets of the uverse caot teract by meas of ay sgal ad the observatos are cofed to the coected compoet where the observer s stuated (Mohaja 14a). It s ot kow f M s smply coected or multply coected. Mafold M s assumed to be Hausdorff, whch esures the uqueess of lmts of coverget sequeces ad corporates our tutve oto of dstct space-tme evets (Josh 1996). Hypersurface: I the Mkowsk space-tme ds dt dx dy dz, the surface t s a three-dmesoal surface wth the tme drecto always ormal to t. Ay other surface costat 1 -dmesoal mafold. If t s also a spacelke surface ths sese. Let S be a there exsts a C map : S M whch s locally oe-oe.e., f there s a eghborhood N for 1 every p S such that restrcted to N s oe-oe, ad s a N, the S s called a embedded sub-mafold of M. A hypersurface S of ay -dmesoal mafold M s defed as a 1 -dmesoal embedded sub-mafold of M. Let V p be the 1 - dmesoal subspace of there exsts a uque vector C as defed o T p of the vectors taget to S at ay p S from whch follows that a Tp ad s orthogoal to all the vectors V p (Mohaja 13d). a a Here s called the ormal to S at p. If the magtude of s ether postve or egatve at all a a b pots of S wthout chagg the sg, the could be ormalzed so that g 1. If a b g ab 1 the the ormal vector s tmelke everywhere ad S s called a spacelke hypersurface. If the ormal s spacelke everywhere o S wth a postve magtude, S s called a a tmelke hypersurface. Fally, S s ull hypersurface f the ormal s ull at S (Mohaja 15). 3. Basc Cocept of Geeral Relatvty The covarat dfferetatos of vectors are defed as; ab A ; A, A (3a) A ; A, A (3b) where sem-colo deotes the covarat dfferetato ad coma deotes the partal dfferetato (Mohaja 14a). By (3b) we ca wrte; A ; ; A ; ; R A, (4) 6

8 Asa Joural of Appled Scece ad Egeerg, Volume 5, No 1/16 where R (4a) ; ; s a tesor of rak four ad called Rema curvature tesor. From (4) we observe that the curvature tesor compoets are expressed regardg the metrc tesor ad ts secod dervatves. From (4a) we get; R. (5) Takg er product of both sdes of (4a) wth g oe gets covarat curvature tesor; R 1 g x x g x x g x x g x x Cotracto of curvature tesor (6) gves Rcc tesor; + g. (6) R g R. (7) Further cotracto of (7) gves Rcc scalar; From whch oe gets Este tesor as; Rˆ g R. (8) G 1 R R (9) where dv G G ;. The space-tme M, g s sad to have a flat coecto f ad oly f; R. (1) Ths s the ecessary ad suffcet codto for a vector at a pot p to rema ualtered after parallel trasported alog a arbtrary closed curve through p. Ths s because all such curves ca be shruk to zero, whch case the space-tme s smply coected (Hawkg ad Ells 1973). The eergy mometum tesor T s defed as; T u u (11) where s the proper desty of matter, ad f there s o pressure. A perfect flud s characterzed by pressure p px, the; 7

9 Asa Joural of Appled Scece ad Egeerg, Volume 5, No 1/16 T pu u pg. (1) The prcple of local coservato of eergy ad mometum states that; T ;. (13) The most approprate tesor of the form requred s the Este s tesor (9); the Este s feld equato ca be wrtte as (Mohaja 14b); 1 8 G c R gr T. (14) where G m kg s s the gravtatoal costat ad 8 c 1 m/s s the velocty of lght. Este troduced a cosmologcal costat for statc uverse solutos as; 1 8 G c R gr g T. (15) 4 I relatvstc ut G = c = 1, hece relatvstc uts (15) becomes; R 1 g R 8 T. (16) It s clear that dvergece of both sdes of (15) ad (16) s zero. For empty space T the R g, so that; R for (17) whch s Este s law of gravtato for empty space. 4. Causal Structure of Space-tme Mafold I Loretza geometry causalty plays a mportat role, as t dsplays a relatvstc terpretato of space-tme for both specal ad geeral relatvty. Causalty also appears as a frutful terplay betwee relatvstc motvatos ad geometrc developmets. Causal spacetme s establshed at the ed of the 197s, after the works of Carter, Geroch, Hawkg, Krohemer, Perose, Sachs, Sefert, Wu ad others (Hawkg ad Sachs 1974). No materal partcle ca travel faster tha the velocty of lght. Hece, causalty fxes the boudary of the space-tme topology. We assume that the tmelke curves to be smooth; wth future-drected taget vectors everywhere strctly tmelke, cludg ts ed-pots. A causal curve s a curve space-tme whch s owhere spacelke. A causal curve s cotuous but ot 8

10 Asa Joural of Appled Scece ad Egeerg, Volume 5, No 1/16 ecessarly everywhere smooth; ts taget vectors are ether tmelke or ull. A causal curve wll requred ed-pots f t ca be exteded as a causal curve ether to the past or the future. If a causal curve ca be exteded deftely ad cotuously to the past the t s called pastextesble. The future-extesble curve s defed smlarly. If a causal curve s both past ad future-extesble the t s called smply extesble (Hawkg ad Perose 197). A evet x chroologcally precedes aother evet y, deoted by x y, f there s a smooth future drected tmelke curve from x to y. If such a curve s o-spacelke the x causally precedes y,.e., y I x s the set of all pots of the space-tme x. The chroologcal future M that ca be reached from x by future drected tmelke curves. We ca thk of of all evets that ca be flueced by what happes at x. Now defed as (fgure 3), I I x y M x y /, ad x y M y x /. I x ad I x I x as the set of a pot x are Oe ca thk of I x as the set of all evets that ca be flueced by what happes at x. The causal future (past) of x ca be defed as; Also J J x y M x y /, x y M y x /. x y ad y z or x y ad y z mples x z. Hece, the closer ad boudary of I x ad past x I I of a pot x are defed respectvely as (Perose 197); x J x ad I x J y, where I s a topologcal boudary ad I s the closure of I. Chroologcal future I x z s Causal future J x y x Cut Null geodesc through x I x Null geodesc J x Chroologcal past 9

11 Asa Joural of Appled Scece ad Egeerg, Volume 5, No 1/16 Fgure 3: Removal of a closed set from the space-tme gves a causal future closed. Evets x ad s are ot causally coected. J x whch s ot Smlarly, the chroologcal (causal) future of ay set I J S S I xs x x J. xs, ad Smlarly, we ca defe the past subsets of space-tme. S M s defed as; The boudary of the future s ull apart from at S tself. If x s the boudary of the future but s ot the closure of S there s a past drected ull geodesc segmet through x lyg the boudary. Hece the boudary of the future of S s geerated by ull geodescs that have a future ed pot the boudary ad pass to the teror of the future f they tersect aother geerator ad the ull geodesc geerators ca have past ed pots oly o S (Hawkg 1994). Causally Covex Set: Let S ad T be ope subsets of a space-tme M, g, wth T S the T s called causally covex S f ay causal curve cotaed S wth edpots T s etrely cotaed T. I partcular, whe ths holds for S M, T s called causally covex. Aga f T s causally covex S ad U s a ope set such that T U S, the T s causally covex U (Mguzz ad Sáchez 8). Future Set ad Past Set: A ope subset F s a future set f I F F defed by I P P I x F but F. The past set P s. The boudary of a future set F s made of all evets x such that x. If x F the of course x F, sce F s a ope set. Achroal Set: A set S M s sad to be achroal f o two pots x, y S may be joed by a pecewse tmelke curve.e., there do ot exst x, y S such that y I x. Let F be a future set, the the boudary of F s a closed, achroal hypersurface. C -mafold that s a 3-dmesoal embedded Doma of Depedece of a Set: The future doma of depedece (the future Cauchy developmet) of a spacelke three-surace S, deoted by D S, s defed as the set of all pots x M such that every past-extedble o-spacelke curve from x tersects S,.e., D S = {x: every past-extesble tmelke curve through x meets S}. It s clear that S D S J S ad S beg achroal, D S I S. The past doma of depedece D S s defed smlarly. The full doma of depedece for S s defed as; DS D S D S (Josh 1993). 1

12 Asa Joural of Appled Scece ad Egeerg, Volume 5, No 1/16 Cauchy Surface: Let S be a closed achroal set. The edge of S s defed as a set of pots x S such that every eghborhood of x cotas y I x ad z I x wth a tmelke curve from z to y whch does ot meet S. A partal Cauchy surface S s defed as a acausal set wthout a edge. So that o o-spacelke curve tersects S more tha oce, ad S s a spacelke hypersurface. 5. The Global Hyperbolc Space-Tme A partally Cauchy surface s called a Cauchy surface S or a global Cauchy surface f DS M.e., f a set S s closed, achroal, ad ts doma of depedece s all of the space-tme, DS M. I aother way, f DS M.e., f every extesble o-spacelke curve tersect S, the S s sad to be a Cauchy surface (fgure 4). For a Cauchy surface S, edge S. The Cauchy developmet s the rego of spacetme that ca be predcted from data o S. Here S must be a embedded topologcal hypersurface ad must be also crossed by ay extesble causal curve (Hawkg 1966a,b). The exstece of a Cauchy hypersurface S mples that M s homeomorphc to t S, ad all Cauchy hypersurfaces are homeomorphc. Every o-spacelke curve M meets S oce ad exactly oce f S s a Cauchy surface. The relatoshp betwee the global hyperbolcty of M ad the oto of Cauchy surface s show fgure 4 (Hawkg ad Ells 1973): p S q Fgure 4: The spacelke hypersurface S s a Cauchy surface the sese that for ay p future of S, all past o-spacelke curves from p tersect S. The same holds for all future-drected curves from ay pot q past of S. Tme fucto s a cotuous fucto t : M R whch creases strctly o ay future-drected causal curve. If the levels t costat are Cauchy hypersurfaces, the t s a Cauchy tme fucto. The space-tme mafold has a Cauchy surface S. 5.1 Globally Hyperbolcty I mathematcal physcs, global hyperbolcty s a certa codto o the causal structure of a space-tme mafold. If M s a smooth coected Loretza mafold wth boudary, we say t 11

13 Asa Joural of Appled Scece ad Egeerg, Volume 5, No 1/16 s globally hyperbolc f ts teror s globally hyperbolc. Perose has called globally hyperbolc space-tmes the physcally reasoable space-tmes (Wald 1984). A space-tme M, g whch admts a Cauchy surface s called globally hyperbolc. A space-tme M, g whch admts a Cauchy surface s called globally hyperbolc. A ope set O s sad to be globally hyperbolc f, ) for every par of pots x ad y O the tersecto of the future of x ad the past of y has compact closure,.e., f a space-tme M, g s sad to be globally hyperbolc f the sets J x J y are compact for all x, y M (.e., o aked sgularty ca exst space-tme topology). I other words, t s a bouded damod shaped rego (damod-compact) ad ) strog causalty holds o O,.e., there are o closed or almost closed tme lke curves cotaed O (fgure 4). The t also satsfes that J x ad J y are closed x, y M. More precsely, cosder two evets x, y of the space-tme M, g, ad let C x, y be the set of all the cotuous curves whch are future-drected ad causal ad coect x wth y (Hawkg ad Ells 1973). Mkowsk space-tme, de Stter space-tme ad the exteror Schwarzschld soluto, Fredma, Robertso-Walker (FRW) cosmologcal solutos ad the steady state models are all globally hyperbolc. The Kerr soluto s ot globally hyperbolc, sce t represets rotatg model,.e., ot a statc model. O the other had at de Stter space-tme ad the Godel uverse are ot globally hyperbolc. The global hyperbolcty of M s closely related to the future or past developmet of tal data from a gve spacelke hypersurface (Josh 1996). The physcal sgfcace of global hyperbolcty comes from the fact that t mples that there s a famly of Cauchy surfaces Σ(t) for globally hyperbolc ope set O. A Cauchy surface for O s a spacelke or ull surface that tersects every tmelke curve O oce ad oly oce. Let x ad y be two pots of O that ca be joed by a tmelke or ull curve, the there s a tmelke or ull geodesc betwee x ad y whch maxmzes the legth of tmelke or ull curves from x to y (Hawkg 1994). 5. Cauchy Horzos of a Set Let S be a partal Cauchy surface. The N D S D S M ad N must be a proper subset of M. The boudary of N M ca be dvded to two portos. Now suppose that the future Cauchy developmet was compact. Ths would mply that the Cauchy developmet would have a future boudary called the Cauchy horzo, H S. Sce the Cauchy developmet s assumed to be compact, the Cauchy horzo wll also be compact. The S H S whch are respectvely called the future ad past Cauchy horzos of S. We ca wrte (Hawkg ad Perose 197); S x / x D S I x D S H, S S D I D. H ad 1

14 Asa Joural of Appled Scece ad Egeerg, Volume 5, No 1/16 H S s defed smlarly. H S I H S I S D S. s a achroal closed set. Also we ca wrte, The Cauchy horzo wll be geerated by ull geodesc segmets wthout past ed pots. Eve though M may ot be globally hyperbolc ad S s ot a Cauchy surface, the rego It D S or It D S for the mafold It N. Thus H S or H S s globally hyperbolc ts ow rght ad the surface S serves as a Cauchy surface Cauchy surface for M (fgure 5). represets the falure of S to be a global If every geodesc ca be exteded to arbtrary values of ts affe parameter, the t s geodescally complete. If a tmelke or causal curve ca be exteded deftely ad cotuously to the past (future), the t s called past-extesble (future-extesble). H S q λ H S D S Pot removed S t = D S Tmelke curve γ γ Pot removed H S H S Fgure 5: The space-tme obtaed by removg a pot from the Mkowsk space-tme s ot globally hyperbolc. The pot q does ot meet S the past. The evet p D S. The Cauchy horzo s the boudary of the shaded rego whch cossts of pots ot 13 D S. I globally hyperbolc space-tmes, there s a fte upper boud o the proper tme legths of o-spacelke curves two chroologcally related evets. Of course there s o lower lmt of legth for such curves except zero, because the chroologcally related evets ca always be joed usg broke ull curves whch could gve a arbtrary small legth curve betwee them. If S s Cauchy surface globally hyperbolc space-tme M, the for ay pot p the future of

15 Asa Joural of Appled Scece ad Egeerg, Volume 5, No 1/16 S, there s a past drected tmelke geodesc from p orthogoal to S whch maxmzes the legths of all o-spacelke curves from p to S (fgure 6). A mportat property of globally hyperbolc space-tme that s relevat for the sgularty theorems s the exstece of maxmum legth o-spacelke geodescs betwee a par of causally related evets. I a complete Remaa mafold wth a postve defte metrc ay two pots ca be joed by a geodesc of mmum legth ad fact such a geodesc eed ot be uque (Josh 1996). (I a sphere paths of great crcles are geodescs. Opposte poles ca be joed by a fte umbers of geodescs.) p I (p) S Fgure 6: The spacelke hupersurface S s a Cauchy surface the sese that for ay p future of S, all past drected o-spacelke curves from p tersect S. 6. Space-tme Sgulartes The exsteces of real sgulartes where the curvature scalars ad destes dverge mply that all the physcal laws break dow. Let us cosder the metrc; ds 1 dt dx dy dz (18) t whch s sgular o the plae t. If ay observer startg the rego t tres to reach the surface t by travelg alog tmelke geodescs, he wll ot reach at t ay fte tme, sce the surface s ftely far to the future. If we put t l t t the (18) becomes (Mohaja 13e); ds dt dx dy dz (19) wth t whch s Mkowsk metrc ad there s o sgularty at all (Clarke 1986). A tmelke geodesc whch, whe maxmally exteded, has o ed pot the regular space-tme ad whch has fte proper legth, s called tmelke geodescally complete. Now we shall dscuss some deftos related wth the sgularty (Clarke 1986). Defto: The geeralzed affe parameter (GAP) legth of a curve :,a M respect to a frame, 14 wth

16 Asa Joural of Appled Scece ad Egeerg, Volume 5, No 1/16 E E, a,1,,3 a at s gve by; E a 3 g 1 Es ds d where s taget vector ad E(s) s defed by parallel propogato alog the curve, ds startg wth a tal value E(). Defto: A curve :,a M s complete f t has fte GAP legth wth respect to some frame E at. If E, the f we take ay other frame E at we have E. Ths s because the correspodg parallel propogated frames satsfy (Mohaja 13e); E L E j j for a costat Loretz matrx L ad hece; E L E, where Sup L 1 j X L. Defto: A curve :,a M s termed extesble f there s o curve : M wth that,b b a such that, a. Ths s equvalet to sayg that there s o pot p M such s p as s a,.e., has o ed pot M. Defto: A space-tme s complete f t cotas a complete extesble curve. By the above deftos we ca say that a space-tme s called complete f t cotas a complete tmelke extesble curve. The Fredma Bg Bag models are geodescally complete, sce the curve defed by (Mohaja 13e); s S t s s Costat, = 1,, 3 s a geodesc whch s complete, havg o edpot M as s St. Mkowsk space s ot complete. The rego r m the Schwarzschld metrc s complete, whle the rego r m s ot a space-tme, sce the metrc s ot defed at r m. 15

17 Asa Joural of Appled Scece ad Egeerg, Volume 5, No 1/16 Defto: A exteso of a space-tme M, g s a sometrc embeddg : M M where M, g s a space-tme ad s oto a proper subset of M. By the above defto, Schwarzschld metrc s ot sgular at r m by Kruskal-Szekeres exteso (Kruskal 196, Szekeres 196). A space-tme s termed extesble f t has a exteso. Defto: A space-tme s sgular f t cotas a complete curve :,a M there s o exteso : M M for whch s extesble. 7. Schwarzschld Sgularty such that The Schwarzschld metrc whch represets the outsde metrc for a star s gve by (Mohaja 13e); 1 m m ds 1 dt 1 dr r d s d () r r If r s the boudary of a star the r r gves the outsde metrc as (). If there s o surface, () represets a hghly collapsed object vz. a black hole of mass m (wll be dscussed later). The metrc () has sgulartes at r = ad r = m, so t represets patches r m or m r. If we cosder the patches r m the t s see that as r teds to zero, the curvature scalar, R R 48m 6 r teds to ad t follows that r = s a geue curvature sgularty.e., space-tme curvature compoets ted to fty (Mohaja 13a). 8. Fredma, Robertso Walker (FRW) Model The FRW model plays a mportat role Cosmology. Ths model s establshed o the bass of the homogeety ad sotropy of the uverse as descrbed above. The curret observatos gve a strog motvato for the adopto of the cosmologcal prcple statg that at large scales the uverse s homogeeous ad sotropc ad, hece, ts large-scale structure s well descrbed by the FRW metrc. The FRW geometres are related to the hgh symmetry of these backgrouds. Due ths symmetry umerous physcal problems are exactly solvable, ad a better uderstadg of physcal effects FRW models could serve as a hadle to deal wth more complcated geometres (Mohaja 13b). I t,r,, coordates the Robertso-Walker le elemet s gve by; dr t r d s d ds dt S (1) 1 kr 16

18 Asa Joural of Appled Scece ad Egeerg, Volume 5, No 1/16 where k s a costat whch deotes the spatal curvature of the three-space ad could be ormalzed to the values +1,, 1. Whe k = the three-space s flat ad (1) s called Este de-stter statc model, whe k = +1 ad k = 1 the three-space are of postve ad egatve costat curvature; these corporate the closed ad ope Fredma models respectvely (fgure 7). Let us assume the matter cotet of the uverse as a perfect flud the by (14) ad (15), solvg (1) we get; 3S 4 S 3p, ad () 3S S 3k 8 S (3) where we have cosdered. If ad p the S. So S = costat ad S dcates the uverse must be expadg, ad S dcates cotractg uverse. The observatos by Hubble of the red-shfts of the galaxes were terpreted by hm as mplyg that all of them are recedg from us wth a velocty proportoal to ther dstaces from us that s why the uverse s expadg. For expadg uverse S, so by () ad (3) we get S. Hece S s a decreasg fucto ad at earler tmes the uverse must be expadg at a faster rate as compared to the preset rate of expaso. But f the expaso be costat rate as lke the preset expaso rate at all tmes the, S(t) k = 1 k = k = 1 t t t t1 t Fgure 7: The behavor of the curve S(t) for the three values k = 1,, +1; the tme t t preset tme ad t t1 s the tme whe S(t) reaches zero aga for k = +1. s the 17

19 Asa Joural of Appled Scece ad Egeerg, Volume 5, No 1/16 S S tt H. (4) Now H mples a global upper lmt for the age of ay type of Fredma models. So the age 1 of the uverse wll be less tha 1 H. The quatty H s called Hubble costat ad at ay gve epoch t measures the rate of expaso of the uverse. By observato H has a value somewhere the rage of 5 to 1 kms 1 Mpc 1. At S =, the etre three-surface shrks to zero volume ad the destes ad curvatures grow to fty. Hece, by FRW models there s a sgularty at a fte tme the past. Ths curvature sgularty s called the bg bag (Islam, Hawkg ad Ells 1973). 9. Cocluso I ths study we have dscussed the global hyperbolc space-tme mafold ad the sgulartes there. Here we have dscussed two types of sgulartes: ) Bg Bag sgularty, whch s foud Fredma, Robertso-Walker s cosmologcal soluto, ad cosders as the begg of the uverse; ) Black hole type sgularty s foud the Schwarzschld soluto, whch s the fal fate of a massve star. I the begg of the study we have provded some elemetary deftos of dfferetal geometry ad topology. The, we have dscussed the basc cocepts of geeral relatvty. We have also dscussed the causal structure of space-tme mafold. The we have dscussed global hyperbolcty to make the paper terestg to the readers. Refereces Beral, A.N. ad Sáchez, M. (3), O Smooth Cauchy Hypersurfaces ad Geroch s Splttg Theorem, Commucatos Mathematcal Physcs, 43(3): Beral, A.N. ad Sáchez, M. (5), Smoothess of Tme Fuctos ad the Metrc Splttg of Globally Hyperbolc Spacetmes, arxv:gr-qc/4111v3 15. Clarke, C.J.S. (1986), Sgulartes: Global ad Local Aspects Topologcal Propertes ad Global Structure of Space-tme (ed. P.G. Bergma ad V. de. Sabbata), Pleum Press, New York. Geroch, R.P. (197), Doma of Depedece, Joural of Mathematcal Physcs, 11: Geroch, R.P.; Krohemer, E.H. ad Perose, R. (GKP) (197), Ideal Pots Space-tme, Proceedgs of the Royal Socety, Lodo, A 37: Hawkg, S.W. (1966a), The Occurrece of Sgulartes Cosmology I, Proceedgs of the Royal Socety, Lodo, A94:

20 Asa Joural of Appled Scece ad Egeerg, Volume 5, No 1/16 Hawkg, S.W. (1966b), The Occurrece of Sgulartes Cosmology II, Proceedgs of the Royal Socety, Lodo, A95: Hawkg, S.W. (1994), Classcal Theory, arxv:hep-th/949195v1 3 Sep Hawkg, S.W. ad Ells, G.F.R. (1973), The Large Scale Structure of Space-tme, Cambrdge Uversty Press, Cambrdge. Hawkg, S.W. ad Perose, R. (197), The Sgulartes of Gravtatoal Collapse ad Cosmology, Proceedgs of the Royal Socety of Lodo, Seres A, Mathematcal ad Physcal Sceces, 314: Hawkg, S.W. ad Sachs, R.K. (1974), Causally Cotuous Spacetmes, Commucatos Mathematcal Physcs, 35: Islam, J.N. (), A Itroducto to Mathematcal Cosmology, Cambrdge Uversty Press, Cambrdge. Josh, P.S. (1996), Global Aspects Gravtato ad Cosmology, d Edto, Claredo Press, Oxford. Kruskal, M.D. (196), Maxmal Exteso of Schwarzschld Metrc, Physcal Revew, 119(5): Leray, J. (1953), Hyperbolc Dfferetal Equatos, The Isttute for Advaced Study, Prceto, N. J. Mguzz, E. ad Sáchez, M. (8), The Causal Herarchy of Spacetmes, arxv:grqc/69119v3. Mohaja, H.K. (13a), Schwarzschld Geometry of Exact Soluto of Este Equato Cosmology, Joural of Evrometal Treatmet Techques, 1(): Mohaja, H.K. (13b), Fredma, Robertso-Walker (FRW) Models Cosmology, Joural of Evrometal Treatmet Techques, 1(3): Mohaja, H.K. (13c), Sgularty Theorems Geeral Relatvty, M. Phl. Dssertato, Lambert Academc Publshg, Germay. Mohaja, H.K. (13d), Mkowsk Geometry ad Space-tme Mafold Relatvty, Joural of Evrometal Treatmet Techques, 1(): Mohaja, H.K. (13e), Space-tme Sgulartes ad Raychaudhur Equatos, Joural of Natural Sceces, 1():

21 Asa Joural of Appled Scece ad Egeerg, Volume 5, No 1/16 Mohaja, H.K. (14a), Geeral Upper Lmt of the Age of the Uverse, ARPN Joural of Scece ad Techology, 4(1): 4 1. Mohaja, H.K. (14b), Upper Lmt of the Age of the Uverse wth Cosmologcal Costat, Iteratoal Joural of Recprocal Symmetry & Theoretcal Physcs, 1(1): Mohaja, H.K. (15), Basc Cocepts of Dfferetal Geometry ad Fbre Budles, ABC Joural of Advaced Research, 4(1): Mohaja, H.K. (16), Global Hyperbolcty Space-tme Mafold, Iteratoal Joural of Professoal Studes, 1(1): Perose, R. (197), Techques of Dfferetal Topology Relatvty, A.M.S. Colloquum Publcatos, SIAM, Phladelpha. Posso E (4). A relatvst s toolkt: the mathematcs of black hole mechacs, Cambrdge Uv. Press, Cambrdge. Sachs, R.K. ad Wu, H. (1977), Geeral Relatvty ad Cosmology, Bullet of the Amerca Mathematcal Socety, 83(6): Sáchez, M. (1), Recet Progress o the Noto of Global Hyperbolcty, arxv: v [gr-qc] 5 Feb 1. Szekeres, P. (196), O the Sgulartes of a Remaa Mafold, Publcatoes Mathematcae, Debrece, 7: Wald, R.M. (1984), Geeral Relatvty, The Uversty of Chcago Press, Chcago.