The Modification of the Oppenheimer and Snyder Collapsing Dust Ball to a Static Ball in Discrete Space-time

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1 The Modfcton of the Oppenhemer nd Snyder Collpsng Dust Bll to Sttc Bll n Dscrete Spce-tme G. Chen Donghu Unversty, Shngh, 060, Chn Eml: gchen@dhu.edu.cn Abstrct. Besdes the sngulrty problem, the fmous Oppenhemer nd Snyder soluton s dscovered to be of defcency n two spects: the nternl Fredmnn spce-tme does not hve the nherent symmetry nd cnnot connect to the externl Schwrzschld spce-tme. So the process of grvttonl collpse descrbed by ths soluton s doubtful. The defcency, together wth the sngulrty problem, result from the mperfecton of the feld theory n contnuous spce-tme, whch s expressed by the nfnte precson functon theory. The spce-tme structure of the Oppenhemer nd Snyder dust bll s founded to be dscrete rther thn contnuous, nd to descrbe the feld theory n dscrete spce-tme t requres functon theory wth fnte precson. Bsed on the order rel number nd ts equvlence clss, whch s defned n the rel number feld, the nfnte precson functon theory s extended to the fnte precson functon theory. The Ensten feld equtons re expressed n the form of fnte precson, nd then the collpsng dust bll soluton n contnuous spce-tme s modfed to sttc bll soluton n dscrete spce-tme. It solves ll the problems of Oppenhemer nd Snyder soluton nd shows tht, wth Plnck length nd Plnck tme s spce-tme quntum, mechnsm to resst the grvttonl collpse could be obtned by the dscretzton of spce-tme. PACS numbers f, 04.0.Cv,04.0.Dw,04.0.Jb. Introducton It ws sserted tht the grvttonl collpse of dust bll results n the formton of grvttonl sngulrty by the Oppenhemer nd Snyder thess publshed n 939 []. On the nnls of generl reltvty, ths s generlly cknowledged mlestone result. Strtng wth ths thess, extensve reserches for the grvttonl collpse nd spce-tme sngulrtes hve been done nd mny theoretcl hypothess hve been put forwrd [-45]. However, when probng nto the soluton, we could fnd ts defcency n two spects: the nternl Fredmnn spce-tme does not hve the nherent symmetry tht t should possess nd cnnot connect to the externl Schwrzschld spce-tme. Therefore, the process of grvttonl collpse descrbed

2 by ths thess s doubtful. In my opnon, the defcences of the Oppenhemer nd Snyder soluton, together wth the sngulrty, demonstrte the mperfecton of the feld theory n contnuous spce-tme, whch s expressed by the nfnte precson functon theory. Further nlyss revels tht the spce-tme structure of the dust bll should be dscrete rther thn contnuous, nd descrbng the feld theory n dscrete spce-tme requres the functon theory wth fnte precson. So, we ntroduce the order rel number nd ts equvlence clss n rel number feld, nd bsed on whch, the nfnte precson functon theory s extended to the fnte precson functon theory. Then, we use the fnte precson functon theory to express the Ensten feld equtons nd study the grvttonl soluton of the dust bll. By the extenson nd dscretzton of the Oppenhemer nd Snyder soluton, the collpsng dust bll n contnuous spce-tme s modfed to sttc bll n dscrete spce-tme. It solves ll the problems of Oppenhemer nd Snyder soluton nd shows tht, wth Plnck length nd Plnck tme s pce-tme quntum, mechnsm to resst the grvttonl collpse could be obtned by the dscretzton of spce-tme.. The problems of the Oppenhemer nd Snyder soluton Ths secton wll dscuss the problems of Oppenhemer nd Snyder soluton on the grvttonl collpse of dust bll n three spects. Frstly, for the soluton n comovng coordntes, the nteror spce-tme geometry of dust bll cn be expressed s Fredmnn metrc [46] dr ds = dt + b ( t ) + r ( dθ + sn θdφ ). () kr Where b (t ) s gven by the prmetrc equtons of cyclod: t η + snη =, b = ( + cosη), () k For Eq.(), wth η defned n [ 0, π ], there s n nherent sngulrty t η = π. Accordng to Brkhoff theorem, the exteror spce-tme geometry of the dust bll cn be expressed s Schwrzschld metrc [47] ( dθ sn θd ) GM dr ds = dt + + r + φ. (3) r GM / r For Eq.(3),there re two sngulrtes, one nherent sngulrty t r = 0, nother coordnte sngulrty t r = GM. As we ll know, the exstence of the sngulrtes reflects the ncompleteness of spce-tme geometry n the Oppenhemer nd Snyder soluton.

3 Secondly, the nteror spce-tme nd exteror spce-tme of the dust bll re relted to ech other by the followng coordnte trnsformtons (for Eq. (),gvng GM r =, k =, s the rdus of dust bll n comovng coordntes,nd M s 3 the grvttonl mss of dust bll ) [48]: r = R( η ) = ( + cosη), (4) t = GM ln ( /(GM ) ) ( /(GM ) ) / / + tn( η / ) + tn( η / ) / ( /(GM ) ) [ η + ( /(4GM ))( η + snη) ] θ = θ, (6) φ = φ (7) (5) From the vewpont of grvttonl collpse,eqs.(4),(5),(6) nd (7) re regrded s the geodesc equtons whch descrbe the evolvement of collpsng sphercl surfce of the dust bll. For the gven θ nd φ, these equtons represent rdl geodesc n the r - t plne, strtng from the mxmum sphercl surfce r = b( 0) = nd termntng t the nherent sngulrty r = b( π ) = 0. For the geodesc, the rdl coordnte r s the functon of coordnte tme t, but for the Schwrzschld metrc descrbed n Eq. (3), r nd t re two ndependent coordnte vrbles, whch form r -t plne for the gven θ nd φ. Of course, the geodesc n the r -t plne s not equvlent to the r -t plne. From Eqs. (4) nd (5) we cn deduce dr dt GM = r Then substtutng Eq. (8) nto (3), we hve r GM r GM (8) 3

4 ds r GM GM r dt + r r GM = + GM dr GM GM r r r GM r ( dθ + sn θdφ ) + r ( dθ + sn θdφ ),, r > GM r < GM (9) Notce tht the tem + GM dr GM GM r r r GM r r GM GM r dt r GM or s tme-lke, nd from Eqs.()nd (4) we know tht the nherent tme t ncreses when r vres from to 0, then Eq. (9) represents keepng shrnkng sphercl surfce rther thn the Schwrzschld spce-tme. Ths mens tht the Schwrzschld spce-tme could not be formed by the process of grvttonl collpse of the dust bll. On the other hnd, supposng the exteror of the dust bll s Schwrzschld spce-tme, the nteror Fredmnn spce-tme, nd the two spce-tme re connected by the collpsng sphercl surfce of the dust bll. The equton of tme-lke rdl geodesc n the exteror Schwrzschld spce-tme cn be deduced s or dr GM GM = dt r β r dr dt o o = β GM r (0) () dr GM Where, dr o =, dt o = dt. Whle the equton of tme-lke GM r r rdl geodesc n the nteror Fredmnn spce-tme cn be deduced s 4

5 or dr dt dr dt = b ( ) kr = η η () (3) Where, dr bdr =, dt = dt.it turns out tht, for Eqs. (0),(),() nd(3), kr there lwys exst such vlues of, β, t nd t b, tht mke the collpsng sphercl surfce of the dust bll locte t r = r ( rb r = ) when t = t ( t = tb ), nd hvng t < tb, r > rb, r > b GM, β = GM η r, GM β η r.the b correspondng tme-lke rdl geodesc enters nto the nteror spce-tme from the exteror spce-tme t the sphercl surfce of r = r, nd then from the nteror spce-tme to the exteror spce-tme t the sphercl surfce of r = rb. The geodesc s contnuous on the sphercl surfce of r = r : dr dt dr o = dt o,nd dscontnuous on the sphercl surfce of r = rb : dr o dt o b dr dt b. Ths mens tht, s the collpsng of the dust bll, the contnuty between the exteror nd nteror spce-tme would be broken. Therefore, n ths crcumstnce, the Fredmnn spce-tme nd the Schwrzschld spce-tme could not be connected together. Thrdly, for equton (), when t = 0, we hve b ( t ) =.Then, from Eq.(), we obtn the expresson of nteror spce geometry of the dust bll s d σ + r + ( dθ sn θdφ ) dr =. (4) κr By the followng demonstrton,we cn come to the concluson: for 0 r,ths spce geometry s prt of the 3-dmenson hypersphere n 4-dmensonl Eucld spce. At frst, we mke the trnsformton from polr coordntes to Crtesn coordntes x = r snθ cosφ, 5

6 nd let x = r snθ snφ, (5) 3 x = r cosθ μ 4 x x = x x + ( x ) =, μ =,,3, 4, =,, 3, (6) κ then, we cn obtn the equvlent form of Eq. (4) s wth d r μ = dx dx = dx dx + (dx ) μ μ 4 σ (7) = κ 4 4 [ r( x )] = x x = ( x We know tht, f nd only f ( ) 0 ) (8) x μ k wth μ =,,3, 4, Eq.(6)expresses 3-dmensonl hypersphere n 4-dmenson Eucld spce. However, from Eq.(8), 4 0 r,we cn derve k ( x ) k, nd for < k, Eq.(4)s just prt of the 3-dmenson hypersphere. It s well known tht spce geometry wth symmetry.e., unformty nd sotropy, cn be such 3-dmenson hypersphere n 4-dmensonl Eucld spce or -dmensonl sphere n 3-dmensonl Eucld spce, nd so on. However, when t = 0, the nteror spce geometry of the dust bll s prt of the 3-dmensonl hypersphere n 4-dmensonl Eucld spce, thus breks the nherent symmetry. In my opnon, the bove-mentoned problems of the Oppenhemer nd Snyder soluton result from the mperfecton of the feld theory n contnuous spce-tme, whch s expressed by the nfnte precson functon theory. To solve these problems nd obtn complete grvttonl soluton of the dust bll, t requres feld theory n dscrete spce-tme, whch cn be expressed by the fnte precson functon theory. Then, the fnte precson functon theory wll be put forwrd n the next secton. 3. Fnte precson functon theory )The order rel number nd ts equvlence clss An order rel number s defned s A = αc. Where C, rel scle coeffcent, s postve nd greter thn, whle α rel number stsfyng C α, { 0, ±, ±,... } > C / /. 6

7 An ' order equvlence clss of order rel number s defned s set of order rel numbers, n whch the bsolute vlue of the dfference between ny two order rel numbers A nd B s less thn or equl to gven order postve rel number. The ' order equvlence clss s denoted s ~ A or ~. Whle the B relton of A nd B s referred s ' order equvlence nd s denoted s A B. Notce tht, wth ny order rel number A = αc s reference pont, dfferent ' order equvlence clss ~ A cn be formed n the rel number felds. An ' order equvlence clss of order rel number cn be expressed s set of '' order equvlence clsses, whle n '' order equvlence clsses cn be expressed s set of ''' order equvlence clsses nd so on, s long s ' '' ''' L. Moreover, the equvlent relton of rel numbers only exsts wthn one equvlence clss tht they belong to. Dfferent equvlence clsses don t hve the equvlent relton. The rel number felds nd the whole equvlence clsses of rel numbers re denoted s R nd R ~ respectvely. ) Functon A functon s defned s mppng between the ndependent vrble nd dependent vrble. The ndependent vrble s clled vrble for short, nd the dependent vrble s regrded s functon. The vrble nd functon re the sets of R ~ 3) Opertons of functon. Lmt nd contnuty Lmt Let f (x) be n order functon of j order vrble x, Α n order rel number, nd x 0 j order rel number. If there exsts j order equvlence 7

8 clss of x 0 nd n order equvlence clss of Α, when x belongs to the j order equvlence clss of x 0, f (x) belongs to the order equvlence clss of Α : f ( x) Α, x0,, j j, (9) x j then, we defne Α s the lmt of f (x) s x pproches x 0. Contnuty Let n order functon f (x) be defned on set of j order rel number of R ~, nd for n element x 0 of the set, the vlue of the order functon s f x ). If for j order equvlence clss of x 0, there exsts n order equvlence clss of f ( x 0 ), when x belongs to the j order equvlence clss of x 0, f (x) belongs to the order equvlence clss of f x ) : ( 0 ( 0 f ( 0 j x) f ( x ), x x0,, j j, (0) then the functon f (x) s contnuous t x 0. b.dfferentl nd dervtve Let f (x) be k order functon of j order vrble x. When x belongs to j order equvlence clss of j order equvlence clss, nd Δ x s the dfference between x nd ny element of j order equvlence clss djcent to the j order equvlence clss t belongs to, then we cll dy k f ( x + Δx) f ( x), k k k () 8

9 the dfferentl of k order functon f (x), nd dx j Δx, j j j () the dfferentl of j order vrble. If for ny Δ x, there exsts ( ) equvlence clss of order rel number f ( x) f ( x) dy f x + Δx f x ( ) ( ) dx Δx, (3) then ( x) f s clled the dervtve of f (x). If f (x) exsts on every pont of rel number set of R ~, then f (x) s clled the dervtve functon or the dervtve of f (x). c. Indefnte ntegrl nd defnte ntegrl Indefnte ntegrl If n order functon s defned n set of j order rel number of R ~, nd f (x) s the dervtve of k order functon F (x), or f ( x) dx the dfferentl of F (x),tht s or F x + Δx F x f x ( ) ( ( ) F ( x) ), Δx k df( x) f ( x) dx, k k k, then F (x) s clled prmry functon of f (x). It s esy to prove tht,f F (x) s prmry functon of f (x), then F (x) + η s lso prmry functon of f (x), where η D k, nd D s ny rel constnt. We cll ll the k order prmry functon F (x) + η of order functon f (x) the ndefnte ntegrl of f (x), denoted s 9

10 k f ( x) dx F(x) + η (4) Defnte ntegrl Let f (x) be n order functon of set of j order rel number of R ~. nd b re two rel number of the set nd < b, [, b] s subset of ths set, contnng ll j ( j j ) order equvlence clss of j order rel numbers, whch re less thn or equl to b nd greter thn or equl to. Every j order equvlence clss of the subset contns set of j ( j j )order equvlence clss, nd the number of ll the j order equvlence clsses n [, b] s ~ ω +: A { ~ j j l = xl }, l = 0,,..., ω. Except the j order equvlence clss contnng or b, we tke ny rel number { } x ~, l =,,..., ω n every j order l x l equvlence clss, nd form rel number serl j ω 0, x ω { x x, x,..., } Δ, j j x0 < x < x <... < xω b. j Then, we defne Δx l xl xl, l =,,..., ω,nd form the k order sum ω f ( x l ) Δxl l= or ω f ( x l ) Δxl. l= If for dfferent rel number serl Δ, the k order sums re k ( k k ) order ω equvlence, they re defned to be defnte ntegrl of f (x) on [, b], expressed s b f ( x) dx k ω k f ( x l ) Δxl f ( x l ) Δxl. (5) l= ω l= The bove-mentoned defntons cn be drectly extended to the multvrble functon. 0

11 4) The reltonshp between the fnte precson functon theory nd the nfnte precson functon theory It s obvous tht the functon theory, founded on the equvlence clsses set of rel numbers, should be mthemtcl theory of fnte precson, nd the now-exstng functon theory, founded on the rel number feld, s mthemtcl theory of nfnte precson. The former functon theory s the extenson of the ltter one. Frstly, nfnte precson functon cn lwys be extended to fnte precson functon. For exmple,consderng contnuous functon of nfnte precson, defned on the ntervl [, b] of rel number felds, wth gven scle coeffcent C, we dscretze ts order vrble nto set, consstng of seres of ( = )order equvlence clsses,nd the number of the equvlence clsses s ω +, two of whch contnng the termnls of the ntervl nd b respectvely. Ether of the two order equvlence clsses, combned wth the djcent order one,consttutes ( = ) order equvlence clss, nd for the other order equvlence clsses n the set, the djcent three ones consttute ( = )order equvlence clss. Accordngly, the vlues of the contnuous functon re dscretzed nd extended to set of order equvlence clss consstng of order equvlence clsses. Thus, we obtn fnte precson functon whose vrbles nd vlues re rel number equvlence clsses. Ths fnte precson functon hs ll the propertes of the nfnte precson one, such s lmt, contnuty, dervtve, dfferentl, ntegrl, nd so on. Secondly, for the vrble of rel number equvlence clss wth gven order, such s the vrble of ( = ) order equvlence clss of 0( = 0 )order rel numbers, when C,we hve ω. Then the fnte precson functon would degenerte nto the nfnte precson functon. On the other hnd, for gven scle coeffcent C, f cn be ny ntegers, then the equvlent relton n fnte precson functon theory s the sme s the equl relton = n nfnte precson functon theory. Bsed on the bove-mentoned processng, we could ccomplsh the followng conclusons: on one hnd, the fnte precson functon theory ncludes the nfnte precson one. On the other hnd, for ny dfferentl equtons of nfnte precson, we cn keep ther forms unchnged nd endow ther functons, vrbles nd operton symbols wth the menng of fnte precson functon theory, therefore mkng them become the dfferentl equtons of fnte precson. Thrdly, wth dfferent scle coeffcent C, order, nd equvlence clsses, fnte precson functon cn possesses dfferent precson or uncertnty. In some condtons, functon of fnte precson or uncertnty cn be degenerted nto functon of nfnte precson or certnty. Thus, the fnte precson functon theory cn descrbe both certnty nd uncertnty systems. Fnlly, s the rnge or domn of the fnte precson functon, the rel number equvlence clsses re generlly the sets of dscrete dots n the rel number xs, whch cn be denoted pproxmtely s contnuum n the specl crcumstnce. Then, the fnte precson functon theory cn not only denote the dscrete system n the generl menng, but lso descrbe the

12 contnuous system n the specl cse. So the unty of dscreteness nd contnuty cn be cheved. 4. A sttc bll n dscrete spce-tme On the bss of bove dscusson, we cn extend the now-exsted feld theory n contnuous spce-tme to the feld theory n dscrete spce-tme by keepng the forms of feld equtons unchnged, nd chngng feld vrble or spce-tme coordnte from the set of R to the set of R ~,or chngng the rnge of vlue of ny n dmenson physcl feld from the set of R n to the set of R ~ n,nd chngng the domn of defnton of the feld.e., the 4 dmenson spce-tme coordntes from the set of 4 R to the set of ~ 4 R, then chngng ll the opertons,defned n the nfnte precson contnuous functon theory, to those defned n the fnte precson dscrete functon theory. In ths wy, the Ensten feld equtons of fnte precson cn be ccomplshed. Therefore, the grvttonl problems of the dust bll cn be solved on the bss of the Ensten feld theory n dscrete spce-tme. For the soluton of the dust bll, consderng tht the Schwrzschld externl soluton s sttc metrc, nd the Fredmnn nternl soluton s dynmc metrc, n order to connect the two metrcs, the only wy s to extend the Fredmnn nternl soluton perodclly nd dscretze the spce-tme. Tht s extendng the domn of η from [ 0, π ], defned n Eq.(), to R nd dscretzng t. Thus,s η ncreses, the nherent tme t wll ncreses monotonously,s long s the metrc fctor b s nvrble n the menng of equvlence, the Fredmnn nternl soluton becomes the equvlent sttc one. Then, from coordnte trnsformtons (4),(5),(6) nd (7), the Fredmnn nternl soluton cn be connected to the dscretzed Schwrzschld externl soluton. As mentoned bove,by mens of extenson nd dscretzton, we cn endow Eqs. (),(),(3), (4),(5),(6) nd (7) wth the menng of fnte precson functon theory, n whch the equl sgn = s regrded s the equvlent sgn, where s gven proper vlue n every equton. Herenfter, we wll use equl sgn = to represent the equvlent sgn n whch cn be of ny nteger. For the soluton of dust bll wth fnte precson, we regrd π s unt element n the menng of ( = ) order equvlence, choosng η = 0 s reference pont, then dscretzng the prmeter η. Tht s letng η = nπ + η C, n =, ±, ±,..., ± N 0,where N

13 n rbtrry postve nteger,c lrge enough, nd for gven n,by chngng the vlue of η to form n order equvlence clss of η. Consderng C η nπ nd substtutng t nto the second Eq. of (), we hve b, nd nto Eqs. (4),(5), we hve r, (6) /, 4πGM ( /(GM ) ) ( /(4GM ) + ) t nτ Then,substtutng τ. ( 7) GM η nπ nd k nto the frst Eq. of(),gves 3 t nτ, τ π. (8) GM When η nπ nd r,from Eqs.(), (),(3),(6) nd(7),we obtn / dt ( GM / ) dt. (9) And from Eqs. (7), (8), we cn derve / t ( GM / ) ( + 4GM / )t. (30) Then, dfferenttng Eq. (30) nd comprng wth (9), we obtn 4GM. (3) By substtutng Eq.(3) nto Eqs. (6),(7),nd (8), we obtn nd r 4GM, (3) t nτ, τ 8πGM (33) t nτ, τ 4 πgm. (34) Here, Eqs. (3), (33) denote the dscretzton of Schwrzschld spce-tme coordntes, nd Eq. (34) denotes the dscretzton of Fredmnn tme coordnte. Herenfter, we wll dscuss the dscretzton of Fredmnn spce coordntes. Consderng tht Fredmnn coordntes re the comovng coordntes of dust mtter, t s obvous tht the dust mtter does not dstrbute n the ponts of Fredmnn coordntes, otherwse for the lne elements relted to the dust mtter, there exst d r 0, d θ 0 nd d φ 0.Ths mens tht the Fredmnn spce possesses zero 3

14 mesure n the menng of order equvlence,nd would not connect to the Schwrzschld spce-tme whch hs the dscrete structure denoted by Eqs. (3),(33). Thus, the dust mtter must possess knd of spce-tme structure wthout nner coordnte. In order to exhbt the structure, substtutng Fredmnn metrc n Eq.() nto Ensten feld equtons, n equton on the dust mtter cn be derved [5], whose fnte precson form s 8 G b& π + k ρb. 3 (35) GM Then, usng Eq.(3) nd k, we hve k 4 3 GM. Consderng tht b nd b & 0, we hve 3 k M ρ, 8πG v q 4π 3 3 vq. (36) Where M, ρ re the mss nd mss densty of the dust mtter respectvely, nd v q the volume of sphere n flt spce wth rdus. It cn be nferred tht the spce-tme structure of the dust mtter s n equvlent flt bll wthout nner coordntes. Therefore the dscrete form for the rdl coordnte of Fredmnn spce cn be derved s r (37) Thus, n the menng of order equvlence, the nternl Fredmnn spce s two-dmensonl sphere n three-dmensonl Eucld spce. We know tht the two-dmensonl sphere possesses the symmetry.e., unformty nd sotropy,nd connects to the externl Schwrzschld spce-tme on the sphercl surfce. By the nlyss of rdl geodesc, smlr to secton, t cn be proved tht the spce-tme geometry s contnuous on the surfce n the menng of order equvlence. Moreover, consderng tht grvttonl felds re represented by spce-tme metrc, nd spce-tme metrcs re denoted s the mppng of spce-tme coordntes, there exsts no spce-tme metrcs.e., no grvttonl felds,n the equvlent dust bll wthout nner coordnte. Ths mens tht the dust bll hs no self-grvttonl ntercton, nd would not produce collpse, thus formng sngulrtes. Fnlly, for Eqs.(3),(33) nd (34), let we hve / 8 ( G ) r M π, (38) 4

15 nd r ( 8πG ) / (39) π, ( ) / t nτ t nτ τ 8πG, (40) τ 4πG (4), ( ) / Where M, r, τ or τ re the mss,rdus, dscrete tme length of dust bll, nd they re of the orders of Plnck mss, Plnck length, Plnck tme respectvely. Then, from Eqs.(3), (33) nd (34), t cn be deduced tht, for dust bll wth the mss of m tmes the Plnck mss, the rdus nd dscrete tme length would be of m tmes the Plnck length nd Plnck tme respectvely. So f we ssume the bll wth mss M to be n elementry prtcle, then mny elementry prtcles could form nto bll wth gret mss. It shows tht, wth Plnck length nd Plnck tme s the spce-tme quntum, mechnsm to resst the grvttonl collpse could be obtned by the dscretzton of spce-tme. 5. Concluson Besdes the sngulrty problem, the defcences of the fmous Oppenhemer nd Snyder dust soluton re dscovered n two spects: the nternl Fredmnn spce-tme does not hve the nherent symmetry nd cnnot connect to the externl Schwrzschld spce-tme. The problems of the soluton result from the mperfecton of the feld theory n contnuous spce-tme, whch s expressed by the nfnte precson functon theory. To solve these problems nd obtn complete grvttonl soluton of the dust bll, t requres feld theory n dscrete spce-tme, whch cn be expressed by the fnte precson functon theory. Bsed on the order rel number nd ts equvlence clss, whch s defned n the rel number feld, the nfnte precson functon theory s extended to the fnte precson functon theory. The Ensten feld equtons re expressed n the form of fnte precson, nd the Oppenhemer nd Snyder collpsng dust bll soluton s modfed to sttc bll soluton. Ths soluton shows tht: ) The spce-tme geometry of the bll hs dscrete structure. b) The mtter of the bll s exsted n equvlent flt bll wthout nner coordntes. Thus t would not gve rse to collpse nd form sngulrtes by the self grvttonl ntercton. c) There exsts bll of Plnck mss, wth Plnck length nd Plnck tme s ts rdus nd elementry tme length respectvely, nd the bll cn be regrded s elementry prtcle. d) m prtcles cn form nto lrge bll, whch hs mss of m tmes the Plnck mss, rdus of m tmes the Plnck length, nd dscrete tme length of m tmes the Plnck tme. As result, wth Plnck length nd 5

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