SLAC-PUB-10766 Simple Analyic Models of Gaviaional Collapse. J. Adle, 1 J. D. Bjoken, 2 P. Chen, 2 and J. S. Liu 3 1 Hansen Laboaoy fo Expeimenal Physics, Sanfod Univesiy, Sanfod, CA 94309, USA 2 Sanfod Linea Acceleao Cene, Sanfod Univesiy, Sanfod, CA 94309, USA 3 Depamen of Physics, Sanfod Univesiy, Sanfod, CA (Daed: Febuay 9, 2005 Mos geneal elaiviy exbooks devoe consideable space o he simples example of a black hole conaining a singulaiy, he Schwazschild geomey. Howeve only a few discuss he dynamical pocess of gaviaional collapse, by which black holes and singulaiies fom. We pesen hee wo ypes of analyic models fo his pocess, which we believe ae he simples available; he fis involves collapsing spheical shells of ligh, analyzed mainly in Eddingon- Finkelsein coodinaes; he second involves collapsing sphees filled wih a pefec fluid, analyzed mainly in Painleve-Gullsand coodinaes. Ou main goal is pedagogical simpliciy and algebaic compleeness, bu we also pesen some esuls ha we believe ae new, such as he collapse of a ligh shell in Kuskal-Szekees coodinaes. PACS numbes: 11.10.Kk, 11.25.Uv, 11.30.Pb, 98.80.Es I. INTODUCTION Black holes and singulaiies ae ceainly some of he mos peculia and ineesing feaues of geneal elaiviy[1]. Fom he pespecive of fundamenal heoy hei popeies ae fa fom compleely undesood, especially hei elaion o quanum heoy[2]; fom he pespecive of asophysics and obsevaional asonomy hey seem moe and moe o play a cenal ole in he physics of objecs anging fom sella o cosmological size[3]. Mos geneal elaiviy exbooks devoe consideable space o discussing he simples example of a black hole conaining a singulaiy, he Schwazschild geomey[3 12], and some also discuss Ke s genealizaion o a oaing sysem, alhough only a few deive i[4, 13]. Mos also explain, a leas qualiaively, why black holes should exis in he eal wold, based on he insabiliy of neuon sas of moe han abou 4 sola masses; even neuon degeneacy pessue is no sufficien o povide sabiliy. A few exs also discuss Schwazschild s ineio soluion fo a consan densiy sa[4, 8], which clealy illusaes some of he same insabiliy feaues as moe ealisic neuon sa models. Bu he acual dynamical pocess of collapse, wheeby a massive body becomes a black hole, is a moe complex dynamical poblem, and is ofen eihe negleced o eaed heuisically and qualiaively. Fo a bief hisoy of gaviaional collapse see efeence[14]. The sudy of gaviaional collapse dynamics began wih he classic seminal wok of Oppenheime and Snyde[15], who se up he equaions and gave a semi-quaniaive discussion of he collapse of a spheical non-adiaing sa. They also found a complee analyic soluion fo he case of a unifom pefec fluid of zeo pessue - now widely efeed o as dus. This example is less aificial han i migh appea since he effec of pessue in a moe ealisic eamen uns ou o be no vey ciical; bu he eamen by Oppenheime and Snyde involves some awkwad algeba in is eamen of he bounday beween he ineio and exeio of he dus sa. Some exbooks discuss collapse much like Oppenheime and Snyde, noably Landau and Lifshiz[5] and Misne Thone and Wheele[3]. In paicula hese exs use a Novikov ype coodinae sysem, in which he dus is co-moving in a bound obi[16]. This coodinae sysem is synchonous and co-moving, and is wondefully simple concepually, bu almos equally awful algebaically. Due o he algebaic difficuly, we believe, his is no he simples way o ea he poblem. Mos ohe exs give a heuisic bu convincing analysis by compaing he moion of a dus paicle jus ouside a collapsing dus sphee wih he moion of he suface; maching his moion is equivalen o maching he exeio and ineio geomeies, which is a ahe obvious bu impoan heoem[17]. Wha we have ied o do in he pesen pape is o analyze dynamical collapse in as simple a way as possible by ou choice of coodinaes; we use Eddingon- Finkelsein[18, 19] ype coodinaes fo infalling incoheen ligh o null dus, and Painleve-Gullsand[20, 21] ype coodinaes fo pefec fluids, including dus. These coodinaes seem almos magically well suied o his pupose, as we hope will become appaen in secions 2 and 5. We use zeo enegy fluid sysems in conas o negaive enegy bound sysems as used in efeences [3] and [5], lagely because he algeba is much simple[4]. This choice seems o us well jusified because he zeo and nonzeo enegy sysems have he same qualiaive behavio as hey appoach he black hole sae as seen fom ouside in ems of sandad Schwazschild ime, and have exacly he same behavio as hey appoach Wok suppoed in pa by he Depamen of Enegy, conac DE-AC02-76SF00515
2 he singulaiy as seen in ems of he pope ime of a falling obseve. Moeove he zeo enegy case does no have he awkwad concepual poblem of he moion befoe he maximum size is eached, since he maximum size occus a an infinie ime in he pas. We heefoe believe he zeo enegy case is moe illusaive and economical of effo. Almos needless o say we have eaed only spheically symmeic collapsing sysems. The Bikhoff heoem guaanees ha a spheically symmeic sysem does no emi gaviaional adiaion; non-spheically symmeic collapse geneally enails gaviaional adiaion, making he poblem vasly moe complex, hough also moe ineesing. To achieve ou goal of simple exbook ype examples of collapse, we mus pay a pice, which is some degee of aificialiy. The main aificialiy is he spheical symmey; also he in-falling shells of ligh o ohe massless maeial of secions 2 and 3 ae no likely o be found in naue; similaly we do no expec o find he pefec fluid sphee suounded by an elasic shell of secion 5, o he zeo pessue pefec fluid of secions 6 and 7. Since ou pupose in he pesen pape is ovely pedagogical, much of wha we obain hee is known and available somewhee in he vas eseach lieaue, alhough obained wih diffeen echniques and diffeen coodinaes. Howeve we believe ou eamen is he simples, boh concepually and algebaically, mainly due o ou coodinae choice. We do no know of any use of Painleve-Gullsand coodinaes in he pesen conex, alhough hey ae becoming moe widely used[22, 23]; as will be seen in secion 6 hese coodinaes combine a Schwazschild ype adial coodinae and a Fiedmann- obeson-walke (FW ype cosmological ime coodinae o give a naual inepolaing sysem fo descibing boh he ineio and exeio of a fluid sphee. While he Eddingon-Finkelsein and Painleve-Gullsand coodinaes seve hei mahemaical pupose quie well, hey have a numbe of dawbacks such as a ime make ha is no well-behaved eveywhee in spaceime, and hey ae no synchonous; we have heefoe also eaed he collapse of a hin shell of ligh by ansfoming o Kuskal-Szekees coodinaes, which ae confomally fla and hus display he causal sucue of he collapse pocess quie clealy[24, 25]. In Kuskal-Szekees coodinaes one has a nice picue fo disinguishing eal wold black holes fom eenal black holes, wih hei associaed womholes and whie hole segmens, none of which may be expeced o acually exis in ou univese. We use houghou his pape a ahe novel echnique fo handling he bounday condiions beween he fluid and he exeio, which we believe is physically clea. Since he pioneeing wok of Oppenheime and Snyde hee has been an enomous amoun of wok on gaviaional collapse and elaed opics, mos dieced o eseach ahe han pedagogy. The pesen pedagogical pape oiginaed fom seveal eseach poblems, one involving a fomaion mechanism fo singulaiy-fee black holes filled wih heavy vacuum, and anohe involving a finie vesion of he sandad o concodance cosmological model. Fo hese we needed analyic collapse models ha wee algebaically simple and moe explici han we found in he lieaue. Ou hope is ha ou mehods and models migh also be of use o sudens and eaches in illusaing moe sandad gaviaional collapse. This pape is oganized as follows: he fis pa (secions 2 o 4 deals wih collapsing shells of incoheen ligh o null dus, which is eniely chaaceized by is null 4-velociy and enegy densiy; he second pa (secions 6 and 7 deals wih sphees of pefec fluid, including some wih nonzeo pessue and some wih non-unifom densiy. In secion 2 we use Eddingon- Finkelsein coodinaes o obain he meic fo he collapse of a hin ligh shell ono a black hole o fom a lage black hole, and he collapse of a hin ligh shell o fom a black hole ab iniio. The ab iniio collapse is amusing in ha i is almos ceainly he simples complee scenaio fo he fomaion of a black hole, albei i a ahe aificial one. In secion 3 we use he esuls of secion 2 o obain he meic fo he collapse of a hick shell of ligh, laye by laye. The densiy pofile of he shell is lagely abiay, and we give one specific example. In secion 4 we ansfom he meic fo he hin ligh shell collapse o confomably fla Kuskal-Szekees coodinaes in ode o display he causal sucue of he geomey in a clea way - nealy equivalen o a Penose diagam; as aleady noed we use hese coodinaes o emphasize he diffeence beween black holes fomed by collapse and eenal Schwazschild black holes. In secion 5 we pesen ou way of doing he classic poblem of collapse of a unifom fluid sphee, using Painleve-Gullsand coodinaes fo maximum simpliciy. We also obain an explici soluion fo a nonzeo pessue fluid wih a linea equaion of sae: p = αρ; o balance he pessue gadien foce a he suface we use he aifice of a hin elasic shell wih angenial pessue insead of he slowly deceasing adial pessue of a eal sella sysem; ou model is essenially a balloon wih unifom inenal densiy and pessue. The sess enegy enso of he suface is descibed by he adius of he sphee, is enegy densiy, and he paamee α; fo α = 0 we ecove he sandad esuls fo he dus ball. In secion 6 we deal wih zeo pessue non-unifom dus sphees, wih he densiy chaaceized by a lagely abiay funcion; again using Painleve-Gullsand coodinaes we build up he sysem laye by laye, o obain he sandad esuls of Oppenheime and Snyde and Landau and Lifshiz ec. We do no allow he dus layes o coss, which could give ise o infinie densiies and supeficial singulaiies[26]. We give one specific example of he meic of a non-unifom sphee; he special case of unifom densiy agees wih he analysis of secion 5. Since his pape is mainly pedagogical we have houghou sacificed beviy and included consideable algebaic deail; only elemenay mahemaical mehods ae used. I is woh menioning ha some simple and faily obvious exensions of he pesen wok can be made. Fis,
3 by a evesal of ime he collapse of ligh shells can be viewed as ougoing adiaion, so we may easily obain esuls like hose of Vaidya fo he geomey of a adiaing body[27]. Similaly he collapse of he pefec fluid can be ime evesed o yield he meic fo a black hole emiing mae, ha is a ype of whie hole. Also i is saighfowad o include a cosmological consan em in he equaions o obain he meic fo a collapsing sysem in an exeio Schwazschild de Sie geomey; due o is somewha moe cumbesome algeba we have no included his in he pesen pape, and leave i as an execise. (See efeence[28]. We will say moe abou fuhe exensions and applicaions in secion 8. II. THIN LIGHT SHELLS The moion of ligh in Schwazschild geomey is algebaically awkwad a and inside he Schwazschild adius when Schwazschild coodinaes ae used, bu i becomes quie simple in Eddingon-Finkelsein (EF coodinaes. In his secion we use EF coodinaes o sudy he moion of ligh, he gowh of a black hole by absopion of ligh, and he ab iniio fomaion of a black hole by a hin shell of ligh. The las pocess is mos likely he simples model fo he fomaion of a black hole by gaviaional collapse. The Schwazschild meic in Schwazschild coodinaes is, wih c = 1, ds 2 = (1 + ud 2 s d2 1 + u 2 dω 2, (1 whee u = / and dω 2 = dθ 2 + sin 2 θdφ 2. The black hole suface is a sphee a he Schwazschild adius (wice he geomeic mass = 2m = 2GM. I is boh an infinie edshif suface whee g 00 = 0 and a null suface o one-way-membane. In ems of he Schwazschild ime coodinae boh ligh and paicles ake an infinie ime o each he black hole suface fom he exeio. Specifically, fo ligh falling adially fom i s = i + ln ( i. (2 Thus ligh neve eaches he suface bu appoaches asympoically wih a chaaceisic ime. I is fo his eason ha we inoduce he EF ime coodinae, in ems of which he suface is eached in a finie ime. The EF ime coodinae is obained fom he Schwazschild ime by a adially dependen shif This leads o he EF fom of meic, s = + g(. (3 ds 2 = d 2 d 2 2 dω 2 + u(d ± d 2, (4 povided ha we choose he ansfomaion funcion g o obey dg d = ± u 1 + u. (5 The soluion o his is g = ln ( 1, (6 which has an infinie sech a =. Hencefoh we efe o he coodinaes and meic fom in Eq.(4 as EF fo any funcion u(,. The EF fom is exemely convenien because adially infalling ligh behaves quie simply if he plus sign in Eq.(4 is chosen. Fo such ligh we se he line elemen equal o zeo, o obain so ha ds 2 = d 2 d 2 + u(d + d 2 = (d + d(d d + ud + ud = 0. (7 d + d = 0, + = cons, infalling ligh, (8 d/d = (1 + u/(1 u, ougoing ligh. (9 Thus he pah of infalling ligh is independen of he meic funcion u, and is he same as in fla space. See Fig. 1. Convesely, ougoing ligh salls whee u = 1 and g 00 = 0, ha is a he infinie edshif suface. singulaiy (a ` ligh shell, + = 0 singulaiy ` (b ligh shell, + = 0 FIG. 1: In (a a hin ligh shell falls ino a black hole o poduce a lage black hole. In (b a hin ligh shell wih fla Minkowski ineio collapses o fom a black hole. We will soon need he Einsein enso fo he EF meic fom Eq.(4. Is nonzeo componens ae easily calculaed o be 8708A1 G 0 0 = u + u 2, G1 0 = u, G0 1 = u, G 1 1 = u + u 2 2 u, G 2 2 = G 3 3 = u u + ü + u 2 u. (10 2 Hee a do denoes a ime deivaive and a pime denoes a adial deivaive. We define he sess enegy enso in ems of he Einsein enso by using he field equaions G α β 8πGT α β. (11 Fo pue Schwazschild geomey he Einsein enso is eveywhee zeo, as is easily veified. We now sudy he fomaion of a black hole by a single hin shell of ligh, and in he nex secion we will conside a hick shell of ligh. Fo boh puposes we begin wih a hin shell of ligh falling ino a Schwazschild black hole wih adius o fom a black hole wih adius
4 as shown in Fig. 1a; he ligh shell obeys + = 0 eveywhee. We may wie he meic in all of spaceime using a sep funcion Θ and is complemen Θ = 1 Θ, as u = Θ( + Θ( +, (12 which is of couse ime dependen. Subsiuing his ino Eq.(55 and Eq.(11 we obain he sess-enegy enso fo he hin ligh shell, which is singula due o he sep funcion, T α β = 8πG 2 δ( + kα k β, (13 whee k α = (1, 1, 0, 0 and k β = (1, 1, 0, 0. The null veco k α coesponds o infalling ligh. Fom he enegy densiy T 0 0 we may calculae he enegy o effecive mass of he ligh shell by going o a ime in he disan pas when he shell was in asympoically fla space, M s = 4π 2 T 0 0d = 2G = M M. (14 This veifies ha he mass of he iniial black hole plus he enegy o effecive mass of he ligh shell equals he mass of he final black hole. The above esuls hold fo he special case when he iniial black hole is eplaced by fla space, o = 0. Noe ha eveyhing above is consisen wih he ineio of a spheical hin ligh shell being fla Minkowski space. This epesens he fomaion of a black hole by he gaviaional collapse of a single hin ligh shell (o ohe massless maeial, as shown in figue 1b; his would seem o be he simples complee example of gaviaional collapse. III. THICK LIGHT SHELLS Ou esuls fom he peceding secion may be used o consuc a model fo he collapse of a hick ligh shell, laye by laye. The iniial sae is a sequence of concenic hin shells wih a fla Minkowski ineio, as shown in Fig. 2a. Each egion of spaceime has a Schwazschild geomey as discussed in he pevious secion. The innemos shell wih enegy m 1 collapses o fom a black hole of mass m 1 = m 1, followed by ohes o fom inemediae black holes of mass m j = m 1 + + m j, ending wih a final black hole wih mass m f and Schwazschild adius = 2m f. Beween he shells he geomey is given by Eq.(4, wih a meic funcion u = 2m j /. (15 The evolving infinie edshif suface, defined by u = 1, is he zigzag line in Fig. 2a. The coninuous analog of he sequence is a family of concenic shells wih a coninuous label λ, which we choose o un fom 0 o 1. The paamee λ plays he same ole as he discee label j: ha is he enegy o effecive mass inside he λ shell is denoed by m(λ. As in Eq.(15 he meic funcion in he ligh shell is u = 2m(λ/. (16 The evolving infinie edshif suface is he smooh line in Fig. 2b. (a (b infinie edshif, u = -1 FIG. 2: In (a a discee sequence of ligh shells foms a black hole; (b shows he coninuous vesion of he same pocess. A labeling scheme, which is convenien fo boh he pesen ligh shell and he fluid sysems o be discussed lae, is o ake he enegy inside he λ shell o be popoional o λ, 8708A2 m(λ = λm f, m f = final mass. (17 The ime a which he λ shell eaches he cene may be chosen almos abiaily as a funcion of λ, subjec o he consains ha i be 0 fo he innemos shell λ = 0 and incease monoonically o he final ime fo he ouemos shell λ = 1, as shown in Fig. 2. We denoe his funcion as F 1 (λ, fo easons ha will become appaen. The equaion of moion fo he shell λ is hus + = F 1 (λ. (18 Fom is definiion F 1 mus have an invese, so we may inve his o obain ( + λ = F, m(λ = m f λ = m f F, (19 wih F (0 = 0 and F (1 = 1. Thus F seves as a densiy pofile funcion, and he meic in he ineio of he ligh shell may be wien as u = 2m f ( + F = F. (20 The infinie edshif suface, defined by u = 1, is hus deemined by ( + = F. (21 Inveing his we obain fo he infinie edshif suface = F 1 (/. (22 This implies ha if he oal duaion of collapse is sufficienly lage hen mus be a posiive monoonic funcion of.
5 Using he meic funcion Eq.(20 we may calculae he Einsein enso and he sess-enegy enso fom Eq.(10 and Eq.(11. This gives fo he ligh shell, T α β = F 8πG 2 kα k β, ligh shell, (23 whee F denoes he deivaive of F wih espec o is agumen, and k α is he null veco defined in Eq.(13. Fom his we obain he enegy densiy and he oal enegy of he ligh shell as ρ = T 0 0 = F 8πG 2, M s = 4π 2 ρd = m f F (1 G = m f G, (24 whee he inegal is again done in he asympoically fla spaceime of he disan pas, veifying ha he enegy of he ligh shell is equal o he final black hole mass. Ou appoach o boundaies and bounday condiions is somewha unohodox; we do no impose bounday condiions pe se on he meic funcion beween egions of spaceime. Insead we use he above soluions (in vacuum and wihin he ligh shell o calculae he sessenegy enso, which is defined by he field equaions and Eq.(10. The esuling sess enegy enso mus be zeo in vacuum, coecly epesen he sess enegy wihin he ligh shell, and descibe he sess on he boundaies. If i does, he soluion makes physical sense. Fo he egion nea he inne and oue boundaies of he ligh shell we may wie he meic funcion as u = { (F/Θ( +, inne suface, (F/ Θ( +, oue suface. (25 Calculaing he sess enegy enso fom Eq.(10 and Eq.(11 we find ha hee is no singula shell a hese boundaies since he elevan singula deivaives cancel; hus he sess-enegy enso associaed wih he boundaies is zeo and he soluion is hus physically easonable. As a specific example le us ake he densiy pofile funcion o be linea, F ( + = +, m(λ = m f ( +. (26 This coesponds o a consan ae of enegy impacing he cene. Fom Eq.(20 he meic funcion in he vaious egions is, 0, Minkowski egion, u = /, Schwazschild egion, (27 ( + /(, wihin ligh shell. The infinie edshif suface, fom Eq.(21, is he linea funcion ( = 1. (28 The slope of his is posiive fo / > 1, ha is when he ligh shell hickness is geae han is Schwazschild adius. Wihin he ligh shell he enegy densiy is, fom Eq.(10 and Eq.(11, T 0 0 = ρ = 8πG 2, ligh shell ineio, (29 which is independen of ime. In Fig. 3 we show a qualiaive skech of some ougoing ligh ays fo a faily geneal ligh shell. The skech is made by noing ha, fom Eq.(9: he slope of ougoing ays is 1 in he Minkowski egion and a lage disances i appoaches 1; he slope is 0 along he infinie edshif suface; finally, he slope is -1 a he cene of he black hole. Thee is a las-ay-ou emied fom he oigin o infiniy, afe which all ougoing ays (as well as paicles ae apped wihin he suface a and evenually fall ino he singulaiy. The suface defined by he las-ayou is hus a global hoizon. This illusaes ha he hoizon and he infinie edshif suface ae quie diffeen inside he ime dependen ligh shell, unlike he siuaion fo he ime independen Schwazschild geomey. infinie edshif las ay ou A B C FIG. 3: Some ougoing ligh ays in he hick ligh shell collapse. The las ay ou (B hoves a he Schwazschild adius and defines a hoizon. As a fuhe applicaion of ou mehods we noe ha ou esuls can be evesed in ime o descibe adiaion being emied by a spheically symmeic sysem such as a sa o black hole [27, 29 31]. To do his we use he minus sign fo he meic in Eq.(4 and un he diagam in Fig.2b upside down. I is hen easy o modify ou algebaic esuls o descibe any easonable ligh shell densiy pofile. This may be useful in sudying he final sages of black hole evapoaion, when he gaviaional field of he adiaion becomes compaable o ha of he black hole and canno be negleced [22]; i migh help in deemining if a black hole adiaes eniely away o vacuum o leaves behind a emnan [32, 33]. 8708A3 IV. THIN LIGHT SHELLS IN KUSKAL-SZEKEES COODINATES The hin ligh shell discussed in secion 2 is pobably he simples model of gaviaional collapse. Howeve
6 in EF coodinaes g 00 is negaive fo he ineio of he black hole. This means ha if one is a coodinae es (d = dθ = dφ = 0 hen he squae of he pope ime ineval ds 2 has he opposie sign of he squae of he coodinae ime ineval d 2, so ha may no be inepeed as a good ime make in ha egion of spaceime. The same is ue of Schwazschild coodinaes. Kuskal- Szekees (KS coodinaes, which ae discussed in many exs, wee developed o solve his poblem[4, 24, 25]. Hee we will obain KS ype coodinaes fo he hin ligh shell collapse of secion 2. By KS coodinaes we mean a sysem in which he, pa of he meic is confomal o fla Minkowski space, wih no singulaiies o zeoes. We fis show how a meic in EF fom, wih u = u(, may be pu ino confomal fom. The meic Eq.(4 may be facoed as follows, wih angula dependence suppessed, ds 2 = d 2 d 2 + u(d + d 2 = (1 + u(d + d(d 1 u 1 + u d 1 u = (1 + ud( + d( σ, σ 1 + u d.(30 The quaniies + and σ ae emed confomal null coodinaes since he line elemen is zeo along lines of consan + o σ; hese hus epesen adially moving ligh ays. We ansfom o ohe null confomal coodinaes by choosing any funcions w( + and v(σ, so ha o map he exeio egion o he w s > 0, v s < 0 quadan we choose γ = 1 and w = 1. Then he ansfomaion and meic funcions ae explicily w s = e + 2, ( vs = e 2 1, H s = 43 e.(35 Noe ha w s and v s ae dimensionless and H s is he squae of a disance. Fo he ineio of he black hole, <, and ouside he ligh shell (see Fig.1b we choose he opposie sign fo he logaihm and γ = ω = 1 in Eq.(34, o obain he same expession as in Eq.(35, which is hus valid houghou he Schwazschild geomey. The expessions Eq.(35 ae simila o he sandad ones used o ansfom beween Schwazschild coodinaes and KS coodinaes, bu diffe in impoan ways. Lines of =cons. map o hypebolae wih w s v s = cons.; in paicula = 0 coesponds o w s v s = 1 and = coesponds o w s v s = 0. The lines = and = boh map o v s = 0, while = maps o w s = 0. (Unlike he case wih Schwazschild coodinaes lines of consan do no map o ays of v s /w s = cons.! See Fig. 4. v =0 A B singulaiy C w collapsing hin ligh shell dw = w ( + d( +, dv = v (σ d(σ,(31 so he meic in ems of w, v is ds 2 = 1 + u w dwdv = Hdwdv. (32 v The meic funcion H may be consideed o be a funcion of and, o w and v. We may also elae he null coodinaes o Loenz-like coodinaes, w = τ + ρ, v = τ ρ, ds 2 = H(dτ 2 dρ 2. (33 Thus τ and ρ may be inepeed as ime and adial coodinaes povided ha H is posiive and has no singulaiies o zeos. Ouside he Schwazschild adius and he ligh shell he funcion σ and a convenien choice fo he funcions w and v ae he following + ( σ( = d = + 2 ln 1, >, w s = γe a(+, v s = ωe a(σ = ωe a( ( 2a 1, H s = 1 γωa 2 e 2a 2a ( 1 2a, (34 whee a, γ, ω = cons. This ansfomaion is chosen o make H s independen of ime. In ode ha H s also be nonsingula and have no zeos we choose a = 1/2, and 8708A4 FIG. 4: Collapse of a hin ligh shell in KS coodinaes, o be compaed wih Fig. 1b in EF coodinaes. Only he spaceime egion o he igh of he line = 0 has physical meaning. Compae he ligh ays A B C o hose in Fig. 3. Thee ae many ways o ansfom fom EF o KS coodinaes fo he Minkowski geomey inside he ligh shell; of couse Minkowski geomey in EF coodinaes is aleady in KS fom, bu no one ha is useful o us. We mus choose a ansfomaion ha joins coninuously wih he ansfomaion Eq.(35 along he bounday line + = 0, and we also demand ha he funcion H be coninuous along ha line. Fo ha space he meic funcion u = 0, so ha fom Eq.(30 σ =. Thus fom Eq.(32 he meic funcion H is H m = 1 w m( + v m(. (36 Equaing his wih H s fom Eq.(35 along + = 0 we obain a diffeenial equaion fo v m v m(2 = 1 4 3 w m(0 e/. (37
7 Tha is, in ems of he agumen, denoed x, v m(x = (cons.xe x/2, v m (x = (cons.(x 2e x/2. (38 Fo coninuiy we choose w m o be he same as in he Schwazschild egion Eq.(35, so wih appopiae consans we aive a w m = e + 2, v m = (1 e 2, 2 H m = 4 3 2 ( e. (39 These funcions ae obviously equal o hose in Eq.(35 along he line + = 0, as desied. Noe ha he pice we pay fo having H s in he Schwazschild egion independen of ime is ha H m in he Minkowski egion is dependen on ime. The naue of he ansfomaion Eq.(39 is bes seen in ems of he mapping of some lines and poins: = maps o v = 0; = maps o w = 0; lines of consan + map o consan w lines; lines of consan map o consan v lines; in paicula + = 0 maps o w = 1 and = = 0 maps o w = v = 1. Finally he oigin = 0 maps o w m = e /2, v m = (1 + 2 e /2, (40 so v m = 1 (1 + ln w m. (41 w m Hencefoh we dop he subscips on w and v. Figue 4 show he complee collapse pocess, wih he Schwazschild and Minkowski geomeies siched ogehe along he line + = 0. Only he egion o he igh of he line coesponding o = 0 has physical meaning. I is eviden fom he figue ha he line v = 0 defines a hoizon, and i is also appaen ha he singulaiy a = 0 does no behave like a ime independen spaial posiion. Noe also ha ρ seves as a adial make, even hough egions of he spaceime have negaive ρ. Figue 4 may be compaed wih he pue o eenal Schwazschild geomey discussed in many exbooks [3, 4] and shown in KS coodinaes in Fig. 5; his shows he maximum analyic exension of he Schwazschild soluion. The egion w < 0, v < 0 is inepeed as a whie hole, and is absen in figue 4; also absen is he egion w < 0, v > 0, inepeed as he ohe side of he womhole, and pa of he ineio egion w > 0, v < 0. This illusaes ha if he fomaion of he black hole by gaviaional collapse is aken ino accoun hen he much-discussed whie hole and womhole egions ae no pesen; hee is no eason o expec ha such egions occu in naue, as emphasized by Wheele and many ohes [17]. v black hole suface whie hole suface 8708A5 Schwazschild exeio #2 BH ineio WH ineio singulaiy, =0 Schwazschild exeio singulaiy, =0 w black hole suface whie hole suface FIG. 5: The pue o eenal Schwazschild geomey in KS coodinaes. The enie spaceime egion shown is given physical meaning in ems of a whie hole egion and a second exeio Schwazschild egion. Compae o Fig. 4. See fo example efeences [3,4]. V. UNIFOM FLUID SPHEES The spheical shells of ligh used in he peceding secions povide vey simple models of gaviaional collapse bu ae ahe aificial and do no appoximae anyhing we expec o find in naue. We now un o a moe ealisic sysem, a sphee of unifom pefec fluid wih a linea equaion of sae. This includes ou appoach o he special case of a dus ball, he oiginal [15] and sill a favoed sysem fo collapse sudies [3, 5, 8]. We believe ou appoach is he simples available, because he Painleve-Gullsand (PG coodinae sysem is emakably well suied o he ask [20, 21]. Some of he mahemaical echniques we used fo ligh shells, such as use of concenic layes and he handling of he bounday condiions, will also pove useful fo fluid sphees. The fluid collapse involves only wo spaceime egions, he fluid ineio, and he Schwazschild exeio. A cucial sep is o find a meic fom (he PG fom which descibes boh egions simply, and in which he moion of he fluid is simple. The spaceime geomey coesponding o a unifom fluid is well known fom cosmology; i is descibed by he Fiedmann-obeson-Walke meic in co-moving coodinaes, which coves all of spaceime fom he big bang onwads [3,4]. Fo simpliciy we conside he spaially fla case, ha is k = 0, fo which he meic is ds 2 = d 2 a( 2 (d 2 c + 2 cdω 2. (42 (This coesponds o zeo enegy collapsing sysem. The co-moving adial coodinae c is dimensionless, while he scale funcion a( is a soluion of he cosmological equaions, wih he dimension of a lengh. To descibe a finie sphee we uncae he adial coodinae a some value. If he fluid has a linea equaion of sae, wih pessue and densiy obeying p = αρ, hen he scale funcion is a powe of, specifically a( = A n 2, A = cons., n = 3(α + 1. (43
8 In paicula dus (o cold mae has negligible pessue so α = 0 and n = 2/3, while adiaion (o ho mae has α = 1/3 and n = 1/2. This is he ange of nomal mae. In he cosmological scenaio ime uns fom he big bang a = 0 o infiniy, bu in he collapse scenaio i will un fom negaive infiniy o = 0. Tha is, a vey lage fluid sphee in he fa disan pas collapses owad zeo size a = 0. To obain he desied fom fo he meic inside he fluid we inoduce a new adial coodinae, = a( c, (44 which is no co-moving, o obain he meic ds 2 = [1 (ȧ/a 2 ]d 2 + 2(ȧ/add d 2 2 dω 2 = [1 (n/ 2 ]d 2 + 2(n/dd d 2 2 dω 2. (45 This conains he single meic funcion n/ and is disincive in having a coss em and g = 1. I has an infinie edshif suface whee g 00 = 0, o n/ = ±1. The minus sign is appopiae since we will deal wih negaive imes. The empy egion exeio o he fluid is descibed by he Schwazschild meic in Eq.(1, which we now wie in he fom ds 2 = (1 ψ 2 d 2 s d2 1 ψ 2 2 dω 2, ψ = ±. (46 To make his compaible wih he meic in Eq.(45 in he fluid we choose a new ime coodinae ha makes g = 1. Taking we find ha s = + g(, (47 ds 2 = (1 ψ 2 d 2 ± 2ψdd d 2 2 dω 2, (48 povided ha g obeys g = ± ψ 1 ψ 2. (49 As expeced his ansfomaion involves an infinie ime sech a he Schwazschild adius, whee ψ 2 = 1. Fo he Schwazschild egion he soluion o Eq.(49 is ( g = 2 + ln. (50 The meic Eq.(48 and he ansfomaion Eq.(50 ae hose obained by Painleve and Gullsand [20, 21]. Boh egions of spaceime ae now descibed by he meic fom Eq.(48, wih ψ allowed o be a funcion of boh and ; specifically { n/, fluid ineio, ψ = (51 /, Schwazschild exeio. We choose he posiive signs in Eq.(49 and negaive sign in Eq.(50 o coespond o collapse duing negaive ime, and efe o he meic fom and coodinaes as genealized Painleve-Gullsand o simply PG. The PG meic has a emakable popey ha is cucial o ou analysis. The geodesic equaions fo adial moion of a paicle in he meic eq(48 lead, afe some algeba, o [ 0 = d2 ds 2 + ψ ( d ] ds 2 1, ψ ψ, 1 = (1 ψ 2 ( d ds 2 + 2ψ d d ds ds (d ds 2. (52 One obvious soluion o he fis is d/ds = 1, = s s 0. (53 Thus coodinae ime and pope ime inevals ae equal fo such a feely falling paicle, and his holds fo any meic funcion ψ. The second equaion in Eq.(52 now becomes quie simple d ds = d = ψ. (54 d In he Schwazschild egion his is he same as he classical Newonian equaion fo a adially falling es paicle of zeo enegy. We will soon need he Einsein enso and he sessenegy enso. Fom he PG meic fom i is saighfowad and only slighly edious o calculae hese. The nonzeo componens of he Einsein enso ae G 0 0 = 2ψψ G 1 1 = 2ψψ ψ2 2, G1 0 = ψ2 2 2 ψ, 2ψ ψ, G 2 2 = G 3 3 = ψ + 2ψψ ψ ψψ ψ 2. (55 As befoe we define he sess-enegy enso via he field equaions G α β = 8πGT α β ; in paicula he enegy densiy is ρ = T 0 0 = 1 8πG (2ψψ + ψ2 2 { 0, exeio, = 3n 2 8πG, ineio. 2 (56 Thus he densiy is unifom, as expeced. We now join he spaceime egions inside and ouside he fluid by demanding ha he meic funcion ψ in Eq.(51 be coninuous acoss he fluid suface bounday. This gives 3/2 + = 0. (57 n The geodesic equaion fo a zeo enegy falling paicle in he Schwazschild exeio egion is 3/2 + 3 2 ( 0 = 0. (58
` 9 This agees wih he moion of he suface Eq.(57 only fo he case of n = 2/3, ha is dus. Thus a feely falling paicle may hove a he falling suface of a dus ball, as i should. Howeve, if he pessue is nonzeo and n < 2/3 hen he suface will fall moe apidly. This is due he sess in he suface laye, as we will see. Figue 6 shows he collapse scenaio in PG coodinaes. singulaiy infinie edshif VI. ZEO PESSUE FLUID SPHEES We nex conside he collapse of a sphee filled wih zeo pessue pefec fluid, ha is a dus ball. This poblem is discussed by Landau and Lifshiz[5] using Lemaie-like coodinaes[34]. Fo simpliciy and physical claiy we use he same echnique as in secion 3 fo consucing hick ligh shells; ha is we build he dus ball laye by laye fom a sequence of hin dus shells. The layes ae pohibied fom cossing o peven supeficial singulaiies due o he consequen infinie densiy[26]. PG coodinaes will again be used, wih he meic fom Eq.(48 conaining a single meic funcion ψ(,. The Einsein enso fo his is given in Eq.(55, and we ecall ha in PG coodinaes we may ake pope ime and coodinae ime inevals o be equal along zeo enegy paicle geodesics. 8708A6 FIG. 6: Collapse of a unifom fluid sphee o fom a black hole in PG coodinaes. I is saighfowad o calculae he sess-enegy enso fo he fluid suface wih he same echnique we used in secion 3. In he viciniy of he suface he meic funcion ψ may be wien singulaiy dus shell, 3/2 + (3 / 2 = 0 ψ = n Θ( 3/2 + n Θ(3/2 +. (59 n This leads o a singula sess-enegy enso T 2 2 = T 3 3 = n 8πG 2 (1 3n ( 2 δ ( /n 2/3.(60 Fo a zeo pessue fluid, α = 0 and n = 2/3, his vanishes as expeced. If he pessue is no zeo i epesens a suface ension, as we will now show. The suface ension of a fluid sphee is elaed o is adius and pessue by τ = p/2; in he pesen case his implies fom Eq.(43 and Eq.(56, τ = p 2 = αρ 2 = n 8πG 2 (1 3n, (61 2 which agees wih Eq.(60. Thus he sess-enegy enso indeed epesens a suface ension ha balances he inenal pessue of he fluid o keep i sable. This is why he suface falls fase han a feely falling paicle. In summay he collapse of a unifom fluid sphee is descibed by he meic funcion in Eq.(48 and Eq.(51, he densiy in Eq.(56, and he suface ension in Eq.(60. Is collapse is qualiaively simila o ha of he ligh shells, bu ahe less aificial. Ou use of suface ension o sabilize he suface is a concepually simple subsiue fo he gadual pessue gadien of a moe ealisic model; such angenial pessues ae also menioned by Singh in ef.[26] and in he efeences conained heein. 8708A7 FIG. 7: A hin shell of dus falls ino a black hole o fom a lage black hole in PG coodinaes. This is he analog of Fig. 1 fo ligh. In analogy wih secion 3 we begin wih a hin shell of dus falling ono a black hole, shown in Fig. 7. The shell is assumed o be vey ligh, so ha m and m diffe by a small m, and he Schwazschild adii diffe by 2 m. As in Secion 5 he meic funcion in he wo Schwazschild egions is { /, iniial Schwazschild egion, ψ = (62 /, final Schwazschild egion. The equaion of he hin dus shell is ha of a geodesic fo a zeo enegy paicle in he Schwazschild geomey, given in Eq.(58. I is impoan ha coodinae and pope ime inevals ae equal along he geodesic, so ha he equaion fo he bounday is a elaion beween he coodinaes and. To obain he sess-enegy enso fo he dus shell we wie he meic in he viciniy of he bounday as ψ = Θ( 3 + 3 2 Θ( 3 + 3. (63 2 whee Θ + Θ = 1 as befoe. Fom he Einsein enso in eq(efpgg his leads, as in secion 3, o a singula enegy
10 densiy fo he dus shell, ρ = T 0 0 = 3 8πG δ( 3/2 + 3. (64 3/2 2 Fom his we may calculae he mass of he shell by going o lage negaive imes when he shell is in nealy fla space. M s = 4π 2 ρd = G ( m G. (65 We hus veify ha he mass of he iniial black hole plus he shell mass is equal o he mass of he final black hole. As in secion 3 we label he shells by he oal mass inside he shell, m(λ = λm f, (69 whee m f is he final mass. The ime a which he λ shell impacs he cene may be chosen ahe abiaily, and we denoe i by h(λ. Fo simpliciy and o avoid densiy singulaiies i is impoan ha he shells do no coss each ohe. This will be ue if h(λ inceases monoonically; ha is, oue shells impac a imes lae han inne shells. Thus we choose h(λ o incease monoonically fom 0 o as λ uns fom 0 o 1. The equaion fo he λ shell is hen he fee fall equaion fo a paicle of zeo enegy, wih he paicle eaching he oigin a h(λ, o 3/2 + 3 2 λ[ h(λ] = 0. (70 (a (b 8708A8 As we will see, he funcion h(λ deemines he dus ball densiy. If he funcion h(λ is specified we can use Eq.(68 and Eq.(69 and Eq.(70 o ge a paameic expession fo he infinie edshif suface; he adius and ime ae given in ems of λ by FIG. 8: In (a a discee sequence of dus shells foms a black hole, and in (b a coninuous vesion of he same pocess foms a black hole. These ae analogs of Fig. 2 fo ligh. This elemenay esul allows us o consuc a ahe geneal dus ball wih non-unifom densiy fom layes of hin shells. A discee sequence of hin shells is shown in Fig. 8a. The iniial sysem is a black hole of vanishingly small mass m 0 wih a hin shell of mass m 1 collapsing ono i a = 0 o give a black hole of mass m 1 = m 0 + m 1, followed by moe shells, and ending wih a final shell impacing he oigin a = o give a final black hole of mass m f and Schwazschild adius = 2m f. In ems of he inemediae masses m j he meic funcion in he egions beween shells is ψ = 2m j /, j-h Schwazschild egion. (66 The infinie edshif suface whee ψ = 1 and g 00 = 0 is he zigzag line in he figue. A coninuous vesion of he dus shell sequence is shown in Figue 8b. The shells ae labelled by a coninuous vaiable λ anging fom 0 o 1, wih he oal geomeic mass inside a shell denoed by m(λ; m(λ is he coninuum analog of m j, and he meic funcion in he fluid egion is hus ψ = 2m(λ/, fluid egion. (67 The infinie edshif suface is whee ψ = 1, o = 2m(λ, infinie edshif suface. (68 = λ, = h(λ 2 λ. (71 3 In paicula fo he cenal shell wih λ = 0, which impacs he cene a = 0, and he oue shell wih λ = 1, which impacs a =, we have fo he infinie edshif suface = = 0, cenal shell, =, = 2 3, ouside shell. (72 Thus he suface will move ouwad wih ime if > 2/3 ha is if he mass impacs he oigin sufficienly slowly. Cuiously he same expession occued fo he ligh shell collapse in Eq.(28. Fo he special case of zeo oal impac ime, = 0, all of he shells impac he cene a = 0. This is coesponds o he unifom densiy fluid sphee in he pevious secion, so h(λ = 0 clealy implies unifom densiy. We can explicily see he elaion beween h(λ and he densiy by solving he shell equaion Eq.(70 fo he mass m(λ inside he shell λ a a lage negaive ime = T, when T >> h(λ, o ge [ m(λ = 2 9(T + h 2 ] 3 2 9T 2 3. (73 Thus in he disan pas he mass is popoional o 3, meaning ha he dus is asympoically unifom. As ime pogesses o smalle negaive values he densiy may deviae moe and moe fom unifom, and may be quie non-unifom a = 0. To obain he meic and ohe popeies of a collapsing dus ball we fis specify a funcion h. Then he shell equaion Eq.(70 gives he shell paamee as a funcion
11 of posiion, ha is λ = λ(,. This is consisen because we allow only one shell o pass hough a given poin. Wih he paamee known as a funcion of posiion he meic in he fluid is given by Eq.(67. Wih his expession fo he shell paamee he shell elaion Eq.(70 is an ideniy in,. A numbe of ineesing popeies of he dus egion may be expessed in ems of he funcion h(λ. (Lae we will conside a specific h(λ o illusae fuhe. Fom he expession Eq.(67 fo he meic funcion ψ and λ(, we may calculae he Einsein enso fom Eq.(55, obaining G 0 0 = λ 2, G1 0 = λ 2, λ = λ, λ = λ. (74 Fo he ohe componens we need o elae he deivaives λ and λ o each ohe, which may be done using he shell elaion Eq.(70. Diffeeniaing Eq.(70 we find λ / = h + 2λdh/dλ, λ λ = h + 2λdh/dλ, λ λ = λ. (75 Thus he aio of deivaives is independen of he funcion h. Wih he use of Eq.(75 i is only slighly edious o show ha, fo any h, G 1 1 = G 2 2 = G 3 3 = 0. (76 Thus he pessue ems of he sess-enegy enso ae zeo, as we expec fo dus. Moeove he enegy densiy of he dus is ρ = T 0 0 = λ, dus egion. (77 8πG2 Fom his we may veify ha he oal mass of he dus ball is he mass of he final black hole, which we do by again going o lage negaive imes when he dus is in nealy fla space, M f = 4π 2 ρd = 2G λ d = m f G. (78 Finally he sess enegy enso on he dus ball suface is zeo accoding o Eq.(61 wih n = 2/3. Thee ae wo specific examples of he funcion h(λ fo which Eq.(55 is easily solved. The fis example is h(λ = / λ. (79 We leave i as an execise o he eade o show ha he meic funcion in he dus is hen / [ ψ = + 2 2 + (8 /3 ] 3/2. (80 In he limi of 0 his gives he same esul as Eq.(51 fo a unifom dus ball. The enegy densiy is [ + 2 + (8 /3 3/2 ] ρ = 8πG 3/2 2 + (8 /3 3/2 1/(2πG 2, fo lage. (81 This is a well-behaved posiive funcion wih no singulaiies o zeos fo negaive. The second example is h(λ = D/ λ, (82 whee D is a consan. This is singula and hus quie diffeen fom he fis example, and is no epesened by Fig.8b: i is singula fo λ 0, meaning ha he cenal dus laye collapsed o he cene in he infinie pas and he oue laye eached he cene a ime D. We again leave i as an execise o show ha he meic funcion is and ha he densiy is ψ = (2/3 + /( /, (83 ρ = D /(4πG 2 3/2. (84 The singulaiy a = 0 is mild in he sense ha he mass inside a sphee goes like 3/2. As 0 he densiy appoaches infiniy as expeced. Thus his example epesens a dus ball wih a mildly singula cenal densiy in he disan pas, which gows songe wih ime unil complee collapse, and epesens a ahe ealisic and amusing sysem [35]. Finally, we biefly noe some popeies of ligh ays in he collapsing dus ball. Fom he PG meic Eq.(48 we may wie he coodinae velociy of ligh as v c = d d = { 1 + ψ, ougoing, 1 + ψ, infalling. (85 We geneally expec ha ψ will be zeo a = 0, excep nea = 0, so v c = ±1 hee; noe ha his is no ue fo he second example consideed above. Also, v c = ±1 fo asympoically lage disances. On he infinie edshif suface ψ = 1 so ha v c = 0 fo ougoing ligh and v c = 2 fo infalling ligh. Fo he cene in he Schwazschild egion v c =. This allows us o make a ough qualiaive skech of some ligh ays in Fig.9. (Fo = 0 Eq.(85 may be solved exacly. Noe ha, as in he case of he hick ligh shell in Secion 3, hee is a las ligh ay ou, whose pah defines a global hoizon inside of which neihe paicles no phoons may escape o infiniy. VII. SUMMAY AND FUTHE STUDY In his pape we have ied o pesen a simple inoducion o he dynamics of gaviaional collapse, which
12 infinie edshif membes of he Gaviy Pobe B heoy goup fo many ciical and simulaing discussions, in paicula Fancis Evei, obe Wagone, and Alex Silbeglei. EFEENCES las ay ou A B C FIG. 9: Some ougoing ligh ays in he collapsing dus ball. No ay can escape fom he cene afe he las ay ou (B. Compae o Fig. ThickLighShellEFHoizon fo ligh shell collapse. we hope can povide a bidge beween basic exbook geneal elaiviy and numeous eseach opics of cuen inees. The ligh shells in EF coodinaes and he fluid sphees in PG coodinaes povide he simples way we know of o sudy collapse. Some simple bu amusing exensions of hese models can be made, as aleady menioned: one may evese he ime and sudy he emission of ligh and mae fom whie holes o ohe spheically symmeic objecs, and i is easy o add a cosmological consan in he fluid collapse. The pesen eseach lieaue is dense wih sudies involving collapse, anging fom fundamenal heoy o asophysical applicaions. A sampling of ecen opics fom he Los Alamos achives xxx.lanl.gov includes he following, which we paaphase: Enopy in collapse o a black hole - whee does he infomaion go? Uniaiy - is collapse consisen wih uniay quanum evoluion? Cosmic censoship - ae all singulaiies suounded by a hoizon? Collapse in conex of sing heoy, ani de Sie space Quanizaion and enopy of he suface of a black hole. Collapse in divese dimensions. Scala enso (o ohe gaviy heoies and collapse Collapse wih a cosmological consan included. Paicle poducion in collapse. ole of pessue (adial o angenial in collapse. Sabiliy of sas agains collapse due o oaion. Gaviaional adiaion fom (non-spheical collapse. VIII. 8708A9 ACKNOWLEDGEMENT This wok was suppoed by NASA gan 8-39225 o Gaviy Pobe B, and by he US Depamen of Enegy unde Conac No. DE-AC02-76SF00515. We hank he [1] See fo example P. Davies (ed., The New Physics, Chape 3 on The enaissance of Geneal elaiviy, C. Will, and Chape 6 on The New Asophysics, M. Longai (Cambidge, 1989. [2] A bief oveview of such concens is given in G. Fase (ed. The New Physics fo he 21s Cenuy, Chape 3 on Gaviy,. J. Adle (Cambidge, 2004 (in pess. [3] C. W. Misne, K. S. Thone, and J. A. Wheele, Gaviaion (Feeman, 1973. [4]. J. Adle, M. Bazin, and M. M. Schiffe, Inoducion o Geneal elaiviy (McGaw Hill, 1975. [5] L. D. Landau, E. M. Lifshiz, The Classical Theoy of Fields, 4h evised English ed., (Buewoh Heinemann, 1999. [6] H. C. Ohanian,. uffini, Spaceime and Gaviaion, (Noon, 1994. [7] I.. Kenyon, Geneal elaiviy, (Oxfod, 1990. [8] S. Weinbeg, Gaviaion and Cosmology, (Wiley, 1972. [9] W. indle, Essenial elaiviy, (Van Nosand and einhold, 1969. [10] B. F. Schuz, A Fis Couse in Geneal elaiviy, (Cambidge, 1985. [11]. Wald, Geneal elaiviy, (U. of Chicago pess, 1984. [12]. D Inveno, Inoducing Einsein s elaiviy, (Claendon, Oxfod, 1995. [13] M. M. Schiffe,. J. Adle, J. Mak, C. Sheffield, J. Mah. Phys. 14, 52 (1973. [14] C. Hillman, A Bief Hisoy of he Concep of Gaviaional Collapse, hp://mah.uc.edu/home/ baez/elwww/hisoy.hml [15] J.. Openheime and H. Snyde, Phys. ev. 56, 455 (1939. [16] I. D. Novikov, docoal disseaion, (Shenbeg Asonomical Ins. Moscow, 1963. [17] B. K. Haison, K. S. Thone, M. Wakano, J. A. Wheele, Gaviaional Theoy and Gaviaional Collapse, (U. of Chicago Pess, 1965. This ahe ealy book is mainly devoed o equilibium configuaions and hei sabiliy, wih dynamical collapse finally discussed in chape 11. [18] A. S. Eddingon, Naue 113, 192 (1924. [19] D. Finkelsein, Phys. ev. D 110, 965 (1958. [20] P. Painleve, C.. Hebd. Acad. Sci. (Pais 173, 677 (1921. [21] A. Gullsand, Akiv. Ma. Ason. Fys. 16, 1 (1922.
13 [22] M. K. Paikh and F. Wilczek, Phys. ev. Le. 85 5042 (2000; axiv: hep-h/9907001. [23] M. Paikh, A Sece Tunnel Though he Hoizon, Gav. eseach Foundaion Essay Compeiion, 1s place. [24] M. D. Kuskal, Phys. ev. 119, 1743 (1960. [25] G. Szekees, Publ. Ma. Debecen 7, 285 (1960. [26] T. P. Singh, Gaviaional Collapse, Black Holes, and Naked Singulaiies, axiv:g-qc/9805066v1. [27] P. C. Vaidya, Poc. Ind. Acad. Sci. A33, 246 (1951 and Phys. ev. 33, 10 (1951. [28] K. Lake, Phys. ev. D 62 027301 (2000; axiv:gqc/0002044. [29] S. W. Hawking, Comm. Mah. Phys. 43 199 (1975. [30] G. W. Gibbons, S. W. Hawking, Phys. ev. D 15, 2752 (1977. [31] S. Hawking,. Penose, The Naue of Space and Time, (Pinceon, 1996. [32]. J. Adle, D. I. Saniago, Mod. Phys. Le. A 14, 1371 (1999. [33]. J. Adle, P. Chen, D. I. Saniago, Gen. el. Gav. 33, 2101 (2001. [34] Fo a sho discussion of elevan coodinae sysems see M. Weinsein, Wokshop on Ligh-Cone Physics: Paicles and Sings, Teno 2001; axiv:gqc/0111027. [35] J. S. Liu, Docoal Thesis, Depamen of Physics, Sanfod Univesiy (2004.