PHYSICAL REVIEW' D VOLUME 50, NUMBER 12 15 DECEMBER 1994

Disk collapse in

Andrew M. Abrahams, ' Stuart I . Shapiro, t and Saul A. Teukolsky~ Center for Radiophysics and Space Research, Cornell University, Ithaca, ¹zv York 14853 {Received 25 May 1994) The radial collapse of a homogeneous disk of collisionless particles can be solved analytically in Newtonian gravitation. To solve the problem in general relativity, however, requires the full machinery of numerical relativity. The collapse of a disk is the simplest problem that exhibits the two most significant and challenging features of strong-Seld gravitation: formation and generation. We carry out dynamical calculations of several diferent relativistic disk systems. We explore the growth of ring instabilities in equilibrium disks, and how they are suppressed by sufhcient velocity dispersion. We calculate waveforms &om oscillating disks, and &om disks that undergo gravitational collapse to black holes. Studies of disk collapse to black holes should also be useful for developing new techniques for numerical relativity, such as apparent horizon boundary conditions for black hole .

PACS number(s): 04.25.Dm, 04.20.Jb, 04.30.Db, 04.70.—s

I. INTRODUCTION but because of conservation of angular momentum this motion. is not dynaxnical. We consider cases in which the The simplest example of gravitational collapse is that disk particles are initially at rest and also in which they of a homogeneous sphere of particles initially at rest. have initial angular motion. In the latter case we focus This collapse solution is analytic both in Newtonian grav- on disks with total J = 0, i.e., with equal numbers of ity and general relativity. In general relativity, this so- corotating and counterrotating particles. lution is known as Oppenheimer-Snyder collapse [1] (the In. Newtonian theory, it is known that equihbrium disks solution is a "piece" of a closed Friedmann imiverse). Be- supported against collapse by rotation alone are unsta- cause of Birkhoff's theorem we know that this solution is ble to ring formation [3]. The disk can be stabilized by nonradiating. Both the particle motion and the gravita- "heating" the disk, that is, converting some of the or- tional field are radially syxnxnetric, i.e., functions of one dered rotational energy into random "thermal" motion spatial variable. [4]. We explore here whether or not a similar result holds The radiating problem which is the simplest analogue in general relativity. to Oppenheixner-Snyder collapse is that of an axisymmet- Our study of disks is geared to analyze two relativis- ric, infinitely thin disk of particles initially at rest. This tic phenomena that do not arise in Newtonian theory: case is simple because the particle motion still depends collapse to black holes and the generation. of gravita- on only one spatial coordinate, although the gravitational tional waves. Since this is the simplest wave generation field now depends on two. If the disk is constructed by problem, disk collapse provides a useful proving ground squashing a homogeneous sphere into a pancake, keep- for testing codes designed to treat gravitational radia- ing the density homogeneous, then the solution is still tion in general relativity Once w. e can evolve disk sys- analytic in Newtonian theory [2]. tems accurately in general relativity, we should be able However, to solve the problem for relativistic gravita- to compute the &om any axisyxnmetric tion requires the full machinery of numerical relativity, source, since the equations for the field are essentially the and has not been addressed until now. Indeed, the dy- same as those for general axisymmetric sources. The ma- namical properties of a disk of collisionless particles has jor problems associated with general axisymmetric col- never been studied in general relativity, although it has lapse are all contained in this test case: formation and been extensively treated in Newtonian theory [3]. In this evolution of black holes and propagation of gravitational paper we tackle the collapse of an axisymmetric collision- waves. less disk of particles in general relativity theory. Particles We have previously studied disk collapse in the context in an axisyxnmetric disk can also have angular motion, of nonlinear scalar gravitation, where many of the tech- niques for handling infinitely thin disks were developed [2]. The basic code for evolving axisymmetric spacetimes in general relativity has been discussed in a number of Also at the Laboratory of Nuclear Studies, Cornell Uni- — versity, Ithaca, NY 14853. Current address: Department of papers [5 7]. We will refer extensively to the equations Physics and Astronomy, University of North Carolina, Chapel in Ref. [6] in the discussion that follows. The organiza- Hill, NC 27599. tion of this paper is as follows. In Sec. II we present ~Also at the Departments of Astronomy and Physics, Corne11 the equations for an evolving, general relativistic disk. University, Ithaca, NY 14853. In Sec. III we discuss analytic test problems including ~Also at the Departments of Physics and Astronomy, Cornell Newtonian and relativistic disk solutions. In Sec. IV we University, Ithaca, NY 14853. give the results of our numerical calculations.

0556-2821/94/50(12)/7282(10)/$06. 00 50 1994 The American Physical Society 50 DISK COLLAPSE IN GENERAL RELATIVITY 7283

II. BASIC EQUATIONS integrated across the equator yields

1 ~ t'sm 8 The particles comprising the disk are ass»med to in- ~ . -„T -~l ~ K ~ ~ Bs rsin8dH Bs(sinH K"„)+ ~ . 2 & & teract exclusively by gravitation; i.e., they obey the rel- sin 8 sin 8 & ) ativistic collisionless Boltzmann equation (Vlasov equa- tion). Accordingly, we have constructed a n»merical code that solves Einstein's equations for the gravitational Beld 1 = B— coupled to matter sources obeying the Vlasov equation. -Ss+ „(r K"s), (4) This is the se~e mean-field, particle simulation code de- scribed in Refs. to study nonspherical gravitational [5,6] where 6 denotes 8 = z/2 + e, e -+ 0. Functions that collapse. The code is designed to handle axisymmetric are symmetric across the equatorial plane, such as K„" systems with no net angular momentum. The present Hence reduces to version ass»mes equatorial symmetry. We solve the field and Kf„are continuous there. Eq. (4) equations in 3+ 1 form following Arnowitt, Deser, and 0 = 0. Now consider Eq. (3). Integrating it gives Misner [8]. We use maximal time slicing and quasi- + vrrit- 1 -„+ isotropic spatial gauge in axisymmetry. The metric, 0 = S„rsinHd8 ——Ks~ ten in spherical-polar coordinates is r

ds' = —a'dt'+ A'(dr + p"dt)'+ A'r'(dH + p'dt)' Since Ks is antisymmetric across the equator, Eq. (5) +B r sin Hdqrs. gives + The matter satisfies the relativistic Vlasov equation, KP+ = —KP = — S,r sinHdH. (6) which we solve by particle simulation in the mean grav- 2 itational field. The basic code is identical to the one described in Refs. [5]. The key equations and definitions Similarly, integration of the Hamiltonian constraint equa- of variables are given in the Appendix of Ref. [6]. The tion [Eq. (A6) of Ref. [6]] leads to equations below are written in terms of the auxiliary vari- ables = B~~2, T = and = lnT. (These were 1 . + 1 1 @ A/B, g —sin 8@s]+ = @ri,s ——— p'r sin HdH. (7) generalized to include net rotation in Ref. [7].) r 4r 8$ Integrating the lapse equation [Ref. [6], Eq. (A7)] gives A. Jn~p conditions 1 . + 1 —sin 8(ag) = avgas s~—— r s[ 4r The disk matter source afFects the metric via jump con- + ditions in the field equations across the equatorial (disk) (p'+ 2S)r sinHdH. (8) plane. These j»mp conditions replace the usual mat- 8 B ter source terms that appear in the field equations. A condition is used to set the value derivation and n»clerical implementation of such a j»mp The boundary Eq. (6) out- condition was presented in Ref. [2] for the simple case of of Ks all along the equatorial plane. In the vacuum, scalar gravity. There the governing equation is a nonlin- side of the equatorial plane, K& is deter~~+ed by inte- ear wave equation with a matter source on the right-hand grating the evolution equation, Eq. (A3) of Ref. [6], as side. usual. As an example of how the j»~p conditions may be When finite difFerencing the Ha~iltonian constraint derived in 3+ 1 general relativity, consider the taro mo- [Eq. (A6) of Ref. [6]], the derivative terms Q s and q s ment»m constraint equations (A4) and (A5) in Ref. [6]: appear in exactly the combination as in Eq. (7). The only place vrhere the matter source term p' appears in 2 8 the Hamiltonian constraint is through this boundary con- " 4') dition. Equation is used in an analogous fashion for sinH sins 8 g T (8) the lapse equation [Ref. [6], Eq. (A7)]. The dynamical equation for g [Ref. [6] Eq. (A2)] and the shift equations [Ref. Eqs. and for P" = —Ss + B„(r K"s),—(2) [6], (A8) (A9)] r and P~ remain unchanged. Note that g, P", and P~ are metric coefBcients and thus must be continuous across the equator. B„(r K"„)+—K~~ B„g = S„— . Bs(sinH K"s). B. Matter sources

Since the particles are confined to the equatorial plane The geodesic equations of motion for the collisionless 8 = m/2 where P = 0, the particle four-velocity compo- matter particles are given by Eqs. (A10)—(A16) of Ref. [6] nent satisfies us = us = 0. Hence Ss = 0. Equation (2) with the following simplifications: us = 0 and 8 = w/2. 7284 ABRAHAMS, SHAPIRO, AND TEUKOLSKY 50

Hence, as in spherical symmetry, only the radial motion III. ANALY'TIC SOLUTIONS AND TESTS is dynamical for an infinitely thin disk. The particles are binned in annuli to determine the Before we consider numerical solutions of the dynam- source terms for the field equations. Equations (A17)— ical equations for disks and their gravitational fields, we (A21) of Ref. [6] lead to the disk sources [9] give here some analytic results that will serve as code + checks and initial data for our evolutions. o— p'r sin 8d8 = ) ' : 2vrrAr ~ + A. Oscillating Newtonian disks Z„— S„rsin 8d8 = : ) 2~re, r ~' (10) 2 As discussed in Ref. [2], there exists a complete an- alytic solution that furnishes a good test of a numeri- cal disk code in the weak-field, slow-motion limit. The Z— Sr sin 8d8 = p'r sin 8d8 : solution describes an oscillating homogeneous spheroid m in Newtonian gravitation in the disk limit (eccentricity . e ~ 1). The surface density of such a disk of mass M u~ (2nrAr ), and radius R is Here m is the particle rest mass related to the total rest mass Mp by m = Mp/N with N the total particle num- 3M &, ber. We obtain Mo f'rom 2~R' I, R') Start with the equation of motion for the semimajor Mo —— o02mrdr, (12) axis R of an oblate homogeneous spheroid [e.g., Eq. (5.6) of Ref. the limit e -+ 1 and find where [11]].Take 3m. GM h o'p = ppr sin 8d8 = (13) (2o) ) 2vrrbr ~' 4 R2 R3' where h is the conserved angular momentum per unit and where po is the rest-mass density. mass of a particle at the surface. Since the motion is C. Hydrodynamical disks homologous, the radius of each particle satisfies a similar equation. Choose h to be a fraction of the equilibrium In the special case in which concentric shells of parti- ( angular momentum hp ——(3m GMRp/4) ~ . Set cles do not cross, collisionless matter may be treated as hydrodynamical dust. Thus, as an alternative to inte- R = RpX(t). (21) grating geodesic equations followed by binning of parti- cles, one could integrate the equations of relativistic hy- Then the radius r of each particle satisfies drodynamics. The hydrodynamical equations have the disadvantage that they are partial difFerential equations r = rpX(t), (PDE's) and not ordinary differential equations (ODE's) such as the geodesic equations. However, they have the where rp is the initial radius. Substituting Eq. (21) into advantage that they produce intrinsically smooth source Eq. (20) we see that, X satisfies the familiar equation profiles, unlike particle descriptions which are stochastic. of an elliptic orbit for a particle with specific angular The basic equations of relativistic hydrodynamics in moment»m h,g = ((M/Rps)i~2 around a fixed central 3+1 Arnowitt-Deser-Misner (ADM) form are given in, mass M,~ = 3vrM/4. The parametric solution for X(t) 1s e.g. , Ref. [10]. For a cold axisymmetric disk they reduce to the continuity and radial Euler equations: X = a(l —e cos u), —1 = P . P Ogo'p + 0~(ro'pv ) 0, (14) t = — —e sinu) — r 2' (u —,2' (24) 1 ~a — = ~~ + ~~ (rZ„v") oB„a+ g„—g„P" + ZB„ln Aa, where we ass»me X = 1 and X = 0 at t = 0. In Eqs. (23) (15) and (24), the semimajor axis, eccentricity, and period are where given by

p 'I e=1 —(, 4R,' P = 2m' i — s (25) I, 3m GM(2 P) ) 18 The radial and tangential particle velocities are given by 50 DISK COLLAPSE IN GENERAL RELATIVITY

X C. Relativistic disk 8) = —f') (26) (3~aM) "' Here we construct a relativistic generalization of the 4'=(X, ~ (27) cold homogeneous Newtonian disk described in Sec. III A. IE 4&~ )I This is useful for studying the dynamical behavior of 6eld where almost nothing It is simple to derive the gravitational wave amplitude disks in the strong region, for an oscillating disk in the quadrupole approximation. is known. in circular equilibri»m. In axisymmetry, in the absence of rotation, there is only Consider a disk of particles The Hamiltonian equation reduces to one polarization and its amplitude is given by [Ref. [6] Eq. (A6)]

——3- V Q= —2z —,P (36) eh+ —2"I„sin 8, (2S) where where we have temporarily restored. a factor of Sz' to the right-hand side. We can obtain an analytic solution if 1 2 3 2m 3 2 —— r pd z = —— r odr = ——MR . (29) we first cast Eq. (36) into Poisson's equation by setting I„=$$ 3 15 p'/@ = 2pN, obtaining Here we have used Using Eqs. and we Eq. (19). (20) (21) V = — find g 4zpN, (37) and then equating to the homogeneous Newtonian ——— —— pN rh+ Msin 8 BOX + ~ + limit 4 X X') density profile for an oblate spheroid in the pancake ~ [12]. We then find (30) g = 1 —4~, (38)

where ON is the Newtonian potential for a Battened B. Kalnajs disk spheroid [12]. In the disk interior, this yields 3zM~ ( When = 1, the above Newtonian disk solution cor- — ——lr g = ~ CN ~ 1 z (interior). (39) responds to a»»iformly rotating disk in dynamical equi- 4Z. ~ 2 Z,'~ libri»~. As mentioned in the Introduction, such a disk is The total mass of the disk M is related to the Newtonian »»stable to the formation of rings but can be stabilized mass MN appearing in 4N by by heating. Kal»ajs [4] has given an analytic prescription for constructing hot homogeneous disks in equilibrium. M = 2MN. (4o) They all have the same surface density o and gravita- tional potential 4 as the cold disk in Sec. IIIA, but Comparing Eqs. (36) and (37) we find that the surface differ in the amount of random motion. In these models, density Eq. (9) is given by the particles have an isotropic velocity distribution in a rotating &arne that moves with angular velocity 0: o =2@o~, (41)

8 cos g = vy —Of' ) (31) where the Using = 8). (32) oN is corresponding Newtonian density given by Eq. (19). The angular motion of a particle in an equilibrium disk = Here v is the magnitude of the isotropic velocity in the is determined by setting dv.,/dt u„= 0 and using Eqs. and of Ref. The result is rotating &arne and g is a random angle about the particle (AS) (A13) [6]. position in the disk plane. The distribution of particle (a,„/a)Bzr 2 velocities is given by (eq»ilibri»rn), (42) B,„/B —a,„/a + 1/r f(v)vdv = 2zK v —v vdv. where B = @z is given by Eqs. (38) and (39). The lapse Here a is found &om the maximal slicing (K,' = BqK,*. = 0) condition, Eq. (A7) of Ref. [6] specialized to eqmhbri»~, v = (020 —0 )(R~ —r ), (34) for which q = 0 = K'-. The source term Z appearing in hp the j»~p condition for the lapse equation is given by Op —— z=~ 2 and K is a normalization constant. Models in this family Br +u are parametrized by the ratio 0/Ae. Cold disks have 0 = Oe. Hot disks with 0/Ao ( 0.816 are stable against where we have used Eq. (A16) of Ref. [6]. Because a ring formation. and u& are interdependent, it is necessary to iterate the 7286 ABRAHAMS, SHAPIRO, AND TEUKOLSKY lapse equation and Eq. (42). As an initial guess we use when it was necessary to extract the waveforms at small 1 + @(exterior) radii, r 10 M, we tested the procedure by making cor- The rest-mass surface density oo defined in Eq. (13) is rections for the Schwarzschild background. These cor- given by rections were of the same order as the inconsistencies in the waveforms extracted at different radii: 10—20%. In 2o.~@ Sec. IVB we discuss further the sources of error in the + u2/gPr (44) extracted waveforms. We also monitor quasilocal mass indicators; for the runs shown here the total mass is nu- fmm which the total rest mass can be computed using merically preserved to within about 3%. Eq. (12). We can obtain a solution to the initial value equations for a nonequilibrium disk at a moment of time symmetry A. Oscillating cold Newtonian disk when the particles are moving at a fraction ( of their equilibrium velocity: As a check on our code, one would Grst like to simulate the collapse of the oscillating homogeneous disk described (equil) XLy = (ted, (45) in Sec. III A. Unfortunately, such a cold configuration is unstable to ring formation. This is the familiar result for The other equations remain unchanged. In the Newto- cold equilibrium disks with 0 = 00 and ( = 1 discussed nian limit, this solution goes over to the cold oscillating above; we find numerically that it also holds for cold os- disk of Sec. IIIA. cillating disks. We can still employ this analytic model to An interesting nonequilibrinm solution is the one in check the Beld solver in our code provided we supply the which ( = 0. This corresponds to the collapse of a homo- unperturbed source function n analytically as a function geneous disk in which all the particles are at rest initially. of time. We then allow the code to solve the Geld equa- This is the pancake analogue of Oppenheimer-Snyder col- tions and thereby determine the gravitational radiation lapse of a homogeneous sphere. In the disk case, however, numerically from the metric and extrinsic curvature. the solution is radiative and is not known analytically. In Here we give results &om a typical evolution of a ho- this case the total mass and rest mass are related by mogeneous disk with ~/Mo —30 and velocity cut- down factor = 0.9. This choice of radius is suffi- 2Mp ( (46) ciently large that the system remains essentially New- 1+ (1+6mMO/5')~/2 tonian throughout its evolution. We place the outer boundary at r —500 and use a mesh with 30 (see Ref. [12]). /Mo radial zones inside the matter, 220 outside, and 16 an- gular zones in the upper hemisphere. In Fig. 1 we com- IV. NUMERICAL RESULTS pare the gravitational waveform extracted at an arbitrary fixed exterior radius of 42 M using a gauge-invariant tech- nique [13], with the analytic prediction Eq. (30). After In this section, we evolutions com- give examples of an initial transient, the waveforms coincide very closely. puted with our fully relativistic disk particle plus gravity The slight noise in the extracted waveform arises kom code. Convergence and other tests of the basic code have interpolation error when the analytic source is mapped been reported on in other — sim- papers [5 7]. For realistic onto the numerical grid. ulation the convergence of the with grids, properties code The excellent agreement between the analytic and nu- the new disk source boundary conditions were consistent merical results here gives confidence in the reliability of with earlier results. the numerical implementation of the jump conditions and In Table I we s»mmarize the cases discussed in the text. the solution of the Geld equations. When possible, we show gravitational waveforms and also make contact with analytic results. The waveforms are extracted with the standard gauge invariant extraction method of Abrahams and Evans [13]. The waveforms B. Oscillating Kalnajs disk are extracted at a Gxed coordinate radius and displayed as a function of coordinate time. (Our radial coordinate To test the particle simulation aspects of our code, we is equivalent to the isotropic Schwarzschild coordinate set up stable equilibrium Kalnajs disks as described in for spherically symmetric spacetimes. ) In certain cases Sec. IIIB. Our code successfully holds these in equilib-

TABLE I. Disk evolution cases. Case Ro/I &/no Fate Oscillating cold Newtonian (analytic source) 30.0 1.0 0.9 Oscillates Oscillating Kalnajs 20.0 0.8 0.9 Oscillates Cold relativistic equilibrium 8.0 1.0 0.99 Rings Hot relativistic near-equilibrium 10.0 0.85 1.0 Collapses and virializes Cold relativistic 1.5 1.0 O.G collapses to black hole 50 DISK COLLA L RPLATIVITY 7287

I 200 radl4 I I I y 0 015 I

evolut»n are shown jjsjonless matatter f+om the hot. dzs Iz escapes m 0 O~O— P~jcles stas ay together an d oscillat~ hornomologo~ ly around . - I Fig. ~ew compare the ex- uted nllmer- p.005 IC to the quad'uP"le formu a ] gs (47) 3

uhjpo p.pp5 expec e nt over sever ooscjjlatjons. zn g of the ln ean- e particie. szmmujatjon sc em e i jncludzng 4 the 6eld and partlcclee jntegratom~ the p~ cle and I I ~ p pfp 500 000 1500 samp ' d the source binn ng algorjthm- . ~aveforms ~ extracte rent radll ag . 15'' level at thzs jd resol«zon eriments ~ t that most of FIG ' 'tational ~avefo rm from a cold oscillating dis ~ erenc, and the difference from t e zjuaaadrupole h an analytic N e~tpnian matte»& The ~ ~ numerically cal«at'on' arise &om 'jnstantaneo~ ropagate d er- generate and propaga ted gravitatio na] wave fo™(so]ig line), . . . ~ ' rors zn s the constrmnt equatlons. To re- extract ed at a radius of 41 m, is comp ared wit h thee analytic ~-zone an d gaauge ejfects~ the ~ave extraction prediction~ q' ) line) ~ The disk has an eq~ hbrium (dotted ns between different radius M a velocity go The wave cut«n multjpole moments. amplitude 's d' h+ is imenensionless vrhi e r units of ~ M. source can cause»ac curacies: t e Partlc e pro zy rap over very sm ail spatial stances. This leads rjism forr several rotationa periods. errors jnln the uted matter sources terms ' comPu ~hose the productzon of gravl'tational radia- e ropagateed jnstantaneol usl ln the arts of the oscillation of a hot ujhbrzn~ dzs ~ fpQIQ — the e ptjcIC con- u a K l ass Mo —20 an v strmnt equatjons. 'c locity dispersion e determined y the hyperbo. evolution eq~ b reducing all partzc e v not respon g u t ons the source lns tantaeomly.a

I I I I I I I j I I I I I I t.=o t=112 20 —,.

15 Y

1

5;::.:. :. ":::".', :.:: .. :

'L'. :.. i':i'i I i i i 0 3'.g. I:. I.i;.I '.k . r i'l«»liiil p ots of the particle positions the evolution ofo an osci a in i L is l radius is t,=224 t=336 ve oci y = 0.9, an d lo t d- rameter 0 00 —0. . or ina tes and time are in u

'. 'J' .: :.': . I-. =. I;4:.4".":41' . 4"t.".'I I:.. . . f.l 0;. : .itii33: i i i i'liiiiliiir l J ~ 4 . 0 5 10 15 20 25 0 5 10 15 7288 ABRAHAMS, SHAPIRO, AND TEUKOLSKY

I I I I I I I 4—i 0.010 ) t to answer is whether this disk is unstable to rings as in the Newtonian theory when the source and gravitational Geld are evolved consistently. In Fig. 4 we show a rela- tivistic disk with = 8. Before the evolution 0.005 Ro/M pro- gresses very far, ring formation begins and by t = 30M is completely dominant. Less relativistic conGgurations behave similarly. This calculation was carried out with 0.000 300 radial by 16 angular zones and 12 000 particles, but r»~~ with as many as 48000 particles did not not slow I down the ring formation.

I I g —0.005 t l I I I I t D. Hot relativistic disk I I —0.010 l As discussed before, ring formation is suppressed by II ql the addition of suKcient random particle motion. Un- I fortunately, only small values of velocity dispersion can 0.0 500.0 1000.0 be used without dissipating the outer region of the disk. Here we consider the evolution of a relativistic disk FIG. 3. Gravitational waveform from the oscillating with Ro/M = 10, and velocity dispersion parameter Kalnajs disk shorn in Fig. 2. The quadrupole waveform ex- 0/Os —0.85 and ( = 1.0. Because of its compactness, tracted at a radius of r/M = 21 (solid line} is compared with this near-equi&ibrium relativistic Kalnajs disk is unsta- the result, Eq. (28} (dashed line). ble to collapse. As shown in Fig. 5 the disk initially collapses and then oscillates about a new equilibrium of This contrasting behavior leads to errors in the extracted about Rs/M = 8.0. At late times it virializes to a static waveforms because extraction relies on delicate cancella- eon&librium state. tions of the near-zone Beld components. However, when In Fig. 6 the gravitational waveform extracted at smoothed out, the waveforms extracted at different radii r/M = 12.0 is compared with the quadrupole formula agree within the accuracy stated above. result. As discussed above, we have evidence that the stochasticity of the particle source is responsible for the C. Cold equilibrium relativistic disk short-time scale discrepancies between the waveforms. This calculation used a 200 radial by 16 angular zones Here we consider eq»i&tbrium disks constructed accord- and 12000 particles. Tmproving the accuracy requires ing to the prescription of Sec. IIIC. The question we wish higher grid resolution and more particles; unfortunately

8— t=o t=16

Y 4

FIG. 4. Snapshots of the particle posi- tions for the evolution of a cold, equilibrium relativistic disk. The initial disk radius is 8— R /Ms= 8.0. The rapid growth of concen t=49 tric rings is apparent.

I I . l I i 8 0 2 4 6 X 50 DISK COLLAPSE IN GENERAL RELATIVITY 7289

I) III) II I I I I ).I I I I I I I I I I 10 '— ) ) ) t=92

'~ ~ .1

FIG. 5. Snapshots of the particle positions I' I I I I I for the evolution of a hot, near-equilbrium relativistic disk. The initial disk radius is

I I I I I I I I I I I I I I ~/M = 10.0 and the velocity dispersion pa- ) ) ) 10— rameter = 0.85. Following t=183 t=274 0/Ao collapse, the disk settles doom to a new equilbrium state.

I I I I I I 0 2 4 6 8 10 0 2 4 6 8 10

the noise in the waveform is a ~N effect so iacreasing radiation from collapse and, perhaps, the first half wave- the number of particles quickly becomes computationally length of a quasi-normal-mode oscillation. ln Fig. 8 we prohibitive. show an estimate of the asymptotic quad upole wave form extracted at r/M = 8.0. After an initial peak rep- resenting radiatioa in the initial data we see an oscillat- E. Cold disk collapse ing signal with total wavelength of approximately 16 M, comparable with that of the most slowly damped E = 2 Fiaally, we consider the collapse of a cold homogeneous quasinormal mode oscillation which has A = 16.8 M. The disk, the disk analogue of Oppenheimer-Snyder collapse energy radiated is less than 0.1'%%uo in the quadrupole mode, in spherical symmetry. Initially, all particle velocities are and we estimate that it is less than 0.01' in higher E equal to zero (( = 0). We consider a very relativistic case with Rp/M = 1.5. (Recall that in isotropic coordinates I I I I I I I —i„' [ t a Schwarzschild black hole has a radius Rp/M = 0.5. 0.010 ) I With such a compact disk, the collapse is quick enough l that an apparent horizon appears before the ring instabil- 1 ity becomes significant. In Fig. 7 we show results &om an evolution carried out with 300 radial by 16 angular zoaes and 24000 particles. The apparent horizon appears at a 0.000 time of about 4.0 M and riag formation is just discernible at the disk center at this time. ~ ~ In order to prolong the numerica, l evolution after the black bole forms and counteract the efFect of throat A stretching" that occurs ia our maximal (and other sin- —0.010 gularity avoiding) time-slicing conditions, we have imple- I mented a moving mesh algorithm that moves ) I the inner I I radial grid zone to track the of the conformal growth I I factor With this method, we are able to evolve the ~l g. tl black hole and preserve constancy of out quasilocal mass —0.020— I I I measures (the Brill and ADM masses) to a few percent 0.0 200.0 400.0 600. for around 15 M aSer hole formation. Evolution beyond 0 this time is prevented by the n»clerical difficulties in inte- grating the particle geodesics on the extremely stretched mesh. FIG. 6. Waveform from the hot relativistic disk shorn in This time is sufhcient, however, to see the pulse of Fig. 5. Wave amplitudes are labeled as in Fig. 3. ABRAHAMS, SHAPIRO, AND TEUKOLSKY

I I I I I I I I I I I I 1 I I I 1 I 1 I t=o

I ~ T'

5 I

~ \ I J v

' 't I:: '1. "l ' 0 'i:: . :!I '3::I:l.::I-:I.k t I FIG. 7. Snapshots of the particle positions

I I for the collapse a cold relativistic Ini- I I I I I I I I I I of disk. tially the radius is Rs/M = 1.5 and the par- t=g ticles are all at rest. The apparent horizon (dashed line) first appears at t 4.0 M.

Y

1 1.5 0 1 1.5

modes. In a future paper [14] we study the waves from horizon area A/16''Mz has a value of 0.95 when it first this case with a somewhat different method. forms, and asymptotes towards the Schwarzschild value We monitor the area of the apparent horizon and its of 1.0 before the calculation terminates. The black hole polar and equatorial circumferences (see Ref. [6] for def- is initially oblate when the apparent horizon forms. The initions). As shown in Fig. 9, the normalized apparent normalized proper circ»mferences oscillate around their Schwarzschild values C,s/4+M = C„/4+M = 1.0 as the black hole radiates off its initial asphericity. This oscil-

I I I I I I I

0.08 1.05 —:, ';C

Q 1 ~ W

0.01 N 'I 0 r tg e 4 r g$ r 'a 'I P 0 ~ ~ 0 —0.01

0.95 I l I I l —0.02— l I I I I I l I I I I 15 l I I I I l I I I I I I I I I I I 5 10 0 5 10 — t. r FIG. 9. Horizon diagnostics for the black hole formation shown in Fig. 7. The proper equatorial circumference, polar FIG. 8. Gravitational waveform from the disk collapse circumference, and area of the apparent horizon normalized to showa in Fig. 7. The quadrupole waveform extracted at their Schwarzschild values C,~/4s M, C„/4s M and, A/16s'M r/M = 8.0 is shown as a function of retarded time. are plotted as functions of time in units of M. 5Q DISK COLLAPSE IN GENERAL RELATIVITY 7291 lation period again is comparable with that of the most ity. Specifically, no general algorithm is currently known slowly d~mped E = 2 quasinormal mode. that can integrate Einstein's equations for long enough after a black hole forms to compute the full gravitational waveform. It may be possible that this problem can be V. CONCLUSIONS solved with suitable apparent horizon boundary condi- tions, "cutting out" the black hole (see e.g., Ref [15)). Vfe have presented here preliminary results kom a new We expect disk collapse to be a useful proving ground relativistic mean-6eld, particle-simulation code that can for developing apparent-horizon boundary conditions and evolve disks of collisionless matter and compute the emit- other black hole evolution techniques in an axisymmetric ted gravitational radiation. The disk analogue to the context. Oppenheimer-Snyder solution, cold homogeneous disk collapse, is an interesting benchmark for numerical rel- ativity codes as it is one of the simplest scenarios that ACKNOWLEDGMENTS encounters the two most computationally challenging fea- tures of relativistic collapse: black hole formation and This work was supported by National Science Founda- evolution, and gravitational wave production and prop- tion Grant No. AST 91-19475 and No. PHY 90-07834 agation. The hydrodynamical formulation of this prob- and by Grand Challenge Grant No. NSF PHY 93-18152/ lem, provided in Sec. IIC, should permit implementa- ASC 93-18152 (ARPA supplemented). Computations tion of disk collapse in codes without collisionless matter were performed at the Cornell Center for Theory and sources. Simulation in Science and Engineering, which is sup- Failure to complete the computation of the full radi- ported in part by the National Science Foundation, IBM ation data in the case of disk collapse to a black hole Corporation, New York State, and the Cornell Research re8ects the fundamental problem in numerical relativ- Institute.

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