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Disk Collapse in General Relativity

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Disk Collapse in General Relativity PHYSICAL REVIEW' D VOLUME 50, NUMBER 12 15 DECEMBER 1994 Disk collapse in general relativity 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: black hole formation and gravitational wave 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 spacetimes. 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 gravitational field &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.
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