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The Violation of Cosmic Censorship VOLUME 66, NUMBER 8 PHYSICAL REVIEW LETTERS 25 FEBRUARY 1991 Formation of Naked Singularities: The Violation of Cosmic Censorship Stuart L. Shapiro and Saul A. Teukolsky Center for Radiophysics and Space Research and Departments of Astronomy and Physics, Cornell University, Ithaca, New York 14853 (Received 7 September 1990) We use a new numerical code to evolve collisionless gas spheroids in full general relativity. In all cases the spheroids collapse to singularities. When the spheroids are su%ciently compact, the singularities are hidden inside black holes. However, when the spheroids are su%ciently large, there are no apparent hor- izons. These results lend support to the hoop conjecture and appear to demonstrate that naked singulari- ties can form in asymptotically flat spacetimes. PACS numbers: 04.20.Jb, 95.30.Sf, 97.60.Lf It is well known that general relativity admits solu- nite. For collisionless matter, prolate evolution is forced tions with singularities, and that such solutions can be to terminate at the singular spindle state. For oblate produced by the gravitational collapse of nonsingular, evolution, the matter simply passes through the pancake asymptotically flat initial data. The cosmic censorship state, but then becomes prolate and also evolves to a hypothesis' states that such singularities will always be spindle singularity. clothed by event horizons and hence can never be visible Does this Newtonian example have any relevance to from the outside (no naked singularities). If cosmic cen- general relativity? We already know that inftnite cylin- sorship holds, then there is no problem with predicting ders do collapse to singularities in general relativity, and, the future evolution outside the event horizon. If it does in accord with the hoop conjecture, are not hidden by not hold, then the formation of a naked singularity dur- event horizons. But what about finite configurations in ing collapse would be a disaster for general relativity asymptotically flat spacetimes? theory. In this situation, one cannot say anything precise Previously, we constructed an analytic sequence of about the future evolution of any region of space con- momentarily static, prolate, and oblate collisionless taining the singularity since new information could spheroids in full general relativity. We found that in the emerge from it in a completely arbitrary way. limit of large eccentricity the solutions all become singu- Are there guarantees that an event horizon will always lar. In agreement with the hoop conjecture, extended hide a naked singularity? No definitive theorems exist. spheroids have no apparent horizons. Can these singu- Counterexamples are all restricted to spherical symme- larities arise from the collapse of nonsingular initial try and typically involve shell crossing, shell focusing, or data? To answer this, we have performed fully relativis- self-similarity. Are these singularities an accident of tic dynamical calculations of the collapse of these spherical symmetry? spheroids, starting from nonsingular initial configur- For nonspherical collapse Thorne has proposed the ation. hoop conjecture: Black holes with horizons form when We find that the collapse of a prolate spheroid with and only when a mass M gets compacted into a region sufticiently large semimajor axis leads to a spindle singu- whose circumference in every direction is C 4aM. If larity without an apparent horizon. Our numerical com- the hoop conjecture is correct, aspherical collapse with putations suggest that the hoop conjecture is valid, but one or two dimensions appreciably larger than the others that cosmic censorship does not hold because a naked might then lead to naked singularities. singularity may form in nonspherical relativistic collapse. For example, consider the Lin-Mestel-Shu instability Wurnerical code. —Our numerical code solves Ein- for the collapse of a nonrotating, homogeneous spheroid stein's equations for the evolution of nonrotating, col- of collisionless matter in Newtonian gravity. Such a lisionless matter in axisymmetric spacetimes. The field configuration remains homogeneous and spheroidal dur- equations are expressed in 3+1 form following Arnowitt, ing collapse. If the spheroid is slightly oblate, the con- Deser, and Misner. We use maximal time slicing and figuration collapses to a "pancake, " while if the spheroid isotropic spatial coordinates. The field equations reduce is slightly prolate, it collapses to a spindle. While in both to a coupled set of nonlinear hyperbolic and elliptic par- cases the density becomes infinite, the formation of a tial diff'erential equations which we solve by finite dif- spindle during prolate collapse is particularly worrisome. ferencing. The matter equations are the geodesic equa- The gravitational potential, gravitational force, tidal tions in the self-consistent gravitational field. The nu- force, and kinetic and potential energies all blow up. merical treatment is a mean-field particle simulation This behavior is far more serious than Inere shell cross- scheme that solves the Vlasov equation in general rela- ing, where the density alone becomes momentarily infi- tivity. It is an extension of our previous relativistic treat- 994 1991 The American Physical Society VOLUME 66, NUMBER 8 PHYSICAL REVIEW LETTERS 25 FEBRUARY 1991 ment for spherical spacetimes and our Newtonian spherical equilibrium clusters; reproducing Oppenheim- method ' for axisymmetric configurations. The code can er-Snyder collapse of homogeneous dust spheres and handle matter velocities approaching the speed of light Newtonian collapse of homogeneous spheroids. ' We and strong gravitational fields, including black holes. It constructed a number of geometric probes to diagnose is designed to treat cases in which the collisionless matter the evolving spacetime. We tracked the Brill mass and collapses to a singularity: Specifically, oblate collapse to outgoing radiation energy flux to monitor mass-energy flat pancakes and prolate collapse to thin spindles. conservation. To confirm the formation of a black hole, We have carried out a large battery of test-bed calcu- we probed the spacetime for the appearance of an ap- lations to ensure the reliability of the code. These tests parent horizon and computed its area and shape when it included the propagation of Iinearized analytic quadru- was present. To measure the growth of a singularity, we pole waves with and without matter sources and non- computed the Riemann invariant I—=R,p»R ~~ at every linear Brill waves in vacuum spacetimes; maintaining spatial grid point. To test the hoop conjecture, we com- equilibria and identifying the onset of instability for puted the minimum equatorial and polar circumferences outside the matter. Typical simulations were performed with a spatial grid | i I 1 I t I I f I I I I I I 2— ( i i of 100 radial and 32 angular zones, and with 6000 test t/M =0 particles. A key feature enabling us to snuggle close to singularities was that the angular grid could fan and the i.5— radial grid could contract to follow the matter. Collapse of collisionless spheroids. —We followed the to— t/M =0 0.5 —- 8— 0 j I I I 2— [ 6— t/M = 7, 7— x a 4— t, 5— 0-'i i. i. l. ~ i t I 0 10— t/M = 23 05 1 1.5 2 6— Equator V) X FIG. 1. Snapshots of the particle positions at initial and late times for prolate collapse. The positions (in units of M) are 4 projected onto a meridional plane. Initially the semimajor axis of the spheroid is 2M and the eccentricity is 0.9. The collapse proceeds nonhomologously and terminates with the formation of a spindle on the an hor- '. axis. However, I I I singularity I I i I apparent 0 I I 1 izon (dashed line) forms to cover the singularity. At t/M =7.7 0 2 4 6 8 10 its area is A/16@M =0.98, close to the asymptotic theoretical Equator limit of 1. Its polar and equatorial circumferences at that time FIG. 2. Snapshots of the = particle positions at initial and are C ~~//4~M 1.03 and C,q /4rrM =0.91. At later times final times for prolate collapse with the same initial eccentrici- these circumferences become equal and approach the expected ty as Fig. 1 but with initial semimajor axis equal to 10M. The theoretical value 1. The minimum exterior polar circumfer- collapse proceeds as in Fig. 1, and terminates with the forma- ence is shown by a dotted line when it does not coincide with tion of a spindle singularity on the axis at t/M=23. The the matter surface. Likewise, the minimum equatorial cir- minimum polar circumference is C'Pl",/4+M=2. 8. There is no cumference, which is a circle, is indicated by a solid dot. Here apparent horizon, in agreement with the hoop conjecture. This Cg'"/4aM=0. 59 and CPg/4zrM=0. 99. The formation of a is a good candidate for a naked singularity, which would vio- black hole is thus consistent with the hoop conjecture. late the cosmic censorship hypothesis. 995 VOLUME 66, NUMBER 8 PHYSICAL REVIEW LETTERS 25 FEBRUARY 1991 collapse of nonrotating prolate and oblate spheroids of beyond the pole at r =5.8M (Fig. 4). In fact, the peak various initial sizes and eccentricities. The matter parti- value of I occurs in the vacuum at r =6.1M. Here the cles are instantaneously at rest at t=0 and the con- exterior tidal gravitational field is blowing up, which is figurations give exact solutions of the relativistic initial- not the case for shell crossing. The absence of an ap- value equations. In the Newtonian limit, these spher- parent horizon suggests that the spindle is a naked oids reduce to homogeneous spheroids. When they are singularity. large (size»M in all directions), we confirm that their When our simulation terminates, I along the axis falls evolution is Newtonian. ' to half its peak value at r =4.5M inside the matter and Figure 1 shows the fate of a typical prolate config- r =6.7M outside the matter.
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