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. The singularity is not a uration that collapses from a highly compact and relativ- point. Rather it is an extended region which includes the istic initial state to a black hole. Note that in isotropic matter spindle, but grows most rapidly in the vacuum ex- coordinates a Schwarzschild black hole on the initial terior above the pole. A t =const slice has a spatial time slice would have radius r =0.5M, corresponding to metric a Schwarzschild radius =2M. Figure 2 depicts the 2 rg + 2dr2+Ppde++2r2 sin 2gdp2 outcome of prolate collapse with the same initial eccen- tricity but from a larger semimajor axis. Here the con- In Aat space A =8=1. At the termination of the simu- figuration collapses to a spindle singularity at the pole lation these quantities have a modest maximum value without the appearance of an apparent horizon. " The 2 =8= 1.7, which occurs at the origin. They decrease spindle consists of a concentration of matter near the monotonically outwards, reaching unity at large dis- axis at r =5M. Figure 3 shows the growth of the tances. However, it is their second derivatives that con- Riemann invariant I at r =6.1M on the axis, just outside tribute to I and these blow up. While A and 8 steadily the matter. Before the formation of the singularity, the grow with time, I diverges much more rapidly. The be- typical size of I at any exterior radius r on the axis is havior is similar to the logarithmic divergence of the -M /r «1. With the formation of the spindle singu- metric in the analytic prolate sequence of Ref. 6. We larity, the value of I rises without bound in the region emphasize that the above characterization of the singu- near the pole. The maximum value of I determined by larity and the behavior of the metric is dependent on the our code is limited only by the resolution of the angular time slicing and may be different for other choices of grid: The better we resolve the spindle, the larger the time coordinate. In principle, the spindle singularity value of I we can attain before the singularity causes the might first occur at the center rather than the pole with a code (and spacetime) to break down. Unlike shell- different time slicing. crossing singularities, where I blows up in the matter in- The absence of an apparent horizon does not neces- terior whenever the matter density is momentarily sarily imply the absence of a global event horizon, al- infinite, the singularity also extends outside the matter though the converse is true. Because such singularities cause our numerical integrations to terminate, we cannot map out a spacetime arbitrarily far into the future,

lO I I I I I i t I I c I I I I I I I t I I I I I I I for- I i which would be necessary to completely rule out the t64 mation of an event horizon. However, we do not think this is at all likely: For collapse from an initially com- pact state (Fig. 1), outward null geodesics turn around

|t 32

I I I I I I I I I I I )0 5 l0 )5 20 25 0 t/M Q~ FIG. 3. Growth of the Riemann invariant I (in units of M ) vs time for the collapse shown in Fig. 2. The simulation was repeated with various angular grid resolutions. Each curve FIG. 4. Profile of I in a meridional plane for the collapse is labeled by the number of angular zones used. We use dots to shown in Fig. 2. For the case of 32 angular zones shown here, show where the singularity has caused the code to become the peak value of I is 24/M and occurs on the axis just outside inaccurate. the matter.

996 VOLUME 66, NUMBER 8 PHYSICAL REVIEW LETTERS 25 FEBRUARY 1991

near the singularity. For collapse from large radius, by celerate the formation of singularities via relativistic contrast (Fig. 2), outward null geodesics are still propa- "pressure regeneration. "' gating freely away from the vicinity of the singularity up The examples considered here have no angular to the time our integrations terminate. It is an interest- momentum. The presence of angular momentum could ing question for future research whether any time slicing prevent an infinitesimally thin spindle singularity from can be found which will be more effective in snuggling forming on the axis. Yet we know that a small amount ' up to the singularity without actually hitting it. Such of angular momentum does not prevent the formation of a slicing would enable one to confirm that all outward a singularity when a Kerr black hole forms. In any case, null geodesics propagate to large distances. it would be disturbing if a collapsing configuration could Further evidence for the nakedness of the singularity become arbitrarily close to singular for arbitrarily small is the similarity of the spindle singularity to the infinite angular momentum. cylinder naked singularity. ' In both cases the proper This research was supported in part by National Sci- length of a given segment of matter along the axis grows ence Foundation Grants No. AST 87-14475 and No. slowly, while its proper circumference and surface area PHY 90-07834 at Cornell University. Computations shrink to zero much more rapidly. Also, the singularity were performed on the Cornell National Supercomput- is an extended region along the axis and not just a point. er Facility. We thank A. Abrahams, M. Choptuik, We have also followed the collapse of an initially ob- C. Evans, and L. S. Finn for helpful discussions. late configuration with the same initial eccentricity and semimajor axis as Fig. 2. Following pancaking, it over- shoots, becomes and forms a black hole. At the prolate, 'R. Penrose, Riv. Nuovo Cimento 1 (Numero Special), 252 q'" time our integrations terminate, we find that 'Eppes 8 (1969). =0.85(4~M ). See, e.g., D. S. Goldwirth, A. Ori, and T. Piran, in Frontiers All of the above results are consistent with the hoop in Numerical Relativity, edited by C. R. Evans, L. S. Finn, conjecture. When black holes form, the minimum polar and D. W. Hobill (Cambridge Univ. Press, Cambridge, 1989), and equatorial circumferences satisfy C™n~4aM. Con- p. 414, for discussion and references. versely, when naked singularities form, the minimum po- K. S. Thorne, in Magic Without Magic: John Archibald lar circumference is much bigger than this value. In all Wheeler, edited by J. Klauder (Freeman, San Francisco, cases where an apparent horizon forms, its area satisfies 1972), p. 1. 4C. C. Lin, L. Mestel, and F. H. Shu, to within numerical accuracy A ~ 16aM, as required Astrophys. J. 142, 1431 (1965). theoretically. ' In every case we find that gravitational 5C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravita- radiation carries away a negligible fraction of («1%) tion (Freeman, San Francisco, 1973), p. 867. the total mass energy by the time a black hole or naked T. Nakamura, S. L. Shapiro, and S. A. Teukolsky, Phys. singularity forms. Rev. D 38, 2972 (1988). In conclusion, we have presented numerical evidence 7R. Arnowitt, S. Deser, and C. W. Misner, in Gravitation: that the hoop conjecture is a valid criterion for the for- An Introduction to Current Research, edited by L. Witten mation of black holes during nonspherical gravitational (Wiley, New York, 1962), p. 227. collapse. We have also found numerical candidates for C. R. Evans, Ph. D. thesis, The University of Texas at Aus- the formation of naked singularities from nonsingular in- tin, 1984 (unpublished). itial configurations. These examples are in contrast with S. L. Shapiro and S. A. Teukolsky, Astrophys. J. 298, 34 (1985); 298, 58 (1985); 307, 575 (1986); Astrophys. (Lett. any cases of singularities which may arise during spheri- J. ), 292, L41 (1985). cal collapse. There the exterior spacetime is always the 'OS. L. Shapiro and S. A. Teukolsky, Astrophys. J. 318, 542 Schwarzschild metric and the Riemann invariant is al- (1987). ways exactly 48M /rP, which is finite outside the matter. ''We searched for both a single global horizon centered on In spherical collapse the singularities can thus only occur the origin as well as a small disjoint horizon around the singu- inside the matter. Here the singularities extend above larity in each hemisphere. the pole into the vacuum exterior. These examples sug- ' Maximal slicing apparently does not hold back the forma- gest that the unqualified cosmic censorship hypothesis tion of prolate spindle singularities. For prolate spheroids the cannot be valid. Newtonian potential diverges only logarithmically as the ec- While the matter treated here has kinetic pressure, it centricity 1, which may explain why a does not plummet is collisionless, not fluid. We do not regard the collision- precipitously near a spindle. '3See, e. D. M. Eardley, Phys. Rev. D 12, 3072 (1975). al properties of the matter as crucial: First, the forma- g., ' See, e.g., Ref. 5, p. 605. Moreover, there is at least one ten- tion of naked singularities should not depend on the par- tative numerical example [T. Nakamura and H. Sato, Prog. ticular details of the fundamental interactions affecting Theor. Phys. 68, 1396 (1982); T. Nakamura, K. Oohara, and matter at high densities. The gravitational field equa- Y. Kojima, Prog. Theor. Phys. Suppl. 90, 57 (1987)] of prolate tions alone should be sufficient to rule out naked singu- Quid collapse (adiabatic index I 2) that appears to be evolv- larities, at least in the vacuum exterior, for true cosmic ing to a singular state without the formation of an apparent censorship. Second, collisional eAects may even ac- horizon. 997