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Research.

Geometry of a Black HoleCollision

Richard A. Matzner,* H. E. Seidel, Stuart L. Shapiro, L. Smarr,W.-M. Suen, Saul A. Teukolsky, J. Winicour

The Binary Black Hole Alliance was formed to study the collision of black holes and theresulting gravitational radiation by computationally solving Einstein's equations for gen-eral relativity. The location of the black hole surface in a head-on collision has beendetermined in detail and is described here. The geometrical features that emerge arepresented along with an analysis and explanation in terms of the spacetime curvatureinherent in the strongly gravitating black hole region. This curvature plays a direct,important, and analytically explicable role in the formation and evolution of the eventhorizon associated with the surfaces of the black holes.

Black holes are small (a black hole of amillion solar masses would be only as largeas the sun), distant objects that have yet tobe observed directly. But definite predic-tions about them come from detailed stud-ies of solutions of Einstein's equations ofgeneral relativity. Fortunately, analyticalmethods (1) are powerful enough to solvethese equations for a single, stationary blackhole. However, this is not true for dynam-ically interacting black holes, believed tounderlie some of the most dramatic phe-nomena in our universe.

The two-body problem in general relativ-ity is still unsolved and is the subject of aNational Science Foundation High-Perform-ance Computing and CommunicationsGrand Challenge project, termed the Binary

R. A. Matzner is professor of physics and director of theCenter for Relativity, University of Texas, Austin, TX78712, USA. H. E. Seidel is senior research scientist atthe National Center for Super Computing Applications(NCSA) and visiting associate professor of physics at theUniversity of Illinois, Urbana, IL 61801, USA. S. L. Shapiroand S. A. Teukolsky are professors of astronomy andphysics at Cornell University, Ithaca, NY 14853, USA. L.Smarr is director of NCSA and professor of astronomyand physics at the University of Illinois, Urbana, IL 61801,USA. W.-M. Suen is associate professor of physics atWashington University, St. Louis, MO 63130, USA. J.Winicour is professor of physics and astronomy at theUniversity of Pittsburgh, Pittsburgh, PA 15260, USA.*To whom correspondence should be addressed.

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Black Hole Alliance. The members of theAlliance include principal investigators ateight universities as well as a number ofassociates, collaborators, students, and post-doctoral fellows. The focus is to solve theproblem as formulated topologically by Ein-stein and Rosen (2) in 1935 and as formu-lated numerically by DeWitt and Misner (3)in 1957. Solution of this 60-year-old prob-lem will require the teraflop supercomputersof the late 1990s.

Supercomputers have advanced inspeed by over 50,000 times since Hahnand Lindquist (4) made the first numericalattack on the problem 30 years ago. Nearly20 years ago, Smarr and Eppley (5) ob-tained the first numerical solution of thehead-on collision of two black holes ofequal mass. They determined that theblack holes did coalesce, radiating gravi-tational waves with energy of approxi-mately 10'- Mc2, where M is the mass ofthe system and c is the speed of light. Thegravitational waveform was similar to thedamped vibrations ("ringing modes") fa-miliar from perturbation calculations ofblack holes (6). However, numerical in-stabilities prevented those early calcula-tions from being used to determine thedetails of the coalescence, which we reporthere.

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The Alliance is committed to a 5-yearinvestigation on the full inspiral and mergerof two black holes orbiting about each oth-er. This fully three-dimensional problem isvastly more complex than the head-on col-lision treated in this article. Nonetheless,we expect the full problem to bear manysimilarities. The Alliance's solutions will bevery important to the LIGO gravitationalwave observatory (7), which is a system oflarge Michelson interferometer detectors ofgravitational radiation presently under con-struction in the United States. LIGO is partof a global network, including the French-Italian VIRGO detector under constructionnear Pisa and other detectors in variousstages of planning.

Einstein's equations are a set of nonlin-ear, coupled partial differential equations inspace and time. The numerical solution forgeneric configurations of black holes is atthe edge of possibility by using recent ad-vances in computer hardware and algo-rithms. We will not describe the details ofthe numerical computations here. Rather,we will present one of the first fruits of thiscollaborative effort: locating the precise sur-faces of black holes in axisymmetric config-urations as they collide, and mapping outtheir geometric structure.

Black Holes

John Wheeler (8) coined the insightful andprovocative name "black holes" in the con-text of general relativity, Einstein's theoryof gravity. However, the major properties ofblack holes can be understood in terms ofNewtonian physics, together with the rela-tivistic principle that nothing can travelfaster than the speed of light. A black holeis an "object" with an escape velocity equal

ct

zx

1+

Fig. 1. Photons emanating from a flashbulb gen-erate a compactified light "cone" in flat space-time. V, vertex (origin of the light rays); 1+, infinity;;, circle representing the last time slice at infinitythrough the expanding sphere of light radiatingfrom V; c, speed of light; t, time.

to the speed of light (the escape velocityVesc is the minimum velocity required tosend an object to infinity). Long agoLaplace (9) noted that the escape velocityfrom the gravitational pull of a sphericalstar of mass M and radius R is

t2GMVesc = R (1)

where G is Newton's gravitational constant.Adding mass to the star (increasing M) orcompressing the star (decreasing R) increas-es vesc* When the escape velocity exceeds c,even light cannot escape and the star be-comes a black hole. (For any known kind ofmatter, the gravitational field at this stage isso strong that the star inevitably collapsesto an infinite density singularity.)

The required radius RBH for a black holeof mass M follows from Eq. 1, with vesC setequal to c:

2GMRBH= 2 (2)

For a solar-mass black hole, M - 2 x 1033g and RBH - 3 km. Remarkably, Eq. 2 holdseven in general relativity, which describesgravitation by curvature of space and timethought of as one entity: spacetime. A blackhole is a region where curvature traps lightfrom escaping to infinity.

To portray this idea, we use spacetimediagrams in which two spatial dimensionsare drawn, as well as the evolution in time(depicted as the vertical direction). Thespacetime trajectories of the photons ema-nating from a flashbulb are shown in Fig. 1.To get a useful picture, we must use the sameunits on all the axes. Thus, we plot timemultiplied by c on the vertical axis. Thenthe path of a light ray, satisfying d(distance)/dt = c, makes an angle of 450 with respect tothe time axis. The third spatial direction (y)is suppressed in the drawing in order toprovide an intelligible perspective. Lightfrom the flash travels outward in a sphericalshell. (At any one time, a t = constant sliceproduces a circle in our diagram, our two-dimensional representation of a sphere.) Toview the entire spacetime, we "compactify"it in the lightlike direction of the radiationtravel, by rescaling the distance so thatpoints at infinity, denoted by I+, are mappedto a finite location (10). With such a rescal-ing, the picture of the expanding shell oflight has a finite outer boundary, corre-sponding to the last spherical time slice atinfinity. The essential advantage gainedhere is that we can represent this last sphereby the circle I in our finite-sized diagram.We will use this picture of a light "cone" inflat spacetime (spacetime with no gravita-tional field) to explain the curved spacetimegeometry for the case of a head-on collisionof two black holes.

Now consider a spherical star clusterthat collapses to a black hole. Figure 2 is thespacetime diagram. The infalling stars even-tually produce a region at the center whereVesc = c. As more stars fall in, this regiongrows. Finally, after all the stars fall in, thesize of this region remains constant. Thestars eventually collapse to an infinite den-sity singularity indicated by the jagged line.The black hole first forms at the center, andas it grows, it surrounds the singularity, inaccord with the hypothesis of cosmic cen-sorship that states that singularities mustnot be visible to distant observers. Theevolution of the surface of the black holetraces out a curved version of a light cone,indicated in Fig. 2 by H.

Except for its curvature, H is analogousto the light cone in Fig. 1. It forms at t = tB(B for "birth") in the diagram, where lightrays from the center of the cluster just missescaping to infinity. These light rays pro-vide the skeletal structure of the black holeanalogous to the way straight lines rule(generate) a cone in Euclidean geometry.As these rays diverge outward from thecenter, the gravitational attraction of theinfalling matter distorts the spacetime sothat they do not expand indefinitely but"hover" at the final black hole radius. Atlate times they therefore define a nondi-verging (and nonconverging) parallel cylin-der (topologically S2 x R1, where S2 is thesphere and R1 is the line), which is gener-ated by the rays.

Imagine repeatedly setting off a flash atthe center. Before the black hole forms, thelight flashes travel out to infinity. The sur-face of the black hole is determined by thatparticular light cone that just misses escap-ing to infinity. This is a subtle concept thatinvolves the complete future history of thesystem. One must first find the specialsphere of light rays that is hovering at latetimes and then trace it back to find its

Fig. 2. Spacetime diagram for a spherical col-lapse to a black hole. H, trace of the evolution ofthe black hole surface or horizon; tB, time atwhich the black hole was "bom."

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birth. To appreciate the reverse time ordernecessary for this construction, imagine adiagram like Fig. 2 but with an additionalhollow spherical shell of matter falling in atlate times. The additional gravitational at-traction from this shell would be sufficientto refocus to a point the nondiverging por-tion of the cone drawn in Fig. 2. Indeed, inthat case, a cone that emerged from a slight-ly earlier flash, and that would otherwiseexpand to infinity, would now be deflectedinward to exactly zero divergence at latetimes and would thus form the actual blackhole surface.

The final equilibrium state formed aftermatter and energy stop falling into the holeis well understood. There are theorems thatstate that when the black hole eventuallysettles down to a stationary configuration,its gravitational field has the analytic formof a Kerr black hole which is uniquely de-termined by its mass and angular momen-

tum. They are called "no hair" theorems toemphasize the simplicity of the black holeequilibrium state. (Electric charge is alsoallowed but is irrelevant for astrophysicallyrealistic black holes.)

Because light cannot escape the blackhole surface, events that occur inside can-not be seen from outside. Hence, the nameblack hole event horizon is used to denotethe evolution of the black hole surface. It isa horizon because we cannot see beyond it.In our computations to be described below,we use the fact that the black hole state isknown at late times to find the event hori-zon at late times. Then our methods inte-grate back into the past to determine thehorizon structure during the earlier dynam-ical part of the evolution.

General relativity allows not only blackholes formed by matter collapse but alsoblack holes that are stable topological struc-tures in spacetime and need have no mattercontent at all. Such black holes do havemass, and the nonlinearity of their gravita-tional field creates curvature that holdsthem together. We will return to such"eternal" black holes below (11-13).

Referring to Fig. 2, we can verify severalkey properties that apply to event horizonsin general: (i) The horizon is generated bylight rays passing through each point in thehorizon, and (assuming cosmic censorship)these generators continue along the horizonforever into the future. (ii) The horizonbegins where these generators meet in thepast. (iii) The cross-sectional area of thehorizon monotonically increases and ap-proaches a constant at late time. (For asingle spherical black hole, the area ap-proaches the value 4+RBH2.)

The collision of two black holes cannotbe treated analytically and requires full-fledged computation. We have simulated thehead-on collision of nonrotating black holesof equal mass to form a single black hole(14). In this simulation, each black hole isformed from the collapse of a spherical ball

of noncolliding particles, analogous to thestars in a spherical galaxy. Each ball collapsesbecause the particles are chosen to have noinitial motion. Thus, there is na centrifugalforce to balance the inward pull of gravity.To mimic a collision, we begin the simula-tion with the two clusters well separated butheaded toward each other with a velocity of0.15c. This initial configuration is shown inthe first frame of Fig. 3 (the initial separationis about five times the radius of the finalblack hole). The two clusters then fall to-ward each other while individually collaps-ing to form black holes. The individual blackholes originate at the center of each clusterand then grow outward. The second frame ofFig. 3 depicts the instant at which these twotidally distorted black holes first touch. Theblack holes then merge and in the processconvert -3 X 10-4 of the system's mass intoenergy in the form of gravitational wavesradiating outward. The last frame of Fig. 3shows the end of the simulation, with asingle black hole encompassing all of thematter. The final black hole is settling into aspherical equilibrium state.

Figure 3 provides a picture, in ordinarythree-dimensional space, of the black holeformation and merger process at three in-stants of time. The complete numerical evo-lution constructs these spatial pictures in analmost continuous way, like the frames of amovie. From these spatial pictures, we canreconstruct a four-dimensional spacetimedescription of the evolution. In a spacetimediagram, each frame of the type shown inFig. 3 would correspond to a horizontal slice,representing all of space at a given instant oftime. We have mapped out the location ofthe event horizon in this computed space-time by numerically propagating light rays[see (15), especially figure 3].

Figure 4 is a spacetime picture showingsome of these light rays superimposed onthe resulting event horizon. It was con-structed by propagating light rays backwardin time from the surface of the final equi-

Fig. 4. Computational constructionof some of the light rays generatingthe horizon for the case shown inFig. 3. The collision axis goes fromleft to right, and the time axis is ver-tical. The inset zooms in on thecaustic and crossover structure atthe birth of the horizon.

Fig. 3. Snapshots of the collision and merger oftwo black holes formed by the collapse of twoballs of particles. The spatial location of the eventhorizon is shown in black. The clock in eachframe shows the fraction of time elapsed duringthe simulation.

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librium black hole with the ideas in (16,17). The blue slice at the top of Fig. 4corresponds to the end of the simulation,the final frame of Fig. 3. The picture repre-sents a calculation of the "pair of pants"description of the event horizon as sketchedby Hawking and Ellis [(18), figure 60] orMisner, Thorne, and Wheeler [(19), figure34.6] more than 20 years ago. We see bytracing the rays backward in time that someof them cross each other and leave thehorizon. The simulation helps reveal theimportant role of these crossovers as theinitial events where the horizon forms. Theinset shows a closer view of the beginningof the horizon and how the rays behavenear those events.

The crossovers are the earliest pointsalong the rays that lie on the horizon. Be-fore crossing, these rays travel in a region ofspacetime to the past of the horizon that isvisible to a distant observer. This is clearfrom the light cone of Fig. 1: any ray con-tinued backward through the vertex V,where it intersects other rays, enters thespacetime region to the past of V, which isvisible to distant observers at I' at timesbefore E. Crossovers are ubiquitous featuresof black hole interactions. The attractivenature of gravity causes even a beam thatinitially has perfectly parallel rays to bendand self-intersect, analogous to the intersec-tion of two flashlight beams from separatedirections. In Fig. 4, a line of such crossoverpoints extends from the "crotch" on the"pair of pants" down along each inside trou-ser "seam," around each bottom, and a smalldistance up each outside "seam." At theendpoints of the line of crossovers, slightlyup the outside of each "leg," are caustics,analogous to a focal point where a singleflashlight beam would converge to infiniteintensity (if it were not for diffraction). Inthe spherically symmetric case, the beamemerging from a point caustic would, if grav-ity were weak, trace out the light cone ofFig. 1 or, if gravity were sufficiently strong,trace out the horizon for the single blackhole of Fig. 2. The tidal effects of the col-liding black holes shift the location of thecaustics and lead to the line of crossovers.We will see below how this comes about.

One feature of Fig. 4 deserves comment.The figure makes it appear that the area ofthe horizon decreases at late times, contraryto property (iii) above. In fact, a carefulcheck of the computed spacetime showsthat the horizon area increases asymptoti-cally to a constant value at late times. Theapparent decrease is an artifact of the coor-dinates chosen to represent the horizon inFig. 4. Figure 10, which is for a differentsimulation and is fully discussed in the sec-tion entitled "Vacuum Black Hole Colli-sions," is a visualization in which this effecthas been rigorously controlled (20).

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vi

Fig. 5. The lower diagram illustrates colliding lightcones in a flat spacetime. The upper diagram illus-trates the event horizon corresponding to the sur-face : at infinity 1+. The compactification distortsspatial directions so that the (ct,x,z) "compass" isonly schematic. In particular, the +x direction rayfrom V1 intersects the +x direction ray from V2 atthe kink in S. In the diagrams, these rays appear tohave a spurious z-motion, which is the price wepay for compactification. Similar comments applyto Figs. 6 and 9. X, crossover line.

Colliding Light Cones

Consider the collision of two black holes interms of the spacetime picture of the colli-sion of two light cones depicted in thelower diagram of Fig. 5. In this compactifiedpicture, the vertices VI and V2 of the lightcones are separated along the z axis, aboutwhich the collision has rotational symme-try. This axisymmetry makes it possible for athree-dimensional (t,x,z) picture to repre-sent all aspects of the collision. Each pointon the x axis corresponds to a circle ofaxisymmetry in the x,y plane, centeredabout the z axis. In the picture, points withthe same t and z but with values ±x lie onthe same circle, and the circle reduces to asingle point on the z axis where x = 0.

Each light cone intersects infinity in asphere. But to discuss a "sphere at latetimes" that resembles the late-time spheri-cal surface of a black hole, we restrict at-tention to the two "outside" hemispheres inwhich the light rays from one vertex reachinfinity without hitting rays from the othervertex. These two hemispheres join togeth-er to form l, the closed two-dimensionalsurface at infinity indicated in Fig. 5. Thesurface M is kinked where the two hemi-spheres join in a circle in the x,y plane. Ourfigure shows two kinks (corresponding tothe two intersections of that circle with the

Fig. 6. Deformation of the crossover line in Fig. 5due to spacetime curvature.

x axis). One of these kinks corresponds tothe two light rays, one from each vertex,traveling in the positive x direction. Be-cause these two rays are parallel, they meetat infinity. The other kink corresponds tothe analogous rays traveling in the negativex direction from the vertices. [The radialcompactification distorts the apparent di-rection of the rays so that the ray in the +xdirection from each vertex appears to havea motion in the z direction. We must drawthem this way so that the parallel rays even-tually intersect at I+, in this case at the kinkin E. This means that we view the (ct,x,z)"compass" in Fig. 5 as only schematic.]

The collision takes place between thehemisphere of rays from each vertex thathead toward each other. The ray in thepositive z direction from V1 collides firstwith the ray in the negative z directionfrom V2. The entire set of colliding rays isindicated in the figure by the parabola,which is the conic section formed by theintersection of the light cones with the x,tplane located midway between them.

In this flat spacetime model, the parab-ola of collision points is the analog of thestrong geometrical interaction occurring inthe collision of two black holes. The analogof the horizon H is the boundary of theregion visible to observers at infinity beforethe time corresponding to the surface E.The analog of an observer in a black holespacetime is an observer outside the lightcone stretching back from ;. We shall callthat structure H in this case also.

To construct H, we trace back all of thelight rays reaching E until they meet one ofthe other rays in either a crossover or acaustic. Obviously, the rays from one out-side hemisphere converge to V1 and thosefrom the other outside hemisphere con-verge to V2.

But what about the rays traced back fromthe kinks on E where the hemispheres join?

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L

Fig. 7. Spatial picture of a generic cusp caustic (C)produced by an aspherical beam. F, fold line; L,light ray.

Two of these are the rays from V1 and V2that travel in the positive x direction. Theseparallel rays intersect at infinity. In fact, allparallel rays in the positive x direction start-ing from points along the z axis between V1and V2 have the same "endpoint" at infin-ity, the kink at the end of the parabolic arc

formed by the colliding outgoing rays. Inthe backward time direction, each of theserays intersects, at x = 0, the opposite ray

traveling in the negative x direction fromthe opposite kink. From these clues, we see

that H corresponds to the two half-conesfrom V1 and V2 together with the two setsof parallel rays sweeping out in the +xdirection as illustrated in the upper diagramof Fig. 5. These rays originate at points of Halong the spatial line on the z axis connect-ing V1 and V2. This part of the z axis is thecrossover line where rays from ±+x directionintersect. Note also the surprising resultthat the parabola formed by the collidingrays is to the future of H and thereforeinvisible to observers "outside the horizon."

Deforming the Picture

Our picture is beginning to resemble thehorizon formed in the simulated collisionbetween two black holes. To make it morerealistic, the gravitational effect of space-time curvature must be included. Thestrong gravitational field near the holescauses light rays to bend inward toward one

another. Tracing the rays generating Hbackward from infinity, this effect causesthem to intersect "sooner" along a curvedspatial crossover line, as depicted in Fig. 6.In this figure, we have also smoothed outthe kinks on E, which has a similar effect as

curvature to cause the crossover line Xwhere H originates to bulge upward. X cor-

responds to a seam on the inside pant legsin the trouser picture of the black holecollision; the shape is bowlegged, and theprofile corresponds to a smooth spatial linewith no special sharpness at the midpoint ofthe seam where the legs join.

To complete the picture, another subtle

X; x

F2

F1

Fig. 8. Spacetime version of Fig. 7, with timedirected vertically.

effect of curvature must be included. Thevertices of the light cones in Fig. 6 are

caustic points corresponding to the focus ofa finite solid angle of rays. These pointcaustics arise from the spherical symmetryof the light cones in our flat spacetimemodel. In reality, point caustics are highlyunlikely to occur in a generic system. Anyslight perturbation of spherical symmetrydestroys the perfect focus in an unstable,unpredictable manner. Such results havebeen made rigorous by using the methods ofcatastrophe theory (21) and nonlinear dy-namics (22). The spatial geometry of theray structure of the elementary stable caus-tics has received a great deal of attentionbecause of their omnipresence in classicaland diffraction optics (23).

In the presence of rotational symmetryabout an axis, as in our simulations, theonly stable caustics consist of cusps andfolds. Figure 7 illustrates a spatial picture ofa typical axisymmetric wavefront converg-ing to a caustic, with the circles of axisym-metry suppressed. The light ray L encoun-

ters a cusp caustic at the point C, but lightrays L1 just below L do not focus at C butalong the fold line F1. Similarly, light raysL2 focus along the fold line F2. Of specialimportance to the structure of horizons, therays L1 and L2 intersect along a crossoverline X before they reach their respectivefold lines. Hence, we expect the point caus-tics of our flat space model to be replaced bycusp caustics, connected by a crossover line.

The major theoretical investigations ofcaustics have typically dealt with steadybright patterns, which result from the time-averaged appearance of an optical frequen-cy beam. The spacetime geometry has re-

ceived relatively little attention, except ingeneral relativity. The pioneering studies ofPenrose (24), which led to the first proofthat singularities result from gravitationalcollapse, were based on the spacetime prop-erties of caustics. Many details of the space-time geometry of the elementary stablecaustics have now been worked out (25).

Figure 8 is the spacetime version of Fig.7 and exhibits some of those details. Theray L traced backward in time along thehorizon encounters a cusp point C. Fromthe spacetime event C, L continues on into

SCIENCE * VOL. 270 * 10 NOVEMBER 1995

the past, but for simplicity we suppress thathere, because the portion of L to the past ofC is not on the horizon but in the visibleregion of spacetime. The horizon beginswhere its generating rays meet, either atcaustics or at crossover points. The foldlines F1 and F2 are also not on the horizonbecause the rays L1 and L2 meet (travelingbackward) along the crossover line X beforethey meet F1 or F2. Only the cusp C at thetip of the crossover line is on the horizonand thus not visible to distant observers.

The entire crossover line X is also on thehorizon; it is where the bulk of the genera-tors originate. Crossover lines, which arisewhen two separate beams of rays intersect,must be spacelike. This means that the linescannot be traced by light rays or particlesmoving at less than c. Their detailed prop-erties have not been studied as extensivelyas caustics, where a single beam focuses.Our computational study of the cusp causticreveals the surprising result that the cross-over line, although spacelike at each point,becomes asymptotically lightlike as it ap-proaches the cusp C. As a result, X joinssmoothly onto the light ray L, as shown inFig. 8. This is the final clue necessary todeform Fig. 6 into the horizon found in thecollision between black holes.

Putting the Pieces TogetherThe qualitative features of the stable caus-tics are preserved by perturbations, whetherdue to the refractive properties of a mediumor to the gravitational curvature of space-time. Even more relevant to the simulationof black holes, they are insensitive to smallnumerical error. Thus, in the axisymmetriccase of a head-on collision, one expects tofind cusps on the horizon. However, thestability is only a local property that de-scribes the generic behavior in the neigh-borhood of the cusp. Pieces of the horizoncan be expected to resemble Fig. 8, but thepieces must be put together. The way to dothis is suggested by a comparison of Figs. 6and 8. By replacing the unstable feature ofFig. 6, the point vertices V1 and V2, each bythe cusp of Fig. 8, we obtain the stablehorizon structure shown in Fig. 9. Figure 9supplies the "key" to the computed blackhole structure in Fig. 4. It shows a cusp C onthe outside of each "trouser leg." The cuspsare connected by a spatial crossover line X,which forms a seam on the trousers. Theseam joins smoothly with the two light rays,analogs of L in Fig. 8, emanating from thecusps (26). Those rays are exceptional. Theother light rays emanate from X in pairs(the circles of axisymmetry being sup-pressed), one up the front of the trousersand one up the back. The final black holeequilibrium state long after the collision isa sphere, represented by a circle (and I is

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almost circular) in our figure. This pro-vides a consistent theoretical model forour computations.A final point. The crossover line drawn

in Fig. 9 is actually a caustic in the exactlyaxisymmetric case, because it is the focalline of a circle of rays in the x,y plane.However, under a perturbation away fromexact axisymmetry there would be defocus-ing, and this line would broaden into atwo-dimensional crossover surface.

Vacuum Black Hole Collisions

We turn now to vacuum black holes, oftencalled "etemal" black holes because they arenot formed from matter collapse but are per-manent features of solutions of the nonlinearEinstein equations. At t = tjintialf we choosea system consisting of two nonrotating blackholes poised near one another. The fielddata are momentarily stationary (the fieldmomenta vanish and tinitial is a moment oftime symmetry of the fields, though as weshall see, the horizon is not time symmetric)(27). We have evolved this system (16, 11-13, 28) into the future from tinitial for a time- 150 GM/c3, and backward in time for -8GM/c3 (where M is the mass of the finalblack hole) (Fig. 10). For a solar mass blackhole, these times are less than 10-3 s, indi-cating the violent nature of the merger.

At late times, the horizon oscillates withdecaying amplitude around a spherical shape(12, 16, 17). When disturbed, a black holewill vibrate and emit gravitational waveswith a characteristic frequency determinedby its mass and spin, just as a bell emitssound waves when struck by a mallet. Asthese ringing modes subside, the hole settlesdown to the known equilibrium solution,which provides us with the black hole sur-face at late times. We then integrate back-

z

x

Une of crossovers

'A1+~~~~~~~~~~~~1Fig. 9. Key for Fig. 4: The horizon formed by themerger of two black holes. The base contains twoversions of the cusp in Fig. 8 joined by a commoncrossover line X.

ward in time (16, 17) to locate the surface atearlier times. Figure 10 shows the horizonstructure in the most interesting epoch,starting from 5 GM/c3 in the past of begin-ning of coalescence up to 20 GM/c3 in thefuture. Figure 10 differs from Fig. 4 in twoimportant ways. First, the black holes in Fig.4 are "bom" and the horizon's origin can bestudied, as discussed above, whereas theblack holes in Fig. 10 exist etemally. Second,Fig. 4 gives a coordinate description of thehorizon, whereas Fig. 10 gives a more intrin-sic visualization of the horizon geometry interms of the time evolution of its polar cir-cumference. This reveals the monotonic in-crease and asymptotic constancy of the area,consistent with the black hole theorems.

Unlike the matter-filled collapse, thetrouser legs in Fig. 10 do not have cusps butcontinue into the past from our diagram. Wedo find a crossover line along the insidetrouser seam, exactly analogous to that inFig. 4 for our matter simulation. Althoughspacetime is symmetric in time about tinitial1the horizon itself is not time-symmetric raysbut expands monotonically into the future.This expansion of the actual rays tracing thehorizon is clearly visible at tinitial in Fig. 10.

From the time symmetry of the system,one might have naively (and incorrectly)concluded that its past evolution describes atime-reversed collision: radiation impingingon a single black hole, blowing it into twoholes that come to a momentarily station-ary pause at the "initial" time, with radia-tion then emitted as the holes merge. Suchexpectations of the dynamics turn out to bewrong. Except in a completely stationarycase, the horizon cannot behave symmetri-cally about tinitial' This is an obvious con-clusion from two theorems by Hawking:The area of a black hole always increases(29) [see property (iii) above], and blackholes cannot bifurcate (30). Generically, anevent horizon is not time symmetric.

Fig. 10. Computer simulation of the collision be-tween two vacuum black holes.

Conclusion

Before having computational tools for study-ing the collision and merger of two blackholes, we had to rely on sketches and intel-ligent guesses to describe the interaction.Now the numerical simulations have re-

vealed new qualitative features of the colli-sion and have pointed the way to simpleanalytic models that reveal the underlyingphysics. Specifically, the computations haveled to the discovery of the exact nature ofthe crossover line and its caustic endpointsat the formation of the horizon.An even more formidable challenge will

be to track the coalescence and merger oftwo black holes in a circular orbit as theyspiral together because of the emission ofgravitational waves. This scenario is astro-physically more realistic than the head-oncollision considered here. However, the totallack of symmetry will require sophisticatednew algorithms and will tax the most pow-

erful computers. Work is in progress to meetthis next challenge.

REFERENCES AND NOTES

1. K. Schwarzschild, Sitzungsber. Dtsch. Akad. Wiss.Berlin, KI. Math. Phys. Tech. 1916,189 (1916); R. P.Kerr, Phys. Rev. Lett. 11, 237 (1963).

2. A. Einstein and N. Rosen, Phys. Rev. 48, 73 (1935).3. See, forexample, L. Smarr, in Quantum TheoryofGrav-

ity, S. Christenson, Ed. (Higer, Bristol, 1984), p. 14.4. S. Hahn and R. W. Lindquist, Ann. Phys. NY29, 304

(1964).5. L. Smarr, Sources of Gravitational Radiation (Cam-

bridge Univ. Press, Cambridge, 1979), p. 245.6. See, for example, S. Chandrasekhar, The Mathe-

matical Theory of Black Holes (Oxford Univ. Press,Oxford, 1983).

7. A. Abramovici et al., Science 256, 325 (1992).8. J. A. Wheeler, Am. Sci. 56, 1 (1968).9. P. S. Laplace, Le Systmer du Monde (Paris, 1795),

vol. II [EngI. transl. (Flint, London, 1809)].10. R. Penrose, Phys. Rev. Lett. 10, 66 (1963).11. P. Anninos, D. Hobill, E. Seidel, L. Smarr, W.-M.

Suen, ibid. 71, 2851 (1993).12. P. Anninos et al., Phys. Rev. D 50, 3801 (1994).13. P. Anninos, D. Hobill, E. Seidel, L. Smarr, W.-M.

Suen, ibid., in press.14. S. Shapiro and S. Teukolsky, ibid. 45, 2739 (1992).15. S. Hughes et al., ibid. 49, 4004 (1994).16. P. Anninos et al., Phys. Rev. Lett. 74, 630 (1995).17. J. Ubson et al., in preparation.18. S. W. Hawking and G. F. R. Ellis, The Large Scale

Structure of Spacetime (Cambridge Univ. Press,Cambridge, 1973).

19. C. W. Misner, K. S. Thome, J. A. Wheeler, Gravita-tion (Freeman, San Francisco, 1973).

20. General relatMty allows a wide choice of the Wnd ofsimultaneity "slices" for the descaiptionof spacetime.Both the simulation shown in Figs. 3 and 4 and thesimulaffon of Fig. 10 use times-for instance, thatshown on the clock in Fig. 3-which are variants oftheso-called maximal slicing. [See, for example, L. Smarrand J. W. York Jr., Phys. Rev. D 17, 1945 (1978); ibid.,p. 2529.]

21. R. Thom, Stabilite Structurele et Morphogenwse(Berqamin, Reading, MA, 1972) [Structural Stabilityand Morphogenesis, English translation (Benjamin,Reading, MA, 1975)].

22. V. I. Amold, Matematicheskie Metody KlassicheskofMekhaniki (Nauka, Moscow, 1974) [MathematicalMethods of Classical Mechanics, English translation(Springer-Veriag, New York, 1978)].

23. M. V. Berry and C. Upstill, Prog. Opt. XVIII, 256(1980).

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i Sm

24. R. Penrose, (1963) reprinted in Gen. Rel. Grav. 12,225 (1980).

25. H. Friedrich and J. M. Stewart, Proc. R. Soc. LondonSer. A 385, 345 (1983).

26. Because the crossover line is spacelike, the "bow-leggedness" of the diagram is to some extent due toour choice of coordinates. A different choice of si-multaneity can change the qualitative shape butmust leave invariant the smooth merger of thespacelike crossover line with the light ray lines leav-ing the cusps.

27. C. Misner, Ann. Phys. NY 24, 102 (1963).28. W.-M. Suen, paper presented at Grand Challenge

Meeting, National Center for Supercomputing Appli-cations (NCSA), University of Illinois at Urbana-

Champaign, 6 to 8 November 1994 (lecture notesavailable from NCSA).

29. S. W. Hawking, in Black Holes, C. DeWitt and B.DeWitt, Eds. (Gordon and Breach, New York, 1973).

30. Commun. Math. Phys. 25,152 (1972).31. We thank D. Eardley, R. Geroch, R. G6mez, J. M.

Stewart, K. Thome, F. Tipler, and M. Visser for helpfulcomments on this work. We are especially grateful toJ. Masso and P. Walker for producing Fig. 10 and forcontnbuting significantly to the analysis in this paper.We also thank M. Blanton, M. Chia, S. Hughes, C.Keeton, P. Walker, and K. Walsh for participating inthe computation and production of Figs. 3 and 4. Thisresearch was funded by the NSF Grand ChallengeGrant PHY93-18152/ASC93-18152 (ARPA supple-

mented) and supported by the following individualgrants: by NSF grant PHY83 10083, Texas TARP-085, and a Cray University Research Grant (R.A.M.);by NCSA and by NSF grant PHY 94-07882 (H.E.S.and L.S.); by NSF grants AST 91-19475 and PHY94-08378 and National Aeronautics and Space Ad-ministration grant NAGW-2364 (S.L.S. and S.A.T.);by NSF grant PHY 94-04788 (W.-M.S.); and by NSFgrant PHY92-08349 (J.W.). Computer time wasmade available by NCSA and the Pittsburgh Super-computing Center (in part through MetaCenter grantMCA94PO1 5), and by the Comell Center for Theoryand Simulation in Science and Engineering (support-ed in part by NSF, IBM Corporation, New York State,and the Comell Research Institute).

* RESEARCH ARTICLES

Geophysics of the Pitman FractureZone and Pacific-Antarctic PlateMotions During the CenozoicSteven C. Cande, Carol A. Raymond, Joann Stock,

William F. Haxby

Multibeam bathymetry and magnetometer data from the Pitman fracture zone (FZ) permitconstruction of a plate motion history for the South Pacific over the past 65 million years.Reconstructions show that motion between the Antarctic and Bellingshausen plates wassmaller than previously hypothesized and ended earlier, at chron C27 (61 million yearsago). The fixed hot-spot hypothesis and published paleomagnetic data require additionalmotion elsewhere during the early Tertiary, either between East Antarctica and WestAntarctica or between the North and South Pacific. A plate reorganization at chron C27initiated the Pitman FZ and may have been responsible for the other right-steppingfracture zones along the ridge. An abrupt (80) clockwise rotation in the abyssal hill fabricalong the Pitman flowline near the young end of chron C3a (5.9 million years ago) datesthe major change in Pacific-Antarctic relative motion in the late Neogene.

The Pacific-Antarctic Ridge is the key linkin the global plate circuit tying the relativemotion of the oceanic plates of the Pacificbasin to the rest of the world (1, 2). Forexample, one of the more astounding con-sequences of rigid plate tectonics is that theaccuracy of models of westem North Amer-ican deformation, including motion on theSan Andreas fault, depends on how wellmagnetic anomalies and fracture zones(FZs) can be reconstructed in the far SouthPacific (3, 4). Because key areas in theSouth Pacific are remote and poorly sur-veyed, circum-Pacific plate reconstructionshave continued to have large uncertaintieswhile uncertainties in other links in theglobal plate motion circuit have been pro-gressively reduced. In this paper we describe

S. C. Cande is at the Scripps Institution of Oceanogra-phy, La Jolla, CA 92093-0215, USA. C. A. Raymond is atthe Jet Propulsion Laboratory, Mail Stop 183-501, Cali-fornia Institute of Technology, Pasadena, CA 91109-8099, USA. J. Stock is at the Seismological Laboratory,Mail Stop 252-21, Califomia Institute of Technology, Pas-adena, CA 91125, USA. W. F. Haxby is at Lamont-Doherty Earth Observatory, Palisades, NY 10964, USA.

a geophysical survey of an FZ near thesouthernmost end of the Pacific-Antarcticridge that enables us to greatly reduce theuncertainties in the global plate circuit.

Plate motions and global tectonics. Sev-eral fundamental questions of circum-Pacif-ic tectonics can be addressed by a betterunderstanding of Pacific-Antarctic platemotions. For the South Pacific, some of themost important unresolved issues involvethe timing and amount of early Tertiaryrelative motion between East and WestAntarctica and between the Lord HoweRise and Campbell Plateau, two of the linksin the plate loop tying the Pacific, Austra-lia, and Antarctic plates together (2, 5). Aglobal hot-spot reference frame also de-pends on the accuracy of the Pacific-Ant-arctic link. Atlantic and Indian Ocean hotspots, when rotated back to the Pacific, do a

poor job of predicting the track of the Ha-waiian hot spot (6-8), raising questions ofthe fixity of hot spots (9). Non-Pacific pa-leomagnetic poles, when rotated back to thePacific, do not agree with Pacific plate pa-

SCIENCE * VOL. 270 * 10 NOVEMBER 1995

leomagnetic poles. This misfit has led to thesuggestion that there may be one or moremissing plate boundaries between the NorthPacific and East Antarctica (10, 1).

These unresolved questions have led toseveral hypotheses. Stock and Molnar (9)proposed that several puzzling aspects ofearly Tertiary Pacific tectonics could beexplained if there was an undiscovered fos-sil spreading center in the Southeast Pacificthat separated the region of the Antarcticplate near the Bellingshausen Basin (re-ferred to as the Bellingshausen plate) fromthe part of the Antarctic plate adjacent toMarie Byrd Land (see Fig. 1). This scenarioprovided a better fit to magnetic anomalydata from chron C30 to chron C25 south ofthe Campbell Plateau; it reduced theamount of unexpected motion in the loopamong the Australia, Antarctic, and Pacificplates in the early Tertiary; and it alsohelped to explain the failure of global re-constructions to predict the bend in theHawaiian-Emperor chain (12).

Numerous tectonic issues can also beaddressed by increasing the resolution ofplate reconstructions. Several studies haveproposed that there was a late Neogenechange in the absolute motion of the Pacif-ic plate, corresponding to a clockwise rota-tion in the separation direction of the Pa-cific and Antarctic plates (13-16). The ageof this event, however, has been difficult toestablish with the pre-existing data sets;estimates of its age vary from less than 9.8Ma (million years ago) (13), 5 Ma (14), 5 to3.2 Ma (15) to 3.86 to 3.4 Ma (16). Itsexact age is of considerable interest becausea number of late Neogene events aroundthe Pacific, such as the onset of Pliocenecompression along the San Andreas fault(15, 17), have been attributed to it.

Recent studies of the Pacific-Antarcticridge have focused on interpreting satelliteradar altimetry data collected by SEASATand GEOSAT, which provide new data onthe location of FZs (18-20). However, al-though images of the gravity field over theseafloor generated from satellite radar al-timetry measurements have provided im-proved locations and trends of FZs and oth-er tectonic features, without shipboard geo-physical measurements in certain critical

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Geometry of a Black Hole CollisionRichard A. Matzner, H. E. Seidel, Stuart L. Shapiro, L. Smarr, W.-M. Suen, Saul A. Teukolsky and J. Winicour

DOI: 10.1126/science.270.5238.941 (5238), 941-947.270Science 

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