Violations of Weak Cosmic Censorship in Black Hole collisions

Tomas Andrade,1, ∗ Pau Figueras,2, † and Ulrich Sperhake3, 4, ‡1Departament de Fı́sica Quàntica i Astrofı́sica, Institut de Ciències del Cosmos,

Universitat de Barcelona, Martı́ i Franquès 1, 08028 Barcelona, Spain2School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom

3DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK4California Institute of Technology, Pasadena, California 91125, USA

We study collisions of boosted rotating black holes in D = 6 and 7 spacetime dimensions with a non-zeroimpact parameter. We find that there exists an open set of initial conditions such that the intermediate stateof the collision is a dumbbell-like horizon which is unstable to a Gregory-Laflamme-type instability. As isusually the case for similar unstable configurations, the evolution of such an instability leads to a pinch off ofthe horizon in finite asymptotic time, thus forming a naked singularity. Hence, this is the first fully genuineviolation of Weak Cosmic Censorship conjecture in higher dimensional asymptotically flat spacetimes.

Introduction.—General Relativity (GR) governs the largescale structure of our Universe. Despite its enormous suc-cess, the fact that singularities can be generically formedin dynamical settings [1] in principle calls into question itspredictive power. Motivated by this, Penrose postulated the“Weak Cosmic Censorship (WCC) Conjecture” [2], whichproposes that singularities formed in dynamical evolutionmust be hidden inside black hole (BH) horizons. This con-stitutes a key link between the singularity theorems and theubiquitousness of astrophysical black holes, for which Pen-rose was recently awarded the Nobel prize. Through ac-tive investigation in over half a century, counter-examplesto WCC in settings beyond astrophysics have been found,which can be regarded as a possibility to access the quantumregime of gravity, at least theoretically. They fall into twomajor categories: i) critical collapse, which involves fine tun-ing of initial data such that a zero mass singularity is formedin 4D and above [3]; ii) death by fragmentation, which re-sults from elongated horizons becoming unstable and even-tually pinching off in finite time. The latter category will bethe main focus of our letter.

The instability which drives the initial stages of thisphenomenon was famously discovered by Gregory andLaflamme (GL) [4], who showed that, in 5D and above, thinenough black strings (and p-branes in general) have expo-nentially growing linear modes that trigger spatial modula-tion along the horizon, in a way that resembles the Rayleighinstability in fluids [5, 6]. Later on, [7] followed the un-stable black string into the non-linear regime by means of anumerical simulation, which provided convincing evidencein favour of the pinching off of the horizon in finite time.Moreover, the authors observed the formation of BH satel-lites joined by thin necks organized in a fractal structure.It is by now understood that not only strings, but all elon-gated horizons which are locally string-like and thin enough,present these unstable modes, e.g., black rings [8], and ultra-spinning BHs [9–14]. The full non-linear evolution of suchunstable configurations has been studied in [15–17], con-cluding in all these scenarios that the horizon pinches off infinite time, in a way similar to the black string case.

All of the above setups have in common that the startingpoint is an unstable configuration. Therefore, one may won-der if the so-obtained singularities are a truly natural featureof the evolution in GR. Along these lines, a new mechanism

for violation of WCC has been recently proposed [18, 19]:the collision of BHs in higher dimensions. What the authorsnoted is that, if the total angular momentum is high enough,an elongated horizon should form as an intermediate, long-lived, state after the collision. After this point, the horizoncould undergo the breaking described above, depending onthe competition between radiation of mass and angular mo-mentum, and the GL instability. The former tends to makethe horizon rounder and shorter, thus taming the effect ofGL. Since radiation is exponentially suppressed with D, andthe growth rates of the GL instability remain relatively con-stant, this mechanism is guaranteed to work if D is suffi-ciently large. On the other hand, this behaviour is absent inhead-on collisions in four or higher dimensions [20, 21], andin grazing BH collisions in 4D [22], none of which exhibitelongated horizons or any signs of censorship violation evenwhen colliding above 90 % of the speed of light. Uncover-ing exactly when and how the proposed phenomenon takesplace, requires fully-fledged numerical GR techniques. Wenote that bar mode instabilities of rotating black holes areknown in 6D and above [13, 14], so elongated horizons aremore likely to exist in D ≥ 6. For a sufficiently large initialangular momentum, these unstable ‘bar-shaped’ black holesdevelop a local GL instability. During the evolution of thisinstability, only about 5% of the total mass of the system isradiated, which is insufficient to prevent the formation of anaked singularity in finite time [17]. Furthermore, Ref. [21]reports that for v≤ 0.6 the radiation emitted in head-on col-lisions is below 1%. All of this considered, we envision thatintermediate velocities, relatively high intrinsic spins and im-pact parameters, and D≥ 6 constitute a favourable region forWCC violation in parameter space.

Here we undertake the task of numerically simulating thecollision of 6D and 7D stable, single-spinning BHs with afinite impact parameter. This initial setting is closely anal-ogous to that of a BH binary merger in 4D, but the dimen-sionality gives rise to striking differences in the evolution aswe now describe. We identify, within a significant propor-tion of the parameter space, configurations that form an elon-gated horizon shortly after the merger, which breaks throughthe further evolution of a local GL instability. This processdoes not require any fine tuning. Moreover, we observe thatthe late time dynamics resemble the end point of the blackstrings and ultra-spinning BHs of [15–17]. In particular,

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we observe the formation of two well-defined generationsof satellites, and the beginning of a third one. Each genera-tion consists of spherical blobs joined by thin tubes, whichcan be approximated by quasi-stationary BHs and strings, re-spectively. The last stage to which we have access consistsof a central rotating black object, joined by a thin horizon totwo non-spinning blobs that move away from each other, seeFig 1. The thin tubes in the middle are stretching due to themotion of the outer blobs, and develop their own satellites.We see that each segment is GL unstable, so it is clear thatthe process will continue until the pinch off.

We extract the gravitational waves produced in the colli-sion, finding two peaks. The first one is simply due to themerger of the BHs. The second peak produces a strongersignal, and we attribute it to the sudden change of angularvelocity and shape of the elongated horizon. Employing atoy model of a D > 4 dimensional binary which undergoesa qualitatively similar change in angular velocity and sepa-ration, we extract the radiated energy and observe a similarpeak; cf. the Supplementary Material for more details. Wefind that the total radiated energy is strongly suppressed (itamounts to less than 1% in 7D), confirming the hypothesisthat there is no mechanism to stop the formation of a nakedsingularity in finite asymptotic time.

Numerical methods.—We solve the vacuum Einsteinequations in D = 6,7 using the CCZ4 formulation as de-scribed in [23, 24]. We restrict ourselves to dynamics in 3+1dimensions by imposing an SO(D− 3) symmetry by meansof the modified cartoon method [13, 25, 26]. The conven-tions for the damping parameters κ1 and κ2, and choice ofgauge evolution are those of [17]. We construct the initialdata by superimposing two boosted Myers-Perry (MP) BHs[27] written in Kerr-Schild (KS) coordinates, as describedin [28]. This yields a mild violation of the constraints inthe initial surface, but in higher dimensions it can be eas-ily remedied as we describe below. We begin with a single,un-boosted MP BH with mass and rotation parameters µ , a,whose relation to the mass and angular momentum are givenin Eq. (9) in the Supplementary Material. We write the met-ric as

gµν = ηµν + f (x)kµ kν , (1)

where η is the Minkowski metric, f is a function of thespace-time coordinates x and kµ is a null vector. The ex-plicit expressions for f and kµ are given in Ref. [27]; seealso the Supplementary Material for details. For this form ofthe metric, we can boost the BH by simply applying the cor-responding Lorentz transformation to each of the vectors kµ

in Eq. (1). Henceforth, we split the coordinates as t,x,y,z,wiwhere ∂wi are “cartoon” directions in the language of [26],and choose a single spin aligned with the z−axis.

Without loss of generality, we place one of the BHsat (x0,y0,0) and give it an initial velocity along thex−direction, which we denote by v. We ease the tracking ofthe dynamics by choosing symmetric data in which the sec-ond BH is positioned at (−x0,−y0,0) with speed (−v,0,0),which gives an orbital angular momentum pointing in thez−direction. Both black holes have intrinsic spin along thez−direction, with spin parameter a. We emphasise that in our

setting the orbital and intrinsic angular momenta are aligned,which allows us to achieve higher values of the total angu-lar momentum. The impact parameter is controlled by y0.Moreover, recall that in D > 4 there is no inspiral phase inbinaries [29], which means that, generically, BHs interactonly if y0 is sufficiently small. While x0 does not play afundamental role, we need to choose it such that the initialBHs are sufficiently far apart so that the constraint violationson the initial surface as a result of simply superposing twoMP BHs are small. In addition, having a sufficiently largex0 allows us to decouple the initial gauge dynamics from thecollision; our gauge conditions result in an adjustment phaselasting about 3 µ1/(D−3).

Our main observable of interest is the location of the ap-parent horizon (AH). Tracking it from first principles en-tails solving a non-linear PDE in two variables at a giveninstant of time. Shortly after the collision of the two BHs,the common AH evolves into a non-star-shaped surface, withincreasingly complex structure as shown in Fig. 1; there ex-ist as yet no algorithms to find the AH for such surfaces. Inthis letter we therefore borrow from the results of [17] whichhave provided convincing evidence in a variety of scenariosthat the location of the AH matches within a few percent witha contour of the conformal factor χ . While there is some de-pendence on the space-time dimension D, this has been ob-served to be mild [15–17]. Motivated by this, we will referto the location of the “apparent horizon” by the location ofthe contour χ = 0.6. We provide further details in the Sup-plementary Material.

We are also interested in the pattern of the gravitationalwaves produced by our collisions. For their extraction, wefollow Ref. [17] and evaluate the appropriate integrals of thecomponents of the Weyl tensor projected on a basis of spher-ical harmonics. We focus on the components Ω′(A)(B) of theWeyl tensor along null rays projected onto a basis of spheri-cal harmonics Y (A)(B)`... according to

Ω′`... = limr→∞ r(D−2)/2

∫dΩ(n)Y

(A)(B)`...

∗Ω′(A)(B) . (2)

Here ` . . . denotes the set of quantum numbers of the spher-ical harmonics and dΩ(n) is the volume element of the n-sphere. We record the relevant tensor harmonics for the 5-sphere in the Supplementary Material, while the n = 4 casecan be found in [17].

We implement the numerical scheme in GRChombo [30,31], using up to twelve levels of refinement with a 2:1 re-finement ratio. We add the extra levels to better resolve thespatial gradients thus keeping the regions of constraint vio-lations far away from the χ = 0.6 surface, which we use asa proxy for the horizon. We ensure that the region containedwithin this contour is covered by at least 40 grid points. Un-acceptably large constraint violations due to insufficient res-olution lead to unphysical effects, such as a rapid swelling ofthe horizon. We have checked the robustness of our resultsby introducing refinement levels at different times, providedthat the extra levels are added whilst the constraint violationregion is sufficiently small. Following [15–17], we applyartificial diffusion in the region χ ≤ 0.075 to keep the con-straint violations under control. We have employed cubic

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computational domains, with side L. The x and y are stan-dard Cartesian coordinates, upon which we impose Sommer-feld boundary conditions at the edges of the domain. At z= 0we impose the regularity conditions as required by the Modi-fied Cartoon Method, while at z = L we impose Sommerfeldboundary conditions. We perform the time evolution usingthe fourth order Runge-Kutta method (RK4). The remainingparameters of our simulations are as in [17].

Results.—For both D = 6,7 we have explored the spaceof parameters a, v, y0 by running some low resolution sim-ulations until we observe the formation of a long bar-likehorizon after the merger. As mentioned above, no fine tun-ing is required in order to achieve such states. Out of theparameters explored, we have selected three study cases tocontinue the evolution further with the aid of higher resolu-tion simulations. We shall discuss in the main text the D = 7case, for which we have been able to follow the late stages ofthe evolution for longer times. We discuss the D = 6 casesin the Supplementary Material, for which we have obtainedqualitatively similar results.

Henceforth we measure time and distance in dimension-less form using the mass parameter µ of a single memberof our BH binary. Specifically, we display time in the formof t = t̃/µ1/(D−3), where t̃ is coordinate time in geometricunits, and likewise for radius r and Cartesian coordinatesx, y, z. Our D = 7 collision is characterized by initial datawith parameters v = 0.5, a = 0.7, y0 = 1.1, x0 = 10. Notethat the spin is within the stability bound amax = 0.74 re-ported in [14]. First, we have performed a simulation in asmall domain of cubic size L = 64, including up to 10 lev-els. This simulation, displayed in Fig. 1, clearly shows thata dumbbell-shaped horizon forms after merger, consisting oftwo spherical blobs joined by a thin tube. Moreover, we ob-serve the appearance of spatial modulation along the hori-zon, characteristic of the GL instability. At the same time,the length of the dumbbell increases making the middle tubethinner, which accelerates the effect of the instability. Con-tinuing the evolution further, we have been able to see theformation of two generations of satellites, e.g., small spher-ical beads joined by thin tubes, closely resembling the latestages of unstable strings and ultra-spinning BHs. Our nu-merics have allowed us to observe the onset of a third gener-ation, in which all string segments present in the second gen-eration, develop their own local GL instability. Note that in7D, the critical value for the width over length ratio of stringswhich are GL unstable is rGL,7D ≡ widht/length ≈ 0.5. Wehave explicitly checked that the string segments present inour first and second generation are well below this value.

In order to extract the gravitational waves, we have in-creased the size of the computational domain to L = 256,adding two levels in order to achieve equal resolutions atthe horizons of the BHs. We have extracted the waves atradii R = 25.0,27.5,30.0 and observed the expected linearbehaviour: all three wave forms match in all channels af-ter the corresponding time shifts. We display the results forthe D = 7 case in Fig. 2 Interestingly, we observe an initialregime in which small waves arrive for which this matchingproperty does not hold. This corresponds to the initial con-straint violations from superposing the two MP BHs and to

FIG. 1. Snapshots of the 7D evolution with v = 0.5, a = 0.7, b =2.2, at times ∆t = t− tmerger. They correspond to: initial condition,merger, bar formation, first generation, second generation, lateststage. We show a spatial length of 10 units on the bottom left panel.

the adjustment of the gauge, and it is unphysical.In Fig. 2, we observe the first physical peak of radiation

produced in the m= 2 channel for `5 = 2,4 at t−tmerger∼ 30,corresponding to the “collision” of the two BHs, when acommon enveloping horizon first appears. As shown in thesecond panel in Fig 1 (see also the first panels in Figs. 5 and6) this common horizon is dumbbell shaped. Furthermore,this resulting BH is rotating since it acquires part of the totalangular momentum of the initial state, whilst some of the an-gular momentum has already been radiated away. At somepoint, which depends on the initial total angular momentum(and the number of dimensions D), the BH stops rotating andthe neck connecting the two bulges starts expanding, produc-ing a second radiation peak; in the D = 7 example of Fig. 2,this happens at around t− tmerger = 7.5. We confirm this pic-ture with a simple model; see Supplementary Material.

Up until and including the second peak, the total radiatedenergy is about 0.01% of the Arnowitt-Deser-Misner (ADM)mass, see Fig 7, which follows from the data in Fig. 2 us-ing eq. (24). We expect radiation emitted at later stages toamount to a smaller fraction, since the dynamics of the outerblobs is quasi-stationary, and the GL cascade taking placein the strings produces very little radiation ([17] reports thatless than 5% of the total ADM mass is radiated during thelast stage of GL in the 6D evolution of ultraspinning BHs).We discuss this issue further in the Supplementary Material,where we provide a crude upper bound. In summary, theeffects of radiation are sufficiently suppressed in D ≥ 6, so

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�-�������FIG. 2. Imaginary part of the projection of the matrix of Weylscalars Ω′AB onto the harmonics (`5,m) = (2,2),(4,2),(4,4) for thecollision in D = 7 (see Supplementary Material). In these plots,the extraction was done at a radius R = 30 from the center of thegrid. The first peak is observed at t− tmerger ∼ 30, correspondingto the initial collision of the two BHs; the oscillations observedbefore this time are due to the constraint violations in the initialdata and initial gauge adjustment. Two large peaks are observed ataround t− tmerger ∼ 37.5, which corresponds to time at which thedumbbell-shaped horizon stops rotating and starts expanding. Be-yond this point, whilst the gravitational wave signal continues todevelop new features corresponding to the onset and further evolu-tion of the GL instability in the neck, their amplitude is never aslarge as that of these two initial peaks. Consequently, the energyloss in absolute value decreases.

that there is no mechanism to stop the violation of WCC inBH collisions provided the initial angular momentum is suf-ficiently large.

Discussion.—By means of numerical GR simulations, wehave been able to establish the violation of the WCC con-jecture in BH collisions for D = 6,7. The driving mecha-nism is the GL instability, and the dynamics unfold in closeparallel to other known cases of horizon pinching off. Wehave observed the formation of two generations of satellitesand the beginning of a third one, which further improves ourunderstanding of the dynamical process. We have extractedthe gravitational waves emitted by the binaries, showing thatthere is a second peak which surpasses the burst due to themerger. After this second peak, we expect the emission ofradiation to be small, paving the way for the GL instability

to break the horizon. Since radiation is further suppressedin higher dimensions, our results strongly indicate that theviolation of WCC we have identified is present in all D≥ 6.However small, the emission of gravitational waves in oursetting has revealed interesting features, and we expect tostudy this issue further – in particular, the radiation producedduring the formation of satellites – in future work.

It is remarkable that the large D effective theory [32–34],out of which the arguments for this mechanism were orig-inally drawn [18, 19], is able to capture the qualitative dy-namics so well. In our context, the main drawback of the ef-fective theory is the non-perturbative character of radiation.In hindsight, perhaps the main reason for its success in pre-dicting the violation of WCC is the relatively small role thatradiation plays already in 6D.

As in all other known cases of WCC violation by frag-mentation of horizons, the regions which we expect to be-come Planckian are “small” at least with respect to the sizedefined by the geometry of the apparent horizon. In the lan-guage of [35], they correspond to mild violations of WCC,which means that any suitable quantum resolution (analogueof the evaporation of a droplet in hydrodynamics) will havea small effect on the subsequent (classical) dynamics. In thissense, although the more strict version of the WCC conjec-ture is violated (naked singularities do appear in the courseof evolution), its spirit is salvaged (the lack of predictabilitythat the violations entail is very small).

The case of D = 5 is special in various ways. Classically,the decays of the angular momentum barrier and the gravi-tational potential for point particles exactly match (∼ r−3).Consistent with this general observation, no “zoom-whirl”orbits (i.e. long inspirals which result in mergers or scatter-ing to infinity) have been found [36]. Moreover, no linear in-stabilities of 5D of single spinning MP BHs have been iden-tified [14, 17], which in turn suggests that in this dimensionblack bars do not exist, as opposed to D ≥ 6 [13]. Thesefacts, and the observation that radiation is less suppressedwith respect to 6D, suggest that the violations of WCC re-ported here, should at least behave quite differently in 5D, ifpresent at all.

An interesting technical point which we have not ad-dressed here is the derivation of the location of the apparenthorizon from first principles. As discussed above, this re-quires parametrizing the shape of a reference surface, whichis challenging in scenarios like ours where spatial modula-tion is constantly changing the shape of the horizon. Obtain-ing the apparent horizon in this way would allow us to accu-rately study entropy production. We leave these interestingissues for future research.

Acknowledgments.—We thank Roberto Emparan formany insightful discussions. Our special thanks are for theentire GRChombo collaboration (www.grchombo.org) fortheir help and support. TA is supported by the ERC Ad-vanced Grant GravBHs-692951. He also thanks QMULfor the hospitality during the early stages of this project.PF is supported by the European Research Council GrantNo. ERC-2014-StG 639022-NewNGR, by a Royal Soci-ety University Research Fellowship (Grant No. UF140319and URF\R\201026) and by a Royal Society Enhancement

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Award (Grant No. RGF\EA\180260). US acknowledgessupport from the European Union’s H2020 ERC Consolida-tor Grant “Matter and strong-field gravity: new frontiers inEinstein’s theory” Grant No. MaGRaTh–646597, the STFCConsolidator Grant No. ST/P000673/1, and the GWverseCOST Action Grant No. CA16104, “Black holes, gravi-tational waves and fundamental physics”. The simulationspresented here were done on the MareNostrum4 cluster atthe Barcelona Supercomputing Centre (Grant No. FI-2020-2-0011 and FI-2020-2-0016), the SDSC Comet and TACCStampede2 clusters through NSF-XSEDE Grant No. PHY-090003, and the Cambridge Service for Data Driven Dis-covery (CSD3), part of which is operated by the Universityof Cambridge Research Computing on behalf of the STFCDiRAC HPC Facility (www.dirac.ac.uk). The DiRAC com-ponent of CSD3 was funded by BEIS capital funding viaSTFC capital grants ST/P002307/1 and ST/R002452/1 andSTFC operations grant ST/R00689X/1. DiRAC is part of theNational e-Infrastructure.

SUPPLEMENTARY MATERIAL

Initial data.—For D even, the metric in Kerr-Schild formfor a single BH is given by Eq. (1) with

f =µr

Π(r)F(r,xi), (3)

Π(r) = Πs(D)i (r2 +a2i ) , (4)

F = 1−s(D)

∑i=1

a2i (X2i +Y

2i )

(r2 +a2i ), (5)

k = dt +s(D)

∑i=1

{r(XidXi +YidYi)+ai(XidYi−XidYi)}r2 +a2i

+zdzr

,

(6)

where k = kµ dxµ , s(D) = (D− 2)/2, and Xi, Yi denote sub-sets of the coordinates that specify the rotation planes; theywill be specified explicitly below. The index i runs from1..(D−2)/2, and corresponds to the number of possible ro-tation planes, in such a way that a non-vanishing ai adds rota-tion on the axis orthogonal to (Xi,Yi). We explicitly displaythe sum over i to emphasise that there is no extra summa-tion of repeated indices. The quantity denoted by r is notan independent coordinate, but can be expressed in terms ofthe Xi, Yi by means of the condition kµ kν ηµν = 0. In or-der to implement the modified cartoon method in D = 6, wechoose ai = (a,0), Xi = (x,w1), Yi = (y,w2) with w1, w2 be-ing isometric directions. This gives the BH rotation along thez-axis. Finally, we use a translation (x′,y′) = (x− x0,y− y0)and a Lorentz boost along x,

t ′′ = tcη + x′sη , x′′ =−tsη + x′cη (7)

with cη = coshη , sη = sinhη and coshη = (1− v2)−1/2.For odd D, we have Eqs. (3)-(5) along with

k = dt +s(D)

∑i=1

{r(XidXi +YidYi)+ai(XidYi−XidYi)}r2 +a2i

(8)

and now the number of rotation planes is s(D) = (D−1)/2.To implement the cartoon method in D = 7, we chooseai = (a,0,0), and Xi = (x,z,w1), Yi = (y,w2,w3), where(w1,w2,w3) parametrize the isometric directions. This givesthe BH rotation along the z-axis.

The mass and angular momentum for a single spinningBH are given by

M =(D−2)ΩD−2

16πGµ, J =

2(D−2)

Ma , (9)

where

ΩD−2 =2π(D−1)/2

Γ[(D−1)/2]. (10)

Apparent horizons and contours of χ .— In this sectionwe provide some more quantitative details of the relation be-tween the apparent horizon (AH) and the contours of theconformal factor χ . This relation is gauge dependant. Wehave observed that it works in traditional 1+ log slicing plusGamma driver gauges used in numerical relativity in a va-riety of contexts, ranging from astrophysical BH binaries in3+1 dimensions to studies of BH instabilities in higher di-mensions [15–17].

As pointed out in the main text, the fact that the AH, thatforms in the collisions that we study in this Letter (see Figs.1, 6, 5), is a non-star-shaped, highly complex and rapidlychanging surface makes the problem of finding it particularlychallenging. One possible way forward is to consider a refer-ence surface with the approximate shape of the AH [37, 38].However, in our case, the reference surface should capturesome of the key features that appear as a consequence of theGL cascade; since the latter develop randomly on differentscales, both in time and space, finding a suitable referencesurface at each time step seems impractical. Alternatively,one could use an intrinsic parametetrization of the surfaceand impose some gauge condition (e.g., harmonic) to fix thereparametrization freedom as in [16]. However, in this refer-ence it was found that the resulting system of equations wasnot very robust in practice in spite of being manifestly el-liptic. Therefore, the problem of parametrizing and findingthe AHs that generically appear in higher dimensional BHphysics remains open. As we will explain below, a promis-ing avenue would be to use certain contours of the evolvedvariable χ as the reference surface. This approach wouldhave the advantage that these contours capture some of thekey features of the AH at any given instant of time and theyare readily available since χ is one of the evolved variables.Implementing this in practice is beyond the scope of this Let-ter and we leave it for future work. In the following, wewill provide quantitative evidence that the χ = 0.6 contourdoes indeed capture the essential features of the AH’s shape.Other contours, such as χ = 0.5 or χ = 0.4, would also becomparatively informative. This insensitivity to the specificvalue of the contour is due to the steep gradients of the metricfields in higher-dimensions.

In Fig. 3 (top) we show the values of χ on the AH near theendpoint of the ultraspinning instability of a MP BH in 6Dwith an initial spin parameter a/µ1/3 = 1.85 reported by one

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FIG. 3. Top: χ (in red) on the AH plotted against the embeddingcoordinate u near the endpoint of the ultraspinning instability of aMP BH in 6D with an initial spin parameter a/µ1/3 = 1.85 (see[16] for more detials); the Z coordinate of the embedding is shownin black and it serves to guide the eye. This figure shows that on theAH, χ ∼ 0.5 on the blobs and χ ∼ 0.3 on the necks. Bottom: AHand several χ contours during the evolution of the GL instability ofa black string in 5D.

of us in [16]. The shape of the AH is shown in black. Thisplot shows that the χ = 0.5 contours capture the ‘bulges‘quite accurately, especially those from the early generationsthat have reached a quasi-stationary state. On the other hand,the χ = 0.3 contours capture the neck regions of the AH. Inthe bottom plot of Fig. 3 we depict the AH together with dif-ferent χ contours at some instant of time during the evolutionof the GL instability of a black string in 5D [39]. This plotshows that indeed these contours of χ capture that key fea-tures of the AH in the evolution of the GL instability, in thiscase of a black string. We want to highlight that in the topand bottom panels in Fig. 3 we consider completely differentsystems, namely a rotating spherical asymptotically flat BHin 6D and a black string in 5D; the fact that in both cases thesame contours of χ capture the shape of the AH makes usconfident that results in this Letter based using the χ = 0.6contour as a proxy for the actual AH are robust.

Finally, in Fig. 4 we compare the χ = 0.6 (in blue), χ =0.5 (in black) and the χ = 0.3 (in red) contours for the 7DBH collision described in the main text. According to theprevious discussion, the χ = 0.6 contour should accuratelycapture the shape of AH in the bulges, especially those ofthe early generations which have already reached a quasi-stationary state. We used this contour of χ to estimate themasses and angular momenta of the big bulges. On the otherhand, the χ = 0.3 contour should capture the shape of theAH on the neck regions. In fact, both contours are very closeto each other in the neck regions and both capture the mostsalient features of the shape of the AH.

6DDD case.—In this section we discuss the results we have

FIG. 4. Comparison of the χ = 0.6 (in blue), χ = 0.5 (in black)and χ = 0.3 (in red) contours for the the 7D BH collision reportedin the main text. The χ = 0.6 contour should track the AH nearthe bulges whilst the χ = 0.3 should do so in the necks. The keyfeatures of the AH are captured by both contours; in particular, inthe neck regions the two contours are very close to each other.

obtained in D = 6. As mentioned in the main text, our resultsare qualitatively similar to the D = 7 case, in that we alsosee strong evidence for the pinching off of the horizon in fi-nite asymptotic time. We have run two detailed simulationsin cubic domains of size L = 64, with collision parameters{v = 0.5, a = 0.6, b = 2.5}, {v = 0.45, a = 0.7, b = 2.5},respectively. Note that both of these values of a are wellwithin the stability regime for rotating MP BHs amax = 0.73[14]. In Figs. 5, 6 we show representative snapshots of χon the z = 0 plane. In the simulation shown in Fig. 5, theformation of the first generation of satellites is clear, and weobserve that the dynamics closely resemble the 7D case. Thecritical dimensionless ratio for GL instability in this dimen-sion is rGL,6D ≡ widht/length ≈ 0.4. Estimating this valuefor t = 13.6 in Fig. 5, rGL ≈ 0.02, showing that this stringis well in the unstable regime. In the dynamics that follow,we again observe the appearance of the central rotating blob,joined to the outer blobs by thin strings. Our computationalresources did not allow us to reach the same stage for thesimulation in Fig. 6. However, we do observe a rapid thin-ning of the neck which should lead to the development ofthe GL instability as in the previous cases. In particular, fort = 15.0 in 6 we estimate for rGL ≈ 0.04, so the strings areunstable.

As emphasised in the main text, there was no fine tuningrequired to find configurations such as these in which theneck thins and the GL instability kicks in. Hence, we cancomfortably claim that the effect that we report is generic.

Gravitational waves in 7DDD.—It is convenient to write themetric on the S5 at infinity such that it has a manifest SO(4)symmetry and one of its U(1)’s is aligned with the U(1) onthe rotation plane of the incoming MP BHs:

ds2 = dχ2 + sin2 χ dφ 2 + cos2 χ dΩ2(3) , (11)

where dΩ2(3) is the metric on the unit S3. As shown in [17],

due to the symmetries of the spacetime, the gravitationalwave signal is contained in the scalar derived tensor harmon-ics. Given the form of the metric (11), a convenient basis for

6

FIG. 5. 6D-A evolution with v = 0.5, a = 0.6, b = 2.5.

FIG. 6. 6D-B evolution with v = 0.45, a = 0.7, b = 2.5.

the scalar harmonics on the S5 is

Y`5,m,`3,... = N eimφ (sin χ)|m|(cos χ)`3Y`3 ...

× 2F1(`3 + |m|− k,k+ 32 , `3 +

22 ; cos

2 χ),(12)

where 2F1 is a hypergeometric function, N is a normalizationfactor, k is a non-negative integer and Y`3 ... are the scalar har-monics on the S3, so that ∆S3Y`3 ... = −`3(`3 + 2)Y `3 ...; thedots in Y`3 ... denote the remaining labels for quantum num-

bers of the harmonics. Regularity of the harmonics implies

`5 = 2k− (`3 + |m|) . (13)

Since the spacetimes that we consider in this paper pre-serve a round S3 inside the S5, to extract the gravitationalwaves we only need to consider harmonics on the S5 that alsopreserve a round S3. Therefore, from now on, in (12) we willconsider harmonics with `3 = 0 and hence we can uniquelylabel them with only two quantum numbers, (`5,m).

Scalar derived traceless tensor harmonics on the n-spherecan be constructed from the scalar harmonics by [40]

S`n,...,`1ab =√

n√(n−1)(`n−1)(`n +n)

×

(∇aS`n,...,`1b +

√`n(`n +n−1)

ngabS`n,...,`1

),

(14)where S`n,...,`1a ≡ 1√

`n(`n+n−1)∇aS`n,...,`1 are the scalar derived

vector harmonics, and gab is the metric on the unit Sn. Thecoefficients are chosen so that the scalar derived tensor har-monics are suitably normalized:∫

Sngac gbd S`n,...,`1ab S

`′n,...,`′1

cd = δ`n`′n...`1`′1 . (15)

To do the extraction, we need to compute the tensor har-monics on the S5 in a certain basis of angular vectors. Forthe metric in (11), we choose the obvious basis,

m(1) =∂

∂ χ, m(2) =

1sin χ

∂∂φ

, m(i) =1

cos χm̂(i) , (16)

where m̂(i), i = 1,2,3 are a basis of angular (unit) vectors onthe S3.

After these preliminaries, we are now ready to list the non-vanishing components of the scalar derived tensor harmonicson the S5 that we have used to extract the gravitational waves:

• (`5,m) = (2,0):

S(2,0)11 =1

π3/2

√370[2cos(2χ)+1

],

S(2,0)22 = −1

π3/2

√370[

cos2(χ)−4],

S(2,0)i j =−δi j

2π3/2

√3

70[

cos(2χ)+3].

(17)

• (`5,m) = (2,2):

S(2,2)11 =1√

70π3/2e2iφ[2cos(2χ)+3

],

S(2,2)12 =i

π3/2

√5

14e2iφ cos(χ) ,

S(2,2)22 = −1

2√

70π3/2e2iφ[

cos(2χ)+9],

S(2,2)i j =δi j√

70π3/2e2iφ sin2(χ) .

(18)

7

• (`5,m) = (4,0):

S(4,0)11 =−1

2√

10π3/2[

cos(2χ)+5cos(4χ)−2],

S(4,0)22 =1

8√

10π3/2[−24cos(2χ)+5cos(4χ)+3

],

S(4,0)i j =δi j

24√

10π3/2[28cos(2χ)+15cos(4χ)−11

].

(19)

• (`5,m) = (4,2):

S(4,2)11 =−1

4√

5π3/2e2iφ[2cos(2χ)+5cos(4χ)−2

],

S(4,2)12 =i√

516π3/2

e2iφ[

cos(χ)−5cos(3χ)],

S(4,2)22 =1

16√

5π3/2e2iφ[2cos(2χ)+5cos(4χ)+13

],

S(4,2)i j =−δi j

4√

5π3/2e2iφ sin2(χ)

[5cos(2χ)+6

](20)

• (`5,m) = (4,4):

S(4,4)11 =1

2√

2π3/2e4iφ sin2(χ)

[2cos(2χ)+3

],

S(4,4)12 =5i

2√

2π3/2e4iφ sin2(χ)cos(χ) ,

S(4,4)22 =−1

4√

2π3/2e4iφ sin2(χ)

[cos(2χ)+9

],

S(4,4)i j =1

2√

2π3/2e4iφ sin4(χ) .

(21)

The harmonics with negative m can be obtained from theones listed above by complex conjugation.

Finally, to do the extraction we compute the componentsof the matrix of Weyl scalars Ω′AB in a certain orthonormalbasis. We construct the latter starting from

m(1) = x∂x + y∂y + z∂z ,

m(2) = xz∂x + yz∂y− (x2 + y2)∂z ,m(3) =− y∂x + x∂y ,

(22)

and carrying out a standard Gram-Schmidt orthonormaliza-tion [41]. The unit vectors on the transverse sphere directionare simply given by

m(î) =1√γww

∂wi . (23)

One can show that, asymptotically, the angular basis vectorsobtained from (22) are aligned with (16).

Radiation peak due to separation.—In order to eluci-date the origin of the second peak we observe in the patternon gravitational radiation, we consider a simple model for ahigher dimensional binary and use the quadrupole formulaof [42]. Since a general expression is only available for evendimensions, we consider the case D = 6 which we expectwill best capture the qualitative features of our setting.

� �� �� �� ��-�������-�������-�������-�������-�������-��������������

�-�������

��/��

� �� �� �� ��-�������-�������-�������-�������-�������-��������������

�-�������

��/��

FIG. 7. Energy loss for our 7D simulation as computed at a radiusR = 30 from the merger point. The large peak at t− tmerger ∼ 37.5is produced when the dumbbell-shaped BH stops rotating and startsexpanding.

The quantity that we can readily compare is the energyloss dE/dt, which is related to the Weyl scalars as [43]

dEdt

=− limr→∞

rd−2

8π

∫Sd−2

(∫ u−∞

Ω′AB(û,r,x)dû)2

dω . (24)

We show our results for the energy loss in D= 7 as computedfrom the Weyl scalars extracted from our numerics in Fig 7.The corresponding quadrupole formula for even D can bewritten as [42]

dEdt

= P(D)

[(D−1)∂

D+22

t Mi j(t)∂D+2

2t Mi j(t)−

∣∣∣∣∂ D+22t Mii(t)∣∣∣∣2].

(25)

where the prefactor is given by

P(D) =22−D(D−3)D

πD−5

2 Γ[D−1

2

](D2−1)(D−2)

G , (26)

and

Mi j =∫

dD−1xT 00(t,x)xix j . (27)

In addition, [19] provided an expression for the emission ofangular momentum in even dimensions, which in our casereduces to

dJi j

dt=

4(D2−1)D+2

P(D)[

∂D+2

2t Q

[ik∂

D2

t Qj]

k

], (28)

where

Qi j = Mi j− 1D−1

δi jMkk . (29)

In order to evaluate these expressions, we make the assump-tion that the energy distribution T 00 is simply a delta functionwith support on two point particles that follow a trajectory ona two-dimensional plane. We parametrize the trajectory inpolar coordinates x1(t) = ρ(t)cosθ(t),x2(t) = ρ(t)sinθ(t).For concreteness, we consider a circular binary which slows

8

FIG. 8. Top: Trajectory for a binary which starts in a circular orbitand experiences a slow down and separation. (Bottom): Energy andangular momentum loss as computed from the quadrupole formulae(25), (28) in 6D.

down and separates along a straight line. This captures someof the qualitative features we observe in our simulations aftermerger. We show the profiles we have chosen for ρ and θ onFig 8 (top) while the obtained pattern for energy and angularmomentum loss is depicted on Fig 8 (bottom). At early timeswe observe a sinusoidal pattern for energy radiation charac-teristic of a circular binary. Interestingly, the later slow downand separation leads to a relatively sharp peak, which resem-bles the second radiation peak we observe in Fig 7. We alsoobserve that the radiated angular momentum follows the pat-tern of the radiated energy averaged over the orbits.

Radiated energy and angular momentum.—We ap-proximate the total initial and final mass and angular mo-mentum according to

Mi = 2γimi , (30)M f = 2γ f m f +mcentral +2mstring , (31)

Ji = γimi[

bivi +22

D−2ai

], (32)

J f =[

γ f m f b f v f +2

D−2mcentralacentral

]. (33)

Here mi is the mass of the BHs in the initial state, with ve-locities vi, spins ai and impact parameter bi – defined as thedistance between the center of the BHs and the plane alongwhich the binary is reflection symmetric, i.e. bi = y0 in thenotation of the main text –, m f is the mass of the BHs atthe extremes of the final dumbbell, which have velocities v fand impact parameter b f (we have checked that they do not

have intrinsic spin), mcentral , acentral are the mass and spin ofthe central BH, and mstring is the mass of each of the stringsjoining the central and outer BHs.

We assume the BHs and strings involved in this calcula-tion to be stationary, in which case we can compute theirproperties from Eq. (9) and (we set G = 1 henceforth)

Mstring =(D−2)ΩD−2

16πGr+L . (34)

The initial parameters are given by (recall µ = 1),

mi = 3.1, vi = 0.5, bi = 2.2, ai = 0.7 . (35)

For the final parameters, we can measure v f and b f by track-ing the position of the outgoing blobs, and extracting themetric components in the region of the corresponding blobs.We find

v f ≈ 0.3, b f ≈ 4.26 . (36)

We see that the outgoing BHs do not rotate, so they can beapproximated Schwarzschild BHs. Their mass is then givenby

m f ≈ 3.1 . (37)

The mass and rotation parameter of the central blob can beestimated by computing Ω = β φ ≈ 1.5, gφφ ≈ 0.25 and com-paring this with the values for a MP BH. We obtain

mcentral ≈ 0.07, J ≈ 0.009, acentral ≈ 0.32 . (38)

Collecting these results, we obtain

Mi ≈ 7.1, Ji ≈ 5.9, M f ≈ 6.6, J f ≈ 4.2 . (39)

Assuming that there could be an uncertainty of 10% in thevalues of m f , v f and b f , we obtain that about 8+10−10% of thetotal mass-energy and 30+19−25% of the angular momentum areradiated.

Numerical tests.—In view of the very high computationalresources needed for the simulations presented in this work(about 2M CPU hours per run), a standard convergence anal-ysis with two additional simulations at higher resolution isprohibitively costly. In order to achieve a practical conver-gence study, we apply two steps that greatly reduce the ex-penses. First, we consider a quantity where the continuumlimit is already known, namely the Hamiltonian constraint.This reduces the number of required resolutions from threeto two. Second, instead of repeating the entire simulation,we instead use the fact that the code dynamically adds reso-lution in the critical regions by adding extra refinement lev-els. More specifically, we have performed the later stagesof the 6D-A simulation twice. In one case, the code adds,as instructed by the tagging criterion, an additional refine-ment level, the 9th level, that covers the black string regionand the satellites. In the second case, we have disallowedthe code from adding this level and instead let the simula-tion continue from its previous checkpoint with a mere 8 re-finement levels. These two instances of our simulation en-able us to analyse the dependence of the Hamiltonian con-straint on resolution in the vicinity of the high curvature re-gion. We note in this context that the constraint violations

9

FIG. 9. Left column: The Hamiltonian constraint for the 6D-A simulation in the xy plane at t = 22. The upper panel shows the constraintobtained using 8 refinement levels only. The constraint with an additional 9th level is shown in the center panel and, amplified by a factorQ2 = 4 for second-order convergence, in the bottom panel. Right column: A zoom-in of each of the left panels shows the constraint in asmaller region around the string. Here, we also display the grid structure in the form of a gray mesh.

10

rapidly decrease in the outer regions of our computationaldomain, as expected from the rapid fall-off of gravity inhigher-dimensional spacetimes.

In Fig. 9, we show the Hamiltonian constraint for the 6D-A simulation in the xy plane at t = 22. In the left column ofthis figure, we display the upper left half of the string onlyfor better image quality; the lower right half is obtained fromreflection across the bottom right corner. Let us first comparethe upper and center panels which are obtained using 8 and9 levels, respectively. We clearly see that the string shapedregion of large constraint violations is thicker for the lowerresolution in the upper panel. The resulting larger constraintviolations also extend into the outer regions as visible in theform of the cloud like structure surrounding the string, espe-cially in the upper half. Both these effects are visibly reducedby the higher resolution around the string in the center panel,although at a small price: the additional refinement bound-aries introduce some numerical noise visible in the form ofthe rectangular structures in the center panel.

In order to obtain a rough estimate of the convergence or-der, we show in the bottom left panel of Fig. 9 the higherresolution result amplified by a factor Q2 = 4 as expected forsecond-order convergence, which is the dominant discretiza-tion order in the prologation operation. This amplificationroughly widens the region of larger constraint violations inthe bottom panel to a level comparable to the top panel. Notethat we only expect this convergence in the immediate vicin-ity of the string, where extra refinement is present in the cen-ter and bottom panels.

In order to obtain a clearer illustration of the rate of con-vergence, we show in the right column of Fig. 9 the sameresult but now for a small region around the string, where weare now also able to display the grid structure. Again, theeffect of the additional resolution near the string in the lowertwo panels is clearly perceptible. The comparable width ofthe string in the top and bottom panel furthermore indicatesconvergence commensurate with 2nd order around the string.

∗ tandrade@icc.ub.edu† p.figueras@qmul.ac.uk‡ U.Sperhake@damtp.cam.ac.uk

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11

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Violations of Weak Cosmic Censorship in Black Hole collisionsAbstract Acknowledgments Supplementary material References