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6AEI-2006-018, LSU-REL-033006

Introduction to dynamical horizons in numerical relativity

Erik Schnetter,1, 2, ∗ Badri Krishnan,2, † and Florian Beyer2, ‡

1Center for Computation and Technology, 302 Johnston Hall,

Louisiana State University, Baton Rouge, LA 70803, USA§

2Max-Planck-Institut fur Gravitationsphysik, Albert-Einstein-Institut, Am Muhlenberg 1, D-14476 Golm, Germany¶

(Dated: May 31, 2006)

This paper presents a quasi-local method of studying the physics of dynamical black holes in nu-merical simulations. This is done within the dynamical horizon framework, which extends the earlierwork on isolated horizons to time-dependent situations. In particular: (i) We locate various kinds ofmarginal surfaces and study their time evolution. An important ingredient is the calculation of thesignature of the horizon, which can be either spacelike, timelike, or null. (ii) We generalize the calcula-tion of the black hole mass and angular momentum, which were previously defined for axisymmetricisolated horizons to dynamical situations. (iii) We calculate the source multipole moments of the blackhole which can be used to verify that the black hole settles down to a Kerr solution. (iv) We also studythe fluxes of energy crossing the horizon, which describes how a black hole grows as it accretes matterand/or radiation.

We describe our numerical implementation of these concepts and apply them to three specific testcases, namely, the axisymmetric head-on collision of two black holes, the axisymmetric collapse of aneutron star, and a non-axisymmetric black hole collision with non-zero initial orbital angular mo-mentum.

PACS numbers: 04.25.Dm, 04.70.Bw, 95.30.Sf, 97.60.Lf,

I. INTRODUCTION

In spite of fundamental advances in our understand-ing of black holes, relatively little is known about themin the fully non-perturbative, dynamical regime of gen-eral relativity. Most of our intuition regarding blackholes comes from studying the stationary, axisymmet-ric Kerr-Newman solutions, and perturbations thereof.This, along with post-Newtonian calculations whichtreat the black hole as a point particle, are usually ade-quate for understanding many astrophysical processesinvolving black holes. However, understanding thegravitational waveforms arising due to, say, the mergerphase of the coalescence of two black holes or the grav-itational collapse of a star, will require us to go beyondperturbation theory and to confront the non-linearitiesand dynamics of the full Einstein equations. This regimemay contain qualitatively new, non-perturbative fea-tures. In this paper, we discuss an important ingredi-ent for understanding this regime, namely, the dynam-ics of the black hole horizon. Numerical simulationsof black holes have greatly improved in the last fewyears. Simulations of the entire merger process, start-ing from the last few orbits of the inspiral right up tothe ringdown have become possible in the past year[1, 2, 3, 4, 5, 6, 7, 8]. It is then important to look for better

∗Electronic address: schnetter@cct.lsu.edu†Electronic address: badri.krishnan@aei.mpg.de‡Electronic address: florian.beyer@aei.mpg.de§URL: http://www.cct.lsu.edu/about/focus/numerical/¶URL: http://numrel.aei.mpg.de/; URL: http://www.aei.mpg.

de/

ways to extract more physical information from simu-lations and to compare results from two different sim-ulations performed using different coordinate systems,gauge conditions etc. This can be a non-trivial task initself, and understanding black hole horizons is a neces-sary ingredient.

Due to their global nature, black hole event hori-zons can only be located once a simulation is completeand we have obtained the full spacetime. In numeri-cal simulations, it is instead common to use marginallytrapped surfaces to locate black holes on a Cauchy sur-face in real time. We use the formalism of dynami-cal horizons [9, 10] to study black holes. Using iso-lated/dynamical horizons, it is shown that marginallytrapped surfaces, while not a substitute for event hori-zons, do have many useful properties and can be usedfruitfully to study black hole physics. Dynamical hori-zons are a significant extension of the isolated horizonframework [11, 12, 13, 14, 15], which models isolatedstationary black holes in an otherwise dynamical space-time. Both these frameworks are, in turn, very closelyrelated to and motivated by the earlier work on trappinghorizons by Hayward [16, 17, 18]. See [19, 20, 21] forreviews. Information obtained from these quasi-localhorizons complements the information obtained fromthe event horizon. Once a simulation is complete andready for post-processing, event horizons are useful forstudying global properties and the causal structure ofthe spacetime, and also phenomena such as the topol-ogy change of the horizon during a black hole coales-cence. Reliable and computationally efficient codes arenow available for locating event horizons (see e.g. [22]).Such information cannot be obtained at the quasi-locallevel, which is instead better for tracking the physical

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parameters and geometry of a black hole in real time.The dynamics of apparent and event horizons have

been numerically studied in the past in detail in axisym-metry (see e.g. [23, 24, 25, 26, 27, 28]). We want to extendthis work to non-axisymmetric and non-vacuum space-times, and we want to emphasise non-gauge-dependentanalysis methods. In particular, we consider the follow-ing applications: (i) We study the behavior of variousmarginally trapped surfaces under time evolution. Thisleads to greater insights about the trapped region of aspacetime. An important ingredient here is the signa-ture of the world tube of marginally trapped surfaces.This world tube is known to be null for isolated hori-zons, and more generally, it can be either spacelike ortimelike; we show that both types occur frequently innumerical simulations. (ii) We give meaningful defini-tions for the angular momentum, mass, and higher mul-tipole moments for the dynamical black hole. The mul-tipole moments capture gauge invariant geometrical in-formation regarding the horizon geometry, and shouldbe useful for understanding fundamental issues such asthe final state of black hole collapse. For example, wewould expect that after a black hole has formed and set-tled down, its multipole moments should be identical tothe source multipoles of a Kerr black hole. We show thatit is, in principle, possible to verify this conjecture and tocalculate the rate at which a black hole approaches equi-librium. (iii) We also describe and implement methodsfor calculating the energy flux falling into the horizon.This gives us detailed information on how black holesgrow as they swallow matter and radiation.

This paper is organized as follows. Section II setsup notation, and summarizes the basic definitions andproperties of trapped surfaces and dynamical horizons.Section III describes the various physical quantities thatwe calculate using dynamical horizons, and also theirnumerical implementation. Section IV presents threeconcrete, well known numerical examples where theseconcepts are applied and finally, section V discussessome open issues and directions for further work. Un-less mentioned otherwise, we use geometrical unitswith G = c = 1, the spacetime signature is (−,+,+,+),all manifolds and fields are assumed to be smooth, andthe Penrose abstract index notation is used throughout.The derivative operator compatible with the spacetimemetric gab is ∇a and, following Wald [29], the Riemann

tensor is defined via (∇a∇b −∇b∇a)ωc = Rabcdωd.

II. BASIC NOTIONS AND DEFINITIONS

A. Trapped surfaces and apparent horizons

Let S be a closed, orientable spacelike 2-surface ina 4-dimensional spacetime (M, gab). The expansion ofany such surface can be defined invariantly without anyreference to a time slicing of the spacetime. Since S issmooth, spacelike, and 2-dimensional, the set of vec-

tors orthogonal to it at any point form a 2-dimensionalMinkowskian vector space. Thus, we can define two lin-early independent, future-directed, null vectors ℓa andna orthogonal to S such that

gabℓanb = −1 . (2.1)

Note that this convention is different from that used in[10]. We shall assume that we know a priori what theoutgoing and ingoing directions on M are. By conven-tion, ℓa will denote an outgoing null normal and na aningoing one. The null normals are specified only up to aboost transformation

ℓa → f ℓa , na → f−1na (2.2)

where f is a, positive definite, smooth function on S. Allphysical quantities must be invariant under this gaugetransformation.

The Riemannian 2-metric qab on S induced by thespacetime metric gab is

qab = gab + ℓanb + naℓb . (2.3)

The tensor qba can be viewed as a projection operator on

to S. The null expansions are

Θ(ℓ) = qab∇aℓb , Θ(n) = qab∇anb . (2.4)

These expansions tell us how the area element of Schanges as it is deformed along ℓa and na respectively.

The shear of ℓa, σ(ℓ)ab, is the symmetric trace-free partof the projection of ∇aℓb:

σ(ℓ)ab = qca qd

b∇(cℓd) −1

2Θ(ℓ)qab . (2.5)

Similarly, the shear of na is

σ(n)ab = qca qd

b∇(cnd) −1

2Θ(n)qab . (2.6)

Note that these definitions only involve derivatives tan-gential to S. Thus ℓa and na can, if necessary, be ex-tended arbitrarily away from S while computing thesequantities.

The closed 2-surface S is said to be a trapped surfaceif both expansions Θ(ℓ) and Θ(n) are strictly negative.This is very different from a sphere in normal flat spacewhich has positive outgoing expansion and negative in-going expansion. This definition was first introducedby Penrose [30], who recognized its importance in theformation of singularities. On a marginal surface, oneof the two null expansions vanish. Of particular in-terest are the marginally outer trapped surfaces (MOTSs),for which the outgoing null rays along ℓa have zero ex-pansion. In addition, we shall mostly deal with futuremarginally outer trapped surfaces (FMOTSs), i.e., MOTSswith Θ(n) < 0.

There are three main reasons why closed trapped sur-faces are important for studying black holes. First, the

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existence of a trapped surface implies the existence ofa singularity in the future [30, 31]. Secondly, they areguaranteed to always lie within the event horizon. Fi-nally, in stationary spacetimes, the null generators of theevent horizon have zero expansion. Thus for stationaryspacetimes, the cross-section of the event horizon is aMOTS.

While trapped and marginally outer trapped surfacesare defined in the full four dimensional spacetime, innumerical relativity, one usually considers trapped sur-faces in conjunction with a foliation of (partial) Cauchysurfaces containing S; it is numerically much easier tolook for closed surfaces on the Cauchy surface ratherthan in the full spacetime manifold. For concreteness,we shall work in the ADM formalism where the rele-vant portion of spacetime is foliated by spacelike sur-faces, and Σ shall denote one of the leaves of this folia-tion. However, it will be obvious that the formalism isapplicable no matter how Einstein’s equations are im-plemented.

The trapped region TΣ on Σ is defined to be the set ofpoints in Σ through which there passes a trapped sur-face contained entirely in Σ. Note that there could bepoints in Σ not contained in TΣ, but through which therepasses a trapped surface not contained in Σ. Thus, TΣ isa subset of the intersection of Σ with the 4-dimensionaltrapped region in the full spacetime. A connected com-ponent of the boundary of TΣ is called an apparent hori-zon (AH). Under suitable regularity conditions, the AHcan be shown to be a MOTS [32, 33]. Thus, an appar-ent horizon is the outermost MOTS on Σ. Due to this“outermost” property, an AH is not a quasi-local objecton Σ. The behavior of AHs under time evolution can bequite irregular. For example, they can “jump” discontin-uously. On the other hand, as we shall soon see, MOTSsare more regular.

B. Dynamical horizons

1. Definition and examples

We can use marginal surfaces to extract physically in-teresting information about the black hole. The key ideais to look not at a single MOTS by itself, but rather aworld tube H of MOTSs constructed by stacking up theMOTSs obtained by time evolution. Such a world tube iscalled a Marginally Trapped Tube (MTT). An MTT is thusa smooth 3-surface foliated by MOTSs.

The existence of MTTs: Numerically, it has been ob-served that marginal surfaces (though not apparenthorizons — see below) usually behave smoothly undertime evolution and produce a smooth MTT. This obser-vation is placed on a more rigorous footing by the re-cent result of Andersson et al. [34], which proves the lo-cal existence of MTTs for a large class of MOTSs. Theirresults require the MOTS to be strictly-stably-outermost.An MOTS S on Σ is said to be strictly-stably-outermost

if there exists an infinitesimal first order outward defor-mation which makes S strictly untrapped. Working witha radial coordinate r on Σ such that S is a level set ofr, and r increases in the outward direction, a sufficient(but not necessary) condition for S to be strictly-stably-

outermost is ∂rΘ(ℓ)(r) > 0 everywhere1 on S. Here itis understood that we obtain Θ(ℓ) as a function of r bycalculating Θ(ℓ) for the constant-r surfaces in the vicin-ity of S. In principle, for an unfortunate choice of r, itmight happen that ∂rΘ(ℓ) < 0 even though there is adifferent choice for which this condition is satisfied. Inany case, this is sufficient for verifying that S is strictly-

stably-outermost.2 This condition, unlike the outermostcondition for an AH, is a quasi-local condition. We havefound in our simulations that most physically interest-ing MOTSs, such as ones which asymptote to the eventhorizon, and also AHs, satisfy this condition quite gen-erally. However, as we shall see, there exist also MOTSswhich are not strictly-stably-outermost. In practice, in-stead of checking ∂rΘ(ℓ) > 0 directly, we look for a sur-face with a small positive (or negative) non-vanishingexpansion, and check that it lies completely outside (orinside) the MOTS.

It is shown in [34] that if a MOTS S is strictly-stably-outermost, then at least locally in time, S is a cross-section of a smooth MTT. More explicitly, this resultshows that given a foliation of the spacetime by Cauchysurfaces Σt, if there is a MOTS S0 on Σ0 which is strictly-stably-outermost, then MOTSs St exist on Σt for −ǫ <

t < ǫ (for sufficiently small ǫ) such that the union⋃

St

is a smooth MTT. The MTT will exist for at least as longas the MOTS remains strictly-stably-outermost. This isa conceptually important result for numerical relativitybecause it shows that a large class of MOTSs behave reg-ularly under time evolution. How is this to be reconciledwith the known fact that AHs can “jump” during a timeevolution? The reason is simply because of the outer-most property. It is possible that a new MOTS can ap-pear on the outside of a given MOTS. The “old” MOTS isthen no longer the globally outermost one even thoughit is locally outermost, and it continues to evolve in aperfectly regular manner, but it is no longer an AH.

There are, as yet, no similar existence proofs forMOTSs which are not strictly-stably-outermost. How-ever, as we shall see later, we find in all the exampleswe have looked at, that MOTSs evolve smoothly evenin this case, forming a regular world tube.

Isolated and dynamical horizons: An MTT is null in equi-librium situations when no matter or radiation is fallinginto it; the rest of the spacetime is still allowed to be

1 More precisely, ∂rΘ(ℓ)(r) ≥ 0 with ∂rΘ(ℓ)(r) > 0 somewhere on S.2 It is harder to show that a MOTS is not strictly-stably-outermost.

This can be done by calculating the signature of the horizon (seebelow) or by calculating the principle eigenvalue of the stability op-erator defined in [34].

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highly dynamical. This situation is formalized by thenotion of an isolated horizon [11, 12, 13, 14, 15]. Us-ing isolated horizons, it has been possible to derive thelaws of black hole mechanics, use it as a basis for thequantum black hole entropy calculations and find unex-pected properties of hairy black holes in Einstein-Yang-Mills theory; see [19] and references therein. Most im-portantly for our purposes, isolated horizons have alsoproved to be useful in numerical relativity. For exam-ple, isolated horizons provide a coordinate invariantmethod of calculating the angular momentum and massof a black hole [35]. They can be used to obtain bound-ary conditions for constructing quasi-equilibrium initialdata sets [36, 37, 38, 39]. They might have a role in wave-form extraction [15]. A pedagogical review of isolatedhorizons from the numerical relativity perspective canbe found in [21].

In this paper, we are more interested in the dynamicalregime when the MTT is not null. A spacelike MTT con-sisting of future-marginally trapped surfaces is calleda Dynamical Horizon (DH). Thus, a dynamical horizonis a spacelike 3-surface equipped with a given foliationby FMOTSs. The properties of a dynamical horizon arestudied in detail in [9, 10, 40]. The case when the horizonis very close to being isolated but still evolving dynam-ically has been studied in [41, 42] and its Hamiltoniantreatment is considered in [43]. Note that the local ex-istence of DHs follows from the local existence of MTTsbecause if Θ(n) < 0 at any given time, it will continueto be strictly negative for at least a short duration. Weelaborate on the spacelike property below.

A timelike MTT will be called a timelike membrane(TLM). A TLM cannot be considered to represent thesurface of a black hole since a time-like surface is nota one-way membrane, and both ingoing and outgoingcausal curves can pass through it. In some instances, weshall use the term “horizon” loosely to refer to a genericmarginal surface or a MTT without any further quali-fiers. The exact meaning should hopefully be clear fromthe context.

An explicit example of a dynamical horizon is pro-vided by the Vaidya spacetime which describes thegravitational collapse of null dust [44, 45, 46]. (Seealso [47] for further examples in spherically symmetry).More generally, figure 1 depicts a dynamical horizon Hbounded by two MOTSs S1 and S2. S is a typical mem-ber of the foliation. The vector τa is the future directedunit timelike normal to H, ra is tangent to H and isthe unit outward pointing spacelike normal to the cross-sections. A fiducial set of null normals is

ℓa =

1√2(τa + ra) , na =

1√2(τa − ra) . (2.7)

As before, Θ(ℓ) = 0 and Θ(n) < 0. The area of a cross-section S will be denoted by AS and its radius by RS :=√

AS/4π. A radial coordinate on H will be denoted byr; the cross sections of H are the constant r surfaces. The3-metric and extrinsic curvature of H will be denoted

PSfrag replacements

τa

S1

S2

nara

ℓaH

S

Ta

RaΣ

FIG. 1: A dynamical horizon H bounded by MOTSs S1 and S2.ℓa is the outgoing null normal, na is the ingoing null normal, ra

is the unit spacelike normal to the cross-sections, and τa is theunit timelike normal to H. Σ is a Cauchy surface intersectingH in a 2-sphere S. Ta is the unit timelike normal to Σ and Ra

is the unit space-like outward pointing vector normal to S andtangent to Σ.

respectively by qab and Kab, and qab is the 2-metric on S.

Figure 1 shows also a Cauchy surface Σ intersectinga dynamical horizon H. This intersection S will alwaysbe assumed to be one of the given cross-sections of H.The unit timelike normal to the horizon is Ta and theunit outward pointing spacelike normal to S within Σ

is Ra. The three metric and extrinsic curvature of Σ aredenoted by qab and Kab respectively. The fiducial set ofnull normals to S arising naturally from Σ are

ℓa =

1√2(Ta + Ra) , na =

1√2(Ta − Ra) . (2.8)

A boost transformation of the form of equation (2.2) con-nects (ℓa, na) and (ℓa, na):

ℓa = f ℓa , na = f−1na . (2.9)

When the horizon settles down and becomes null, aninfinite boost ( f → ∞) is required to go from (ℓa, na) to(ℓa, na).

We conclude this sub-section with a short summary ofsome basic properties of a dynamical horizon:

Topology: The cross-sections of a DH can be eitherspherical or toroidal [9, 10, 16, 34]. Toroidal topol-ogy is possible only in exceptional cases when

σ(ℓ)ab, the scalar curvature R of S, LℓΘ(ℓ), Rabℓb,

and ζa (defined in section III) all vanish on S [10].We shall therefore always take the cross-sections tobe spherical. There are no similar results for cross-sections of TLMs. However, we use an apparenthorizon tracker which can only locate sphericalAHs [48] and therefore all observed MOTSs havespherical topology.

Second Law: The area of the cross-sections of a DH in-creases along ra [9, 10]. Thus, if we choose a timeevolution vector field ta for which t · r > 0, thenthe area of the dynamical horizon will increasein time, and this result can be called the second

5

law for dynamical horizons. Similarly, the areaof a TLM decreases if Θ(n) < 0, and increases ifΘ(n) > 0.

Foliation and Uniqueness: Any given spacelike MTTcannot have more than one distinct dynamicalhorizon structure on it [40]. This means that a DHcan have one, and only one foliation by FMOTSs.This further implies that if a Cauchy surface Σ

does not intersect a given DH in one of the pre-ferred cross-sections, then the intersection can-not be a MOTS at all. Thus, different choices ofCauchy surfaces in general lead to different dy-namical horizons. There are however some con-straints on the location of dynamical horizons andtrapped surfaces as proved by Ashtekar and Gal-loway [40]. For example, they show that given adynamical horizon H (along with a mild generic-ity assumption), there cannot be any trapped sur-faces (and therefore no DHs) contained entirely inthe past domain of dependence of H. See also[46, 49] for further discussion.

2. The signature of a MTT

As discussed above, MTTs have been shown to existfor a large and physically interesting class of MOTSs,and this is borne out in a large number of numeri-cal simulations where MOTSs are located and evolvedsmoothly. How many of these MTTs are actually dy-namical horizons? In other words, when is a MTT space-like? The first result in this direction was obtained byHayward [16] (see also [35]). Using the Raychaudhuriequation for ℓa, it can be shown that an MTT is space-like if α < 0, null if α = 0 and timelike if α > 0, where

α ≡σ(ℓ)abσab

(ℓ) + Rabℓaℓb

LnΘ(ℓ). (2.10)

In writing this expression, it is assumed that ℓa andna are extended off H geodetically, so that LnΘ(ℓ) ismeaningful. The term in the numerator is strictly pos-itive in the case of dynamical horizons if the matterfields satisfy, say, the null energy condition. It vanishesfor isolated horizons. The denominator is negative forthe Vaidya spacetime and also for the stationary Kerr-Newman family. This captures the notion that as wego inside the black hole, the outgoing null rays becomemore and more converging. Assuming that the numer-ator of Eq. (2.10) is nowhere vanishing on H, the hy-pothesis that H is spacelike is equivalent to LnΘ(ℓ) < 0.As shown by Ben-Dov [50], this last condition is notsatisfied for all MTTs; in Oppenheimer-Snyder collapse[51], there exists a timelike world tube of FMOTSs withLnΘ(ℓ) > 0.

The issue of the signature has been considered in [34].There it is shown that if a MOTS S is strictly stably out-

ermost, and if the quantity σ(ℓ)abσab(ℓ) + Rabℓ

aℓb is non-

zero somewhere on S (and assuming the null energy con-dition), then the MTT containing S is spacelike in aneighborhood of S. This result is stronger than Hay-ward’s result (Eq. (2.10)) and it shows clearly that thespacelike case is physically the most interesting because

σ(ℓ)abσab(ℓ) + Rabℓ

aℓb will not vanish in a non-stationary

situation. It also shows, somewhat surprisingly, thateven if matter or radiation is falling into a black holeonly in the form of say, a single narrow beam from a par-ticular direction, the entire MTT is spacelike. One mightnaively have thought that the MTT would be spacelikeonly on portions where the energy flux is non-zero, andnull otherwise. This is not the case because of the ellipticnature of the equations governing the deformations of a

MOTS. 3

In all the examples we present later, it turns out thatMOTSs form in pairs, i.e., just after a MOTS S0 appearsinitially, it bifurcates into “outer” and “inner” MTTs,Hout and Hin respectively. The initial MOTS S0 is thecommon cross section of Hout and Hin, and the unionHtot = Hout

⋃

Hin forms a single smooth manifold, as faras we can tell numerically (though a more detailed anal-ysis of the differentiability of Htot is required). In par-ticular, the area of the cross-sections is a differentiableand monotonic function on this manifold. Furthermore,Hout is spacelike, even on the initial cross-section S0.This means that the inner MTT Hin is, by continuity, ini-tially spacelike in an open neighborhood of S0. How-ever, in some cases Hin soon acquires a mixed signatureand becomes more and more timelike, and ends up asa TLM. We strongly suspect that such a bifurcation is ageneral phenomenon whenever a new MOTS is formed.The MOTSs on the inner MTT are not strictly-stably-outermost and thus Hin is not required to be spacelikeaccording to the results of [34].

There is one configuration where the existence of aninner MTT is plausible. Figure 2 shows two MOTSsS(1),(2) surrounded by a common MOTS Sout; Θ(ℓ) van-ishes on all these surfaces. Let us assume that S(1), S(2),and Sout are all strictly-stably-outermost and that de-forming S(1) and S(2) outward yields strictly untrapped

surfaces S′(1) and S′

(2). Similarly, suppose that deforming

Sout inwards gives a strictly trapped surface S′out. Then,

since Θ(ℓ) must change sign somewhere between S′out

and S′(1) or S′

(2), it is plausible that there is a MOTS Sin in

the intermediate region inside Sout and outside S1 andS2. This argument is supported by a recent result bySchoen [52] which shows the existence of a MOTS be-

3 The Oppenheimer-Snyder case studied in [50] does not satisfy thehypotheses of these theorems because for this case, the matter fieldshave a discontinuity at the surface of the star. Further examples inspherical symmetry are studied in [47] where the matter fields aresmooth.

6

PSfrag replacements

SoutSin

S(1)

S(2)

ra

FIG. 2: Two MOTSs S(1) and S(2) surrounded by a commonMOTS Sout. Spheres lying just inside these FMOTSs must havenegative outgoing expansion. Thus, there must be a innertrapped horizon Sin inside Sout which encloses S(1) and S(2).

tween a trapped (in our case S′out) and an untrapped sur-

face (in our case S′(1)

⋃

S′(2)). It might be possible to ex-

tend this proof to rigorously prove the existence of Sinin our case, and to check whether it is topologically asphere. S(1), S(2), and Sout are cross sections of a dynam-ical horizon while Sin is a cross-section of an MTT, notnecessarily a dynamical horizon.

III. APPLICATIONS

This section discusses some possible applications ofdynamical horizons. These ideas are illustrated usingconcrete numerical examples later in Section IV.

A. Computing the signature of a MTT

From a numerical standpoint, it is more convenientto deduce the signature of H by directly calculating theinduced metric qab, rather than from Eq. (2.10) by cal-culating LnΘ(ℓ) which requires extensions of ℓa and na

away from the horizon. The signature of H is then de-termined by the sign of the determinant of qab whichis gauge independent; note that the determinant is it-self gauge dependent. To calculate qab we find a frameea(i)

(i = 1, 2, 3) on H, i.e., three smooth vector fields on

H which are pointwise linearly independent. We thensimply need to compute the determinant of the matrix

q(i)(j) := gabea(i)e

a(j) . (3.1)

We construct a frame on H as follows. Let (t, xi)(i = 1, 2, 3) be the spacetime coordinates on M used inthe numerical simulation. The MTT H is topologically

I × S2 (I some interval in R) so that we can assume co-ordinates (r, θ, φ) on it. Here (θ, φ) are standard coor-dinates on S2 and r is a radial coordinate. We can use

the time coordinate t as the radial coordinate r on H byconsidering H to be embedded into the spacetime M bymeans of the map

F(r, θ, φ) = (t = r, xi = Fi(r, θ, φ)) . (3.2)

The maps Fi are known as soon as the MOTSs are foundby the AH tracker. As a frame on H we choose

e(1) = ∂θ , e(2) =1

sin θ∂φ, e(3) = ∂r. (3.3)

Hence, e(3) connects a point on a MOTS at a certain in-stant of time with a corresponding point on the MOTSat the next instant of time. Note that this choice of framebreaks down at the poles of the sphere. To apply for-mula (3.1), the frame (3.3) on H must be pushed forwardto M by means of the embedding F in the standard way:

e(3) = (1, ∂rF1, ∂r F2, ∂r F3). (3.4)

This enables us to calculate q(i)(j) using the 3-metric on

the Cauchy surface, and the lapse and shift.Having calculated the matrix q(i)(j) and assuming

its determinant to be positive, we can easily calculatethe unit vector ra. It is simply the outward pointingunit spacelike vector which is a linear combination of(e(1), e(2), e(3)), and is orthogonal to e(1) and e(2). Thisconstruction of ra will also work in the timelike case, butnot in the null case where q(i)(j) becomes degenerate.

B. Angular momentum and mass

Let ϕa be a rotational vector field on H tangent to

each cross-section.4 The angular momentum of a cross-section S associated with ϕa is given by

J(ϕ)S = − 1

8π

∮

SKabϕa rbd2V . (3.5)

We refer to [10] for a justification for this formula. The

interpretation of J(ϕ)S as angular momentum is most

clear cut when ϕa is a rotational symmetry on H, i.e.,when LϕKab = 0 and Lϕqab = 0. See [35] for a methodof finding Killing vectors suitable for numerical imple-mentation. Booth and Fairhurst have shown that thisformula also arises from a Hamiltonian calculation [43].

As we shall see below, J(ϕ)S is also gauge invariant when

ϕa is only divergence free, and not necessarily a sym-

metry vector. However, J(ϕ)S may not be meaningful for

more general ϕa.

4 This means that ϕa is tangent to S, has closed integral curves, andis normalized so that its integral curves have an affine length of 2π,and it vanishes at exactly two points on S.

7

If a cross-section S has radius RS and angular momen-

tum J(ϕ)S , we can meaningfully talk about the mass:

M(ϕ)S =

1

2RS

√

R4S + 4(J

(ϕ)S )2 . (3.6)

This mass has the same dependence on the area andangular momentum as in the Kerr solution. There is ameaningful balance law for the mass and furthermore, itsatisfies a physical process version of the first law [9, 10].

Equation (3.5) uses the metric qab and Kab and extrin-sic curvature of the dynamical horizon. It is more con-venient to recast this in terms of the metric qab and ex-trinsic curvature Kab of the partial Cauchy surface Σ (seefigure 1). It is also more convenient to work with the nullnormals (ℓa, na) defined in equation (2.8). It is clear that(ℓa, na) must be related to the old null normals (ℓa, na)by a boost transformation, i.e., there must exist a posi-tive function f on S such that

ℓa = f ℓa and na = f−1na . (3.7)

After some simple algebra, equation (3.5) can be writtenas:

J(ϕ)S = − 1

8π

(∮

SKabRa ϕb d2V +

∮

SLϕ ln f d2V

)

.

The second integral vanishes precisely when ϕa is di-vergence free, i.e., when ϕa is a symmetry of the areaelement on S. In this case:

J(ϕ)S = − 1

8π

∮

SKabRa ϕb d2V . (3.8)

In particular, this will be true when ϕa is a symmetryof the metric qab, but the divergence free condition ismuch weaker than this. For example, following [19],we can always construct a divergence free vector fieldon a 2-sphere even in the absence of axisymmetry asfollows. Let h be any smooth function on S, and g an-

other smooth function satisfying ǫab∂ah∂bg = 0, whereǫab is the volume form on S. It is easy to check explic-itly that the following vector field is automatically di-vergence free:

ϕa = gǫab∂bh . (3.9)

The integral curves of ϕa are the level curves of h. In par-ticular, if h is chosen to be a geometric quantity such as,say, the curvature R, and g chosen such that ϕa has affinelength 2π, then ϕa will coincide with an axial Killingvector, if it exists. Therefore, ϕa can be viewed as an er-satz axial symmetry vector field even in the absence ofaxisymmetry.

However, we haven’t as yet satisfactorily imple-mented the above construction due to numerical diffi-culties arising from errors in taking derivatives of thescalar curvature. Furthermore, the ϕa coming from eq.

(3.9) may not look like a rotational vector field; in par-ticular it may vanish at more than just two points on the

sphere even when S is close to axisymmetry.5 This iswork in progress. The results presented below all usethe method described in [35] of finding Killing vectorsbased on the Killing transport equations. This reducesthe problem of finding Killing vectors on a sphere tothe diagonalization of a 3 × 3 matrix, and integrating a1-dimensional ordinary differential equation. We havefound this method to be quite reliable for the cases whenthe horizon is sufficiently close to axisymmetry, evenin cases when the coordinate system is not adapted tothe axial symmetry. Thus, it works well for the head-on collision and axisymmetric neutron star collapse, butonly at very early and late times for a non-axisymmetricblack hole collision. This caveat only affects the exam-ple of section IV B. It is important to keep in mind thatthis Killing transport method is not reliable for check-ing whether the horizon is close to axisymmetry; this re-quires an independent calculation of Lϕ qab to verify thatit is sufficiently small. Finally, we emphasize that thismethod is also not guaranteed to produce a divergencefree rotational vector field; this must also be checked in-dependently.

C. Multipole moments

The notion of multipole moments play a very impor-tant role in Newtonian gravity and classical electrody-namics. Let us focus on classical electrodynamics inMinkowski space with axisymmetric charge and currentdistributions ρ and ja respectively, given on a sphere S ofradius RS. Let (θ, φ) be coordinates on S; ρ and ja, beingaxisymmetric, are functions only of θ. The electric mul-tipoles En and magnetic multipoles Bn are respectivelydefined as

En = RnS

∮

ρPn(cos θ)d2V , (3.10)

Bn = −Rn+1S

∮

S

(

~j ×~∂Pn(cos θ))

· n d2V , (3.11)

where Pn is the nth Legendre polynomial, ∂ denotes thestandard derivative operator on a sphere, and n is theunit outward normal to the sphere. For black holes, theanalogs of the electric and magnetic multipole momentsare respectively the mass and angular momentum mul-tipole moments. Motivated by this analogy, there ex-ist meaningful definitions of the source multipole mo-ments for an isolated horizon [53]. Roughly speaking,these definitions correspond to taking the moments ofthe free data on an axisymmetric isolated horizon, andknowledge of these moments is sufficient to constructthe entire horizon geometry.

5 We thank Ivan Booth for this comment.

8

For dynamical horizons, we can generalize the con-struction of [53] to construct a set of multipole momentswhich capture the geometry of a dynamical horizon atany instant of time, and which are furthermore equal tothe isolated horizon multipole moments when the blackhole is isolated. The analog of charge density is (propor-tional to) the scalar curvature on S:

ρS =1

8πMSR , (3.12)

and the angular momentum current is

ja = − 1

8πqc

aKcbRb . (3.13)

The moments of these quantities will give the desired

multipole moments. We could also use qcaKcbrb instead

of qcaKcbRb above; the two expressions are related by a

boost transformation. Just as for angular momentum,the final expressions for the multipole moments givenbelow will be boost invariant if the ϕa used in their def-inition is divergence free. To define the moments, weneed a preferred coordinate system on S so that we candefine the preferred spherical harmonics.

The construction of the preferred coordinate system(θ, φ) on S is the same as given in [53]: φ ∈ [0, 2π) isthe affine parameter along ϕa and ζ := cos θ ∈ [−1, 1] isdefined by the condition

Daζ =1

R2S

ǫbaϕa . (3.14)

The freedom to add a constant to ζ is removed by requir-

ing its integral over S to vanish:∮

S ζ d2V = 0. When ap-plied to a Kerr black hole, these invariant coordinatesturn out to be the same as the usual Boyer-Lindquist(θ, φ) coordinates.

The mass and angular multipole moments are then re-spectively:

Mn =Rn

S MS

8π

∮

S

RPn(ζ)

d2V , (3.15)

Jn = −Rn+1S

8π

∮

S

ǫab(∂bPn(ζ))KacRc

d2V

=Rn−1

S

8π

∮

SP′

n(ζ)Kab ϕaRb d2V (3.16)

where P′n(ζ) = dPn(ζ)/dζ. We have used equation (3.14)

to obtain the final expression for Jn above. This formclarifies the relation of Jn to the angular momentum andalso demonstrates the gauge invariance of Jn when ϕa isdivergence free. Using the Gauss-Bonnet theorem, it istrivial to check that M0 = MS and J1 = JS. J0 vanishesbecause we do not consider any topological defects. Fur-thermore, these expressions are well suited for numeri-cal computation because they involve only quantities onthe Cauchy surface and an integral over the MOTS.

D. The energy and angular momentum fluxes

Hawking’s area theorem shows that if matter satis-fies the null energy condition, then the area of the eventhorizon can never decrease. This is one of the centralresults of black hole physics, and it leads to the classicalpicture of the black hole growing inexorably as it swal-lows matter and radiation. Therefore, one might expectthere to be a balance law relating the increase in area tofluxes of matter and radiation crossing the event hori-zon. However, the teleological nature of event horizonsis again a problem; there cannot exist any such local bal-ance law for the area of the event horizon. A clear ex-ample is seen in the Vaidya spacetime where the eventhorizon is formed in flat space and its area increases inanticipation of matter falling into the black hole at a latertime; see [19] for a discussion.

For DHs, it is possible to obtain an exact balance lawfor the area increase [9, 10]; i.e., given two cross-sectionsS1 and S2 with radii R1 and R2 respectively, and withS2 lying to the outside of S1, the increase in the radiusis given by the sum of the energy flux due to matter

(F (m)) and gravitational radiation (F (g)), both of whichare manifestly positive.:

R2 − R1

2= F (m) + F (g) , (3.17)

where

F (m) =

∫

H

√2Tabτa

ℓbdR d2V , (3.18)

F (g) =1

8π

∫

H

|σ(ℓ)|2 + |ζ|2

dR d2V . (3.19)

Here |σ(ℓ)|2 := σ(ℓ)abσab(ℓ), |ζ|2 := ζaζa where ζa is a vector

on S defined as

ζa :=√

2qabrc∇cℓb , (3.20)

and d2V is the natural geometric volume element on H.

The extra factors of 2 and√

2 in the above equations ascompared to the corresponding equations in [10], arisebecause of our normalization convention ℓ · n = −1; [10]uses ℓ · n = −2.

See [10] for additional reasons why F (g) has the rightproperties to be viewed as the flux of gravitational radi-ation. Equation (3.17) is an exact statement about blackholes in full non-linear general relativity, and it is theanalog of the Bondi mass balance law at null infinity.

From a numerical point of view, F (g) is inconve-nient to calculate, especially when the horizon is settlingdown and is close to being null. First of all, we have di-rect access only to the fiducial null normals (ℓa, na) de-fined in eq. (2.8) and not to (ℓa, na) themselves. The twosets of null normals are related to each other by a boost

transformation ℓa = f ℓa, n = f−1na. Under this trans-formation, σℓ = f σℓ. Similarly, it is easy to show that

9

ζa = f 2κa − ωa , (3.21)

where

κa = qabℓ

c∇c ℓb and ωa = qabnc∇c ℓb . (3.22)

Here κa and ωa are tangent to the cross-sections of theDH. When the DH approaches equilibrium, f → ∞.

However, the value of F (g) itself remains finite. Allfields with a bar remain finite even when the horizonbecomes null even though f diverges While this is nota problem analytically, this does cause numerical errorsin the transition to equilibrium when we multiply a verysmall quantity on the horizon with a very large one. Thisis consistent with the results of [41] where it is foundthat |σ(ℓ)|2 is the most important when the horizon is

close to equilibrium.Let t be the time coordinate used to label the Cauchy

surfaces. Using this coordinate, we can identify the di-

vergence of various terms appearing in F (g). We start

by rewriting F (g) as:

F (g) =1

8π

∫

H

|σ(ℓ)|2 + |ζ|2 dR

dtd2V dt . (3.23)

The integrand on the right hand side can be expandedas

(

|σ(ℓ)|2 + |ζ|2)

R =

R f 4|κ|2 + R f 2(|σ(ℓ)|2 − ω · κ) + R|ω|2 . (3.24)

Let us look at the various terms in this expression. First,ωa can be shown to be equal to the angular momentumcurrent; for an axial symmetry vector ϕa, the angularmomentum is simply the integral of ϕaωa over the crosssection of the MTT. Thus, ωa need not vanish even whenthe MTT becomes an isolated horizon. The |ω|2 term inthe flux can, in some sense, be viewed as the flux of ro-tational energy entering the horizon. Now consider κa.

For an isolated horizon, ℓb∇b ℓa ∝ ℓa because in this case

ℓa is guaranteed to be geodetic. This implies κa = 0.On the dynamical horizon side, we can choose suitableextensions of ℓa (and na) away from the MTT so thatκa = 0. The shear σ(ℓ) on the other hand contains most

of the non-trivial information about the radiation fallinginto the black hole. It vanishes on an isolated horizonas it should, and it is independent of any extensions ofℓa, na away from the MTT. Therefore, in the examples ofsection IV, we shall usually plot σ(ℓ) to show the energy

flux falling into the horizon.The angular momentum also obeys a balance law sim-

ilar to equation (3.17):

J2 − J1 = J (m)ϕ + J (g)

ϕ (3.25)

where

J (m)ϕ = −

∫

∆HTabτaϕbd3V , (3.26)

J (g)ϕ = − 1

16π

∫

∆HPabLϕqabd3V (3.27)

where Pab := Kab − Kqab. Unlike the energy flux F (g),

the angular momentum flux J (g) is not positive definite.

Also, J (g) vanishes when ϕa is an axial Killing vectoron H. Thus, angular momentum is conserved in the ax-isymmetric vacuum case, as it should be.

IV. EXAMPLE NUMERICAL SIMULATIONS

In this section, we apply the ideas discussed in theprevious sections to three concrete numerical simula-tions: i) A head-on collision of two black holes startingwith Brill-Lindquist initial data; ii) A non-axisymmetricblack hole collision using puncture initial data with non-vanishing linear momentum and iii) Axisymmetric col-lapse of a neutron star. Each of these three cases is quitewell known in the numerical relativity literature, and allhave been well studied. This section aims to further ex-plore these examples using the tools described in Sec-tion III.

A. Head-on collision with Brill-Lindquist data

The Brill-Lindquist initial data [54] for binary blackholes represent initial data for two non-spinning blackholes without any orbital angular momentum. Thereader can consult a review on initial data, such as [55]for details. Here we simply note that these initial dataare conformally flat and time-symmetric:

qab = ψ4δab , Kab = 0 . (4.1)

The manifold Σ is R3 with two points removed (the

punctures). The only equation to be solved is the flatspace Laplace equation for the conformal factor:

∆ψ = 0 . (4.2)

Let d denote the shortest distance between the twopunctures as measured with respect to the fictitious flatbackground metric δab; the physical proper distance be-tween the punctures is actually infinite. It was shown in[54] that each of the punctures is actually an asymptot-ically flat region. The total ADM mass of the commonasymptotic region is

mADM = 2α(1) + 2α(2) , (4.3)

10

and the ADM masses of the two punctures are

mADM

(1) = 2α(1) +2α(1)α(2)

d(4.4)

mADM

(2) = 2α(2) +2α(1)α(2)

d. (4.5)

(4.6)

These are exact results, irrespective of the distance d be-tween the punctures. In the next two sub-sections, welook at two different regimes (i) the far limit when d islarge and (ii) the merger of the two holes starting fromrelatively small values of d.

1. The far limit

Before presenting the results from the numerical evo-lution of this data, it is instructive to look at a specialcase which is amenable to analytic treatment, namely,in the far limit where the separation between the holesis very large: d ≫ α(1), α(2). In this case, there aretwo MOTSs surrounding each of the punctures with-out any common MOTS surrounding them. The angu-lar momenta of the two black holes are trivially zerobecause the extrinsic curvature vanishes. What aboutthe mass? Should mADM

(1) and mADM

(2) be identified with the

masses of the black holes? There are three difficultieswith this. First, these ADM masses also include contri-butions from radiation present in the respective asymp-totic regions. Secondly, if this identification is correct,mADM

(i) (i = 1, 2) is supposed to be the mass of the black

hole for all values of d, even when the two black holesare very close to each other. Shouldn’t the mass of theblack holes in this regime also include, say, contribu-tions from the tidal distortions produced by the otherhole? Finally, the strategy of using the asymptotic re-gions to define black hole masses is not applicable gen-erally, say in the case when there are matter fields and

the topology of Σ is just R3, or in Misner data [56] where

the two black holes do not have their own individualasymptotic regions.

From the isolated/dynamical horizon perspective,since the black holes have zero angular momentum,from equation (3.8), the irreducible mass is the correct

measure of mass in this case: m(i) =√

a(i)/16π where

a(i) is the area of the MOTS around each of the punc-tures. Let us then calculate the mass of the black holes asa power series in 1/d. To simplify calculations, put theorigin of coordinates at the location of the first punctureand the other puncture on the z-axis at (0, 0, d). Intro-duce the usual spherical coordinates (r, θ, φ) so that theconformal factor becomes explicitly

φ(r, θ) = 1+α(1)

r+

α(2)

r

(

1 − 2d cos θ

r+

d2

r2

)− 12

. (4.7)

We see that due to axisymmetry, there is no dependenceon φ. Let the surface of the FMOTS around the origin begiven by the equation r = h(θ). In the limit when d →∞, the initial data reduces to Schwarzschild in isotropiccoordinates so that the horizon is located at r = α(1).

Higher order effects can also be explicitly calculated.

It turns out [57] that up to O(d−3), the location of theMOTS is given by

r = α(1) −α(1)α(2)

d+

α(1)α(2)

d(α(2) − α(1) cos θ)

−α(1)α(2)

3

(

α2(2) − 3α(1)α(2) cos θ

+5

7α2(1)P2(cos θ)

)

+O(d−4) (4.8)

where P2 is the second Legendre polynomial. Using this

result, the horizon mass m(i) =√

a(i)/16π can be calcu-

lated and, somewhat surprisingly, the mass is the sameas the ADM mass even up to third order:

m(1) = 2α(1) +2α(1)α(2)

d+O(d−4) . (4.9)

This relation was verified numerically for a sequence ofBL data with different values of d. However, we did nothave sufficient resolution to estimate the leading orderdeviation between m(1) and mADM

(1) . Similarly, the shear

of the horizon vanishes up to third order indicating thatthe individual horizons are isolated to an excellent ap-proximation. As we shall see below, the individual hori-zons are isolated even for relatively small values of donce the common MOTS has formed.

2. Numerical results for the merger phase

We performed a numerical evolution starting withBrill-Lindquist initial data. Working in units wherethe total ADM mass is unity, the punctures were lo-cated at z = ±0.5, and the individual black holes hadequal masses. Thus 2α(1) = 2α(2) = 0.5. The domainhad an explicit octant symmetry and extended up tox, y, z = 96. Near the outer boundary the spatial res-olution was h = 1.6, and near the punctures we usedmesh refinement to increase the resolution successivelyup to h = 0.0125, so that the individual horizon diam-eters contained initially 32 grid points. We used fourthorder accurate spatial differencing operators, and a thirdorder Runge–Kutta time integrator.

We excised [58] coordinate spheres with a radius ofre = 0.0625 about the punctures from the domain, cor-responding to a diameter of 10 grid points. We usedthe AEI BSSN formulation [58, 59] for time evolution,using the boundary conditions also described in [58].These boundary conditions are known to be incompati-ble with the Einstein equations. We used a 1 + log slic-ing condition [60] starting from α = 1, and a zero shift.

11

-0.5

0

0.5

1

-1.5 -1 -0.5 0 0.5 1 1.5

x

z

Horizon shapes at t=1

individual horizonsinner horizonouter horizon

FIG. 3: Coordinate shapes of the horizons at t = 1 in the xzplane. A common horizon has formed, and the inner and outercommon horizons have already separated. Compare figure 2.

This makes both the individual and the outer commonhorizon grow in coordinate space. We used the Cactusframework [61, 62], the Carpet mesh refinement driver[63, 64], and the CactusEinstein infrastructure. We lo-cated the apparent horizon surfaces with J. Thornburg’sAHFinderDirect [48, 65].

In this setup, the apparent horizon has two discon-nected components in the initial data, and a commonMOTS forms shortly after t = 0.5. The individual hori-zons are null up to numerical errors (consistent withthe result on the smallness of σ(ℓ) in the far limit), andtheir masses are essentially constant up to numerical er-ror. As discussed in section II B 2 and figure 2, the com-mon MOTS forms initially as a single surface but thenbifurcates: an outer horizon which is strictly-stably-outermost, and an inner one which becomes strictly un-trapped on being deformed inwards. Figure 3 showsthe shapes of the individual and the inner and outercommon MOTSs at time t = 1, where the inner andouter common MTTs have already noticeably separated.As expected, the outer MTT is purely spacelike whilethe inner MTT, being spacelike initially, becomes partlytimelike quickly. Figure 4 shows the horizon world tubemetric signature at t = 0.6 and t = 1. At later times,the outer MTT tends to become null (as expected), whilethe inner MTT becomes completely timelike, and thenbecomes so distorted at about t = 1.2 that it cannot bereliably tracked any more. This coordinate distortion isalready evident in figure 3, and the horizon discretisa-tion used in the apparent horizon finder is inaccuratenear the neck of the inner horizon [48]. Figure 5 shows

the time evolution of the masses M =√

AS/16π of theindividual and the common horizons (in this case, theangular momentum vanishes identically). If M∞ is theasymptotic value of the mass of the outermost horizonat late times, then MADM − M∞ is, in principle, a reliableway of estimating the amount of energy radiated awayto infinity in the form of gravitational waves. This dif-

-1

0

1

2

3

4

5

6

7

0 π/2 π

det q

θ

Horizon metric determinant at t=0.6

individual horizoninner horizonouter horizon

(a) t = 0.6

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 π/2 π

det q

θ

Horizon metric determinant at t=1

individual horizoninner horizonouter horizon

(b) t = 1

FIG. 4: Determinant of the horizon world tube’s three-metricvs. latitude θ at t = 0.6 and t = 1. The individual MTTs arenull, i.e., det q = 0 (up to numerical errors). The commonouter MTT is spacelike (i.e., det q > 0) and it tends to nullat late times. The inner common MTT is partially timelike att = 0.6; later it becomes completely timelike.

ference could be used as a consistency check on other es-timates using the extracted waveforms at large distancesfrom the black holes. However, our emphasis in this pa-per is on the dynamics of the merger and not on longduration stable evolutions. Our simulations do not lastlong enough to estimate M∞ reliably.

Another feature of the horizons, shown in figure 5, isthat while the common outer MTT increases in area asexpected, the area of the common inner MTT decreasesmonotonically. This is explained as follows. Initially,when the common MOTS is just formed, by continu-ity with the outer MTT, the inner MTT is spacelike for avery short duration (much before t = 0.6) and it is thusa DH for this duration. However, this DH is being tra-

12

0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99

1

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

M

t

Irreducible mass

inner horizonouter horizon

FIG. 5: Irreducible mass vs. time for the individual and thecommon MTTs. The outer common MTT grows and accretesmass, while the inner MTT shrinks and loses mass.

versed in the inwards direction (i.e., along −ra) so that itsarea appears to decrease. Shortly after its formation, theinner MTT becomes partly timelike and later fully time-like. Recall that for a TLM, the area decreases if Θ(n) < 0.Thus, both the spacelike and timelike portions of the in-ner MTT contribute to its monotonic area decrease. Thisbehavior of the outer MTT is roughly similar to whatwas found in [47] for spherically symmetric horizons;however due to spherical symmetry, the horizons in [47]did not have any cross sections of mixed signature.

Figure 6 demonstrates how the common outer ap-parent horizon grows. The energy flux vanishes at thepoles, and the shear (but not the total flux) is maximumat the equator. The horizon is spacelike all the time, butit becomes increasingly isolated at late times as it ap-proaches equilibrium. Thus the rate of area increase be-comes smaller and the fluxes also becomes correspond-ingly smaller.

Let us now consider the higher mass multipoles Mn

(all the Jns vanish identically). Here, since all quanti-ties are symmetric with respect to a reflection about theequatorial plane, Mn = 0 for odd n. Figure 7 plots themass quadrupole moment M2 and also M4 and M6 ofthe outer and inner common MTTs as a function of time.We expect that the black hole should eventually settledown to a Schwarzschild solution by radiating away allof its higher multipole moments. Clearly, for the outerMTT, M2, M4 and M6 all become smaller with time, ap-proaching zero. However, the run did not last longenough for us to obtain the asymptotic fall-off rate. Itis interesting to note that, as far as we can tell, the mul-tipole moments for the inner MTT do not vanish asymp-totically. This tells us that the spacetime near the innerMTT is not close to Schwarzschild even at late times.At even later times, all the inner horizons presumablycease to exist (see next paragraph) and the spacetime ap-proaches Schwarzschild everywhere.

We conclude this section with some remarks on the

0

0.0015

0.003

0.0045

0.006

0.0075

0 π/2 π 0

0.05

0.1

0.15

0.2

0.25

Ene

rgy

flux

d2 E /

dA d

t

She

ar |σ

|2

θ

Outer horizon at t=0.6

d2E / dA dt|σ|2

0.001

0.01

0.1

1

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

Tot

al e

nerg

y flu

x dE

/dt

t

Outer horizon

dE/dt

FIG. 6: Energy flux and shear |σ(ℓ)|2 through the outer com-

mon horizon vs. latitude θ at t = 0.6, and the total energy fluxvs. time. The shear vanishes at the poles and the black holesettles down exponentially.

eventual fate of the inner MTT. First of all, as expected,the outer MTT eventually settles down and approachesfuture timelike infinity. The inner MTT shrinks and ap-proaches the two individual horizons which are essen-tially stationary. It is interesting to speculate on how, ifat all, the inner MTT will merge with the two individualMTTs. Does the inner MTT “pinch off” into two indi-vidual horizons? If the inner MTT is indeed the one pre-dicted by [52], then it has a priori curvature bounds. Ifthese curvature bounds are maintained in the limit, thenthe inner horizon cannot pinch off. It is more likely thatthe two individual MTTs merge first with each other andthen later, perhaps also with the inner MTT. It wouldbe interesting to investigate this question further. Ifthe inner MTT does indeed merge smoothly with thetwo individual MTTs, then the set of all MTTs in thiscase would form one single smooth 3-manifold. Fur-thermore, the area of the cross-section of this manifoldwould be monotonic in the outward direction – travers-ing this manifold in the outward direction means going

13

-6

-4

-2

0

2

4

6

8

10

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

Mul

tipol

e m

omen

ts

t

Outer horizon

M2M4M6

-5

0

5

10

15

20

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

Mul

tipol

e m

omen

ts

t

Inner horizon

M2M4M6

FIG. 7: Some mass multipole moments vs. time for the innerand outer MTTs for the head-on collision. The multipole mo-ments for the outer horizon all approach their Schwarzschildvalues (i.e., 0) but the inner horizon does not seem to do so.

forward in time on the individual and outer MTTs, andbackward in time on the inner MTT.

We are not able to settle these issues numerically in aconclusive manner because the inner MTT becomes sodistorted at late times that the AH tracker is no longerable to track it. This is because the AH tracker can onlylocate star-shaped surfaces and, as is clear from figure3, the inner MTT will not necessarily be star-shaped atlater times. Furthermore, our gauge choice in which weallow the outer MTT to grow in coordinate space, makesthe inner MTT shrink and therefore harder to resolve atlater times.

B. Non-axisymmetric black hole collision

The head-on collision described above does not incor-porate any effects of angular momentum. In this section,we remove the restriction of axisymmetry by taking ini-tial configurations in which the black holes are orbiting

around each other. We use the so called “puncture” dataintroduced by Brandt and Brugmann [66], which is ageneralization of the Brill-Lindquist construction. Thedata is still taken to be conformally flat, but now nolonger assumed to be time symmetric.

We performed a numerical evolution of puncture ini-tial data corresponding to the innermost stable circularorbit as predicted in [67]. This model was also studiedas “QC-0” with the Lazarus perturbative matching tech-nique [68, 69] and later in [2, 3, 4, 5, 70]. In our setup,the punctures were located at x = ±1.168642873, andtheir mass parameters were m = 0.453, and their mo-menta were py = ±0.3331917498. The domain had anexplicit rotating quadrant symmetry and extended upto x, y, z = 10. Near the outer boundary the spatial res-olution was h = 0.4, and near the punctures we usedmesh refinement to increase the resolution successivelyup to h = 0.025, so that the individual horizon diam-eters contained initially 16 grid points. We used fourthorder accurate spatial differencing operators, and a thirdorder Runge–Kutta time integrator.

We excised [58] coordinate spheres with a radius ofre = 0.075 about the punctures from the domain, corre-sponding to a diameter of 6 grid points. We used againthe AEI BSSN formulation [58, 59] for time evolution, a1+ log slicing condition [60] starting from a lapse that isone at infinity and zero at the punctures, and a Γ drivershift condition, starting from a rigid co-rotation with anangular velocity of ω = 0.06. We also used a drift cor-recting shift term similar to [71, 72] to keep the individ-ual horizons centered about their initial locations.

As previously, we used the Cactus framework [61, 62],the Carpet mesh refinement driver [63, 64], and theCactusEinstein infrastructure. We solved the initialdata equation with M. Ansorg’s TwoPuncture solver[73], and we located the apparent horizon surfaces withJ. Thornburg’s AHFinderDirect [48].

This setup contains two initially separated horizonsthat rotate around each other for a fraction of an orbitbefore a common horizon forms [68, 70]. Its ADM massis MADM = 1.00788, the initial proper horizon separationis L ≈ 4.99 MADM, and the horizons have initially the an-

gular momentum J ≈ 0.78 M2ADM and angular velocity

Ω ≈ 0.17/MADM. The common apparent horizon formsat about t = 17.5, which we verified through pretracking[74].

Figure 8 shows the shape of the various MOTSs at atime t = 18, a short while after the common horizon hasformed. The qualitative behavior of the various MTTs isexactly the same as in section IV A 2. Figure 10 showsthe irreducible mass of the outer and inner MTTs as afunction of time. Again, the behavior is qualitatively thesame as we saw in the head-on collision.

Figure 11 shows the flux of gravitational wave energyfalling into the outer horizon at t = 18.4 and also the

shear |σ(ℓ)|2 at the same time, for the outer and individ-

ual horizons. The 2-d contour plots of the shear |σ(ℓ)|2

14

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-3 -2 -1 0 1 2 3

y

x

Horizon shapes at t=18

individual horizoninner horizonouter horizon

FIG. 8: Coordinate shapes of the MOTSs at t = 18 for the non-axisymmetric black hole collision. Note that the individualhorizons are locked in place through the co-rotating coordi-nate system and through an adaptive shift condition.

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

ππ/20

det q

θ

Horizon metric determinant at t=18

individual horizoninner horizonouter horizon

FIG. 9: Determinant of the MTT three-metric at t = 18. Asin the head-on case, the outer MTT is purely spacelike whilethe inner MTT is partly spacelike and partly timelike. At latertimes, it becomes purely timelike. The individual MTTs arenull at this time.

and the total flux on the horizon shows in detail howgravitational radiation is falling into the horizon. Un-like in the head-on case (fig. 6), the shear and the fluxare now no longer axisymmetric. Therefore, the flux isno longer constant along the φ direction but its maximastill lie on the equator. The shear on the other hand, nowhas its maximum on the poles and its minima lie on theequator. It would be interesting to further investigate

the behavior of |σ(ℓ)|2 and the energy flux as a function

of time and for different physical situations to gain a bet-ter understanding of how a black hole grows.

Let us now turn to the rotational vector ϕa on theouter horizon and the quantities such as angular mo-mentum, mass, and multipole moments associated withit. The simulation presented here was run only up tot ≈ 19.4, and the final black hole has not settled downsufficiently, and has not attained axisymmetry at this

0.76

0.78

0.8

0.82

0.84

0.86

0.88

0.9

17.5 18 18.5 19

M

t

Irreducible mass

inner horizonouter horizon

FIG. 10: A plot of the irreducible mass Mirr =√

A/16π asa function of time for the outer an inner MTTs in the non-axisymmetric black hole collision. As expected, the outer MTThas increasing area while the inner MTT shrinks.

φ

Energy flux d2E / dA dt at t=18.4 0.01 0.0075 0.005 0.0025

2π3π/2ππ/20π

π/2

0

φ

Shear |σ|2 at t=18.4 0.1 0.075 0.05

0.025

2π3π/2ππ/20π

π/2

0

FIG. 11: Energy flux through the common horizon and shear

|σ ¯(ℓ)|2 on the horizon at t = 18.4 in (θ, φ)-coordinates. The

spacetime is not axisymmetric, and since it contained two in-spiralling black holes, there is no mirror symmetry across thex − z or y − z planes either. The energy flux shows this asym-metry clearly.

15

0

5

10

15

20

ππ/20

θ

Lie derivative of the two-metric at t=18

|Lθ qab||qab|

FIG. 12: Lie derivative of the two-metric Lϕqab at t = 18 onthe φ = 0 line. The two-metric qab is also shown for compar-ison. The quantity shown in the plots are actually the norms√

∑ab(Lϕqab)2 and

√

∑ab(qab)2 in the coordinate system (θ, φ)

on the horizon. The vector field ϕ is Killing on the equator (seemain text), but not everywhere. This shows that the horizon isnot (yet) axisymmetric. We expect it to become axisymmetricat later times. Note that we have only shown the plots alongthe φ = 0 curve and we do not have axisymmetry here.

point. Figure 12 shows the Lie derivative of the 2-metricLϕ qab on the horizon at t = 18, where ϕa is the Killingvector candidate found by the algorithm presented in[35]. It is clear that Lϕ qab is very far from 0 at this time.This means that the angular momentum, mass, and mul-tipole moments associated with this ϕa are not mean-ingful at this point. This is to be expected, since the fi-nal black hole should attain axisymmetry only on a timescale set by the quasi-normal mode ringdown, whichhas a period of 15.9MADM in this case. It is interestingto see that our Killing vector field candidate is indeedKilling on the equator. This is by construction, since wechoose the Killing vector field candidate by an integralalong the equator; see [35]. However, the vector field ϕa

is far from Killing away from the equator.

A word of caution is due here regarding the Killingvector finding algorithm of [35]. First of all, the algo-rithm only produces a candidate for a Killing vector, andan independent check is required to see whether Lϕ qabis sufficiently small or not. Furthermore, as mentionedpreviously, this method reduces the problem of finding aKilling vector on a sphere to diagonalizing a 3× 3 matrixfollowed by integrating a 1-dimensional ODE. In partic-ular, the method requires that one of the eigenvalues ofthis matrix is sufficiently close to unity. While this is finewhen the horizon is exactly axisymmetric, the subtletyarises when the horizon is only approximately axisym-metric. It is not clear how close the eigenvalue mustbe to unity for the horizon to be regarded as approxi-mately axisymmetric. Work is in progress to understandthis better and to also investigate an alternate method offinding an appropriate ϕa as discussed in section III B,

which is guaranteed to produce a divergence free vec-tor.

C. Axisymmetric gravitational collapse

1. The initial configuration

Up to now, all of our examples have involved onlyvacuum spacetimes. In this section, we present an exam-ple of the gravitational collapse of a neutron star to forma black hole in an axisymmetric spacetime. These sim-ulations were performed using the Whisky code whichdeals with the matter terms of the Einstein equations inthe framework of the Cactus toolkit. Thus, the Whisky

code solves the conservation equations for the stress en-ergy tensor Tab and for the matter current density Ja:

∇aTab = 0 , ∇a Ja = 0 . (4.10)

For details about the Whisky code and the implementa-tion of the above equations, we refer the reader to [75]and references therein. Here we shall restrict ourselvesto describing the initial stellar configuration which isone of the configurations studied in [75].

The neutron star is modeled as a uniformly rotatingball of perfect fluid. The equation of state is taken to be aK = 100, Γ = 2 polytrope so that the pressure p and rest-

mass density ρ are related according to p = KρΓ. Theequilibrium configuration is determined by the massMNS, central density ρc, and the angular momentumJNS; when necessary, the subscript NS is used in order toavoid any confusion with previously defined symbols.The model we take is the one denoted as “D4” in [75]which has MNS = 1.86M⊙, ρc = 1.934× 1015 g cm−3, and

JNS = 0.543M2NS

. This leads to a ratio of polar to equa-torial coordinate radii of 0.65, a circumferential equa-torial radius of 14.22 km, and a rotational frequencyof 1295.34Hz. This equilibrium configuration turns outto be dynamically unstable. In practice, the instabilityis induced by uniformly reducing the pressure slightlythroughout the star.

2. Numerical results

We simulated the above system on a grid with an ex-plicit rotating octant symmetry. The outer boundarywas at x, y, z = 150, and the grid spacing near the outerboundary was h = 3. We used mesh refinement to in-crease our spatial resolution in the center of the domainto h = 0.375 at the initial time, and progressively intro-duced more mesh refinement levels to increase the cen-tral resolution up to h = 0.046875 as the neutron starcollapsed, based on the maximum density in the star[76, 77]. We also apply third order Kreiss–Oliger dissipa-tion [78] to the spacetime (but not the hydrodynamics)variables.

16

0

0.5

1

1.5

2

2.5

3

3.5

4

130 140 150 160 170 180 190 200 0

0.5

1

1.5

2

2.5

3

3.5

4A

real

rad

ius

R

Avg

. coo

rd. r

adiu

s r

t

Horizon radii

areal radius Rcoordinate radius r

FIG. 13: The average coordinate radius and the area radius asa function of time for the outer and inner MTTs for the neutronstar collapse. The inner horizon is not to be trusted after t ≈140 due to lack of resolution, since its coordinate radius hasbecome very small by that time.

We find an apparent horizon starting at about t = 130;this time is mainly dependent on the details of howthe collapse is induced and has no intrinsic meaning.The horizon is born with an irreducible mass of aboutMirr = 1.51 and an angular momentum of J = 0.89(a = 0.38), giving it a total mass of MH = 1.54. Sometime after t = 185, a singularity forms in the spacetime,and the simulation aborts because we do not use exci-sion inside the apparent horizon. As before, a pair ofMOTSs is formed, an outer and an inner one. The outerMTT is spacelike, has increasing area, and tends to nullat late times. In this case, the inner MTT remains space-like. However, its area decreases because we are travers-ing it in the inward direction; in other words, the timeevolution vector ta is such that at the inner MTT, t · r < 0so that that the area decreases along ta. Our gauge con-ditions are such that the outer horizon grows in coordi-nate space while the inner horizon shrinks. After aboutt = 140, the inner horizon is so small that we do nothave enough resolution to track it beyond that time. Seefigure 13. The areal radius of the outer MTT increasesbut not as rapidly as the coordinate radius; it levels offat later times. The area radius of the inner horizon de-creases initially and shows an increase at later times, butthis is probably just a numerical artefact due to poor res-olution at later times.

Figure 14 shows the determinant of the metric on theMTTs. The outer MTT is initially spacelike, which is con-sistent with its growing, and exponentially approachesnull at late times. After about t = 160, the simula-tion cannot distinguish the horizon world tube signa-ture from null any more. As an example we also showthe determinant as a function of the latitude θ at t ≈ 138,and the average value of the determinant over the hori-zon as a function of time. The inner MTT is also space-like and becomes more and more null at least as long as

1e-05

0.0001

0.001

0.01

0.1

1

10

130 140 150 160 170 180 190 200

aver

age

of d

et q

t

Horizon metric determinant

outer horizoninner horizon

0.004 0.006 0.008 0.01

0.012 0.014 0.016 0.018 0.02

0.022 0.024

0 π/2 π

det q

θ

Horizon metric determinant at t=138.24

outer horizoninner horizon

FIG. 14: Average of the determinant of the horizon worldtube’s three metric vs. time, and vs. latitude θ at t = 138.24for the inner and outer horizons for the neutron star collapse.

we are able to track it reliably.

Figure 15 shows the outer horizon has grown at t =155 to an irreducible mass of Mirr = 1.80 and an angu-lar momentum of J = 1.93 (a = 0.55), giving it a totalmass of MH = 1.87. For comparison, the correspond-ing ADM quantities are MADM = 1.86 and JADM = 1.88(a = 0.54). Because the spacetime is axially symmet-ric, gravitational waves cannot carry away angular mo-

mentum. That means that the spin a = J/M2 is ap-proximately correct at late times. Unlike in the non-axisymmetric black hole collision discussed earlier, thepresent case is explicitly axisymmetric and there are noproblems with locating the rotational symmetry vector.

Figure 16 shows the mass quadrupole moment M2

and the angular momentum octopole moment of theouter and inner MTTs as a function of time. Given thatwe know the asymptotic values of the area and angu-lar momentum of these MTTs (the ADM values), we canalso calculate the expected values of M2 and J3 at latetimes. The plots clearly show that the values of M2 andJ3 approach the Kerr values at later times (though thismatching is not exact, presumably due to numerical er-rors). Also note that M2 is noisy. We have observed suchnoise only in simulations that include matter, and we

17

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

130 140 150 160 170 180 190 200

M

t

Total mass

outer horizoninner horizon

ADM mass

-0.5

0

0.5

1

1.5

2

130 140 150 160 170 180 190 200

J

t

Angular momentum

outer horizoninner horizon

ADM J

FIG. 15: The total mass MH ,, and angular momentum J as afunction of time for the outer and inner MTTs for the neutronstar collapse.

find that this noise is much improved by using artificialdissipation on the spacetime variables (which we do).The angular momentum multipoles seem unaffected.

V. DISCUSSION

In this article, we have applied the dynamical hori-zon formalism to numerical simulations of black holespacetimes. The main theme in this formalism is totake trapped surfaces seriously as a way of describingblack hole physics. Marginally trapped surfaces behavemore regularly that one might have expected previously,and they are useful for extracting interesting physicalinformation about the horizon. We have shown howthe mass, angular momentum, multipole moments, andthe flux of energy due to in-falling gravitational radi-ation and matter can be calculated in a coordinate in-dependent way (given a particular time slicing of ourspacetime). We have implemented these ideas numeri-cally and shown three concrete examples. In these ex-amples, we see how the black hole is formed, how itgrows, and how it settles down to an isolated Kerr blackhole. We have also seen that the dynamical horizon for-

-2-1.8-1.6-1.4-1.2

-1-0.8-0.6-0.4-0.2

0

130 140 150 160 170 180 190 200

M2

t

Mass quadrupole moment

outer horizonM2 for Kerr

-14

-12

-10

-8

-6

-4

-2

0

130 140 150 160 170 180 190 200

J 3

t

Angular momentum octupole moment

outer horizonJ3 for Kerr

FIG. 16: Horizon mass quadrupole moment M2 and angularmomentum octopole moment J3 vs. time for the neutron starcollapse. For comparison, the values for a Kerr black hole withthe same ADM mass and angular momentum as the initialdata are also shown.

malism is valuable for exploring the geometry of thetrapped region. It allows us to classify various types oftrapped surfaces which might appear during the courseof a gravitational collapse or a black hole coalescence.Finally, these ideas can also be viewed as a set of di-agnostic tools which allow us to keep track of what isgoing on during the course of a numerical simulation,and whether numerical results make sense and satisfysome basic, but non-trivial properties in the strong fieldregion.

Some suggestions for future work:

i. As mentioned in the text, the calculation of the ax-ial vector ϕa for non-axisymmetric cases is not yetsatisfactory. We have used the method suggestedin [35] which works well enough at early and latetimes, when the horizon is approximately axisym-metric. However, in general, the result is not guar-anteed to be divergence free and thus the angularmomentum not guaranteed to be gauge invariant.We have not yet implemented the generalizationdescribed in section III B satisfactorily; this is workin progress.

18

ii. The accuracy of the numerical examples that we haveshown decreases with time, and this is a commonfeature of most present day black hole numericalsimulations. Thus, we have not been conclusivelyable to prove that the black hole settles down toKerr (though there are strong indications that thisdoes happen). We have not been able to extract therate at which equilibrium is reached, thereby ex-tending Price’s law (see [79] and e.g. [80]) to moregeneral situations, but this is, in principle possibleand requires more stable and accurate simulations.Similarly, we have not been able to accurately cal-culate the asymptotic value of the black hole massM∞. The difference MADM − M∞ is, in principle,a reliable estimate of the amount of energy radi-ated to infinity. While the ADM mass is hard tocalculate reliably during the simulation because ofthe finite grid and low resolution in the asymp-totic region, it can usually be calculated accuratelyfrom the initial data themself. Calculating M∞ andunderstanding this estimate of the radiated energyrequires more accurate and stable runs, applied todiverse and realistic initial data. The results of [41]could also be used to study the approach to equi-librium.

iii. It would be useful numerically to have a gauge con-dition which ensures that the horizon stays at thesame coordinate location at all times. While suchconditions are not difficult to find in the isolatedcase, dynamical situations are harder. Given thelocation of an outer MOTS at a particular instant oftime, the results and methods of [34] can be usedto predict the location of the MOTS at the next in-stant by solving an elliptic equation on the MOTS.This could be used to construct appropriate gaugeconditions and evolution schemes which take thehorizon geometry into account [49, 81].

iv. What happens to the inner horizon of figures 2, 3,and 8? As described in section IV A 2, the eventualfate of these inner MTTs and the two individualhorizons is not yet known, and would be inter-esting to investigate further. This requires simu-lations with higher resolution near the inner hori-zons, different gauge conditions, and perhaps also

AH trackers capable of handling non-star-shapedsurfaces, and perhaps also higher genus surfaces.

v. Can the methods of [34] be extended for MOTSswhich are not strictly-stably-outermost? In this re-gard, it would be interesting to study the stabil-ity operator LΣ introduced in [34]. For a strictly-stably-outermost MOTS, the principle eigenvalueof LΣ turns out to be strictly positive and this isan important ingredient in the existence results.A numerical computation of the eigenvalues ofthis operator, especially during the transition be-tween inner and outer MTTs and for the inner non-spacelike MTTs might lead to further insights.

Acknowledgments

We are grateful to Lars Andersson and AbhayAshtekar for many valuable suggestions and fruit-ful discussions. We also thank Ivan Booth, SergioDain, Steve Fairhurst, Greg Galloway, Ian Hawke, SeanHayward, Jan Metzger, Denis Pollney, Reinhard Prix,Jonathan Thornburg, and Robert Wald for useful discus-sions.

As always, our numerical calculations would havebeen impossible without the large number of peoplewho made their work available to the public: we usedthe Cactus framework [61, 62] and the CactusEinstein

infrastructure [82] with a number of locally developedthorns, such as the initial data solver TwoPunctures

by M. Ansorg, the mesh refinement criteria set up viaWhiskyCarpetRegrid by C. D. Ott and I. Hawke, andthe horizon finder AHFinderDirectby J. Thornburg. Wealso used the general relativistic hydrodynamics codeWhisky [83] developed by the authors of [75], and theinitial data generator RNSID by N. Stergioulas, whichwere both developed during the EU training network“Sources of Gravitational Waves”. The code uses rou-tines of the LAPACK [84, 85] and BLAS [86] libraries fromthe Netlib Repository [87], the Numerical Recipes [88],and the UMFPACK [89] library. The numerical simulationswere performed on the Peyote Beowulf Cluster at theAEI. ES was partly funded by the DFG’s special researchcentre SFB TR/7 “Gravitational Wave Astronomy”. Thiswork was supported by the Albert–Einstein–Institutand the Center for Computation & Technology at LSU.

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