Horizons in a binary black hole merger II: Fluxes, multipole moments and stability

Daniel Pook-Kolb,1, 2 Ofek Birnholtz,3 José Luis Jaramillo,4 Badri Krishnan,1, 2 and Erik Schnetter5, 6, 7

1Max-Planck-Institut für Gravitationsphysik (Albert Einstein Institute), Callinstr. 38, 30167 Hannover, Germany2Leibniz Universität Hannover, 30167 Hannover, Germany

3Department of Physics, Bar-Ilan University, Ramat-Gan 5290002, Israel4Institut de Mathématiques de Bourgogne (IMB), UMR 5584, CNRS,

Université de Bourgogne Franche-Comté, F-21000 Dijon, France5Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada

6Physics & Astronomy Department, University of Waterloo, Waterloo, ON N2L 3G1, Canada7Center for Computation & Technology, Louisiana State University, Baton Rouge, LA 70803, USA

We study in detail the dynamics and stability of marginally trapped surfaces during a binaryblack hole merger. This is the second in a two-part study. The first part studied the basic geometricaspects of the world tubes traced out by the marginal surfaces and the status of the area increaselaw. Here we continue and study the dynamics of the horizons during the merger, again for the head-on collision of two non-spinning black holes. In particular we follow the spectrum of the stabilityoperator during the course of the merger for all the horizons present in the problem and implementsystematic spectrum statistics for its analysis. We also study more physical aspects of the merger,namely the fluxes of energy which cross the horizon and cause the area to change. We construct anatural coordinate system on the horizon and decompose the various fields appearing in the flux,primarily the shear of the outgoing null normal, in spin weighted spherical harmonics. For each ofthe modes we extract the decay rates as the final black hole approaches equilibrium. The late partof the decay is consistent with the expected quasi-normal mode frequencies, while the early partdisplays a much steeper fall-off. Similarly, we calculate the decay of the horizon multipole moments,again finding two different regimes. Finally, seeking an explanation for this behavior, motivated bythe membrane paradigm interpretation, we attempt to identify the different dynamical timescalesof the area increase. This leads to the definition of a “slowness parameter” for predicting the onsetof transition from a faster to a slower decay.

I. INTRODUCTION

In classical general relativity, black holes are perfectabsorbers. They grow inexorably by absorbing matterand/or radiation from their surroundings. Emission ofelectromagnetic or gravitational radiation occurs due tointeractions of the black hole with surrounding spacetimeor matter. Gravitational waves are emitted due to non-stationarities and non-linearities of the spacetime metricin the region around the black hole. Black holes have anadditional special feature which does not hold for otherphysical objects, namely a very special set of equilibriumstates determined by only two parameters in astrophys-ical contexts. In other words, astrophysical black holeswithin standard general relativity have no hair. Normalphysical objects reach equilibrium by both absorbing andemitting, but black holes do not have that luxury. Notonly must they only absorb, but they must absorb veryselectively so that the absorbed radiation precisely can-cels any hair it might initially have.

This picture applies to a binary black hole merger.When the final remnant black hole is initially formed,its horizon is highly distorted but its final state is thatof a simple Kerr black hole. This process of reachingequilibrium from its initial state at formation must fol-low the process of selective absorption mentioned above.This process of reaching equilibrium is often referred to asthe black hole “radiating away its hair”. This is accuratewhen one considers a sufficiently large spacetime region

containing the black hole; after all, it is not just the hori-zon that reaches equilibrium, but rather the spacetimeitself in a neighborhood of the horizon. However, “radi-ating away hair” is not an apt description for the horizonitself in classical general relativity.

The issue of how a black hole knows precisely howmuch radiation to absorb at any given time, is an im-portant one in general relativity. From a mathematicalperspective, it touches on the question of the stabilityof the Kerr black hole in full non-linear general relativ-ity. From a theoretical physics viewpoint, any deviationsof the final state from Kerr might indicate support foralternate theories of gravity. As we have argued in theprevious paragraph, this issue of the final state is inti-mately connected with the in-falling energy flux throughthe horizon. One important goal of analytic or theoreticalstudies is thus to discover universalities in the approachto equilibrium of a black hole horizon in full non-lineargeneral relativity. These universalities might be reflectedin the rates of exponential or power-law decay. Gravi-tational wave observations of binary black hole mergersoffer opportunities for testing these predictions observa-tionally.

A useful way of approaching these problems is via thestudy of marginally trapped surfaces. These are specialspherical surfaces for which outgoing light rays have van-ishing convergence. These surfaces are well suited for de-scribing not only stationary black holes, but also binarymergers and other dynamical processes involving blackholes. The entire process of merger and approach to equi-

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librium can be understood in terms of marginally trappedsurfaces. Recent numerical studies have discovered newgeometric and topological features of marginally trappedsurfaces in binary black hole mergers. These include theirbehavior under time evolution, the status of the area in-crease law, and the presence of topological features suchas cusps and knots. These numerical results rely on a newmethod for locating marginally outer trapped surfaces[1, 2], and the physical results are based on the formal-ism of quasi-local horizons. This formalism is based onthe world tube of marginally trapped surfaces and it pro-vides a coherent way of studying various aspects of blackhole physics quasi-locally [3–9]. For our purposes, it isimportant that there exist exact flux formulae for thesehorizons within full general relativity, which quantify theamount of energy and radiation crossing the horizon, andrelate it to the change in horizon area [10, 11]. The fluxdue to gravitational radiation is positive definite and al-ways causes the area to increase. This is analogous tothe well known Bondi mass-loss formula at null infinityin the Bondi-Sachs framework describing the energy car-ried away by gravitational radiation.

It turns out that in these astrophysical situations, thefluxes falling through the horizon are highly correlatedwith the fluxes at infinity which can be observed by grav-itational wave detectors [12–16]. This might appear sur-prising at first glance since the horizons are causally dis-connected from observers outside the event horizon. How-ever, in these astrophysical situations the source of the in-falling radiation and the outgoing radiation are one andthe same, namely non-linearities and non-stationarities inthe spacetime region near (but outside) the black holes.Thus, a better understanding of the horizon fluxes mighthelp us to quantify these correlations better. Eventually,one might be able to observationally infer properties ofspacetime regions hidden behind event horizons.

The goal of this paper is to study, via numerical simu-lations, horizon fluxes in binary black hole mergers, andthe approach to equilibrium. The basic scenario outlin-ing how marginally trapped surfaces merge has been es-tablished in [1, 2]. The present series of papers followsup on these results by studying physical and geometricalproperties of marginally trapped surfaces and their timeevolution. The first paper (henceforth paper I) has stud-ied basic properties of these world tubes including theirsignature and the status of the area increase law. Thegoal here is to study in detail physical aspects of theseworld tubes. These include energy fluxes across the worldtubes, their decay rates as the final black hole approachesequilibrium, the evolution of the horizon multipole mo-ments, and their stability properties. While we often referto paper I (and the reader might benefit by having a copyof that paper at hand), this paper is meant to be mostlyself-contained.

The plan for the rest of this paper is as follows. Sec. IIsets up notation and briefly summarizes some of the ba-sic notions and results that we shall use later. Paper Ihas already summarized the main definitions and con-

cepts of quasi-local horizons that we employ. Here weshall summarize results pertaining to the horizon fluxes,the stability operator and the multipole moments. Espe-cially important will be the construction of an invariantcoordinate system on the horizon which will be used todecompose various fields on the horizon. Sec. III discussesthe stability of the various MOTSs. The stability hererefers to the properties of a MOTS under small outwarddeformations, and is governed by an elliptic operator.The horizon will be stable if this operator is invertible,i.e. when its spectrum does not contain zero. This leadsus then to analyze the spectral properties of the opera-tor, yielding what might be called the stability spectrumof the MOTS and pushing forward the study of the fullMOTS-spectral problem formulated in [17–19] in partic-ular introducing a discussion in terms of spectrum statis-tics.

Sec. IV addresses the question of why the area changes,namely due to the flux of gravitational radiation acrossthe horizon. The most important part of the radiationflux is the shear which, just like the gravitational ra-diation observed by gravitational wave detectors, is asymmetric tracefree tensor, except that it lives on thehorizon. The horizon, being a non-null surface, also hasanother contribution to the flux from a vector field onthe horizon. We study the multipolar decomposition ofboth of these contributions. We then connect the decayrate of the flux to the quasi-normal mode frequencies as-sociated with the final black hole. Sec. V presents theevolution of the horizon multipole moments. The multi-pole moments capture the deviation of the horizon from asimple Schwarzschild geometry (or Kerr, if the black holeshad been rotating). Thus, the evolution of the multipolemoments in time tells us about how the two individualblack holes become increasingly distorted, and how the fi-nal black hole approaches equilibrium. This is, of courseclosely connected with the fluxes discussed in Sec. IV.Sec. VI offers a tentative explanation for why we havetwo regimes in the approach to equilibrium. It shows thatthe non-linear effects dominate in the steep decay regimeat early times, while the later time is consistent withlinear behavior. Sec. VII concludes by discussing openquestions and possible directions for future work. Themathematical issues discussed in Sec. III (namely spec-tral theory) are quite different from the topics of Secs. IVand V (fluxes, multipole moments, quasi-normal modes,and non-linearities); they can thus be read quite inde-pendently of each other.

II. BASIC NOTIONS

A. Marginally trapped surfaces and dynamicalhorizons

The basic notions of marginally trapped surfaces anddynamical horizons were already summarized in paper I.Several review articles on the subject are also available

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[3–5, 7–9]. We shall therefore be very brief with the basicdefinitions. The focus will be on the flux laws, multipolemoments and the stability operator.

Let spacetime be modeled as a 4-dimensional manifoldM equipped with a Lorentzian metric gab with signature(−,+,+,+). We shall only consider vacuum spacetimes.Let ∇a be the derivative operator compatible with gab.Let S be a closed 2-dimensional spacelike manifold im-mersed in M . S is taken to be orientable and of sphericaltopology. Let q̃ab, �̃, and Da be the intrinsic Riemannianmetric on S, the volume 2-form, and the correspondingderivative operator, respectively. The intrinsic scalar cur-vature of S will be denotedR, its area AS , and the Lapla-cian on S is ∆S .

The outgoing and ingoing future directed null-normalsto S will be denoted by `a and na respectively. We willtie the normalizations of the null normals together byrequiring ` · n = −1. Finally, given a complex null vectorma tangent to S satisfying m · m̄ = 1, we obtain a null-tetrad (`, n,m, m̄).

The expansions Θ(`) and Θ(n) of `a and na are respec-

tively

Θ(`) = q̃ab∇a`b , Θ(n) = q̃ab∇anb . (1)

The shears σ(`) and σ(n) of `a and na, respectively, are

σ(`) = mamb∇a`b , σ(n) = mamb∇anb . (2)

We shall usually not need σ(n) in this paper, and thus weshall often refer to σ(`) just as the shear σ.

The other important field is the connection 1-form onthe normal bundle of S:

ωa = −nbqca∇c`b . (3)

It can be shown that ωa relates to the angular momentumassociated with S (see e.g. [11, 20]). In this paper weconsider only non-spinning black holes. Thus while wewill occasionally mention ωa where appropriate, all ofour results have ωa = 0.S is said to be a future-marginally-outer-trapped sur-

face if Θ(`) = 0 and Θ(n) < 0. If Θ(n) > 0, then S issaid to be past-marginally-outer-trapped. A surface satis-fying only Θ(`) = 0 with no restriction on Θ(n) is calleda marginally outer trapped surface, or MOTS in short.

It is clear that a MOTS is a geometric concept ina spacetime, and makes no reference to any spacelikeCauchy surfaces or time coordinate. Nevertheless, onecan think of a Cauchy surface as a convenient meansof locating a MOTS: They can be located on a spacelikeCauchy surface Σ equipped with a 3-metric and extrinsiccurvature, and well known numerical methods exist forthis. The canonical choice of null normals for S immersedin Σ is

`a =1√2

(T a +Ra) , na =1√2

(T a −Ra) . (4)

Here Ra is the unit spacelike normal to S (and tangentto Σ), while T a is the unit timelike normal to Σ. We use

a numerical method recently developed in [1, 2], capa-ble of locating highly distorted surfaces; our implemen-tation is available at [21]. This method is an extensionof the widely used method developed in [22–26]. Our nu-merical calculation use Einstein Toolkit [27, 28]. We useTwoPunctures [29] to set up initial conditions and anaxisymmetric version of McLachlan [30] to solve the Ein-stein equations, which uses Kranc [31, 32] to generateefficient C++ code. Results in this paper are obtainedfrom simulations with spatial resolutions 1/∆x = 480running until Tmax = 20M and 1/∆x = 60 running un-til Tmax = 50M, where M := MADM/1.3 is our simu-lation time unit. For brevity, we will occasionally statesimulation times using lowercase t := T/M. Here M isa suitable mass scale in the problem. Further details ofthe simulation specific to our problem are detailed in [2].

The initial configuration is the same as that used in pa-per I and in [2]. We use the Brill-Lindquist construction[33], i.e. the initial data is conformally flat and time sym-metric. The initial data has two non-spinning black holeswith vanishing linear momentum. The “bare masses” arem1 = 0.5 and m2 = 0.8 with the total ADM mass beingMADM = 1.3. The initial separation d0 is d0/MADM = 1.At the initial time, there are two disjoint horizons S1and S2 with S2 being the larger one. The common hori-zon forms at a time Tbifurcate shortly after the simulationstarts and splits into inner and outer surfaces, Sinner andSouter, respectively. The world tubes of these horizons areshown in Fig. 1 of paper I.

The 3-dimensional world tube traced out by theMOTSs is taken as a bonafide geometric object in its ownright and we attempt to understand its physical and geo-metric properties. The pioneering work by Hayward [34]was an important step in this direction. Another impor-tant aspect is a detailed study of the case when the worldtube is null, i.e. just like the stationary Schwarzschildand Kerr solutions, the black hole is not absorbing mat-ter/energy and not increasing in area. This can be viewedas an approximation in suitable physical situations (anexcellent approximation in many cases), or as the limit-ing case asymptotically as the black hole reaches equi-librium. The basic definition of a non-expanding horizonand its extensions to an isolated horizon has been sum-marized in paper I. A detailed understanding of this casehas been achieved and an extensive literature on isolatedhorizons is available (see e.g. [20, 35–44]). For the dy-namical case, we need to consider a general world tubeof arbitrary signature which will be called a dynamicalhorizon. Additional qualifiers such as timelike or space-like, and future and past (depending on the sign of Θ(n))will be included as required.

B. Variations and the stability operator

Given a MOTS S on a Cauchy surface Σ and a choiceof lapse and shift, i.e. a time evolution vector, considerthe behavior of the MOTS under time evolution. If the

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MOTS were to evolve smoothly under this time evolu-tion, it would trace out a smooth 3-dimensional worldtube. In the well known stationary solutions, e.g. theSchwarzschild or Kerr black holes, the event horizons arefoliated by MOTSs. If the world tube does exist also infully dynamical situations, then it is possible to formu-late black hole physics and thermodynamics in variousphysical scenarios. Seminal work by Hayward in 1994 in-troduced the notion of trapping horizons [34] and showedhow one could formulate the laws of black hole thermody-namics in this framework for dynamical black holes. Sim-ilarly, horizon fluxes were studied in [10, 11] and shownto be manifestly positive definite. In this early work onthis topic, it was usually assumed that this smooth worldtube exists in full non-linear general relativity. This wasa reasonable assumption, especially given the fact thatMOTSs were already widely used in numerical relativityfor locating and extracting physical black hole param-eters [45]. In these numerical simulations the apparenthorizons were generally found to evolve smoothly. Themathematical conditions under which a MOTS evolvessmoothly were found in 2005 [46–48]. A central role inthese proofs is played by the stability operators associ-ated with a MOTS and their eigenvalues, which we nowdescribe.

The starting point here is the notion of the variation ofa MOTS [49]. One chooses a vector field Xa along whichS is to be varied, thereby obtaining a family of surfacesSλ at least for small values of λ. Starting with a pointp on S, varying λ yields a curve with Xa as the tangentvector at p; Sλ=0 is identified with S itself. Variationstangent to S do not play an important role here, andwe take Xa to be orthogonal to S. Given this family Sλdepending smoothly on λ, one can consider variationsof geometric quantities on S. For a MOTS, the quantityof interest is the expansion Θ(`). For each Sλ, we definenull normals just as for S itself. The expansion can becomputed for each value of λ and then differentiated. Thisdefines the variation of Θ(`) along X

a, which is denotedδXΘ(`). This is not be confused with usual derivatives ofΘ(`). In particular, δψXΘ(`) 6= ψδXΘ(`) when ψ is nota constant. This leads to the definition of the stabilityoperator L acting on functions ψ : S → R as

L(X)[ψ] := δψXΘ(`) . (5)

Since Xa is orthogonal to S, given a choice of the nullnormals (`a, na), we can write

Xa = b`a + cna , (6)

where b and c are functions on S. We see then that there isnot just a single stability operator, but several dependingon the normal direction. This is why we label the stabilityoperator L(X) with X.

One case is well known and easy to understand, namelywhen Xa is along `a. This should just be the Raychaud-huri equation, and indeed, setting Θ(`) = 0 and assumingspacetime to be vacuum leads to

L(`)[ψ] = δψ`Θ(`) = −2 |σ|2 ψ . (7)

Clearly, if ψ is positive, then this variation will be neg-ative. Moreover, this variation is linear in ψ and doesnot involve any derivatives. The other component of thevariation is along na; it will be convenient to consider theoutgoing direction −na instead. This turns out to lead toa second order elliptic operator:

L(−n)[ψ] = (−∆S + 2ωaDa)ψ

+

(1

2R+Daωa − ωaωa

)ψ . (8)

The presence of the first derivative causes this operatorto be non-self-adjoint. We will have ω = 0 in this paper,whence this simplifies to a self-adjoint operator

L(−n)[ψ] =

(−∆S +

1

2R)ψ . (9)

We have seen that the variation along `a is “negative”.On the other hand, since −∆S has positive eigenvalues,the variation along −na is seen to be positive if R ispositive (this shall not always be the case in this paper).

In numerical simulations, MOTSs are found on Cauchysurfaces in the course of a time evolution. Thus, if S lieson a spacelike Cauchy surface Σ, and if Ra is the unitoutgoing spacelike vector normal to S, then it is naturalto look at variations along Ra. This leads to the stabilityoperator associated with Σ:

LΣ[ψ] :=√

2 δψRΘ(`) , (10)

where we used the freedom to choose a factor of√

2 tosimplify the following expressions. We label this stability

operator by Σ instead of L(√

2R) to emphasize the connec-tion with the Cauchy surface. Since Ra = (`a − na)/

√2,

we have (setting ωa = 0)

LΣ[ψ] = δψ`a−ψnaΘ(`) =

(−∆S +

1

2R− 2 |σ|2

)ψ .

(11)Since LΣ and L

(−n) are elliptic operators on a compactmanifold, they have a discrete spectrum. In general thesespectra are complex (due to the first derivative term in-volving ωa). However the eigenvalue with smallest realpart can be shown to be real, and is known as the prin-cipal eigenvalue Λ0. The corresponding eigenfunction φ0can be chosen to be positive. We note that the eigen-values do not depend on the scaling of the null normals.If the null-normals are rescaled according to ` → f`,n→ f−1n, then L(−n) undergoes a similarity transforma-tion: L(−n) → fL(−n)f−1. The eigenfunctions of L(−n)are scaled by f but its eigenvalues are unaffected.

We now summarize some results and their connectionto properties of the various horizons that we have alreadyencountered in paper I. First we need a definition.

Definition 1 (Strictly-Stably-Outermost). A MOTS Sis said to be strictly-stably-outermost along a directionXa normal to S if there exists some ψ ≥ 0 such thatδψXΘ(`) ≥ 0, and δψXΘ(`) does not vanish everywhere.

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This turns out to be equivalent to the principal eigen-value being positive definite: Λ0 > 0. If Λ0 > 0 thenwe can choose ψ to be the lowest eigenfunction, and thecondition δψXΘ(`) > 0 follows. The converse is shown in[47]. The principal eigenvalue itself depends on the direc-tion of Xa: it is largest for Xa = −na, and decreases asXa turns towards `a. Two results are important for ourpurposes:

• Starting with a MOTS on Σ, it evolves smoothlyin time as long as LΣ is invertible, i.e. none ofits eigenvalues vanish. As a special case, this holdsif Λ0 > 0 whence all other eigenvalues also havepositive real parts.

The signature is also restricted if Λ0 > 0:

• Let S be a strictly-stably-outermost MOTS. Theworld tube, i.e. the dynamical horizon, generated bythe time evolution of S is spacelike if |σ|2 is non-zero somewhere on S.

In our simulation, this scenario applies for the individualdynamical horizons and for the outer common horizon.All of these turn out to be strictly-stably-outermost and,as we saw in paper I, they are all spacelike. The innerhorizon is, as in other aspects, much more interesting. Ithas Λ0 < 0, and as we saw in paper I, its signature is notrestricted to be spacelike. The spectra of LΣ and L

(−n)

will be described in detail in Sec. III.

C. Invariant coordinates on an axisymmetrichorizon

For physical applications to be studied below, it willbe important to decompose various fields on the horizonswhich have topology S2 × R. These fields will be scalar,vector and second rank tensors. For a given MOTS S,some important geometric fields of interest are the in-trinsic curvature scalar R, the rotational 1-form ωa andthe shear. Thus, it is very important to have a canonicalnotion of scalar, vector and tensor spherical harmonicsor equivalently, spin weighted spherical harmonics. Dif-ferent choices of spherical coordinates (θ, φ) on a MOTSwill in general yield different multipolar decompositions.On an axisymmetric horizon, it turns out to be possi-ble to construct an invariant coordinate system following[50].

We exploit the manifest axisymmetry present in ourcalculations, i.e. the existence of an axial vector ϕa whichpreserves the 2-metric qab on the horizon. For an axisym-metric surface S of spherical topology S with area AS andradius RS =

√AS/4π, we construct a coordinate system

(θ, φ) adapted to ϕa. We assume that ϕa vanishes at pre-cisely 2 points (the poles), and has closed integral curves.The coordinate φ is the affine parameter along φa, takento be in the range [0, 2π); we still need to fix the pointswith φ = 0, which we shall do shortly. Second, the analog

of cos θ is a coordinate ζ defined as follows:

Daζ =4π

AS�̃baϕ

b ,

∮

Sζ dA = 0 . (12)

It follows obviously that Daζ is orthogonal to ϕa and itsintegral curves are the lines of longitude connecting thetwo poles. Fix any one of these curves, and set φ = 0 onit; this specifies φ completely. It is then straightforwardto show that the 2-metric on S can be written as

ds2q = R2S

(dζ2

F+ Fdφ2

), (13)

where

F (ζ) =4πϕaϕ

a

AS, (14)

and it can be shown that −1 < ζ < 1 so that we can setcos θ = ζ.

We can now write the spin weighted spherical harmon-ics in terms of (θ, φ). It is important to note that the or-thogonality relationships between the spherical harmon-ics continue to hold with the natural volume element onS: in the volume element for the metric in Eq. (13), thefactors of F cancel out. Thus, the volume element is iden-tical to that of a fictitious canonical round 2-sphere met-ric

q(0)ab = R

2S(dθ2 + sin2 θdφ2

). (15)

Spherical harmonics, including the spin weighted spheri-cal harmonics, can be constructed in the usual way, butnow using this canonical metric. Finally, a natural choicefor the null vector m is

m =RS√

2

(dζ√F

+ i√Fdφ

). (16)

Thus, we have a complete null tetrad where (`, n) is givenby Eq. (4) and m is given here.

Having constructed the preferred coordinates on agiven MOTS, let us now look at its time evolution andlet H be the dynamical horizon. For most of our results,the invariant coordinates described above suffice: at eachinstant of time, we can locate the axisymmetric MOTS,construct the invariant coordinate system, calculate therelevant physical quantity in this coordinate system, andthen consider it as a function of time. There is no needto explicitly consider the problem of identifying points atdifferent instants of time. In future work, when we do nothave axisymmetry, this issue will be especially importantif we wish to have a canonical notion of time evolutionon H. Even in this paper, it will be useful to clarify whatone means by time evolution on H.

Let us label the MOTSs on H by a parameter λ (whichin our case can just be the time coordinate of the nu-merical evolution) and let us consider a vector field Xa

tangent to H. In principle it need not necessarily be or-thogonal to the MOTSs. The role of Xa is to evolve ge-ometric fields from one MOTS to the next. In order to

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talk about “time evolution” of fields and multipole mo-ments on a dynamical horizon, it is necessary to have acanonical choice of Xa. One obvious choice is to take Xa

such that it preserves the foliation of H by MOTSs, andis orthogonal to the MOTSs. We shall call this vectorfield V a. For concreteness, take H to be spacelike every-where so that we have a unit spacelike normal r̂a to eachMOTS. Then, orthogonality of V a to the MOTSs implies

V a = ar̂a (17)

with a being a function on H. V a preserves the foliationif we can choose V a∂aλ = 1, and this naturally restrictsa. We also require V a to preserve the axial symmetry ϕa:LϕV a = 0.

There are many situations where the above choice ofV a as evolution vector is not appropriate, and we needto add a shift vector Na tangent to S:

Xa = ar̂a +Na . (18)

An obvious example is when we have spinning blackholes, so that we might need to add an angular velocityterm:Xa = ar̂a+Ωϕa. Even for non-spinning black holes,it might be natural to have a non-vanishing shift vector.A general construction for Xa satisfying certain naturalconditions is given in [51] to determine the “lapse” and“shift” for Xa as we move from one MOTS to the next.Let us briefly summarize the construction, specializingonly later to the case when each MOTS is axisymmet-ric. An important condition, it turns out, is to chooseXa such that it preserves divergence free vector fields.The MOTSs are changing in area and thus the volume2-form �̃ab is varying in time. We can think of this vari-ation as being composed of i) an overall, homogeneouschange corresponding to the overall area change, and ii)inhomogeneous variations on smaller scales which aver-age away to zero on each MOTS. It turns out that theright condition is to choose Xa such that

LX(�̃abAS

)= 0 . (19)

Note that the quantity �̃ab/AS integrates to unity andcontains the local inhomogeneous fluctuations in the areaelement on S. Since this construction uses only invari-antly defined geometric structures on H, the axial sym-metry vector ϕa is preserved, i.e.

LXϕa = 0 . (20)

From the previous two equations and Eq. (12), it followsthat ζ is preserved as well: LXζ = 0. Thus we constructthe preferred coordinates (θ, φ) as above on each MOTSand then we simply take Xa such that ζ (or equivalentlyθ) remains fixed. We would still have the freedom to add ashift in the ϕ direction, but for non-spinning black holes,we can choose the shift to be completely in the ζ direc-tion. In our case, it turns out that this construction leadsto a non-zero shift vector in the ζ direction.

D. Fluxes, balance laws and multipole moments

We conclude this section by summarizing the flux lawfor spacelike dynamical horizons and the notion of mul-tipole moments. The reason the area of a horizon in-creases is, of course, due to in-falling radiation and mat-ter. The same applies to angular momentum, mass andhigher multipole moments. This can be seen as a “phys-ical process” version of the first law of black hole ther-modynamics. For spacelike dynamical horizons it is pos-sible to derive exact expressions for these fluxes in fullnon-linear general relativity. Since H is spacelike, it isequipped with a unit timelike normal τ̂a, and each leafof H has a unit spacelike normal r̂a tangent to H. Then,a choice of null normals defined by H is

̂̀a = 1√2

(τ̂a + r̂a) , n̂a =1√2

(τ̂a − r̂a) . (21)

This is analogous to Eq. (4), but the two choices are dif-ferent and related by a scaling. Let Ai and Af be theinitial and final areas respectively of a (not necessarilyinfinitesimal) portion ∆H of a spacelike dynamical hori-zon and let ∆R = Rf − Ri be the change in the arearadius. Then, in vacuum spacetimes,

∆R =1

4π

∫

∆H

(σ̂abσ̂

ab + 2ξ̂aξ̂a)NR d

3V . (22)

Here σ̂ is the shear of the outgoing null normal ̂̀a,ξ̂a = q̃abr̂c∇c ̂̀b, and NR is a suitable lapse function. Theintegrand in this expression is manifestly positive defi-nite. The important point here is that we have identified

the shear and the vector ξ̂a as the relevant fields whichcarry energy across H. We have already written the shearas a complex field σ of spin weight 2, and we can simi-

larly write ξ̂a as a complex field ξ̂ = ξ̂ama of spin weight1. The identification of σ as an important part of the en-ergy flux is similar to the flux across null surfaces [52]; seealso [16, 48, 53–55]. The presence of the additional spin

weight 1 field ξ̂ occurs because we are here dealing with

non-null surfaces. It is also worth noting that ξ̂ becomesnumerically difficult to calculate asH approaches equilib-rium and becomes null (τ̂a and r̂a are ill-behaved in thelimit). Below we shall study the decomposition of σ intomodes of spin weight 2, and their time evolution. Notethat we shall use `a defined in Eq. (4) and not Eq. (21)for computing the shear and ξ.

Also of importance for us in this paper will be thenotion of multipole moments [50] for axisymmetric hori-zons, analogous to the well known Geroch-Hansen mul-tipole moments at infinity [56, 57]. These were first de-fined for isolated horizons where it can be shown that thetwo-dimensional scalar curvature R and the rotational1-form ωa characterize the geometry of an isolated hori-zon. Thus, by considering multipole moments of thesefields, one can characterize the horizon geometry com-pletely with a set of multipole moments (see also [58]

7

for an alternate set of moments). These multipole mo-ments continue to be useful even in dynamical cases [51].Specifically, since we are dealing with non-spinning blackholes, we only need to consider R, which lead to the massmultipole moments of an axisymmetric MOTS S:

Il =1

4

∮

SRYl,0(ζ) d2V . (23)

Here ζ is the invariant coordinate defined in Eq. (12), andYl,0 is the corresponding spherical harmonic. It is clearthat the lowest moment I0 is just a topological invari-ant, and for spherical topology I0 =

√π. Furthermore,

I1 can be shown to vanish identically from the definitionof the coordinate system (in effect these invariant coordi-nates automatically place us in the center of mass of thesystem). Non-trivial information is obtained from l = 2onwards, i.e. from the mass quadrupole, octupole etc.

III. THE SPECTRUM OF THE STABILITYOPERATOR

In this section we describe the spectrum of the stabil-ity operator for the various horizons. We consider mostlyLΣ, and L

(−n) briefly (both have qualitatively similar fea-tures). We will break up the discussion into three partsconsidering in turn the principal eigenvalue, a selection ofthe next eigenvalues, and then finally a statistical analy-sis of the higher eigenvalues.

A. The principal eigenvalues

Beginning with the principal eigenvalues of LΣ, wehave already mentioned that for S1, S2, and Souter, Λ0 isalways positive. Souter is born with Λ0 = 0, but it imme-diately becomes positive and remains so. At early timesfor S1 and S2, and at late times for Souter, two things hap-pen: i) the flux |σ|2 is small and thus the differences be-tween LΣ and L

(−n) are small. ii) The scalar curvature Rhas only small variations, and thus the spectrum of L(−n)

is almost the same as that of the Laplacian on a roundsphere, with a shift corresponding to the value of the cur-vature. Thus, in this limit where R ≈ 2/R2 = 1/2M2irrwith R being the area radius, and Mirr being the irre-ducible mass, the eigenvalues are labeled by two quantumnumbers (l,m) and will be approximately1

Λl,m ≈1

4M2irr(1 + l(l + 1)) . (24)

The state l is (2l + 1)-fold degenerate in this limit. Inthe general but still axisymmetric case, the degeneracy

1 Note that, for l = 0, this expression can be justified withoutassuming spherical symmetry (cf. Appendix B).

between states of different |m| is broken, with m beingthe label for the angular modes. The fundamental angularmode m = 0 will in general not be degenerate, while wefind a 2-fold degeneracy (±m) for the higher modes withm 6= 0 due to axisymmetry. The principal eigenvaluesof LΣ are shown in Fig. 1 for all the horizons, whereasFig. 2 shows the principal eigenvalues of L(−n). The maindifference with LΣ is that Sinner has positive principaleigenvalue for a short duration.

Most of this analysis does not apply to Sinner. Just likeSouter, the inner horizon Sinner is born with Λ0 = 0. How-ever unlike Souter, it becomes negative thereafter. Sinneris therefore unstable – it is not strictly stably outermostand there are thus no outward deformations which couldmake it strictly untrapped. It is also far too distorted forEq. (24) to be even a rough approximation to its spec-trum. We see from the right panel of Fig. 1 that Λ0 forthe inner horizon apparently diverges to −∞ at Ttouchwhere it has a cusp (though of course we cannot reallyprove this numerically).

This divergence, if it indeed exists, can be understoodas follows. Given the structure of the stability operator,it is tempting to interpret it as the Hamiltonian of aquantum particle living on a sphere. The Laplacian isthe analog of the kinetic energy while the other terms inL(−n) and LΣ can be viewed as a potential. The groundstate energy is then the analog of Λ0. This analogy canbe extended also for spinning black holes where ωa isnon-vanishing [18, 19]. Then, the ground state energywill diverge to −∞ only if the potential also divergesto −∞. Of course, just because the potential divergesat a point does not mean the ground state energy alsodiverges; the hydrogen atom being the classic example.Whether or not Λ0 → −∞ depends on the details ofhow R diverges at the cusp2. For L(−n), the potential isjust R/2, which is partially negative for Sinner near thecusp [1], and it diverges at Ttouch. For LΣ, the potentialalso contains |σ|2 which complicates matters somewhat.However, since |σ|2 is non-negative and comes with anegative sign, we see that the potential will still diverge.A detailed investigation of this mathematical questionwill take us too far afield from the goals of our numericalstudy here, and thus we will postpone this to future work.

There is one result for Houter that will be importantfor us later, namely its approach to equilibrium. Havingcomputed Λ0 for Houter at all times, we can ask how itapproaches the equilibrium result of Eq. (24). For l = 0,we must have Λ0 → 14M2irr at late times whence we cancompare 4M2irrΛ0 with unity. This is shown in Fig. 3 on alogarithmic scale. We see clearly a steep initial decay justafter Tbifurcate, followed by a shallower decay and oscilla-tions. We observe a transition between the two regimes

2 This can be studied using the Lieb-Thirring inequality which re-lates the negative eigenvalues to the negative part of the poten-tial (see e.g. [59]). In quantum mechanics, this inequality playsa critical role in mathematically proving that matter is stable.

8

0 1 2 3 4 5 6 7 8

T/M

−1.0

−0.5

0.0

0.5

Λ0M

2

SouterSinnerS1S2

0 2 4 6 8

T/M

10−3

100

103

106

109

Λ0M

2

Ttouch

SouterS1S2Sinner (×−1)

FIG. 1: The principal eigenvalue Λ0 of LΣ for all the horizons. Except Sinner, all the horizons have positive Λ0. Thisis easily seen in the left panel. For Sinner, Λ0 shows a cusp at Ttouch. This is shown in the right panel on a

logarithmic scale (we plot −Λ0 for Sinner because of the logarithmic scale).

0 1 2 3 4 5 6 7 8

T/M

−1.0

−0.5

0.0

0.5

Λ(−

n)

0M

2

SouterSinnerS1S2

FIG. 2: The principal eigenvalue for L(−n) for thevarious horizons. These values turn out to be somewhatlarger than the corresponding values for LΣ. Thus, the

bifurcation between Sinner and Souter occurs at apositive value of Λ0. Thus, Sinner has positive principaleigenvalue for a short duration, and it does not cease to

exist when Λ0 crosses zero.

0 10 20 30 40

T/M

10−5

10−3

10−1

1−

Λ0/(1/4M

2 irr)

Souter

FIG. 3: Comparison of Λ0 with the perturbative result(B4). Around T ∼ 10M, the curve changes from a

steep to a more shallow exponential decay.

at ≈ 10M. This is our first encounter with this kind ofbehavior, and we shall see this same pattern repeatedlynumerous times in this paper. We shall study this behav-ior quantitatively in detail for other geometric fields onHouter in the following sections.

B. Low eigenvalues

For S1, S2 and Souter, all the higher eigenvalues mustbe positive since Λ0 > 0. Also for Sinner, apart from Λ0,all other eigenvalues must be positive till Ttouch. The rea-son is that at Tbifurcate, Λ0 = 0 and all the other eigenval-ues are positive definite. Since the evolution is smooth,the other eigenvalues must remain positive as long asSinner exists. If any of these eigenvalues were to crosszero, Sinner would cease to exist. Fig. 4 therefore showsthe next eigenvalue Λ1. It turns out to be positive withpossibly a cusp at Ttouch. This is shown in the secondpanel of Fig. 4. We see that the graph of Λ1 as a functionof time appears to be forming a cusp at Ttouch, though weare not numerically able to resolve this. The precise valueof Λ1 at the cusp is of interest. If this were to be neg-ative, then it means that Λ1 vanishes before Ttouch andtherefore Sinner does not exist near the cusp. This seemsunlikely since we find Sinner very shortly after Ttouch. Itseems more reasonable to assume that Sinner exists atall times around Ttouch and our numerical methods arenot able to locate it. This implies that the value of Λ1at Ttouch should be non-negative. It would be interestingto prove (or disprove) this conjecture. In any event, Λ1is still far from vanishing at the last time before Ttouchwhen it is located, indicating that it must exist for atleast a short time longer. Similarly, at the first time itis located after Ttouch, Λ1 is similarly positive indicatingthat it must have existed for at least a short time earlier.Interestingly, the two lowest degenerate eigenvalues withangular modes m = ±1 are positive before Ttouch, whileafter Ttouch the lowest m = ±1 eigenvalues become nega-

9

2 4 6 8

T/M

0.2

0.3

0.4

Λ1M

2forS i

nner

Ttouch

5.45 5.50 5.55 5.60

T/M

0.258

0.260

0.262

0.264

0.266

Λ1M

2forS i

nner

Ttouch

FIG. 4: The second eigenvalue Λ1 for Sinner with angular mode m = 0. The second panel shows a close-up nearTtouch. The graph appears to show cusp-like behavior at Ttouch.

2 4 6 8

T/M

−200

−150

−100

−50

0

Λl,m

M2

Ttouch

Λ0,0

Λ0,0

Λ0∗,±1

stability spectrum of Sinner

FIG. 5: The negative eigenvalues for Sinner. After Ttouch,two new (degenerate) negative eigenvalues appear for

the m = ±1 angular modes.

tive. We chose to label these as new eigenvalues withoutrelabeling the higher m = ±1 ones. That is, instead ofthe usual Λ1,1 < 0 < Λ2,1 < . . . we assign the labelsΛ0∗,1 < 0 < Λ1,1 < . . .. This is shown in Fig. 5. TheseΛ0∗,±1 eigenvalues are seen to increase much more rapidlythan Λ0 itself but, as far as we are able to track Sinner,none of these eigenvalues cross zero and Sinner continuesto exist.

C. Global behavior of the spectrum

The higher eigenvalues of LΣ are shown3 in Fig. 6. The

top panels show the spectra for S1 and S2. At early timeswe have the behavior predicted by Eq. (24). The largerblack hole, i.e. S2, has smaller eigenvalues for the same

3 The spectrum of L(−n) has similar global properties, except thatwe obtain slightly larger values corresponding to |σ|2, and inaccordance with the general results in [47]. We have chosen toshow just the principal eigenvalue, cf. Fig. 2.

value of l. A multiplet structure is apparent here. As weget closer to the merger, the states with different m areno longer degenerate, analogous to the splitting of en-ergy levels of a quantum system in an external field. Thestates with ±m remain degenerate due to axisymmetry.For generic configurations (including spins, non-zero or-bital angular momentum etc.), this symmetry would thennot be present and the ±m states would not be degener-ate.

As we approach Ttouch, the energy levels are seen tocross and it becomes more difficult to distinguish thestates with different l, though the multiplet structurewith splitting can still be identified. The apparent hori-zon has the opposite behavior. It approaches this sim-ple spectrum at late times when it settles down to aSchwarzschild black hole. The multiplet structure hereis again apparent.

The inner horizon Sinner apparently shows no suchsimplicity. Nevertheless, some spectroscopy-like analysisseems possible. In particular, a transfer of states betweendifferent multiplets seems to happen, with a migration ofstates from l → l + 2. This can be understood in termsof tidal coupling. Specifically, at around T ∼ 3M, Sinneris sufficiently deformed. It structures itself into two wellidentified lobes that ultimately pinch at Ttouch. The sys-tem starts to effectively behave as a binary, dramaticallyillustrated by the eigenfunctions which situate themselvesin either one or the other lobe (illustrated in Fig. 7). Thetwo components of this “quasi-binary” interact tidally(l = 2) inducing this coupling in the spectrum levels.

In summary, this kind of non-trivial coupling betweenlevels results in a completely different multiplet restruc-turing after Ttouch (e.g. the two lowest multiplets are sin-glets, as a consequence of the loss of states to higher lev-els). Globally, there turns out to be a further complexityfor the inner horizon that suggests the need to resortto other systematic tools to probe its underlying struc-ture. Looking further ahead to future work when we con-sider more generic configurations without axisymmetry,the spectrum will be complex and yet more complicated.

10

It will not be possible to investigate each eigenvalue indetail. We must then resort to a statistical analysis ofthe spectrum, from which we can extract valuable infor-mation. The remainder of this section can be seen as aprecursor to the more complicated case.

1. Crossing of energy levels

Still in a spectroscopic spirit, a clearly evident fea-ture of the spectra shown in Fig. 6, including that ofSinner, is the crossing of eigenvalue levels. This is verysignificant, since it is not the generic situation for realself-adjoint operators (of the class we are studying) de-pending on a single parameter, time t in our case. Thevariation of the Hamiltonian with time typically leadsto level repulsion, whereas level-crossing requires two pa-rameters [60]. This can be accounted for in terms of thecorresponding classical dynamics, if the operator is un-derstood as a classical Hamiltonian on a phase space.It turns out that for generic classical Hamiltonian sys-tems, namely non-integrable (or chaotic in rough terms),level-crossing translates into an over-determined condi-tion which generically admits no solution if only one pa-rameter is available. As a result, eigenvalues repel, some-thing that quantum-mechanically corresponds to cou-pling of the levels and the impossibility of defining quan-tum numbers.

On the contrary, when the underlying classical mo-tion is integrable, the eigenvalue curves indeed can(quasi-)cross4. Levels do not interact and evolve inde-pendently, quantum numbers can be tracked and clus-tering can happen due to the absence of level repulsion.In our present case, the corresponding classical systemis not only integrable, but our problem is actually sepa-rable5 as a consequence of axisymmetry. The latter is astronger (non-generic) feature that implies integrability[60]. From this perspective, nothing distinguishes Sinnerfrom the other horizons. In summary, for the four spec-tra shown in Fig. 6, level-crossing is a strong indicationof classical integrability and in our case a confirmationof the a priori knowledge about the separability of thesystem.

4 Actual crossing requires a stronger condition, namely separabil-ity, whereas in general integrable systems level lines can approachto extremely narrow separations but can then ultimately repel[60].

5 An interesting consequence of the separability of our eigenvalueproblem, as a consequence of axisymmetry, is the crossing ofnodal lines of the eigenfunctions. This is not the generic situa-tion even for integrable system (c.f. e.g. [61]), and follows fromseparability in two-dimensions in an orthogonal coordinate sys-tem. This is illustrated in Fig. 7 for two eigenfunctions of Sinner.

2. Spectrum statistics

The spectrum of a given MOTS stability operator isof course purely deterministic and can be efficiently cal-culated numerically. The underlying system, black holesin standard classical general relativity, do not have anyquantum aspects. However, we have found it useful tothink of the spectral problem as being associated withthe Hamiltonian of a quantum particle living on theMOTS. We shall now push this analogy further to thehigher eigenvalues and borrow techniques from quantummechanics. In the present self-adjoint setting the oper-ators L(−n) and LΣ can be seen (cf. sec 4.4. in [18]) asthe quantum Hamiltonian Ĥ corresponding to a classicalHamiltonian function H(p, q) = qabpapb +

12 R(q) on the

cotangent bundle T ∗S. Much insight can be gained theninto the actual MOTS spectrum from semi-classical con-siderations connecting the quantum system defined by Ĥto the underlying classical Hamiltonian system [60, 62–65]. Tools and concepts from the study of quantum chaoswill be adapted to the present MOTS setting. Differenteigenvalue-level statistics can be devised to address dis-tinct aspects of the spectrum. We will focus here on thesmall scale aspects of the spectrum, i.e. the interactionbetween adjacent levels.

For the higher eigenvalues, a statistical perspectiveon the distribution of eigenvalues can reveal importantstructural features of the underlying geometric object.This approach parallels the research program initiatedby Wigner [66] to undertake the understanding of thespectral properties of complex heavy nuclei in terms ofstatistical ensembles, leading to Dyson’s random-matrixmodels [67–69]. Later, these tools have been also sys-tematically employed in the setting of quantum chaos,exploring the subtle interplay between the quantum andthe underlying semi-classical system. Here we will focuson the application to our spectra of a short-range correla-tion in the spectrum, namely the ‘nearest neighbor spac-ing distribution’ P (S) which we describe shortly. Thisspectral statistic accounts for the fine-scale structure ofthe spectrum and in particular it is sensitive to the clus-tering or repulsion between the energy levels.

An important point is a need to remove “trivial” degen-eracies due to symmetries. In our case these degeneraciescorrespond to the ±m degeneracy. We do not want thedistribution P (S) to be dominated by this degeneracy,and thus they must be removed at the very start of theanalysis. Eigenvalues can then be ordered as

Λo < Λ1 ≤ Λ2 ≤ . . . ≤ Λn ≤ . . . , (25)

where the non-degeneracy of Λo has been taken into ac-count.

Prior to the introduction of spectral statistics, we per-form a normalization of the spectrum by setting its aver-age level density to unity. Specifically, we first introducea function N(Λ) counting the number of eigenvalues Λi

11

0 2 4 6 8

T/M

0

5

10

15

20

25

30

35

40

Λl,m

M2

l = 0l = 1l = 2

l = 3

l = 4

l = 5

l = 6m = 0

m = ±6l = 7

l = 8

Ttouch

stability spectrum of S1

0 2 4 6 8

T/M

0

5

10

15

20

25

30

35

40

Λl,m

M2

l = 0l = 1l = 2l = 3l = 4l = 5l = 6l = 7

l = 8

m = 0

m = ±8

l = 9

l = 10

l = 11

l = 12

l = 13

Ttouch

stability spectrum of S2

0 2 4 6 8 10 12 14 16 18 20

T/M

0

5

10

15

20

25

30

35

40

Λl,m

M2

l = 0l = 1l = 2l = 3l = 4l = 5l = 6l = 7l = 8l = 9l = 10

l = 11

l = 12m = 0

m = ±12l = 13

l = 14

l = 15

Ttouch

stability spectrum of Souter

2 4 6 8

T/M

0.0

2.5

5.0

7.5

10.0

12.5

15.0

17.5

20.0Λl,m

M2

l = 1l = 2l = 3l = 4l = 5l = 6l = 7l = 8

l = 9

l = 10

l = 11

l = 12

l = 13

l = 14

l = 15

Ttouch

stability spectrum of Sinner

FIG. 6: The stability spectrum of the various horizons. As expected, S1 and S2 (top two panels) start off with asimple spectrum corresponding to Eq. (24) and become more complicated near the merger. The spectrum for Souter

in the bottom-left panel shows the opposite behavior. The bottom right panel shows the positive part of thespectrum for Sinner.

below a certain value Λ as

N(Λ) =∑

i

Θ(Λ− Λi) , (26)

where Θ(y) = 1 − H(y), with H(y) the Heaviside func-tion. The counting function N(Λ) has a staircase struc-ture. The level density (density of states) is then defined

as

ρ(Λ) =dN

dΛ. (27)

We can write N(Λ) as

N(Λ) = Nav(Λ) +Nfl(Λ) . (28)

12

FIG. 7: Visualization of two eigenfunctions ψl,m of LΣfor Sinner at a time T = 5.35M before

Ttouch ≈ 5.53781M. The top row shows the functionvalues in blue (< 0) and red (> 0) on Sinner, while the

lower two rows show the function values and signchanges, respectively, as functions of (θ, φ) on a sphere.

The dotted line indicates the θ-coordinate of the“waist” visible in the first row. From the bottom row itis clear that the extended green regions are only close to

zero but still contain structure.

Here Nav(Λ) is a monotonically increasing smooth func-tion; it is the secular part of N(Λ) interpolating the stepsin N(Λ). Nfl(Λ) is the fluctuating part accounting for thedifference with respect to the secular increase. The “un-folding” of the spectrum is a “rectification” of the lattersuch that secular level density is 1. In particular, by in-troducing x = Nav(Λ), for the “unfolded spectrum”

xi = Nav(Λi) , (29)

we obtain an average level density of unity in the newvariable

ρav(x) =dNavdx

=dNavdΛ

dΛ

dx=dNavdΛ

(dNavdΛ

)−1= 1 .

(30)We focus here on the fine scale features in the spectrum,in terms of the distribution of separations between adja-cent eigenvalues in Eq. (25). Nearest-neighbor spacings

Si are calculated in the unfolded spectrum as

Si = xi+1 − xi . (31)

The probability of finding a spacing between S and S+dSis given by P (S)dS and, because of using the unfoldedspectrum, the average spacing 〈S〉 is unity:

〈S〉 =∫P (S)SdS = 1 . (32)

Since P (S) measures the correlation between adjacenteigenvalues, P (S) is said to be a “short-range level” cor-relation measure. We shall calculate P (S) for the sta-bility spectrum and attempt to interpret the result as arepresentative of a particular universality class. As a triv-ial example of such a universality class, consider the so-called “picket fence” distribution (namely a Dirac delta)centered at unity:

P (S) = δ(S − 1) . (33)

It is clear that such a distribution characterizes a per-fectly regular spectrum.

More interestingly, for real Laplacian-like operators asin Eqs. (9) and (11), P (S) presents a universality behav-ior according to the type of classical motion, ‘integrable’versus ‘chaotic’, of the corresponding classical Hamilto-nian:

i) “Integrable” classical motion: In this case we obtaina Poisson distribution

P (S) = e−S . (34)

This corresponds to a distribution showing a ten-dency to cluster since P (0) 6= 0. Moreover, thelevels Λ(t) cross6. In particular, crossing happensfor separable systems. The associated degeneracyis accounted by a non-vanishing P (0) and quantumnumbers can be assigned to levels in a straightfor-ward manner.

ii) “Chaotic” classical motion: This is the so-called

6 They can actually repel at an exponentially small scale [60].

13

Wigner surmise 7:

P (S) =π

2Se−

πS2

4 . (36)

This behavior displays repulsion between eigenval-ues since P (0) = 0. The eigenvalue curves (generi-cally) do not cross [60], and therefore they do notdegenerate. Level crossing requires two parameters.Therefore close levels couple and repel, with thestrength of the coupling given by the minimum en-ergy difference between the two repelling eigenvaluecurves. No “quantum numbers” can be assigned tosuch levels.

It is important to keep in mind that any of this behav-ior becomes evident only after the “trivial” degeneraciesdue to symmetries have been eliminated. Long-range cor-relations can be studied with other spectral statistics (cf.e.f. [65]), such as the number variance Σ(L) or the spec-tral rigidity ∆(L), presenting also universality in certainregimes (small L in this case). We postpone this to a laterstudy.

We are now ready to apply the above formalism to thestability spectrum. We start by mapping the spectrumto the “unfolded” spectrum where the average spacingbetween neighboring levels is normalized to 1. For thiswe first determine the average Nav(Λ) of the spectrumlevel-counting function N(Λ). Fig. 8 shows in panel (a)the step-wise N(Λ) for all four horizons at a time veryclose to Ttouch. In particular, we note the nice agreementwith Weyl’s law (see Appendix A) at large eigenvalues.

Then defining the unfolded levels as xi = Nav(Λi),we can construct the distribution of the nearest-neighbordistance variable, Si = xi+1 − xi. First we notice that ifonly eigenvalues with a fixed m are considered, then weobtained a perfectly regular distribution correspondingto a “picket fence” centered at S = 1 given in Eq. (33).This case is shown in panel (c) of Fig. 8. This is non-generic behavior, resulting from axisymmetry where mis the only preserved quantum number for all times. Thedistribution is dominated by this degeneracy and we are

7 Very interestingly, the Wigner surmise appears also in the set-ting of the Gaussian Orthogonal Ensemble (GOE) universalityclass in random matrices. More generally, the Bohigas-Giannoni-Schmit conjecture (cf. e.g. [65]), the eigenvalues correspondingto a chaotic classical system obey the same universal statisticsof level spacings as those Gaussian random matrices [67–69]. Inparticular, real time-reversal symmetric systems follow GaussianOrthogonal Ensemble (GOE) statistics, whereas (complex) non-time-reversal symmetric Hamiltonians are associated with theGaussian Unitary Ensemble (GUE). Other “more exotic” non-time-reversal systems, appearing for instance in spin systems, arerelated to the Gaussian Symplectic Ensemble (GSE). For com-pleteness, we present here the universal P (S) distributions forGUE and GSE statistics

PGUE(S) =32

πS2e−

π4S2

π , PGSE(S) =218

36π3S3e−

64S2

9π . (35)

not able to infer any relevant non-trivial structure. To fixthis, consider now all eigenvalues with the ±m symmetryremoved. The resulting histogram for P (S) is shown inFig. 8, panels (b) and (e). As expected, a Poisson dis-tribution is obtained for both Souter and Sinner despitetheir very different appearance in Fig.6. This is a conse-quence of the underlying classical integrability. The effectof level-crossing is apparent in the non-vanishing value ofP (0), indicating the generic occurrence of degeneracies.

Finally, we comment on the oscillations of the eigenval-ues visible in Fig. 6. For example, near T ≈ 9M, we seefrom the bottom-left panel of the figure that the eigen-values with the same l (but different m) are apparentlyalmost degenerate. Remarkably at this time, the spec-trum is in fact very close to that of a round sphere – thevarious oscillation modes of the MOTS conspire near thistime to produce a nearly round sphere for a short dura-tion. Panel (f) of Fig. 8 shows the distribution P (S) atthis time. This is very close to a quasi-picket-fence dis-tribution centered at S = 0 in. As we shall explain later,this behavior is consistent with the observed evolution ofthe horizon multipoles in Fig. 16.

Regarding Sinner, we note that the P (S) statistic doesnot capture many specific features of the spectrum. Thisincludes, for example, the multiplet reorganization be-tween different levels, which is not a short-correlationeffect. Addressing this requires the implementation ofstatistics for long-range correlations among spectrum lev-els, such as the number variance Σ(L) or the spectralrigidity ∆(L), and will be done somewhere else. Finally,the present spectrum statistics analysis could have beenanticipated from the a priori knowledge of the system sep-arability. The interest therefore lies in providing a bench-mark for future comparison with generic binary mergerswhere separability will be lost and, presumably, classicalintegrability will also disappear.

IV. HORIZON SHEAR AND FLUXES

Paper I has provided a detailed understanding of howthe area increases. Now we turn our attention to whythe area increases, i.e. because of the in-falling flux ofradiation (and potentially matter fluxes if we had matterfields). Recall here the expression for the area flux givenin Eq. (22). There are two contributions, the first beingthe familiar shear term. This is analogous to the wellknown outgoing radiation at least in the sense that theshear is a field of spin weight 2. It has been observed tobe closely correlated with the News tensor at null infinity[16]. The second term involving ξ has no correspondingcounterpart at null infinity (this is not surprising giventhat the dynamical horizon is not null). Being a vectorfield, ξ = ξam

a has spin weight +1.The dominant term in the flux is the shear. Let us

therefore consider the 2-dimensional integral of |σ|2 overthe various MOTSs; let us call this the shear flux. Theresult is shown in Fig. 9. The shear-flux increases for S1

14

0 10 20 30 40 50 60 70

Λ

0

200

400

Ncounting functions near Ttouch

SouterSinnerS1S2

(a) Function N for the four horizons. Thedotted lines show Weyl’s law.

0 1 2 3 4

s

0.0

0.5

1.0

P

Souter at T ≈ Ttouch, m ≥ 0

e−s

(b) Distribution P for Souter showing aPoisson-like shape.

0 1 2 3 4

s

0

2

4

P

Sinner at T ≈ Ttouch, m = 0

(c) “Picket-fence” distribution whenconsidering a fixed m spectrum.

45 46 47 48 49 50

Λ

300

310

320

330

340

N

counting functions near Ttouch

SouterNavΛA/4π

(d) Close-up of (a) showing Nav.

0 1 2 3 4

s

0.0

0.5

1.0P

Sinner at T ≈ Ttouch, m ≥ 0

e−s

(e) Poisson-like distribution for Sinner.

0 1 2 3 4

s

0

2

4

P

Souter at T = 9.0M, m ≥ 0

e−s

(f) “Quasi-picket-fence” at T = 9M.

FIG. 8: Construction and examples of the spectrum statistics. See text for details.

and S2, while it decreases for Souter. The dip in the shear-flux for Souter near T ≈ 13M is because of an oscillationin the dominant l = 2 mode of the shear as we shall seebelow. This is to be compared with Fig. 10 of paper Ishowing the corresponding dip in the plot of the rate ofchange of the area as a function of time. For the inner-common horizon Sinner, the shear-flux increases rapidlyin the beginning and soon reaches a plateau. It is note-worthy that there is no discontinuity across the mergerwhen Sinner develops a cusp and then self-intersections.

Being a symmetric tracefree tensor, we expand σ inspherical harmonics of spin weight +2. We have alreadyconstructed in Sec. II C a preferred coordinate system(θ, φ) which exploits the axisymmetry of the problem.These coordinates can obviously also be used for ourneeds in this section, i.e. expanding spin weight 2 fields.For the complex scalar σ we get

σ(θ, φ, t) =

∞∑

l=2

l∑

m=−lσlm(t)2Ylm(θ, φ) . (37)

Here 2Ylm are spin-weighted spherical harmonics and σlmare the mode amplitudes. This decomposition can be car-ried out for all of the horizons in our problem, namelythe two individual and the two common horizons. Fur-thermore, since we have explicit axisymmetry with σ in-dependent of φ, we will only have the m = 0 modes andwe will drop the index m in σl,m.

Fig. 10 shows |σl| for the two individual horizons, forl = 2, 3, . . . , 12. As the figure shows, the mode amplitudes

0 5 10 15 20

T/M

10−4

10−3

10−2

10−1

100

101

∫|σ|2

dA

Ttouch SouterSinnerS1S2

FIG. 9: The integral of |σ|2 := σabσab for the outgoingnormal `a given in Eq. (4). The dashed and dotted linesare for the individual horizons while the solid lines are

for the two common horizons.

decrease monotonically as the mode index l increases, sothat the l = 2 mode dominates. Similarly, as expected,the shear generally increases with time, indicating largerfluxes as we approach the merger. This is confirmed bythe integrals of |σ|2 over S1 and S2 shown in Fig. 9.

Fig. 11 shows the shear modes for the inner and outerhorizons. These have a number of interesting featuresworth pointing out. Consider first the shear on the ap-parent horizon which is expected to be correlated with

15

0 2 4 6 8

T/M

10−12

10−9

10−6

10−3

100|σ

l|

Ttouch

decompositon of σ(ℓ) of S1l = 2l = 3l = 4l = 5l = 6l = 7l = 8l = 9l = 10l = 11l = 12

0 2 4 6 8

T/M

10−12

10−9

10−6

10−3

100

|σl|

Ttouch

decompositon of σ(ℓ) of S2l = 2l = 3l = 4l = 5l = 6l = 7l = 8l = 9l = 10l = 11l = 12

FIG. 10: The mode decomposition of the shear for the two individual black holes. See text for details.

5 10 15

T/M

10−7

10−5

10−3

10−1

|σl|

Ttouch

decompositon of σ(ℓ) of Souterl = 2l = 3l = 4l = 5l = 6l = 7l = 8l = 9l = 10l = 11l = 12

2 4 6 8

T/M

10−3

10−2

10−1

100

|σl|

Ttouch

decompositon of σ(ℓ) of Sinnerl = 2l = 3l = 4l = 5l = 6l = 7l = 8l = 9l = 10l = 11l = 12

FIG. 11: Shear modes for the common horizons. The left panel shows |σl| for the outer common horizon and theright panel shows the mode coefficients for the inner horizon. See text for further discussion.

the post-merger gravitational waveform measured in thewavezone far away from the source. It was observed in[15] that the horizon multipole moments (which will bediscussed below) fall-off exponentially with decay ratesconsistent with the quasi-normal mode frequencies of thefinal black hole. Moreover, it was shown that the fall-off ofthe multipole moments is well explained by the presenceof two exponentially damped modes. This is consistentwith [70] which observed that the post-merger waveformis well explained by the quasi-normal modes, includingthe higher overtones. Motivated by these results, we con-sider a model for the shear amplitude |σl(t)| with twoexponentially damped modes:

σl(t) = A(1)l e

α(1)l t +A

(2)l e−iα(2)l t . (38)

Here we take α(1)l to be real, and α

(2)l to be complex

because, as shown below, at early times the shear doesnot show any oscillations, while at later times it exhibitsdamped oscillations. When one mode falls off much morerapidly than the other, a simplified piecewise-exponentialmodel can be used:

σl(t) = A(1)l e

α(1)l t , 0 < t < t(1) , (39)

σl(t) = A(2)l e−iα(2)l t , t > t(2) . (40)

Again, the early part is just exponentially damped, whilethe later part is an exponentially damped oscillation. Wedo not necessarily choose t(1) = t(2). In practice, we findthat one of the modes is rapidly decaying with an initiallylarger amplitude, and a second mode which is longer livedbut with lower initial amplitude. This simplified modelwith suitably chosen transition times t(1,2) will thereforesuffice for our purposes. Before presenting the best fitvalues of the decay rates, it is instructive to look at someof the fits to the individual modes in Fig. 12. For this fig-ure and the following fitting results, our simulation withthe lower resolution of 1/∆x = 60 and Tmax = 50Mwas used in order to obtain late time data for the outerhorizon Houter. It is clear from these plots that the modeamplitudes have qualitatively different fall-offs at earlyand late times with the transition occurring roughly be-tween T = 8M and T = 10M. It is also clear thataccurate values of α

(1)l , α

(2)l respectively will be obtained

by taking t(1) as small as possible, and t(2) as large aspossible; we take t(1) = 4 and t(2) = 20. Finally, the fits

of the imaginary part =(α(2)l ) are obtained by consider-ing the local maxima of |σl| after t(2), and the real part

16

0 10 20 30 40 50

T/M

10−5

10−4

10−3

10−2

10−1

100|σ

l|T (1)

T (2)

shear modes of Souter (l = 2)

shear modeearly dampinglate damping

0 10 20 30 40 50

T/M

10−6

10−5

10−4

10−3

10−2

10−1

100

|σl|

T (1)

T (2)

shear modes of Souter (l = 3)

shear modeearly dampinglate damping

0 10 20 30 40 50

T/M

10−6

10−4

10−2

100

|σl|

T (1)

T (2)

shear modes of Souter (l = 4)

shear modeearly dampinglate damping

0 10 20 30 40 50

T/M

10−7

10−5

10−3

10−1

101

|σl|

T (1)

T (2)

shear modes of Souter (l = 5)

shear modeearly dampinglate damping

0 10 20 30 40 50

T/M

10−7

10−5

10−3

10−1|σ

l|T (1)

T (2)

shear modes of Souter (l = 6)

shear modeearly dampinglate damping

0 10 20 30 40 50

T/M

10−8

10−6

10−4

10−2

100

|σl|

T (1)

T (2)

shear modes of Souter (l = 7)

shear modeearly dampinglate damping

FIG. 12: Fits of the shear mode for l = 2, 3, . . . , 7. The curves in blue show the shear amplitude, the orange dottedline shows the exponential fit at early times (before T (1) = 4M), and the dashed green line is the exponential fit at

late times (after T (2) = 20M). In each case we see a clear transition from steep decay to a slower decay rate.

Before looking at the best fit values obtained for theparameters in the above model, it will be useful to keepin mind the values of the standard quasi-normal modefrequencies for a Schwarzschild black hole. Quasi-normalmodes are defined in the framework of perturbation the-ory, and they are solutions which are purely outgoing atthe horizon and at infinity [71, 72]. This condition leadsto a discrete set of complex frequencies labeled just bythe mass of the black hole (for spinning and charged blackholes, these would be determined by the mass, spin andcharge). The complex frequencies are labeled by threeintegers (n, l,m): (l,m) are the usual angular quantumnumbers while n = 1, 2, . . . is the overtone index for theradial wave-function. For a Schwarzschild black hole weonly need to consider m = 0. Some values of the imag-inary part of the frequency are shown in Table. I. Sim-ilarly, it will be useful to know the real part of the fre-quency of the lowest (n = 1) overtone for different valuesof l. For l = 2, 3, . . . 7 these are given in Table II. De-tailed data files are available at [73], based on [74, 75].It is useful to note that the imaginary frequency for agiven overtone index n is fairly insensitive to the valueof l, but for a given l, the higher overtones are dampedmore rapidly.

At late times, we fit separately for the oscillatory and

TABLE I: Some values of the imaginary Schwarzschildquasi-normal-mode frequencies for different (n, l) (taken

from [73]).

n = 1 n = 2 n = 3 n = 4l = 2 −0.0890 −0.2739 −0.4783 −0.7051l = 3 −0.0927 −0.2813 −0.4791 −0.6903l = 4 −0.0942 −0.2843 −0.4799 −0.6839l = 5 −0.0949 −0.2858 −0.4803 −0.6786l = 6 −0.0953 −0.2866 −0.4806 −0.6786l = 7 −0.0955 −0.2872 −0.4807 −0.6773

TABLE II: Some values of the lowest overtone (n = 1)of the real Schwarzschild QNM frequency for

l = 2, 3, . . . , 7 taken from [73].

l = 2 l = 3 l = 4 l = 5 l = 6 l = 70.3737 0.5994 0.8092 1.0123 1.2120 1.4097

damped parts. We fit =(α(2)l ) by looking at the local max-ima of |σl| and fitting them to a straight line (on a loga-rithmic scale), while we fit

17

to depend sensitively on the time t(1) in Eq. (39). Thechoice t(1) = 4 was made to roughly minimize these vari-ations. Similarly, to get accurate values we choose to uset(2) = 20.

Let us now look at best fit values of the exponents.The best fit values for α(1) and α(2) are shown in Tab. IIIscaled with the ADM mass set to unity. Comparing thebest fits for the real and imaginary parts of α(2) withTable II and the first column of I, we find consistencyover all the 6 modes considered. This leads us to believethat at late times the shear modes are associated withthe fundamental overtone of the quasi-normal modes.

Things are not so clear with α(1). Recent work hasfound that in binary black hole merger waveforms, theimmediate post-merger signal is consistent with thehigher overtones of the quasi-normal modes [70, 76–78].It is thus tempting to think that α(1) should be connectedwith the higher overtones. However, comparing the bestfit values of α(1) in Table III with the complex frequen-cies for the higher overtones given in Table I, we find nocompelling evidence here. It is possible that a combina-tion of these higher overtones could be considered, butwe shall not attempt to do so here.

TABLE III: Fits of the shear modes based on thepiecewise-exponential model of Eqs. (39) and (40). We

show the coefficients α(1)l of early times (T < 4M) and

α(2)l for late times (T > 20M). For l < 7, we estimate

the errors of

18

0 5 10 15

T/M

10−5

10−3

10−1∫ξaξ a

dA

Ttouch

Souter

0 2 4 6 8

T/M

10−4

10−2

100

∫ξaξ a

dA

Ttouch

S1S2

FIG. 13: The behavior of the integral of |ξ|2 for H1, H2 and Houter. The behavior is qualitatively similar to |σ|2.

0 5 10 15

T/M

10−5

10−4

10−3

10−2

10−1

100

|ξ̄ l|

Ttouch

modes of ξ̄(ℓ) for Souterl = 1l = 2l = 3l = 4l = 5l = 6l = 7l = 8l = 9l = 10

FIG. 14: The mode amplitudes of ξ̄ for Houter forl = 1, 2, . . . 10. Again the qualitative behavior is similar

to the shear modes.

and after t(2) = 20. The best fits are shown graphicallyin Fig. 17. As before, we see clear evidence for the tworegimes: a steep initial decay followed by damped oscilla-tions. The best fit values are shown in Table IV. For thelate time behavior, we again get good agreement with thefundamental quasinormal mode frequencies and dampingtimes. Again, the case for identifying the early time steepdecay with any of the higher overtones is not very con-vincing.

VI. THE SLOWNESS PARAMETER

We have now seen, from apparently very different per-spectives, that we have two distinct post-merger regimesfor the outer horizon Houter. The first is immediately af-ter its formation, at Tbifurcate, where we see a rapid ap-proach to equilibrium. Thus, the stability spectrum be-comes very close to that of a round 2-sphere, the shearmodes and multipoles decay rapidly to zero. This regimeis followed by a much slower decay where oscillations inthe various fields are easily visible. The decay rates and

TABLE IV: Fits of the multipole moments based onEqs. (39) and (40) where we chose T (1) = 4M andT (2) = 20M. For l < 7, we estimate the errors of

19

0 1 2 3

proper distance/M

10−8

10−6

10−4

10−2

100

|I l|

S 1,S 2

touch

multipoles of S1 and S2I2I3I4I5I6I7I8

0 1 2 3 4 5 6 7

T/M

10−8

10−6

10−4

10−2

100

|I l|

Ttouch

multipoles of S1 and S2I2I3I4I5I6I7I8

FIG. 15: Multipoles of the two individual horizons as functions of time and separation between the black holes. Ineach case, the solid line refers to the multipoles of the smaller black hole S1, and the dotted line refers to the larger

black hole S2.

0 5 10 15 20

T/M

10−4

10−3

10−2

10−1

100

|I l| T t

ouch

multipoles of Souter and SinnerI2I3I4I5I6

68 70 72 74 76 78 80 82 84

area/M2

10−4

10−3

10−2

10−1

100

|I l|

bifurcation

Sinner Souter

multipoles of Sinner and Souter

I2I3I4I5I6I7

FIG. 16: Multipoles of the inner and outer common horizons as functions of time for l = 2, 3, . . . 6. The left panelshows the moments of the inner and outer horizons as functions of time. The tight panel treats the inner and outer

MOTSs as forming a single surface with the area as a coordinate.

ing the idea of black hole spectroscopy [85], i.e observa-tionally testing the black hole no-hair theorem using theringdown modes [86] (cf. also possible caveats to this in[87]).

Turning now to the properties of Houter, we have seenthat the rapid decay rates immediately after the mergerare not consistent with any single higher overtone. Thisdoes not rule out the possibility that several modes couldbe combined to accurately reproduce the decay functionthat we observe, but we shall not attempt to do so here.Furthermore, even if the immediate post-merger regimeis non-perturbative, it does not imply that the quasi-normal modes have no role to play: several modes couldbe present and could be coupled due to non-linear ef-fects. Here we wish to address this question in a differentway, namely by looking at evolution equations on Houter,identifying non-linear terms, and attempting to quantifytheir importance. We first need to identify which geo-metric quantities one should consider. In principle, thisquestion is closely tied to the free data on H, i.e. the

independent geometric fields that must be specified onH so that we can construct the spacetime in a neighbor-hood of H. This has been studied in [44]. As expected,the extrinsic curvatures of each MOTS in the null direc-tion are part of this free data. Our starting point willbe an equation we have encountered in paper I, namelythe evolution of the expansion Θ(V ) of the time evolutionvector V a in the membrane paradigm interpretation. Asin paper I, in terms of the null normals from Eq. (4), thetime evolution vector is V a = b`a + cna, and the vectororthogonal to Houter is W a = b`a − cna. We define alsoκ(V ) = −nbV a∇a`b. The qualitative average evolutionof the H can be understood in terms of two dynamicalmechanisms simultaneously in place. Each of these mech-anisms has an associated time scale. Following [88] whichdefined a slowness parameter using different timescales(though in a different context), and along the lines in[12, 13], we start from the equation ruling the evolution

20

0 10 20 30 40 50

T/M

10−4

10−3

10−2

10−1

100

101

|I l|

T (1)

T (2)

multipoles of Souter (l = 2)

multipolesearly dampinglate damping

0 10 20 30 40 50

T/M

10−5

10−4

10−3

10−2

10−1

100

101

|I l|

T (1)

T (2)

multipoles of Souter (l = 3)

multipolesearly dampinglate damping

0 10 20 30 40 50

T/M

10−5

10−3

10−1

101

|I l|

T (1)

T (2)

multipoles of Souter (l = 4)

multipolesearly dampinglate damping

0 10 20 30 40 50

T/M

10−5

10−3

10−1

101

|I l|

T (1)

T (2)

multipoles of Souter (l = 5)

multipolesearly dampinglate damping

0 10 20 30 40 50

T/M

10−7

10−5

10−3

10−1

101|I l

|T (1)

T (2)

multipoles of Souter (l = 6)

multipolesearly dampinglate damping

0 10 20 30 40 50

T/M

10−7

10−5

10−3

10−1

101

|I l|

T (1)

T (2)

multipoles of Souter (l = 7)

multipolesearly dampinglate damping

FIG. 17: Fits of the multipole moments for l = 2, 3, . . . , 7. The plots are similar to Fig. 12. The blue curves are themultipole moments of the apparent horizon as functions of time, and fits at early and late times are also shown.

0 2 4 6 8 10 12 14 16 18 20

T/M

0.0

0.5

1.0

P2

FIG. 18: The slowness parameter as a function of time.

of the expansion Θ(V ) encountered in Sec. VI of paper I:

LV Θ(V ) + Θ2(V ) = −κ(V )Θ(V ) + σ(V )ab σ

ab(W ) +

1

2Θ2(V )

+GabVaW b + (LV ln c)Θ(V )

+Da (cDab− bDac+ 2bc ωa) . (42)Here Gab is the Einstein tensor. Introducing the notionof a “deformation rate tensor” of S along V a (cf. e.g. [5])

Θ(V )ab =

1

2qdaq

dbLV qcd = σ(V )ab +

1

2qabΘ

(V ) , (43)

and analogously for W a, then using Θ(W ) = −Θ(V ), weeasily get

Θ(V )ab Θ

ab(W ) = σ

(V )ab σ

ab(W ) −

1

2Θ2(V ) . (44)

Eq. (42) can be cast as

LV Θ(V ) = −κ(V )Θ(V ) + Θ(V )ab Θab(W )+GabV

aW b + (LV ln c)Θ(V )+Da (cDab− bDac+ 2bc ωa) . (45)

Focusing on the leading terms of the right-hand-side weidentify two distinct driving mechanisms: a linear decayterm given by the κ(V )Θ(V ) and a non-linear term con-trolled by the deformation rate tensor of the intrinsicgeometry of the surface. We expect the linear regime tobe characterized by a suppression of strong variations inthe area element, and therefore a negligible value of its“acceleration”. This translates into a vanishing of the lefthand side in (45) as a signature of linearity. Introducinga “decay timescale” τ as

1

τ2=

1

AS

∮

Sκ(V )Θ(V )dA , (46)

and an “oscillation timescale” T controlled by the defor-

21

mation rate terms

1

T 2=

1

AS

∮

SΘ

(V )ab Θ

ab(W )dA

=1

AS

∮

S

(σ

(V )ab σ

ab(W ) −

1

2Θ2(V )

)dA , (47)

we define an instantaneous slowness parameter P [12, 13,88] as the ratio of the two time scales

P =T

τ. (48)

Transition to the linear regime would be marked by the“decay” and “oscillating” terms becoming commensurateand therefore P becoming of order one.

Admittedly, unlike in [88], the identification of the timescales with pure decay and oscillation is not so clear cuthere. We have seen that the shear also decays exponen-tially in time. In any event, regardless of this interpre-tation, the ratio P captures the ratio of the non-linearto linear term in Eq. (45). When P is close or exceedsunity, then the non-linear term will have a correspond-ingly smaller effect8. It is fairly straightforward to calcu-late this quantity for Houter, and the result is shown inFig. 18. It is clear that early times after the merger, Pis small indicating a larger effect of the non-linearities,while it gets close to unity at ≈ 8M. The non-linear ef-fects thus are not expected to dominate after this time,consistent with our observations of the spectrum, shear,and multipole moments. Appendix D briefly considersthe connection between the slowness parameter devel-oped here and the quality factor of a resonator [90]; inparticular expressed in terms of quasi-normal mode fre-quency and damping time [91].

VII. CONCLUSIONS

In this series of two papers we have studied in detailthe properties of marginally trapped surfaces in a head-on collision of two non-spinning black holes. Even in thissimple and otherwise well studied case, we find interest-ing geometric and physical behavior. Paper I has consid-ered the status of the area increase law and the associ-ated geometric properties. Here in the second paper, wehave studied the stability, the time evolution of fluxesacross the horizon and the multipole moments. We haveshown that the stability spectrum can be used to obtain

8 It is interesting to look at P from the perspective of thefluctuation-dissipation theorem [89] in statistical mechanics. Inrough terms, such a theorem states that (crucially, in the linearregime, near equilibrium), the relaxation rate and the fluctua-tions in a system satisfying a detailed balance are commensurate.In this sense, P ∼ 1 would mark the transition to a linear regimein which oscillations(/fluctuations) of the system equal its decayrate. Before linearity, there is no reason for this relation to hold.

greater insights into the merger process. We have shownthat the decay of fluxes and multipole moments for thefinal common horizon is consistent with the quasi-normalmode decay time. However, closer to Tbifurcate, the timewhen the common horizon is formed, the decay turns outto be much steeper. This holds for all the modes of theshear and for the various multipole moments as well. Theconsistency with the quasi-normal mode decay times isnot understood from first principles, but it is consistentwith the idea of a strong correlation between fields on thehorizon and the usual gravitational waveform observed atinfinity. We have explored two potential explanations ofthe faster decay just after Tbifurcate. The first is the pres-ence of higher overtones of the fundamental quasi-normalmode, and the second in terms of the slowness parame-ter. Both of these could potentially explain the behavior.As far as the horizons are concerned, estimates of thedecay rates of the shear modes and multipoles favor theslowness parameter.

Future work will consider more generic initial configu-rations allowing for the black holes to be spinning, andfor generic orbits. It should be possible to extend ournumerical methods for locating MOTSs to these generalsituations. This would allow us to tackle interesting ques-tions of interest from both astrophysical and mathemati-cal viewpoints. For example, do the fluxes and multipolemoments generically decay at the rate consistent withthe quasi-normal modes of the final spinning black hole?Is the early decay consistent with the higher overtonesand does the slowness parameter still provide a viableexplanation? On the mathematical side, the stability op-erator becomes non self-adjoint, and the question of sta-bility and zero-crossings of the eigenvalues become muchmore interesting and complex. This leads to deep con-nections with the spectral theory of non-self adjoint op-erators which will be explored in forthcoming work.

ACKNOWLEDGMENTS

We thank Abhay Ashtekar, Ivan Booth and Lamis AlSheikh for valuable comments and suggestions. Researchat Perimeter Institute is supported in part by the Gov-ernment of Canada through the Department of Innova-tion, Science and Economic Development Canada andby the Province of Ontario through the Ministry of