A hybrid method for understanding black-hole mergers: head-on case
aa r X i v : . [ g r- q c ] N ov Hybrid method for understanding black-hole mergers: Head-on case
David A. Nichols ∗ and Yanbei Chen † Theoretical Astrophysics 350-17, California Institute of Technology, Pasadena, California 91125, USA (Dated: November 15, 2010)Black-hole-binary coalescence is often divided into three stages: inspiral, merger and ringdown.The post-Newtonian (PN) approximation treats the inspiral phase, black-hole perturbation (BHP)theory describes the ringdown, and the nonlinear dynamics of spacetime characterize the merger.In this paper, we introduce a hybrid method that incorporates elements of PN and BHP theories,and we apply it to the head-on collision of black holes with transverse, anti-parallel spins. Wecompare our approximation technique with a full numerical-relativity simulation, and we find goodagreement between the gravitational waveforms and the radiated energy and momentum. Ourresults suggest that PN and BHP theories may suffice to explain the main features of outgoinggravitational radiation for head-on mergers. This would further imply that linear perturbations toexact black-hole solutions can capture the nonlinear aspects of head-on binary-black-hole mergersaccessible to observers far from the collision.
PACS numbers: 04.25.Nx, 04.30.-w, 04.70.-s
I. INTRODUCTION
Even prior to the complete numerical-relativity simula-tions of black-hole-binary mergers [1–4], black-hole colli-sions were thought to take place in three stages: inspiral(or infall), merger, and ringdown. During inspiral, thespeed of the holes is sufficiently low and the separationof the bodies is large enough that the system behaveslike two separate particles that follow the post-Newtonian(PN) equations of motion. Eventually, the black holes be-come sufficiently close that the dynamics given by the PNexpansion significantly differ from those of full relativity.This stage is the merger, during which the two black holesbecome a single, highly distorted, black hole. The mergerphase is brief; the strong deformations lose their energyto gravitational radiation, and the system settles downto a weakly perturbed, single black hole. The ringdownphase describes these last small oscillations of the blackhole.Of the three stages of binary-black-hole coalescence,merger remains the most inaccessible to analytical tools.Nevertheless, full numerical relativity is not the onlytechnique to have success investigating merger. Histori-cally, most analytical investigations of the merger phasearise from trying to extend the validity of perturbativetechniques, particularly black-hole-perturbation (BHP)and PN theories. Those researchers working from a BHPapproach try to push the approximation to hold at ear-lier times, whereas those employing a PN method at-tempt to stretch the technique to hold later into merger.Alternatively, one can see if there are exact, nonlinearanalytical models whose dynamics can represent variousaspects of black-hole-binary mergers. Rezzolla, Macedo,and Jaramillo [5] recently took this latter approach in ∗ Electronic address: [email protected] † Electronic address: [email protected] their study of anti-kicks from black-hole mergers. In theirwork, they showed that they could relate the curvatureanisotropy on the past apparent horizon of a Robinson-Trautman spacetime to the kick velocity (computed fromthe Bondi momentum). Through appropriate tuning ofthe initial data, they could recover kick velocities found innumerical-relativity simulations of unequal-mass, inspri-aling black holes. While this type of approach is interest-ing and has proved successful, our work focuses on usingperturbative approaches (and we will, therefore, take amore comprehensive look at the prior use of perturbativemethods to understand mergers).From the BHP side, Price and Pullin [6], initially, andmany collaborators, subsequently, (see, e.g., [7]) devel-oped the “close-limit approximation” (CLA). This tech-nique begins with initial data containing two black holesthat satisfy the vacuum Einstein equations and splits itinto a piece representing the final, merged black hole andperturbations upon that black hole. The exact form ofthe initial data varies in the CLA, but for head-on colli-sions of black holes, it typically involves some variationof Misner [8], Lindquist [9], or Brill-Lindquist [10] time-symmetric, analytic, wormhole-like solutions. To investi-gate the late stages of an inspiral, the CLA usually beginsfrom non-time-symmetric, but conformally flat, multipleblack-hole initial data set forth by Bowen and York [11].Independent of the initial data, however, the CLA trans-lates the original problem of merger into a calculationinvolving BHP theory. The CLA does not allow for avery large separation of the black holes; as a result, onlythe very end of the merger is captured in this process.Moreover, “junk radiation” appears in the waveform be-cause the initial data do not describe the binary black-hole-merger spacetime in both the wave zone and thenear zone. Unlike in full numerical-relativity simulationswhere the junk radiation leaves the grid during the wellunderstood inspiral phase, in the CLA the junk radiationappears during the merger stage. This radiation (bothfrom the absence of waves in the initial data and fromerrors in the near-zone physics), therefore, is difficult todisentangle from the physical waveform.The Lazarus Project (see e.g. [12]) followed roughlythe same approach as the CLA, but it used even morerealistic black-hole-binary initial data for its CLA calcu-lation; namely, its initial data came from a numerical-relativity simulation just prior to merger. At the sametime, however, because the initial data is now numeri-cal, one loses the analytical understanding of the proper-ties of spacetime near merger. More recently, Sopuerta,Yunes, and Laguna [13] applied the CLA in combina-tion with PN flux formulae to obtain an estimate of thegravitational recoil from unequal-mass binaries (includ-ing binaries with small eccentricity [14]). They proposedusing more realistic initial data in the CLA, which LeTiec and Blanchet [15] ultimately carried out. Le Tiecand Blanchet chose to use the 2PN metric (keeping onlythe first post-Minkowski terms) as initial data for theCLA, and they applied it with considerable success toinspirals of unequal-mass black-hole binaries in a paperwith Will [16]. Despite the improved initial data, thisapproach does not eliminate the problem of junk radi-ation discussed above. It would be of interest to see ifeven more realistic initial data, such as that of Johnson-McDaniel, Yunes, Tichy, and Owen [17] would lead toimproved results within the CLA.From the PN side, Buonanno et al., [18], as well asDamour and Nagar [19] take a different approach to un-derstanding the physics of merger. Using the Effective-One-Body (EOB) method [20], they study the dynamicsof the system until roughly the beginning of the mergerphase. To obtain a complete waveform, they attacha ringdown waveform by smoothly fitting quasinormalmodes to the EOB inspiral and plunge waveform. Whenthey calibrate the two free parameters of this model tonumerical-relativity waveforms, the EOB results matchnumerical-relativity waveforms precisely. In this method,they fit the PN dynamics and the ringdown at the lightring of the EOB particle motion; it is not immediatelyapparent, however, what this feature tells about the na-ture of merger.As a result, there remains a need to develop simple ana-lytical models that help reveal the behavior of spacetimeduring merger. Toward this end, it is helpful to delvedeeper into the question of what exactly is the merger.First, the inspiral-merger-ringdown classification is basedon the validity of the PN expansion and that of BHP.The inspiral, in other words, is just the set of times forwhich the PN expansion holds on the whole spatial do-main (to a given level of accuracy). Correspondingly, theringdown is the times for which BHP works everywherethroughout space. Merger, in this picture, is just the gapbetween those times during which PN and BHP theoriesgive accurate results.In this paper, we propose that we can push each ap-proximation technique beyond its current range of use, aslong as we do not apply it to all of space at a given time.We observe that at any time, there is a region outside a certain radius from the center of mass in which BHPapplies. While this seemingly runs contrary to the com-mon notion that PN theory is the natural description ofthe weak-field region of a black-hole-binary spacetime, ablack-hole metric in the limit of radii much larger thanthe mass and binary separation is identical to that ofPN in the same region. If the PN expansion appliesto the remaining portion of the spacetime (within theregion where BHP holds), then BHP and PN theoriescould cover the entire physics of black-hole-binary coa-lescence. While it is somewhat unreasonable to supposethat PN theory truly applies to the strong-field regionof a binary-black-hole spacetime, revisiting Price’s treat-ment of non-spherical stellar collapse suggests that thismay not be essential.In Price’s 1972 paper [21], he addresses, among otherissues, why aspherical portions of stellar collapse quicklydisappear when, in fact, one could plausibly argue thecontrary. Namely, if any non-spherical perturbationsasymptote to the horizon (from the perspective of anobserver far away), they would remain visible to thisobserver indefinitely. Price realizes, however, that theexterior of a collapsing star is just the Schwarzschild so-lution (due to Birkhoff’s theorem, up to small perturba-tions), and perturbations on the Schwarzschild spacetimeevolve via a radial wave equation in an effective poten-tial. Moreover, he notes that when the surface of thestar passes through the curvature effective potential ofthe Schwarzschild spacetime, the gravitational perturba-tions induced by the star redshift. Finally, because theeffective potential reflects low-frequency perturbations,the spacetime distortions produced by the stellar interiorbecome less important, and observers outside the star ul-timately see it settle into a spherical black hole in a finiteamount of time. Most importantly, this argument doesnot depend strongly upon the physics of the stellar inte-rior; as long as there is gravitational collapse to a blackhole, this idea holds.In this paper, we adopt this idea, but we replace thestellar physics of the interior with a PN, black-hole-binary spacetime (see Fig. 1). While in Price’s case, thedivision of spacetime into two regions comes naturallyfrom tracking the regions of space containing the starand vacuum, in our case the split is somewhat more ar-bitrary; one simply needs to find a region in which bothPN and BHP theories hold, to some level of accuracy.How we choose the boundary between the two regionsand the quantities that we evolve are topics that will bediscussed in greater detail in Sec. II.To test the above idea in this paper, we study a head-oncollision of two black holes with transverse, anti-alignedspins and compare the waveforms and energy-momentumflux obtained from our approximation method with theequivalent quantities from full numerical simulations.Specifically, we organize this paper as follows. In Sec. II,we give a more detailed motivation for our model, andwe then present the mathematical procedure we use inour method, for an equal-mass, head-on collision of black m o s tl yd i r ec t p a r t H i n f a ll (cid:144) i n s p i r a l L t r a n s iti on H m e r g e r L m o s tl y t a il H r i ngdo w n L i n t e r i o r P o s t - N e w t on i a n exteriorBH perturbation A BCD egf -
50 0 50 100 150 200050100150200250300 r * t FIG. 1: (Color online) This figure depicts a spacetime dia-gram of a black-hole collision, modeled after Price’s descrip-tion of stellar collapse. We choose the trajectory of the twoholes as a way to separate the spacetime into an interiorand an exterior region. The exterior region is a perturbed,black-hole spacetime, whereas the interior is that of a post-Newtonian (PN) black-hole-binary system [shaded in yellow(light gray)]. The red (dark gray) region of the diagram showsthe place at which the effective potential of the black hole issignificantly greater than zero. This formalism allows us todivide the waveform into three sections: inspiral (or infall),which extends from the beginning of the binary’s evolutionuntil when the l = 2 effective potential of the exterior BHPspacetime starts to be exposed; merger, which extends fromthe end of inspiral to when the majority of the exterior poten-tial is revealed; and ringdown, which represents the remain-der. We overlay the even-parity, ( l = 2, m = 0) mode of thewaveform. holes. In Sec. III, we present an explicit calculation forthe head-on collision of two black holes with transverse,anti-aligned spins, and we compare waveforms, radiatedenergy and radiated linear momentum, from our modelwith the equivalent quantities from a full numerical-relativity simulation. In Sec. IV we discuss how ourmethod can help interpret the waveform during merger,and finally, in Sec. V, we conclude. We will use geometri-cal units ( G = c = 1) and Einstein summation conventionthroughout this paper. II. A DETAILED DESCRIPTION OF THEMETHODA. Further Motivation
Before going into the details of our procedure, it isworth spending some time discussing why our specificimplementation of PN and BHP theories will help avoidsome of the difficulties that arose in other methods in theintroduction and noting the limitations and assumptionsthat underlie our approach.It is certainly hard to argue that existing orders ofPN (up to v in the metric, for near-zone dynamics [22])and BHP (up to second order for Schwarzschild, see [23]for a gauge-invariant formulation) theories are accuratein the whole space, simultaneously. Nevertheless, it isplausible to argue that these approximation techniquescover different spatial regions at different times in a waysuch that each theory is either valid to a reasonable levelof accuracy or occupying a portion of spacetime that willnot influence physical observables where it fails. Usingan approach of this type, we aim to get the most out ofthe approximation methods.Specifically, we find that the following procedure givesgood agreement with the waveform of a numerical-relativity simulation presented in Sec. III. First, we havethe reduced mass of the binary follow a plunging geodesicin the Schwarzschild spacetime. Then, we divide this tra-jectory in half to make a coordinate radius (and thus acoordinate sphere) that passes through the centers of theblack holes. The set of all the coordinate spheres de-fines a time-like surface in spacetime. Finally, we applyPN theory within this time-like surface and BHP on theexterior. The two theories must agree on this time-likesurface, which we will subsequently call the shell.Matching PN and BHP theories on this shell has cer-tain advantages. Because BHP theory relies upon a mul-tipole expansion, this makes it necessary to treat the PNinterior in terms of multipoles of the potentials. For one,this is useful, because physical observables like the radi-ated energy and momentum very often do not need manymultipoles to find accurate results. (In fact, in our exam-ple in Sec. III, we see that the quadrupole perturbationsalone suffice.) Second, a multipole expansion may alsobe helpful for the convergence of the approach. For twopoint particles, for example, each multipole componentof the Newtonian potential U ( l ) N at the location of theparticles satisfies U ( l ) N < ∼ M/R , where M is the mass ofthe binary and R is the distance from the center of mass.This is small for much of the infall, when R ≫ M , andeven when the binary reaches what will be the peak of theeffective potential of the merged black hole, U ( l ) N ∼ / B. Details of the Implementation
The procedure that we follow can be broken down,more or less, into five steps: (i) we describe relevant physics in the PN interior; (ii) we match the metric ofthe PN interior to the BHP metric through a boundary;(iii) we explicitly construct the boundary between the PNand BHP spacetimes; (iv) we evolve the metric pertur-bations in the exterior Schwarzschild spacetime; (v) weextract the waveforms and compute the radiated energyand momentum. We shall devote a subsection to each ofthese topics below.Before we do so, however, it will be helpful to clar-ify our notation regarding the different coordinates weuse for the two metrics and the matching shell. In thePN coordinate system, we use Minkowski coordinates(
T, X, Y, Z ) and spherical-polar coordinates ( R, Θ , Φ)within spatial slices. We will only consider linear per-turbations to Minkowski in the harmonic gauge. For ourBHP, we employ Schwarzschild coordinates ( t, r, θ, ϕ ),and similarly, we only examine linear perturbations tothe Schwarzschild spacetime. As we will show in Sec.II B 3, we can match these descriptions when
R, r ≫ M .This procedure is accurate up to terms of order ( M/R ) in the monopole part of the metric and M/R in thehigher-multipole portions, assuming that we relate thetwo coordinate systems by T = t , Θ = θ ,
Φ = ϕ , R = r − M . (1)The identification above allows us to label every pointin spacetime by two sets of coordinates, ( t, r, θ, ϕ ) and( t, R, θ, ϕ ), where R = r − M . Because our programrelies upon applying PN theory in an interior region andBHP on the exterior, it is therefore natural to talk abouta coordinate shell at which we switch between PN andBHP descriptions of the space-time. On this shell, we canuse either the Minkowski or Schwarzschild coordinates.In keeping with the notation above, we shall denote theseparation of the binary by A ( t ) = 2 R ( t ) in PN coordi-nates and a ( t ) = 2 r ( t ) in Schwarzschild coordinates. Fi-nally, we will denote the radial coordinate on the bound-ary by adding a subscript s to the PN or Schwarzschildcoordinate radii [e.g., R s ( t ) = A ( t ) / r s ( t ) = a ( t ) / BHP r=R+MMatching ShellA(t) ≡ s (t) = 2(r s (t)−M) ≡ a(t) − 2MR PN FIG. 2: This figure shows, at a given moment in time, theSchwarzschild and PN radial coordinates, the binary separa-tion, and the position of the shell where we match the twotheories.
While we introduce a new method of matching PN andBHP theories, the idea of combining PN and BHP ap-proximations is not new. In fact, it is at the core of theEffective-One-Body formalism of Buonanno and Damour
PN Spacetime Matching Shell Perturbed Schwarzschild SpacetimeCoordinates ( t, R, θ, ϕ ) ( t, R s ( t ) , θ, ϕ ) or ( t, r s ( t ) , θ, ϕ ) ( t, r, θ, ϕ ), r = R + M Binary Separation A ( t ) A ( t ) or a ( t ) a ( t )Matching Radius R ( t ) = A ( t ) / R s ( t ) = a ( t ) / − M or r s ( t ) = A ( t ) / M r ( t ) = a ( t ) / [20]. In the EOB description, however, they match thepoint-particle Hamiltonians of PN and BHP theories,rather than joining the spacetime geometry. It wouldbe interesting, as a future study, to see whether one cancombine our procedure with that of the EOB to producea geometrical EOB approach. Le Tiec and Blanchet [15],on the other hand performed a more accurate matchingbetween PN and BHP in their close-limit calculation with2PN initial data, but their matching only takes place ona single spatial slice of initial data. It would also be in-teresting to extend their higher-order approach to ourprocedure as well.
1. The PN Interior Solution
For our method, we will need to describe the metric ofthe PN spacetime in the interior, which we do at leadingNewtonian order: ds = − (1 − U N ) dt − w i dtdX i + (1 + 2 U N ) δ ij dX i dX j . (2)Here U N is the Newtonian potential and w i is the grav-itomagnetic potential, and the index i runs over X , Y ,and Z . Our notation follows [25] [the above takes theresults of Eq. (2.1) of that paper]. We then expand theNewtonian potential, U N , into multipoles, keeping onlythe lowest multipoles necessary to complete the calcula-tion. In this paper the monopole and quadrupole piecessuffice (the dipole piece can always be made to vanish), U N ≈ U (0) N + U (2) N . (3)The quadrupole piece can be expressed as a term with-out angular dependence times a spherical harmonic, as isdone below, U (2) N = U ,mN Y ,m ( θ, ϕ ) . (4)We shall follow the same procedure with the gravitomag-netic potential, although here we will, temporarily, keepthe dipole term, w i ≈ w (1) i + w (2) i . (5)We will be able to remove the dipole term through gaugetransformations, but this discussion is much simpler on acase-by-case basis. When we write the gravitomagnetic potential, w , in spherical polar coordinates, we will beable to remove the radial component through a gaugetransformation. We will, therefore, consider just the θ and ϕ components of w , and when writing it in indexnotation, we will denote them with Latin letters fromthe beginning of the alphabet, (e.g. a, b = θ, ϕ ). We canthen expand the components w (2) a in terms of two vectorspherical harmonics, w (2) a = w ,m (e) ∇ a Y ,m ( θ, ϕ ) + w ,m (o) ǫ a b ∇ b Y ,m ( θ, ϕ ) . (6)Here ∇ a is the covariant derivative on a 2-sphere, and ǫ a b is the Levi-Civita tensor (with nonzero components ǫ θ ϕ = 1 / sin θ and ǫ ϕ θ = − sin θ ). A convenient abbre-viation for the two spherical harmonics above is w (2) a = w ,m (e) Y ,ma + w ,m (o) X ,ma . (7)The two terms are denoted by (e) and (o) as a shorthand for even and odd, because they transform as ( − l and ( − l +1 under parity transformations, respectively.The odd- and even-parity vector spherical harmonics aregiven explicitly by X l,mθ = − θ ∂Y lm ∂ϕ , X l,mϕ = sin θ ∂Y lm ∂θ (8)and Y l,mθ = ∂Y lm ∂θ , Y l,mϕ = ∂Y lm ∂ϕ , (9)respectively. These are the only parts of the PN metricthat will be necessary for our approach.
2. Matching to Perturbed Schwarzschild
We then note that the Schwarzschild metric takes theform ds = − (cid:18) − Mr (cid:19) dt + (cid:18) − Mr (cid:19) − dr + r d Ω , (10)where the last piece is the metric of a 2-sphere. We willuse r without any subscript to denote the Schwarzschildradial coordinate. When M ≪ r the Schwarzschild met-ric takes the form ds ≈ − (cid:18) − Mr (cid:19) dt + (cid:18) Mr (cid:19) dr + r d Ω . (11)By making the coordinate transformation R = r − M ,and identifying M/R with the monopole piece of theNewtonian potential, U (0) N , then one can find that theSchwarzschild metric takes the form of the Newtonianmetric in spherical coordinates, ds ≈ − (1 − U (0) N ) dt + (1 + 2 U (0) N ) × ( dR + R d Ω) . (12)This similarity between the PN and Schwarzschild met-rics suggests a way to match the two at the boundary. Wewill assume that the monopole piece of the Newtonianpotential becomes the M/r term in the Schwarzschildmetric. For the remaining pieces of the Newtonian met-ric (namely U (2) N and w (2) i ) we will translate them di-rectly into the Schwarzschild metric after performingany needed gauge transformations to make such a directmatch reasonable.The original works on perturbations of theSchwarzschild spacetime are those of Regge andWheeler [26] for the odd-parity perturbations and Zerilli[27] for the even-parity perturbations. Moncrief [28]then used a variational principle to show that onecan derive quantities from the metric perturbations ofRegge, Wheeler and Zerilli that satisfy a well-posed,initial-value problem in any coordinates that deviatefrom Schwarzschild at linear order in perturbationtheory. These quantities are related to the gravitationalwaves at infinity, and they satisfy a one-dimensionalwave equation in a potential. We follow Moncrief’sapproach in computing these so-called gauge-invariantmetric perturbation functions, but for our notation, weuse that of a recent review article by Ruiz et al. [29].Both the even-parity [transform as ( − l under parity]and odd-parity [transform as ( − l +1 under parity] per-turbations are not very difficult to find. By writing thePN metric in spherical-polar coordinates, ds = − (1 − U (0) N − U (2) N ) dt + (1 + 2 U (0) N + 2 U (2) N ) × ( dR + R d Ω) − w (2) b dtdx b , (13)where dx b = dθ, dϕ , one can see that the even-parityperturbations are nearly diagonal in the metric. In fact,at leading Newtonian order, it is exactly diagonal, be-cause the non-diagonal term coming from w ,m (e) arises ata higher PN order. We will show this explicitly in Sec.III. For this reason, we only consider the diagonal metriccomponents in the discussion below.The even-parity metric perturbations in Schwarzschildare often denoted( h l,mtt ) (e) = H l,mtt Y l,m , ( h l,mrr ) (e) = H l,mrr Y l,m , (14)( h l,mθθ ) (e) = r K l,m Y l,m , ( h l,mϕϕ ) (e) = r sin θK l,m Y l,m , a specialization of Eqs. (57)-(59) of Ruiz et al. Thus, bymatching the two metrics, one can see that H ,mtt = H ,mrr = K ,m = 2 U ,mN . (15) The odd-parity term is somewhat simpler, because thereis only one metric perturbation in Schwarzschild to matchat leading order,( h l,mtθ ) (o) = h l,mt X l,mθ , ( h l,mtϕ ) (o) = h l,mt X l,mϕ , (16)Eq. (61) of Ruiz et al. From this, one can find that h ,mt = − w ,m (o) . (17)The matching procedure thus gives a way to produceperturbations in the Schwarzschild spacetime.
3. The Boundary Shell
We now must find a boundary region where one canmatch a PN metric expanded in multipoles with a per-turbed Schwarzschild metric. For head-on collisions, wefind that the boundary can be a spherical shell whoseradius varies in time as the binary evolves. We can mo-tivate where this boundary should be with a few simplearguments, but the true test of the matching idea willcome from comparisons with exact waveforms from nu-merical relativity.We know that at early times and for larger separationsof the black holes, the PN spacetime is valid around thetwo holes; thus it is not unreasonable to suppose thatthe shell should have a radius equal to half the binaryseparation. Later in the evolution, BHP will be valid ev-erywhere, so the shell should asymptote to the horizon ofthe merged hole (as seen by outside observers). The tra-jectory of the shell should be smooth throughout the en-tire process, as well. Finally, the boundary should mimicthe reduced-mass motion of the system, which physicallygenerates the gravitational waves.A simple way to achieve this quantitatively is insteadof having the motion of the reduced mass follow thePN equations of motion, we impose that it undergoesplunging geodesic motion in the Schwarzschild space-time. Given that the exterior spacetime is a perturbedSchwarzschild, and that we are matching the two approx-imations on a shell that passes through the centers of thetwo black holes, it is just as reasonable to use a trajectoryin the Schwarzschild spacetime. Moreover, at large sepa-rations, the motion of the reduced mass of the system inboth Schwarzschild and PN are quite similar; we primar-ily choose the geodesic in Schwarzschild for its behaviorat late times. For completeness, we write down the dif-ferential equation we use to find the motion of a radialgeodesic in Schwarzschild. Since we think of the blackholes as point particles residing in the PN coordinatesystem, we write the evolution of the binary separation A ( t ), measured in the PN coordinates, dA ( t ) dτ = − p B − (1 − M/A ( t )) ,dtdτ = B (cid:18) − MA ( t ) (cid:19) − , (18)where B is a positive constant ( B = 1 − M/A (0) for ahead-on plunge from rest). This expression can be foundin many sources; see, for example [30]. The coordinateradius of the shell in the PN spacetime is just half thedistance A ( t ), R s ( t ) = 12 A ( t ) . (19)Since the PN and Schwarzschild radii are related by R = r − M , one can find that in the Schwarzschild coor-dinates, the position of the shell is given by: r s ( t ) = 12 A ( t ) + M . (20)In the Schwarzschild coordinates, since A ( t ) goes to2 M at late times, the radius of the shell asymptotesto the horizon. More specifically, let us first define δr s ( t ) = r s ( t ) − M and then note that at late times,the trajectory approaches the speed of light as it fallstoward the horizon. In terms of the tortoise coordi-nate, r ∗ = r + 2 M log[ r/ (2 M ) − v = t + r ∗ = const . . Writing this with respect to thevariable δr s ( t ), we find − t M ∼ δr s ( t )2 M + log (cid:18) δr s ( t )2 M (cid:19) . (21)We neglected the constant value of v because doing sohas no affect on finding the scaling of r s ( t ) at late times t . The equation above has a solution in terms of theLambert W function, W ( x ), given by δr s ( t ) ∼ W ( e − t/ (2 M ) ) . (22)Because we are interested in the behavior at large t , e − t/ (2 M ) is small, and we can use the leading-orderterm in the Taylor series for the Lambert W function, W ( x ) ∼ x + O ( x ). We find that δr s ( t ) ∼ e − t/ (2 M ) . (23)Thus, matching the spacetimes at half the PN separa-tion of the binary and having the separation of the binaryevolve via a Schwarzschild geodesic in the PN coordinatesystem makes the shell track the PN reduced-mass mo-tion at early times, but still head to the horizon at latetimes. We illustrate these different behaviors by plottingthe full trajectory of the shell r s ( t ), the trajectory of ashell that follows a plunging Newtonian orbit (which wedenote by r s, ( N ) ( t ) and which represents the behavior ofthe shell at early times), and e − t/ (2 M ) (the late-time be-havior of the shell) in Fig. 3. We choose the initial valuesof the trajectory to conform with those of the numericalsimulation with which we compare in Sec. III B. The up-per and lower insets show how the shell’s trajectory con-verges to the Newtonian behavior (at early times) andthe expected exponential decay at late times.While choosing the trajectory of the shell might havea slightly ad hoc feel, in future work we will develop aframework that determines the shell motion consistentlythrough radiation reaction. r / M s (t)r s,(N) (t)e −t/(2M) FIG. 3: (Color online) This figure shows the trajectory of theboundary shell as the solid blue (black) curve labeled by r s ( t ).The other two curves show the early- and late-time behaviorof the shell. The red (gray) dashed curve labeled by r s, ( N ) ( t )shows the trajectory of a shell that follows the Newtonianequations of motion for a plunge from rest. The green (lightgray) dashed and dotted curve [denoted by e − t/ (2 M ) ] showsthe exponential convergence to the horizon at the rate ex-pected in a Schwarzschild spacetime. The upper inset showshow the shell agrees with a Newtonian plunge from rest atearly times, and the lower inset shows how the shell convergesexponentially to the horizon at the expected rate.
4. Black-Hole Perturbations
One can then take the odd- and even-parity metricperturbations from the second subsection and transformthem into two quantities, the Regge-Wheeler and Zerillifunctions, respectively, that each satisfy a simple waveequation. We first treat the even-parity perturbations.Eqs. (63)-(65) of Ruiz et al. show how to take metricperturbations and transform them into the even-parity,gauge-invariant Zerilli function. Substituting our Eq.(15) into those three of Ruiz, we findΨ ,m (e) = 2 r (cid:26) U ,mN + r − M r + 3 M × (cid:20)(cid:18) − Mr (cid:19) U ,mN − r∂ r U ,mN (cid:21)(cid:27) , (24)where ∂ r is just the radial derivative with respect to theSchwarzschild radial variable. The odd-parity perturba-tions come directly from applying our Eq. (17) to Eq.(67) of Ruiz et al. This gives thatΨ ,m (o) = 2 r (cid:18) ∂ r w ,m (o) − r w ,m (o) (cid:19) . (25)The odd- and even-parity perturbations then evolveaccording to the Regge-Wheeler and Zerilli equations re-spectively, ( ∂ t − ∂ r ∗ + V l (e , o) ( r ))Ψ l,m (e , o) = 0 , (26)where r ∗ = r + 2 M log[ r/ (2 M ) −
1] is the tortoise coor-dinate. The potentials can be expressed most conciselyvia the expression V l (e , o) ( r ) = (cid:18) − Mr (cid:19) (cid:18) λr − Mr U l (e , o) ( r ) (cid:19) , (27)where λ = l ( l + 1) and U l (o) ( r ) = 1 , U l (e) ( r ) = Λ(Λ + 2) r + 3 M ( r − M )(Λ r + 3 M ) . (28)Here Λ = ( l − l + 2) / λ/ −
1. These expressionsfollow Eqs. (5.3)-(5.6) of [15].In our procedure, we find it easiest to evolve the Regge-Wheeler-Zerilli equations using a characteristic method.To do so, we define u = t − r ∗ and v = t + r ∗ and seethat the evolution equation becomes ∂ Ψ l,m (e , o) ∂u∂v + V l (e , o) Ψ l,m (e , o) . (29)We will now discuss how we evolve our Regge-Wheeler-Zerilli functions with the aid of Fig. 1.We must provide data in two places, the matching shell(in Fig. 1 it is the lower-left curve labeled by the points Aef gD ) and the initial outgoing characteristic (the linelabeled by AB on the lower right). Once we do this, how-ever, we can determine the Regge-Wheeler-Zerilli func-tions within the quadrilateral (with the one curved side) ABCD . We already discussed how we determine theshell in Sec. II B 3, and the data we provide along thatcurve are just the Regge-Wheeler [Eq. (25)] or Zerilli [Eq.(24)] functions restricted to that curve. The data wemust provide along AB are less well determined. If ourcomputational grid extended to past null infinity, thenwe could impose a no ingoing wave condition. At finitetimes, we can still impose this boundary condition, but itleads to a small spurious pulse of gravitational radiationat the beginning of our evolution. To limit the effects ofthis, we keep the shell at rest until the junk radiation dis-sipates, and then we begin our evolution. At this point,the data along the line AB more closely represent thoseof a binary about to begin its plunge.With these data, we can then evolve the Regge-Wheeler-Zerilli equations numerically, using the second-order-accurate, characteristic method described byGundlach et al. in [31]. The essence of this methodis that one can use the data at a point plus those atone step ahead in u and v , respectively, to get the nextpoint advanced by a step ahead in both u and v . Ex-plicitly, if one defines Ψ N = Ψ l,m (e , o) ( u + ∆ u, v + ∆ v ),Ψ W = Ψ l,m (e , o) ( u + ∆ u, v ), Ψ E = Ψ l,m (e , o) ( u, v + ∆ v ), andΨ S = Ψ l,m (e , o) ( u, v ), then one has thatΨ N = Ψ E + Ψ W − Ψ S − ∆ u ∆ v V l (e , o) ( r c )(Ψ E + Ψ W )+ O (∆ u ∆ v, ∆ u ∆ v ) , (30) where r c is the value of r at the center of the discretizedgrid. Because our boundary data do not always lie on oneof the grid points in the ( u, v )-plane, we must interpolatethe bottom point Ψ l,m (e , o) ( u, v ′ ) to fall at the same valueof v as the next boundary point at Ψ l,m (e , o) ( u + ∆ u, v ).As long as we do this interpolation with a quadratic or apolynomial of higher degree, it does not seem to influencethe second-order convergence of the method. Finally, wecan extract the Regge-Wheeler-Zerilli functions from theline BC in Fig. 1 as they propagate toward future nullinfinity.
5. Waveforms and Radiated Energy-Momentum
As we mentioned at the end of the previous section, it iseasy to find the Regge-Wheeler-Zerilli functions from theexterior of our computational grid; this is useful, becausethese functions are directly related to the gravitationalwaveform h , asymptotically. For radii much larger thanthe reduced gravitational wavelength, r ≫ λ GW / (2 π ),one has that h + − ih × = 12 r X l,m s ( l + 2)!( l − h Ψ l,m (e) + i Ψ lm (o) i − Y lm , (31)where − Y lm is a spin-weighted spherical harmonic. Theabove comes from Eq. (84) of Ruiz et al., which alsocontains a discussion about the spin-weighted harmonicsin an appendix. One can then substitute this into theusual expressions for the energy and momentum radiatedby gravitational waves, dEdt = lim r →∞ r π I | ˙ h + − i ˙ h × | d Ω , (32) dP i dt = lim r →∞ r π I n i | ˙ h + − i ˙ h × | d Ω , (33)(where n i is a unit vector, a dot represents a time deriva-tive, and d Ω is the volume element on a 2-sphere). Alengthy, but straight-forward calculation done by Ruiz etal. shows that dEdt = 164 π X l.m ( l + 2)!( l − (cid:16) | ˙Ψ l,m (e) | + | ˙Ψ l,m (o) | (cid:17) (34)[their Eq. (91)]. For the components of the momentumwe are interested in (in the xy -plane) combining theirEqs. (86)-(88) and (93) and using their definition P + = P x + iP y gives dP + dt = − π X l.m ( l + 2)!( l − h ia l,m ˙Ψ l,m (e) ˙¯Ψ l,m +1(o) + b l +1 ,m +1 (cid:16) ˙Ψ l,m (e) ˙¯Ψ l +1 ,m +1(e) + ˙Ψ l,m (o) ˙¯Ψ l +1 ,m +1(o) (cid:17)i . (35)The bar denotes complex conjugation. The coefficients a l.m and b l,m are given by their Eqs. (41) and (42), whichwe reproduce here a l.m = p ( l − m )( l + m + 1) l ( l + 1) (36) b l,m = 12 l s ( l − l + 2)( l + m )( l + m − l − l + 1) (37)With the framework now in place, we are prepared tomake a comparison with numerical relativity. III. HEAD-ON COLLISION OF SPINNINGBLACK HOLES WITH TRANSVERSE,ANTI-PARALLEL SPINS
In this section, we discuss the specific example of ahead-on collision of equal-mass black holes with trans-verse, anti-parallel spins. We will specialize the generalframework presented in Sec. II to the current configura-tion in the first subsection and then make the comparisonwith numerical relativity in the second.
A. The Hybrid Model for the Head-on Collision
We will mimic the configuration used in the numerical-relativity simulation for ease of comparison. We thuschoose our two black holes, labeled by A and B , to havemasses M A = M B = M/
2, to start with initial separa-tion X A = A (0) / − X B [ A (0) = 7 . M in the numer-ical simulations and Y A = Y B = Z A = Z B = 0] andto have their spins along ± Z axis, respectively (so that S ZA = 0 . M A and S ZB = − . M B and all other compo-nents of the spins are zero). Though they initially fallfrom rest, as in the numerical simulation, we will denotetheir speeds by V A and V B .
1. Even-Parity Perturbations
As we argued in Sec. II, the even-parity perturbationwill only rely upon the Newtonian potential, which hasthe familiar form, U N = M A R A + M B R B . (38)Here R A and R B denote the distance from the centersof black holes A and B in the PN coordinates. We thenexpand the Newtonian potential, U N , into multipoles,keeping only the monopole and quadrupole pieces (thedipole piece vanishes), U N = U (0) N + U (2) N (39)= MR + M A ( t ) R r π " Y , − − r Y , + Y , . Y l,m are the usual scalar spherical harmonics. One canthen see that the nonzero coefficients of the sphericalharmonics are U , ± N = r π M A ( t ) R = − r U , N . (40)After applying the transformation of the PN andSchwarzschild radial coordinates, R = r − M , (and simi-larly A ( t ) = a ( t ) − M ) one finds that U , ± N = r π M ( a ( t ) − M ) r − M ) = − r U , N . (41)One can then substitute this into Eq. (24) to find theZerilli function,Ψ , ± = r π M a ( t ) r (cid:18) − M r (cid:19) = − r
32 Ψ , . (42)We have only kept terms to linear order in M/r in thiscalculation, because we only use Newtonian physics tocalculate the gravitational potential. At the boundary of r s ( t ) = a ( t ) /
2, the perturbation is constant at leadingorder, and varying only at higher orders.Ψ , ± (cid:12)(cid:12)(cid:12) shell = r π M (cid:18) − M a ( t ) (cid:19) , (43)Ψ , (cid:12)(cid:12)(cid:12) shell = − r π M (cid:18) − M a ( t ) (cid:19) . (44)
2. Odd-Parity Perturbations
The calculation with the gravitomagnetic potentialis slightly more difficult, because it involves additionalgauge transformations. The gravitomagnetic potential isgiven by w i = ǫ ijk S j N kA R A + M A V iA R A + ǫ ijk S j N kB R B + M B V iB R B . (45)These results appear, for example, in Eq. (6.1d) of [25].The new symbols N kA and N kB represent unit vectorspointing from the centers of the two black holes in the PNcoordinates, and V iA and V iB are the velocities of the twoblack holes. Expanding the gravitomagnetic potential toleading order in A ( t ) and simplifying the trigonometricportions of the equations below, we see that w x = M V A ( t ) sin θ cos ϕ R − A ( t ) S sin θ sin 2 ϕ R , (46) w y = − A ( t ) S (1 + 3 cos 2 θ − θ cos 2 ϕ )8 R , (47) w z = 0 , (48)The variables S and V are just the magnitudes of thespin and velocity of each black hole, respectively. For0this equal-mass collision, the spins have the same mag-nitude, and the velocities of the holes do, as well. Wemust then convert the gravitomagnetic potential intospherical-polar coordinates, w R = M V A ( t ) sin θ cos ϕ R − A ( t ) S sin θ sin ϕ R , (49) w θ = M V A ( t ) sin 2 θ cos ϕ R − A ( t ) S cos θ sin ϕ R , (50) w ϕ = − M V A ( t ) sin θ sin 2 ϕ R + A ( t ) S cos ϕ (5 sin θ − θ )8 R . (51)There is a dipole term in the component w R of the grav-itomagnetic potential, and this term will not evolve ac-cording to the Regge-Wheeler-Zerilli equation. One canremove it via the small gauge transformation to the met-ric, ˆ h αβ = h αβ − ξ α | β − ξ β | α , (52)where the bar refers to a covariant derivative with respectto the background metric (in this case flat space). Recallthat the metric components, h ti , are related to the grav-itomagnetic potential, w i , by h ti = − w i . If we make agauge transformation where the only nonzero componentof ξ µ is ξ t = 2 M A ( t ) V sin θ cos ϕR − A ( t ) S sin θ sin ϕR , (53)this has several important effects. For one, it eliminatesˆ h tr , and it introduces a term, − ξ t | t = − M sin θ cos ϕR (cid:18) V − M A ( t )2 R (cid:19) , (54)into h tt . This term, however, is of 1PN order, and, sincewe are considering only the leading Newtonian physics,we will drop it. Then, it turns the remaining pertur-bation into the sum of odd- and even-parity quadrupoleperturbations. Letting b = θ, ϕ , one has thatˆ h tb = − A ( t ) SR r π X , b − X , − b ) , (55) − r π
15 4
M A ( t ) VR Y , b − r Y , b + Y , − b ! . If one were to include the even-parity, vector-harmonicterm in the Zerilli function, one would need to takeits time derivative. This means it enters as a next-to-leading-order effect, and we can ignore that term in ourleading-order treatment. Thus, the relevant perturbationof the gravitomagnetic potential is w , = r π A ( t ) S R = − w , − . (56) Finally, we make the transformation to the Schwarzschildradial coordinate, R = r − M (and similarly for A ( t ) = a ( t ) − M ), to find that w , = r π a ( t ) − M ) S r − M ) = − w , − . (57)We can then find the Regge-Wheeler function from Eq.(25) which isΨ , = − r π a ( t ) Sr = − Ψ , − . (58)As before, we keep only terms linear in M/r . At theboundary, the odd-parity perturbation isΨ , (cid:12)(cid:12)(cid:12) shell = − Sa ( t ) r π − Ψ , − (cid:12)(cid:12)(cid:12) shell . (59)
3. Energy and Momentum Fluxes
Finally, because we only have quadrupole perturba-tions, the expressions for the energy and momentumfluxes greatly simplify. The energy flux, for the l = 2modes (taking into account that the m = ± m = ± E = 38 π (cid:20) (cid:16) ˙Ψ , (cid:17) + 2 (cid:16) ˙Ψ , (cid:17) (cid:21) , (60)and the momentum flux is given by˙ P y = 1 π ˙Ψ , ˙Ψ , . (61)We have also used the fact that Ψ , = p / , ± inthis head-on collision. B. Comparison with Numerical Relativity
In this section we compare the results of our head-on collision of spinning black holes (with transverse,anti-parallel spins) with the equivalent results froma numerical-relativity simulation (see the paper byLovelace et al. [32] for a complete description of the sim-ulation). Although the paper by Lovelace et al. dealtmostly with using the Landau-Lifshitz pseudotensor todefine a gauge-dependent 4-momentum and an effectivevelocity to help develop intuitive understanding of black-hole collisions, they also investigated the gauge-invariantgravitational waveforms and radiated energy-momentum(calculated from the gravitational waves at large radii).We will not attempt to study any of these Landau-Lifshitz quantities in this work, and, instead, we will justlook into the gauge-invariant radiated quantities. Specifi-cally, for our comparison, we focus on the waveforms (the1 l = 2 modes of the gravitational waves) and the radiatedenergy and momentum.In Figs. 4 and 5, we compare, respectively, the even-parity perturbation Ψ (2 , and the odd-parity perturba-tion Ψ (2 , from our method with the equivalent quan-tities from the numerical simulation S1 featured inLovelace et al. (Since the l = 2, m = − , − , r = 3 M , in the hybrid method.This is the peak of the effective potential, and due tothe low-frequency opacity of this potential, much of theinfluence of the boundary data is hidden within the po-tential after this time (and the waveform is due mostlyto the quasinormal modes of the final black hole). Be-fore this time the match is not exact (as a result of junkradiation in the numerical simulation and the differencebetween the time coordinates), but the Newtonian orderperturbations do quite a good job of exciting quasinormalmodes of a reasonable amplitude. −60 −40 −20 0 20 40 60 80−0.1−0.0500.050.10.15 t/M Ψ ( , ) e v en Numerical RelativityHybrid Method
FIG. 4: (Color online) The blue (dark gray) dashed curve isΨ , from our hybrid method, whereas the black solid curveis the same quantity in full numerical relativity. The red (lightgray) dashed vertical line corresponds to the retarded time atwhich the shell in the hybrid method reaches the light ring ofthe final black hole, r = 3 M . We shift the numerical-relativitywaveform so that the peaks of the numerical and hybrid Ψ , waveforms align. The even- and odd-parity waveforms in the hybridmethod are the pieces of Ψ (2 , and Ψ (2 , restricted tothe outer boundary of the characteristic grid, labeled by BC in Fig. 1. We found these perturbations throughthe procedure described in Sec. II B, applied to the spe-cific binary parameters described in Sec. III A. For thenumerical-relativity waveforms, we chose to present themin terms of the even- and odd-parity perturbation func-tions Ψ (2 , and Ψ (2 , , as well. To find these perturba-tion functions from the numerical simulations, we first −60 −40 −20 0 20 40 60 80−0.02−0.015−0.01−0.00500.0050.010.0150.020.0250.03 t/M Ψ ( , ) odd Numerical RelativityHybrid Method
FIG. 5: (Color online) The blue (dark gray) dashed curve isΨ (2 , from our hybrid method, whereas the black solid curveis the same quantity in full numerical relativity. The red (lightgray) dashed vertical line corresponds to the retarded time atwhich the shell in the hybrid method reaches the light ring ofthe final black hole, r = 3 M . We shift the numerical-relativitywaveform so that the peaks of the numerical and hybrid Ψ , waveforms align. twice integrated the Weyl scalar Ψ with respect to timeto get the waveforms h + and h × (since − Ψ = ¨ h + − i ¨ h × ,at large radii, where a dot denotes a time derivative).One can relate them to the gravitational waveforms h + and h × by Eq. (31), at large r . In the case of the l = 2 perturbations shown here, rh (2 , = √ (2 , and rh (2 , × = √ (2 , . We compared the h + and h × founddirectly from the numerical simulation through extrac-tion of the Regge-Wheeler and Zerilli functions from met-ric coefficients in the numerical code [33], and the twoprocedures gave essentially identical results.In order to make the comparison between our hy-brid method and the full numerical-relativity waveforms,we must shift the numerical waveforms by a constant.Specifically, we choose this constant so that the peaks ofthe exact and approximate waveforms of Ψ (2 , match (ata time that we set to be t = 0). We add this constant shiftin time, because there is no clear relationship between thecoordinate time at which the waveform in our code beginsand the same coordinate time in the numerical-relativitysimulation. Trying to find a relationship between thesetimes is complicated by the fact that the hybrid methodevolves on a characteristic grid, whereas the numerical-relativity simulation solves an initial-value problem ina gauge that changes as the black holes move together.Nevertheless, because both the numerical and the hy-brid method use asymptotically flat coordinates, at largeradii, the time coordinates move at the same rate. This,in turn, means that it is only necessary to shift the timecoordinates rather than rescaling them. As an interestingaside, Owen [34] found that this agreement between the2time coordinates in the numerical simulation and pertur-bation theory appears to extend even into the near zone,when he observed that multipole moments of the hori-zon oscillate at the quasinormal mode frequencies of theblack hole. See the end of Sec. III of that paper for adiscussion of why that might be the case.We also compute the momentum flux, and we show theaccumulation of the velocity of the final black hole in Fig.6, for both our method and the full numerical-relativitysimulation. For our hybrid method, we use just the l = 2modes of the waveform to compute the momentum flux,˙ P y , our Eq. (61). We then find the velocity of the finalblack hole as a function of time by computing v y ( t ) = − M Z tt ˙ P y ( t ′ ) dt ′ , (62)where we introduce an extra minus sign to account forthe fact that the black-hole’s velocity is opposite that ofthe momentum carried by the gravitational waves. Forthe numerical waveform, we show the equivalent velocitycomputed from the full Weyl scalar, Ψ . For the numer-ical simulations, one typically computes˙ P y = lim r →∞ r π I sin θ sin φ (cid:12)(cid:12)(cid:12)(cid:12)Z t −∞ Ψ dt ′ (cid:12)(cid:12)(cid:12)(cid:12) d Ω , (63)where Ψ is the Weyl scalar extrapolated to infinity, and d Ω is the surface-area element on a unit sphere. This ex-pression appears in a variety of sources [see, for example,Eq. (29) of the paper by Ruiz et al.]. We then can com-pute the velocity of the final black hole in the numericalsimulations through Eq. (62), as we did for the hybridmethod. Again, we perform the same time-shifting pro-cedure as we did with the waveforms. The kick we find isremarkably close; 22 km/s for the numerical simulationand 25 km/s for our hybrid method.The radiated energy does not agree quite as well dueto the fact that the even-parity perturbation is some-what larger than the equivalent numerical quantity (andit is the dominant contribution to the energy flux). Nev-ertheless, the results agree within a factor of two; thenumerical simulation shows that roughly 0 . . −60 −40 −20 0 20 40 60 800510152025 t/M v k i ck ( k m / s ) Numerical RelativityHybrid Method
FIG. 6: (Color online) The blue (dark gray) dashed curveis the velocity of the final black hole as a function of time(inferred from the gravitational radiation) from our hybridmethod, using only the l = 2 modes of the wave. The blacksolid curve is the equivalent quantity in numerical relativity,computed from the full Weyl scalar, ψ . The red (light gray)dashed vertical line corresponds to the retarded time at whichthe shell in the hybrid method reaches the light ring of thefinal black hole, r = 3 M . As before, the numerical-relativityvelocity is shifted so that the peaks of the numerical and hy-brid Ψ , waveforms align. total energy that will be radiated in the head-on colli-sion, has escaped. This poses a small problem for EOBapproaches, because as described in the introduction tothis paper, Sec. I, one must choose a point at which tostop the EOB inspiral-and-plunge waveform and match itto a set of quasinormal modes to obtain a full waveform.For inspiraling black holes, there is a natural point to dothis: when the frequency of the inspiral-plunge dynamicsapproaches the quasinormal mode frequencies of the finalblack hole that will be formed. For a head-on collision,however, there is no analogous frequency at which onecan match. We will, therefore, reserve any comparisonsbetween our method and that of EOB for future work,when we extend our method to inspiraling, black-holebinaries. IV. THE THREE STAGES OF BLACK-HOLEMERGERS
In addition to producing reasonably accurate full wave-forms, our approach also provides a possible interpreta-tion of the infall, merger and ringdown stages of a binary-black-hole merger. As shown in Fig. 1, before the shellreaches point e and enters the strong-field region [the red(dark gray) area, in which the l = 2, even-parity, effec-tive potential exceeds 1 / e , wavesthat scatter off the effective potential (and thereby prop-agate within the light cone) become more significant.These waves often are called tail waves. Although PNwaveforms do include the tail part, the fact that thehigher-order PN terms that contain the tail dominateover the lower-order terms [24] does not bode well forthe ability of the PN series to easily capture this ef-fect. Nevertheless, we are able to associate this mixtureof direct and tail portions in the waveform to merger.In our model, this stretch of the waveform is related tothe retarded times when the shell is passing through thestrong-field region of spacetime (points e , f , and g in thediagram).Finally, after the shell passes through the potential, thedetails of the perturbation no longer become important,as was found by Price in his stellar collapse model. Be-cause waves do not efficiently propagate through the bar-rier, the gravitational waveform associated with points g through D should arise from before and while the shellpasses through the effective potential (not after). Thislast piece is that of ringdown.There is one subtlety to note about our interpretationof ringdown that might arise if the final black hole is aKerr black hole. Mino and Brink [37] and Zimmermanand Chen [38] showed that mergers that lead to a Kerrblack hole can emit waves at integer multiples of the hori-zon frequency that decay at a rate proportional to thesurface gravity. These modes come from a calculation inthe near-horizon limit of a Kerr black hole, and from thevantage point of observers far away, these waves wouldappear to be coming from the horizon. These modeshave a sufficiently high frequency that they could pen-etrate the effective potential of a Kerr black hole andcontribute to the ringdown portion of the gravitationalwave. Nevertheless, if we expand our description of theringdown phase to include these horizon modes, our in-terpretation holds more or less as described above. V. CONCLUSION
In this paper, we show, by examining the head-on col-lisions of spinning black holes, that a combination of PNand BHP theories gives a gravitational waveform thatmatches well with that of full numerical-relativity sim- ulations. We were able to do this not by applying theapproximation methods to distinct times in the evolu-tion of the system, but by choosing regions of spacein which the two methods work and finding that thewaveform from black-hole-binary collisions can be pro-tected from lack of convergence in these approximations.Specifically, our method lumps all monopole pieces of aPN black-hole-binary spacetime into the Schwarzschildmetric and treats the higher multipoles as perturbationsof that Schwarzschild that evolve via a wave equation.Moreover, since PN and BHP theories describe the wave-form, this suggests that much of the nonlinear dynamicsappearing in the gravitational waves of a head-on black-hole-binary merger can be well approximated by linearperturbations of the Schwarzschild solution.Our approach certainly cannot replace full numericalsimulations. For one, we must test its validity for differ-ent kinds of coalescence by comparison with fully nonlin-ear numerical results. Nevertheless, we are hopeful thatour method maybe be useful for gaining further under-standing of the spacetime of black-hole-binary mergersand for producing templates of gravitational waveformsfor data analysis. To move towards these goals, we wouldneed to make several modifications to our method (whoseimplementation we leave for future work). Most of thesechanges revolve around finding a way to treat inspirals ofblack-holes binaries within our method. The most neces-sary addition would be finding a way to consistently treatradiation reaction within the formalism. This feature isessential for capturing the correct inspiral and plunge dy-namics. Also important for describing realistic physics ofthe ringdown would be to analyze the problem in a Kerrbackground. Each of these problems requires significantwork, so we leave them for future studies.
Acknowledgments
We thank Geoffrey Lovelace and Uli Sperhake forsupplying waveforms and energy-momentum fluxes fromtheir numerical simulations; we thank Lee Lindblom,Mark Scheel and B´ela Szil´agyi for advice on solving waveequations with characteristic methods. We thank DrewKeppel for his input in discussions during the early stageof this work, and we thank Kip S. Thorne and YasushiMino for discussing related aspects of black-hole physicswith us. This work has been supported by NSF GrantsNo. PHY-0601459, PHY-0653653 and CAREER GrantPHY-0956189, by the David and Barbara Groce start-upfunds at the California Institute of Technology, and bythe Brinson Foundation. D.N.’s research was supportedby the David and Barbara Groce Graduate Research As-sistantship at the California Institute of Technology. [1] F. Pretorius, Phys. Rev. Lett. , 121101 (2005). [2] M. Campanelli, C. O. Lousto, P. Marronetti, and Y. Zlo- chower, Phys. Rev. Lett. , 111101 (2006).[3] J. G. Baker, J. Centrella, D.-I. Choi, M. Koppitz, and J.van Meter, Phys. Rev. Lett. , 111102 (2006).[4] M. A. Scheel, M. Boyle, T. Chu, L. E. Kidder, K. D.Matthews, and H. P. Pfeiffer, Phys. Rev. D , 024003(2009).[5] L. Rezzolla, R. P. Macedo, and J. L. Jaramillo, Phys.Rev. Lett. , 221101 (2010).[6] R. H. Price and J. Pullin, Phys. Rev. Lett. , 3297(1994).[7] G. Khanna, J. Baker, R. J. Gleiser, P. Laguna, C. O.Nicasio, H.-P. Nollert, R. Price, and J. Pullin, Phys. Rev.Lett. , 3581 (1999).[8] C. W. Misner, Phys. Rev. , 1110 (1960).[9] R. W. Lindquist, J. Math. Phys. , 938 (1963).[10] D. R. Brill and R. W. Lindquist, Phys. Rev. , 471(1963).[11] J. M. Bowen and J. W. York, Phys. Rev. D , 2047(1980).[12] J. Baker, M. Campanelli, and C. O. Lousto, Phys. Rev.D , 044001 (2002).[13] C. F. Sopuerta, N. Yunes, and P. Laguna, Phys. Rev. D , 124010 (2006).[14] C. F. Sopuerta, N. Yunes, and P. Laguna, Astrophys. J. , L9 (2007).[15] A. Le Tiec and L. Blanchet, Class. Quant. Grav. ,045008 (2010).[16] A. Le Tiec, L. Blanchet, and C. Will, Class. Quant. Grav. , 012001 (2010).[17] N. K. Johnson-McDaniel, N. Yunes, W. Tichy, B. J.Owen, Phys. Rev. D , 124039 (2009).[18] A. Buonanno, Y. Pan, H. P. Pfeiffer, M. A. Scheel, L. T.Buchman, and L. E. Kidder, Phys. Rev. D , 124028(2009).[19] T. Damour and A. Nagar, Phys. Rev. D , 081503(R) (2009).[20] A. Buonanno and T. Damour, Phys. Rev. D , 064015(2000).[21] R. H. Price, Phys. Rev. D, , 2419 (1972).[22] L. Blanchet and G. Faye, Phys. Rev. D , 062005 (2001).[23] A. Garat and R. H. Price, Phys. Rev. D , 044006(2000).[24] E. Racine, A. Buonanno, and L. Kidder, Phys. Rev. D , 044010 (2009).[25] J. D. Kaplan, D. A. Nichols, and K. S. Thorne, Phys.Rev. D , 124014 (2009).[26] T. Regge and J. A. Wheeler, Phys. Rev. , 1063(1957).[27] F. J. Zerilli, Phys. Rev. D , 2141 (1970).[28] V. Moncrief, Ann. Phys. (N.Y.) , 323 (1974).[29] M. Ruiz, M. Alcubierre, D. N´u˜nez, and R. Takahashi,Gen. Rel. Grav. , 1705 (2008).[30] H. Stephani. General Relativity: An Introduction to theTheory of the Gravitational Field, nd ed . CambridgeUniversity Press, Cambridge, 1990, Sec. 22.2.[31] C. Gundlach, R. H. Price, and J. Pullin, Phys. Rev. D , 883 (1994).[32] G. Lovelace, Y. Chen, M. Cohen, J. D. Kaplan, D. Kep-pel, K. D. Matthews, D. A. Nichols, M. A. Scheel, andU. Sperhake, Phys. Rev. D , 064031 (2010).[33] O. Rinne, L. T. Buchman, M. A. Scheel, H. P. Pfeiffer,Class. Quant. Grav. , 075009 (2009).[34] R. Owen, Phys. Rev. D , 084012 (2009).[35] R. J. Gleiser, C. O. Nicasio, R. H. Price, and J. Pullin,Phys. Rev. Lett. , 4483 (1996).[36] C. K. Mishra and B. R. Iyer, arXiv:1008.4009.[37] Y. Mino and J. Brink, Phys. Rev. D78