We consider the initial data problem for several black holes in vacuum with
arbitrary momenta and spins on a three space with punctures. We compactify the
internal asymptotically flat regions to obtain a computational domain without
inner boundaries. When treated numerically, this leads to a significant
simplification over the conventional approach which is based on throats and
isometry conditions. In this new setting it is possible to obtain existence and
uniqueness of solutions to the Hamiltonian constraint.Comment: 4 pages, LaTeX (RevTeX), minor changes, improved presentation, to
  appear in PR

Brandt, Steven

Bruegmann, Bernd

English

arXiv

Brandt, S.

Brügmann, B.

MPG.PuRe

VOLUME 78, NUMBER 19 P H Y S I C A L R E V I E W L E T T E R S 12 MAY 1997andtaincantInnian3606A Simple Construction of Initial Data for Multiple Black HolesSteven Brandt* and Bernd Brügmann†Max-Planck-Institut für Gravitationsphysik, Schlaatzweg 1, 14473 Potsdam, Germany(Received 11 February 1997)We consider the initial data problem for several black holes in vacuum with arbitrary momentaspins on a three space with punctures. We compactify the internal asymptotically flat regions to oba computational domain without inner boundaries. When treated numerically, this leads to a signifisimplification over the conventional approach which is based on throats and isometry conditions.this new setting it is possible to obtain existence and uniqueness of solutions to the Hamiltoconstraint. [S0031-9007(97)03144-X]PACS numbers: 04.20.Ex, 04.25.Dm, 04.70.Bw, 95.30.Sflygerocasrtldasee.atoe-edtsedtricu-al-nye-ntBinary black hole spacetimes are one of the great chlenges for numerical general relativity, even if no mater sources are present. Here we consider the probof finding initial data for several black holes in vacuumwith arbitrary momenta and spins. In general relativitinitial data on a hypersurface cannot be specified freebecause the Einstein equations give rise to four equatiothree momentum constraints, and the Hamiltonian costraint that the initial data has to satisfy. The purposethis Letter is to introduce a novel approach which is sinificantly simpler than the conventional method basedthroats and conformal imaging.In all that follows we will assume vacuum, that thmetric is conformally flat, and that the extrinsic curvatuis tracefree. A convenient form of the constraintsgeneral relativity can be obtained by rescaling the physithree-metricgphab and its extrinsic curvatureKphab by aconformal factorc,gphab ­ c4gab , Kphab ­ c22Kab . (1)The momentum constraint becomes=aKab ­ 0 , (2)and the Hamiltonian constraint becomes an elliptic eqution for the scalar fieldc ,Dc 118 KabKabc27 ­ 0 , (3)where the covariant derivatives are defined by the flmetric gab , which is also used to raise and lower indice(see [1,2]).In order to obtain black hole vacuum data, one hto introduce a nontrivial topology. The first calculationwere performed by Einstein and Rosen [3] in thework on point particles in general relativity. Variouconstructions for black holes based on Einstein-Rosbridges and “wormholes” were given in, e.g., [4–7]. Thspatial slice typically consists of two or more copies ofR3with several spheres removed and identifications of tvarious spherical inner boundaries. In this way seveasymptotically flat regions are obtained that are connecby bridges or “throats.”0031-9007y97y78(19)y3606(4)$10.00al-t-em,ly,ns,n-of-onefala-atsssirenehealedThe simplest example derives from the Schwarzschispacetime in quasi-isotropic coordinates. Considereda problem onR3 minus the pointr ­ 0, the constraintequations (2) and (3) are solved byc ­ 1 1m2r, Kab ­ 0 , (4)where m is the mass andr the isotropic radius. Tomake contact with the throat picture, recall that therexists an isometry given byr ! m2y4r which leaves thecoordinate spherer ­ my2 invariant and which maps theentire exterior asymptotically flat space into that spherConsequently, there exists a second asymptotically flregion nearr ­ 0. Equivalently, one can represent thissolution to the constraints on a space consisting of twcopies of R3 with a sphere excised and appropriatidentification at the spheres.For N black holes and nonvanishing extrinsic curvature, York and others [8] have developed a sophisticatmethod to solve the constraints for two asymptotically flaspaces that are connected by as many throats (i.e., excispheres) as there are black holes, and that are isomecopies of each other. Note that there are explicit soltions to the momentum constraint (2) that characterizesingle black hole with given momentumPa, and spinSa.For example,KabPS ­32r2fPanb 1 Pbna 2 sgab 2 nanbdPcncg13r3seacdScndnb 1 ebcdScndnad, (5)wherena is the radial normal vector. Since the conformametric is flat, we will use in what follows either standardspherical or Cartesian coordinates, i.e.,r ­ sx2 1 y2 1z2d1y2 andna ­ xayr.By the method of images it is possible to obtain an infinite series based on (5) forKab which solves the momen-tum constraint and satisfies an isometry condition at anumber of spheres [6,8]. Given such a solution, what rmains to be done is to solve the Hamiltonian constrai(3), which is an elliptic equation onR3 minus several© 1997 The American Physical SocietyVOLUME 78, NUMBER 19 P H Y S I C A L R E V I E W L E T T E R S 12 MAY 1997etireeynuhisnlsetnslnrtserrspheres, with the inner boundary given by the isomeand the outer boundary determined by asymptotic flatnThree independent numerical implementations ofabove scheme have been given and compared in the detive paper on three-dimensional black hole initial dataCook, Choptuik, Dubal, Klasky, Matzner, and Oliveira [9For the Hamiltonian constraint, they consider a nonlear block full approximation storage multigrid scheme fCadez coordinates, a successive over-relaxation schin Cartesian coordinates, and a multiquadratics approaAll these approaches are greatly complicated by the pence of the inner spherical boundaries and the isomcondition.Let us return to the Schwarzschild solution of thconstraints on a “punctured”R3. As noted by Misnerand Wheeler [5] and studied in detail by Brill anLindquist [7] (MWBL), the Schwarzschild solution to thconstraints generalizes trivially toN black holes for timesymmetry,c ­ 1 1NXi­1msid2j$r 2 $rsidj, Kab ­ 0 , (6)where msid characterizes the mass of theith black hole(i.e., the ADM mass isPmsid) and $rsid is the location ofthe ith black hole. For regularity of the conformal factothe MWBL solution is considered on a singleR3 with thepoints $r ­ $rsid removed. We refer to the$rsid as punctures.The isometry present in the Schwarzschild solution is loalthough there still exist minimal surfaces characterizithe throats [7].Let us now discuss the method that we propoto find data for multiple black holes with arbitrarboosts and spins. The idea is to compactify the interasymptotically flat regions in order to obtain a simpdomain of integration. Such compactification bringsissues of regularity of the fields (e.g., [10,11]), which waddress below.As before, we consider vacuum spacetimes, the meis conformally flat, and the extrinsic curvature is tracefreAs the spatial slice we choose a singleR3 with Npunctures as in the MWBL data. First, we solve tmomentum constraint (2) by settingKab ­NXi­1KabPSsid , (7)where each term is defined by (5) with its own orig$rsid, momentum $Psid, and spin $Ssid. These parametercorrespond to the ADM quantities in the limit that thseparation of the holes is very large. Equation (7) defithe solution to the momentum constraint that we actuause, and is not just the starting point for the methodimages that is usually invoked to distortKab to obtain anisometric solution (cf. [12], where the same simplificatioarises for a trapped surface boundary condition atinner boundary).tryss.hefini-by].n-oremech.es-tryedr,st,ngseallepetrice.eneesllyofntheGiven Kab as defined in (7), we proceed to solve theHamiltonian constraint, (3). We rewrite the conformafactor in terms of functionsa andu given byc ­1a1 u ,1a­NXi­1msid2j$r 2 $rsidj. (8)On the puncturedR3, the Laplacian of1ya is zero,so that the Hamiltonian constraint equation become(cf. [11] for a single asymptotic region and vanishinglinear momentum)Du 1 bs1 1 aud27 ­ 0 , (9)b ­18 a7KabKab . (10)To complete the definition of the problem, we haveto specify boundary conditions foru. For asymptoticflatness at infinity we requireu 2 1 ­ Osr21d for largedistances to the punctures.The key question that remains is what condition wwant to impose onu close to the punctures. As it turnsout, to build in asymptotically flat regions as are presenin the MWBL data near the punctures,it suffices to solve(9) everywhere onR3 without any points excised.Thiscompletes the statement of our proposal.Let us discuss existence and uniqueness of solutioto the modified Hamiltonian constraint equation (9) onR3. Since in this case the topology is trivial, we canshow existence and uniqueness of aC2 solution byrepeating the proof given in [13] for the conventionaHamiltonian constraint equation onR3. By definition,both a and b are proportional toj$r 2 $rsidj near theformer punctures and are thereforeC0, despite the factthat KabKab goes asj$r 2 $rsidj26 at these points. Fromthe maximum principle and the outer boundary conditiowe obtain that there exists at most oneC2 solution, and,in particular, thatu $ 1 and 1 1 au $ 1. Referring to[13] for the definition of weighted Sobolev spaces, fothe existence part of the proof we require thatu [ Mps,dandbs1 1 aud27 [ Mps22,d12 with p . 3 for the normands $ 3 characterizing differentiability. The falloff of(9) is the same as in the standard case. Interior poinlike the $rsid do not affect the weightd. Because of theloss of differentiability at the punctures inb, we find thatin our casep . 3 ands ­ 3. (In comparison, [14] usesp ­ 2 but requiress $ 4.) The proof proceeds as in [13]with minor changes in the algebra and due care whenevit matters thata and b may vanish. So, althoughuis only C2 at the $rsid, there exists a unique solution forthe conformal factorc ­ u 1 1ya on the puncturedR3determined by our proposal foru on the unpuncturedR3.Given a solutionu, we can demonstrate that eachpuncture represents the “point at infinity” for anotheasymptotically flat spacetime (cf. [7]). Hence, solving3607VOLUME 78, NUMBER 19 P H Y S I C A L R E V I E W L E T T E R S 12 MAY 1997l.es,yye-re-te-eofrealdt,sfse,enneot(9) on R3 involves a particular compactification ofNout of N 1 1 asymptotically flat regions, one of whichis distinguished by our choice ofKab ; see below. Weperform a coordinate inversion through a sphere nethe ith puncture, r̄ ­ a2yr, under which the metrictransforms asds2 ­ c4sdr2 1 r2dV2d ­ c̄4sdr̄2 1 r̄2dV2d,(11)with c̄ ­ c rya (which is not an isometry). Settinga ­ msidy2 wheremsid is the bare mass of the puncturwe are considering, we obtain for our choice ofc , (8),thatc̄ ­ 1 1m̄sid2r̄1 Oµ1r̄2∂, (12)m̄sid ­ msidµus$rsidd 1Xjfiimsjd2j$rsid 2 $rsjdj∂. (13)Therefore, the metric becomes flat as we approachpunctures.To show asymptotic flatness at the punctures, it rmains to be shown that̄Kab ­ Osr̄22d. In the physicalvariables, we just have the coordinate transformationr̄ ­a2yr, Lbc ­ ≠xby≠x̄c ­ sa2yr̄2dLbc where Lbc ­ dbc 22nbnc. In the unphysical variables, we also have to takinto account the transformation ofc . With Kab ­ c̄2K̄phabandK̄phab ­ sLKphdab , we obtainK̄ab ­µar̄∂6sLKdab . (14)Therefore, for our choice ofKab, (5), the momentum termof orderr22 is mapped to order̄24 while the spin termof order r23 is mapped to order̄ 23. This observationextends to multiple hole data withKab defined in (7).Hence the black hole puncture data are asymptotically flat the punctures.In fact, we also learn that the black hole seen frothe region near the puncture appears to be unboosbecause the boost term is obtained from ther24 termin the untransformed space. This term is always zeroour space by construction. In the conventional methoimposing the isometry leads to terms which go asr24 inKab . These terms may not be included in our metho(without modification), becauseb would not be regular atthe punctures if they were.Since the initial data are asymptotic to Schwarzschdata near the punctures, and since small spherical surfacentered at the puncture of Schwarzschild data are outrapped surfaces (with outside referring to the asymptoregion away from the punctures), this indicates that tregion near the punctures is inside black holes. As[7], several punctures may be hidden behind a commhorizon, which we confirmed numerically but will notdiscuss here.We now come to the numerical implementation of ouproposal. We found that a full approximation storag3608arethee-eatmted,ind,dildcester-ticheinonremultigrid method built around a nonlinear Gauss-Seiderelaxation scheme [15] performs very well for (9) on afinite Cartesian grid with a Robin boundary conditionNote that the standard representation ofDu by centeredfinite differences is only first order at the punctures. Oncould use a different prescription at the puncture pointbut in numerical tests we found that the correspondingllower rate of convergence is contained in a surprisinglsmall neighborhood of the puncture points, whether thosare part of the grid or not.The multigrid program is part ofBAM, a bifunctionaladaptive mesh package for elliptic and hyperbolic problems in three-dimensional numerical relativity (see [16] fothe hyperbolic part; we do not use adaptivity here). Sincin general the Hamiltonian constraint has to be solved numerically, it is important to provide at least one efficiennumerical implementation. There is no reason to believthat the multigrid method is the only good method. However, when available, multigrid methods have proven to bamong the best performers. We consider the absenceirregular boundaries in our method to be a valuable featusince it makes a straightforward multigrid implementationpossible.As a test, let us compute the correction to the conformfactor for two equal mass black holes boosted towarone another when we keep only terms of smallP (themomentum of each hole) andL (the coordinate positionsof the holes along thez axis being6L). Considerb tobe of ordere, and consider the MWBL solution to bethe zeroth order solution to the Hamiltonian constrainu ­ us0d 1 eus1d ­ 1 1 eus1d. We now haveeDus1d ­ 2bs1 1 ad27. (15)The ADM mass to first order ine isMADM ­ 212pIr­`=aµ1a∂dSa112pZbs1 1 ad27 dV­ 2m 1 mµPm∂2∑1150µLm∂222435µLm∂4∏. (16)This equation predicts the correction to the ADM masresulting from the full nonlinear solve to within about5%whenPym # 1.0 andLym # 0.2 and the grid is a cubewhose sides have a length of about 16m and are resolvedwith 67 zones.This calculation provides one test for the correctness othe solution of the data set, at least for the ADM masof the spacetime for perturbative cases. This is valuablbecause the code cannot be tested for the case whKab ­ 0 since then the solution is trivially the MWBLsolution.As a general 3D strong field example we present iTable I data comparable to the A2B8 data set of [9]. Notthat the data sets produced by the two methods are nVOLUME 78, NUMBER 19 P H Y S I C A L R E V I E W L E T T E R S 12 MAY 1997eobitltintthysaeorronnlols,-erc-ts,lI.-,TABLE I. Results for the A2B8 data set of [9]. All unitsare in terms ofms2d, h is the grid spacing,n the number ofgrid points, d is the residual,t the truncation error,s theconvergence rate, andMADM is the ADM mass of the box.A2B8, box of size 32ms1d ­ 2 ms2d ­ 1$rs1d ­ s0, 0, 4d $rs2d ­ s0, 0, 24d$Ps1d ­ s15, 0, 0d $Ps2d ­ s215, 0, 0d$Ss1d ­ s220, 20, 0d $Ss2d ­ s0, 20, 20dh n kdk2 ktk2 s MADM4.000 113 1.159e-08 3.813e-03 4.029 9.2092.000 193 5.350e-08 1.984e-03 22.119 10.0971.000 353 2.141e-07 5.400e-03 0.838 10.6010.500 673 2.582e-07 1.216e-02 1.121 10.8330.250 1313 1.552e-07 2.222e-02 1.430 10.9170.125 2593 6.629e-08 3.383e-02 1.712 10.942identically the same. They are equally generic, but eahas a different gravitational wave content.There are two punctures with different mass parametof order one, and with comparatively large values fboosts and spins. The results are obtained for a singlecentered around the origin for various grid spacings. Earun was performed for the same multigrid parameters suthat the values for the residua reflect the efficiency wwhich the discretized equations are solved at differeresolutions. We chose to make the residua much smathan the truncation error estimate, which turns outbe in a range compatible with the still rather large grspacing. The convergence ratess are based onu atthree resolutions. As expected, we found in this aother examples that the convergence rate at the puncpoints was one order less than elsewhere, but we afound that convergence was only affected very close topunctures. As the resolution increases, the mass estimconverges with a rate comparable tos.To summarize, we have introduced a new setup for tinitial data problem for multiple black holes with arbitrarmass, momentum, and spin in three dimensions. Coparing with the wormhole constructions of black holethe key difference is how the various asymptotically flregions are defined on a single copy ofR3. Instead ofusing an isometry condition at interior spheres, we rmove the inner boundary by compactification and look fa solution to a modified Hamiltonian constraint equatioon R3, for which we have existence and uniqueness.corresponding proof should be possible for the isometwormhole data, although to our knowledge, such a prohas not been given yet.There are two obvious numerical advantages compato the wormhole approach. First, note that the solutiKab to the momentum constraint is all we need; i.esince we do not impose the isometry condition there ischrsroxchchhntleroddurelsoheateem-,t-rnAicofedn.,oneed for the method of images. Second, in the solutioof the Hamiltonian constraint we avoid the numericacomplications due to an inner boundary. Compared tthe standard Cartesian (Cadez) method, one (two) leveof sophistication less are required for an implementationwhich should make general black hole initial data morewidely available.Finally, note that black hole puncture data has already been successfully evolved in the case of 3DSchwarzschild initial data [17,16], which is possible evenwhen the punctures are part of the numerical grid sincthere are no physical singularities on the initial slice. Foa long term evolution of black holes, one may start withdata obtained by either the throat or puncture constrution, and then cut out the interior regions at the apparenhorizon to avoid the physical singularities on future slice[12,18].It is a pleasure to thank Helmut Friedrich, Alan RendallBernd Schmidt, Ed Seidel, and Paul Walker for helpfudiscussions. The computations were performed at the AEin Potsdam.*Electronic address: brandt@aei-potsdam.mpg.de†Electronic address: bruegman@aei-potsdam.mpg.de[1] A. Lichnerowicz, J. 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Brandt, inMultigrid Methods,edited by W. Hackbuschand U. Trottenberg, Springer Lecture Notes in Mathematics No. 960 (Springer-Verlag, Berlin, 1982); W. Press,S. Teukolsky, W. Vetterling, and B. Flannery,NumericalRecipes in C(Cambridge University Press, Cambridge,1992).[16] B. Brügmann, Phys. Rev. D54, 7361 (1996).[17] P. Anninos, K. Camarda, J. Massó, E. Seidel, W. Suenand J. Towns, Phys. Rev. D52, 2059 (1995).[18] E. Seidel and W. Suen, Phys. Rev. Lett.69, 1845 (1992).3609

A simple construction of initial data for multiple black holes

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