Constant Crunch Coordinates for Black Hole Simulations Adrian P. Gentle,1,∗ Daniel E. Holz,2,† Arkady Kheyfets,3,‡ Pablo Laguna,4,§ Warner A. Miller,1,¶ and Deirdre M. Shoemaker4,∗∗ 1Theoretical Division (T-6, MS B288), Los Alamos National Laboratory, Los Alamos, NM 87544 2Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106 3Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205 4Department of Astronomy and Astrophysics and Center for Gravitational Physics and Geometry, Penn State University, State College, PA 16802 (Dated: February 7, 2008) We reinvestigate the utility of time-independent constant mean curvature foliations for the nu- mericalsimulationofasinglespherically-symmetricblackhole. Eachspacelikehypersurfaceofsuch afoliationisendowedwiththesameconstantvalueofthetraceoftheextrinsiccurvaturetensor,K. Of the three families of K-constant surfaces possible (classified according to their asymptotic be- haviors),wesingleoutasub-familyofsingularity-avoidingsurfacesthatmaybeparticularlyuseful, 1 and provide an analytic expression for the closest approach such surfaces make to the singularity. 0 We then utilize a non-zero shift to yield families of K-constant surfaces which (1) avoid the black 0 hole singularity, and thus the need to excise the singularity, (2) are asymptotically null, aiding in 2 gravity wave extraction, (3) cover the physically relevant part of the spacetime, (4) are well be- n haved (regular) across the horizon, and (5) are static under evolution, and therefore have no “grid a stretching/sucking”pathologies. Preliminarynumericalrunsdemonstratethatwecanstablyevolve J asinglespherically-symmetricstaticblackholeusingthisfoliation. Wewish toemphasizethatthis 8 coordinatization produces K-constant surfaces for a single black hole spacetime that are regular, static and stable throughout their evolution. 2 v 3 1 1 I. CONSTANT CRUNCH SURFACES 5 0 In this paper, we address a single question: Is there a numerically-viable coordinatization of a Schwarzschild 0 0 black hole spacetime foliated by hypersurfaces of constant (not necessarily zero) mean extrinsic curvature? In other / words, can we coordinatize the Schwarzschild spacetime with constant mean extrinsic curvature (Tr(K)=constant) c hypersurfaces so as to bound the growth of metric components and their gradients? We demonstrate here that the q - singleshiftfreedomyieldsaspacetimemetricthatisstatic,andthereforeboundsthegrowthintimeofsuchgradients. r A more complete analysis of the stability of our coordinatization, and a more thorough canvassing of the parameter g : space, will appear elsewhere. [1] Our foliation is consistent with that of Iriondo et al. [2], who provided a generic v constant mean curvature (CMC) foliation of the Reissner-Nordstro¨m spacetime for the purpose of finding trapped i X surfaces. In this paper we focus on the utility of CMC slicings for the numericalsimulationof black holes, in support of the emerging field of gravity-waveastrophysics. r a The trace of the extrinsic curvature tensor (Tr(K) = Ka = K) at a point on a spacelike hypersurface measures a thefractionalrateofcontractionof3-volumealongaunitnormaltothesurface. Itrepresentstheamountof“crunch” the 3-surface is experiencing at the point, at a given time. If all the observers throughout a spacelike hypersurface movingintimeorthogonaltothesurfaceexperiencethesameamountofcontractionperunitpropertime,wesaythat the surface is a K-surface or a “constant crunch” surface. In this paper we examine foliations of a single spherically- symmetric,staticblackholewhereeachspacelikehypersurfacehasthesameconstantvalueofthe extrinsiccurvature, K. Generic K-surface foliations have found greatutility in the numerical simulation of cosmologicalspacetimes. [3] In addition to decoupling the three momentum constraintequations from the Hamiltonian constraint,these surfaces (in the case of compact or W-model universes)provide a convenientcosmologicaltime parameter (K, or York, time). [4] Furthermore, for such cosmological spacetimes one has powerful existence and uniqueness theorems. [5, 6] Extensive ∗Electronicaddress: [email protected] †Electronicaddress: [email protected] ‡Electronicaddress: [email protected] §Electronicaddress: [email protected] ¶Electronicaddress: [email protected] ∗∗Electronicaddress: [email protected] 2 work into the characteristics of these surfaces for Schwarzschild spacetimes has been done by Brill et al., [7] and foundational work into their use in numerical relativity was done by D. Eardley et al.. [8] More recently Pervez et al. [9] provideda foliation partially coveringthe Schwarzschildspacetime with K-surfaces,with K ranging from to ,andIriondoet al.[2]providedagenericconstantmeancurvaturefoliationoftheReissner-Nordstro¨mspacet−im∞e ∞ for the purpose of finding trappedsurfaces. In this paper we build upon the workof these investigatorsby examining the utility of these surfaces for numerical relativity in support of gravity-wavedetectors. AlthoughsurfacesofconstantKwerethoroughlyinvestigateddecadesago,theiruseincurrentnumericalsimulations of black holes is conspicuously absent (apart from the use of maximal (K = 0) surfaces). [10] One reason why these slicing methods have not been more fully developed is that they lag in time close in, to avoid crashing into the singularity, while they simultaneously evolve forward normally at the outer edge of the grid to allow for wave extraction. Thistension,manyfear,willunavoidinglyleadtounboundedgrowthinthemetricandextrinsiccurvature components in the intermediate region, as is indeed found in maximal slicing. This computational concern has been referredto by the numerical relativity community as “gridstretching” or “gridsucking.” We show in this paper that aproperchoiceofradialshiftcanyieldaconstantcrunchfoliationofaspherically-symmetricblackholewithoutsuch pathologies. In fact, we foliate a Schwarzschild black hole such that the 3-metric and extrinsic curvature are both bounded and static (i.e. unchanging in time). Tonumericallyevolveablackhole3-spaceintime itis desirableto havea foliation,andits coordinatization,which satisfy the following four properties: 1. Avoids black hole singularities or facilitates their excision. 2. Possessesasymptotically null hypersurfaces to aid in radiation extraction. 3. Minimizes steep gradients in the lapse, shift, 3-metric and extrinsic curvature tensor. 4. Maximizes the future development of the initial data for the purpose of gravity-waveextraction. As a first step towards achieving these goals for systems containing multiple black holes, we explore the families of K-surfaces in the Schwarzschild spacetime, and find a CMC foliation satisfying the above properties. In the next section we construct the K-surfaces outside and inside the horizon. In section III we explore the properties of the K-surfaces, dwelling in particular on their approaches to the singularity. We also examine and illustrate the three families of K-surfaces. In section IV we derive a metric for Schwarzschild whose constant time slices are K-surfaces. We restrict our attention to a subfamily of K-surfaces – surfaces which, when generalized to the colliding black hole spacetimes, support the gravity wave detection problem. We also present some preliminary numericalsimulations using constant-crunchcoordinates. We conclude withgeneralcomments onthe applicabilityof K-surfaces to numerical calculations of more general black hole spacetimes. II. CONSTRUCTION OF CONSTANT CRUNCH SURFACES Over three decades ago Eardley and Smarr [8] carried out a generic classification of the spacetimes that could be simulatednumerically,andinvestigatedthelimitationsthatthepresenceofsingularitieswouldimpose. Intheirpaper they argued that CMC slicings are particularly useful for numerical purposes. In particular, they demonstrated this explicitly byconstructingnumericalsolutionstoa widearrayofdustcollapsemodels. Inasimilarvein,Brilletal.[7] exploredthe nature ofCMC slices ofthe Schwarzschildspacetime,and they alsopresentedsomenumericalexamples. In the present work,we explore the numerical utility of CMC slicings in the case of single black hole spacetimes. For the sake of clarity, we will commence with a re-derivationof the equations governingthe CMC surfaces starting from Schwarzschildcoordinates. Fromthere, we will explore specific properties of the surfaces, paying particularattention to implications for numerical relativity. We wish to find a spacelike hypersurface in the Schwarzschild spacetime such that every point on the surface has the same constant value of the trace of the extrinsic curvature tensor. We have at our disposal the specification of the initial-value data, as well as the freedom to choose the lapse and shift throughout the evolution. To begin, let us take the standard coordinate system of a single black hole spacetime of mass M in Schwarzschildcoordinates: ds2 = B(r)dt2+C(r)dr2 +r2 dθ2+sin2θdϕ2 , (1) − with B(r)=(1 2M/r) and C(r)=1/B(r). It will be convenien(cid:0)t to treat separa(cid:1)tely the regions inside and outside − of the horizon. We will find that the two are related by an isometry. 3 t r=2M r=R(t) ξ n t=T(r) ξ n r FIG. 1: To construct CMC slices, at each point we introduce a unit normal vector n and the unit tangent vector ξ to the surface. Thelocal light cones are depicted bythelight dashed lines. A. Outside the Horizon (r>2M) Outside the horizon, CMC surfaces will be labeled by t = T(r) (Fig. 1). The requirement that the trace of the extrinsic curvature be constant throughout this surface yields a first order differential equation for T(r), determined by examining the behavior of the normals to the surface. The normal n to the spacelike hypersurface T is given by, n=N (t T(r))=n dt+n dr =N (dt T′dr), (2) 0 t r 0 ∇ − − where N is a normalizationconstant and primes denote differentiation with respect to r. The normalizationis fixed 0 by demanding that n n = 1 (3) · − = gttn n +grrn n (4) t t r r 1 1 = N2 + T′2 . (5) 0 −B C (cid:18) (cid:19) Therefore, 1 N = , (6) 0 −√C BT′2 − and T′ n = , (7) r √C BT′2 − 1 n = . (8) t −√C BT′2 − The contravariantcomponents of the normal are given by T′ nr = grrn = , (9) r C√C BT′2 − 1 nt = gttn = . (10) t B√C BT′2 − The trace of the extrinsic curvature is the fractional rate of contraction of 3-volume per unit proper time along the normal, namely 1 d K = nα = r2nr . (11) − ;α −r2dr (cid:0) (cid:1) 4 Substitution of nr into Eq. (11) yields a second-order ordinary differential equation for T. Integrating this equation, we find Br2T′ H = +J, (12) √C BT′2 (cid:18) − (cid:19) with H an integration constant and J an indefinite integral given by r 1 J = K√BCr2dr = Kr3. (13) 3 Z Alongthe surfacethe rateofchangeofpropertime,dτ, withproperdistance,ds, isrelatedtothe slope ofthe surface T, dτ B = T′. (14) ds C r From Eq. (12) we find dτ 2 (H J)2 = − . (15) ds (H J)2+Br4 (cid:18) (cid:19) − B. Inside the Horizon (r<2M) FindingtheK-constantslicesofEq.(1)withinthehorizonissimilartothecalculationdoneintheprevioussection; however, as the roles of time and space coordinates reverse within the horizon, we will find it useful to parameterize our spacelike surface as a function of coordinate t, and look for K-constant surfaces of the form (Fig. 1) r =R(t). (16) The normal n to the spacelike hypersurface R is given by n=N (R(t) r)=n dt+n dr =N (R˙dt dr), (17) 0 t r 0 ∇ − − with differentiation with respect to t denoted by dots and the N a normalization constant fixed by: 0 n n = 1 (18) · − = gttn n +grrn n (19) t t r r 1 1 = N2 R˙2 . (20) 0 C − B (cid:18) (cid:19) We have, therefore, 1 N = − , (21) 0 CR˙2 B − and p 1 n = , (22) r CR˙2 B − p R˙ n = − . (23) t CR˙2 B − The contravariantcomponents of the normal are given bpy 1 nr = grrn = , (24) r C CR˙2 B − nt = gttn = p R˙ . (25) t B CR˙2 B − p 5 FromEq.(11),weonceagainfindthatfixingthe traceoftheextrinsiccurvaturegivesusasecond-orderdifferential equation for R(t), namely, 2CB′R˙2 B B′+C′R˙2+2CR¨ 2 − K = − , (26) CR CR˙2 B − 2 C(cid:16)R˙2 B 3/2 (cid:17) − − p (cid:16) (cid:17) which can be simplified to d BR2 K R2R˙ = . (27) −dt CR˙2 B! − Paralleling the approach from the last section, we introdupce an integration constant, H, and an indefinite integral J (given by Eq. (13)), to obtain the first integral: BR2 H = +J. (28) CR˙2 B! − From Eq. (27) we find that the “proper velocity” alopng the surface, ds/dτ = C/BR˙, results in the same equation both inside and outside of the horizon (Eq. (15)). This can be rewritten as p ds 2 B R4 =1 + . (29) dτ (H J)2 (cid:18) (cid:19) − The spacelike K-surfaces obtained from the first integrals, Eqs. (15) and (29), differ only by an isometry, 1 T′ . (30) ⇐⇒ R˙ III. PROPERTIES OF THE K-SURFACES The spatial metric of a K-surface outside of the horizon is given by ds2 = dℓ2+r2dΩ2 (31) = C BT′2 dr2+r2dΩ2. (32) − Within the horizon it becomes (cid:0) (cid:1) ds2 = dℓ2+r2dΩ2 (33) = CR˙2 B dt2+r2dΩ2. (34) − (cid:16) (cid:17) These two expressions differ by the isometry of Eq. (30). Using Eq. (15) we can rewrite them in terms of H and K: r4 ds2 = dr2+r2dΩ2. (35) (H J)2+Br4 − From this we arrive at the scalar curvature of the K-surface: 2 6H2 (3)R= K2 + . (36) −3 r6 Similarly, by using Eqs. (15) and (29), the extrinsic curvature associated with observers moving on world lines orthogonalto the K-slices are also expressible in terms of K and H: 1 2H Krˆ = K + , (37) rˆ 3 r3 Kθˆ = Kφˆ = 1K H. (38) θˆ φˆ 3 − r3 6 SSSSSScccccchhhhhhwwwwwwaaaaaarrrrrrzzzzzzsssssscccccchhhhhhiiiiiilllllldddddd KKKKKKrrrrrruuuuuusssssskkkkkkaaaaaallllll------SSSSSSzzzzzzeeeeeekkkkkkeeeeeerrrrrreeeeeessssss 20 2 15 1 HHHHHHHHHHHH t 10 v 0 5 -1 0 -2 0 1 2 3 4 5 -2 -1 0 1 2 20 2 16 12 1 8 HHHHHHSSSSSS t 4 v 0 0 -1 -4 -8 -2 0 1 2 3 4 5 -2 -1 0 1 2 20 2 16 12 1 8 SSSSSSSSSSSS t 4 v 0 0 -1 -4 -8 -2 0 1 2 3 4 5 -2 -1 0 1 2 r u FIG. 2: An example of the three families of spacelike K-surfaces for K = 0.1. The first row (HH) depicts a representative − horizon-to-horizon surface using H = 1.25, which corresponds to Rmin 1.816 . This spacelike hypersurface is represented − ≈ both in Schwarzschild coordinates (left column) and in Kruskal-Szekeres coordinates (right column). The middle two graphs are the horizon-to-singularity (HS) surfaces using H = 1.43. The bottom two graphs represents a typical singularity-to- − singularity (SS) surface. We have used H = 1.25 to generate this SS K-surface. The HH and SS surfaces are close to their − critical radii (Rc 1.5646) and therefore appear flattened,as described in thetext. ≈ The K-surfaces are therefore parametrized by two constants: the trace of the extrinsic curvature tensor, K, and the constant of integration, H. In addition one must fix a single point on the surface, t = T(r ), which amounts to o o setting a time translation parameter. As can be seen in Eqs. (36)–(38), the constant H controls the variation of the intrinsic and extrinsic curvatures over the K-surface. To elucidate the nature of the K-surfaces, we numerically integrate Eqs. (15) and (29). We find that within the horizon there are 3 classes of K-constant surfaces, differentiated by their asymptotic behavior. The singularity- singularity surfaces (SS) begin at the singularity aligned with the null surface, reach up towards the horizon, and then fall back, reaching the singularity along the null cone. The horizon-horizon surfaces (HH), which we have also dubbed “horizon-hugging”surfaces, asymptote to the horizon (r 2M for t in Schwarzschild),dipping down towards the singularity in between. This feature was previously→remarked u|p|o→n b∞y Brill et al. [7]. The asymptotes convergetowardanullsurfaceatthehorizon. Finally,thehorizon-singularity(HS)surfacesbeginatthehorizon,and asymptote in to the singularity. Representative surfaces for the value K = 0.1 are shown in Fig. 2. We integrate the HH and HS surfaces across the horizoninto the regionr >2M by imposi−ng continuity of the surface and its first derivative at the horizon. Because of the isometry, Eq. (30), the surfaces outside of the horizon are characteristically similar to those on the inside; in particular, both sets are null at their asymptotes. We have chosen to use the acronym HH for the horizon-to-horizon hypersurfaces, in lieu of referring to them as “regular”hypersurfaces[7], as eachof the three types of K-surfaces are,in a strict sense, regular. In particular,each surface asymptotes to a null surface, be it at the singularity or the horizon. Observers on such a surface, or more precisely, observers that are time synchronized throughout the surface, are never seen crossing the horizon, nor do they ever reach the singularity! Outside the horizon, every HH and HS K-surface (K = 0) asymptotes for large r 6 to null infinity. K-surfaces corresponding to positive values of K asymptote to past null infinity, and asymptote to future null infinity for K <0. To gain a qualitative understanding of the K-constant foliation, it is useful to analyze Eq. (29) as an energy 7 15(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) 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10(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)Past HH and SS 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5(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) Future HS (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) 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(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) -5 (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) 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(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) -15 -5 -4 -3 -2 -1 0 1 2 3 4 5 K FIG. 3: Phase space diagram of the classification of K–surfaces. The K–surfaces have been classified into 3 groups according to their asymptotic behavior (see text). Which class a particular surface falls into depends upon the values of K and H, as detailedinthefigure. Forthepurposesofnumerics,thesurfacesofinterestaretobefoundinthe“FutureHHandSS”wedge, with K <0 (see section III). conservation equation for a particle of unit total energy (E =1) moving in the potential 1 2M r4 V(r)= − − r . (39) H(cid:0) 1Kr(cid:1)3 2 − 3 By using this energy equation we can determine the c(cid:0)losest appr(cid:1)oach to the singularity, R , of a given HH K- min constant surface. Two conditions must be satisfied to determine R . First, the closest approach occurs when min R˙ =(ds/dτ)=0, which is equivalent to demanding V(R )=1. (40) min This condition leads to a sixth-order polynomial in R : min K2 HK R6 +R4 2 M + R3 +H2 =0. (41) 9 min min− 3 min (cid:18) (cid:19) Second, the solution for this surface at R(t ) = R must be concave, so as to rule out the SS surfaces (which min min bend towards the singularity rather than the horizon). This is enforced by demanding: (R 2) 3+2R +KR2 2 1 R¨(t ) = min− − min min Rmin − 0. (42) min |R˙(tmin)=0 h R3 q i ≥ min The solution of these two conditions, as shown in Fig. 3, gives rise to the emergence of two critical values for H, namely H , for a given value of K. In addition, an HH surface can be made to approach arbitrarily closely to the ± singularity at r = 0 by choosing an appropriately large positive value of K. Negative values of K tend to “hug the horizon.” For fixed K and H we know how to compute how closely a CMC surface comes to the singularity. But, for a given valueofK,whatvalueofH givestheclosestoverallapproachtothesingularity? Wecandeterminethecriticalvalues for R and H, given by R and H respectively, by looking at the point where the first and second time derivatives c ± of R(t) vanish. H occurs along the lower boundary of the contour plot in Fig. 3, and corresponds to the K-surface − that reaches down the furthest towards the singularity for a given value of K. The vanishing of R¨ leads to the |R˙=0 following equation for R : c 2 (R 2) 3+2R +KR2 1 =0, (43) c− − c c R − (cid:18) r c (cid:19) 8 Critical Value of R 2 1.5 R 1 0.5 0 -10 -5 0 5 10 K FIG.4: Critical value of Rc =R− as a function of K for Schwarzschild coordinates and M =1. Onecan readily seethatthe K-surface “hugs” thehorizon for large negative values of K. which can be rewritten as a 4th order polynomial in R c K2R4 2K2R3+4R2 12R +9=0. (44) c − c c − c One musttake carein examining the roots ofthis equation, asthere are moresolutions to Eq.(44)thanthere are for Eq. (43). Nevertheless this equation gives two distinct real roots, depending on the sign of K: 1 1 8 2 16 √χ R = 3 χ+ + + , (45) c|K>0 2 − 2s − K2 − √χ K2√χ 2 1 1 8 2 16 √χ R = + 3 χ+ + + , (46) c|K<0 2 2s − K2 − √χ K2√χ 2 where ξ 32+108K2+243K4+27K2 16+56K2+81K4, (47) ≡ 8 16+36K2 2p1/3ξ1/3 χ 1 + + . (48) ≡ − 3K2 3 21/3K2ξ1/3 3K2 When K =0 we see from Eq. (43) that R =3/2. The regular K-surfaces are thus bounded between the horizon at c R =2M andR =R M. UsingEq.(28),andsettingR˙ =0,weobtain,forthecaseofablackholeinSchwarzschild + − c coordinates B R2 KR3 H = ± ± + ±, (49) ± B 3 ± − with p 2M B = 1 . (50) ± − R (cid:18) ±(cid:19) We therefore have 8 H = M3K. (51) + 3 For large values of K one can show that the critical value of R, R , depends upon the sign of K. In particular, − | | 2 1K−2 for K 1 R − 8 ≪− . (52) c −→( 94 1/3 K−2/3 for K ≫1 This in turn gives the following asymptotic value(cid:0)s f(cid:1)or H : m 8K 1 for K 1 H− −→ −−93 − 2K for K ≪1− . (53) (cid:26) 2K ≫ 9 IV. CONSTANT CRUNCH COORDINATES: A SPACETIME METRIC FOR A K-SURFACE FOLIATION OF THE SCHWARZSCHILD BLACK HOLE. A number of features of the K-constant surfaces presented in the previous sections seem particularly well suited to the numerical analysis of generic black hole spacetimes. First, the surfaces asymptote to a null surface, making them effective for gravity wave extraction. Second, the K-surfaces naturally avoid the crushing singularity. Finally, for large negative values of K the surfaces “hug the horizon.” This last feature, illustrated in Fig. 4, allows one to focus attention on the region relevant for gravity wave generation—the region outside the horizon. InthissectionwegenerateaK-constantfoliationfortheSchwarzschildblackholethat,inadditiontotheproperties just mentioned, also has regular and static metric and extrinsic curvature components. To generate this K-constant slicing we use the coordinate transformation t¯ = t T(r), (54) − ρ = r. (55) Under this transformation, the metric from Eq. (1) becomes 2M (J H) ρ4 ds2 = (1 )dt¯2+2 − dt¯dρ+ dρ2+ρ2dΩ2. (56) − − ρ (H −J)2+(1− 2Mρ )ρ4 (H −J)2+(1− 2Mρ )ρ4 q The constant t¯slices of this metric are K-constant surfaces. It is to be noted that Eq. (56) agrees with Eq. (53) of Iriondo et al. (for the case of constant K and vacuum). [2] However,in order to regularize the g metric component ρρ at the throat, we add the isotropic-like radial transformation [11]: 1 1 r¯= R +ρ+ R ρ+ρ2 , (57) min min 2 −2 − (cid:18) (cid:19) p with R the minimum coordinate location of the throat, given by Eq. (41). This coordinate representation of a min black hole spacetime provides a foliation with K-constant, H-constant spacelike hypersurfaces. Each hypersurface is metricallyequivalenttoallothers—thesurfacesareindependentoft¯,andhencestatic. Inaddition,the hypersurfaces areasymptotically null(T′(r) 1 asr ). Furthermore,the lapse, the shift, and allofthe 3-metricand extrinsic → →∞ curvature components are regular and well behaved, as illustrated in Fig. 5. Inadditiontorestrictingourselvesto the singularity-avoidingfamily (HH)ofK-surfaces,twoadditionalconditions onthe K-surfacesaredemanded by the nature of our problem– the eventualsimulationofthe gravity-waveemission from two interacting black holes. First, the foliations must asymptote at large r¯to future null infinity. Therefore, we must restrict our attention to negative values for the trace of the extrinsic curvature tensor, K. Second, we require thatsuchnegativeK hypersurfacesenterthefuturesingularityregion. Thiswillensurepropercoverageoftherelevant regionjustabovethefuturehorizon,whichispreciselywherethegravitywavesareproduced. However,weexpectthe initial-data formulation for such surfaces to be involved,and this may guide our choices even more systemically. The two additional requirements limit us to the relatively narrow wedge of Fig. 3, formed by restricting to the “Future HH and SS” shaded region with K <0. A representative constant-crunch foliation generated by Eq. (56) is shown in Fig. 6. The avoidance of “grid stretching” is accomplished by a suitable choice of shift vector. To illustrate the non-zero shift we show the K = 1, H 3.11 foliation of Schwarzschild in Fig. 7, with the explicit misalignment of the r = constant line segment a−nd ≈ − the normal vector. Finally, we describe several preliminary numerical experiments using K-surfaces. A full treatment of a single black hole using K-constant foliations will be presented elsewhere [1]. Here we present several sample evolutions demonstrating the utility of constant mean curvature slicings. Figures 8 and 9 display results from the simplest possible test of K-constant foliation of the Schwarzschild geometry; the domain is taken to be a thin shell close to the horizon(in this case r [1,5]), analytic Dirichlet conditions are applied at both boundaries of the computational ∈ domain, and analytic lapse and shift conditions obtained from Eqs. (56) and (57) are used. The figures represent the singularity-avoiding foliation K = 1 and H = 3, for which the evolution was found to be stable and accurate − − oververy longtimescales. Using 50gridpoints, the code succesfully ranbeyondt=50,000M while maintaininghigh accuracy. The fractional error in the metric components was typically 1 2%. − Fig. 8 shows the convergenceof the mean fractionalerrorin the metric component a¯=√gr¯r¯ as a function of time. Eachcurvehasbeen rescaledby a factorof4p,where the number ofgridpoints is givenby 400/2p,with p=0,1,2,3. Fort>2the solutionisapproximatelysecondorderaccurate. Initially,noisegeneratedonthe innerboundarycauses fluctuations whose magnitude is largely independent of the number of grid points. Fig. 9 shows snapshots of the 10 − g−r −r α 24 2 20 1.6 16 1.2 12 0.8 8 4 0.4 0 0 1 10 1 10 β−r K−r −r 0.5 0 0 -0.5 -0.1 -1 -1.5 -0.2 -2 -2.5 -0.3 -3 -3.5 -0.4 1 10 1 10 Kθθ K = K−r −r + 2 Kθθ 0.12 0 0.08 0.04 -0.1 0 -0.04 -0.2 1 −r−r−r−r−r−r 10 1 −r−r−r−r−r−r 10 FIG.5: Theradialbehaviorofthevariouscomponentsofthespacetimemetric(Eq.(56)),undertheisotropic-liketransformation given in Eq. (57), for M = 1, K = 0.1, H = 1.0 and R¯min 0.47922. From upper left to lower right we plot the radial metriccomponent(gr¯r¯),lapse(α¯),ra−dialshift(β−r¯),diagonalcom≈ponentsoftheextrinsiccurvaturetensor(Kr¯r¯andKθθ¯¯=Kφφ¯¯), and in the lower right frame the trace of the extrinsic curvature tensor (K = Kr¯r¯+Kθθ¯¯+Kφφ¯¯) as a consistency check. All of the functions are regular and well behaved. The growth of the lapse and shift for large r¯is expected, as the surfaces become asymptotically null. We show an exploded view of the behavior of the radial shift near the throat to emphasize that the shift changes sign and becomes positive before reaching r¯=R¯min. SScchhwwaarrzzsscchhiilldd KKrruusskkaall--SSzzeekkeerreess 20 2 15 1 10 t v 0 5 -1 0 -5 -2 2 3 4 5 6 7 8 -2 -1 0 1 2 r u FIG. 6: The constant crunch coordinate foliation of the Schwarzschild spacetime. Weshow thefoliation generated byEq. (56) for M =1, K = 0.1, H = 1.25 and Rmin 1.816, in Schwarzschild and Kruskal-Szekerescoordinates. − − ≈ Hamiltonianconstraintatvarioustimes in the evolution. Qualitatively,a wavewhich is triggeredby truncationerror is seen to propagate outwards from the inner boundary. The amplitude reduces rapidly, before growing once more as it is reflected off the outer boundary. The magnitude and speed of propagation of the wave quickly decay as the wavemovesbackinto the domain,leavinga static solutionwhichis stablebeyondt=50,000M. The numericalerror which initially propagates through the domain is caused entirely by the analytic Dirichlet boundary conditions, and can be largely eliminated by the use of more realistic conditions. [1] The numerical runs presented here evolve a K = 1 singularity-avoiding hypersurface that asymptotes to null − infinity, entering the past singularityregion. This is notofthe classof K-surfacesemphasizedin this paper for use in numerical relativity. Ideally, we would have preferred presenting the evolution of a K = 1 HH surface that enters the future singularity region. However, we were unable to find an HH surface in the fut−ure singularity region that