Generating Binary Black Hole Initial Data

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1 Generating Binary Black Hole Initial Data Gregory B. Cook Wake Forest University May 2, 2003 Abstract A formalism for constructing initial data representing black-hole binaries in quasi-equilibrium is developed. If each black hole is assumed to be in quasi-equilibrium, then conditions on the shear and expansion of the outgoing null rays at each black hole yield a set of boundary conditions for the initial data variables. This formalism should allow for the construction of completely general quasi-equilibrium black hole binary initial data. Initial numerical results using this approach will be examined and compared with previous numerical and post-newtonian results. Collaborators: Harald Pfeiffer & Saul Teukolsky (Cornell

2 Motivation Black hole binaries are among the most likely sources for early detection with LIGO, VIRGO, GEO,... Available computed waveforms should increase chance of detecting collision events. Quasi-Equilibrium Binary Data General Relativity doesn t permit true equilibrium for astrophysical binary systems. When the bodies are sufficiently far apart, the timescale for orbital decay is much larger than the orbital period. If the orbit is nearly circular (quasi-circular then there is a corotating reference frame in which the binary appears to be at rest. Quasi-equilibrium gives us a physical condition to guide us in fixing boundary conditions and data that is not otherwise constrained. Greg Cook (WFU Physics 1

3 The Decomposition Lapse : α Spatial metric : γ ij αn µ δt t + δt Shift vector : β i Extrinsic Curvature : K ij n µ t µ δt t ds 2 Time vector : t µ = αn µ + β µ = α 2 dt 2 + γ ij (dx i + β i dt(dx j + β j dt β µ β µ δt γ µν = g µν + n µ n ν K µν = 1 2 γα µ γβ ν L ng αβ Constraint equations R + K 2 K ij K ij = 16πρ j (K ij γ ij K = 8πj i S µν j µ T µν γ α µ γβ ν T αβ γ ν µ nα T να ρ n µ n ν T µν = S µν + 2n (µ j ν + n µ n ν ρ Evolution equations t γ ij = 2αK ij + i β j + j β i t K ij = i j α + α[ Rij 2K il K l j + KK ij ] 8πS ij + 4πγ ij (S ρ + β l l K ij + K il j β l + K jl i β l Greg Cook (WFU Physics 2

4 Generalized Conformal/TT Initial-Data Decomposition[13] γ ij = ψ 4 γ ij : K ij = ψ 10 Ã ij ψ 4 γ ij K : Ã ij 1 σ ( LV ij + M ij σ ψ 6 σ : 2 ψ 1 8 ψ R 1 12 ψ5 K ψ 7 Ã ij Ã ij = 2πψ 5 ρ j ( LV ij ( LV ij j ln σ 2 3 σψ6 i K = σ j M ij + 8π σψ 10 j i M ij is symmetric-tracefree, but not divergenceless. The variable V i incorporates the solution of the constraints and the decomposition of M ij ij into M T T. σ = 1 Conf-TT Method (Method A σ = 1 Phys-TT Method (Method B M ij 1 2 αũij σ 2 α Conformal Thin Sandwich(TS[16] γ ij = ψ 4 γ ij : K ij = ψ 10 Ã ij ψ 4 γ ij K : Ã ij 1 2 α Conf. Thin Sandwich [ũ ij t γ ij ] (( Lβ ij ũ ij α ψ 6 α : 2 ψ 1 8 ψ R 1 12 ψ5 K ψ 7 Ã ij Ã ij = 2πψ 5 ρ j ( Lβ ij ( Lβ ij j ln α 4 3 αψ6 i K = α j ( 1 αũij + 16π αψ 10 j i 2 (ψ 7 α 1 8 ψ7 α R 5 12 ψ11 αk ψ 1 αãijãij ψ 5 β i i K α and β i are the lapse and shift. ũ ij is symmetric-tracefree. = 2πψ 11 αk(ρ + 2S ψ 5 t K Greg Cook (WFU Physics 3

5 Degrees of Freedom Kinematical variables Lapse α : 1 degree of freedom Shift β i : 3 degrees of freedom Initial-data variables Metric γ ij : 6 degrees of freedom Extrinsic curvature K ij : 6 degrees of freedom Decomposition of initial-data variables γ ij = ψ 4 γ ij ψ : 1 constrained DOF 3 spatial gauge DOF γ ij : 2 dynamical DOF Freely Specifiable K ij = ψ 10 [ 1 σ ( LV ij + M ij ]+ 1 3 ψ 4 γ ij K K σ V i : M ij : ( LV ij i V j + i V j 1 3 γij k V k 3 constrained DOF j M ij = ( j ( LX ij 1 σ 2 dynamical DOF : 1 temporal gauge DOF : Defn. of TT decomp. Freely Specifiable Greg Cook (WFU Physics 4

6 Traditional Black-Hole Data Conformal flatness and maximal slicing γ ij = f ij (flat M ij = 0 i ( LV ij Bowen-York solution[4] = 0 Analytic solutions for Ãij K = 0 2 ψ + σ = ψ 7 Ã ij Ã ij = 0 Three general solution schemes Conformal Imaging-[7] Puncture Method-[5] Inversion symmetry inner-bc No inner-bc: singular behavior factored out Apparent Horizon BC-[14] Apparent horizon inner-bc All methods can produce very general configurations of multiple black holes, but are fundamentally limited by choices for γ ij and Bowen-York Ãij. Greg Cook (WFU Physics 5

7 Better Black-Hole Data What is wrong with traditional BH initial data? Results disagree with PN predictions for black holes in quasi-circular orbits. There is no control of the initial wave content. Spinning holes are not represented well. How do we construct improved BH initial data? We must carefully choose the initial dynamical degrees of freedom [in γ ij and M ij ] initial temporal and spatial gauge degrees of freedom [in γ ij and K] boundary conditions on the constrained degrees of freedom [in ψ and V i ] TT decomposition weight function [ σ] so as to conform to the desired physical content of the initial data. - For black holes in quasi-circular orbits, we can use the principle of quasi-equilibrium to guide our choices. - Quasi-equilibrium is a dynamical concept and we can simplify our task by choosing a decomposition of the initial-data variables that has connections to dynamics. Greg Cook (WFU Physics 6

8 Conformal Thin-Sandwich Decomposition αn µ δt n µ β µ t µ δt β µ δt t + δt γ ij = ψ 4 γ ij t K ij = ψ 10 2 α [ ( Lβ ij ũ ij] ψ 4 γ ij K Hamiltonian Const. 2 ψ 1 8 ψ R 1 12 ψ5 K ψ 7 Ã ij Ã ij = 2πψ 5 ρ Momentum Const. j ( Lβ ij ( Lβ ij j α = 4 3 αψ6 i K + α ( 1 j + 16π αψ 10 j i Const. Tr(K eqn. [ 2 (ψ 7 α (ψ 7 1 α 8 ψ R ] ψ5 K ψ 7 Ã ij Ã ij ψ 5 β i i K = 2πψ 5 K(ρ + 2S ψ 5 t K Ã ij 1 2 α [( Lβ ij ũ ij] Constrained vars : ψ and β i Freely specified : γ ij, ũ ij, K, and t K ũ ij and β i have a simple physical interpretation, unlike M ij and V i. ũ ij = 0 Quasi-equilibrium t K = 0 Greg Cook (WFU Physics 7

9 Ham. & Mom. const. eqns., & Const Tr(K eqn. from Conf. TS + ũ ij = t K = 0 Equations of Quasi-Equilibrium Eqns. of Quasi-Equilibrium With γ ij = f ij and K = 0, these equations have been widely used to construct binary neutron star initial data[2, 12, 3, 15]. Binary neutron star initial data require: boundary conditions at infinity compatible with asymptotic flatness and ( corotation. ψ r = 1 β i r = Ω φ i α r = 1 compatible solution of the equations of hydrostatic equilibrium. ( Ω Binary black hole initial data require: a means for choosing the angular velocity of the orbit Ω. with excision, inner boundary conditions are needed for ψ, β i, and α. Gourgoulhon, Grandclément, & Bonazzola[10, 11]: Black-hole binaries with γ ij = f ij & K = 0, inversion-symmetry, and Killing-horizon conditions on the excision boundaries. Solutions require constraint violating regularity condition imposed on inner boundaries! Greg Cook (WFU Physics 8

10 Constructing Regular Binary Black Hole QE ID Why does the GGB approach have problems? Inversion-symmetry demands α = 0 on the inner boundaries. Inversion-symmetry BC s are not compatible with ( Lβ ij = 0 and ũ ij = 0, unless t is a true Killing vector. [conjecture] How do we proceed? ] Ã ij 1 2 α [( Lβ ij ũ ij Eliminate dependence on inversion symmetry by letting the physical condition of quasi-equilibrium dictate the boundary conditions. Find a method that allows for general choices of γ ij & K. Approach Demand that the excision(inner boundary be an apparent horizon. Demand that the apparent horizon be in quasi-equilibrium. Greg Cook (WFU Physics 9

11 The Inner Boundary s i i τ τ h ij γ ij s i s j k µ 1 2 (n µ + s µ Σ ḱ µ n µ k µ ḱ µ 1 2 (n µ s µ Extrinsic curvature of S embedded in s i spacetime S Σ µν 1 2 hα µ hβ ν L kg αβ Σ µν 1 2 hα µ hβ ν L ḱ g αβ Extrinsic curvature of S embedded in Σ H ij 1 2 hk i hl j L sγ kl Projections of K ij onto S Σ ij = 1 2 (H ij J ij Σ ij = 1 2 (H ij + J ij J ij J i h k i hl j K kl h k i sl K kl J h ij J ij = h ij K ij Expansion of null rays θ h ij Σ ij = 1 2 (H J θ h ij Σij = 1 2 (H + J σ ij Shear of null rays Σ ij 1 2 h ijθ σ ij Σ ij 1 2 h ij θ Greg Cook (WFU Physics 10

12 AH and QE Conditions on the Inner Boundary The quasi-equilibrium inner boundary conditions start with the following assumptions: 1. The inner boundary S is a (MOTS: marginally outer-trapped surface θ = 0 2. The inner boundary S doesn t move: L ζ τ = 0 and D i L ζ τ h j i j L ζ τ = 0 t µ = αn µ + β µ ζ µ αn µ + β s µ β β i s i β s µ s µ β µ 3. The inner boundary S remains a MOTS: L ζ θ = 0 4. The horizons are in quasi-equilibrium: σ ij = 0 and no matter is on S t µ n µ ζ µ β µ s µ Greg Cook (WFU Physics 11

13 AH/Quasi-Equilibrium Boundary Conditions θ = ψ 2 [ hij ] i s j + 4 s k k ln ψ ψ 2 J 2 L ζ θ = 1 2 [θ(θ + 1 θ ] K + E (β + α 1 2 [θ( 1 2 θ 1 2 θ K + D + 8πT µν k µ ḱ ν] (β α + θs i i α D h ij (D i + J i (D j + J j 1 22 R E σ ij σ ij + 8πT µν k µ k ν σ ij = 1 ( (H ij 1 2 h ijh 1 β 2 α 1 2 ψ 4 α { } D(i β j 1 2 h ij Dk β k 1 2 [ h ik hjl ũ kl 1 2 h ij hkl ũ kl ] s k k ln ψ = 1 4 ( h ij i s j ψ 2 J β i = αψ 2 s i + β i 0 = D (i β j 1 2 h ij Dk β k Greg Cook (WFU Physics 12

14 Defining the Spin of the Black Hole The spin parameters β i can be defined by demanding that the time vector associated with quasi-equilibrium in the corotating frame must be null, forming the null generators of the horizon. k µ (n µ + s µ = k µ = [1, αs i β i] This vector k µ is null for any choice of α & β i. In the frame where a black hole is not spinning, the null time vector has components t µ = [1, 0]. Corotating Holes Corotating holes are at rest in the corotating frame, where we must pose boundary conditions. So, [ k µ = 1, αs i β i] = [1, 0] Thus we find β i = αs i β i = 0 Irrotational Holes Irrotational holes are at rest in the inertial frame. With the time vectors in the inertial and corotating frames related by k µ = = + Ω t t φ [ 1, αs i β i] = [1, Ω( / φ i ] Thus we find ( i β i = αs i + Ω β i = Ωξ i φ ξ i ( φ i & D (i ξ j 1 2 h ij Dk ξ k = 0 Greg Cook (WFU Physics 13

15 The Orbital Angular Velocity For a given choice of the Lapse BC, γ ij and K, we are still left with a family of solutions parameterized by the orbital angular velocity Ω. Except for the case of a single spinning black hole, it is not reasonable to expect more than one value of Ω to correspond to a system in quasiequilibrium. GGB[10, 11] have suggested a way to pick the quasi-equilibrium value of Ω: Ω is chosen as the value for which the ADM energy E ADM equals the Komar mass M K. Komar mass M K = 1 4π γ ij ( i α β k K ik d 2 S j Acceptable definition of the mass only for stationary spacetimes. ADM energy E ADM = 1 16π γ ij k (G k i δk i Gd2 S j Acceptable definition of the mass for arbitrary spacetimes. G ij γ ij f ij Greg Cook (WFU Physics 14

16 The Lapse BC & QE So far, nothing has fixed a boundary condition on the lapse α. One possibility[9] is to recall that θ θ is a Lorentz invariant and so to consider L ζ θ = 0 as a quasi-equilibrium condition. L ζ θ = 0 J s i i α = ψ 2 (J 2 JK + Dα D ψ 4 [ h ij ( D i J i ( D j J j 1 22 R + 2 D2 ln ψ] This conditions is satisfied for stationary solutions, but seems to be degenerate with the other QE boundary conditions. To see this, note that the stationary maximal slicings of Schwarzschild form a 1-parameter family: ds 2 = α = β R = C R 2 dr 2 1 2M R + + R 2 d 2 Ω C2 R 4 1 2M R + C2 R 4 1 2M R + C2 R 4 K i j = C R α S = C 4M 2 Greg Cook (WFU Physics 15

17 Summary of QE Formalism γ ij = ψ 4 γ ij K ij = ψ 10 Ã ij γij K Ã ij = ψ6 2α ( Lβ ij t γ ij = 0 2 ψ 1 8 ψ R 1 12 ψ5 K ψ 7 Ã ij Ã ij = 0 j ( Lβ ij ( Lβ ij j ln αψ 6 = 4 3 α i K [ 2 1 (αψ (αψ R ψ4 K ψ 8 A ij A ij] = ψ 5 β i i K t K = 0 s k k ln ψ S = 1 4 ( h ij i s j ψ 2 J S θ = 0 β i S = αψ 2 s i S αψ 2 s i S + Ωξ i S corotation irrotation L ζ θ = 0 σ ij = 0 α S = unspecified by QE ψ r = 1 ( β i r = Ω φ α r = 1 i The only remaining freedom in the system is the choice of the lapse boundary condition, the initial spatial and temporal gauge, and the initial dynamical( wave content found in α S, γ ij and K. Greg Cook (WFU Physics 16

18 Results Corotation γ ij = f ij : Maximal Slicing: (αψ r = 0 αψ = 1 2 (αψ r = αψ 2r γ ij = f ij : Eddington-Finkelstein Slicing: (αψ r = 0 αψ = 1 2 (αψ r = αψ Compared with Effective-One-Body PN[8] Inversion-Symmetric HKV[11] Irrotation γ ij = f ij : Maximal Slicing: (αψ r = 0 αψ = 1 2 (αψ r = αψ 2r γ ij = f ij : Eddington-Finkelstein Slicing: (αψ r = 0 αψ = 1 2 (αψ r = αψ Compared with Effective-One-Body PN[8] Conformal Imaging[6] Puncture Method[1] Greg Cook (WFU Physics 17

19 Constructing Evolutionary Sequences Given quasi-equilibrium data for binary black holes at various separations, how do we connect them together to form an evolutionary sequence? There is the freedom to set the overall mass scale for each solution. Let e(s, j(s, and ω(s denote the numerical solutions along a sequence parameterized by some measure of the separation s. Let χ(s define a mass scaling along the same sequence. Define E ADM (s χ(sm(s J(s χ 2 (sj(s Ω(s χ 1 (xω(s Choose the mass scale to preserve de ADM = ΩdJ. Integrating from s 1 to s 2 gives: χ(s 2 = χ(s 1 exp { } s 2 m ωj s 1 m 2ωj ds Greg Cook (WFU Physics 18

20 Results: E b /µ vs J/µm E b /µ CO: EF - d(αψ/dr=0 CO: EF - αψ = 1/2 CO: EF - d(αψ/dr = αψ IR: EF - d(αψ/dr=0 IR: EF - αψ = 1/2 IR: EF - d(αψ/dr = αψ CO: MS - d(αψ/dr = 0 CO: MS - αψ = 1/2 CO: MS - d(αψ/dr = (αψ/2r IR: MS - d(αψ/dr = 0 IR: MS - αψ = 1/2 IR: MS - d(αψ/dr = (αψ/2r J/µm Greg Cook (WFU Physics 19

21 Results: E b /µ vs l/m E b /µ CO: EF - d(αψ/dr=0 CO: EF - αψ = 1/2 CO: EF - d(αψ/dr = αψ IR: EF - d(αψ/dr=0 IR: EF - αψ = 1/2 IR: EF - d(αψ/dr = αψ CO: MS - d(αψ/dr = 0 CO: MS - αψ = 1/2 CO: MS - d(αψ/dr = (αψ/2r IR: MS - d(αψ/dr = 0 IR: MS - αψ = 1/2 IR: MS - d(αψ/dr = (αψ/2r l/m Greg Cook (WFU Physics 20

22 Results: E b /µ vs mω E b /µ CO: EF - d(αψ/dr=0 CO: EF - αψ = 1/2 CO: EF - d(αψ/dr = αψ IR: EF - d(αψ/dr=0 IR: EF - αψ = 1/2 IR: EF - d(αψ/dr = αψ CO: MS - d(αψ/dr = 0 CO: MS - αψ = 1/2 CO: MS - d(αψ/dr = (αψ/2r IR: MS - d(αψ/dr = 0 IR: MS - αψ = 1/2 IR: MS - d(αψ/dr = (αψ/2r mω Greg Cook (WFU Physics 21

23 Results: J/µm vs mω 5 J/µm CO: EF - d(αψ/dr=0 CO: EF - αψ = 1/2 CO: EF - d(αψ/dr = αψ IR: EF - d(αψ/dr=0 IR: EF - αψ = 1/2 IR: EF - d(αψ/dr = αψ CO: MS - d(αψ/dr = 0 CO: MS - αψ = 1/2 CO: MS - d(αψ/dr = (αψ/2r IR: MS - d(αψ/dr = 0 IR: MS - αψ = 1/2 IR: MS - d(αψ/dr = (αψ/2r mω Greg Cook (WFU Physics 22

24 Results: J/µm vs l/m J/µm CO: EF - d(αψ/dr=0 CO: EF - αψ = 1/2 CO: EF - d(αψ/dr = αψ IR: EF - d(αψ/dr=0 IR: EF - αψ = 1/2 IR: EF - d(αψ/dr = αψ CO: MS - d(αψ/dr = 0 CO: MS - αψ = 1/2 CO: MS - d(αψ/dr = (αψ/2r IR: MS - d(αψ/dr = 0 IR: MS - αψ = 1/2 IR: MS - d(αψ/dr = (αψ/2r l/m Greg Cook (WFU Physics 23

25 Results: mω vs l/m mω CO: EF - d(αψ/dr=0 CO: EF - αψ = 1/2 CO: EF - d(αψ/dr = αψ IF: EF - d(αψ/dr=0 IR: EF - αψ = 1/2 IR: EF - d(αψ/dr = αψ CO: MS - d(αψ/dr = 0 CO: MS - αψ = 1/2 CO: MS - d(αψ/dr = (αψ/2r IR: MS - d(αψ/dr = 0 IR: MS - αψ = 1/2 IR: MS - d(αψ/dr = (αψ/2r l/m Greg Cook (WFU Physics 24

26 Results: ISCO E b /M irr vs E ADM /M irr E b /M irr CO: EF - d(αψ/dr=0 CO: EF - αψ = 1/2 CO: EF - d(αψ/dr = αψ IR: EF - d(αψ/dr=0 IR: EF - αψ = 1/2 IR: EF - d(αψ/dr = αψ CO: MS - d(αψ/dr = 0 CO: MS - αψ = 1/2 CO: MS - d(αψ/dr = (αψ/2r IR: MS - d(αψ/dr = 0 IR: MS - αψ = 1/2 IR: MS - d(αψ/dr = (αψ/2r E ADM /M irr - 1 Greg Cook (WFU Physics 25

27 Results: ISCO E b /M irr vs ΩM irr E b /M irr CO: HKV CO: EOB 1PN CO: EOB 2PN CO: EOB 3PN A CO: EF - d(αψ/dr=0 (E ADM CO: EF - d(αψ/dr=0 (E b CO: EF - αψ = 1/2 (E ADM CO: EF - αψ = 1/2 (E b CO: EF - d(αψ/dr = αψ (E ADM CO: EF - d(αψ/dr = αψ (E b CO: MS - d(αψ/dr=0 CO: MS - αψ = 1/2 CO: MS - d(αψ/dr = αψ/2r IR: IVP-conf IR: IVP-punct IR: EOB 1PN IR: EOB 2PN IR: EOB 3PN A IR: EF - d(αψ/dr=0 (E ADM IR: EF - d(αψ/dr = 0 (E b IR: EF - αψ = 1/2 (E ADM IR: EF- αψ = 1/2 (E b IR: EF - d(αψ/dr = αψ (E ADM IR: EF - d(αψ/dr = αψ (E b IR: MS - d(αψ/dr=0 IR: MS - αψ = 1/2 IR: MS - d(αψ/dr = αψ/2r Ω M irr Greg Cook (WFU Physics 26

28 Results: ISCO E b /M irr vs J/M 2 irr CO: HKV CO: EOB 1PN CO: EOB 2PN CO: EOB 3PN A CO: EF - d(αψ/dr=0 (E ADM CO: EF - d(αψ/dr=0 (E b CO: EF - αψ = 1/2 (E ADM CO: EF - αψ = 1/2 (E b CO: EF - d(αψ/dr = αψ (E ADM CO: EF - d(αψ/dr = αψ (E b CO: MS - d(αψ/dr=0 CO: MS - αψ = 1/2 CO: MS - d(αψ/dr = αψ IR: IVP-conf IR: IVP-punct IR: EOB 1PN IR: EOB 2PN IR: EOB 3PN A IR: EF - d(αψ/dr=0 (E ADM E b /M irr IR: EF - d(αψ/dr = 0 (E b IR: EF - αψ = 1/2 (E ADM IR: EF - αψ = 1/2 (E b IR: EF - d(αψ/dr = αψ (E ADM IR: EF - d(αψ/dr = αψ (E b IR: MS - d(αψ/dr=0 IR: MS - αψ = 1/2 IR: MS - d(αψ/dr = αψ/2r J/M 2 irr Greg Cook (WFU Physics 27

29 Results: ISCO J/M 2 irr vs ΩM irr J/M 2 irr CO: HKV CO: EOB 1PN CO: EOB 2PN CO: EOB 3PN A CO: EF- d(αψ/dr=0 (E ADM CO: EF - d(αψ/dr=0 (E b CO: EF - αψ = 1/2 (E ADM CO: EF - αψ = 1/2 (E b CO: EF - d(αψ/dr = αψ (E ADM CO: EF - d(αψ/dr = αψ (E b CO: MS - d(αψ/dr=0 CO: MS - αψ = 1/2 CO: MS - d(αψ/dr = αψ/2r IR: IVP-conf IR: IVP-punct IR: EOB 1PN IR: EOB 2PN IR: EOB 3PN A IR: EF - d(αψ/dr=0 (E ADM IR: EF - d(αψ/dr = 0 (E b IR: EF - αψ = 1/2 (E ADM IR: EF - αψ = 1/2 (E b IR: EF - d(αψ/dr = αψ (E ADM IR: EF - d(αψ/dr = αψ (E b IR: MS - d(αψ/dr=0 IR: MS - αψ = 1/2 IR: MS - d(αψ/dr = αψ/2r Ω M irr Greg Cook (WFU Physics 28

30 References [1] T. W. Baumgarte. Innermost stable circular orbit of binary black holes. Phys. Rev. D, 62:024018/1 8, July [2] T. W. Baumgarte, G. B. Cook, M. A. Scheel, S. L. Shapiro, and S. A. Teukolsky. General relativistic models of binary neutron stars in quasiequilibrium. Phys. Rev. D, 57: , June [3] S. Bonazzola, E. Gourgoulhon, and J.-A. Marck. Numerical models of irrotational binary neutron stars in general relativity. Phys. Rev. Lett., 82: , Feb [4] J. M. Bowen and J. W. York, Jr. Time-asymmetric initial data for black holes and black-hole collisions. Phys. Rev. D, 21: , Apr [5] S. Brandt and B. Brügmann. A simple construction of initial data for multiple black holes. Phys. Rev. Lett., 78: , May [6] G. B. Cook. Three-dimensional initial data for the collision of two black holes II: Quasicircular orbits for equal mass black holes. Phys. Rev. D, 50: , Oct [7] G. B. Cook, M. W. Choptuik, M. R. Dubal, S. Klasky, R. A. Matzner, and S. R. Oliveira. Threedimensional initial data for the collision of two black holes. Phys. Rev. D, 47: , Feb [8] T. Damour, E. Gourgoulhon, and P. Grandclément. Circular orbits of corotating binary black holes: Comparison between analytical and numerical results. Phys. Rev. D, 66:024007/1 15, July [9] D. M. Eardley. Black hole boundary conditions and coordinate conditions. Phys. Rev. D, 57: , Feb Greg Cook (WFU Physics 29

31 [10] E. Gourgoulhon, P. Grandclément, and S. Bonazzola. Binary black holes in circular orbits. I. A global spacetime approach. Phys. Rev. D, 65:044020/1 19, Feb , 14 [11] P. Grandclément, E. Gourgoulhon, and S. Bonazzola. Binary black holes in circular orbits. II. Numerical methods and first results. Phys. Rev. D, 65:044021/1 18, Feb , 14, 17 [12] P. Marronetti, G. J. Mathews, and J. R. Wilson. Binary neutron-star systems: From the Newtonian regime to the last stable orbit. Phys. Rev. D, 58:107503/1 4, Nov [13] H. P. Pfeiffer and J. W. York, Jr. Extrinsic curvature and the Einstin constraints. Phys. Rev. D, 67:044022/1 8, Feb [14] J. Thornburg. Coordinate and boundary conditions for the general relativistic initial data problem. Class. Quantum Gravit., 4: , Sept [15] K. Uryū and Y. Eriguchi. New numerical method for constructing quasiequilibrium sequences of irrotational binary neutron stars in general relativity. Phys. Rev. D, 61:124023/1 19, June [16] J. W. York, Jr. Conformal thin-sandwich data for the initial-value problem of general relativity. Phys. Rev. Lett., 82: , Feb Greg Cook (WFU Physics 30

Initial Data for Black-Hole Binaries

Initial Data for Black-Hole Binaries Gregory B. Cook Wake Forest University June 11/1, 004 Abstract We will examine the current state of our efforts to generate astrophysically realistic initial data for