Initial Data for Black-Hole Binaries

Size: px

Start display at page:

Download "Initial Data for Black-Hole Binaries"

Barrie Simon
5 years ago
Views:

1 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 black-hole binaries. Collaborators: Harald Pfeiffer (Caltech) & 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 Time vector : t µ = αn µ + β µ = α dt + 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 γα µ γβ ν L ng αβ Constraint equations R + K 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 + n (µ j ν) + n µ n ν ρ Evolution equations t γ ij = αk ij + i β j + j β i t K ij = i j α + α[ Rij K 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)

4 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 α [ ( Lβ) ij ũ ij] ψ 4 γ ij K Hamiltonian Const. ψ 1 8 ψ R 1 1 ψ5 K ψ 7 Ã ij Ã ij = πψ 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. [ (ψ 7 α) (ψ 7 1 α) 8 ψ R ] ψ5 K ψ 7 Ã ij Ã ij ψ 5 β i i K Ã ij 1 α [( Lβ) ij ũ ij] Constrained vars : ψ, β i, and α ψ 6 α Freely specified : γ ij K ũ ij t γ ij and t K = πψ 5 K(ρ + S) ψ 5 t K t γ ij = 0 Quasi-equilibrium t K = 0 Greg Cook (WFU Physics) 3

5 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[, 9, 3, 10]. 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)

6 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[, 9, 3, 10]. 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[7, 8]: 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) 4

7 The Inner Boundary s i i τ τ h ij γ ij s i s j k µ 1 (n µ + s µ ) Σ ḱ µ n µ k µ ḱ µ 1 (n µ s µ ) Extrinsic curvature of S embedded in s i spacetime S Σ µν 1 hα µ hβ ν L kg αβ Σ µν 1 hα µ hβ ν L ḱ g αβ Extrinsic curvature of S embedded in Σ H ij 1 hk i hl j L sγ kl Projections of K ij onto S Σ ij = 1 (H ij J ij ) Σ ij = 1 (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 (H J) θ h ij Σij = 1 (H + J) σ ij Shear of null rays Σ ij 1 h ijθ σ ij Σ ij 1 h ij θ Greg Cook (WFU Physics) 5

8 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. The inner boundary S remains a MOTS: L ζ θ = 0 t µ = αn µ + β µ ζ µ αn µ + β s µ β β i s i ζ µ is null on the AH and the chosen form is a gauge choice. 3. The horizons are in quasi-equilibrium: σ ij = 0 and no matter is on S β s µ s µ β µ t µ n µ ζ µ β µ s µ Greg Cook (WFU Physics) 6

9 AH/Quasi-Equilibrium Boundary Conditions θ = ψ [ hij ] i s j + 4 s k k ln ψ ψ J L ζ θ = 1 [θ(θ + 1 θ ] + 1 K) + E (β + α) 1 [θ( 1 θ 1 θ + 1 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 R σ ij = 1 ( (H ij 1 h ijh) 1 β ) α 1 ψ 4 α E σ ij σ ij + 8πT µν k µ k ν { } D(i β j) 1 h ij Dk β k 1 [ h ik hjl ũ kl 1 h ij hkl ũ kl ]

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

11 AH/Quasi-Equilibrium Boundary Conditions θ = ψ [ hij ] i s j + 4 s k k ln ψ ψ J L ζ θ = 1 [θ(θ + 1 θ ] + 1 K) + E (β + α) 1 [θ( 1 θ 1 θ + 1 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 R σ ij = 1 ( (H ij 1 h ijh) 1 β ) α 1 ψ 4 α E σ ij σ ij + 8πT µν k µ k ν { } D(i β j) 1 h ij Dk β k 1 [ h ik hjl ũ kl 1 h ij hkl ũ kl ] s k k ln ψ = 1 4 ( h ij i s j ψ J) β i = αψ s i + β i 0 = D (i β j) 1 h ij Dk β k t ln ψ = [ Dk β k k + 4β Dk ln ψ 1 h kl ũ kl + ] θ + (β α)h Greg Cook (WFU Physics) 7

12 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 h ij Dk ξ k = 0 Greg Cook (WFU Physics) 8

13 The Lapse BC & QE So far, nothing has fixed a boundary condition on the lapse α. One possibility[6] 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 (J JK + D)α D ψ 4 [ h ij ( D i J i )( D j J j ) 1 R + D ln ψ]

14 The Lapse BC & QE So far, nothing has fixed a boundary condition on the lapse α. One possibility[6] 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 (J JK + D)α D ψ 4 [ h ij ( D i J i )( D j J j ) 1 R + D 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 = α = β R = C R dr 1 M R + + R d Ω C R 4 1 M R + C R 4 1 M R + C R 4 K i j = C R α S = C 4M Greg Cook (WFU Physics) 9

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[7, 8] 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 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 G)d S j Acceptable definition of the mass for arbitrary spacetimes. G ij γ ij f ij Greg Cook (WFU Physics) 10

16 Summary of QE Formalism γ ij = ψ 4 γ ij K ij = ψ 10 Ã ij γij K Ã ij = ψ6 α ( Lβ) ij t γ ij = 0 ψ 1 8 ψ R 1 1 ψ5 K ψ 7 Ã ij Ã ij = 0 j ( Lβ) ij ( Lβ) ij j ln αψ 6 = 4 3 α i K [ 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 J) S θ = 0 β i S = αψ s i S αψ 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) 11

17 Results Corotation γ ij = f ij : Maximal Slicing: (αψ) r = 0 αψ = 1 (αψ) r = αψ r γ ij = f ij : Eddington-Finkelstein Slicing: (αψ) r = 0 αψ = 1 (αψ) r = αψ r Compared with Effective-One-Body PN[5] Inversion-Symmetric HKV[8] Irrotation γ ij = f ij : Maximal Slicing: (αψ) r = 0 αψ = 1 (αψ) r = αψ r γ ij = f ij : Eddington-Finkelstein Slicing: (αψ) r = 0 αψ = 1 (αψ) r = αψ r Compared with Effective-One-Body PN[5] Conformal Imaging[4] Puncture Method[1] Greg Cook (WFU Physics) 1

18 Corotating; Maximal Slicce; QE-BC; E b /µ vs J/µm -0.0 CO: MS - d(αψ)/dr = 0 CO: MS - αψ = 1/ CO: MS - d(αψ)/dr = (αψ)/r E b /µ J/µm Greg Cook (WFU Physics) 13

19 Corotating; Maximal Slicce; Comparison; E b /µ vs J/µm CO: MS - d(αψ)/dr = (αψ)/r CO: HKV - GGB CO: EOB - 3PN CO: EOB - PN CO: EOB - 1PN E b /µ J/µm Greg Cook (WFU Physics) 14

20 Irrotational; Maximal Slicce; QE-BC; E b /µ vs J/µm IR: MS - d(αψ)/dr = 0 IR: MS - αψ = 1/ IR: MS - d(αψ)/dr = (αψ)/r E b /µ J/µm Greg Cook (WFU Physics) 15

21 Irrotational; Maximal Slicce; Comparison; E b /µ vs J/µm IR: MS - d(αψ)/dr = (αψ)/r IR: Conf. Imaging/Eff. Pot. IR: EOB - 3PN IR: EOB - PN IR: EOB - 1PN E b /µ J/µm Greg Cook (WFU Physics) 16

22 Maximal Slicce; Comparison of ISCO; E b /M irr vs ΩM irr E b /µ CO: HKV CO: EOB 1PN CO: EOB PN CO: EOB 3PN A CO: MS - d(αψ)/dr=0 CO: MS - αψ = 1/ CO: MS - d(αψ)/dr = αψ/r IR: IVP-conf IR: EOB 1PN IR: EOB PN IR: EOB 3PN A IR: MS - d(αψ)/dr=0 IR: MS - αψ = 1/ IR: MS - d(αψ)/dr = αψ/r Ω m Greg Cook (WFU Physics) 17

23 Maximal Slicce; Comparison of ISCO; E b /M irr vs J/M irr E b /µ CO: HKV CO: EOB 1PN CO: EOB PN CO: EOB 3PN A CO: MS - d(αψ)/dr=0 CO: MS - αψ = 1/ CO: MS - d(αψ)/dr = αψ/r IR: IVP-conf IR: EOB 1PN IR: EOB PN IR: EOB 3PN A IR: MS - d(αψ)/dr= IR: MS - αψ = 1/ 3.4IR: MS - d(αψ)/dr 3.6 = αψ/r J/µm Greg Cook (WFU Physics) 18

24 Open Questions Is the physics of the corotating and irrotational models correct? Do the corotating black holes have the correct angular momentum? Is the angular momentum of the irrotational holes nearly zero? How do we make a physically motivated choice for γ ij? Greg Cook (WFU Physics) 19

25 References [1] T. W. Baumgarte. Innermost stable circular orbit of binary black holes. Phys. Rev. D, 6:04018/1 8, July [] 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., 8:89 895, Feb [4] 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 [5] T. Damour, E. Gourgoulhon, and P. Grandclément. Circular orbits of corotating binary black holes: Comparison between analytical and numerical results. Phys. Rev. D, 66:04007/1 15, July [6] D. M. Eardley. Black hole boundary conditions and coordinate conditions. Phys. Rev. D, 57:99 304, Feb [7] E. Gourgoulhon, P. Grandclément, and S. Bonazzola. Binary black holes in circular orbits. I. A global spacetime approach. Phys. Rev. D, 65:04400/1 19, Feb , 10 [8] P. Grandclément, E. Gourgoulhon, and S. Bonazzola. Binary black holes in circular orbits. II. Numerical methods and first results. Phys. Rev. D, 65:04401/1 18, Feb , 10, 1 [9] 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 [10] K. Uryū and Y. Eriguchi. New numerical method for constructing quasiequilibrium sequences of irrotational binary neutron stars in general relativity. Phys. Rev. D, 61:1403/1 19, June Greg Cook (WFU Physics) 0

Black-Hole Binary Initial Data: Getting the Spin Right

Black-Hole Binary Initial Data: Getting the Spin Right Gregory B. Cook Wake Forest University October 5, 2005 Collaborators: Harald Pfeiffer[7] (Caltech), Jason D. Grigsby (WFU), & Matthew Caudill (WFU)