Toward Astrophysical Black-Hole Binaries Gregory B. Cook Wake - PowerPoint PPT Presentation

Toward Astrophysical Black-Hole Binaries Gregory B. Cook Wake Forest University Mar. 29, 2002 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 a complete set of boundary conditions for all initial data variables can be developed. This formalism should allow for the construction of completely general quasi-equilibrium black hole binary initial data. [5] Related LANL preprint. . . Collaborators: Harald Pfeiffer & Saul Teukolsky (Cornell)

Motivation • How do we go about constructing improved initial-data sets that more acurately represent astrophysical compact binary systems? • How do we define astrophysically realistic data? Focus Issues • Which decomposition of the constraints will be used? • How do we choose boundary conditions so that the constraints are well-posed and yield solutions with the desired physical content? • What choices for the spatial and temporal gauge are compatible with the desired physical content? • How do we fix the remaining freely specifiable data so as to yield the desired physical content? – Greg Cook – (WFU Physics) 1

The 3 + 1 Decomposition Lapse : α Spatial metric : γ ij αn µ δt Shift vector : β i Extrinsic Curvature : K ij t + δt Time vector : t µ = αn µ + β µ t µ δt n µ t d s 2 = − α 2 d t 2 + γ ij (d x i + β i d t )(d x j + β j d t ) 2 γ α µ γ β β µ K µν = − 1 γ µν = g µν + n µ n ν ν L n g αβ β µ δt Constraint equations Evolution equations R + K 2 − K ij K ij = 16 πρ ¯ ∂ t γ ij = − 2 αK ij + ¯ i β j + ¯ ∇ ∇ j β i K ij − γ ij K � � � = 8 πj i R ij − 2 K iℓ K ℓ ¯ ∂ t K ij = − ¯ i ¯ ¯ ∇ ∇ ∇ j α + α j + KK ij j � − 8 πS ij + 4 πγ ij ( S − ρ ) S µν ≡ γ α µ γ β ν T αβ j µ ≡ − γ ν µ n α T να + β ℓ ¯ j β ℓ + K jℓ ¯ ℓ K ij + K iℓ ¯ i β ℓ ∇ ∇ ∇ ρ ≡ n µ n ν T µν T µν = S µν + 2 n ( µ j ν ) + n µ n ν ρ – Greg Cook – (WFU Physics) 2

“Traditional” Black-Hole Data Conformal flatness and maximal slicing  Bowen-York solution[3]   ∆ L X i = 0   ˜ Analytic solutions for ˜ A ij  ⇒ γ ij = f ij (flat) ˜     ⇒ (conformal tracefree extrinsic curvature) K = 0    8 ψ − 7 ˜ A ij = 0  ˜ A ij ˜ ∇ 2 ψ + 1   Three general solution schemes Conformal Imaging-[6] Puncture Method-[4] No inner-BC: Inversion singular symmetry behavior inner-BC factored out Apparent Horizon BC-[11] All methods can produce very general Apparent configurations of multiple black holes, but horizon are fundamentally limited by choices for inner-BC γ ij and Bowen-York ˜ A ij . ˜ – Greg Cook – (WFU Physics) 3

“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 γ ij and ˜ A ij • initial dynamical degrees of freedom [in ˜ T T ] • initial temporal and spatial gauge degrees of freedom [in ˜ γ ij and K ] • boundary conditions on the constrained degrees of freedom [in ψ and X i ] 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) 4

Conformal Thin-Sandwich Decomposition[13] αn µ δt t + δt γ ij = ψ 4 ˜ γ ij t µ δt  n µ u i K ij = ψ − 10 u ij ≡ ∂ t ˜ ˜ γ ij (˜ i = 0) t L β ) ij − ˜  � u ij � 3 γ ij K (˜ + 1 2˜ α β µ α ≡ ψ − 6 α ˜ β µ δt  12 ψ 5 K 2 + 1 8 ψ − 7 ˜ A ij = − 2 πψ 5 ρ ∇ 2 ψ − 1 ˜ 8 ψ ˜ A ij ˜ R − 1 Hamiltonian Const. ∆ L β i − (˜ L β ) ij ˜ αψ 6 ˜ � u ij � ˜ ∇ i K + ˜ α ˜ αψ 10 j i α = 4 1 Momentum Const. ∇ j ˜ 3 ˜ ∇ α ˜ + 16 π ˜ j ˜ A ij ≡ � L β ) ij − ˜ u ij � u ij and β i have a simple physical interpretation, ˜ (˜ 1 ˜ 2˜ α unlike ˜ A ij T T and X i . Constrained vars : ψ and β i  u ij = 0 ˜  u ij , K , and ˜ Quasi-equilibrium ⇒ Freely specified : ˜ γ ij , ˜ α ∂ t K = 0 (Const. on α )  12 ψ 5 K 2 + 7 � ∇ 2 ( αψ ) − α 8 ψ − 7 ˜ A ij ˜ 8 ψ ˜ A ij ˜ 1 R + 5 Const. Tr( K ) eqn. � = ψ 5 β i ˜ + 2 πψ 5 K ( ρ + 2 S ) ∇ i K – Greg Cook – (WFU Physics) 5

Equations of Quasi-Equilibrium u ij = 0 , and K = 0 , With ˜ γ ij = f ij , ˜  Ham. & Mom. const.  ⇒ Eqns. of these equations have been widely used  eqns. from Conf. TS Quasi-Equilibrium to construct binary neutron star initial + Const. Tr( K ) eqn.   data[1, 10, 2, 12]. Binary neutron star initial data require: • boundary conditions at infinity compatible with asymptotic flatness and � ∂ i corotation. � β i | r →∞ = Ω ψ | r →∞ = 1 α | r →∞ = 1 ∂φ • 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, Grandcl´ ement, & Bonazzola[8, 9]: Black-hole u ij = 0 , K = 0 , “inversion-symmetry”, binaries with ˜ γ ij = f ij , ˜ and “Killing-horizon” conditions on the excision boundaries. “Solutions” require constraint violating regularity condition imposed on inner boundaries! – Greg Cook – (WFU Physics) 6

Constructing Regular Binary Black Hole QE ID Why does the GGB approach have problems? • Inversion-symmetry demands ˜ α = 0 & K = 0 on the inner boundary. � u ij � A ij ≡ L β ) ij − ˜ ˜ (˜ 1 • It is hard to move beyond ˜ γ ij = f ij . 2˜ α How do we proceed? • Find a method that allows for general choices of ˜ γ ij & K . ⋆ Eliminate dependence on inversion symmetry by letting the physical condition of quasi-equilibrium dictate the boundary conditions. Approach • Demand that the excision (inner) boundary be an apparent horizon . • Demand that the apparent horizon be in quasi-equilibrium. – Greg Cook – (WFU Physics) 7

The Inner Boundary ¯ ∇ i τ s i ≡ | ¯ ∇ τ | h ij ≡ γ ij − s i s j k µ ≡ 2 ( n µ + s µ ) 1 √ k µ ≡ 2 ( n µ − s µ ) ´ n µ 1 √ ´ k µ k µ Σ Extrinsic curvature of S embedded in spacetime s i 2 h α µ h β Σ µν ≡ − 1 ν L k g αβ S ´ 2 h α µ h β Σ µν ≡ − 1 ν L ´ k g αβ Extrinsic curvature of S embedded in Σ 2 h k i h ℓ H ij ≡ − 1 1 j L s γ kℓ Σ ij = 2 ( J ij + H ij ) √ ´ 1 Σ ij = 2 ( J ij − H ij ) √ Projections of K ij onto S Expansion of null rays J ij ≡ h k i h ℓ Shear of null rays j K kℓ θ ≡ h ij Σ ij = 1 2 ( J + H ) σ ij ≡ Σ ij − 1 √ 2 h ij θ J i ≡ h k i s ℓ K kℓ θ ≡ h ij ´ ´ 2 h ij ´ σ ij ≡ ´ 1 Σ ij − 1 Σ ij = 2 ( J − H ) ´ θ J ≡ h ij J ij = h ij K ij √ – Greg Cook – (WFU Physics) 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 → β ⊥ s µ β µ s µ 2. The inner boundary S doesn’t move: L ζ τ = 0 and ˆ ∇ i L ζ τ ≡ h j i ¯ ∇ j L ζ τ = 0 → ζ µ ≡ αn µ + β ⊥ s µ t µ = αn µ + β µ β ⊥ ≡ β i s i 3. The inner boundary S remains a MOTS[7]: ζ µ n µ t µ L ζ θ = 0 and L ζ ´ θ = 0 → s µ β µ 4. The horizons are in quasi-equilibrium: σ ij = 0 and no matter is on S → – Greg Cook – (WFU Physics) 9

Evolution of the Expansions � � 2 ´ 1 θ ( θ + 1 1 L ζ θ = θ − 2 K ) + E ( β ⊥ + α ) √ √ 2 � k ν � 2 ´ 2 K ) + D + 8 πT µν k µ ´ 1 θ ( 1 2 θ − 1 1 + θ − ( β ⊥ − α ) √ √ 2 + θs i ¯ ∇ i α, � � L ζ ´ θ (´ ´ 2 K ) + ´ − 1 θ + 1 1 θ = 2 θ − E ( β ⊥ − α ) √ √ 2 � k ν � ´ 2 ´ 2 K ) + ´ D + 8 πT µν k µ ´ 1 θ ( 1 θ − 1 1 − 2 θ − ( β ⊥ + α ) √ √ 2 θs i ¯ − ´ ∇ i α, Incorporates the constraint and evolution equations h ij ( ˆ ∇ i + J i )( ˆ ∇ j + J j ) − 1 2 ˆ D ≡ R of GR, the Gauss–Codazzi–Ricci equations governing the embedding of S in the spatial hypersurface, and h ij ( ˆ ´ ∇ i − J i )( ˆ ∇ j − J j ) − 1 2 ˆ D ≡ R the demand that S remain at a constant coordinate σ ij σ ij + 8 πT µν k µ k ν E ≡ location. These equations incorporate no assumption σ ij + 8 πT µν ´ k µ ´ k ν of quasi-equilibrium. ´ E ≡ σ ij ´ ´ Terms that vanish because we demand S be a MOTS, remain a MOTS, or because we demand the horizon to be in equilibrium are in RED . – Greg Cook – (WFU Physics) 10