Getting The Spin Right In Black-Hole Binaries
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1 Getting The Spin Right In Black-Hole Binaries Gregory B. Cook Wake Forest University July 25, 2005 Abstract We will take a detailed look at the issues involved in setting the spin of a black hole during the construction of initial data. Theory seems to provide sufficient freedom that any physically allowed spin can be chosen. The more difficult issue is finding a way to choose the specific spin that we want. I will present some new results that explore the spins of individual black holes in both corotating and irrotational binary systems. Collaborators: Harald Pfeiffer[3] (Caltech), Jason D. Grigsby (WFU), & Matthew Caudill (WFU)
2 Motivation Black hole binaries are among the most likely sources for early detection with LIGO, VIRGO, GEO, TAMA,... Available computed waveforms should increase chance of detecting collision events. Quasiequilibrium 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. Quasiequilibrium gives us a physical condition to guide us in fixing boundary conditions and data that is not otherwise constrained.
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 + αh Rij 2K il K l j + KK ij i 8πS ij + 4πγ ij (S ρ) + β l l K ij + K il j β l + K jl i β l
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 2 α h ( Lβ) ij ũ iji ψ 4 γ ij K Hamiltonian Const. 2 ψ 1 8 ψ R 1 12 ψ5 K ψ 7 à ij à 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. h 2 (ψ 7 α) (ψ 7 1 α) 8 ψ R i ψ5 K ψ 7 à ij à ij ψ 5 β i i K à ij 1 2 α h( Lβ) ij ũ iji Constrained vars : ψ, β i, and α ψ 6 α Freely specified : γ ij K ũ ij t γ ij and t K = 2πψ 5 K(ρ + 2S) ψ 5 t K 8 < t γ ij = 0 Quasiequilibrium : t K = 0
5 Measuring & Setting the Spin Spin is only rigorously defined at spatial/null infinity. Must use quasi-local definition: e.g. Brown & York[2] or Ashtekar & Krishnan[1] S = 1 K ij ξ i s j hd 2 x 8π BH Spin is fixed primarily by value of K ij : suggests using the shift at the black hole to fix the spin. If we assume that the black hole is in approximate equilibrium (isolated), then we can connect the shift vector to the null congruence that defines the black hole s horizon.
6 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
7 AH and QE Conditions on the Inner Boundary The quasiequilibrium inner boundary conditions start with the following assumptions: 1. The inner boundary S is a (MOTS): marginally outer-trapped surface θ = 0 2. The horizons are in quasiequilibrium: σ ij = 0 and no matter is on S Raychaudhuri s equation implies that MOTS initially evolves along k µ. L k θ = 1 2 θ2 σ µν σ µν ω µν ω µν R µν k µ k ν = 0 3. The time evolution vector is null on S: t µ k µ S = 0 k µ 1 2 (n µ + s µ ) t µ = αn µ + β µ 9 = ; = α S = β i s i S β S
8 AH/Quasiequilibrium Boundary Conditions θ = ψ 2 h hij i i s j + 4 s k k ln ψ ψ 2 J 2 σ ij = 1 (H ij 1 2 h ijh) 1 β «2 α 1 2 ψ 4 α n o 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 ]
9 AH/Quasiequilibrium Boundary Conditions θ = ψ 2 h hij i i s j + 4 s k k ln ψ ψ 2 J 2 σ ij = 1 (H ij 1 2 h ijh) 1 β «2 α 1 2 ψ 4 α n o 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
10 AH/Quasiequilibrium Boundary Conditions θ = ψ 2 h hij i i s j + 4 s k k ln ψ ψ 2 J 2 σ ij = 1 (H ij 1 2 h ijh) 1 β «2 α 1 2 ψ 4 α n o 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 t ln ψ = 1 h Dk β k k + 4β Dk ln ψ h kl ũ kl + i 2θ (α β )H
11 Defining the Spin of the Black Hole The spin parameters β i can be defined by demanding that the time vector associated with quasiequilibrium in the corotating frame must be null, forming the null generators of the horizon. k µ (n µ + s µ ) = k µ = h1, αs i β ii 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, h k µ = 1, αs i β ii = [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 0 t t φ h 1, αs i β ii = [1, Ω 0 ( / φ) i ] Thus we find «i β i = αs i + Ω 0 β i = Ω 0 ξ i φ ξ i ( φ )i & D (i ξ j) 1 2 h ij Dk ξ k = 0
12 Corotating; Maximal Slice; Comparison; E b /µ vs J/µm CO: MS - d(αψ)/dr = (αψ)/2r CO: HKV - GGB CO: EOB - 3PN CO: EOB - 2PN CO: EOB - 1PN E b /µ J/µm
13 Irrotational; Maximal Slice; Comparison; E b /µ vs J/µm IR: MS - d(αψ)/dr = (αψ)/2r IR: IVP Conf.Imaging/Eff.Pot. IR: EOB - 3PN IR: EOB - 2PN IR: EOB - 1PN E b /µ J/µm
14 Maximal Slice; Comparison of ISCO; E b /M irr vs ΩM irr E b / m CO: QE CO: HKV-GGB CO: PN EOB CO: PN standard IR: QE IR: IVP conf IR: PN EOB IR: PN standard mω 0
15 Maximal Slice; Comparison of ISCO; E b /M irr vs J/M 2 irr E b / m CO: QE CO: HKV-GGB CO: PN EOB CO: PN standard IR: QE IR: IVP conf IR: PN EOB IR: PN standard J / m 2
16 Maximal Slice; Comparison of ISCO; J/M 2 irr vs ΩM irr J / m CO: QE CO: HKV-GGB CO: PN EOB CO: PN standard IR: QE IR: IVP conf IR: PN EOB IR: PN standard mω 0
17 Corotation: Flat-KV Spin; S/M 2 irr vs mω S/M CO: CK - d(αψ)/dr = (αψ)/2r CO: CK - d(αψ)/dr=0 CO: CK - αψ = 1/ mω 0
18 Corotation: Flat-KV Spin Error; (S S Kerr )/S Kerr vs mω CO: CK - d(αψ)/dr = αψ CO: CK - d(αψ)/dr=0 CO: CK - αψ = 1/2 (S - S Kerr )/S Kerr mω 0
19 Irrotation: Flat-KV Spin; S/M 2 irr vs mω S/M IR: CK - d(αψ)/dr = (αψ)/2r IR: CK - d(αψ)/dr=0 IR: CK - αψ = 1/ mω 0
20 Irrotation: Flat-KV Spin Error; S/S Kerr vs mω IR: CK - d(αψ)/dr = αψ IR: CK - d(αψ)/dr=0 IR: CK - αψ = 1/2 S/S Kerr mω 0
21 Searching for Killing Vectors In general, a 2-surface will not have any Killing vectors. We searched for the any Killing vectors or the closest approximation there is to one Use the Killing transport equations[4]: v a D a ξ b = v a L ab v a D a L bc = 2 R cba d ξ d v a
22 Searching for Killing Vectors When we have orbital plane reflection symmetry we find a true Killing field on the surface! Can use this instead of the Conformal Killing vector in the computation of the quasilocal spin.
23 Killing Vectors on BH Surface
24 Killing Vectors on BH Surface
25 Killing Vector Differences on BH Surface
26 Killing Vector Differences on BH Surface
27 Killing Vector Differences on BH Surface
28 Killing Vector Differences on BH Surface
29 Killing Vector Differences on BH Surface
30 Corotation: Flat-KV Spin; S/M 2 irr vs mω S/M CO: CK - d(αψ)/dr = (αψ)/2r CO: CK - d(αψ)/dr=0 CO: CK - αψ = 1/2 CO: KV - d(αψ)/dr = αψ CO: KV - d(αψ)/dr=0 CO: KV - αψ = 1/ mω 0
31 Corotation: Flat-KV Spin Error; (S S Kerr )/S Kerr vs mω CO: CK - d(αψ)/dr = αψ CO: CK - d(αψ)/dr=0 CO: CK - αψ = 1/2 CO: KV - d(αψ)/dr = αψ CO: KV - d(αψ)/dr=0 CO: KV - αψ = 1/2 (S - S Kerr )/S Kerr mω 0
32 Killing Vector Differences on BH Surface
33 Killing Vector Differences on BH Surface
34 Killing Vector Differences on BH Surface
35 Killing Vector Differences on BH Surface
36 Killing Vector Differences on BH Surface
37 Irrotation: Flat-KV Spin; S/M 2 irr vs mω S/M IR: CK - d(αψ)/dr = (αψ)/2r IR: CK - d(αψ)/dr=0 IR: CK - αψ = 1/2 IR: KV - d(αψ)/dr = αψ IR: KV - d(αψ)/dr=0 IR: KV - αψ = 1/ mω 0
38 Irrotation: Flat-KV Spin Error; S/S Kerr vs mω IR: CK - d(αψ)/dr = αψ IR: CK - d(αψ)/dr=0 IR: CK - αψ = 1/2 IR: KV - d(αψ)/dr = αψ IR: KV - d(αψ)/dr=0 IR: KV - αψ = 1/2 S/S Kerr mω 0
39 True Irrotation * The spin of the Irrotational blacks seem much too large. Choice of using Ω 0 in boundary condition for irrotational black holes is probably wrong. Choose magnitude of β i to make spin vanish.
40 Killing Vector Differences on BH Surface
41 Killing Vector Differences on BH Surface
42 Killing Vector Differences on BH Surface
43 Killing Vector Differences on BH Surface
44 Killing Vector Differences on BH Surface
45 True Irrotation: True-KV Spin; S/M 2 irr vs mω S/M TI: CK - d(αψ)/dr = (αψ)/2r TI: CK - d(αψ)/dr=0 TI: CK - αψ = 1/2 TI: KV - d(αψ)/dr = αψ TI: KV - d(αψ)/dr=0 TI: KV - αψ = 1/ mω 0
46 True Irrotation: True-KV Spin Error; S/S Kerr vs mω S/S Kerr TI: CK - d(αψ)/dr = αψ TI: CK - d(αψ)/dr=0 TI: CK - αψ = 1/2 TI: KV - d(αψ)/dr = αψ TI: KV - d(αψ)/dr=0 TI: KV - αψ = 1/ mω 0
47 True Irrotation; Maximal Slice; Comparison; E b /µ vs J/µm IR: MS - d(αψ)/dr = (αψ)/2r TI: MS - d(αψ)/dr = (αψ)/2r IR: IVP Conf.Imaging/Eff.Pot. IR: EOB - 3PN IR: EOB - 2PN IR: EOB - 1PN E b /µ J/µm
48 Maximal Slice; Comparison of ISCO; E b /M irr vs ΩM irr E b / m CO: QE CO: HKV-GGB CO: PN EOB CO: PN standard IR: QE IR: IVP conf IR: PN EOB IR: PN standard TI: QE mω 0
49 Maximal Slice; Comparison of ISCO; E b /M irr vs J/M 2 irr E b / m CO: QE CO: HKV-GGB CO: PN EOB CO: PN standard IR: QE IR: IVP conf IR: PN EOB IR: PN standard TI: QE J / m 2
50 Maximal Slice; Comparison of ISCO; J/M 2 irr vs ΩM irr J / m CO: QE CO: HKV-GGB CO: PN EOB CO: PN standard IR: QE IR: IVP conf IR: PN EOB IR: PN standard TI: QE mω 0
51 References [1] A. Ashtekar and B. Krishnan. Dynamical horizons and their properties. Phys. Rev. D, 68:104030/1 25, [2] J. D. Brown and J. W. York, Jr. Quasilocal energy and conserved charges derived from the gravitational action. Phys. Rev. D, 47: , [3] G. B. Cook and H. P. Pfeiffer. Excision boundary conditions for black hole initial data. Phys. Rev. D, 70:104016/1 24, [4] O. Dreyer, B. Krishnan, D. Shoemaker, and E. Schnetter. Introduction to isolated horizons in numerical relativity. Phys. Rev. D, 67:024018/1 14, Jan
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