BINARY BLACK HOLES IN CIRCULAR ORBITS: AN HELICAL KILLING VECTOR - PowerPoint PPT Presentation

✬ ✩ BINARY BLACK HOLES IN CIRCULAR ORBITS: AN HELICAL KILLING VECTOR APPROACH Eric Gourgoulhon Silvano Bonazzola Philippe Grandcl´ ement Phys. Rev. D. 65 , 044020 (2002). Phys. Rev. D. 65 , 044021 (2002). ✫ ✪

✬ ✩ 3+1 FORMALISM (VACUUM CASE) Orthogonal projection of Einstein’s equations on spatial hypersurfaces = ⇒ split space-time into space AND time. ds 2 = − N 2 − N i N i � dt 2 + 2 N i dtdx i + γ ij dx i dx j � • Hamiltonian constraint : R + K 2 − K ij K ij = 0 • Momentum constraints : D j K ij − D i K = 0 • Evolution equations : ∂K ij R ij − 2 KikK k � � − L � N K ij = − D i D j N + N j + KK ij ∂t ∂γ ij ∂t − L � N γ ij = − 2 NK ij ✫ ✪

✬ ✩ QUASI-STATIONARITY • approximate but 4D space-time representing two black holes in exact circular orbits • Valid when τ orb . << τ grav . • Rigorous definition of Ω. • Circular orbit due to radiation. ✫ ✪

✬ ✩ HELICAL KILLING VECTOR ⇒ Helical Killing vector � Circular orbits = l . Advance δt in time ⇐ ⇒ Rotation of δϕ = Ω δt . Inertial coordinates : � ∂ � ∂ � α � α l α = + Ω ∂t ∂ϕ = const. Corotating coordinates : t x i’ � ∂ � α • such as l α = . ∂t • coordinate t is ignorable. n α � ∂ α � i l N • corotating shift β i = N i + Ω . ∂ϕ v α • functions N and γ ij are the same. Σ t α B ✫ ✪

✬ ✩ ADDITIONAL HYPOTHESIS Gauge choice : Maximum slicing K = 0 . Conformal flatness approximation : γ ij = Ψ 4 f ij . ✫ ✪

✬ ✩ ELLIPTIC EQUATIONS We solve 5 of the 10 Einstein’s equations : ∆Ψ = − Ψ 5 A ij ˆ ˆ A ij • Hamiltonian constraint : 8 A ij � ¯ ∆ β i + 1 D i ¯ D j β j = 2 ˆ ¯ D j N − 6 N ¯ � • Momentum constraints : D j ln Ψ 3 • Trace of ∂K ij ∆ N = N Ψ 4 ˆ A ij − 2 ¯ A ij ˆ D j ln Ψ ¯ D j N : ∂t A ij = Ψ 4 K ij . with ˆ A ij = Ψ − 4 K ij and ˆ 1 A ij = ⇒ ˆ 2 N ( Lβ ) ij Definition of K = ( Lβ ) ij is the conformal Killing operator : ( Lβ ) ij = ¯ D i β j + ¯ D j β i − 2 3 ¯ D k β k f ij Set of 5 non-linear, highly-coupled, elliptic equations. ✫ ✪

� � ✬ ✩ CHOICE OF THE TOPOLOGY R × Misner-Lindquist. (t,r 1 θ 1 φ 1 , ) , a 1 a 2 I(P) r 1 P 3 I R I P I II I(P) a 1 a 2 I(P) (t,r θ φ 2 , , ) r 2 2 2 P 3 I R ✫ ✪

✬ ✩ ISOMETRY Mapping from M I to M II . I M I − → M II ( t, x I , y I , z I ) − → ( t, x II = x I , y II = y I , z II = z I ) � t, a 2 � ( t, r 1 , θ 1 , ϕ 1 ) − → r 1 , θ 1 , ϕ 1 1 � t, a 2 � ( t, r 2 , θ 2 , ϕ 2 ) − → r 2 , θ 2 , ϕ 2 2 Hypothesis : the 4-metric is isometric . ∂I µ ∂I ν ∂x β g µν ( I ( P )) = g αβ ( P ) ∂x α Consequence : solve only on M I with boundary conditions on the throats. ✫ ✪

✬ ✩ BOUNDARY CONDITIONS ON THE THROATS  N | S i = 0  N ( I ( P )) = ± N ( P ) = ⇒ ( or ∂ r N | S i = 0)   β r | S i = 0   − a 2   β r ( I ( P )) r 2 β r ( P )  ∂ θ β r | S i  = 0 =         β θ ( I ( P )) β θ ( P ) = ⇒ ∂ ϕ β r | S i = 0 =    β ϕ ( I ( P )) β ϕ ( P )  ∂ r β θ � = 0   =   � S i    ∂ r β ϕ | S i  = 0  � ∂ r Ψ + 1 Ψ ( I ( P )) = a � r Ψ ( P ) = ⇒ 2 a Ψ = 0 � � S i Consequence: K ij ( I ( P )) = − ∂I i ∂I j K kl ( P ) ∂x k ∂x l The throats are apparent horizons . ✫ ✪

✬ ✩ STATE OF ROTATION � β not completely fixed by isometry. � ⇒ � Corotating black holes = = 0. β � � S i Properties : • Analogy with rigidity theorem . • Throats are Killing horizons . ✫ ✪

✬ ✩ ISOMETRY AND REGULARITY Rigidity implies isometry except : ∂ r β θ � ∂ r β ϕ | S i = 0 S i = 0 and � To have a regular K :  ( Lβ ) ij ˆ A ij  =  ⇒ ( Lβ ) ij � = S i = 0 . 2 N � � N | S i = 0   So to have RIGIDITY , REGULARITY and ISOMETRY one must have : � � = 0 β � � S i � ∂ r � = 0 β � � S i In this framework it is impossible to have REGULARITY for non-corotating ✫ black holes. ✪

✬ ✩ REGULARIZATION OF THE SHIFT One solves for � β , using Dirichlet-type boundary condition : � � β = 0 � � S i At each iteration one modifies the shift vector by : β new = � � β + � β cor � β cor is chosen so that : � � β new = 0 � � S i � ∂ r � = 0 . β new � � S i At the end of a calculation : • if � β cor → 0 : exact solution. • if � β cor is small : approximate solution. • else not a solution ! ✫ ✪

✬ ✩ BOUNDARY CONDITIONS AT INFINITY To recover Minkowski space-time : N → 1 when r → ∞ Ψ → 1 when r → ∞ β → Ω ∂ � when r → ∞ ∂ϕ ✫ ✪

✬ ✩ DETERMINATION OF Ω Ω only present in the boundary condition for � β . One can solve for ANY value of Ω (example : Ω = 0 = ⇒ Misner-Lindquist). � r − 1 � SUPPLEMENTARY CONDITION : the O part of the metric when ( r → ∞ ) is identical to Schwarzschild . Ψ ∼ 1 + M ADM and N ∼ 1 − M K A priori : 2 r r ⇒ Ψ 2 N ∼ 1 + α One chooses the ONLY Ω such that : M K = M ADM ⇐ r 2 Justifications : • exact stationary asymptotical space-times. • Newtonian limit = ⇒ virial theorem. • True for binary neutron stars. ✫ ✪

✬ ✩ CONSTRUCTION OF A SEQUENCE Each configuration depends on 2 parameters : • radius of the throats a . • separation D a . Existence of a scaling factor : only one sequence . a is chosen so that : � dM ADM � = Ω . � dJ � sequence ✫ ✪

✬ ✩ AREA OF THE HORIZONS First law of thermodynamics for binary black holes : dM ADM = Ω dJ + 1 8 π ( κ 1 dA 1 + κ 2 dA 2 ) Consequence : along the sequence dA = 0. Horizon area is constant : quasi-static evolution (Second law of thermodynamics for black holes). ✫ ✪

✬ ✩ NUMERICAL METHODS Basic features : • Multi-domain : two sets of spherical coordinates. • Compactification : exact treatment of spatial infinity. • Spectral decomposition : spherical harmonics and Tchebychev polynomials. ✫ ✪

✬ ✩ NORM OF � β cor Correction (21*17*16) -1 10 Error on J (21*16*17) Correction (33*21*20) Error on J (33*21*20) -2 10 Relative error -3 10 -4 10 10 15 20 25 30 Separation parameter D/a ⇒ � J ∞ = J S ⇐ β cor = 0 ✫ ✪

✬ ✩ SMARR FORMULA -1 10 Relative error on Smarr formula -2 10 -3 10 J infinity (21*17*16) J throats (21*17*16) J infinity (33*21*20) J throats (33*21*20) -4 10 10 15 20 25 Separation parameter D/a M − 2Ω J = − 1 S i − 1 � � Ψ 2 ¯ Ψ 2 ¯ D i Nd ¯ D i Nd ¯ S i 4 π 4 π S 1 S 2 ✫ ✪

✬ ✩ AREA OF THE HORIZONS Relative change of the irreducible mass M ir 21*17*16 33*21*20 1e-03 1e-04 0.1 0.2 0.05 0.075 0.125 0.15 0.175 Orbital velocity Ω M ir � � A 1 A 2 M ir = 16 π + 16 π ✫ ✪

✬ ✩ LAPSE IN THE ORBITAL PLANE ISCO configuration ✫ ✪

✬ ✩ CONFORMAL FACTOR IN THE ORBITAL PLANE ISCO configuration ✫ ✪

✬ ✩ K XY IN THE ORBITAL PLANE ISCO configuration ✫ ✪

✬ ✩ SEQUENCE : VARIATION OF M ADM Comparison Numerical results <-> 3-PN EOB Total energy along a sequence 0,995 Grandclement et al. 2001, 33x21x20 Grandclement et al. 2001, 21x17x16 EOB 3-PN a4=4.67 corot, Damour et al. 2001 EOB 3-PN a4=4.67 irrot, Damour et al. 2000 0,99 Cook 1994, Pfeiffer et al. 2000, irrot M ADM / M ir 0,985 0,98 0 0,05 0,1 0,15 Ω M ir ✫ ✪

✬ ✩ SEQUENCE : VARIATION OF J Comparison Numerical results <-> 3-PN EOB Total angular momentum along a sequence 1 Grandclement et al. 2001, 33x21x20 Grandclement et al. 2001, 21x17x16 EOB 3-PN a4=4.67 corot, Damour et al. 2001 EOB 3-PN a4=4.67 irrot, Damour et al. 2000 Cook 1994, Pfeiffer et al. 2000, irrot 0,9 2 J / M ir 0,8 0 0,05 0,1 0,15 Ω M ir ✫ ✪

✬ ✩ POSITION OF THE ISCO 33*21*20 21*17*16 3−PN, corotating, ω s =0 (Damour et al. 2001) Binding energy (M−M ir ) / M ir 3PN, S=0, ω s =−9 (Damour et al. 2000) −0.015 Conformal imaging S=0 (Pfeiffer et al. 2000) Conformal imaging S=0.08 (Pfeiffer et al. 2000) Conformal imaging S=0.17 (Pfeiffer et al. 2000) −0.02 Puncture S=0 (Baumgarte 2000) −0.025 −0.03 0.05 0.1 0.15 0.2 0.25 0.3 Orbital velocity Ω M ir ✫ ✪

✬ ✩ PLAUSIBLE EXPLANATION Main difference with IVP : extrinsic curvature tensor IVP HKV choice given by the shift vector 3 1 Ψ 4 K ij = 2 N ( Lβ ) ij Ψ 2 K ij = � P i n j + P j n i − ( f ij − n i n j ) P k n k � 2 r 2 Indications that IVP does not produce real circular orbits : • Plunge even for pre-ISCO initial conditions = ⇒ the real ISCO is further away . • One must impose 0 . 55Ω to maintain the black holes in the corotating frame = ⇒ Ω is to big . ✫ ✪