Trapping horizons in constrained evolutions of excised black holes - PowerPoint PPT Presentation

Trapping horizons in constrained evolutions of excised black holes Eric Gourgoulhon and Jos´ e Luis Jaramillo Laboratoire de l’Univers et de ses Th´ eories (LUTH) Observatoire de Paris 92195 Meudon, France Instituto de Astrof´ ısica de Andaluc´ ıa (IAA-CSIC) Granada, Spain Based on collaboration with: initial data: Marcus Ansorg, Silvano Bonazzola, Sergio Dain, Badri Krishnan, Fran¸ cois Limousin, Guillermo Mena-Marug´ an evolution analysis: Isabel Cordero-Carri´ on, J.M. Ib´ a˜ nez, J´ erˆ ome Novak, Nicolas Vasset From geometry to numerics ( General Relativity Trimester IHP ), Paris, 22 November 2006

Plan of the talk 1. General Objective : BH evolutions 2. Methodology : a) Dynamical trapping horizons b) Constrained evolution scheme c) Excision = ⇒ Specific objective : BH inner boundary conditions 3. Antecedents : Isolated horizons and BH Initial data 4. Results : Boundary conditions for dynamical trapping horizons 5. Conclusions

General objective Main goal Evolution of black hole spacetimes, with emphasis in the study and control of the properties of the horizon since Event Horizon global... Local characterization of the black hole horizon by means of the dynamical and trapping horizons , based on the notion of marginally trapped surface (“apparent horizons”...) ...a priori vs. a posteriori analysis (cf. talk by B. Krishnan) (Dreyer et al. 03, Baiotti et al. 05, Schnetter et al. 06, Schnetter & Krishnan 06...) Motivations Binary black hole evolutions and Gravitational Waves Geometrical properties of trapping horizons

Methodology

Geometric inner boundary conditions : 3+1 notation { Σ t } 3+1 slicing of spacetime n µ timelike unit normal to Σ t t µ = Nn µ + β µ evolution vector N lapse function β µ shift vector γ µν = g µν + n µ n ν spatial 3-metric K µν = − 1 2 L n γ µν extrinsic curvature

Dynamical trapping Horizons ( see previous talks by S. Hayward and B. Krishnan ) based on (marginally) trapped surfaces S t s µ unit normal vector to S t , in Σ t ℓ µ outgoing null vector k µ ingoing null vector, k µ ℓ µ = − 1 q µν = γ µν − s µ s ν induced metric on S t θ ( ℓ ) ≡ q µν ∇ µ ℓ ν = 0 Vanishing (outgoing) expansion θ ( k ) ≡ q µν ∇ µ k ν < 0 Negative (ingoing) condition Quasi-local horizon world-tube H of “apparent horizons”, i.e. : 1. H ≈ S 2 × R foliated by future marginally trapped 2-surfaces ( θ ( k ) < 0 and θ ( ℓ ) = 0 ) 1.a Future outer trapping horizon (FOTH) (Hayward 1994) : L k θ ( ℓ ) < 0 1.b Dynamical horizon (DH) (Ashtekar & Krishnan 2002) : H spacelike.

Characterizations of the equilibrium situation Non-expanding horizons (NEH) (H´ cek 1973) : Null limit of a a´ iˇ FOTH (Hayward ; Booth & Fairhurst 06) = ⇒ intrinsic geometry invariant : L ℓ q = 0 . H null hypersurface ( ℓ null normal) θ ( ℓ ) = 0 Isolated Horizons (IH) (Ashtekar et al. 1999) : NEH, − T αµ ℓ µ future directed Extrinsic geometry (induced connection) invariant : [ L ℓ , ˆ ∇ ] = 0 Intermediate level : Weakly Isolated Horizons (WIH) • NEH ∇ α ℓ β = ω α ℓ β , invariant : • ˆ ∇ vertical components, ˆ ˆ with κ ( ℓ ) = ω µ ℓ µ L ℓ ω = 0 ⇔ ∇ κ ( ℓ ) = 0 , Slowly Evolving Horizons (SEH) (Booth & Fairhurst 2004) : Geometrical characterization of a FOTH near equilibrium

Fully-constrained evolution scheme I : conformal decomposition Conformal decomposition of ( γ ij , K ij ) on Σ t : 3-metric γ ij := f ij + h ij γ ij = Ψ 4 ˜ γ ij , ˜ with ˜ γ ij unimodular : det(˜ γ ij ) = det( f ij ) ( f ij background flat metric) Extrinsic curvature A ij + 1 K ij = Ψ ζ ˜ 3 Kγ ij where � � A ij = Ψ 4 − ζ D j β i − 2 D i β j + ˜ γ ij + ˙ ˜ ˜ ˜ D k β k ˜ γ ij ˜ 2 N 3

Fully-constrained evolution scheme II : Equations Constraints + trace of evolution equations (with a choice for K and ˙ K ) 3 ˜ R D k ˜ ˜ D k Ψ − S Ψ [Ψ , N, β i , K, ˜ 8 Ψ = γ, ... ] D k β i + 1 D i ˜ D k β k + 3 ˜ D k ˜ ˜ ˜ R i k β k S β [Ψ , N, β i , K, ˜ = γ, ... ] 3 D k ˜ ˜ D k N + 2 ˜ D k ln Ψ ˜ γ, ˙ D k N S N [ N, Ψ , β i , K, ˜ = K, ... ] Evolution equations (trace-less part) + generalized Dirac’s gauge : γ ki = 0 (Bonazzola et al. 04) D k ˜ ∂ 2 ˜ γ ij − N 2 γ ij ˜ γ ij − 2 L β γ ij = S ij γ [ N, Ψ , β i , K, ˜ Ψ 4 ∆˜ ∂t + L β L β ˜ γ, ... ] ˜ ∂t 2

Elliptic part : re-scaled coupled PDE (cf. talk by D. Walsh) Rescaling : N = ˜ Nψ a ˜ R 8 Ψ+ 1 Lβ ) ij − 1 32Ψ 5 − 2 a ˜ 12 K 2 Ψ 5 = 0 , ˜ N − 2 (˜ Lβ ) ij (˜ • ∆Ψ − ∆ β i + 1 D i ˜ D k β k + ˜ k β k − ˜ Lβ ) ik ˜ ˜ ˜ N − 1 (˜ D k ˜ R i • N 3 D k Ψ = 4 Lβ ) ik ˜ 3Ψ a ˜ − ( a − 6)Ψ − 1 (˜ N ˜ D i K , ∆ ˜ ˜ N + 2( a + 1) ˜ D k lnΨ ˜ D k ln ˜ • N � a � R + a − 4 12 Ψ 4 K 2 + a ( a + 1) ˜ + ˜ ˜ D k lnΨ ˜ N D k lnΨ 8 − a + 8 32 Ψ 4 − 2 a ˜ Lβ ) ij = Ψ 4 − a β k ˜ N − 1 (˜ Lβ ) ij (˜ D k K . No obvious (...possible ?) choice of a for applying a maximum principle...

Specific problem : Excision Method and Inner Boundary Conditions Excision method We remove a sphere S t for the integration domain Σ t . We enforce this surface to coincide with a spatial slice of the horizon H . Then... Inner Boundary conditions : Elliptic part : Ψ , β ⊥ , V i , N Hamiltonian constraint : Ψ . Momentum constraint : β = β ⊥ s − V , with β ⊥ = β i s i and V i s i = 0 . Prescription of ˙ K : N . γ ij (see talk by J. Novak) Evolution (hyperbolic) part : ˜ Study of the characteristics.

(Technical) Antecedents

Initial Data in instaneous quasi-equilibrium Isolated Horizon boundary conditions 1) Marginally trapped surface θ ( ℓ ) = 0 β ⊥ = N 2) Coordinate system adapted to the Horizon ( t µ tangent to H ) σ ( ℓ ) = L ℓ q − 1 3,4) Quasi-equilibrium condition 2 θ ( ℓ ) q = 0 � � 2 ˜ D a ˜ V b + 2 ˜ D b ˜ V a − ( 2 ˜ D c V c ) ˜ q ab = 0 5) A fifth BC (foliation...) Freedom to choose slicing... (Cook et al. 02,04, JLJ, Mena& Gourgoulhon 04) Constant surface gravity prescription κ ( ℓ ) = κ o = const determines a unique solution of the CTS system, when combined with IH BCs : 1), 2), 3) and 4). No solution if κ o = κ Kerr ( a, J ) (JLJ, Ansorg & Limousin 06)

Prescription of the ingoing null normal expansion Reasons for θ ( k ) < 0 S t future marginally trapped surface Increasing area in the evolution of the marginally trapped surfaces No self-intersections in the evolution of S t (Andersson, Mars & Simon 06) k α = 1 2 ( n α − s α ) : CTT isol. hor. initial data analysis (K=0), with ˆ s i ≤ Ψ 6 · θ (ˆ ˜ D i ˜ k ) ≤ 0 = ⇒ existence and uniqueness of the solution (Dain, JLJ & Krishnan 05) Prescription of θ (ˆ k ) = const < 0 Existence and uniqueness of the solution to CTS equations (rescaled quantity Ψ 6 · θ (ˆ k ) not a good parameter...) (JLJ, Ansorg & Limousin 06)

Initial data of binary black holes (Cook et al. 04, Ansorg 05, Caudill et al. 06, Limousin & JLJ [in prep. ; extension of Grandcl´ ement et al. 02] ) Corotating BHs Maximal slicing, conformal flatness Quasi-Killing hellical vector : ˙ γ = 0 and M ADM = M Komar ˜

Inner Boundary conditions in the Dynamical case

Uniqueness and existence results of dynamical trapping horizons Result 1 (Ashtekar & Galloway 05) Given a DH H , the foliation by marginally trapped surfaces is unique ⇒ Unique vector h : h = N n + b s Result 2 (Andersson, Mars & Simon 05) Given a marginally trapped surface S 0 in a Cauchy hypersurface Σ , to each 3+1 foliation (Σ t ) t ∈ R it corresponds a unique DH H containing S 0 and sliced by MTSs S t ⊂ Σ t Results 1 and 2 ⇒ Non-uniqueness of the S 0 evolution

Adapted coordinate system Norm of h and type of H � b 2 − N 2 � • Definition : C := 1 2 h · h = 1 2 h is spacelike ⇐ ⇒ C > 0 ⇐ ⇒ b − N > 0 h is null ⇐ ⇒ C = 0 ⇐ ⇒ b − N = 0 h is timelike ⇐ ⇒ C < 0 ⇐ ⇒ b − N < 0 . In a FOTH C ≥ 0 (Hayward) , and sign of C is global on S t (Booth & Fairhurst 06) Coordinate system adapted to the horizon : t tangent to H Remember decomposition of the shift vector : β = β ⊥ s − V , t tangent to H ⇔ β ⊥ = b In this case, h = ∂ t + V and β ⊥ − N > 0

Trapping horizon boundary conditions Apparent horizon condition θ ( ℓ ) = 0 (Thornburg 87, Dain 04, Maxwell 04...) : s i ˜ s j − Ψ 3 K = 0 D i Ψ + ˜ s i Ψ + Ψ − 1 K ij ˜ s i ˜ 4˜ D i ˜ Definition of the trapping horizon : L h θ ( ℓ ) = 0 (Eardley 98...) � 2 D a − 2 L a 2 D a + A � ( β ⊥ − N ) = B ( β ⊥ + N ) − 2 D a L a := K ij s i q j with a 2 R − 2 D a L a − L a L a − 4 πT µν ( n µ + s µ )( n ν − s ν ) A := 1 R : Ricci scalar of the metric q on S t 2 σ (ˆ ab σ (ˆ ℓ ) ab + 4 πT µν ( n µ + s µ ) ( n ν + s ν ) ℓ ) B := 1 � �� � :=ˆ ℓ