Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Applications of Dynamical Horizons in Numerical Relativity E. Schnetter 2 B. Krishnan 1 F. Beyer 1 1 Max Planck Institut für Gravitationsphysik Albert Einstein Institut D-14476 Golm, Germany 2 Center for Computation and Technology Louisiana State University Baton Rouge, LA 70803, USA Paris, November 22, 2006 E. Schnetter, B. Krishnan, F . Beyer DHs and NR
Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Outline Motivation and Background 1 Trapped Surfaces The trapping boundary Dynamical horizons 2 Horizon Multipole Moments 3 Example Numerical Simulations 4 Head-on Collision with Brill-Lindquist Data Axisymmetric Gravitational Collapse E. Schnetter, B. Krishnan, F . Beyer DHs and NR
Motivation and Background Dynamical horizons Trapped Surfaces Horizon Multipole Moments The trapping boundary Example Numerical Simulations Summary Definition of a trapped surface For a sphere in flat space q ab ∇ a ℓ b > 0 Outgoing light rays are diverging: Θ ( ℓ ) = ˜ q ab ∇ a n b < 0 Ingoing light rays are converging: Θ ( n ) = ˜ For a trapped surface, both sets of null rays are converging: Θ ( ℓ ) < 0 and Θ ( n ) < 0 Trapped surfaces are signatures of black holes: Existence of trapped surface = ⇒ singularity in future Trapped surfaces lie inside the event horizon For cross sections of stationary EHs Θ ( ℓ ) = 0, Θ ( n ) < 0 Future Marginally Outer Trapped Surface (FMOTS): Θ ( ℓ ) = 0, Θ ( n ) < 0 E. Schnetter, B. Krishnan, F . Beyer DHs and NR
Motivation and Background Dynamical horizons Trapped Surfaces Horizon Multipole Moments The trapping boundary Example Numerical Simulations Summary The trapping boundary Trapping boundary is boundary of region containing trapped surfaces There are spherically symmetric trapped surfaces right up to the Schwarzschild event horizon i + I + E S � i 0 I − E. Schnetter, B. Krishnan, F . Beyer DHs and NR
Motivation and Background Dynamical horizons Trapped Surfaces Horizon Multipole Moments The trapping boundary Example Numerical Simulations Summary The trapping boundary i + I + H E i 0 r = 0 v = 0 i − Vaidya: sph. symmetric trapped surfaces only up to H Suggestion by Eardley: event horizon is the trapping boundary E. Schnetter, B. Krishnan, F . Beyer DHs and NR
Motivation and Background Dynamical horizons Trapped Surfaces Horizon Multipole Moments The trapping boundary Example Numerical Simulations Summary The trapping boundary We can look for marginally trapped surfaces on non-symmetric surfaces using apparent horizon finders In Vaidya we can push marginally trapped surfaces arbitrarily close to the EH Marginal surfaces can also extend into flat region Recent analytic proof by Ben-Dov 2 horizon flat region 1.5 MS 1 0.5 0 z -0.5 -1 -1.5 -2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x E. Schnetter, B. Krishnan, F . Beyer DHs and NR
Motivation and Background Dynamical horizons Trapped Surfaces Horizon Multipole Moments The trapping boundary Example Numerical Simulations Summary Evolution of MTSs in Time It is observed numerically that MTSs evolve smoothly in time Apparent horizons may jump due to outermost condition Smooth world tube of MTSs is a Marginally Trapped Tube MTT shown to exist if MTS is strictly stably outermost Andersson et al , PRL 95 111102 (2005) Untrapped Linear outward deformation Trapped makes S untrapped: δ f r Θ ( ℓ ) > 0 for f ≥ 0 In practice we look for surfaces with Θ ( ℓ ) = ǫ > 0 and check that it lies outside S E. Schnetter, B. Krishnan, F . Beyer DHs and NR
Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Definitions Dynamical Horizon: Spacelike MTT with Θ ( n ) < 0 Outermost MTT usually forms a DH MTT with | σ ( ℓ ) | 2 � = 0 or T ab ℓ a ℓ b � = 0 somewhere are spacelike if they are SSO (Andersson et al.) Timelike Membrane: Timelike MTT Cannot be the black hole surface Inner MTTs might form timelike membranes Isolated Horizon: Null MTT (BH in equilibrium) Other cases: MTTs with mixed signature also possible E. Schnetter, B. Krishnan, F . Beyer DHs and NR
Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Basic Properties Cross section is topologically S 2 r a for a DH Area increases along ˆ Consequence of Θ ( ℓ ) = 0 and Θ ( n ) < 0 Area increases in time if t . r > 0 Area decreases for a TLM Foliation of DH is unique (Ashtekar & Galloway, gr-qc/0503109) Implies that changing Σ leads to different DH Other restrictions on occurence of MTS in presence of DH Event horizon is probably the boundary of the trapped region (Eardley 1998, Schnetter & Krishnan 2006, Ben-Dov 2006) E. Schnetter, B. Krishnan, F . Beyer DHs and NR
Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Multipole Moments We are interested in source multipole moments for black hole In classical electrodynamics we have charge and current multipole moments for sources For a black hole we have mass and angular momentum multipole moments M n and J n J 0 vanishes by absence of monopole charges (here NUT charge) M 0 is mass and J 1 is angular momentum In Kerr, M 0 and J 1 determine all higher moments In Schwarzschild, only M 0 � = 0 E. Schnetter, B. Krishnan, F . Beyer DHs and NR
Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Multipole Moments Given rotational Killing vector ϕ a construct coordinate system ( θ, φ ) on S φ ∈ [ 0 , 2 π ) is affine parameter along ϕ a ζ = cos θ ∈ [ − 1 , 1 ] is defined by D a ζ = 1 � ǫ ba ϕ a , ˜ ˜ ζ = 0 R 2 S S Use spherical harmonics for ( θ, φ ) to define multipoles M n = R n S M S � � � RP n ( ζ ) d 2 V ˜ 8 π S J n = R n − 1 K ab ϕ a R b d 2 V � S P ′ n ( ζ ) ¯ 8 π S E. Schnetter, B. Krishnan, F . Beyer DHs and NR
Motivation and Background Dynamical horizons Horizon Multipole Moments Example Numerical Simulations Summary Multipole Moments Properties of the multipole moments M n and J n are coordinate independent They characterize geometry of DH at any given time Need only data on MTS to calculate them Coincide with corresponding isolated horizon formulae (Ashtekar et. al, CQG 21 2549 (2004)) Useful for characterizing rate of approach to Kerr E. Schnetter, B. Krishnan, F . Beyer DHs and NR
Motivation and Background Dynamical horizons Head-on Collision with Brill-Lindquist Data Horizon Multipole Moments Axisymmetric Gravitational Collapse Example Numerical Simulations Summary Brill-Lindquist Initial Data Describes head on collision for two BH case Σ is R 3 with two “punctures” Time symmetric: ¯ K ab = 0 q ab = ψ 4 δ ab Conformally flat: ¯ ∆ ψ = 0, ψ → 1 as r → ∞ ψ = 1 + α 1 + α 2 2 r 1 2 r 2 Single BH case is Schwarzschild in isotropic coordinates m ADM = 2 α 1 + 2 α 2 and punctures are asymptotic regions Take units such that m ADM = 1 E. Schnetter, B. Krishnan, F . Beyer DHs and NR
Motivation and Background Dynamical horizons Head-on Collision with Brill-Lindquist Data Horizon Multipole Moments Axisymmetric Gravitational Collapse Example Numerical Simulations Summary Brill-Lindquist Initial Data Equal mass: 2 α 1 = 2 α 2 = 0 . 5 punctures initially at z = ± 0 . 5 Explicit octant symmetry and extent upto x , y , z = 96 4 th order spatial differencing and 3 rd order Runge-Kutta Use mesh refinement: h = 1 . 6 at boundary and h = 0 . 0125 at horizon Horizon diameter contains 32 points initially About 10 grid points excised around punctures AEI BSSN formulation (inconsistent boundary conditions!) 1 + log slicing with α = 1 initially; zero shift Common MTS forms at t ≈ 0 . 5 We use Jonathan Thornburg’s AHFinderDirect . E. Schnetter, B. Krishnan, F . Beyer DHs and NR
Motivation and Background Dynamical horizons Head-on Collision with Brill-Lindquist Data Horizon Multipole Moments Axisymmetric Gravitational Collapse Example Numerical Simulations Summary Horizon Shapes Horizon shapes at t=1 individual horizons 1 inner horizon outer horizon 0.5 x 0 -0.5 -1.5 -1 -0.5 0 0.5 1 1.5 z E. Schnetter, B. Krishnan, F . Beyer DHs and NR
Motivation and Background Dynamical horizons Head-on Collision with Brill-Lindquist Data Horizon Multipole Moments Axisymmetric Gravitational Collapse Example Numerical Simulations Summary Signature of MTT Horizon metric determinant at t=0.6 7 individual horizon 6 inner horizon outer horizon 5 4 det q 3 2 1 0 -1 π /2 π 0 θ E. Schnetter, B. Krishnan, F . Beyer DHs and NR
Motivation and Background Dynamical horizons Head-on Collision with Brill-Lindquist Data Horizon Multipole Moments Axisymmetric Gravitational Collapse Example Numerical Simulations Summary Signature of MTT Horizon metric determinant at t=1 0.6 individual horizon 0.4 inner horizon outer horizon 0.2 0 det q -0.2 -0.4 -0.6 -0.8 -1 π /2 π 0 θ E. Schnetter, B. Krishnan, F . Beyer DHs and NR
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