Black Hole and fundamental fields in Numerical Relativity Hirotada - PowerPoint PPT Presentation

Seminar @ Crete Center for Theoretical Physics on 29th, April, 2014 Black Hole and fundamental fields in Numerical Relativity Hirotada Okawa HO, H. Witek, V. Cardoso, arXiv:1401.1548[gr-qc] (2014) accepted for PRD, HO and V. Cardoso in preparation(2014), Lecture Notes for NR, http://blackholes.ist.utl.pt/nrhep2 (How to start NR).

Outline of the talk • Introduction – BH and Matter system – Stability of BHs – Numerical Relativity • Basic equations – Decomposition of Einstein equations – Initial data construction • Results – Bound states – Gravitational “Magnus” effect • Summary

Introduction

Astrophysical Black Holes 4 C + 29 . 30 D ∼ 287 Mpc M BH ∼ 10 8 M ⊙ • Many galaxies may have massive BHs (10 6 − 10 10 M ⊙ ) at their center by observations. • It is not clear how they become so massive. 1. Direct collapse of supermassive star (enough source??) Blue: X-ray 2. Merger of BHs by galaxy collisions Gold: Optical (enough time to grow??) Pink: Radio Credit by NASA The formation process of a (massive) BH is still open problem. Matter around the BH can be related to such high energy phenomina?

Black Hole + Matter System 0.2 (2a) type II • Binary Neutron Star coales- 0.15 0.1 cence gives matter around BH 0.05 Dh(t)/m 0 system(a few solar Mass BH) 0 with particular EOS. -0.05 -0.1 -0.15 • It’s important to know what H3-27 T4 -0.2 could happen through the inter- 5 10 15 20 action between BH and matter. t (ms) Hotokezaka, Kyutoku, HO, Shibata, Kiuchi(2011)

Stability of Black Hole Stability of Schwarzschild BH (Regge&Wheeler ’57, Vishveshwara ’70) • Schwarzschild BH is stable. • Amplitude of perturbations will decay in time.

Stability of Black Hole Stability of Schwarzschild BH (Regge&Wheeler ’57, Vishveshwara ’70) • Schwarzschild BH is stable. • Amplitude of perturbations will decay in time. Superradiance (Penrose ’64, Zel’dovich ’71, Misner ’72) • Matter field around a Kerr BH • Amplification of some modes by scattering in the ergo region of Kerr BH ergo-region • Superradiant condition ω < mΩ H Kerr BH

Black Hole BOMB Black Hole BOMB (Press&Teukolsky ’72, Cardoso et al ’04, Dolan ’07) • Matter field around a Kerr BH with Mirror • Amplification of some modes by scattering in the ergo region of Kerr BH • Superradiant condition ω < m Ω H • Subsequent amplification of modes Mirror ergo-region Kerr BH

Black Hole BOMB Black Hole BOMB (Press&Teukolsky ’72, Cardoso et al ’04, Dolan ’07) • Matter field around a Kerr BH with Mirror • Amplification of some modes by scattering in the ergo region of Kerr BH • Superradiant condition ω < m Ω H • Subsequent amplification of modes Instability? Mirror ergo-region What could be the Mirror? Kerr BH

Massive fields as Natural Mirror Massive field (Damour et al ’76, Detweiler ’80, Zouros &Eardley ’79) Arvanitaki&Dubovsky ’11 � ✷ − µ 2 � Φ = 0 • Photon mass bound Pani et al. (2012) Application to Astrophysics • Graviton mass bound Brito et al. (2013) • In Scalar-Tensor theory Cardoso et al. (2013)

Nonlinear Evolution on Fixed Kerr Background ✞ ☎ a µ 2 � � Bose Nova (Yoshino&Kodama ’12) Potential : V ( φ ) = f 2 1 − cos Φ f a ✝ ✆ 0.002 0 F E F E and F J • Energy and angular momentum -0.002 -0.004 F J fluxes toward the horizon -0.006 -0.008 -0.01 • Negative energy flux 0 200 400 600 800 1000 → Superradiance t / M ✄ � 1 2 µ 2 S Φ 2 Scalar&Vector fields (Witek et al. ’12) Massive field : ✂ ✁ 1 • Beating pattern is different by the extraction radii like string vibration. (Next) Evolution with back-reaction

Topics for Numerical Relativity Gravitational Wave Detection Collapse to BH BH Collision Rotating Star Binary BH Binary NS Gravity in Holography in AdS Higher Dimension (In)Stability by Collapse in AdS matter fields AdS/CFT Correspondence Binary in Scalar-Tensor Theory Beyond GR Cosmology

Binary Black Hole/Black Hole Collision Campanelli et al. ’07 0.3 0.2 • BH spins cause the Gravita- 0.1 tional Recoil (Kick) by emit- 0 ting Gravitational Waves. -0.1 -0.2 -0.3 -3 -2 -1 • High energy BH collisions Gravitational Recoil 0 1 3 show different trajectories. 2 2 1 0 -1 3 -2 -3 Vitor’s talk scatter 1.025 non-prompt merger prompt merger 5 (y 2 -y 1 ) / 2m 0 1.02 2 A / 16 m 0 1.015 0 1.01 -5 1.005 -10 -5 0 5 10 15 20 25 30 1 Shibata et al. ’08 (x 2 -x 1 ) / 2m 0 0 50 100 150 Sperhake et al. ’09 4D high energy BH collision t / m 0

NR in Anti de Sitter Spacetimes Chesler&Yaffe ’10 Maliborski&Rostworowski ’13 Shock wave collision in AdS Collapse to a BH in AdS • Application to heavy ion collisions in Characteristic formalism. • BH formation in AdS. They claim that any small amplitude of scalar field collapses to a BH.

Basic equations

Action Action � R 2 ∇ ρ Φ ∗ ∇ ρ Φ − µ 2 � d x 4 √− g 2 κ − 1 � S 2 Φ ∗ Φ − V (Φ) S = , κ = 8 πG c 4 . Einstein-Hilbert part Matter part: complex scalar field ✞ ☎ R µν − 1 Einstein’s equations 2 g µν R = κT µν Numerical Relativity ✝ ✆ Energy momentum tensor for a scalar field � 1 � � � ∇ ρ Φ ∗ ∇ ρ Φ + µ 2 T µν = ∇ ( µ Φ ∗ ∇ ν ) Φ − g µν S Φ ∗ Φ + V (Φ) 2 Φ − d V � � ∇ ρ ∇ ρ − µ 2 Klein-Gordon equation dΦ = 0 S

ADM formalism • General Relativity : 4 ( N ) -dimensional spacetimes • ADM formalism : Arnowitt-Deser-Misner(1962), re-printed in arxiv:gr-qc/0405109 p i = − ∂ H q i = ∂ H @ Cf. Hamiltonian formalism ˙ ˙ ∂q i , ∂p i q i : coordinates, p i : momenta H : Hamiltonian, @ Spacetimes = Evolution of 3D spaces(3+1 decomposition) a 3-dimensional Cabbage 2-dimensional leaves

ADM decomposition t+ ∆ t 3D spatial metric β a γ ab ≡ g ab + n a n b Projection tensor t a t n a ⊥ a b = δ a b + n a n b Extrinsic curvature 3D hypersurface K ab ≡ − 1 2 α ( ∂ t γ ab − D b β a − D a β b ) n a is a timelike normal vector defined as n a = (1 /α, β i /α ) , n a n a = − 1 . ✞ ☎ Einstein’s equations Projection of Einstein’s equations ✝ ✆ ( G ab − κT ab ) n a n b = 0 , G ab ≡ R ab − 1 2 g ab R = κT ab i ( G cb − κT cb ) n b = 0 , ⊥ c ⊥ c i ⊥ d j ( G cd − κT cd ) = 0 .

Einstein’s equations Hamiltonian Constraint ρ :energy density, j i :current density (3) R + K 2 − K ij K ij G ab n a n b κT ab n a n b = → = 2 κρ, D j K j ⊥ c i G cb n b ⊥ c i κT cb n b = = → i − D i K κj i . All quantities are spatial. Momentum Constraints (They must be satisfied on each 3D hypersurface.)

Einstein’s equations Hamiltonian Constraint ρ :energy density, j i :current density (3) R + K 2 − K ij K ij G ab n a n b κT ab n a n b = → = 2 κρ, D j K j ⊥ c i G cb n b ⊥ c i κT cb n b = = → i − D i K κj i . All quantities are spatial. Momentum Constraints (They must be satisfied on each 3D hypersurface.) ⊥ c i ⊥ d j G cd = ⊥ c i ⊥ d j κT cd − → Evolution equations β k D k K ij + K kj D i β k + K ki D j β k − D j D i α ∂ t K ij = S ij + ρ − S � � �� R ij − 2 K ik K k + j + K ij K − κ α γ ij , 3 S ij ≡⊥ c i ⊥ d j T cd , S ≡ γ ij S ij . γ ij and K ij are variables to evolve in ADM system. ∂ t γ ij = − 2 αK ij + D j β i + D i β j (Definition of K ij ) ✄ � However, the evolution is unstable in ADM formalism. → BSSN formalism ✂ ✁

BSSN formalism We define new variables χ − 1 ˜ γ ij = γ ij , det˜ γ ij ≡ 1 , to make evolutions stable. A ij + 1 Shibata&Nakamura ’95, χ − 1 ˜ 3 χ − 1 ˜ K ij = γ ij K, Baumgarte&Shapiro ’98 ˜ Γ i γ ij . ≡ − ∂ j ˜ BSSN evolution equations 2 � � 3 χ � αK − ∂ i β i � ∂ t − β i ∂ i χ = , γ jl ∂ i β l − 2 γ il ∂ j β l + ˜ � � − 2 α ˜ ∂ t − β l ∂ l γ ij ∂ l β l , γ ij ˜ = A ij + ˜ 3 ˜ A ij + 1 3 K 2 + κ � � � � A ij ˜ ˜ ∂ t − β i ∂ i K = α 2 (2 ρ + S ) , � ˜ � χ [ αR ij − D i D j α − ακS ij ] T F ∂ t − β l ∂ l A ij = A il ∂ j β l − 2 A lj ∂ i β l + ˜ + α � K ˜ A ij − 2 ˜ A il ˜ � + ˜ ˜ A l A ij ∂ l β l , j 3 � ˜ γ jk ∂ j ∂ k β i + 1 Γ j ∂ j β i + 2 γ ij ∂ j ∂ k β k − ˜ � ˜ ∂ t − β j ∂ j Γ i Γ i ∂ j β j = ˜ 3 ˜ 3 ∂ j χ A jk − 3 A ij − 2 − 2 κα � � − 2 ˜ ˜ jk ˜ ˜ A ij ∂ j α + 2 α Γ i γ ij ∂ j K χ j i . 3 ˜ 2 χ

Evolution of a scalar field Conjugate momentum for a scalar field φ = φ R + iφ I Π C ≡ − n µ ∇ µ Φ C , where C denotes the real part and imaginary part of a complex scalar field. Energy density T ab n a n b = 1 � � � � Π 2 R + Π 2 I + µ 2 Φ 2 R + Φ 2 + D i Φ R D i Φ R + D i Φ I D i Φ I ρ = , S I 2 i T ab n b = Π R D i Φ R + Π I D i Φ I . − ⊥ a = j i Evolution equations for a scalar field Setting V (Φ) = 0 , � � ∂ t − β l ∂ l Φ C = − α Π C , � � ∂ t − β l ∂ l αK Π C − D i αD i Φ C − αD i D i Φ C + αµ 2 Π C = S Φ C .