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

4C + 29.30 D ∼ 287Mpc MBH ∼ 108M⊙ Blue: X-ray Gold: Optical Pink: Radio Credit by NASA

• Many galaxies may have

massive BHs (106 − 1010M⊙) at their center by observations.

• It is not clear how they become so massive.
• 1. Direct collapse of supermassive star

(enough source??)

• 2. Merger of BHs by galaxy collisions

(enough time to grow??)

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
• 0.15
• 0.1
• 0.05

0.05 0.1 0.15 0.2 5 10 15 20 Dh(t)/m0 t (ms) (2a) type II H3-27 T4

Hotokezaka, Kyutoku, HO, Shibata, Kiuchi(2011)

• Binary

Neutron Star coales- cence gives matter around BH system(a few solar Mass BH) with particular EOS.

• It’s important to know what

could happen through the inter- action between BH and matter.

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

• Superradiant condition

ω < mΩH 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 < mΩH
• 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 < mΩH
• Subsequent amplification of modes

Mirror ergo-region Kerr BH

Instability?

What could be the Mirror?

Massive fields as Natural Mirror

Massive field

(Damour et al ’76, Detweiler ’80, Zouros &Eardley ’79)

Arvanitaki&Dubovsky ’11

• ✷ − µ2

Φ = 0

Application to Astrophysics

• Photon mass bound Pani et al. (2012)
• Graviton mass bound Brito et al. (2013)
• In Scalar-Tensor theory Cardoso et al. (2013)

Nonlinear Evolution on Fixed Kerr Background

✞ ✝ ☎ ✆

Bose Nova (Yoshino&Kodama ’12) Potential : V (φ) = f 2

aµ2

1 − cos Φ

fa

• 0.01
• 0.008
• 0.006
• 0.004
• 0.002

0.002 200 400 600 800 1000 FE and FJ t/M FE FJ

• Energy and angular momentum

fluxes toward the horizon

• Negative energy flux

→ Superradiance ✄ ✂

• ✁

Scalar&Vector fields (Witek et al. ’12) Massive field :

1 2µ2 SΦ2

1

• Beating pattern is different by the extraction radii like string vibration.

(Next) Evolution with back-reaction

Topics for Numerical Relativity

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

Binary Black Hole/Black Hole Collision

• BH spins cause the Gravita-

tional Recoil (Kick) by emit- ting Gravitational Waves.

• High

energy BH collisions show different trajectories. Vitor’s talk

• 3
• 2
• 1

1 2 3

• 3
• 2
• 1

1 2 3

• 0.3
• 0.2
• 0.1

0.1 0.2 0.3

Campanelli et al. ’07

Gravitational Recoil

1 1.005 1.01 1.015 1.02 1.025 50 100 150 A / 16 m0

2

t / m0

• 5

5

• 10
• 5

5 10 15 20 25 30 (y2-y1) / 2m0 (x2-x1) / 2m0

scatter non-prompt merger prompt merger

Sperhake et al. ’09 Shibata et al. ’08

4D high energy BH collision

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
• f scalar field collapses to a BH.

Basic equations

Action

Action S =

• dx4√−g

R

2κ − 1 2∇ρΦ∗∇ρΦ − µ2

S

2 Φ∗Φ − V (Φ)

• ,

κ = 8πG c4 . Einstein’s equations Rµν− 1

2gµνR = κTµν

✞ ✝ ☎ ✆

Numerical Relativity

Energy momentum tensor for a scalar field Tµν = ∇(µΦ∗∇ν)Φ − gµν

1

2

• ∇ρΦ∗∇ρΦ + µ2

SΦ∗Φ

• + V (Φ)
• Klein-Gordon equation
• ∇ρ∇ρ − µ2

S

• Φ − dV

dΦ = 0

Einstein-Hilbert part Matter part: complex scalar field

ADM formalism

• General Relativity : 4(N)-dimensional spacetimes
• ADM formalism : Arnowitt-Deser-Misner(1962),

re-printed in arxiv:gr-qc/0405109

@ Cf. Hamiltonian formalism ˙ pi = −∂H

∂qi ,

˙ qi = ∂H

∂pi

H : Hamiltonian, qi : coordinates, pi : momenta @ Spacetimes = Evolution of 3D spaces(3+1 decomposition)

a 3-dimensional Cabbage 2-dimensional leaves

ADM decomposition

t+∆t

βa

na

ta

t 3D hypersurface

na is a timelike normal vector defined as na = (1/α, βi/α), nana = −1.

3D spatial metric

γab ≡ gab + nanb

Projection tensor ⊥a

b= δa b + nanb

Extrinsic curvature

Kab ≡ − 1

2α (∂tγab − Dbβa − Daβb)

Einstein’s equations Gab ≡ Rab − 1

2gabR = κTab

✞ ✝ ☎ ✆ Projection of Einstein’s equations (Gab − κTab) nanb = 0, ⊥c

i (Gcb − κTcb) nb = 0,

⊥c

i⊥d j (Gcd − κTcd) = 0.

Einstein’s equations

Hamiltonian Constraint

ρ:energy density, ji:current density

Gabnanb = κTabnanb →

(3)R + K2 − KijKij

= 2κρ, ⊥c

i Gcbnb

= ⊥c

i κTcbnb

→ DjKj

i − DiK

= κji. Momentum Constraints

All quantities are spatial.

(They must be satisfied on each 3D hypersurface.)

Einstein’s equations

Hamiltonian Constraint

ρ:energy density, ji:current density

Gabnanb = κTabnanb →

(3)R + K2 − KijKij

= 2κρ, ⊥c

i Gcbnb

= ⊥c

i κTcbnb

→ DjKj

i − DiK

= κji. Momentum Constraints

All quantities are spatial.

(They must be satisfied on each 3D hypersurface.)

Evolution equations ⊥c

i⊥d j Gcd =⊥c i⊥d j κTcd −

→ ∂tKij = βkDkKij + KkjDiβk + KkiDjβk − DjDiα + α

• Rij − 2KikKk

j + KijK − κ

• Sij + ρ − S

3 γij

• ,

γij and Kij are variables to evolve in ADM system. ∂tγij = −2αKij +Djβi +Diβj

(Definition of Kij) Sij ≡⊥c

i⊥d j Tcd, S ≡ γijSij.

However, the evolution is unstable in ADM formalism. →

✄ ✂

• ✁

BSSN formalism

BSSN formalism

We define new variables to make evolutions stable.

Shibata&Nakamura ’95, Baumgarte&Shapiro ’98

γij = χ−1˜ γij, det˜ γij ≡ 1, Kij = χ−1 ˜ Aij + 1 3χ−1˜ γijK, ˜ Γi ≡ −∂j˜ γij. BSSN evolution equations

• ∂t − βi∂i
• χ

= 2 3 χ αK − ∂iβi ,

• ∂t − βl∂l
• ˜

γij = −2α ˜ Aij + ˜ γil∂jβl + ˜ γjl∂iβl − 2 3 ˜ γij∂lβl,

• ∂t − βi∂i
• K

= α

• ˜

Aij ˜ Aij + 1 3K2 + κ 2 (2ρ + S)

• ,
• ∂t − βl∂l

˜

Aij = χ [αRij − DiDjα − ακSij]T F +α K ˜ Aij − 2 ˜ Ail ˜ Al

j

• + ˜

Alj∂iβl + ˜ Ail∂jβl − 2 3 ˜ Aij∂lβl,

• ∂t − βj∂j

˜

Γi = ˜ γjk∂j∂kβi + 1 3 ˜ γij∂j∂kβk − ˜ Γj∂jβi + 2 3 ˜ Γi∂jβj −2 ˜ Aij∂jα + 2α

• ˜

Γi

jk ˜

Ajk − 3 2 ∂jχ χ ˜ Aij − 2 3 ˜ γij∂jK

• − 2κα

χ ji.

Evolution of a scalar field

Conjugate momentum for a scalar field φ = φR + iφI ΠC ≡ −nµ∇µΦC, where C denotes the real part and imaginary part of a complex scalar field. Energy density ρ = Tabnanb = 1 2

• Π2

R + Π2 I + µ2 S

• Φ2

R + Φ2 I

• + DiΦRDiΦR + DiΦIDiΦI
• ,

ji = − ⊥a

i Tabnb = ΠRDiΦR + ΠIDiΦI.

Evolution equations for a scalar field Setting V (Φ) = 0,

• ∂t − βl∂l
• ΦC

= −αΠC,

• ∂t − βl∂l
• ΠC

= αKΠC − DiαDiΦC − αDiDiΦC + αµ2

SΦC.

Gauge conditions

Puncture Gauge Conditions

Alcubierre+(2003)

“Standard numerical simulations” in 3+1 black hole spacetimes adopt puncture gauges for their stabilities. 1+log slicing condition

∂t − βi∂i α = −2αK,

avoiding black hole singularities. Γ-driver shift conditions

• ∂t − βj∂j
• βi

= −ηββi + ηΓ˜ Γi, where ηβ and ηΓ are arbitrary constants. α, βi, ˜ γij, χ, ˜ Aij, K, ˜ Γi, ΦR, ΦI, ΠR and ΠI are defined as variables to evolve in BSSN formalism.

Initial Value Problem

• Initial data must satisfy the constraints.

R + K2 − KijKij = 2κρ, DjKj

i − DiK = κji.

• We have 16 variables but only 4 constraints.

γij, Kij, Φ, Π.

• At first, we set 12 variables in the situation which we want to
• realize. Then, we solve the constraint equations(Cook ’00).
• The constraints can be rewritten to better form by conformal

transformation (γij = ψ4˜ γij). → ˜ γij, ψ, Kij, Φ and Π. Constraints in Initial Value Problem ˜ △ψ − 1 8ψ ˜ R − 1 8ψ5K2 + 1 8ψ5KijKij = −πψ5 Π∗Π + ∂iΦ∗∂iΦ + µ2

SΦ∗Φ

• DjKj

i − DiK

= 4π (Π∗∂iΦ + Π∂iΦ∗)

Results

Initial Data for Spherical scalar clouds

Ansatz Kij = 0, Φ = 0, ψ = 1 + M0 2r + u00 r Y00 , ΠR = A00 2π e−r2/w2ψ−5/2.

• Cf. Schwarzschild BH in isotropic coordinates

ds2 = −

1 − M0

2r

1 + M0

2r

2

dt2 +

• 1 + M0

2r

4

ηijdxidxj

• 1 -0.5

0.5 1

• 1
• 0.5

0.5 1

• 1
• 0.5

0.5 1 Scalar field(spherical)

• 1 -0.5

0.5 1

• 1
• 0.5

0.5 1

• 1
• 0.5

0.5 1 z Schwarzschild BH x y z

Hamiltonian constraint △ψ + πΠ2ψ−5 = 0 − → u′′

00 + A2 00r

√ 4π e− 2r2

w2 = 0

Solution u00 → 0 as r → ∞ u00 = A2

00w3

16 √ 2

• Erf

√ 2r w

• − 1

Constraint violating Initial Data

1e-10 1e-08 1e-06 1e-04 1e-02 1e+00 0.1 1 10 100 ||H M0

2||

x/M0 t/M0= 0 t/M0=100 t/M0=200 t/M0=300 1e-10 1e-08 1e-06 1e-04 1e-02 1e+00 0.1 1 10 100 ||H M0

2||

x/M0 t/M0= 0 t/M0=100 t/M0=200 t/M0=300

Constraint-Satisfying ID Constraint-Violating ID

HO, Witek, Cardoso ’14

• Hamiltonian

constraint near the horizon grows in time.

• Constraint-Violating

Initial Data decreases the apparent horizon area.

Massive fields around the BH

• 1 -0.5

0.5 1

• 1
• 0.5

0.5 1

• 1
• 0.5

0.5 1 Scalar cloud(dipole)

• 1 -0.5

0.5 1

• 1
• 0.5

0.5 1

• 1
• 0.5

0.5 1 z Schwarzschild BH x y z

• 0.05

0.05 (rex / M0) 11 rex = 30 rex = 40 1000 2000 3000 4000 ( t - rex ) / M0

• 0.05

0.05 (rex / M0) 11 rex = 60 1000 2000 3000 4000 ( t - rex ) / M0 rex = 80

HO, Witek, Cardoso ’14

Beating pattern of scalar field

• Scalar field oscillates for a long time by different pattern at different radius.
• Rotating BH also have bound states during nonlinear evolution.
• Final state of BH: with scalar hair? (Cf. Herdeiro&Radu, ’04)

(Normal) Magnus Effect

Difference of Cross Sections

Cross section of absorption is larger for Counter-Rotating particles.

Bardeen et al.(1970), Thorne(1974)

Numerical Results for anti-Magnus Effect

• 0.16
• 0.14
• 0.12
• 0.1
• 0.08
• 0.06
• 0.04
• 0.02

0.02 0.04 20 40 60 80 100 yBH/M0 t/M0 a/M0=0.0, A0M0=0.11 a/M0=0.3, A0M0=0.11 a/M0=0.5, A0M0=0.11 a/M0=0.5, A0M0=0.13 a/M0=0.5, A0M0=0.15

• 0.1
• 0.05

0.05 0.1 (sU-sL)/M0 1e-12 1e-10 1e-08 1e-06 1e-04 20 40 60 80 100 |dPy/dt| (t-Rext)/M0 dPSF

y/dt, Rext/M0=70

dPGW

y/dt, Rext/M0=70

Top: BH puncture position as a function of time, Middle: Difference of proper distance from BH to Upper/Lower scalar cloud edge, Bottom: Momentum by Scalar and Gravitational waves.

Summary

• We introduced Numerical Relativity for investigating BH +

Matter(Complex Scalar Field) system.

• Initial data are very important for NR simulations, which we

should solve the initial value problem.

• We obtained analytical solutions in special cases with the scalar

field around the non-rotating BH. In other cases, we numerically solved the constraints.

• Gravitational “Magnus” effect would be considered for the

interaction between BH and matter. Thank you very much for your attention. More information: http://blackholes.ist.utl.pt Lecture notes for NR: http://blackholes.ist.utl.pt/nrhep2