An Overview of 3D Black Hole Simulations Pablo Laguna Center for - - PowerPoint PPT Presentation

An Overview of 3D Black Hole Simulations

Pablo Laguna Center for Gravitational Physics and Geometry Center for Gravitational Wave Physics Penn State University 29 October 2002

1

Introduction

• Good news: A picture is emerging about how to evolve spacetimes containing black hole

singularities.

1

Introduction

• Good news: A picture is emerging about how to evolve spacetimes containing black hole

singularities. ⋆ Communication between mathematical and numerical relativity. ⋆ Broader range of numerical techniques. ⋆ Access to larger and faster computers. ⋆ Approximate methods motivated by data analysis needs.

1

Introduction

• Good news: A picture is emerging about how to evolve spacetimes containing black hole

singularities. ⋆ Communication between mathematical and numerical relativity. ⋆ Broader range of numerical techniques. ⋆ Access to larger and faster computers. ⋆ Approximate methods motivated by data analysis needs.

• Bad news: No simulations of BBH orbits yet.

2

Efforts

• AEI-Mexico

(Alcubierre, Diener, Husam, Koppitz, Pollney, Seidel, Takahashi)

• Brownsville

(Campanelli, Lousto)

• Cornell-Caltech

(Kidder, Limblom, Pfeiffer, Scheel, Shoemaker, Teukolsky)

• GSFC-NCS

(Baker, Brown, Centrella, Choi)

• Illinois-Bowdoin

(Baumgarte, Shapiro, Yo)

• LSU

(Calabrese, Lehner, Neilsen, Pullin, Sarbach, Tiglio)

• Oakland

(Garfinkle)

• Penn State

(Br¨ ugmann, Jansen, Kelly, Laguna, Smith, Sperhake, Tichy)

• Pittsburgh

(Gomez, Szilagyi, Winicour)

• Texas

(Anderson, Bonning, Hawley, Matzner, Noble)

• UBC

(Choptuik, Pretorious)

3

This talk ...

• 3+1 Formulations

⋆ Generalized Einstein-Christoffel Hyperbolic ⋆ Conformal-Transverse-Traceless

3

This talk ...

• 3+1 Formulations

⋆ Generalized Einstein-Christoffel Hyperbolic ⋆ Conformal-Transverse-Traceless

• Gauge Conditions

3

This talk ...

• 3+1 Formulations

⋆ Generalized Einstein-Christoffel Hyperbolic ⋆ Conformal-Transverse-Traceless

• Gauge Conditions
• Black Hole Singularity

⋆ Excision ⋆ Punctures

3

This talk ...

• 3+1 Formulations

⋆ Generalized Einstein-Christoffel Hyperbolic ⋆ Conformal-Transverse-Traceless

• Gauge Conditions
• Black Hole Singularity

⋆ Excision ⋆ Punctures

• Boundary Conditions

3

This talk ...

• 3+1 Formulations

⋆ Generalized Einstein-Christoffel Hyperbolic ⋆ Conformal-Transverse-Traceless

• Gauge Conditions
• Black Hole Singularity

⋆ Excision ⋆ Punctures

• Boundary Conditions
• Numerical methods

⋆ Finite Differences ⋆ Spectral Methods ⋆ Finite Elements

3

This talk ...

• 3+1 Formulations

⋆ Generalized Einstein-Christoffel Hyperbolic ⋆ Conformal-Transverse-Traceless

• Gauge Conditions
• Black Hole Singularity

⋆ Excision ⋆ Punctures

• Boundary Conditions
• Numerical methods

⋆ Finite Differences ⋆ Spectral Methods ⋆ Finite Elements

• Results

⋆ Single BH evolutions ⋆ Waveforms from Lazarus and Grazing collisions ⋆ Current status on partial orbit

4

ADM Formulation

Arnowitt, Deser and Misner in Gravitation ed. L. Witten (1962); York in Sources of Gravitational Radiation ed. L.L. Smarr (1979)

Variables: gij : spatial metric Kij : extrinsic curvature α : lapse function βi : shift vector Evolution equations: ∂ogij = −2 α Kij ∂oKij = −∇i∇jα + α Rij + α K Kij − 2 α Kik Kk

j

Note: ∂o ≡ ∂t − Lβ

5

Kidder-Scheel-Teukolsky Formulation

Kidder, Scheel and Teukolsky, PRD 62, 084032 (2000)

Field variables: gij : spatial metric Pij = Kij + ˆ z gij K Mkij = 1 2

• ˆ

k dkij + ˆ e d(ij)k + gij

• ˆ

aglmdklm + ˆ bglmdlmk

• + gk(i
• ˆ

cglmdj)lm + ˆ dglmdlmj)

• dijk

≡ ∂kgij Q = ln (α g−σ) βi : shift vector Evolution equations: ∂ogij = . . . ∂oPij ∝ gkl∂k Mlij + . . . ∂oMkij ∝ ∂k Kij + . . .

6 0.5 1 1.5 2 t/M

10

• 16

10

• 8

10

||C

2+CiC i||

18x8x15 24x8x15 32x8x15 40x8x15

2000 4000 6000 8000 t/M

10

• 20

10

• 16

10

• 12

10

• 8

||C

2+CiC i||

18x8x15 24x8x15 32x8x15

7

BSSN Formulations

Baumgarte and Shapiro, PRD 59, 024009 (1999); Shibata and Nakamura, PRD 52, 5428 (1995)

Variables: Φ = 1 6 ln √g ˆ gij = (√g)−2/3 gij = e−4 Φ gij K = gij Kij ˆ Aij = e−4 Φ (Kij − 1 3gij K)

• Γi

= ˆ gjk Γi

jk

α : lapse function βi : shift vector

8

Evolution equations: ∂oΦ = −1 6α K ∂oˆ gij = −2 α ˆ Aij ∂oK = −∇i∇iα + α ˆ Aij ˆ Aij + 1 3α K2 ∂o ˆ Aij = e−4Φ (−∇i∇jα + α Rij)T F + α K ˆ Aij − 2 α ˆ Ail ˆ Al

j

∂o Γi = −2 ˆ Aij∂jα + 2 α Γi

jk ˆ

Ajk + 12 α ˆ Aij∂jΦ − 4 3αˆ gij∂jK − 2 ˆ Aij ∂jα

8

Evolution equations: ∂oΦ = −1 6α K ∂oˆ gij = −2 α ˆ Aij ∂oK = −∇i∇iα + α ˆ Aij ˆ Aij + 1 3α K2 ∂o ˆ Aij = e−4Φ (−∇i∇jα + α Rij)T F + α K ˆ Aij − 2 α ˆ Ail ˆ Al

j

∂o Γi = −2 ˆ Aij∂jα + 2 α Γi

jk ˆ

Ajk + 12 α ˆ Aij∂jΦ − 4 3αˆ gij∂jK − 2 ˆ Aij ∂jα −

• χ + 2

3

• ∂lβl
• Γi − ˆ

gjk Γi

jk

• Note:

χ helps to reverse the sign of ∂lβl Γi term

Yo, Baumgarte and Shapiro gr-qc/0209066

9

1000 2000 3000

t/M

10

−18

10

−16

10

−14

10

−12

10

−10

10

−8

10

−6

10

−4

∆Krms

N1 N2 N3 N4

1000 2000 3000 4000

t/M

10

−18

10

−16

10

−14

10

−12

10

−10

10

−8

10

−6

10

−4

∆Krms

A1 A2

10

Conformal-Traceless-Mixed Formulation

Laguna and Shoemaker, CQG 19, 3679 (2002)

Variables: Φ = 1 6 ln √g ˆ gij = (√g)−2/3 gij = e−4 Φ gij

• Γi

= ˆ gjk Γi

jk

ˆ Ai

j

= (√g)a Ai

j = e6 a Φ Ai j

ˆ K = (√g)k K = e6 k Φ K N = (√g)n α = e6 n Φ α βi : shift vector

11

Evolution equations: ∂oΦ = −1 6N K e−6 (n+k) Φ ∂oˆ gij = −2 N ˆ Aij e−6 (n+a) Φ ∂o ˆ K = −∇i∇iα e6 n Φ + N ˆ Ai

j ˆ

Aj

i e−6 (n+2 a−k) Φ

+ 1 3(1 − 3 k) N ˆ K2 e−6 (n+k) Φ ∂o ˆ Ai

j

=

• −∇i∇jα + α Ri

j

T F e6 a Φ + (1 − a) N ˆ K ˆ Ai

j e−6 (n+k) Φ

∂o Γi =

• 8 k N

K ˆ gij ∂jΦ − 4 3 N ˆ gij ∂j K

• e−6 (n+a) Φ

+

• 2 N ˆ

Ajk Γi

jk − 2 ˆ

Aij∂jN + 12(1 + n) N ˆ Aij ∂jΦ

• e−6 (n+a) Φ

− 1 3(2 + 3 χ) ∂lβl

• Γi − ˆ

gjk Γi

jk

11

Evolution equations: ∂oΦ = −1 6N K e−6 (n+k) Φ ∂oˆ gij = −2 N ˆ Aij e−6 (n+a) Φ ∂o ˆ K = −∇i∇iα e6 n Φ + N ˆ Ai

j ˆ

Aj

i e−6 (n+2 a−k) Φ

+ 1 3(1 − 3 k) N ˆ K2 e−6 (n+k) Φ ∂o ˆ Ai

j

=

• −∇i∇jα + α Ri

j

T F e6 a Φ + (1 − a) N ˆ K ˆ Ai

j e−6 (n+k) Φ

∂o Γi =

• 8 k N

K ˆ gij ∂jΦ − 4 3 N ˆ gij ∂j K

• e−6 (n+a) Φ

+

• 2 N ˆ

Ajk Γi

jk − 2 ˆ

Aij∂jN + 12(1 + n) N ˆ Aij ∂jΦ

• e−6 (n+a) Φ

− 1 3(2 + 3 χ) ∂lβl

• Γi − ˆ

gjk Γi

jk

• Natural choices: n = −1, a = 1, k = 1/3, χ = 2/3

12

13

Gauge Conditions

Slicing conditions

• ∂tα = −α2 f(α) (K − Ko)
• ∂tα = −∇iβi − α K
• ∂2

t α = −α2 f(α) ∂tK

Shift conditions

• ∂tβi = λ ∂t

Γi

• ∂2

t βi = F ∂t

Γi − η ∂tβi

20 40 60 80 t (M) −0.6 −0.3 0.3 0.6 ψ20

even (M)

3D(131

3x0.183)

fit 30 60 90 120 150 180 −0.6 −0.3 0.3 0.6 2D(300x29) 3D(131

3x0.183)

3D(131

3x0.183)

Alcubierre, Br¨ ugmann, Deiner, Koppits, Pollney, Seidel, Takahashi, gr-qc0206072

14

Boundary Conditions

f = fo + u(r − v t) r

14

Boundary Conditions

f = fo + u(r − v t) r + h(t) rn

15

Excision

Excision

16

Black Holes via Punctures

Alcubierre, Brugmann, Deiner, Koppits, Pollney, Seidel, Takahashi, gr-qc0206072

˜ gij = Ψ−4

BL Ψ4

• δij = O(1)

˜ Kij = Ψ−4

BL Ψ4

• ˆ

Aij = O(r3) where Ψo = u + ΨBL ΨBL = 1 + m1 2 r1 + m2 2 r2 ˆ Aij = 3 2 r2

• ni P j + nj P i −
• δij − ni nj

δkl nk P l

17

Numerical Methods

Finite Differences ∂xφ(xi) = φ(xi+1) − φ(xi−1) 2 ∆x + O(∆x2)

value ?

18

Numerical Methods

Spectral Methods φ(xi) =

• n=1...N

an ψ(xi) Finite Elements

19

Waveforms

20 40 60 80 100

t/M

−0.045 −0.03 −0.015 0.015 0.03

Re[r ψ4]

ISCO, r*=31M, z−axis, m=+2

T=10M T=9M T=0M

Baker, Campanelli, Lousto, Takahashi, astro-ph/0202469

10 20 30 40 t/M −0.4 −0.2 0.2 0.4 ψ22

even/M

0.30m 0.24m 0.20m

Alcubierre, et. al; Phys.Rev.Lett. 87 (2001) 271103

20

Conclusions

• There has been substantial progress in numerical relativity.
• Physics content of numerical results is increasing.
• Few (couple) orbits evolutions are around the corner.
• Formal mathematical input has become an important tool.

20

Conclusions

• There has been substantial progress in numerical relativity.
• Physics content of numerical results is increasing.
• Few (couple) orbits evolutions are around the corner.
• Formal mathematical input has become an important tool.

We need:

• To be more open to consider alternatives to finite differences.
• Better outer boundary conditions.
• Efficient elliptic solvers for gauge conditions and constrained evolutions.
• Adaptive or Fixed Mesh Refinements
• Larger and faster computers.