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: g ij : spatial metric K ij : extrinsic curvature α : lapse function β i : shift vector Evolution equations: ∂ o g ij = − 2 α K ij −∇ i ∇ j α + α R ij + α K K ij − 2 α K ik K k ∂ o K ij = j Note: ∂ o ≡ ∂ t − L β

5 Kidder-Scheel-Teukolsky Formulation Kidder, Scheel and Teukolsky, PRD 62 , 084032 (2000) Field variables: g ij : spatial metric P ij = K ij + ˆ z g ij K � � � � �� 1 ˆ ag lm d klm + ˆ bg lm d lmk cg lm d j ) lm + ˆ dg lm d lmj ) M kij = k d kij + ˆ e d ( ij ) k + g ij ˆ + g k ( i ˆ 2 d ijk ∂ k g ij ≡ ln ( α g − σ ) Q = β i : shift vector Evolution equations: ∂ o g ij = . . . g kl ∂ k M lij + . . . ∂ o P ij ∝ ∂ o M kij ∂ k K ij + . . . ∝

6 -8 0 10 10 -12 10 i || i || -8 2 +C i C 2 +C i C 10 18x8x15 ||C ||C 24x8x15 18x8x15 32x8x15 24x8x15 -16 10 32x8x15 40x8x15 -16 10 -20 10 0 0.5 1 1.5 2 0 2000 4000 6000 8000 t/M t/M

7 BSSN Formulations Baumgarte and Shapiro, PRD 59, 024009 (1999); Shibata and Nakamura, PRD 52, 5428 (1995) Variables: 1 6 ln √ g Φ = ( √ g ) − 2 / 3 g ij = e − 4 Φ g ij g ij ˆ = g ij K ij K = e − 4 Φ ( K ij − 1 ˆ A ij = 3 g ij K ) Γ i g jk � Γ i � = ˆ jk α : lapse function β i : shift vector

8 Evolution equations: − 1 ∂ o Φ = 6 α K − 2 α ˆ ∂ o ˆ g ij = A ij A ij + 1 −∇ i ∇ i α + α ˆ A ij ˆ 3 α K 2 ∂ o K = e − 4Φ ( −∇ i ∇ j α + α R ij ) T F ∂ o ˆ A ij = α K ˆ A ij − 2 α ˆ A il ˆ A l + j Γ i − 2 ˆ A ij ∂ j α + 2 α � Γ i jk ˆ A jk ∂ o � = A ij ∂ j Φ − 4 A ij ∂ j α 12 α ˆ g ij ∂ j K − 2 ˆ + 3 α ˆ

8 Evolution equations: − 1 ∂ o Φ = 6 α K − 2 α ˆ ∂ o ˆ g ij = A ij A ij + 1 −∇ i ∇ i α + α ˆ A ij ˆ 3 α K 2 ∂ o K = e − 4Φ ( −∇ i ∇ j α + α R ij ) T F ∂ o ˆ A ij = α K ˆ A ij − 2 α ˆ A il ˆ A l + j Γ i − 2 ˆ A ij ∂ j α + 2 α � Γ i jk ˆ A jk ∂ o � = A ij ∂ j Φ − 4 A ij ∂ j α 12 α ˆ g ij ∂ j K − 2 ˆ + 3 α ˆ � � ∂ l β l � � χ + 2 Γ i − ˆ g jk � Γ i � − jk 3 Note: Γ i term χ helps to reverse the sign of ∂ l β l � Yo, Baumgarte and Shapiro gr-qc/0209066

9 −4 10 −4 10 −6 10 A1 −6 10 A2 N1 −8 10 N2 −8 10 N3 −10 10 N4 −10 10 ∆ K rms ∆ K rms −12 10 −12 10 −14 −14 10 10 −16 −16 10 10 −18 −18 10 10 0 1000 2000 3000 0 1000 2000 3000 4000 t/M t/M

10 Conformal-Traceless-Mixed Formulation Laguna and Shoemaker, CQG 19, 3679 (2002) Variables: 1 6 ln √ g Φ = ( √ g ) − 2 / 3 g ij = e − 4 Φ g ij g ij ˆ = Γ i g jk � Γ i � = ˆ jk ( √ g ) a A i j = e 6 a Φ A i A i ˆ = j j ( √ g ) k K = e 6 k Φ K ˆ K = ( √ g ) n α = e 6 n Φ α N = β i : shift vector

11 Evolution equations: − 1 K e − 6 ( n + k ) Φ 6 N � ∂ o Φ = − 2 N ˆ A ij e − 6 ( n + a ) Φ ∂ o ˆ g ij = −∇ i ∇ i α e 6 n Φ + N ˆ ∂ o ˆ A i j ˆ A j i e − 6 ( n +2 a − k ) Φ K = 1 K 2 e − 6 ( n + k ) Φ 3(1 − 3 k ) N ˆ + � � T F ∂ o ˆ A i −∇ i ∇ j α + α R i e 6 a Φ = j j (1 − a ) N ˆ K ˆ A i j e − 6 ( n + k ) Φ + � � g ij ∂ j Φ − 4 g ij ∂ j � Γ i e − 6 ( n + a ) Φ ∂ o � 8 k N � = K ˆ 3 N ˆ K � � A ij ∂ j Φ 2 N ˆ A jk � Γ i jk − 2 ˆ A ij ∂ j N + 12(1 + n ) N ˆ e − 6 ( n + a ) Φ + 3(2 + 3 χ ) ∂ l β l � � 1 Γ i − ˆ g jk � Γ i � − jk

11 Evolution equations: − 1 K e − 6 ( n + k ) Φ 6 N � ∂ o Φ = − 2 N ˆ A ij e − 6 ( n + a ) Φ ∂ o ˆ g ij = −∇ i ∇ i α e 6 n Φ + N ˆ ∂ o ˆ A i j ˆ A j i e − 6 ( n +2 a − k ) Φ K = 1 K 2 e − 6 ( n + k ) Φ 3(1 − 3 k ) N ˆ + � � T F ∂ o ˆ A i −∇ i ∇ j α + α R i e 6 a Φ = j j (1 − a ) N ˆ K ˆ A i j e − 6 ( n + k ) Φ + � � g ij ∂ j Φ − 4 g ij ∂ j � Γ i e − 6 ( n + a ) Φ ∂ o � 8 k N � = K ˆ 3 N ˆ K � � A ij ∂ j Φ 2 N ˆ A jk � Γ i jk − 2 ˆ A ij ∂ j N + 12(1 + n ) N ˆ e − 6 ( n + a ) Φ + 3(2 + 3 χ ) ∂ l β l � � 1 Γ i − ˆ g jk � Γ i � − jk Natural choices: n = − 1 , a = 1 , k = 1 / 3 , χ = 2 / 3

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13 Gauge Conditions 0.6 Slicing conditions 0.3 • ∂ t α = − α 2 f ( α ) ( K − K o ) 0 • ∂ t α = −∇ i β i − α K 2D(300 x 29) 3 x 0.183) −0.3 3D(131 3 x 0.183) 3D(131 t α = − α 2 f ( α ) ∂ t K • ∂ 2 −0.6 even (M) 0 30 60 90 120 150 180 0.6 ψ 20 Shift conditions 0.3 • ∂ t β i = λ ∂ t � Γ i 0 t β i = F ∂ t � Γ i − η ∂ t β i • ∂ 2 3 x 0.183) −0.3 3D(131 fit −0.6 0 20 40 60 80 t (M) Alcubierre, Br¨ ugmann, Deiner, Koppits, Pollney, Seidel, Takahashi, gr-qc0206072

14 Boundary Conditions f o + u ( r − v t ) f = r

14 Boundary Conditions f o + u ( r − v t ) + h ( t ) f = r n r

15 Excision Excision

16 Black Holes via Punctures Alcubierre, Brugmann, Deiner, Koppits, Pollney, Seidel, Takahashi, gr-qc0206072 Ψ − 4 BL Ψ 4 g ij ˜ = o δ ij = O (1) ˜ Ψ − 4 BL Ψ 4 o ˆ A ij = O ( r 3 ) K ij = where Ψ o = u + Ψ BL 1 + m 1 + m 2 Ψ BL = 2 r 1 2 r 2 � � δ ij − n i n j � δ kl n k P l � 3 n i P j + n j P i − ˆ A ij = 2 r 2

17 Numerical Methods Finite Differences φ ( x i +1 ) − φ ( x i − 1 ) + O (∆ x 2 ) ∂ x φ ( x i ) = 2 ∆ x value ?

18 Numerical Methods Spectral Methods Finite Elements � φ ( x i ) = a n ψ ( x i ) n =1 ...N

19 Waveforms 0.4 ISCO, r*=31M, z−axis, m=+2 0.2 0.03 even /M 0 ψ 22 0.30m 0.015 −0.2 0.24m 0.20m Re[r ψ 4 ] −0.4 0 0 10 20 30 40 t/M −0.015 T=10M −0.03 T=9M T=0M −0.045 20 40 60 80 100 t/M Baker, Campanelli, Lousto, Takahashi, astro-ph/0202469 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.