Black holes on supercomputers: Numerical relativity applications to - PowerPoint PPT Presentation

Extrinsic curvature Def.: K αβ = −⊥∇ β n α ∇ β n α is not symmetric, but ⊥∇ β n α and, thus, K αβ is! One can show that L n γ αβ = n µ ∇ µ γ αβ + γ µβ ∇ α n µ + γ αµ ∇ β n µ = − 2 K αβ K αβ = − 1 2 L n γ αβ Two interpretations of K αβ → embedding of Σ t in M U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 22 / 195

The projections of the Riemann tensor ⊥ µα ⊥ νβ ⊥ γρ ⊥ σδ R ρσµν = R γδαβ + K γα K δβ − K γβ K δα Gauss Eq. ⊥ µα ⊥ νβ R µν + ⊥ µα ⊥ νβ n ρ n σ R µρνσ = R αβ + KK αβ − K µβ K αµ contracted R + 2 R µν n µ n ν = R + K 2 − K µν K µν scalar Gauss eq. ⊥ γρ n σ ⊥ µα ⊥ νβ R ρσµν = D β K γα − D α K γβ Codazzi eq. n σ ⊥ νβ R σν = D β K − D µ K µβ contracted ⊥ αµ ⊥ νβ n σ n ρ R µρνσ = 1 α L m K αβ + K αµ K µβ + 1 α D α D β α ⊥ µα ⊥ νβ R µν = − 1 α L m K αβ − 2 K αµ K µβ − 1 α D α D β α + R αβ + KK αβ α γ µν D µ D ν α + R + K 2 + K µν K µν R = − 2 α L m K − 2 Here L is the Lie derivative and m µ = α n µ = ( ∂ t ) µ + β µ Summation of spatial tensors: ignore time indices; µ, ν, . . . → m , n , . . . U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 23 / 195

Decomposition of the Einstein equations R αβ − 1 2 Rg αβ + Λ g αβ = 8 π T αβ � � 1 2 ⇔ R αβ = 8 π T αβ − D − 2 g αβ T + D − 2 Λ g αβ Energy momentum tensor ρ = T µν n µ n ν energy density j α = − T µν n µ ⊥ να momentum density S αβ = ⊥ µα ⊥ νβ T µν , S = γ µν S µν stress tensor T αβ = S αβ + n α j β + n β j α + ρ n α n β , T = S − ρ Lie derivative L m = L ( ∂ t − β ) L m K ij = ∂ t K ij − β m ∂ m K ij − K mj ∂ i β m − K im ∂ j β m L m γ ij = ∂ t γ ij − β m ∂ m γ ij − γ mj ∂ i β m − γ im ∂ j β m U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 24 / 195

Decomposition of the Einstein equations Definition: L m γ ij = − 2 α K ij ⊥ µα ⊥ νβ projection: � � S − ρ L m K ij = − D i D j α + α ( R ij + KK ij − 2 K im K mj )+ 8 πα 2 D − 2 γ ij − S ij − D − 2 Λ γ ij Evolution equations n µ n ν projection R + K 2 − K mn K mn = 2 Λ + 16 πρ Hamiltonian constraint ⊥ µα n ν projection D i K − D m K mi = − 8 π j i Momentum constraint U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 25 / 195

Well-posedness Consider a field φ evolved with a first-order system of PDEs The system has a well posed initial value formulation ⇔ There exists some norm and a smooth function F : R + × R + → R + such that || φ ( t ) || ≤ F ( || φ ( 0 ) || , t ) || φ ( 0 ) || Well-posed systems have unique solutions for given initial data There can still be fast growth, e.g. exponential Strong hyperbolicity is necessary for well-posedness The general ADM equations are only weakly hyperbolic Details depend on: gauge, constraints, discretization Sarbach & Tiglio, Living Reviews Relativity 15 (2012) 9; Gundlach & Martín-García, PRD 74 (2006) 024016; Reula, gr-qc/0403007 U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 26 / 195

The BSSN system Goal: modify ADM to get a strongly hyperbolic system Baumgarte & Shapiro, PRD 59 (1998) 024007, Shibata & Nakamura, PRD 52 (1995) 5428 Conformal decomposition, trace split, auxiliary variable 1 K = γ ij K ij φ = 4 ( D − 1 ) ln γ, γ ij = e 4 φ γ ij γ ij = e − 4 φ γ ij ˜ ⇔ ˜ A ij = e − 4 φ � � K ij = e 4 φ � � ˜ 1 ˜ 1 K ij − ⇔ D − 1 γ ij K A ij + D − 1 ˜ γ ij K Γ i = ˜ ˜ γ mn ˜ Γ i mn Auxiliary constraints γ mn ˜ γ = det ˜ ˜ γ ij = 1 , ˜ A mn = 0 U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 27 / 195

The BSSN equations 2 ( D − 1 ) ( ∂ m β m − α K ) ∂ t φ = β m ∂ m φ + 1 γ m ( i ∂ j ) β m − γ ij ∂ m β m − 2 α ˜ γ ij = β m ∂ m ˜ 2 ∂ t ˜ γ ij + 2 ˜ D − 1 ˜ A ij γ mn D m D n α + α ˜ A mn ˜ ∂ t K = β m ∂ m K − e − 4 φ ˜ D − 1 α K 2 1 A mn + + 8 π 2 D − 2 α [ S + ( D − 3 ) ρ ] − D − 2 α Λ A m ( i ∂ j ) β m − A ij ∂ m β m + α K ˜ ∂ t ˜ A ij = β m ∂ m ˜ A ij + 2 ˜ D − 1 ˜ A ij − 2 α ˜ A im ˜ 2 A mj � TF + e − 4 φ � α R ij − D i D j α − 8 πα S ij Γ i = β m ∂ m ˜ Γ i + Γ i ∂ m β m + ˜ γ mn ∂ m ∂ n β i + D − 3 ∂ t ˜ D − 1 ˜ 2 γ im ∂ m ∂ n β n D − 1 ˜ + 2 ˜ A im [ 2 ( D − 1 ) α∂ m φ − ∂ m α ]+ 2 α ˜ mn ˜ Γ i A mn − 2 D − 2 γ im ∂ m K − 16 πα j i D − 1 α ˜ Note: There are alternative versions using χ = e − 4 φ or W = e − 2 φ U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 28 / 195

The BSSN equations In the BSSN equations we use jk = ˜ Γ i Γ i jk + 2 ( δ ik ∂ j φ + δ ij ∂ k φ − ˜ γ im ∂ m φ ) γ jk ˜ R ij = ˜ R ij + R φ ij γ mn ˜ R φ ij = 2 ( 3 − D )˜ D i ˜ D m ˜ γ mn ∂ m φ ∂ n φ ) D j φ − 2 ˜ γ ij ˜ D n φ + 4 ( D − 3 )( ∂ i φ ∂ j φ − ˜ γ ij ˜ Γ m + ˜ R ij = − 1 ˜ γ m ( i ∂ j ) ˜ Γ m ˜ γ mn [ 2 ˜ m ( i ˜ Γ j ) kn + ˜ im ˜ γ mn ∂ m ∂ n ˜ Γ k Γ k 2 ˜ γ ij + ˜ Γ ( ij ) m + ˜ Γ kjn ] D i D j α = ˜ D i ˜ γ mn ∂ m φ ∂ n α D j α − 2 ( ∂ i φ ∂ j α + ∂ j φ ∂ i α ) + 2 ˜ γ ij ˜ The constraints are D − 1 K 2 − ˜ A mn ˜ H = R + D − 2 A mn − 16 πρ − 2 Λ = 0 M i = ˜ D m ˜ D − 1 ∂ i K + 2 ( D − 1 )˜ A mi − D − 2 A mi ∂ m φ − 8 π j i = 0 U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 29 / 195

2.1.2 Generalized Harmonic formulation U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 30 / 195

The Generalized Harmonic (GH) formulation Harmonic gauge: choose coordinates such that � x α = ∇ µ ∇ µ x α = − g µν Γ α µν = 0 4-dim. version of Einstein equations R αβ = − 1 2 g µν ∂ µ ∂ ν g αβ + . . . Principal part of wave equation ⇒ Manifestly hyperbolic Problem: Start with spatial hypersurface t = const . Does t remain timelike? Solution: Generalize harmonic gauge Garfinkle, APS Meeting (2002) 12004, Pretorius, CQG 22 (2005) 425, Lindblom et al, CQG 23 (2006) S447 H α = ∇ µ ∇ µ x α = − g µν Γ α → Source functions µν U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 31 / 195

The Generalized harmonic formulation Any spacetime in any coordinates can be formulated in GH form! Problem: find the corresponding H α Promote H α to evolution variables Einstein field equations in GH form: 2 g µν ∂ µ ∂ ν g αβ = − ∂ ν g µ ( α ∂ β ) g µν − ∂ ( α H β ) + H µ Γ µ 1 αβ � � − Γ µ να Γ ν 2 1 µβ − D − 2 Λ g αβ − 8 π T µν − D − 2 Tg αβ with constraints C α = H α − � x α = 0 Still principal part of wave equation !!! Manifestly hyperbolic Friedrich, Comm.Math.Phys. 100 (1985) 525, Garfinkle, PRD 65 (2002) 044029, Pretorius, CQG 22 (2005) 425 U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 32 / 195

Constraint damping in the GH system One can show that C α | t = 0 = 0 , ∂ t C α | t = 0 = 0 ⇔ The ADM H = 0, M i = 0 Bianchi identities imply evolution of C α : � C α = −C µ ∇ ( µ C α ) − C µ � � � � 1 2 T µα − 8 π D − 2 Tg µα + D − 2 Λ g µα In practice: numerical violations of C µ = 0 ⇒ unstable modes Solution: add constraint damping 2 g µν ∂ µ ∂ ν g αβ = − ∂ ν g µ ( α ∂ β ) g µν − ∂ ( α H β ) + H µ Γ µ αβ − Γ µ 1 να Γ ν µβ � � 2 1 � 2 n ( α C β ) − λ g αβ n µ C µ � − D − 2 Λ g αβ − 8 π T µν − D − 2 Tg αβ − κ Gundlach et al, CQG 22 (2005) 3767 E.g. Pretorius, PRL 95 (2005) 121101 : κ = 1 . 25 / m , λ = 1 U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 33 / 195

Summary GH formulation Specify initial data g αβ , ∂ t g αβ at t = 0 which satisfy the constraints C µ = ∂ t C µ = 0 Constraints preserved due to Bianchi identities Alternative first-order version of GH formulation Lindblom et al, CQG 23 (2006) S447 Auxiliary variables → First-order system Symmetric hyperbolic system → constraint-preserving boundary conditions Used for spectral BH code SpEC Caltech, Cornell, CITA U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 34 / 195

2.1.3 Characteristic formulation U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 35 / 195

Characteristic coordinates Consider advection equation ∂ t f + a ∂ x f = 0 dx Characteristics: curves C : x → at + x 0 ⇔ dt = a df dt | C = ∂ f ∂ t + ∂ f dx dt | C = ∂ f ∂ t + a ∂ f ∂ x = 0 ⇒ f constant along C ∂ x U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 36 / 195

Characteristic “Bondi-Sachs” formulation Here: D = 4 , Λ = 0 Foliate spacetime using characteristic surfaces; light cones Bondi, Proc.Roy.Soc.A 269 (1962), 21; Sachs, Proc.Roy.Soc.A 270 (1962), 103 “ u = t − r , v = t + r ” → double null, ingoing or outgoing outgoing null timelike foliation U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 37 / 195

Characteristic “Bondi-Sachs” formulation Write metric as ds 2 = V e 2 β r du 2 − 2 e 2 β dudr + r 2 h AB ( dx A − U A du )( dx B − U B du ) 2 h AB dx A dx B = ( e 2 γ + e 2 δ ) d θ 2 + 4 sin θ sinh ( γ − δ ) d θ d φ + sin 2 θ ( e − 2 γ + e − 2 δ ) d φ 2 Introduce tetrad k , ℓ, m , ¯ m such that g ( k , ℓ ) = 1 , g ( m , ¯ m ) = 1 and all other products vanish The Einstein equations become 4 hypersurface eqs.: R µν k µ k ν = R µν k µ m ν = R µν m µ ¯ m ν = 0 2 evolution eqs.: R µν m µ m ν = 0 1 trivial eq.: R µν k µ ℓ ν = 0 3 supplementary eqs.: R µν ℓ µ m ν = R µν ℓ µ ℓ ν = 0 U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 38 / 195

Integration of the characteristic equations Provide initial data for γ, δ on hypersurface u = const β, V , U A at u Integrate hypersurface eqs. along r → → 3 “constants” of integration M i ( θ, φ ) Evolve γ, δ using evolution eqs. → 2 “constants” of integration → complex news ∂ u c ( u , θ, φ ) Evolve the M i through the supplementary eqs. U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 39 / 195

Summary characteristic formulation Naturally adapted to the causal structure of GR Clear hierarchy of equations → isolated degrees of freedom Problem: caustics → breakdown of coordinates Well suited for symmetric spacetimes, planar BHs Solution for binary problem? Recent investigation: Babiuc, Kreiss & Winicour, arXiv:1305.7179 [gr-qc] Application to characteristic GW extraction Babiuc, Winicour & Zlochower, CQG 28 (2011) 134006 Reisswig et al, CQG 27 (2010) 075014 U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 40 / 195

Direct methods Use symmetry to write line element, e.g. ds 2 = − a 2 ( µ, t ) dt 2 + b 2 ( µ, t ) d µ 2 − R 2 ( µ, t ) d Ω 2 May & White, PR 141 (1966) 1232 Energy momentum tensor T 00 = − ρ ( 1 + ǫ ) , T 11 = T 22 = T 33 = 0 Lagrangian coords. GRT ENSOR , M ATHEMATICA ,... ⇒ Field equations: a ′ = ... b ′ = ... ¨ R = ... U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 41 / 195

Further reading 3+1 formalism Gourgoulhon, gr-qc/0703035 Characteristic formalism Winicour, Liv. Rev. Rel. 15 2012 2 Numerical relativity in general Alcubierre, “ Introduction to 3+1 Numerical Relativity ”, Oxford University Press Baumgarte & Shapiro, “ Numerical Relativity ”, Cambridge University Press Well-posedness, Einstein eqs. as an Initial-Boundary-Value problem Sarbach & Tiglio, Liv. Rev. Rel. 15 (2012) 9 U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 42 / 195

2.2. NR beyond 4D U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 43 / 195

A list of tasks NR in 3+1 dimensions: O ( 100 ) cores, Gb of memory Each extra dimension can introduce a factor of O ( 100 ) ⇒ reduce D to 3 + 1 dimensions; symmetries Three approaches: Dimensional reduction to 3 + 1 GR + quasi matter C ARTOON type methods Simplify line element using symmetry Outer boundary conditions: regularization, background subtraction U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 44 / 195

2.2.1 Dimensional reduction U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 45 / 195

Conventions Reduce D dimensions to d ; typically d = 3 + 1 = 4 Indices A , B , C , . . . = 0 . . . D − 1 : D dimensional spacetime α, β, γ, . . . = 0 . . . d − 1 : d dimensional base spacetime a , b , c , . . . = d . . . D − 1 : D − d dimensional fibre Symmetry: SO ( D − d + 1 ) ⇒ rotations in D − d + 1 space dimensions ⇔ on S D − d sphere Typically: SO ( D − 3 ) symmetry, S D − 4 sphere U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 46 / 195

General formalism Cho, Phys. Lett. B 186 (1987) 38 Cho & Kim, J. Math. Phys. 30 (1987) 1570 Zilhão, arXiv:1301.1509 [gr-qc] The general D metric can be written ds 2 = g AB dx A dx B g µν + e 2 κ 2 g ab B a µ B b ν dx µ dx ν + 2 e κ B a µ g ab dx µ dx b + g ab dx a dx b � � = Comments e , κ are coupling and scale parameters; they’ll eventually drop out This metric is completely general! We used a special case of this: ADM 3+1 decomposition! U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 47 / 195

General formalism Assumption: g AB admits m Killing vectors ξ ( i ) = ξ a ( i ) ∂ a ⇒ L ξ ( i ) g AB = 0 Def.: dual form ξ ( j ) = ξ ( j ) a d x a such that ξ ( j ) a ξ a ( i ) = δ i j Def.: F abc ≡ − ξ ( i ) b ∂ c ξ a ( i ) Then ⇒ L ξ ( i ) g AB = 0 implies ∂ a g bc = F d ab g dc + F d ac g db ∂ a B b µ = − F bad B d µ ∂ a g µν = 0 U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 48 / 195

General formalism Def.: D µ ≡ ∂ µ − e κ B d µ ∂ d F a µ b ≡ e κ∂ b B a µ = − e κ F abc B c µ F a µν ≡ − e κ � ∂ µ B a ν − ∂ ν B a µ + e κ ( F abc − F acb ) � The covariant derivatives are defined as ∇ σ T a µ b ν ≡ D σ T a µ b ν + F a σ c T c µ b ν − F c σ b T a µ c ν + Γ µ λσ T a λ b ν − Γ λ νσ T a µ b λ ∇ c T a µ b ν ≡ ∂ c T a µ b ν + Γ a dc T d µ b ν − Γ d bc T a µ d ν Here, Γ µ λσ and Γ a dc are the connections associated with g µν and g ab . Note: ∇ σ g µν = ∇ c g ab = 0 , but ∇ σ g ab � = 0! U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 49 / 195

General formalism A tedious but straightforward calculation gives us the Ricci tensor as R ab = R ab − 1 4 g cd ∇ µ g cd ∇ µ g ab + 1 2 g cd ∇ µ g ac ∇ µ g bd + 1 σµ − 1 4 g µν g ρσ g bc g ad F c ρν F d 2 ∇ µ ∇ µ g ab R µ a = e κ R ac B c µ + 1 2 g ρσ ∇ ρ ( g ac F c σµ ) + 1 4 g cd ∇ ρ g cd g ρσ F e σµ g ae + 1 g cd ∇ µ g da � � 2 ∇ c R µν = R µν + 2 e κ B c ( µ R ν ) c − e 2 κ 2 R cd B c µ B d ν − 1 2 g ρσ g cd F c σµ F d ρν − 1 − 1 4 g cd g ab ∇ µ g ca ∇ ν g db − 1 � g cd ∇ µ g cd � 2 ∇ c F c µν 2 ∇ ν R = R ( g µν ) + R ( g ab ) − 1 4 g cd g ρσ g µν F d σµ F c ρν − ∇ µ � g cd ∇ µ g cd � − 1 4 g ca g bd ∇ µ g cd ∇ µ g ab − 1 4 g cd g ab ∇ µ g cd ∇ µ g ab U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 50 / 195

Case: SO ( D − d + 1 ) symmetry, ⇔ S D − d sphere In practice: SO ( D − 3 ) , S D − 4 sphere (e.g. D = 6 ⇒ SO ( 3 ) , S 2 ) S n sphere: ( n + 1 ) n / 2 Killing vectors ξ ( i ) E.g. S 2 sphere: ∂ φ , sin φ∂ θ + cot θ cos φ∂ φ , cos φ∂ θ − cot θ sin φ∂ φ Rotations around x , y , z axes Killing’s equation L ξ ( i ) g AB = 0 implies L ξ ( i ) B a L ξ ( i ) g ab = 0 , L ξ ( i ) g µν = 0 µ = 0 , Consequences: g ab = e 2 ψ ( x µ ) h ab with h ab = metric on S D − d with unit radius g µν = g µν ( x µ ) in adapted coordinates [ ξ ( i ) , B µ ] = 0 for n ≥ 2 (only vector field commuting with all KVs: 0) ⇒ B a µ = 0 U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 51 / 195

Case: SO ( D − d + 1 ) symmetry, ⇔ S D − d sphere With these consequences we get ( D − d − 1 ) − e 2 ψ [( D − d ) ∂ µ ψ ∂ µ ψ + ∇ µ ∂ µ ψ ] � � R ab = h ab R µ a = 0 R µν = R µν − ( D − d )( ∇ ν ∂ µ ψ − ∂ µ ψ ∂ ν ψ ) ( D − d − 1 ) e − 2 ψ − 2 ∇ µ ∂ µ ψ � R = R + ( D − d ) − ( D − d + 1 ) ∂ µ ψ ∂ µ ψ � The D dimensional vacuum Einstein equations R µν thus become e 2 ψ [( D − d ) ∂ µ ψ ∂ µ ψ + ∇ µ ∂ µ ψ ] = ( D − d − 1 ) R µν = ( D − d )( ∇ ν ∂ µ ψ − ∂ µ ψ ∂ ν ψ ) i.e. the d dimensional Einstein equations plus quasi-matter terms U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 52 / 195

Regularity of the variables Note: One of the d − 1 spatial coordinates x , y , z , . . . is a radius Without loss of generality we choose y ⇒ Computational domain: x , z , . . . ∈ R , y ≥ 0 Analysing our equations for some analytically known data, e.g. Brill-Lindquist, shows that e 2 ψ = 0 at y = 0 Solution: Use instead ζ = e − 4 φ y 2 e 2 ψ where e − 4 φ is the BSSN conformal factor. With that we get the BSSN equations with matter terms... U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 53 / 195

BSSN source terms for d=4 Note: We set d = 4, i , j , . . . = 1 , 2 , 3 and use χ ≡ e − 4 φ γ yy ζ − 1 γ mn ∂ m η ∂ n χ − χ ˜ 4 π ( ρ + S ) Γ y = ( D − 5 ) χ ˜ − 2 D − 7 y + D − 6 χ γ mn ∂ m ζ ∂ n ζ 4 ζ ˜ ζ 2 ˜ D − 4 ζ y 2 4 γ ym � � γ mn ( χ ˜ D m ∂ n ζ − ζ ˜ D m ∂ n χ ) + ( D − 4 ) ˜ χ + 1 2 ζ ˜ ζ ∂ m ζ − ∂ m χ y � K ζ � 2 γ ym − KK ζ − K 2 ˜ γ mn ∂ m χ ∂ n χ 3 − 1 y ∂ m χ + D − 1 ζ + K 4 ˜ − ( D − 5 ) ζ 2 χ 3 8 πχ S TF � Γ y = 1 2 χ y ζ ( δ y ( j ∂ i ) ζ − ζ ˜ ij ) + 1 2 χ ∂ i χ ∂ j χ − ˜ D i ∂ j χ + χ ζ ˜ ij D i ∂ j ζ D − 4 2 � TF � � γ ym ∂ m χ − χ ˜ χ + 1 γ mn ∂ n χ 2 χ ˜ γ ij ˜ ζ ∂ m ζ − ˜ γ ij y ∂ m χ − 2 ζ 2 ∂ i ζ ∂ j ζ � K ζ � ˜ ζ + K − A ij 3 U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 54 / 195

BSSN source terms for d=4 � � � � K ζ ζ ∂ i K ζ − K ζ 16 π j i γ ym ˜ D − 4 = 2 δ y i + 2 χ ∂ i χ + 1 1 + 2 ζ − ˜ A mi ζ ∂ i ζ 3 ∂ i K y ζ � � γ nm ˜ 1 ζ ∂ n ζ − 1 − ˜ A mi χ ∂ n χ The matter evolution is given by 3 ζ∂ m β m + 2 ζ β y ∂ t ζ = β m ∂ m ζ − 2 α K ζ − 2 y 3 K ζ ∂ m β m + 2 β y γ ym 3 ζ ( ∂ t − L β ) K − χζ ˜ ∂ t K ζ = β m ∂ m K ζ − 2 y K ζ − 1 y ∂ m α � γ yy − 1 γ ym ( 5 − D ) χ ζ ˜ + ( 4 − D ) χ ˜ − 1 γ mn ∂ m α ( χ∂ n ζ − ζ∂ n χ )+ α 2 ˜ y ∂ m ζ y 2 γ ym ζ ˜ χ + 2 D − 7 y ∂ m χ + 6 − D γ mn ∂ m ζ ∂ n ζ + 2 D − 7 γ mn ∂ m ζ ∂ n χ ζ ˜ ˜ 2 4 4 K 2 ζ + 1 − D γ mn ∂ m χ ∂ n χ + ( D − 6 ) ζ + 2 D − 5 KK ζ + D − 1 9 ζ K 2 ζ χ ˜ 4 3 � D m ∂ n ζ ) + χζ ˜ Γ y γ mn ( ζ ˜ D m ∂ n χ − χ ˜ + 1 2 ˜ y U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 55 / 195

Regularization at y = 0 Note that the previous equations contain divisions by y , E.g. : β y γ ym y , ˜ y ∂ m ζ, . . . These can all be regularized! E.g. : Symmetry of a vector across y = 0 implies β y ( − y ) = − β y ( y ) We can therefore Taylor expand β y around y = 0 as β y ( y ) = b 1 y + O ( y 2 ) β y y = b 1 = ∂ y β y ⇒ lim y → 0 Similar tricks work for all such terms see Zilhão et al, PRD 81 (2010) 084052 U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 56 / 195

2.2.2 C ARTOON methods U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 57 / 195

Dimensional reduction through C ARTOON Originally developed for axisymmetry around z in 3+1 dimensions Alcubierre et al, IJMPD 10 (2001) 273 Coordinates ( z , x , y ) ↔ ( z , ρ, θ ) where x = ρ cos φ, y = ρ sin φ Killing vector ∂ φ = x ∂ y − y ∂ x Extend 2D grid by ghostzones for derivatives; rotate, interpolate U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 58 / 195

Dimensional reduction through C ARTOON Yoshino & Shibata. PRD 80 (2009) 084025 Scalar: ψ ( z , x , y ) = ψ ( z , ρ, 0 ) Vector: Express v x ( z , x , y ) , v y ( z , x , y ) through v ρ ( z , ρ ) , v φ ( z , ρ ) and replace those through the relation on the xy plane v x ( z , ρ, 0 ) = v ρ ( z , ρ ) , v y ( z , ρ, 0 ) = ρ v φ ( z , ρ ) ρ v x ( z , ρ, 0 ) − y ⇒ v x ( z , x , y ) = x ρ v y ( z , ρ, 0 ) v y ( z , x , y ) = y ρ v x ( z , ρ, 0 ) + x ρ v y ( z , ρ, 0 ) Likewise for tensors: T zz like scalar, T zx , T zy like vector � 2 � 2 � � y T yy ( z , ρ, 0 ) − 2 xy x T xx ( z , x , y ) = T xx ( z , ρ, 0 ) + ρ 2 T xy ( z , ρ, 0 ) ρ ρ � 2 � 2 � � y T yy ( z , ρ, 0 ) + 2 xy x T yy ( z , x , y ) = T xx ( z , ρ, 0 ) + ρ 2 T xy ( z , ρ, 0 ) ρ ρ ρ 2 [ T xx ( z , ρ, 0 ) − T yy ( z , ρ, 0 )] + x 2 − y 2 T xy ( z , x , y ) = xy T xy ( z , ρ, 0 ) ρ 2 U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 59 / 195

C ARTOON in D=5 with SO ( 3 ) symmetry Cartesian coordinates: ( w , x , y , z ) each hypersurface w = const is spher. symmetric 3 Killing vectors ξ 1 = y ∂ z − z ∂ y , ξ 2 = z ∂ z − x ∂ z , ξ 3 = x ∂ y − y ∂ x x 2 + y 2 + z 2 � Use data in xw plane, set r = Scalar: ψ ( w , x , y , z ) = ψ ( w , r , 0 , 0 ) Vector, Tensor fields: ... cf. Yoshino & Shibata, PRD 80 (2009) 084025 ⇒ effective 2+1 Cartesian code U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 60 / 195

A modified C ARTOON method Shibata & Yoshino, PRD 81 (2010) 104035 Yoshino & Shibata, PTPS 189 (2011) 269 For larger D , C ARTOON ghostzones require considerable memory Solution: Trade derivatives � z 2 + � i w 2 Coordinates: ( x , y , z , w i ) , i = 1 . . . D − 4, ρ = i Symmetry: SO ( D − 3 ) , i.e. Rotations in w i Scalar: ψ ( x , y , z , w i ) = ψ ( x , y , ρ, 0 ) ⇒ ∂ w i ψ = ∂ ( x , y ) ∂ w i ψ = ∂ z ∂ w i = 0 , ∂ w i ∂ w j ψ = ∂ z ψ z δ ij where ( x , y ) stands for either x or y U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 61 / 195

A modified C ARTOON method Vector: β z ( x , y , z , w i ) = z ρ β z ( x , y , ρ, 0 ) β w i ( x , y , z , w i ) = w i ρ β z ( x , y , ρ, 0 ) ⇒ ∂ w i β z = ∂ ( x , y ) β w i = ∂ z β w i = ∂ ( x , y ) ∂ w i β z = ∂ z ∂ w i β z = ∂ ( x , y ) ∂ ( x , y ) β w i = ∂ ( x , y ) ∂ z β w i = ∂ w j ∂ w k β w i = 0 ∂ ( x , y ) β z ∂ w j β w i = β z ∂ ( x , y ) ∂ w j β w i = z δ ij , δ ij , z ∂ w i ∂ w j β z = ∂ z ∂ w j β w i = � ∂ z β z − β z � δ ij z z 2 U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 62 / 195

A modified C ARTOON method Tensors: � � T zz ( x , y , z , w i ) = z 2 1 − z 2 ρ 2 T zz ( x , y , ρ, 0 ) + T ww ( x , y , ρ, 0 ) ρ 2 T w i w i ( x , y , z , w i ) = w 2 1 − w 2 � � ρ 2 T zz ( z , y , ρ, 0 ) + i i T ww ( x , y , ρ, 0 ) ρ 2 T zw i ( x , y , z , w i ) = zw i ρ 2 [ T zz ( x , y , ρ, 0 ) − T ww ( x , y , ρ, 0 )] T w i w j ( x , y , z , w i ) = w i w j ρ 2 [ T zz ( x , y , ρ, 0 ) − T ww ( x , y , ρ, 0 )] where T ww ≡ T w 1 w 1 = T w 2 w 2 = . . . which are all equal ⇒ ∂ w i T zz = ∂ w j T w i w i = ∂ ( x , y ) T zw i = ∂ z T zw i = ∂ ( x , y ) ∂ w i T zz = ∂ z ∂ w i T zz = ∂ ( x , y ) ∂ w j T w i w i = ∂ z ∂ w j = ∂ ( x , y ) ∂ ( x , y ) T zw i = ∂ ( x , y ) ∂ z T zw i = ∂ z ∂ w i T zz = 0 U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 63 / 195

A modified C ARTOON method Tensors continued: for i � = j we also have ∂ ( x , y ) T w i w j = ∂ z T w i w j = ∂ w k T w i w j = ∂ ( x , y ) ∂ ( x , y ) T w i w j = ∂ ( x , y ) ∂ z T w i w j = ∂ z ∂ z T w i w j = ∂ ( x , y ) ∂ w k T w i w j = ∂ z ∂ w k T w i w j = 0 The non-zero derivatives appearing in the BSSN eqs. are ∂ ( x , y ) T zz − ∂ ( x , y ) T ww ∂ w j T zw i = T zz − T ww δ ij , ∂ ( x , y ) ∂ w j T zw i = δ ij , z z � � ∂ z T zz + 2 ( T ww − T zz 1 ∂ w i ∂ w j δ ij , z z ∂ w j w k T w i w i = 2 ( T zz − T ww ) δ ik δ ij + ∂ z T ww δ jk , z 2 z � � ∂ z T zz − ∂ z T ww − T zz − T ww ∂ z ∂ w j T zw i = 1 δ ij , z z ∂ w k ∂ w l T w i w j = T zz − T ww ( δ il δ jk + δ ik δ jl ) for i � = j z 2 U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 64 / 195

A modified C ARTOON method Plugging these into the D dim. equations enables us to work on a genuine d dim. hypersurface with no ghost zones Note: There are additional fields, e.g. T zw i , but these are only required on the ( xyz ) hyper plane! Note: There are divisions by z ⇒ regularization at z = 0 required! cf. y = 0 in the dimensional reduction U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 65 / 195

Further reading Dimensional reduction Zilhão, arXiv:1301.1509 Modified Cartoon Yoshino & Shibata, PTPS 189 (2011) 269 U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 66 / 195

2.3. Initial data, Gauge, Boundaries U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 67 / 195

2.3.1. Initial data U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 68 / 195

Analytic initial data Schwarzschild, Kerr, Tangherlini, Myers Perry,... e.g. Schwarzschild in isotropic coordinates: � 2 � 4 [ dr 2 + r 2 ( d θ 2 + sin 2 θ d φ 2 )] ds 2 = − � dt 2 + M − 2 r 1 + M � M + 2 r 2 r Time symmetric N BH initial data: Brill-Lindquist, Misner 1960s Problem: Finding initial data for dynamic systems Goals 1) Solve constraints 2) Realistic snapshot of physical system This is mostly done using the ADM 3+1 split U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 69 / 195

The York-Lichnerowicz split We work in D = 4 Conformal metric: γ ij = ψ 4 ¯ γ ij Lichnerowicz, J.Math.Pures Appl. 23 (1944) 37 York, PRL 26 (1971) 1656, PRL 28 (1972) 1082 Note: in contrast to BSSN we do not set ¯ γ = 1 Conformal traceless split of the extrinsic curvature K ij = A ij + 1 3 γ ij K A ij = ψ − 10 ¯ A ij = ψ − 2 ¯ ⇔ A ij A ij U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 70 / 195

Bowen-York data By further splitting ¯ A ij into a longitudinal and a transverse traceless part, the momentum constraint simplifies significantly Cook, Living Review Relativity (2000) 05 Further assumptions: vacuum, K = 0, γ ij = f ij , ¯ ψ | ∞ = 1 where f ij is the flat metric in arbitrary coordinates. Conformal flatness, asymptotic flatness, traceless Then there exists an anlytic solution to the momentum constraint ¯ 3 P i n j + P j n i − ( f ij − n i n j ) P k n k � � A ij = 2 r 2 + 3 � ǫ kil S l n k n j + ǫ kjl S l n k n i � r 3 where r is a coordinate radius and n i = x i r Bowen & York, PRD 21 (1980) 2047 U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 71 / 195

Properties of the Bowen York solution The momentum in an asymptotically flat hypersurface associated with the asymptotic translational and rotational Killing vectors ξ i ( a ) is Π i = 1 K ji − δ ji K ξ i ( a ) d 2 A j � � � 8 π ∞ ⇒ . . . ⇒ P i and S i are the physical linear and angular momentum of the spacetime The momentum constraint is linear ⇒ we can superpose Bowen-York data. The momenta then simply add up Bowen-York data generalizes (analytically!) to higher D Yoshino, Shiromizu & Shibata, PRD 74 (2006) 124022 U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 72 / 195

Puncture data Brandt & Brügmann, PRL 78 (1997) 3606 The Hamiltonian constraint is now given by A mn = 0 ¯ 8 ψ − 7 ¯ A mn ¯ ∇ 2 ψ + 1 Ansatz for conformal factor: ψ = ψ BL + u , where ψ BL = � N m i r i | is the Brill-Lindquist conformal factor, i = 1 2 | � r − � i.e. the solution for ¯ A ij = 0. There then exist unique C 2 solutions u to the Hamiltonian constraints The Hamiltonian constraint in this form is further suitable for numerical solution e.g. Ansorg, Brügmann & Tichy, PRD 70 (2004) 064011 U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 73 / 195

Properties of the puncture solutions r i are bare mass and position of the i th BH. m i and � In the limit of vanishing Bowen York parameters P i = S i = 0, the puncture solution reduces to Brill Lindquist data � 4 � γ ij dx i dx j = ( dx 2 + dy 2 + dz 2 ) m i 1 + � i 2 | � r − � r i | The numerical solution of the Hamiltonian constraint generalizes rather straightforwardly to higher D Yoshino, Shiromizu & Shibata, PRD 74 (2006) 124022 Zilhão et al, PRD 84 (2011) 084039 Punctures generalize to asymptotically de-Sitter BHs Zilhão et al, PRD 85 (2012) 104039 using McVittie coordinates McVittie, MNRAS 93 (1933) 325 U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 74 / 195

2.3.2. Gauge U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 75 / 195

The gauge freedom β i Remember: Einstein equations say nothing about α, Any choice of lapse and shift gives a solution This represents the coordinate freedom of GR β i Physics do not depend on α, So why bother? The performance of the numerics DO depend strongly on the gauge! U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 76 / 195

What goes wrong with bad gauge? U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 77 / 195

What goes wrong with bad gauge? U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 78 / 195

What goes wrong with bad gauge? U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 79 / 195

What goes wrong with bad gauge? U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 80 / 195

Ingredients for good gauge Singularity avoidance Avoid slice stretching Aim at stationarity in comoving frame Well posedness of system Generalize “good” gauge, e .g. harmonic Lots of good luck! Bona et al, PRL 75 (1995) 600, Alcubierre et al. , PRD 67 (2003) 084023, Alcubierre, CQG 20 (2003) 607, Garfinkle, PRD 65 (2001) 044029 U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 81 / 195

Moving puncture gauge Gauge was a key ingredient in the Moving puncture breakthroughs Campanelli et al, PRL 96 (2006) 111101 Baker et al, PRL 96 (2006) 111102 Variant of 1 + log slicing and Γ -driver shift Alcubierre et al, PRD 67 (2003) 084023 Now in use as ∂ t α = β m ∂ m α − 2 α K and ∂ t β i = β m ∂ m β i + 3 4 B i ∂ t B i = β m ∂ m B i + ∂ t ˜ Γ i − β m ∂ m ˜ Γ i − η B i or ∂ t β i = β m ∂ m β i + 3 Γ i − ηβ i 4 ˜ U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 82 / 195

Moving puncture gauge continued Some people drop the advection derivatives β m ∂ m . . . η is a damping parameter or position-dependent function Alic et al, CQG 27 (2010) 245023, Schnetter, CQG 27 (2010) 167001, Müller et al, PRD 82 (2010) 064004 Modifications in higher D : Dimensional reduction Zilhão et al, PRD 81 (2010) 084052 ∂ t α = β m ∂ m α − 2 α ( η K K + η K ζ K ζ ) C ARTOON Yoshino & Shibata, PTPS 189 (2011) 269 ∂ t β i = 2 ( D − 2 ) v 2 D − 1 long B i ∂ t B i = ∂ t ˜ Γ i − η B i Here η K , η K ζ , v long are parameters U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 83 / 195

Gauge conditions in the GH formulation How to choose H µ ? → some experimentation... Pretorius’ breakthrough � H t = − ξ 1 α − 1 α η + ξ 2 n µ ∂ µ H t with ξ 1 = 19 / m , ξ 2 = 2 . 5 / m , η = 5 where m = mass of 1 BH Caltech-Cornell-CITA spectral code: Initialize H α to minimize time derivatives of metric, adjust H α to harmonic and damped harmonic gauge condition Lindblom & Szilágyi, PRD 80 (2009) 084019 , with Scheel, PRD 80 (2009) 124010 The H α are related to lapse and shift: n µ H µ = − K − n µ ∂ µ ln α γ µ i H µ = − γ mn Γ i mn + γ im ∂ m ( ln α ) + 1 α n µ ∂ µ β i U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 84 / 195

2.3.3. Boundaries U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 85 / 195

Inner boundary: Singularity treatment Cosmic censorship ⇒ horizon protects outside We get away with it... Moving Punctures UTB, NASA Goddard ’05 Excision: Cut out region around singularity Caltech-Cornell, Pretorius U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 86 / 195

Moving puncture slices: Schwarzschild Wormhole → Trumpet slice = stationary 1+log slice Hannam et al, PRL 99 (2007) 241102, PRD 78 (2008) 064020 Brown, PRD 77 (2008) 044018, CQG 25 (2008) 205004 Gauge might propagate at > c , no pathologies Natural excision Brown, PRD 80 (2009) 084042 U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 87 / 195

Outer boundary: Asymptotically flat case Computational domains often don’t extend to ∞ Outgoing Sommerfeld conditions Assume f = f 0 + u ( t − r ) where f 0 = asymptotic value r n ∂ t u + ∂ r u = 0 + x i ∂ t f + n f − f 0 r ∂ i f = 0 r Use upwinding, i.e. one-sided, derivatives! U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 88 / 195

Non-asymptotically flat case: de Sitter In McVittie coordinates: r → ∞ ⇒ ds 2 = − dt 2 + a ( t ) 2 ( r 2 + r 2 d Ω 2 2 ) where a ( t ) = e Ht , � H = Λ / 3 Radial null geodesics: dt = ± adr We expect: f = f 0 + a u ( t − a r ) r n a ( t ) ∂ r f + n f − f 0 1 ⇒ ∂ t f − ∂ t f 0 + r a ( t ) − H ( f − f 0 ) = 0 Zilhão et al, PRD 85 (2012) 104039 U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 89 / 195

Anti de Sitter Much more complicated! Time-like outer boundary ⇒ affects interior AdS metric diverges at outer boundary U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 90 / 195

Anti de Sitter metric Maximally symmetric solution to Einstein eqs. with Λ < 0 D − � D − 1 Hyperboloid X 2 0 + X 2 i = 1 X 2 i embedded in D + 1 dimensional flat spacetime of signature − − + . . . + Global AdS X 0 = L cos τ X d = L sin τ cos ρ , cos ρ X i = L tan ρ Ω i , for i = 1 . . . D − 1, Ω i hyperspherical coords. ⇒ ds 2 = cos 2 ρ ( − d τ 2 + d ρ 2 + sin 2 ρ d Ω 2 L 2 D − 2 ) where 0 ≤ ρ < π/ 2 , − π < τ ≤ π Outer boundary at ρ = π/ 2 U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 91 / 195

Anti de Sitter metric continued Poincaré coordinates � i = 1 ( x i ) 2 − t 2 � z 2 + L 2 + � D − 2 1 X 0 = 2 z X i = Lx i z for i = 1 . . . D − 2 � z 2 − L 2 + � D − 2 i = 1 ( x i ) 2 − t 2 � 1 X D − 1 = 2 z X d = Lt z � i = 1 ( dx i ) 2 � ⇒ ds 2 = L 2 − dt 2 + dz 2 + � D − 2 z 2 where z > 0, t ∈ R Outer boundary at z = 0 e.g. Ballón Bayona & Braga, hep-th/0512182 U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 92 / 195

AdS spacetimes: Outer boundary AdS boundary: ρ → π/ 2 (global) z → 0 (Poincaré) AdS metric becomes singular ⇒ induced metric determined up to conformal rescaling only gl ∼ − d τ 2 + d Ω D − 2 ds 2 Global: P ∼ − dt 2 + � D − 2 Poincaré: ds 2 i = 1 d ( x i ) 2 ⇒ Different topology: R × S D − 2 and R D − 1 The dual theories live on spacetimes of different topology U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 93 / 195

Regularization methods Decompose metric into AdS part plus deviation Bantilan & Pretorius, PRD 85 (2012) 084038 Factor out appropriate factors of the bulk coordinate Chesler & Yaffe, PRL 106 (2011) 021601 Heller, Janik & Witaszczyk, PRD 85 (2012) 126002 Factor out singular term of the metric Bizo´ n & Rostworowski, PRL 107 (2011) 031102 Regularity of the outer boundary may constrain the gauge freedom Bantilan & Pretorius, PRD 85 (2012) 084038 U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 94 / 195

Further reading Initial data construction Cook, Liv. Rev. Rel. 3 (2000) 5 Pfeiffer, gr-qc/0510016 U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 95 / 195

2.4 Discretization of the equations U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 96 / 195

Finite differencing Consider one spatial, one time dimension t , x Replace computational domain by discrete points x i = x 0 + i dx , t n = t 0 + n dt Function values f ( t n , x i ) ∼ f n , i U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 97 / 195

Derivatives and finite derivatives Goal: represent ∂ m f ∂ x m in terms of f n , i Fix index n ; Taylor expansion: i dx 2 + O ( dx 3 ) f i − 1 = f i − f ′ i dx + 1 2 f ′′ f i = f i i dx 2 + O ( dx 3 ) f i + 1 = f i + f ′ i dx + 1 2 f ′′ Write f ′ i as linear combination: f ′ i = Af i − 1 + Bf i + Cf i + 1 Insert Taylor expressions and compare coefficients on both sides 2 Adx 2 + 1 0 = 1 2 Cdx 2 ⇒ 0 = A + B + C , 1 = ( − A + B ) dx , ⇒ A = − 1 1 2 dx , B = 0 , C = 2 dx i = f i + 1 − f i − 1 ⇒ f ′ + O ( dx 2 ) 2 dx Higher order accuracy → more points; works same in time U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 98 / 195

Mesh refinement ∼ 1 M 3 Length scales : BH Wavelength ∼ 10 ... 100 M ∼ 100 ... 1000 M Wave zone Critical phenomena Choptuik ’93 First used for BBHs Brügmann ’96 Available Packages: Paramesh MacNeice et al. ’00 Carpet Schnetter et al. ’03 SAMRAI MacNeice et al. ’00 U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 99 / 195

Berger-Oliger mesh refinement Goal: Update from t to t + dt Refinement criteria: numerical error, curvature,... Here for 1 + 1 dimensions U. Sperhake (DAMTP, University of Cambridge) Black holes on supercomputers: Numerical relativity applications to astrophysics and 07/01/2013 100 / 195