Numerical Relativity

Transcription

1 Numerical Relativity Mark A. Scheel Walter Burke Institute for Theoretical Physics Caltech July 7, 2015 Mark A. Scheel Numerical Relativity July 7, / 34

2 Outline: Motivation 3+1 split and ADM equations Well-posedness and hyperbolicity Evolution system 1: BSSN: Equations and gauge choices Evolution system 2: Generalized Harmonic: Equations and gauge choices Handling singularities inside black holes Current status of NR Mark A. Scheel Numerical Relativity July 7, / 34

3 Motivation: Many gravitational wave sources (binaries, supernovae) involve nonlinear gravity. To learn these from LIGO, must solve Einstein s equations to connect source properties with measured gravitational waves. S 2 M2 S 1 M1 h(t) t/m The only way to solve nonlinear, dynamical, strong-field Einstein equations is numerically. Numerical Relativity Mark A. Scheel Numerical Relativity July 7, / 34

4 Statement of the problem: Einstein s Eqs. G µν = 8πT µν have time/space mixed up. Metric g µν unknown analytically in most cases. Goal Rewrite G µν = 8πT µν to solve for the metric later given data now. Mark A. Scheel Numerical Relativity July 7, / 34

5 Analogy: Maxwell s equations Constraints E = 4πρ B = 0 Evolution t E = B 4πJ t B = E t = 2 t = 1 t = 0 Figure: C. Reisswig Mark A. Scheel Numerical Relativity July 7, / 34

6 Analogy: Maxwell s equations Constraints E = 4πρ B = 0 Evolution t E = B 4πJ t B = E t = 2 t = 1 t = 0 Figure: C. Reisswig Note: Evolution eqs. preserve constraints: t ( B) = t B = ( E) = 0 t ( E 4πρ) = t E 4π t ρ = ( B 4πJ) 4π t ρ = 4π( t ρ + J) = 0 Mark A. Scheel Numerical Relativity July 7, / 34

7 Analogy: Maxwell s equations Constraints E = 4πρ B = 0 Evolution t E = B 4πJ t B = E t = 2 t = 1 t = 0 Figure: C. Reisswig Note: Evolution eqs. preserve constraints: t ( B) = t B = ( E) = 0 t ( E 4πρ) = t E 4π t ρ = ( B 4πJ) 4π t ρ = 4π( t ρ + J) = 0 Goal: Do the same for Einstein s Eqs. G µν = 8πT µν Mark A. Scheel Numerical Relativity July 7, / 34

8 Arnowitt-Deser-Misner (1962) 3+1 split Assume spacetime split into slices. Let n µ be normal vector to slice. n µ n µ = 1. t = 2 t = 1 t = 0 Figure: C. Reisswig Mark A. Scheel Numerical Relativity July 7, / 34

9 Arnowitt-Deser-Misner (1962) 3+1 split Assume spacetime split into slices. Let n µ be normal vector to slice. n µ n µ = 1. Define 3-metric γ µν = g µν + n µ n ν. Lapse function α: Proper time / coord time. Shift vector β µ : How coords move. t = 2 t = 1 t = 0 Figure: C. Reisswig Mark A. Scheel Numerical Relativity July 7, / 34

10 Arnowitt-Deser-Misner (1962) 3+1 split Assume spacetime split into slices. Let n µ be normal vector to slice. n µ n µ = 1. Define 3-metric γ µν = g µν + n µ n ν. Lapse function α: Proper time / coord time. Shift vector β µ : How coords move. t = 2 t = 1 t = 0 Figure: C. Reisswig In coordinates: β µ = (0, β i ) n µ = α 1 (1, β i ) ds 2 = α 2 dt 2 + γ ij (dx i + β i dt)(dx j + β j dt) α,β,µ,ν,... go from 0 to 3. i,j,k,... go from 1 to 3. Mark A. Scheel Numerical Relativity July 7, / 34

11 Arnowitt-Deser-Misner (1962) 3+1 split Define extrinsic curvature: spatial projection of gradient of normal: K µν = γ λ µγ ρ ν (λ n ρ) In terms of 3-metric, lapse, shift: K ij = 1 2α ( tγ ij β j;i β i;j ) (β i;j is spatial covariant deriv, i.e. cov. deriv wrt g ij.) Mark A. Scheel Numerical Relativity July 7, / 34

12 Arnowitt-Deser-Misner (1962) equations Evolution equations t γ ij = 2αK ij + β j;i + β i;j [ ] t K ij = α ;ij + α (3) R ij + KK ij 2K im K m j 8πS ij 4π(ρ S)γ ij + β m K ij;m + 2β m (;ik j)m (3) R + K 2 K ij K ij = 16πρ Hamiltonian constraint K i j;i K ;j = 8πS j Momentum constraint 12 evolved variables: γ ij, K ij. 12 evolution equations 4 constraints 4 free variables: α, β i. Matter terms: ρ n µ n ν T µν S µ γ µ ν n λ T νλ S µν γ µ ρ γ ν λ T ρλ S γ µν T µν Mark A. Scheel Numerical Relativity July 7, / 34

13 Free evolution: Use evolution equations to evolve forward in time. Check that constraints are still satisfied. Solve constraints at t = 0. Figure: C. Reisswig Mark A. Scheel Numerical Relativity July 7, / 34

14 Just put equations on a computer and solve them? ADM equations developed in 1960s (ADM), 1970s (York). First attempt at binary BH simulation 1964 (Hahn, Lindquist) In summary, the numerical solution of the Einstein field equations presents no insurmountable difficulties. First successful binary BH simulation 2005 (Pretorius). Mark A. Scheel Numerical Relativity July 7, / 34

15 Just put equations on a computer and solve them? ADM equations developed in 1960s (ADM), 1970s (York). First attempt at binary BH simulation 1964 (Hahn, Lindquist) In summary, the numerical solution of the Einstein field equations presents no insurmountable difficulties. First successful binary BH simulation 2005 (Pretorius). Why is it so difficult? ADM equations are not well-posed. ADM evolution equations amplify small errors in constraints. Black holes have physical singularities inside. Coordinates don t mean anything. You need to choose them. Almost all choices are bad. Mark A. Scheel Numerical Relativity July 7, / 34

16 Well-posedness Hadamard 1902: A problem is well-posed if and only if: A solution exists. The solution is unique. The solution depends continuously on initial and boundary data. Mark A. Scheel Numerical Relativity July 7, / 34

17 Well-posedness Hadamard 1902: A problem is well-posed if and only if: A solution exists. The solution is unique. The solution depends continuously on initial and boundary data. Example 1 (Hadamard 1923): 2 t u 2 xu = 0, x [0, 1] Initial data: u = 0, t u = sin(2πnx) (2πn) P, P 1. Bdry conditions: u = 0 at x = 0, 1. Mark A. Scheel Numerical Relativity July 7, / 34

18 Well-posedness Hadamard 1902: A problem is well-posed if and only if: A solution exists. The solution is unique. The solution depends continuously on initial and boundary data. Example 1 (Hadamard 1923): 2 t u 2 xu = 0, x [0, 1] Initial data: u = 0, t u = sin(2πnx) (2πn) P, P 1. Bdry conditions: u = 0 at x = 0, 1. Solution: u(x, t) = sin(2πnx) sin(2πnt) (2πn) P +1. For n, initial data 0 and u(x, t) 0. = well-posed Mark A. Scheel Numerical Relativity July 7, / 34

19 Well-posedness Example 2: 2 t u+ 2 xu = 0, x [0, 1] only change is sign Initial data: u = 0, t u = sin(2πnx) (2πn) P, P 1. Bdry conditions: u = 0 at x = 0, 1. Mark A. Scheel Numerical Relativity July 7, / 34

20 Well-posedness Example 2: 2 t u+ 2 xu = 0, x [0, 1] only change is sign Initial data: u = 0, t u = sin(2πnx) (2πn) P, P 1. Bdry conditions: u = 0 at x = 0, 1. Solution: u(x, t) = sin(2πnx)sinh(2πnt) (2πn) P +1. For n, initial data 0 but u(x, t). = ill-posed In other words, small perturbation at t = 0 produces arbitrarily large solution at given finite time. Mark A. Scheel Numerical Relativity July 7, / 34

21 Hyperbolicity of PDEs Can write any PDE system as set of 1st order PDEs: t u + A i i u = B. u is a vector of variables, A i are matrices, B is a vector of RHS terms. A i and B can depend on u but not its derivatives. Mark A. Scheel Numerical Relativity July 7, / 34

22 Hyperbolicity of PDEs Can write any PDE system as set of 1st order PDEs: t u + A i i u = B. u is a vector of variables, A i are matrices, B is a vector of RHS terms. A i and B can depend on u but not its derivatives. Example: scalar wave equation in 1D: 2 t ψ 2 xψ = 0 Define Φ x ψ Π t ψ. then t ψ = Π, t Π + x Φ = 0, t Φ + x Π = 0, u = ψ Π Φ A x = B = Π 0 0 Mark A. Scheel Numerical Relativity July 7, / 34

23 Hyperbolicity of PDEs t u + A i i u = B. Pick any spatial unit vector n i. Then n i A i is the characteristic matrix in direction n i. This matrix can have eigenvalues and eigenvectors: e â (n i A i ) = v (â) e â (no sum on â). e â is the âth eigenvector. v (â) is the âth eigenvalue. Eigenvalues vâ are also called characteristic speeds. From eigenvectors can form uâ = u e â, which are called characteristic fields. Mark A. Scheel Numerical Relativity July 7, / 34

24 Hyperbolicity of PDEs t u + A i i u = B. System is Weakly hyperbolic if all eigenvalues of n i A i are real. Usually ill-posed. Well-posedness depends on details B. Strongly hyperbolic if it is weakly hyperbolic and there is a complete set of eigenvectors, independent of the solution and n i. Well-posed. Only characteristic fields corresponding to negative eigenvalues need bdry conditions. Mark A. Scheel Numerical Relativity July 7, / 34

25 Hyperbolicity of PDEs Example: scalar wave in 1d n x A x = Eigenvalues v (0) = 0, v (1) = 1, v ( 1) = Eigenvectors e 0 = 0, e 1 = 1, e 1 = 0 1 Char fields uˆ0 = ψ, u ±ˆ1 = Π ± Φ Strongly hyperbolic. t B.C. on Π + Φ B.C. on Π Φ Mark A. Scheel Numerical Relativity July 7, / 34 x

26 Hyperbolicity of PDEs ADM evolution equations are only weakly hyperbolic. Usually ill-posed. No indication of which variables require BCs. Fortunately, there are a few alternatives to ADM that work. We will discuss two: BSSN Generalized harmonic Mark A. Scheel Numerical Relativity July 7, / 34

27 BSSN Formulation Shibata/Nakamura 1995, Baumgarte/Shapiro 1999 Define new variables γ ij e 4φ γ ij Ā ij e 4φ K ij 1 3 γ ijk Γ i γ jk Γi jk Demand det γ ij = 1, ln γ = 12φ Demand Āi j = 0 Mark A. Scheel Numerical Relativity July 7, / 34

28 BSSN Formulation Shibata/Nakamura 1995, Baumgarte/Shapiro 1999 Define new variables γ ij e 4φ γ ij Ā ij e 4φ K ij 1 3 γ ijk Γ i γ jk Γi jk Demand det γ ij = 1, ln γ = 12φ Demand Āi j = 0 Then can write R ij = R φ ij + R ij, where R φ ij = terms with 2nd derivs of φ R ij = 1 2 γlm γ ij,lm + γ k(i Γk,j) + terms without derivs Now evolve γ ij, Āij, φ, K, and Γ i instead of γ ij, K ij. Mark A. Scheel Numerical Relativity July 7, / 34

29 BSSN Hyperbolicity BSSN is strongly hyperbolic for pre-chosen α, β i (Sarbach+ 02, Nagy+04). BSSN with dynamical α, β i strongly hyperbolic if shift is not too large (too close to 1) (Gundlach+ 06). Mark A. Scheel Numerical Relativity July 7, / 34

30 Gauge (coordinate) conditions for ADM/BSSN Einstein s Eqs. do not determine lapse α or shift β i. Lapse and shift equivalent to coordinate choice. How to choose coordinates? Simplest choice: α = 1, β i = 0. Bad. Why? In vacuum, t K = Āij Ā ij K2, so K at late times. Since t ln γ = K, if K, then γ 0. Coordinate singularity. Mark A. Scheel Numerical Relativity July 7, / 34

31 Gauge conditions: Maximal slicing Maximal slicing: K = t K = 0. Why is this a gauge choice? ADM: t K β i K,i = γ ij α ;ij + α [ K ij K ij + 4π(ρ + S) ] Figure: C. Ott. Mark A. Scheel Numerical Relativity July 7, / 34

32 Gauge conditions: Maximal slicing Maximal slicing: K = t K = 0. Why is this a gauge choice? ADM: t K β i K,i = γ ij α ;ij + α [ K ij K ij + 4π(ρ + S) ] K = t K = 0 = γ ij α ;ij = α [ K ij K ij + 4π(ρ + S) ] Figure: C. Ott. Mark A. Scheel Numerical Relativity July 7, / 34

33 Gauge conditions: Maximal slicing Maximal slicing: K = t K = 0. Why is this a gauge choice? ADM: t K β i K,i = γ ij α ;ij + α [ K ij K ij + 4π(ρ + S) ] K = t K = 0 = γ ij α ;ij = α [ K ij K ij + 4π(ρ + S) ] Properties: Singularity avoiding Collapse of the lapse Grid stetching Figure: C. Ott. Mark A. Scheel Numerical Relativity July 7, / 34

34 ADM/BSSN Gauge conditions: More lapse choices Harmonic slicing: t = α α t = 0 Reduces to t α β k α,k = α 2 K Choose initial lapse; this determines lapse at t > 0 Shift still unspecified. Mark A. Scheel Numerical Relativity July 7, / 34

35 ADM/BSSN Gauge conditions: More lapse choices Harmonic slicing: t = α α t = 0 Reduces to t α β k α,k = α 2 K Choose initial lapse; this determines lapse at t > 0 Shift still unspecified. In practice, put arbitrary function f(α) on RHS: t α β k α,k = α 2 f(α)k f(α) = 1 harmonic slicing f(α) = 0 geodesic slicing f(α) = maximal slicing f(α) = 2 α 1+log slicing Called 1+log because for β i = 0, t α = 2αK. So α = 1 + ln γ. 1+log slicing & variations used in most BSSN codes. Mark A. Scheel Numerical Relativity July 7, / 34

36 BSSN Gauge conditions: shift choices Demand t Γi = 0: 0 = t Γi = γli β k,kl + γ kj β i,kj + terms without 2nd derivs of shift Elliptic eq. for shift, similar to minimal distortion. Mark A. Scheel Numerical Relativity July 7, / 34

37 BSSN Gauge conditions: shift choices Demand t Γi = 0: 0 = t Γi = γli β k,kl + γ kj β i,kj + terms without 2nd derivs of shift Elliptic eq. for shift, similar to minimal distortion. Easier to use Gamma driver to drive Γ i to zero: t β i = 3 4 Bi t B i = t Γi 1 2M Bi Gamma driver & variations used in most BSSN codes. Mark A. Scheel Numerical Relativity July 7, / 34

38 Generalized Harmonic Formulation Different approach than ADM/BSSN Basic idea: Choose coordinates x µ satisfying x µ = 0 harmonic coordinates Mark A. Scheel Numerical Relativity July 7, / 34

39 Generalized Harmonic Formulation Different approach than ADM/BSSN Basic idea: Choose coordinates x µ satisfying x µ = H µ (x µ, g µν ) generalized harmonic coordinates You are free to choose H µ Mark A. Scheel Numerical Relativity July 7, / 34

40 Generalized Harmonic Formulation Different approach than ADM/BSSN Basic idea: Choose coordinates x µ satisfying x µ = H µ (x µ, g µν ) generalized harmonic coordinates You are free to choose H µ Then: 1 Einstein equations become wave equations for g µν : g λρ λ ρ g µν + 2 (µ H ν) + (ΓΓ terms) = (matter terms) Mark A. Scheel Numerical Relativity July 7, / 34

41 Generalized Harmonic Formulation Different approach than ADM/BSSN Basic idea: Choose coordinates x µ satisfying x µ = H µ (x µ, g µν ) generalized harmonic coordinates You are free to choose H µ Then: 1 Einstein equations become wave equations for g µν : g λρ λ ρ g µν + 2 (µ H ν) + (ΓΓ terms) = (matter terms) 2 Constraints C µ H µ x µ involve only 1st derivatives of g µν. Mark A. Scheel Numerical Relativity July 7, / 34

42 Generalized Harmonic Formulation Different approach than ADM/BSSN Basic idea: Choose coordinates x µ satisfying x µ = H µ (x µ, g µν ) generalized harmonic coordinates You are free to choose H µ Then: 1 Einstein equations become wave equations for g µν : g λρ λ ρ g µν + 2 (µ H ν) + (ΓΓ terms) = (matter terms) 2 Constraints C µ H µ x µ involve only 1st derivatives of g µν. Can add constraint damping (Gundlach+ 2005, Pretorius 2005). g λρ λ ρ g µν =... C µ =... Mark A. Scheel Numerical Relativity July 7, / 34

43 Generalized Harmonic Formulation Different approach than ADM/BSSN Basic idea: Choose coordinates x µ satisfying x µ = H µ (x µ, g µν ) generalized harmonic coordinates You are free to choose H µ Then: 1 Einstein equations become wave equations for g µν : g λρ λ ρ g µν + 2 (µ H ν) + (ΓΓ terms) = (matter terms) 2 Constraints C µ H µ x µ involve only 1st derivatives of g µν. Can add constraint damping (Gundlach+ 2005, Pretorius 2005). g λρ λ ρ g µν =... + γ 0 Q λ µνc λ C µ =... +???? Mark A. Scheel Numerical Relativity July 7, / 34

44 Generalized Harmonic Formulation Different approach than ADM/BSSN Basic idea: Choose coordinates x µ satisfying x µ = H µ (x µ, g µν ) generalized harmonic coordinates You are free to choose H µ Then: 1 Einstein equations become wave equations for g µν : g λρ λ ρ g µν + 2 (µ H ν) + (ΓΓ terms) = (matter terms) 2 Constraints C µ H µ x µ involve only 1st derivatives of g µν. Can add constraint damping (Gundlach+ 2005, Pretorius 2005). g λρ λ ρ g µν =... + γ 0 Q λ µνc λ C µ =... + γ 0 t C µ Mark A. Scheel Numerical Relativity July 7, / 34

45 Generalized Harmonic Formulation Different approach than ADM/BSSN Basic idea: Choose coordinates x µ satisfying x µ = H µ (x µ, g µν ) generalized harmonic coordinates You are free to choose H µ Then: 1 Einstein equations become wave equations for g µν : g λρ λ ρ g µν + 2 (µ H ν) + (ΓΓ terms) = (matter terms) 2 Constraints C µ H µ x µ involve only 1st derivatives of g µν. Can add constraint damping (Gundlach+ 2005, Pretorius 2005). g λρ λ ρ g µν =... + γ 0 Q λ µνc λ C µ =... + γ 0 t C µ 3 Strongly hyperbolic with char speeds on light cone, indep. of H µ. Mark A. Scheel Numerical Relativity July 7, / 34

46 Generalized Harmonic: specification of coordinates Different way of choosing coordinates than ADM/BSSN: ADM/BSSN: Choose α, β i GH: Choose H α Relation between two methods: t α β k α,k = α(h t β k H k + αk) t β i β k β i,k = αγ ij [ α(h j + γ kl Γ jkl ) α,j ] Choice of H α determines evolution of lapse and shift. Expect any ADM/BSSN gauge approachable by choice of H α. Mark A. Scheel Numerical Relativity July 7, / 34

47 Generalized Harmonic: How to choose H? Damped harmonic gauge works well for BHs: Spatial coords x i obey damped wave eqn. γ 1/2 /α driven towards unity. To accomplish this, ( ) H µ = η 0 ln γ 1/2 α n µ µ 0 1 α γ µkβ k η 0 and µ 0 are constants. Mark A. Scheel Numerical Relativity July 7, / 34

48 Generalized Harmonic: How to choose H? Damped harmonic gauge works well for BHs: Spatial coords x i obey damped wave eqn. γ 1/2 /α driven towards unity. To accomplish this, ( ) H µ = η 0 ln γ 1/2 α n µ µ 0 1 α γ µkβ k η 0 and µ 0 are constants. For even more damping, [ choose µ 0 = η 0 = ln γ1/2 α ] 2 Mark A. Scheel Numerical Relativity July 7, / 34

49 Handling singularities inside black holes Method 1: Excision Problem: black holes contain physical singularities. Horizon Here be Dragons Mark A. Scheel Numerical Relativity July 7, / 34

50 Handling singularities inside black holes Method 1: Excision Problem: black holes contain physical singularities. Horizon Excision: Solve equations only in the exterior region. Horizon Excision Boundary Here be Dragons Mark A. Scheel Numerical Relativity July 7, / 34

51 Handling singularities inside black holes Method 1: Excision Problem: black holes contain physical singularities. Horizon Excision: Solve equations only in the exterior region. Causality: interior does not affect exterior. If all characteristic speeds subluminal (like in generalized harmonic), no boundary condition needed. Horizon Excision Boundary Here be Dragons Mark A. Scheel Numerical Relativity July 7, / 34

52 Apparent horizons Event horizon (EH) Boundary of region where photons can escape to infinity. Nonlocal in time. Apparent horizon (AH) Smooth closed surface of zero null expansion. µ k µ = 0 Local in time. Time Space k s Theorem: if an AH exists, it cannot be outside an EH. grey=eh; red,green=ah Mark A. Scheel Numerical Relativity July 7, / 34

53 Handling singularities inside black holes Method 2: Moving punctures What are punctures? "other universe" Singularity Horizon Excision Initial Data Puncture Initial Data our universe Singularity Mark A. Scheel Numerical Relativity July 7, / 34

54 Handling singularities inside black holes Method 2: Moving punctures What are punctures? "other universe" Singularity Horizon Excision Initial Data Embedding Diagram A Singularity Puncture Initial Data our universe B C Numerical Grid B A C Mark A. Scheel Numerical Relativity July 7, / 34

55 Handling singularities inside black holes Method 2: Moving punctures What are punctures? "other universe" Singularity Horizon Excision Initial Data Embedding Diagram A Singularity Puncture Initial Data our universe B C Numerical Grid B A C Singularity moves through the grid during evolution! Key to success is carefully-chosen coordinate conditions. Mark A. Scheel Numerical Relativity July 7, / 34

56 Current status of numerical relativity About a dozen working codes in existence Form of Equations: BSSN Z4c Generalized Harmonic Singularity Treatment: Moving Punctures Excision Numerical Methods: Finite Differencing Spectral Mark A. Scheel Numerical Relativity July 7, / 34

57 Current status of numerical relativity About a dozen working codes in existence Most codes Form of Equations: BSSN Z4c Generalized Harmonic Singularity Treatment: Moving Punctures Excision Numerical Methods: Finite Differencing Spectral Mark A. Scheel Numerical Relativity July 7, / 34

58 Current status of numerical relativity About a dozen working codes in existence Most codes BAM Form of Equations: BSSN Z4c Generalized Harmonic Singularity Treatment: Moving Punctures Excision Numerical Methods: Finite Differencing Spectral Mark A. Scheel Numerical Relativity July 7, / 34

59 Current status of numerical relativity About a dozen working codes in existence Most codes BAM Form of Equations: BSSN Z4c Generalized Harmonic Singularity Treatment: Moving Punctures Excision Numerical Methods: Finite Differencing Spectral Princeton, AEI Harmonic Code Mark A. Scheel Numerical Relativity July 7, / 34

60 Current status of numerical relativity About a dozen working codes in existence Most codes BAM Form of Equations: BSSN Z4c Generalized Harmonic Singularity Treatment: Moving Punctures Excision Numerical Methods: Finite Differencing Spectral Princeton, AEI Harmonic Code SXS Collaboration (SpEC) Mark A. Scheel Numerical Relativity July 7, / 34

61 Current status of numerical relativity About a dozen working codes in existence Most codes BAM Form of Equations: BSSN Z4c Generalized Harmonic Singularity Treatment: Moving Punctures Excision Numerical Methods: Finite Differencing Spectral Princeton, AEI Harmonic Code SXS Collaboration (SpEC) Comparing different codes improves confidence in results. Mark A. Scheel Numerical Relativity July 7, / 34

62 Current status of numerical relativity BSSN and GH codes can run 3D GR simulations for Binary black holes Binary neutron stars BH/NS binaries Supernovae... and compute waveforms. Challenges: Simulate more orbits for binaries. Extreme parameters (e.g. large spins) still difficult. Better accuracy/resolution. Interpretation of results (coordinates are just coordinates!) Include more physics in NSNS/BHNS/CCSNe simulations. Neutrinos Magnetic fields Nuclear reaction rates Mark A. Scheel Numerical Relativity July 7, / 34

63 BBH example Large hole spin 0.91 Small hole spin 0.3 Color = Vorticity (a measure of spin) Mass ratio 6 Event Horizon: Andy Bohn Movie: Curran Muhlberger Mark A. Scheel Numerical Relativity July 7, / 34

64 Simulation catalogs S 2 S 1 Goal: cover 7D parameter space of BBHs. M2 M1 (179 simulations, SXS collaboration) Movie: Haroon Khan Several groups have catalogs. Mark A. Scheel Numerical Relativity July 7, / 34

65 We have not covered these important topics: Initial data (solving constraints) Boundary conditions Finding (apparent and event) horizons Extracting gravitational waves from simulations Measuring spins, masses, energy, angular momentum Hydrodynamics Numerical methods (see hands-on exercises) Mark A. Scheel Numerical Relativity July 7, / 34