A review of numerical relativity and black-hole collisions

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1 A review of numerical relativity and black-hole collisions U. Sperhake DAMTP, University of Cambridge Mons Meeting 2013: General Relativity and beyond 18 th July 2013 U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

2 Overview Introduction, motivation Foundations of numerical relativity Formulations of Einstein s eqs.: 3+1, BSSN, GHG, characteristic Initial data, Gauge, Boundaries Technical ingredients: Discretization, mesh refinement,... Applications and Results of NR Gravitational wave physics High-energy physics Appendix. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

3 1. Introduction, motivation U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

4 The Schwarzschild solution Einstein 1915 General relativity: geometric theory of gravity Schwarzschild 1916 ds 2 = ( 1 2M ) r dt 2 + ( 1 2M r ) 1 dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ) Singularities: r = 0: physical r = 2M: coordinate Newtonian escape velocity v = 2M r. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

5 Evidence for astrophysical black holes X-ray binaries e. g. Cygnus X-1 (1964) MS star + compact star Stellar Mass BHs M Stellar dynamics near galactic centers, iron emission line profiles Supermassive BHs M AGN engines. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

6 Conjectured BHs Intermediate mass BHs M Primordial BHs M Earth Mini BHs, LHC TeV Note: BH solution is scale invariant!. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

7 Research areas: Black holes have come a long way! Astrophysics Gauge-gravity duality Fundamental studies GW physics High-energy physics Fluid analogies. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

8 General Relativity: Curvature Curvature generates acceleration geodesic deviation No force!! Description of geometry Metric Connection Riemann Tensor g αβ Γ α βγ R α βγδ. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

9 The metric defines everything Christoffel connection Γ α βγ = 1 2 gαµ ( β g γµ + γ g µβ µ g βγ ) Covariant derivative α T β γ = α T β γ + Γ β µαt µ γ Γ µ γαt β µ Riemann Tensor R α βγδ = γ Γ α βδ δγ α βγ + Γα µγγ µ βδ Γα µδ Γµ βγ Geodesic deviation, Parallel transport,.... Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

10 How to get the metric? Train cemetery Uyuni, Bolivia Solve for the metric g αβ U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

11 How to get the metric? The metric must obey the Einstein Equations Ricci-Tensor, Einstein Tensor, Matter Tensor R αβ R µ αµβ G αβ R αβ 1 2 g αβr µ µ T αβ Matter Einstein Equations Solutions: Easy! Trace reversed Ricci G αβ = 8πT αβ Take metric Calculate G αβ Use that as matter tensor Physically meaningful solutions: Difficult! U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

12 Solving Einstein s equations: Different methods Analytic solutions Symmetry assumptions Schwarzschild, Kerr, FLRW, Myers-Perry, Emparan-Reall,... Perturbation theory Assume solution is close to known solution g αβ Expand ĝ αβ = g αβ + ɛh (1) αβ + ɛ2 h (2) αβ +... linear system Regge-Wheeler-Zerilli-Moncrief, Teukolsky, QNMs, EOB,... Post-Newtonian Theory Assume small velocities expansion in v c N th order expressions for GWs, momenta, orbits,... Blanchet, Buonanno, Damour, Kidder, Will,... Numerical Relativity. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

13 2. Foundations of numerical relativity U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

14 A list of tasks Target: Predict time evolution of BBH in GR Einstein equations: 1) Cast as evolution system 2) Choose specific formulation 3) Discretize for computer Choose coordinate conditions: Gauge Fix technical aspects: 1) Mesh refinement / spectral domains Construct realistic initial data 2) Singularity handling / excision 3) Parallelization Start evolution and waaaaiiiiit... Extract physics from the data U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

15 2.1 Formulations of Einstein s equations U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

16 The Einstein equations R µν 1 2 Rg µν + Λg µν = 8πT µν ( ) R µν = 8π T µν 1 D 2 Tg µν + 2 D 2 Λg µν In this form no well-defined mathematical character hyperbolic, elliptic, parabolic? Coordinate x α on equal footing; time only through signature of g αβ Well-posedness of the equations? Suitable for numerics? Several ways to identify character and coordinates Formulations U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

17 2.1.1 ADM like D formulations U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

18 3+1 Decomposition NR: ADM 3+1 split Arnowitt, Deser & Misner 62 York 79, Choquet-Bruhat & York 80 Spacetime = Manifold (M, g) Hypersurfaces Scalar field t : M R such that t = const defines Σ t 1 form dt, vector t dt, t = 1 Def.: Timelike unit vector: n µ α(dt) µ Lapse: α = 1/ dt Shift: β µ = ( t ) µ αn µ Adapted coordinate basis: t = αn + β, i = i x. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

19 3+1 Decomposition Def.: A vector v α is tangent to Σ t : dt, v = (dt) µ v µ = 0 Projector: α µ = δ α µ + n α n µ For a vector tangent to Σ t one easily shows n µ v µ = 0 µ αv µ = v α Projection of the metric γ αβ := µ α ν βg µν = g αβ + n α n β γ αβ = αβ For v α tangent to Σ t : g µν v µ v ν = γ µν v µ v ν Adapted coordinates: x α = (t, x i ) we can ignore t components for tensors tangential to Σ t γ ij is the metric on Σ t First fundamental form. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

20 3+1 decomposition of the metric In adapted coordinates, we write the spacetime metric ( α g αβ = 2 + β m β m ) β j ( α g αβ 2 α 2 β j ) = α 2 β i γ ij α 2 β i β j β i ds 2 = α 2 dt 2 + γ ij (dx i + β i dt)(dx j + β j dt) Gauge variables: Lapse α, Shift vector β i For any tensor tangent in all components to Σ t we raise and lower indices with γ ij : S ij k = γ jm S i mk etc. γ ij U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

21 Projections and spatial covariant derivative ( p For an arbitrary tensor S of type q ), its projection is ( S) α 1...α p β1...β q = α 1 µ1... αp µ p ν 1 β1... νq β q S µ 1...µ p ν1...ν q Project every free index For a tensor S on Σ t, its covariant derivative is DS := ( S) D ρ S α 1...α p β1...β q = α 1 µ1... αp µ p ν 1 β1... νq β q σ ρ σ S µ 1...µ p ν1...ν q One can show that D = is torsion free on Σ t if is on M ( γ) ijk = 0 metric compatible is unique in satisfying these properties. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

22 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 = 2K αβ K αβ = 1 2 L nγ αβ Two interpretations of K αβ embedding of Σ t in M U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

23 The projections of the Riemann tensor µ α ν β γ ρ σ δ R ρ σµν = R γ δαβ + K γ αk δβ K γ βk δα Gauss Eq. µ α ν β R µν + µα ν βn ρ n σ R µ ρνσ = R αβ + KK αβ K µ βk αµ contracted R + 2 R µν n µ n ν = R + K 2 K µν K µν scalar Gauss eq. γ ρn σ µ α ν β R ρ σµν = D β K γ α D α K γ β Codazzi eq. n σ ν β R σν = D β K D µ K µ β contracted αµ ν βn σ n ρ R µ ρνσ = 1 α L mk αβ + K αµ K µ β + 1 α D αd β α µ α ν β R µν = 1 α L mk αβ 2K αµ K µ β 1 α D αd β α + R αβ + KK αβ R = 2 α L mk 2 α γµν D µ D ν α + R + K 2 + K µν K µν Here L is the Lie derivative and m µ = αn µ = ( t ) µ + β µ Summation of spatial tensors: ignore time indices; µ, ν,... m, n,... U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

24 Decomposition of the Einstein equations R αβ 1 2 Rg αβ + Λg αβ = 8πT αβ ( ) R αβ = 8π T αβ 1 D 2 g αβt + 2 D 2 Λg αβ Energy momentum tensor ρ = T µν n µ n ν j α = T µν n µ ν α energy density momentum density S αβ = µ α ν βt µν, S = γ µν S µν stress tensor T αβ = S αβ + n α j β + n β j α + ρn 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) A review of numerical relativity and black-hole collisions 07/18/ / 159

25 Decomposition of the Einstein equations Definition: L m γ ij = 2αK ij µ α ν β projection: L m K ij = D i D j α+α(r ij +KK ij 2K im K m j)+8πα Evolution equations [ S ρ D 2 γ ij S ij ] 2 D 2 Λγ ij n µ n ν projection R + K 2 K mn K mn = 2Λ + 16πρ Hamiltonian constraint µ αn ν projection D i K D m K m i = 8πj i Momentum constraint U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

26 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) ; Reula, gr-qc/ Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

27 The BSSN system Goal: modify ADM to get a strongly hyperbolic system Baumgarte & Shapiro, PRD 59 (1998) , Shibata & Nakamura, PRD 52 (1995) 5428 Conformal decomposition, trace split, auxiliary variable φ = 1 4(D 1) ln γ, K = γij K ij γ ij = e 4φ γ ij = e 4φ γ ij ) à ij = e (K 4φ ij 1 D 1 γ ijk Γ i = γ mn Γ i mn Auxiliary constraints γ = det γ ij = 1, γ mn à mn = 0 ) K ij = e (Ãij 4φ + 1 D 1 γ ijk U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

28 The BSSN equations t φ = β m m φ 1 2(D 1) ( mβ m αk ) t γ = β m m γ ij + 2 γ m(i i) β m 2 D 1 γ ij m β m 2αÃij t K = β m m K e 4φ γ mn D m D n α + αãmn à mn + 1 D 1 αk 2 + 8π 2 D 2α[S + (D 3)ρ] D 2 αλ t à ij = β m m à ij + 2Ãm(i i) β m 2 D 1Ãij m β m + αk Ãij 2αÃimÃm j +e 4φ ( αr ij D i D j α 8παS ij ) TF t Γ i = β m m Γ i + 2 D 1 Γ i m β m + γ mn m n β i + D 3 D 1 γim m n β n +2Ãim [2(D 1)α m φ m α]+2α Γ i mnãmn 2 D 2 D 1 α γim m K 16παj i Note: There are alternative versions using χ = e 4φ or W = e 2φ U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

29 The BSSN equations In the BSSN equations we use Γ i jk = Γ i jk + 2(δi k j φ + δ i j k φ γ jk γ im m φ) R ij = R ij + R φ ij R φ ij = 2(3 D) D i Dj φ 2 γ ij γ mn Dm Dn φ+4(d 3)( i φ j φ γ ij γ mn m φ n φ) R ij = 1 2 γmn m n γ ij + γ m(i j) Γm + Γ m Γ(ij)m + γ mn [2 Γ k m(i Γ j)kn + Γ k im Γ kjn ] D i D j α = D i Dj α 2( i φ j α + j φ i α) + 2 γ ij γ mn m φ n α The constraints are H = R + D 2 D 1 K 2 Ãmn à mn 16πρ 2Λ = 0 M i = D m à m i D 2 D 1 ik + 2(D 1)Ãm i m φ 8πj i = 0 U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

30 2.1.2 Generalized Harmonic formulation U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

31 The Generalized Harmonic (GH) formulation Appendix. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

32 2.1.3 Characteristic formulation U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

33 The characteristic formulation Appendix. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

34 Direct methods Use symmetry to write line element, e.g. ds 2 = a 2 (µ, t)dt 2 + b 2 (µ, t)dµ 2 R 2 (µ, t)dω 2 May & White, PR 141 (1966) 1232 Energy momentum tensor T 0 0 = ρ(1 + ɛ), T 1 1 = T 2 2 = T 3 3 = 0 Lagrangian coords. GRTENSOR, MATHEMATICA,... Field equations: a =... b =... R =... U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

35 Numerical relativity in D > 4 dimensions Needed for many applications: TeV gravity, AdS/CFT, BH stability Reduction to a 3+1 problem Diagnostics: Wave extraction, horizons Talk H.Witek U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

36 Further reading 3+1 formalism Gourgoulhon, gr-qc/ Characteristic formalism Winicour, Liv. Rev. Rel 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) A review of numerical relativity and black-hole collisions 07/18/ / 159

37 2.2. Initial data, Gauge, Boundaries U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

38 Initial data U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

39 Analytic initial data Schwarzschild, Kerr, Tangherlini, Myers Perry,... e.g. Schwarzschild in isotropic coordinates: ds 2 = M 2r M+2r dt2 + ( 1 + M 2r ) [r 2 + r 2 (dθ 2 + sin 2 θdφ 2 )] 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) A review of numerical relativity and black-hole collisions 07/18/ / 159

40 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 γ ijk A ij = ψ 10 Ā ij A ij = ψ 2 Ā ij U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

41 Bowen-York data By further splitting Āij 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 Ā ij = 3 [ Pi n 2r 2 j + P j n i (f ij n i n j )P k ] n k + 3 ( ɛkil S l n k n r 3 j + ɛ kjl S l n k ) n i where r is a coordinate radius and n i = x i r Bowen & York, PRD 21 (1980) 2047 U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

42 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 8π ( K j i δ j ik ) ξ i (a) d 2 A j... 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) U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

43 Puncture data Brandt & Brügmann, PRL 78 (1997) 3606 The Hamiltonian constraint is now given by 2 ψ ψ 7 Ā mn Ā mn = 0 Ansatz for conformal factor: ψ = ψ BL + u, where ψ BL = N m i i=1 is the Brill-Lindquist conformal factor, 2 r r i i.e. the solution for Āij = 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) U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

44 Properties of the puncture solutions m i and r i are bare mass and position of the i th BH. In the limit of vanishing Bowen York parameters P i = S i = 0, the puncture solution reduces to Brill Lindquist data ( γ ij dx i dx j = 1 + ) 4 i (dx 2 + dy 2 + dz 2 ) m i 2 r r i The numerical solution of the Hamiltonian constraint generalizes rather straightforwardly to higher D Yoshino, Shiromizu & Shibata, PRD 74 (2006) Zilhão et al, PRD 84 (2011) Punctures generalize to asymptotically de-sitter BHs Zilhão et al, PRD 85 (2012) using McVittie coordinates McVittie, MNRAS 93 (1933) 325 U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

45 Gauge U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

46 The gauge freedom Remember: Einstein equations say nothing about α, β i Any choice of lapse and shift gives a solution This represents the coordinate freedom of GR Physics do not depend on α, So why bother? β i The performance of the numerics DO depend strongly on the gauge! U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

47 What goes wrong with bad gauge? U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

48 What goes wrong with bad gauge? U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

49 What goes wrong with bad gauge? U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

50 What goes wrong with bad gauge? U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

51 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) , Alcubierre, CQG 20 (2003) 607, Garfinkle, PRD 65 (2001) Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

52 Moving puncture gauge Gauge was a key ingredient in the Moving puncture breakthroughs Campanelli et al, PRL 96 (2006) Baker et al, PRL 96 (2006) Variant of 1 + log slicing and Γ-driver shift Alcubierre et al, PRD 67 (2003) Now in use as t α = β m m α 2αK and t β i = β m m β i Bi t B i = β m m B i + t Γ i β m m Γ i ηb i or t β i = β m m β i Γ i ηβ i. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

53 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) , Schnetter, CQG 27 (2010) , Müller et al, PRD 82 (2010) Modifications in higher D: Dimensional reduction Zilhão et al, PRD 81 (2010) t α = β m m α 2α(η K K + η Kζ K ζ ) CARTOON Yoshino & Shibata, PTPS 189 (2011) 269 t β i = D 1 2(D 2) v 2 long Bi t B i = t Γi ηb i Here η K, η Kζ, v long are parameters U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

54 Boundaries U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

55 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. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

56 Moving puncture slices: Schwarzschild Wormhole Trumpet slice = stationary 1+log slice Hannam et al, PRL 99 (2007) , PRD 78 (2008) Brown, PRD 77 (2008) , CQG 25 (2008) Gauge might propagate at > c, no pathologies Natural excision Brown, PRD 80 (2009) U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

57 Outer boundary Appendix. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

58 Further reading Initial data construction Cook, Liv. Rev. Rel. 3 (2000) 5 Pfeiffer, gr-qc/ U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

59 2.3 Discretization of the equations U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

60 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) A review of numerical relativity and black-hole collisions 07/18/ / 159

61 Derivatives and finite derivatives Goal: represent m f x m in terms of f n,i Fix index n; Taylor expansion: f i 1 = f i f i dx f i dx 2 + O(dx 3 ) f i = f i f i+1 = f i + f i dx f i dx 2 + O(dx 3 ) 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 0 = A + B + C, 1 = ( A + B)dx, 0 = 1 2 Adx Cdx 2 A = 1 2dx, B = 0, C = 1 2dx f i = f i+1 f i 1 2dx + O(dx 2 ) Higher order accuracy more points; works same in time U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

62 Mesh refinement 3 Length scales : BH 1 M Critical phenomena Choptuik 93 First used for BBHs Brügmann 96 Wavelength Wave zone Available Packages: Paramesh MacNeice et al. 00 Carpet Schnetter et al. 03 SAMRAI MacNeice et al M M. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

63 Berger-Oliger mesh refinement Goal: Update from t to t + dt Refinement criteria: numerical error, curvature,... Here for dimensions U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

64 Berger-Oliger mesh refinement Goal: Update from t to t + dt Refinement criteria: numerical error, curvature,... Here for dimensions U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

65 Berger-Oliger mesh refinement Goal: Update from t to t + dt Refinement criteria: numerical error, curvature,... Here for dimensions U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

66 Berger-Oliger mesh refinement Goal: Update from t to t + dt Refinement criteria: numerical error, curvature,... Here for dimensions U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

67 Berger-Oliger mesh refinement Goal: Update from t to t + dt Refinement criteria: numerical error, curvature,... Here for dimensions U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

68 Berger-Oliger mesh refinement Goal: Update from t to t + dt Refinement criteria: numerical error, curvature,... Here for dimensions U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

69 Berger-Oliger mesh refinement Goal: Update from t to t + dt Refinement criteria: numerical error, curvature,... Here for dimensions U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

70 Alternative discretization schemes Spectral methods: high accuracy, efficiency, complexity Caltech-Cornell-CITA code SpEC Applications to moving punctures still in construction e.g. Tichy, PRD 80 (2009) Also used in symmetric asymptotically AdS spacetimes e.g. Chesler & Yaffe, PRL 106 (2011) Finite Volume methods Finite Element methods D. N. Arnold, A. Mukherjee & L. Pouly, gr-qc/ C. F. Sopuerta, P. Sun & J. Xu, CQG 23 (2006) 251 C. F. Sopuerta & P. Laguna, PRD 73 (2006) U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

71 Further reading Numerical methods Press et al, Numerical Recipes, Cambridge University Press. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

72 3 Results from BH evolutions U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

73 3.1 BHs in GW physics U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

74 Gravitational waves Weak field limit: g αβ = η αβ + h αβ Trace reversed perturbation h αβ = h αβ 1 2 hη αβ Vacuum field eqs.: h αβ = 0 Apropriate gauge h αβ = h + h 0 0 h h where k σ = null vector GWs displace particles eikσxσ. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

75 Gravitational wave detectors Accelerated masses GWs Weak interaction! Laser interferometric detectors. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

76 The gravitational wave spectrum U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

77 Some targets of GW physics Confirmation of GR Hulse & Taylor 1993 Nobel Prize Parameter determination of BHs: M, S Optical counter parts Standard sirens (candles) Mass of graviton Test Kerr Nature of BHs Cosmological sources Neutron stars: EOS. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

78 Free parameters of BH binaries Total mass M Relevant for GW detection: Frequencies scale with M Not relevant for source modeling: trivial rescaling Mass ratio q M 1 M 2, η M 1M 2 (M 1 +M 2 ) 2 Spin: S 1, S 2 (6 parameters) Initial parameters Binding energy E b Orbital ang. momentum L Alternatively: frequency, eccentricity Separation Eccentricity U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

79 BBH trajectory and waveform q = 4, non-spinning binary; 11 orbits US, Brügmann, Müller & Sopuerta 11 Trajectory Quadrupole mode U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

80 Morphology of a BBH inspiral Thanks to Caltech, Cornell, CITA U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

81 Matched filtering BH binaries have 7 parameters: 1 mass ratio, 2 3 for spins Sample parameter space, generate waveform for each point NR + PN Effective one body Ninja, NRAR Projects GEO 600 noise chirp signal U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

82 Template construction Stitch together PN and NR waveforms EOB or phenomenological templates for 7-dim. par. space. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

83 Template construction Phenomenological waveform models Model phase, amplitude with simple functions Model parameters Create map between physical and model parameters Time or frequency domain Ajith et al, CQG 24 (2007) S689, PRD 77 (2008) , CQG 25 (2008) , PRL 106 (2011) ; Santamaria et al, PRD 82 (2010) , Sturani et al, arxiv: [gr-qc] Effective-one-body (EOB) models Particle in effective metric, PN, ringdown model Buonanno & Damour PRD 59 (1999) , PRD 62 (2000) Resum PN, calibrate pseudo PN parameters using NR Buonanno et al, PRD 77 (2008) , Pan et al, PRD 81 (2010) , PRD 84 (2012) ; Damour et al, PRD 77 (2008) , PRD 78 (2008) , PRD 83 (2011) U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

84 Going beyond GR: Scalar-tensor theory of gravity Brans-Dicke theory: 1 parameter ω BD ; well constrained Bergmann-Wagoner theories: Generalize ω = ω(φ) No-hair theorem: BHs solutions same as in GR e.g. Hawking, Comm.Math.Phys. 25 (1972) 167 Sotiriou & Faraoni, PRL 108 (2012) Circumvent no-hair theorem: Scalar bubble Healey et al, arxiv: [gr-qc] Circumvent no-hair theorem: Scalar gradient Berti et al, arxiv: [gr-qc]. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

85 3.2. High-energy collisions of BHs U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

86 The Hierarchy Problem of Physics Gravity other forces Higgs field µ obs 250 GeV = µ 2 Λ 2 where Λ GeV is the grand unification energy Requires enormous finetuning!!! Finetuning exist: = Or E Planck much lower? Gravity strong at small r? BH formation in high-energy collisions at LHC Gravity not measured below 0.16 mm! Diluted due to... Large extra dimensions Arkani-Hamed, Dimopoulos & Dvali 98 Extra dimension with warp factor Randall & Sundrum 99 U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

87 Stages of BH formation Matter does not matter at energies well above the Planck scale Model particle collisions by black-hole collisions Banks & Fischler, gr-qc/ ; Giddings & Thomas, PRD 65 (2002) U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

88 Does matter matter? Hoop conjecture kinetic energy triggers BH formation Einstein plus minimally coupled, massive, complex scalar filed Boson stars Pretorius & Choptuik, PRL 104 (2010) γ = 1 γ = 4 BH formation threshold: γ thr = 2.9 ± 10 % 1/3 γ hoop Model particle collisions by BH collisions U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

89 Does matter matter? Perfect fluid stars model γ = ; BH formation below Hoop prediction East & Pretorius, PRL 110 (2013) Gravitational focussing Formation of individual horizons Type-I critical behaviour Extrapolation by 60 orders would imply no BH formation at LHC Rezzolla & Tanaki, CQG 30 (2013) U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

90 Experimental signature at the LHC Black hole formation at the LHC could be detected by the properties of the jets resulting from Hawking radiation. BlackMax, Charybdis Multiplicity of partons: Number of jets and leptons Large transverse energy Black-hole mass and spin are important for this! ToDo: Exact cross section for BH formation Determine loss of energy in gravitational waves Determine spin of merged black hole. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

91 D = 4: Initial setup: 1) Aligned spins Orbital hang-up Campanelli et al, PRD 74 (2006) BHs: Total rest mass: M 0 = M A, 0 + M B, 0 Boost: γ = 1/ 1 v 2, M = γm 0 Impact parameter: b L P U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

92 D = 4: Initial setup: 2) No spins Orbital hang-up Campanelli et al, PRD 74 (2006) BHs: Total rest mass: M 0 = M A, 0 + M B, 0 Boost: γ = 1/ 1 v 2, M = γm 0 Impact parameter: b L P U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

93 D = 4: Initial setup: 3) Anti-aligned spins Orbital hang-up Campanelli et al, PRD 74 (2006) BHs: Total rest mass: M 0 = M A, 0 + M B, 0 Boost: γ = 1/ 1 v 2, M = γm 0 Impact parameter: b L P U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

94 D = 4: Head-on: b = 0, S = 0 Total radiated energy: 14 ± 3 % for v 1 US et al, PRL 101 (2008) About half of Penrose 74 Agreement with approximative methods Flat spectrum, GW multipoles Berti et al, PRD 83 (2011) U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

95 D = 4: Grazing: b 0, S = 0, γ = 1.52 Radiated energy up to at least 35 % M Immediate vs. Delayed vs. No merger US et al, PRL 103 (2009) U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

96 D = 4: Scattering threshold b scat for S = 0 b < b scat Merger b > b scat Scattering Numerical study: b scat = 2.5±0.05 v M Shibata et al, PRD 78 (2008) (R) Independent study US et al, PRL 103 (2009) , arxiv: γ = : χ = 0.6, 0, +0.6 (anti-aligned, nonspinning, aligned) Limit from Penrose construction: b crit = M Yoshino & Rychkov, PRD 74 (2006) U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

97 D = 4: Scattering threshold and radiated energy S 0 US et al, arxiv: At speeds v 0.9 spin effects washed out E rad always below 50 % M U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

98 D = 4: Absorption For large γ: E kin M If E kin is not radiated, where does it go? Answer: 50 % into E rad, 50 % is absorbed US et al, arxiv: U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

99 3.3 Fundamental properties of BHs U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

100 Stability of AdS m = 0 scalar field in as. flat spacetimes Choptuik, PRL 70 (1993) 9 p > p BH, p < p flat m = 0 scalar field in as. AdS Bizoń & Rostworowski, PRL 107 (2011) Similar behaviour for Geons Dias, Horowitz & Santos 11 D > 4 dimensions Jałmużna et al, PRD 84 (2011) D = 3: Mass gap: smooth solutions Bizoń & Jałmużna, arxiv: Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

101 Stability of AdS Pulses narrow under successive reflections Buchel et al, PRD 86 (2012) Non-linearly stable solutions in AdS Dias et al, CQG 29 (2012) , Buchel et al, arxiv: , Maliborski & Rostworowski arxiv: Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

102 Bar mode instability of Myers-Perry BH MP BHs (with single ang.mom.) should be unstable. Linearized analysis Dias et al, PRD 80 (2009) (R) U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

103 Non-linear analysis of MP instability Shibata & Yoshino, PRD 81 (2010) Myers-Perry metric; transformed to Puncture like coordinate Add small bar-mode perturbation Deformation η := 2 (l 0 l π/2 ) 2 +(l π/4 l 3π/4 ) 2 l 0 +l π/2 U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

104 Cosmic Censorship in D = 5 Pretorius & Lehner, PRL 105 (2010) Axisymmetric code Evolution of black string... Gregory-Laflamme instability cascades down in finite time until string has zero width naked singularity. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

105 Cosmic Censorship in D = 4 de Sitter Zilhão et al, PRD 85 (2012) Two parameters: MH, d Initial data: McVittie type binaries McVittie, MNRAS 93 (1933) 325 Small BHs : d < d crit merger d > d crit no common AH Large holes at small d: Cosmic Censorship holds U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

106 Further reading Reviews about numerical relativity Centrella et al, Rev. Mod. Phys. 82 (2010) 3069 Pretorius, arxiv: Sperhake et al, arxiv: Pfeiffer, CQG 29 (2012) Hannam, CQG 26 (2009) Sperhake, IJMPD 22 (2013) U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

107 Appendix U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

108 A.1 Generalized Harmonic formulation U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

109 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 gµν 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 Source functions H α = µ µ x α = g µν Γ α µν U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

110 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: 1 2 gµν µ ν g αβ = ν g µ(α β) g µν (α H β) + H µ Γ µ αβ ( ) Γ µ ναγ ν µβ 2 D 2 Λg αβ 8π T µν 1 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) , Pretorius, CQG 22 (2005) 425 U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

111 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 [8π µ T µα 1 D 2 Tg µα + 2 D 2 Λg µα In practice: numerical violations of C µ = 0 unstable modes Solution: add constraint damping 1 2 gµν µ ν g αβ = ν g µ(α β) g µν (α H β) + H µ Γ µ αβ Γµ ναγ ν µβ ( ) 2 D 2 Λg αβ 8π T µν 1 D 2 Tg αβ κ [ 2n (α C β) λg αβ n µ ] C µ Gundlach et al, CQG 22 (2005) 3767 E.g. Pretorius, PRL 95 (2005) : κ = 1.25/m, λ = 1 U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

112 Gauge conditions in the GH formulation How to choose H µ? some experimentation... Pretorius breakthrough α 1 H t = ξ 1 α + ξ η 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) , with Scheel, PRD 80 (2009) The H α are related to lapse and shift: n µ H µ = K n µ µ ln α γ µi H µ = γ mn Γ i mn + γ im m (ln α) + 1 α nµ µ β i U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

113 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) A review of numerical relativity and black-hole collisions 07/18/ / 159

114 A.2 Characteristic formulation U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

115 Characteristic coordinates Consider advection equation t f + a x f = 0 Characteristics: curves C : x at + x 0 df dt C = f t + f dx x dt C = f t + a f x dx dt = a = 0 f constant along C U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

116 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) A review of numerical relativity and black-hole collisions 07/18/ / 159

117 Characteristic Bondi-Sachs formulation Write metric as ds 2 = V e2β r du 2 2e 2β dudr + r 2 h AB (dx A U A du)(dx B U B du) 2h AB dx A dx B = (e 2γ + e 2δ )dθ sin θ sinh(γ δ)dθdφ + sin 2 θ(e 2γ + e 2δ )dφ 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) A review of numerical relativity and black-hole collisions 07/18/ / 159

118 Integration of the characteristic equations Provide initial data for γ, δ on hypersurface u = const Integrate hypersurface eqs. along r β, V, U A at u 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.. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

119 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: [gr-qc] Application to characteristic GW extraction Babiuc, Winicour & Zlochower, CQG 28 (2011) Reisswig et al, CQG 27 (2010) U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

120 A.3 Boundaries U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

121 Outer boundary: Asymptotically flat case Computational domains often don t extend to Outgoing Sommerfeld conditions Assume f = f 0 + u(t r) r n t u + r u = 0 t f + n f f 0 r + x i r if = 0 where f 0 = asymptotic value Use upwinding, i.e. one-sided, derivatives! U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

122 Non-asymptotically flat case: de Sitter In McVittie coordinates: r ds 2 = dt 2 + a(t) 2 (r 2 + r 2 dω 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 t f t f a(t) r f + n f f 0 r a(t) H(f f 0) = 0 Zilhão et al, PRD 85 (2012) U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

123 Anti de Sitter Much more complicated! Time-like outer boundary affects interior AdS metric diverges at outer boundary U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

124 Anti de Sitter metric Maximally symmetric solution to Einstein eqs. with Λ < 0 Hyperboloid X X 2 D D 1 i=1 X 2 i embedded in D + 1 dimensional flat spacetime of signature Global AdS X 0 = L cos τ cos ρ, X d = L sin τ cos ρ X i = L tan ρ Ω i, for i = 1... D 1, Ω i hyperspherical coords. ds 2 = L2 cos 2 ρ ( dτ 2 + dρ 2 + sin 2 ρ dω 2 D 2 ) where 0 ρ < π/2, Outer boundary at ρ = π/2 π < τ π. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

125 Anti de Sitter metric continued Poincaré coordinates X 0 = 1 2z [ z 2 + L 2 + D 2 i=1 (x i ) 2 t 2 ] X i = Lx i z for i = 1... D 2 X D 1 = 1 2z X d = Lt z [ z 2 L 2 + D 2 i=1 (x i ) 2 t 2 ] ds 2 = L2 z 2 [ dt 2 + dz 2 + D 2 i=1 (dx i ) 2 ] where z > 0, t R Outer boundary at z = 0 e.g. Ballón Bayona & Braga, hep-th/ U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

126 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 Global: ds 2 gl dτ 2 + dω D 2 Poincaré: ds 2 P dt2 + D 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) A review of numerical relativity and black-hole collisions 07/18/ / 159

127 Regularization methods Decompose metric into AdS part plus deviation Bantilan & Pretorius, PRD 85 (2012) Factor out appropriate factors of the bulk coordinate Chesler & Yaffe, PRL 106 (2011) Heller, Janik & Witaszczyk, PRD 85 (2012) Factor out singular term of the metric Bizoń & Rostworowski, PRL 107 (2011) Regularity of the outer boundary may constrain the gauge freedom Bantilan & Pretorius, PRD 85 (2012) Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

128 A.4 Diagnostics U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

129 The subtleties of diagnostics in GR Successful numerical simulation Numbers for grid functions Typically: Spacetime metric g αβ and time derivative or ADM variables γ ij, K ij, α, β i Challenges Coordinate dependence of numbers Gauge invariants Global quantities at, computational domain finite Extrapolation Complexity of variables, e.g. GWs Spherical harmonics Local quantities: meaningful? Horizons AdS/CFT correspondence: Dictionary U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

130 Newton s gravitational constant Note: We wrote the Einstein equations for Λ = 0 as R αβ 1 2 g αβr = 8πGT αβ The (areal) horizon radius of a static BH in D dimensions then is r D 3 s = 16πGM (D 2)Ω D 2, where Ω D 2 = D 1 2π 2 Γ( D 1 2 ) is the area of the D 2 hypersphere The Hawking entropy formula is S = A AH 4G But Newton s force law picks up geometrical factors: F = (D 3)8πG Mm (D 2)Ω D 2 r D 2 ˆr See e.g. Emparan & Reall, Liv. Rev. Rel. 6 (2008) U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

131 Global quantities Assumptions Asymptotically, the metric is flat and time independent The expressions also refer to Cartesian coordinates ADM mass = Total mass-energy of spacetime 1 M ADM = 4Ω D 2 G lim r S r γγ mn γ kl ( n γ mk k γ mn )ds l Linear momentum of spacetime P i = 1 8πG lim r S r γ(k m i δ m ik )ds m Angular momentum in D = 4 J i = 1 8π ɛ il m lim r S r γx l (K n m δ n mk )ds n By construction, these are time independent! U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

132 Apparent horizons By Cosmic censorship, existence of an apparent horizon implies an event horizon Consider outgoing null geodesics with tangent vector k µ Def.: Expansion Θ = µ k µ Def.: Apparent horizon = outermost surface where Θ = 0 On a hypersurface Σ t, the condition for Θ = 0 becomes ˆD m s m K + K mn s m s n, where s i = unit normal to the (D 2) dimensional AH surface e.g. Thornburg, PRD 54 (1996) 4899 U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

133 Apparent horizons continued Parametrize the horizon by r = f (ϕ i ), where r is the radial and ϕ i are angular coordinates Rewrite the condition Θ = 0 in terms f (ϕ i ) Elliptic equation for f (ϕ i ) This can be solved e.g. with Flow, Newton methods Thornburg, PRD 54 (1996) 4899, Gundlach, PRD 57 (1998) 863 Alcubierre et al, CQG 17 (2000) 2159, Schnetter, CQG 20 (2003) 4719 Irreducible mass M irr = AAH 16πG 2 BH mass in D = 4: M 2 = Mirr 2 + S2 (+P 2 ), 4Mirr 2 where S is the spin of the BH, Christodoulou, PRL 25 (1970) 1596 U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

134 Gravitational waves in D = 4: Newman Penrose Construct a Tetrad n α = Timelike unit normal field Spatial triad u, v, w through Gram-Schmidt orthogonalization E.g. starting with u i = [x, y, z], v i = [xz, yz, x 2 y 2 ], w i = ɛ i mnv m w n l α = 1 2 (n α + u α ), k α = 1 2 (n α u α ), m α = 1 2 (v α + iw α ) l k = 1 = m m, all other products vanish Newman-Penrose scalar Ψ 4 = C αβγδ k α m β k γ m δ In vacuum, C αβγδ = R αβγδ For more details, see e.g. Nerozzi, PRD 72 (2005) , Brügmann et al, PRD 77 (2008) U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

135 Analysis of Ψ 4 Multipolar decomposition: Ψ 4 = l,m ψ lm(t, r)y 2 lm (θφ), where ψ lm = 2π π 0 0 Ψ 4Y 2 lm sin θdθdφ [ Radiated energy: de r dt = lim 2 r 16π ] t Ψ 4d t 2 dω [ Momentum: dp i dt = lim r r 2 16π Ω l i Ω ] t Ψ 4d t 2 dω, where l i = [ sin θ cos φ, sin θ sin φ, cos θ] Angular mom.: djz dt = lim r { r 2 16π Re [ Ω see e.g. Ruiz et al, GRG 40 (2008) 2467 ( ) ( t φ Ψ t ) ]} ˆt 4d t Ψ 4d tdˆt dω U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

136 The AdS/CFT dictionary: Fefferman-Graham coords. AdS/CFT correspondence Vacuum expectation values T ij of the field theory given by quasi-local Brown-York stress-energy tensor Brown & York, PRD 47 (1993) 1407 Consider asymptotically AdS metric in Fefferman-Graham coordinates ds 2 = g µν dx µ dx ν = L2 r 2 (dr 2 + γ ij dx i dx j ), where γ ij (r, x i ) = γ (0)ij + r 2 γ (2)ij + + r D γ (D)ij + h (D)ij r D log r 2 + O(r D+1 ), Note: This asymptotes to Poincaré coordinates as r 0 U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

137 The AdS/CFT dictionary: Fefferman-Graham coords. Here, the γ (a)ij, h (D)ij are functions of x i, logarithmic terms only appear for even D, powers of r are exclusively even up to order D 1 Vacuum expectation values of CFT momentum tensor for D = 4 is { T ij = 4L3 16πG γ (4)ij 1 8 γ (0)ij[γ(2) 2 γkm (0) γln (0) γ (2)klγ (2)mn ] } 1 2 γ (2)i m γ (2)jm γ (2)ijγ (2) where γ (n) Tr(γ (n)ij ) = γ ij (0) γ (n)ij de Haro et al, Commun.Math.Phys. 217 (2001) 595; also for other D Note: γ (2)ij is determined by γ (0)ij CFT freedom given by γ (4)ij U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

138 AdS/CFT: Renormalized stress-tensor Again: Brown-York stress-tensor as the VEVs of the field theory Divergencies in T ab = δs eff δγ ab Regularize by adding boundary curvature invariants to S eff Balasubramanian & Kraus, Commun.Math.Phys. 208 (1999) 413 Foliate D dimensional spacetime into timelike hypersurfaces Σ r homoemorphic to the boundary ds 2 = α 2 dr 2 + γ ab (dx a + β a dr)(dx b + β b dr) ˆn µ = outward pointing normal vector to the boundary Θ µν = 1 2 ( µˆn ν + ν ˆn µ ) Extrinsic curvature (like ADM) U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

139 AdS/CFT: Renormalized stress-tensor Including counter terms, for ADS 5 : T µν = 1 [ 8πG Θ µν Θγ µν 3 L γµν L 2 Gµν] where G µν is the Einstein tensor of the induced metric γ µν Note: Applying this to the global ADS 5 metric gives T µν 0 Casimir energy of quantum field theory on S 3 R Other D: cf. Balasubramanian & Kraus, Commun.Math.Phys. 208 (1999) 413 AdS/CFT Dictionary for additional fields, see e.g. Skenderis, CQG 19 (2002) 5849 de Haro et al, Commun.Math.Phys. 217 (2001) 595 U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

140 Further reading Isolated and dynamical horizons Ashtekar & Krishnan, Liv. Rev. Rel. 7 (2004) 10 U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

141 A.5 BHs in Astrophysics U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

142 Evidence for astrophysical black holes X-ray binaries e. g. Cygnus X-1 (1964) MS star + compact star Stellar Mass BHs M Stellar dynamics near galactic centers, iron emission line profiles Supermassive BHs M AGN engines. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

143 Correlation of BH and host galaxy properties Galaxies ubiquitously harbor BHs BH properties correlated with bulge properties e. g. J.Magorrian et al., AJ 115, 2285 (1998) U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

144 SMBH formation Most widely accepted scenario for galaxy formation: hierarchical growth; bottom-up Galaxies undergo frequent mergers BH merger. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

145 Gravitational recoil Anisotropic GW emission recoil of remnant BH Bonnor & Rotenburg 61, Peres 62, Bekenstein 73 Escape velocities: Globular clusters 30 km/s dsph km/s de km/s Giant galaxies 1000 km/s Ejection / displacement of BH Growth history of SMBHs BH populations, IMBHs Structure of galaxies. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

146 Kicks from non-spinning BHs Max. kick: 180 km/s, harmless! González et al., PRL 98, (2009) U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

147 Spinning BHs: Superkicks Superkick configuration: Kicks up to v max km/s Campanelli et al., PRL 98 (2007) González et al. PRL 98 (2007) Suppression via spin alignment and Resonance effects in inspiral Schnittman, PRD 70 (2004) Bogdanovicź et al, ApJ 661 (2007) L147 Kesden et al, PRD 81 (2010) , ApJ 715 (2010) 1006 U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

148 Even larger kicks: superkick and hang-up Lousto & Zlochower, arxiv: [gr-qc] Superkicks Hangup Moderate GW generation Large kicks Strong GW generation No kicks. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

149 Superkicks and orbital hang-up Maximum kick about 25 % larger: v max km/s Distribution asymmetric in θ; v max for partial alignment Supression through resonances still works Berti et al, PRD 85 (2012) U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

150 Spin precession and flip X-shaped radio sources Merrit & Ekers, Science 297 (2002) 1310 Jet along spin axis Spin re-alignment new + old jet Spin precession 98 Spin flip 71 Campanelli et al, PRD 75 (2006) Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

151 Jets generated by binary BHs Palenzuela et al, PRL 103 (2009) , Science 329 (2010) 927 Non-spinning BH binary Einstein-Maxwell equtions with force free plasma Electromagnetic field extracts energy from L jets Optical signature: double jets U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

152 A.6. BH Holography U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

153 Large N and holography Holography BH entropy A Hor For a Local Field Theory entropy V Gravity in D dims local FT in D 1 dims Large N limit Perturbative expansion of gauge theory in g 2 N loop expansion in string theory N: # of colors g 2 N: t Hooft coupling U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

154 The AdS/CFT conjecture Maldacena, Adv.Theor.Math.Phys. 2 (1998) 231 strong form : Type IIb string theory on AdS 5 S 5 N = 4 super Yang-Mills in D = 4 Hard to prove; non-perturbative Type IIb String Theory? weak form : low-energy limit of string-theory side Type IIb Supergravity on AdS 5 S 5 Some assumptions, factor out S 5 General Relativity on AdS 5 Corresponds to limit of large N, g 2 N in the field theory E. g. Stationary AdS BH Thermal Equil. with T Haw in dual FT Witten, Adv.Theor.Math.Phys. 2 (1998) 253 U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

155 The boundary in AdS Dictionary between metric properties and vacuum expectation values of CFT operators. E. g. T αβ operator of CFT transverse metric on AdS boundary. The boundary plays an active role in AdS! Metric singular! U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159

156 Collision of planar shockwaves in N = 4 SYM Dual to colliding gravitational shock waves in AADS Characteristic study with translational invariance Chesler & Yaffe PRL 102 (2009) , PRD 82 (2010) , PRL 106 (2011) Initial data: 2 superposed shockwaves ds 2 = r 2 [ dx + dx + dx ] + 1 r 2 [dr 2 + h(x ± )dx 2 ±] U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/ / 159