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) A review of numerical relativity and black-hole collisions 07/18/2013 23 / 159
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) A review of numerical relativity and black-hole collisions 07/18/2013 24 / 159
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) A review of numerical relativity and black-hole collisions 07/18/2013 25 / 159
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) A review of numerical relativity and black-hole collisions 07/18/2013 26 / 159
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 ˜ ⇔ ˜ 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) A review of numerical relativity and black-hole collisions 07/18/2013 27 / 159
The BSSN equations 2 ( D − 1 ) ( ∂ m β m − α K ) ∂ t φ = β m ∂ m φ − 1 γ m ( i ∂ i ) β m − γ ij ∂ m β m − 2 α ˜ γ = 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 ∂ i ) 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) A review of numerical relativity and black-hole collisions 07/18/2013 28 / 159
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) A review of numerical relativity and black-hole collisions 07/18/2013 29 / 159
2.1.2 Generalized Harmonic formulation U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 30 / 159
The Generalized Harmonic (GH) formulation → Appendix U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 31 / 159
2.1.3 Characteristic formulation U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 32 / 159
The characteristic formulation → Appendix U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 33 / 159
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 3 3 = 0 Lagrangian coords. GRT ENSOR , M ATHEMATICA ,... ⇒ Field equations: a ′ = ... b ′ = ... ¨ R = ... U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 34 / 159
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 07/18/2013 35 / 159
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) A review of numerical relativity and black-hole collisions 07/18/2013 36 / 159
2.2. Initial data, Gauge, Boundaries U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 37 / 159
2.2.1. Initial data U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 38 / 159
Analytic initial data Schwarzschild, Kerr, Tangherlini, Myers Perry,... e.g. Schwarzschild in isotropic coordinates: [ r 2 + r 2 ( d θ 2 + sin 2 θ d φ 2 )] ds 2 = − M − 2 r M + 2 r dt 2 + 1 + M � � 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) A review of numerical relativity and black-hole collisions 07/18/2013 39 / 159
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) A review of numerical relativity and black-hole collisions 07/18/2013 40 / 159
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) A review of numerical relativity and black-hole collisions 07/18/2013 41 / 159
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) A review of numerical relativity and black-hole collisions 07/18/2013 42 / 159
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) A review of numerical relativity and black-hole collisions 07/18/2013 43 / 159
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) A review of numerical relativity and black-hole collisions 07/18/2013 44 / 159
2.2.2. Gauge U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 45 / 159
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) A review of numerical relativity and black-hole collisions 07/18/2013 46 / 159
What goes wrong with bad gauge? U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 47 / 159
What goes wrong with bad gauge? U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 48 / 159
What goes wrong with bad gauge? U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 49 / 159
What goes wrong with bad gauge? U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 50 / 159
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) A review of numerical relativity and black-hole collisions 07/18/2013 51 / 159
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) A review of numerical relativity and black-hole collisions 07/18/2013 52 / 159
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) A review of numerical relativity and black-hole collisions 07/18/2013 53 / 159
2.2.3. Boundaries U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 54 / 159
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) A review of numerical relativity and black-hole collisions 07/18/2013 55 / 159
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) A review of numerical relativity and black-hole collisions 07/18/2013 56 / 159
Outer boundary → Appendix U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 57 / 159
Further reading Initial data construction Cook, Liv. Rev. Rel. 3 (2000) 5 Pfeiffer, gr-qc/0510016 U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 58 / 159
2.3 Discretization of the equations U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 59 / 159
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 07/18/2013 60 / 159
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) A review of numerical relativity and black-hole collisions 07/18/2013 61 / 159
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) A review of numerical relativity and black-hole collisions 07/18/2013 62 / 159
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) A review of numerical relativity and black-hole collisions 07/18/2013 63 / 159
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) A review of numerical relativity and black-hole collisions 07/18/2013 64 / 159
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) A review of numerical relativity and black-hole collisions 07/18/2013 65 / 159
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) A review of numerical relativity and black-hole collisions 07/18/2013 66 / 159
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) A review of numerical relativity and black-hole collisions 07/18/2013 67 / 159
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) A review of numerical relativity and black-hole collisions 07/18/2013 68 / 159
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) A review of numerical relativity and black-hole collisions 07/18/2013 69 / 159
Alternative discretization schemes Spectral methods: high accuracy, efficiency, complexity Caltech-Cornell-CITA code SpEC http://www.black-holes.org/SpEC.html Applications to moving punctures still in construction e.g. Tichy, PRD 80 (2009) 104034 Also used in symmetric asymptotically AdS spacetimes e.g. Chesler & Yaffe, PRL 106 (2011) 021601 Finite Volume methods Finite Element methods D. N. Arnold, A. Mukherjee & L. Pouly, gr-qc/9709038 C. F. Sopuerta, P . Sun & J. Xu, CQG 23 (2006) 251 C. F. Sopuerta & P . Laguna, PRD 73 (2006) 044028 U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 70 / 159
Further reading Numerical methods Press et al, “ Numerical Recipes ”, Cambridge University Press U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 71 / 159
3 Results from BH evolutions U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 72 / 159
3.1 BHs in GW physics U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 73 / 159
Gravitational waves Weak field limit: g αβ = η αβ + h αβ Trace reversed perturbation ¯ h αβ = h αβ − 1 2 h η αβ ⇒ Vacuum field eqs.: � ¯ h αβ = 0 Apropriate gauge ⇒ 0 0 0 0 0 h + h × 0 e ik σ x σ ¯ h αβ = 0 h × − h + 0 0 0 0 0 where k σ = null vector GWs displace particles U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 74 / 159
Gravitational wave detectors Accelerated masses ⇒ GWs Weak interaction! Laser interferometric detectors U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 75 / 159
The gravitational wave spectrum U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 76 / 159
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 U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 77 / 159
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 1 M 2 η ≡ M 2 , ( M 1 + M 2 ) 2 Spin: � S 1 , � S 2 (6 parameters) Initial parameters Binding energy E b Separation Orbital ang. momentum L Eccentricity Alternatively: frequency, eccentricity U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 78 / 159
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 07/18/2013 79 / 159
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 07/18/2013 80 / 159
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 07/18/2013 81 / 159
Template construction Stitch together PN and NR waveforms EOB or phenomenological templates for ≥ 7-dim. par. space U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 82 / 159
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) 104017, CQG 25 (2008) 114033, PRL 106 (2011) 241101; Santamaria et al, PRD 82 (2010) 064016, Sturani et al, arXiv:1012.5172 [gr-qc] Effective-one-body (EOB) models Particle in effective metric, PN, ringdown model Buonanno & Damour PRD 59 (1999) 084006, PRD 62 (2000) 064015 Resum PN, calibrate pseudo PN parameters using NR Buonanno et al, PRD 77 (2008) 026004, Pan et al, PRD 81 (2010) 084041, PRD 84 (2012) 124052; Damour et al, PRD 77 (2008) 084017, PRD 78 (2008) 044039, PRD 83 (2011) 024006 U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 83 / 159
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) 081103 Circumvent no-hair theorem: Scalar bubble Healey et al, arXiv:1112.3928 [gr-qc] Circumvent no-hair theorem: Scalar gradient Berti et al, arXiv:1304.2836 [gr-qc] U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 84 / 159
3.2. High-energy collisions of BHs U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 85 / 159
The Hierarchy Problem of Physics Gravity ≈ 10 − 39 × other forces µ 2 − Λ 2 � Higgs field ≈ µ obs ≈ 250 GeV = where Λ ≈ 10 16 GeV is the grand unification energy Requires enormous finetuning!!! Finetuning exist: 987654321 123456789 = 8 . 0000000729 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 07/18/2013 86 / 159
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/9906038; Giddings & Thomas, PRD 65 (2002) 056010 U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 87 / 159
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) 111101 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 07/18/2013 88 / 159
Does matter “matter”? Perfect fluid “stars” model γ = 8 . . . 12; BH formation below Hoop prediction East & Pretorius, PRL 110 (2013) 101101 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) 012001 U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 89 / 159
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 U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 90 / 159
D = 4: Initial setup: 1) Aligned spins Orbital hang-up Campanelli et al, PRD 74 (2006) 041501 2 BHs: Total rest mass: M 0 = M A , 0 + M B , 0 √ 1 − v 2 , Boost: γ = 1 / M = γ M 0 Impact parameter: b ≡ L P U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 91 / 159
D = 4: Initial setup: 2) No spins Orbital hang-up Campanelli et al, PRD 74 (2006) 041501 2 BHs: Total rest mass: M 0 = M A , 0 + M B , 0 √ 1 − v 2 , Boost: γ = 1 / M = γ M 0 Impact parameter: b ≡ L P U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 92 / 159
D = 4: Initial setup: 3) Anti-aligned spins Orbital hang-up Campanelli et al, PRD 74 (2006) 041501 2 BHs: Total rest mass: M 0 = M A , 0 + M B , 0 √ 1 − v 2 , Boost: γ = 1 / M = γ M 0 Impact parameter: b ≡ L P U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 93 / 159
� D = 4: Head-on: b = 0 , S = 0 Total radiated energy: 14 ± 3 % for v → 1 US et al, PRL 101 (2008) 161101 About half of Penrose ’74 Agreement with approximative methods Flat spectrum, GW multipoles Berti et al, PRD 83 (2011) 084018 U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 94 / 159
� 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) 131102 U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 95 / 159
D = 4: Scattering threshold b scat for � S = 0 ⇒ b < b scat Merger b > b scat ⇒ Scattering b scat = 2 . 5 ± 0 . 05 Numerical study: M v Shibata et al, PRD 78 (2008) 101501(R) Independent study US et al, PRL 103 (2009) 131102, arXiv:1211.6114 γ = 1 . 23 . . . 2 . 93: χ = − 0 . 6 , 0 , + 0 . 6 (anti-aligned, nonspinning, aligned) Limit from Penrose construction: b crit = 1 . 685 M Yoshino & Rychkov, PRD 74 (2006) 124022 U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 96 / 159
D = 4: Scattering threshold and radiated energy � S � = 0 US et al, arXiv:1211.6114 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 07/18/2013 97 / 159
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:1211.6114 U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 98 / 159
3.3 Fundamental properties of BHs U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 99 / 159
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 Bizo´ n & Rostworowski, PRL 107 (2011) 031102 Similar behaviour for “Geons” Dias, Horowitz & Santos ’11 D > 4 dimensions Jałmu˙ zna et al, PRD 84 (2011) 085021 D = 3: Mass gap: smooth solutions Bizo´ n & Jałmu˙ zna, arXiv:1306.0317 U. Sperhake (DAMTP, University of Cambridge) A review of numerical relativity and black-hole collisions 07/18/2013 100 / 159
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