Alternatives to the ADM equations Unfortunately the ADM eqs. do not seem to work in NR !! Weak hyperbolicity: Nearby initial data can diverge From each other super-exponentially Many alternative formulations have been suggested Two successful families so far ADM based formulations: BSSN Shibata & Nakamura ’95, Baumgarte & Shapiro ‘99 Generalized harmonic formulations Choque-Bruhat ’62, Garfinkle ’04, Pretorius ‘05 32
BSSN One can easily change variables. E.g. wave equation: 2 F c G 0 ∂ − ∂ = 2 u c u 0 t x ∂ − ∂ = ⇔ tt xx F G 0 ∧ ∂ − ∂ = x t BSSN: rearrange degrees of freedom 1 ~ 4 ( ) e ln det − φ γ = γ φ = γ ij ij 12 ~ 1 & # ij K ij K 4 = γ A e K K − φ = − γ $ ! ij ij ij 3 ~ ~ % " ~ ~ i mn i im Γ = γ Γ = −∂ γ mn m Shibata, Nakamura ’95, Baumgarte, Shapiro ‘99 33
The BSSN equations 34
Generalized harmonic (GHG) Harmonic gauge: choose coordinates so that x 0 µ α ∇ ∇ = µ 4-dim. Version of Einstein equations 1 (no second derivatives!!) R g µ ν g ... = − ∂ ∂ + αβ µ ν αβ 2 Principal part of wave equation Generalized harmonic gauge: H : g x µ ν = ∇ ∇ α αν µ 1 1 ( ) R g µ ν g ... H H ⇒ = − ∂ ∂ + − ∂ + ∂ αβ µ ν αβ α β β α 2 2 Still principal part of wave equation!!! 35
The gauge in GHG i Relation between and lapse and shift : H α β α 1 ( ) i H n K µ = − − ∂ α − β ∂ α 0 i µ 2 α 1 1 ( ) i ik i k i mn i H µ ⊥ = γ ∂ α + ∂ β − β ∂ β − γ Γ µ k 0 k mn 2 α α Auxiliary constraint C : H g g µ µ ν = − Γ + ∂ γ γ µ γ µ νγ Requires constraint damping Gundlach et al. 2005 36
3.2. Gauge choices 37
The gauge freedom i Remember: Einstein equations say nothing about α , β Any choice of lapse and shift gives a solution This represents the coordinate freedom of GR i Physics do not depend on α , β So why bother? Avoid coordinate singularities! Stop the code from running into the physical singularity No full-proof recipe, but Singularity avoiding slicing Use shift to avoid coordinate stretching 38
What goes wrong with bad gauge? 39
What goes wrong with bad gauge? 40
What goes wrong with bad gauge? 41
What goes wrong with bad gauge? 42
How do we get good gauge? Target: Avoid singularities and instabilities Methods: Geometric ideas, mathematical structure of equations Harmonic coords. Maximal slicing, Choquet-Bruhat‘62 min.distortion shift Analytic studies Smarr, York ‘78 43
How do we get good gauge? Target: Avoid singularities and instabilities Methods: Geometric ideas, mathematical structure of equations Harmonic coords. Maximal slicing, Choquet-Bruhat‘62 min.distortion shift Analytic studies Smarr, York ‘78 Driver conditions Balakrishna et.al.’96 44
How do we get good gauge? Target: Avoid singularities and instabilities Methods: Geometric ideas, mathematical structure of equations Harmonic coords. Maximal slicing, Choquet-Bruhat‘62 min.distortion shift Analytic studies Smarr, York ‘78 Driver conditions Balakrishna et.al.’96 ~ 1+log, -driver Γ AEI Aim: Stationarity 45
How do we get good gauge? Target: Avoid singularities and instabilities Methods: Geometric ideas, mathematical structure of equations Harmonic coords. Maximal slicing, Choquet-Bruhat‘62 min.distortion shift Analytic studies Smarr, York ‘78 Special case Driver conditions Bona-Massó family Balakrishna Bona, Massó ‘95 et.al.’96 Special case ~ 1+log, -driver Γ Avoid singularities AEI Alcubierre ‘03 Aim: Stationarity 46
How do we get good gauge? Target: Avoid singularities and instabilities Methods: Geometric ideas, mathematical structure of equations Harmonic coords. Maximal slicing, Choquet-Bruhat‘62 min.distortion shift Analytic studies Smarr, York ‘78 Special case Driver conditions Bona-Massó family Balakrishna Bona, Massó ‘95 et.al.’96 Generalized harmonic Special Garfinkle ‘04 case Pretorius ‘05 ~ 1+log, -driver Γ gauge sources Avoid singularities AEI Relation to i α , Alcubierre ‘03 β Aim: Stationarity Moving punctures UTB, Goddard ‘06 47
3.3. Initial data 48
Initial data problem Two problems: Constraints, realistic data ~ 4 York-Lichnerowicz split γ = ψ γ ij ij 1 K A K = + γ ij ij ij 3 Rearrange degrees of freedom Conformal transverse traceless York, Lichnerowicz Physical transverse traceless O’Murchadha, York Thin sandwich Wilson, Mathews; York Conformal flatness: Kerr is NOT conformally flat! Non-physical GWs: problematic for high energy collisions! 49
2 families of initial data Generalized analytic solutions: 4 M ' + $ ( ) Isotropic Schwarzschild: 2 2 2 2 2 ds dt 1 dr r d = − + + Ω % " 2 r & # N Time-symmetric, -holes: ⇒ Brill-Lindquist, Misner (1960s) Spin, Momenta: Bowen, York (1980) ⇒ Brandt, Brügmann (1997) Punctures ⇒ Excision Data: IH boundary conditions on excision surface Meudon group; Cook, Pfeiffer; Ansorg Quasi-circular: Effective potential PN parameters helical Killing Vektor 50
3.4. Mesh refinement 51
Mesh-refinement 3 Length scales: BH 1 M ≈ Wavelength 10 M ≈ Wave zone 100 M ≈ Mesh Refinement! Choptuik ’93 AMR, Critical phenomena Stretch coords.: Fish-eye Lazarus, AEI, UTB FMR, Moving boxes: Berger-Oliger BAM Brügmann’96 Carpet Schnetter et.al.’03 52
Mesh-refinement AMR: Control resolution via curvature Paramesh: MacNeice et.al.’00, Goddard Modified Berger-Oliger: Pretorius, Choptuik ’05 SAMRAI: OpenGR UT Austin Refinement boundaries: reflections, stability Tapered boundaries Lehner, Liebling, Reula ‘05 53
3.5. Singularity treatment 54
Singularities: Excision Cosmic censorship: horizon is causal boundary Unruh ’84 cited in Thornburg ‘87 Grand Challenge: Causale differencing “Simple Excision” Alcubierre, Brügmann ‘01 Dynamic “moving” excision Pitt-PSU-Texas PSU-Maya Pretorius LEAN (U.S.’06) Combined with “Dual coordinate frame” Caltech-Cornell Mathematic properties: Wealth of literature 55
Singularities: Excision Cosmic censorship: horizon is causal boundary Unruh ’84 cited in Thornburg ‘87 Grand Challenge: Causale differencing “Simple Excision” Alcubierre, Brügmann ‘01 Dynamic “moving” excision Pitt-PSU-Texas PSU-Maya Pretorius LEAN (U.S.’06) Combined with “Dual coordinate frame” Caltech-Cornell Mathematic properties: Wealth of literature 56
4. Extracting physics 57
Basic assumptions Extracting physics in NR is non-trivial !! Newtonian quantities are not always well-defined !! We assume that the ADM variables Lapse α i Shift β 3-metric γ ij Extrinsic curvature K ij are given on each hypersurface Σ t Even when using other formulations, the ADM variables are straightforward to calculate 58
Global quantities ADM mass: Global energy of the spacetime 1 ( ) ij kl M lim dS = ∫ γ γ γ γ − γ ADM ik , j ij , k l 16 r π → ∞ S r Total angular momentum of the spacetime 1 ( ) m m P lim K K dS = ∫ γ − δ i i i m 8 r π → ∞ S r 1 ( ) m ! n n J lim x K K dS = ε ∫ γ − δ m m i i ! n 8 r π → ∞ S r By construction all of these are time-independent !! 59
Local quantities Often impossible to define !! Isolated horizon framework Ashtekar and coworkers Calculate apparent horizon irreducible mass, momenta associated with horizon A M AH = irr 16 π Total BH mass Christodoulou 2 S 2 2 2 M M P = + + irr 2 4 M irr Binding energy of a binary: E M M M = − − b ADM 1 2 60
Gravitational waves Most important result: Emitted gravitational waves (GWs) Newman-Penrose scalar C n m n m α β γ δ Ψ 4 = αβγδ Complex 2 free functions ⇒ GWs allow us to measure Radiated energy E rad P rad , J Radiated momenta rad Angular dependence of radiation h , + h Predicted strain × 61
Angular dependence r Waves are normally extracted at fixed radius ex ( ) t , θ , ⇒ Ψ = Ψ φ 4 4 θ , are viewed from the source frame !! φ Decompose angular dependence X ψ ` m ( t ) Y − 2 Ψ 4 = ` m ( θ , φ ) ` m Ψ 4 ( t ) = A ` m ( t ) e i � ( t ) Modes Spin-weighted spherical harmonics 62
5. A brief history 63
A brief history of BH simulations Pioneers: Hahn, Lindquist ’60s, Eppley, Smarr et.al. ‘70s Grand Challenge: First 3D Code Anninos et.al. ‘90s Further attempts: Bona & Massó, Pitt-PSU-Texas, … AEI-Potsdam Alcubierre et al. PSU: first orbit Brügmann et al. ‘04 __________ __________ Codes unstable __________ __________ __________ _____ Breakthrough: Pretorius ’05 “GHG” UTB, Goddard ’05 “Moving Punctures” Currently: codes, a.o. 10 BAM Brügmann ≈ LEAN Sperhake ‘07 64
6. Animations 65
Animations Lean Code representative for other codes Extrinsic curvature tr K Apparent horizon 66
Animations Re[ ] Ψ 4 Angular dependence ⇒ Spherical harmonics ` = 2 , m = 2 dominant 67
Animations Event horizon of binary inspiral and merger BAM Thanks to Marcus Thierfelder 68
7. Results on black-hole binaries 69
Free parameters of BH binaries Total mass M ADM M ➢ Relevant for detection: Frequencies depend on ADM ➢ Not relevant for source modeling: trivial rescaling M M M 1 q , 1 2 Mass ratio = η = 2 M ( ) M M + 2 1 2 S 1 , ~ ~ Spin S 2 Initial parameters Binding energy Separation E b Orbital angular momentum Eccentricty L Alternatively: frequency, eccentricity 70
7.1. Non-spinning equal-mass holes 71
The BBH breakthrough Simplest configuration GWs circularize orbit quasi-circular initial data ⇒ Pretorius PRL ‘05 BBH breakthrough Initial data: scalar field Radiated energy R [ M ] 25 50 75 100 = ex 4.7 3.2 2.7 2.3 E [ % M ] = Eccentricity e 0 ... 0 . 2 = 72
Non-spinning equal-mass binaries 3 . 6 % M Total radiated energy: ADM 98 % mode dominant: > ` = 2 , m = 2 73
The merger part of the inspiral Buonanno, Cook, Pretorius ’06 (BCP) merger lasts short: 0.5 – 0.75 cycles Eccentricity small 0 . 01 ≈ non-vanishing Initial radial velocity 74
Comparison with Post-Newtonian Goddard ‘07 14 cycles, 3.5 PN phasing φ , φ PN Match waveforms: Accumulated phase error 1 rad Buonanno, Cook, Pretorius ’06 (BCP) 3.5 PN phasing 2 PN amplitude 75
Comparison with Post-Newtonian Jena ‘07 18 cycles 1 rad phase error < 6 th order differencing !! Amplitude: % range Cornell/Caltech & Buonanno 30 cycles 0 . 02 rad phase error ≈ Effective one body (EOB) RIT First comparison with spin; not conclusive yet 76
Zoom whirl orbits Pretorius & Khurana ‘07 1-parameter family of initial data: linear momentum Fine-tune parameter ⇒ ”Threshold of immediate merger” Analogue in gedodesics ! Reminiscent of ”Critical phenomena” Similar observations by PSU j 0 . 78 2 Max. spin for L ≈ M fin = 77
7.2. Unequal masses 78
Unequal masses Still zero spins Astrophysically much more likely !! Symmetry breaking Anisotropic emission of GWs Certain modes are no longer suppressed Mass ratios 6 Stellar sized BH into supermassive BH 10 ≈ 3 Intermediate mass BHs 10 ≈ 3 Galaxy mergers 1 ... 10 ≈ Currently possible numerically: 1 ... 10 ≈ 79
Gravitational recoil Anisotropic emission of GWs radiates momentum recoil of remaining system ⇒ Leading order: Overlap of Mass-quadrupole with octopole/flux-quadrupole Bonnor & Rotenburg ’61, Peres ’62, Bekenstein ‘73 Merger of galaxies Merger of BHs ⇒ Recoil ⇒ BH kicked out? ⇒ 80
Gravitational recoil Escape velocities Globular clusters 30 km/s dSph 20 − 100 km/s dE 100 − 300 km/s Giant galaxies 1000 km/s ≈ Merrit et al ‘04 Ejection or displacement of BHs has repercussions on: Structure formation in the universe BH populations IMBHs via ejection? Growth history of Massive Black Holes Structure of galaxies 81
Kicks of non-spinning black holes Simulations PSU ’07, Goddard ‘07 Parameter study Jena ‘07 Target: Maximal Kick Mass ratio: M / M 1 ... 4 = 1 2 150,000 CPU hours Maximal kick 178 km/s M / M 3 for ≈ 1 2 Convergence 2 nd order E 3 %, J 25 % ≈ ≈ rad rad 0 . 45 ... 0 . 7 Spin 82
Features of unequal-mass mergers Berti et al ‘07 Distribution of radiated energy More energy in higher modes Odd modes suppressed ` for equal masses Important for GW-DA 83
Mass ratio 10:1 In preparation: González, U.S., Brügmann Mass ratio ; q 10 Bam = 4 th order convergence Astrophysically likely configuration: Sesana et al. ‘07 Test fitting formulas for spin and kick! 84
4 2 (Fitchett ‘83 Kick: v 1 . 2 10 1 4 ( 1 0 . 93 ) = × η − η − η Gonzalez et al. ’07) V~62 km/s 85
E Δ 2 Radiated energy: 0 . 5802 (Berti et al. ’07) = η M Δ E/M~0.004018 86
Final spin: (Damour and Nagar 2007) a F /M F ~0.2602 87
7.3. Spinning black holes 88
Spinning holes: The orbital hang-up ~ E rad J , ↑ Spins parallel to more orbits, larger ↑ L ⇒ ↑ rad ~ E rad J , Spins anti-par. to fewer orbits smaller ↑ L ⇒ ↓ ↓ rad UTB/RIT ‘07 no extremal Kerr BHs 89
Spin precession and flip X-shaped radio sources Merritt & Ekers ‘07 Jet along spin axis Spin re-alignment new + old jet ⇒ Spin precession 98 ° Spin flip 71 ° UTB, Rochester ‘06 90
Recoil of spinning holes Kidder ’95: PN study with Spins = “unequal mass” + “spin(-orbit)” Penn State ‘07: SO-term larger a 0 . 2 ,..., 0 . 8 = m extrapolated: v 475 km/s = v 440 km/s AEI ’07 : One spinning hole, extrapolated: = v 454 km/s UTB-Rochester: = 91
Super Kicks Side result RIT ‘07, Kidder ’95: maximal kick predicted for v 1300 km/s ≈ Test hypothesis González, Hannam, US, Brügmann & Husa ‘07 Use two codes: Lean, BAM a 0.75 Generates kick for spin v 2500 km/s ≈ = 92
Super Kicks Side result RIT ‘07, Kidder ’95: maximal kick predicted for v 1300 km/s ≈ Test hypothesis González, Hannam, US, Brügmann & Husa ‘07 Use two codes: Lean, BAM a 0.75 Generates kick for spin v 2500 km/s ≈ = v 4000 km/s Extrapolated to maximal spin = RIT ‘07 v 10000 km/s Highly eccentric orbits = PSU ‘08 93
What’s happening physically? Black holes “move up and down” 94
A closer look at super kicks Physical explanation: “Frame dragging” Recall: rotating BH drags objects along with its rotation 95
A closer look at super kicks Physical explanation: “Frame dragging” Recall: rotating BH drags objects along with its rotation Thanks to F. Pretorius 96
How realistic are superkicks? Observations BHs are not generically ejected! ⇒ Are superkicks real? Gas accretion may align spins with orbit Bogdanovic et al. v kick = v kick ( ~ S 1 , ~ Kick distribution function: S 2 , M 1 /M 2 ) Analytic models and fits: Boyle, Kesden & Nissanke, AEI, RIT, Tichy & Marronetti,… Use numerical results to determine free parameters 7-dim. Parameter space: Messy! Not yet conclusive… v 500 km/s EOB study only 12% of all mergers have ⇒ > Schnittman & Buonanno ‘08 97
7.4. Numerical relativity and data analysis 98
The Hulse-Taylor pulsar Hulse, Taylor ‘93 Binary pulsar 1913+16 GW emission Inspiral Change in period Excellent agreement with relativistic prediction 99
The data stream: Strong LISA source SMBH binary 100
Recommend
More recommend
