Thesis Intro Harmonic IBVP Numerical Simulation of Binary Black Hole Spacetimes and a Novel Approach to Outer Boundary Conditions Jennifer Seiler Ph.D. Thesis Disputation Leibniz Universität Hannover February 5, 2010 Jennifer Seiler ❥❡s❡❅❛❡✐✳♠♣❣✳❞❡ CP SBP Boundaries 2nd Order
Thesis Intro Harmonic IBVP Outline Summary of Dissertation 1 Results Background 5 2 Robust Stability Tests Gravitational Waves Convergence Black Holes Black Holes Numerical Relativity Conclusions Numerical Methods 6 Publications Well-posedness 7 Other work Harmonic Decomposition 8 3 Kicks & Spins Initial Boundary Value Problem 4 Detection Summation By Parts Summary SAT and the Energy Method 9 Constraint Preservation Jennifer Seiler ❥❡s❡❅❛❡✐✳♠♣❣✳❞❡ CP SBP Boundaries 2nd Order
Thesis Intro Harmonic IBVP Summary of Dissertation Numerical evolutions of symmetric and asymmetric binary black hole mergers in to explore the parameter space of binary black hole inspirals: Establish bounds on phenomenological formulae for the final spin and recoil velocity of merged black holes from arbitrary initial data parameters Focus on gravitational-wave emission to quantify how much spin effects contribute to the signal-to-noise ratio and to the relative event rates for the representative ranges in masses and detectors Analytical inspiral-merger-ringdown gravitational waveforms from black-hole (BH) binaries with non-precessing spins by matching a post-Newtonian description of the inspiral to our numerical calculations Constraint-preserving boundary conditions for the BSSN evolution system Well-posed constraint-preserving outer boundary conditions for the Harmonic evolution system . . . Jennifer Seiler ❥❡s❡❅❛❡✐✳♠♣❣✳❞❡ CP SBP Boundaries 2nd Order
Thesis Intro Harmonic IBVP GW BH NR FD, MoL, & 3+1 WP General Relativity and Gravitational Waves Coaction between matter and curvature is described by the Einstein Equations: G µν = 8 π T µν Black holes (BH) = Vacuum ( T µν = 0) Gravitational Waves (GW) = finite deviation from Minkowski spacetime: g µν = η νµ + h νµ , | h νµ | ≪ 1 . Linearized field equations in GR � ¯ t + ∇ 2 )¯ h µν = 16 π T µν = ( − ∂ 2 h µν = 0 . Jennifer Seiler ❥❡s❡❅❛❡✐✳♠♣❣✳❞❡ CP SBP Boundaries 2nd Order
Thesis Intro Harmonic IBVP GW BH NR FD, MoL, & 3+1 WP Gravitational waves Gravitational radiation accompanies quadrupolar acceleration of any massive objects as cross-polarized transverse quadrupolar ripples in spacetime will radiate out longitudinally from this system, giving a metric perturbation h ij = h + ( e + ) ij − h × ( e × ) ij Indirect observation: binary pulsar PSR 1913+16 Hulse-Taylor – Nobel Prize 1993. Jennifer Seiler ❥❡s❡❅❛❡✐✳♠♣❣✳❞❡ CP SBP Boundaries 2nd Order
Thesis Intro Harmonic IBVP GW BH NR FD, MoL, & 3+1 WP Gravitational waves The coupling between matter and geometry is very weak. R αβ − 1 2 Rg αβ = kT αβ s 2 k = 8 π G ≃ 2 × 10 − 43 c 4 m · kg Gravitational waves are small features, difficult to detect. Unobstructed by intervening matter Excellent probe into regions opaque to EM radiation. Jennifer Seiler ❥❡s❡❅❛❡✐✳♠♣❣✳❞❡ CP SBP Boundaries 2nd Order
Thesis Intro Harmonic IBVP GW BH NR FD, MoL, & 3+1 WP Gravitational waves Currently there are many ground based detectors online which are designed to detect such passing gravitational waves (LIGO, VIRGO, TAMA, GEO). Even for binary black hole inspiral and merger, the signal strength is likely to be much less than the level of any detector noise. A technique used for this purpose is matched filtering , in which the detector output is cross-correlated with a catalog of theoretically predicted waveforms. Therefore, chances of detecting a generic astrophysical signal depend on the size, scope, and accuracy of the theoretical signal template bank. The generation of such a template bank requires many models of the GW emitted from compact binary systems. Jennifer Seiler ❥❡s❡❅❛❡✐✳♠♣❣✳❞❡ CP SBP Boundaries 2nd Order
Thesis Intro Harmonic IBVP GW BH NR FD, MoL, & 3+1 WP Binary black holes NGC2207 and IC2163 Optical, radio, and x-ray astronomy have provided us with abundant evidence that many galaxies contain SMBHs in their central nuclei. The loudest astrophysical signals in terms of SNR. Known examples among galactic binaries. Supermassive – 10 6 − 10 9 M ⊙ . Low frequency sources – space-based detector (LISA) Formation processes for stellar mass binaries: Collapse within a binary neutron star system. Capture within a dense region, eg. globular cluster. Jennifer Seiler ❥❡s❡❅❛❡✐✳♠♣❣✳❞❡ CP SBP Boundaries 2nd Order
Thesis Intro Harmonic IBVP GW BH NR FD, MoL, & 3+1 WP Binary black holes “Chirp” signal from a binary inspiral 20.0 Black holes captured → highly elliptical orbits. horizon ν = 0.1 ISCO Radiation of gravitational energy 10.0 → circularisation of orbits. → inspiral (PN) q’ y 0.0 Decay of orbit leading to −10.0 → plunge (NR) → merger (NR) Single perturbed BH remnant −20.0 −20.0 −10.0 0.0 10.0 20.0 q’ x → exponential ringdown to axisymmetric (Kerr) BH. Jennifer Seiler ❥❡s❡❅❛❡✐✳♠♣❣✳❞❡ CP SBP Boundaries 2nd Order
Thesis Intro Harmonic IBVP GW BH NR FD, MoL, & 3+1 WP Numerical Relativity R αβ − 1 2 g αβ R = 8 π T αβ The Einstein equations are a hyperbolic set of nonlinear wave equations for the geometry As such, they are most conveniently solved as an initial-boundary-value problem: Assume the geometry is known at some initial time t 0 . Evolve the data using the Einstein equations. Prescribe consistent boundary conditions at some finite radius r 0 . Geometry specified on an initial data slice: metric g ab specifies the intrinsic geometry of the slice. extrinsic curvature determines the embedding in 4D space. Evolution equations are integrated using standard numerical methods, eg. Runge-Kutta. The equations are differentiated in space on a discrete computational grid using finite differencing methods Jennifer Seiler ❥❡s❡❅❛❡✐✳♠♣❣✳❞❡ CP SBP Boundaries 2nd Order
Thesis Intro Harmonic IBVP GW BH NR FD, MoL, & 3+1 WP Finite Differencing discretize our continuum intial data and solve the spatial derivatives in our PDEs. ( i − 1 = 2 ) h x , 0 ≤ i ≤ N x , x i finite difference approach using Taylor series expansions dx | x + h 2 dx 2 | x + h 3 d 2 f d 3 f f ( x ) + h df f ( x + h ) = dx 3 | x + . . . 2 6 dx | x − h 2 dx 2 | x − h 3 d 2 f d 3 f f ( x ) − h df f ( x − h ) = dx 3 | x + . . . 2 6 f ( x + h ) − f ( x − h ) df − 1 6 f ′′′ ( ζ ) h 2 , = dx 2 h Fourth order: − f ( x + 2 h ) + 8 f ( x + h ) − 8 f ( x − h ) + f ( x − 2 h ) df = dx 12 h replace PDE with an algebraic equation on a discrete grid Jennifer Seiler ❥❡s❡❅❛❡✐✳♠♣❣✳❞❡ CP SBP Boundaries 2nd Order
Thesis Intro Harmonic IBVP GW BH NR FD, MoL, & 3+1 WP Method of Lines FD the spatial derivatives of the PDE leaving the time derivatives continuous. This leads to a coupled set of ODEs for the time dependence of the variables u = ( u ij ) at the spatial grid points, ∂ t u = f ( t , u ) ODE integrator to integrate these ODEs forward in time. � u n + 1 − u n � = O (∆ t p + 1 ) Jennifer Seiler ❥❡s❡❅❛❡✐✳♠♣❣✳❞❡ CP SBP Boundaries 2nd Order
Thesis Intro Harmonic IBVP GW BH NR FD, MoL, & 3+1 WP Complications of Numerical Relativity The initial-boundary-value problem needs to be well-posed. Choice of geometrical variables → strongly hyperbolic evolution system. Evolution of the coordinates needs to be carefully considered. The BH centers are physical singularities: Treated as “punctures” by choice of gauge. Excised by imposing a boundary condition around the singularity. It is only within the last 5 years that this problem has been solved: Pretorius (2005), Campanelli et al. (2005), Baker et al. (2005). Jennifer Seiler ❥❡s❡❅❛❡✐✳♠♣❣✳❞❡ CP SBP Boundaries 2nd Order
Thesis Intro Harmonic IBVP GW BH NR FD, MoL, & 3+1 WP Einstein equations in 3+1 form The Einstein equations are manifestly covariant Need to reformulate as a Cauchy problem We have ten equations and ten independent components of the four metric g µν , the same number of equations as unknowns. Only six of these ten equations involve second time-derivatives of the metric. The other four equations, thus, are not evolutions equations. We call these our constraint equations . There are a number of non-unique aspects of the 3+1 decomposition Choice of evolution variables Choice of gauge Binary black hole codes currently use either a harmonic formulation, or a modified (“conformal traceless” or “BSSN”) ADM system. Jennifer Seiler ❥❡s❡❅❛❡✐✳♠♣❣✳❞❡ CP SBP Boundaries 2nd Order
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