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Spin Interactions in Binary Black Hole Initial Data

Spin Interactions in Binary Black Hole Initial Data. Scott Hawley*, Richard Matzner, Michael Vitalo Center for Relativity University of Texas at Austin. Physical System. York-Lichnerowitz conformal-traceless method. Background metric and extrinsic curvature

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Spin Interactions in Binary Black Hole Initial Data

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  1. Spin Interactions in Binary Black Hole Initial Data Scott Hawley*, Richard Matzner, Michael Vitalo Center for Relativity University of Texas at Austin

  2. Scott Hawley, Numerical Relativity 2005, Nov 4 Physical System • York-Lichnerowitz conformal-traceless method. • Background metric and extrinsic curvature • Related to the physical quantities by...

  3. Scott Hawley, Numerical Relativity 2005, Nov 4 Physical System, cont’d • We solve the constraint equations.... ...solve for f, wi

  4. Scott Hawley, Numerical Relativity 2005, Nov 4 Boundary Conditions • Dirichlet at inner boundary (edge of excision mask): • Robin at outer boundary:

  5. Scott Hawley, Numerical Relativity 2005, Nov 4 Multigrid Overview 2:1 grid spacing 2:1 grid spacing • Traditional relaxation methods of solving elliptic problems are extremely inefficient at eliminating long-wavelength components of error, but are very efficient at eliminating short-wavelength components. • They can thus be regarded as smoothing operators (for the error) • Multigrid (Brandt 1978), an O(N) method, basically* works by applying smoothing operators successively at different resolutions to “smooth away” different wavelength-components of the error. The “V-Cycle” Smooth Smooth Restrict Smooth Smooth Prolong SOLVE Prolong (special interp) Restrict *also modify RHS, & prolong doesn’t overwrite

  6. Scott Hawley, Numerical Relativity 2005, Nov 4 Choice of Background: “Binary Kerr-Schild” Matzner, Huq, Shoemaker, 1998 • Kerr-Schild metric is • This suggests a superposition of N BHs via...

  7. Scott Hawley, Numerical Relativity 2005, Nov 4 How ‘Close’ is Background?

  8. Scott Hawley, Numerical Relativity 2005, Nov 4 Binding Energy • Contribution due to spin is... (Wald 1972) We’ll see...

  9. Scott Hawley, Numerical Relativity 2005, Nov 4 Code Details • Elliptic solver. Parallel multigrid code “TEXMEX” • Handles excision (Hawley & Matzner 2004) • In-house, standalone code, designed as a “black box” elliptic solver: input background/guess, output quantities which satisfy constraints • Vertex-centered • Similar to other multigrid solvers (e.g., Brown, Pretorius, Brügmann) • but with unique prolongation/restriction scheme near inner boundary • Smoothing via Newton-Gauss-Seidel, using “rainbow” (like red-black) ordering • Inner boundary values supplied via ...see next slide • SOR on coarsest-grid solves (switches to NGS near target tol.) • Includes separate, 4th- or 2nd-order independent residual evaluator • Use Thornburg’s AH finder for post-process analysis • Have Carpet interface, but it’s out of date • Still testing FMR, Fisheye...

  10. Scott Hawley, Numerical Relativity 2005, Nov 4 Handling Inner Boundaries (i.e. Excision) Definition: Inner boundary points are those points which are immediately interior to a circle of radius rex. Here we show a fine grid and a coincident coarse grid: Restriction scheme: Use weighted restriction everywhere except when doing so would make use of fine-grid inner boundary data. Instead of using IB data, “just copy”. Filling coarse-grid inner boundary values: copy where possible, otherwise via weighted multi-directional extrapolation (inward) from nearby fine grid points

  11. Scott Hawley, Numerical Relativity 2005, Nov 4 Test: Schwarzschild Background

  12. Scott Hawley, Numerical Relativity 2005, Nov 4 Test: BBH Convergence

  13. Scott Hawley, Numerical Relativity 2005, Nov 4 Results... a1 a2 q d • Study binding energy as function of BBH spins • Typical runs: • Spins a1=a2=0.5, d=10M. Hold q2 const, vary q1 • 5133 grids (4 multigrid levels: 653, 1293, 2563 & 5133) • 32 Processors. Bkgrnd takes few mins, solver 3 hrs. • Domain: -15M to 15M, i.e. Dx=M/17. • Excision radius 0.9M • MADM evaluated at 12M

  14. Scott Hawley, Numerical Relativity 2005, Nov 4 Conformal Factor f • a1=a2=0.5 • d=10M • q1 = q2 = 0

  15. Scott Hawley, Numerical Relativity 2005, Nov 4 Binding Energy vs. Spin Orientation Error in bkgnd script? Green

  16. Scott Hawley, Numerical Relativity 2005, Nov 4 Future Work Mesh refinement. 2-level works well: 3-level is OK... • Fix angle defn. Verifyl-3 dependence by watching how amplitude of cosine scales with d. Find where Wald’s eq. breaks down. • Fisheye trans. is coded, untested. • Carpet interface needs rewriting • Parallel performance...? (Mike) • Break grids in two... (Meghann) • Use in constrained evolutions • Force Mike to find ISCOs?

  17. Scott Hawley, Numerical Relativity 2005, Nov 4 Conclusions • Elliptic solver solves constraints to 2nd order • Binding energy measurements agree with “S dot S” part of Wald’s formula • Our use of of rotation angle phi is flawed (error in background-definition script?) • Solver shows promise for several future “physics” applications

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