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Spin Dependence in Computational Black Hole Initial Data

Spin Dependence in Computational Black Hole Initial Data. cf. gr-qc/0604100. Scott Hawley* † , Richard Matzner*, Michael Vitalo* *Center for Relativity, University of Texas at Austin † After Aug 1: Chemistry & Physics Dept, Belmont University, Nashville TN. Overview.

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Spin Dependence in Computational Black Hole Initial Data

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  1. Spin Dependence in Computational Black Hole Initial Data cf. gr-qc/0604100 Scott Hawley*†, Richard Matzner*, Michael Vitalo* *Center for Relativity, University of Texas at Austin †After Aug 1: Chemistry & Physics Dept, Belmont University, Nashville TN

  2. Scott Hawley, New Frontiers in NR, July '06 Overview • Superpose two Kerr-Schild BHs, solve constraints • Define Binding Energy Eb • Rotate one hole’s spin axis, study how Eb varies

  3. Scott Hawley, New Frontiers in NR, July '06 Initial Data Problem • York-Lichnerowitz conformal-traceless method • Background is “superposed Kerr-Schild” • Not conformally flat • In present work, holes are not moving • Happy to let interested parties survey param. space! • Solve four constraint equations for {f,wi } numerically • Finite difference techniques • Excise apparent horizon interiors -- Dirichlet (f=1) inner BC’s • Robin outer boundary conditions • Parallel multigrid (O(n)) elliptic solver “TEXMEX” • Standalone code. Only use Cactus in post-processing (Thornburg’s AHFinderDirect) • Present work is unigrid. FMR almost finished. Cactus interface in the works.

  4. Scott Hawley, New Frontiers in NR, July '06 Form of Initial Data Problem • York-Lichnerowitz conformal-traceless method: • Use some “initial guess” / “background” metric and extrinsic curvature which don’t quite satisfy constriants • Related to the physical quantities by...

  5. Scott Hawley, New Frontiers in NR, July '06 Initial Data Problem cont’d • We then solve the constraint equations.... ...Solve for f, wi numerically via multigrid method

  6. Scott Hawley, New Frontiers in NR, July '06 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, New Frontiers in NR, July '06 Boundary Conditions • Dirichlet at inner boundary (edge of excision mask): • Robin at outer boundary:

  8. Scott Hawley, New Frontiers in NR, July '06 Multigrid (Elliptic Solver) Overview 2:1 grid spacing 2:1 grid spacing • Multigrid (Brandt 1978), an O(N) method, basically* works by applying relaxation/smoothing operators successively at different resolutions to “smooth away” different wavelength-components of the error. *also modify RHS, & prolong doesn’t overwrite The “V-Cycle” Smooth Smooth Restrict Smooth Smooth Prolong SOLVE Prolong (special interp) Restrict

  9. Scott Hawley, New Frontiers in NR, July '06 How ‘Close’ is Background?

  10. Scott Hawley, New Frontiers in NR, July '06 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, New Frontiers in NR, July '06 Configurations Studied • 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

  12. Scott Hawley, New Frontiers in NR, July '06 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 *new* Carpet interface • Still testing FMR, Fisheye...

  13. Scott Hawley, New Frontiers in NR, July '06 Conformal Factor f • a1=a2=0.5 • d=10M • q1 = q2 = 0

  14. Scott Hawley, New Frontiers in NR, July '06 Binding Energy • Contribution due to spin is... (Wald 1972) We’ll see...

  15. Scott Hawley, New Frontiers in NR, July '06 Test: BBH Convergence

  16. Scott Hawley, New Frontiers in NR, July '06 Result: BE vs. 1, for “yz series” BHs at x=+/-5m Spins a1=a2=0.5 Domain: +/-15m 5133 grid points • -cos dependence agrees w/ Wald’s term.

  17. Scott Hawley, New Frontiers in NR, July '06 Binding Energy vs. Spin Orientation

  18. Scott Hawley, New Frontiers in NR, July '06 Additional Effect: Due to Mass Quadrupole Moment • In terms of the Newtonian potential for Kerr: • In xz series, BH at x=+d/2 “sees” different latitudes of hole at x=-d/2 as it rotates • In yz series, same latitude is always exposed, thus no effect.

  19. Scott Hawley, New Frontiers in NR, July '06 Varying Spin Magnitude a1 • Cut spin in half  B1  B1/2, B2  B2/4 • As expected

  20. Scott Hawley, New Frontiers in NR, July '06 Effects of Varying BBH Separation d • Plot curve fit coefficients vs. d • Expect B0~d-1, B1 ~ B2 ~ d-3.Not what we find: ?  B0

  21. Scott Hawley, New Frontiers in NR, July '06 FMR - Single BH Convergence

  22. Scott Hawley, New Frontiers in NR, July '06 FMR BBH gxx f

  23. Scott Hawley, New Frontiers in NR, July '06 FMR BBH p.2 • Outer boundaries now “way out” at +/- 36m

  24. Scott Hawley, New Frontiers in NR, July '06 FMR BBH Convergence

  25. Scott Hawley, New Frontiers in NR, July '06 Conclusions • Binding energy measurements agree with “S dot S” part of Wald’s formula • Additional effect due to mass quadrupole moment, varies w/ spin a as expected. • Variations with separation do not behave as expected. Suspect outer boundary trouble throwing off ADM mass measurments • Solver shows promise for several future “physics” applications • Read the paper! gr-qc/0604100

  26. Scott Hawley, New Frontiers in NR, July '06 Future Work • Verifyd-3 dependences by pushing outer boundaries “way out” (Also: Fisheye trans. is coded, untested) • Survey more parameters: • Rotate both holes (not just one) • Study spin-orbit coupling • Find ISCOs? • (re)Write Carpet interface • Use in constrained evolutions • Parallel performance...?

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