# Presentation: 555570

## 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
- Related to the physical quantities                 by...

- Scott Hawley, Numerical Relativity 2005, Nov 4

## Physical System, cont’d

- We solve the constraint equations....

- ...solve for **, *w*i

- Scott Hawley, Numerical Relativity 2005, Nov 4

## Boundary Conditions

- Dirichlet at inner boundary (edge of excision mask):
- Robin at outer boundary:

- Scott Hawley, Numerical Relativity 2005, Nov 4

## Multigrid Overview

- 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.

- Smooth

- Smooth

- SOLVE

- Smooth

- Smooth

- Restrict

- Restrict

- Prolong (special interp)

- _The “V-Cycle”_

- Prolong

- *also modify RHS, & prolong doesn’t overwrite

- Scott Hawley, Numerical Relativity 2005, Nov 4

## Choice of Background: “Binary Kerr-Schild”

- Kerr-Schild metric is
- This suggests a superposition of *N* BHs via...

- Matzner, Huq, Shoemaker, 1998

- Scott Hawley, Numerical Relativity 2005, Nov 4

## How ‘Close’ is Background?

- Scott Hawley, Numerical Relativity 2005, Nov 4

## Binding Energy

- Contribution due to spin is...  (Wald 1972)
- We’ll see...

- 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...

- 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 *r**ex*. 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

- Scott Hawley, Numerical Relativity 2005, Nov 4

## Test: Schwarzschild Background

- Scott Hawley, Numerical Relativity 2005, Nov 4

## Test: BBH Convergence

- Scott Hawley, Numerical Relativity 2005, Nov 4

## Results...

- Study binding energy as function of BBH spins
- Typical runs:
- Spins a1=a2=0.5,  d=10M.  Hold 2 const, vary 1
- 5133 grids  (4 multigrid levels: 653, 1293, 2563 & 5133)
- 32 Processors.  Bkgrnd takes few mins, solver 3 hrs.
- Domain: -15M to 15M, i.e. x=M/17.
- Excision radius 0.9M
- MADM evaluated at 12M

- Scott Hawley, Numerical Relativity 2005, Nov 4

## Conformal Factor 

- a1=a2=0.5
- d=10M
- 1 = 2 = 0

- Scott Hawley, Numerical Relativity 2005, Nov 4

## Binding Energy vs. Spin Orientation

- Green

- Error in bkgnd script?

- Scott Hawley, Numerical Relativity 2005, Nov 4

## Future Work

- Mesh refinement.  2-level works well:

- 3-level is OK...

- Fix angle defn. Verify *l*-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?

- 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

- Scott Hawley, Numerical Relativity 2005, Nov 4