Numerical Relativity at UT Brownsville

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Motivation

Until relatively recently, full-3D numerical simulations of strong-field gravitational wave sources, such as black-hole binaries, were extremely short-lived, being killed by numerical instabilities well before a single binary orbital period. Even head-on and grazing collisions could not be fully simulated long enough to extract the full leading-order gravitational waveforms. Lazarus was designed to make the most of disparate resources in computational GR:

Far Limit: Binary initial data from e.g. Post-Newtonian calculations Full Numerical: Evolve metric ($\gamma$ij, Kij) using 3+1 scheme (ADM, BSSN ...) Close Limit: At late times, extract radiation from Kerr ($\Psi$4, ∂tΨ4) background, and evolve via the Teukolsky equation

In the FN-CL transition, one way to separate the radiation from the background is to use the Newman-Penrose Weyl scalars --- contractions of the Weyl curvature tensor with combinations of vectors from a null tetrad. With the correct tetrad, $\Psi$₂ will describe the underlying black hole, while $\Psi$₄ will contain the outgoing radiation.

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Numerical tetrad → Kinnersley tetrad

For perturbations of Kerr, we should use the Kinnersley tetrad to calculate outgoing radiation $\Psi$₄^Kin. However, in 3+1 evolution, we only have a coordinate tetrad.

For exact Kerr, can simply transform from coordinate to Kinnersley tetrad. Then the Kinnersley Weyl scalars are just linear combinations of the coordinate scalars: $\Psi$_i^Kin = ∑_j κ^2-i F_ij Ψ_j^num where $\kappa$ (spin-boost) and $F$_ij (mixing terms) are known functions of $(M,\; a)$ and $(r$_BL, θ_BL).

For evolved space-time, we use the same transformations as above to get the Kinnersley $\Psi$₄^Kin from the numerical $\Psi$₄^num.

Disadvantage: $\kappa$ & $F$_ij require knowledge of Kerr background parameters and coordinates. We want a more general way to get to Kinnersley.

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Numerical → quasi-Kinnersley → Kinnersley

The Kinnersley tetrad is transverse: for perturbations of Kerr, the ``longitudinal'' Weyl scalars vanish: $\Psi$₁^Kin = Ψ₃^Kin.

We can find transverse frames for a space-time by finding the eigen-2-forms of the Weyl tensor. Project onto the numerical hypersurface --- $$C_ab ≡ C_ambn τ^m τⁿ --- to get:

$$C^a _b σ^b = λ σ^a
Decompose this on a basis of spatial unit vectors to get: $$Q V = λ V, where $$Q is a (3 x 3) complex matrix constructed from $\Psi$_i^num.

The three $\lambda$ define three transverse frames; only one $\lambda$ will be analytic near $$S = 1; it defines the quasi-Kinnersley (qK) frame. [Notes: In fact, this $\lambda \; =\; 2\; \Psi$₂^Kin. Also, as long as $|$S - 1| < 1, the qK $\lambda$ will simply be the largest one.]

The corresponding unit e-vector $$V yields new null tetrad -- a member of the qK frame. This will differ from Kinnersley tetrad only by a spin-boost.

We can use the new tetrad to recalculate Weyl scalars: $\Psi$_i^Kin. The new qK Weyl scalars are ``closer'' to Kinnersley:

$\Psi$₂^Kin = Ψ₂^qK , Ψ₄^Kin = μ^{- 2} Ψ₄^qK.

Remaining issue: How can we determine the correct spin-boost $\mu$?

Partial solution: For exact Kerr, starting with the numerical $\Psi$_i^num and finding the correct ($\lambda ,$V), we still need $\mu =\kappa$ to rotate to the Kinnersley tetrad. Use this exact-Kerr $\mu$ to transform the numerical $\Psi$₄^qK; much simpler than old transformation -- no mixing between scalars.

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This allows us to extract information about the background Kerr solution, and through this, to identify the radiation present. These techniques will enable us to validate previous Lazarus results, and approach more strongly perturbed situations in a robust manner.

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A more detailed description of old and new Lazarus, with some preliminary results, can be seen in a recent (September 2005) talk [PDF] given at a Penn State Sources & Simulations Seminar.