Towards a wave-extraction method for numerical relativity. II. The quasi-Kinnersley frame

The Newman-Penrose formalism may be used in numerical relativity to extract coordinate-invariant information about gravitational radiation emitted in strong-field dynamical scenarios. The main challenge in doing so is to identify a null tetrad appropriately adapted to the simulated geometry such that Newman-Penrose quantities computed relative to it have an invariant physical meaning. In black hole perturbation theory, the Teukolsky formalism uses such adapted tetrads, those which differ only perturbatively from the background Kinnersley tetrad. At late times, numerical simulations of astrophysical processes producing isolated black holes ought to admit descriptions in the Teukolsky formalism. However, adapted tetrads in this context must be identified using only the numerically computed metric, since no background Kerr geometry is known a priori. To do this, this paper introduces the notion of a quasi-Kinnersley frame. This frame, when space-time is perturbatively close to Kerr, approximates the background Kinnersley frame. However, it remains calculable much more generally, in space-times nonperturbatively different from Kerr. We give an explicit solution for the tetrad transformation which is required in order to find this frame in a general space-time.

Figure 1

A transverse frame which is also a quasi-Kinnersley frame: the $\ell$ vector of the transverse frame sees the two principal null directions $z_1$ and $z_2$ as conjugate pair. As the space-time approaches a Petrov type D one, $z_1$ and $z_2$ will converge and, in particular, they converge to $\ell$.

Figure 2

A transverse frame which is not a quasi-Kinnersley frame: the $\ell$ vector of the frame sees the two principal null directions $z_2$ and $z_3$ as conjugate pair. When $z_1$ and $z_2$ converge, they do not converge to $\ell$.

Figure 3

A representation of a function giving the three eigenvalues of the Weyl tensor as a function of $\mathcal{S}$ in the region $|\mathcal{S}-1|<2$. The front lateral axis is the real part of $\mathcal{S}-1$, while the other lateral axis is its imaginary part. The vertical axis is the modulus of the eigenvalue, and the function itself is clearly triple-valued at most points. The figure demonstrates explicitly that the moduli of the eigenvalues do not equal one another except on the branch lines of the underlying complex function, where of course the eigenvalues themselves are equal.