Degeneracy measures for the algebraic classification of numerical spacetimes

We study the issue of algebraic classification of the Weyl curvature tensor, with a particular focus on numerical relativity simulations. The spacetimes of interest in this context, binary black hole mergers, and the ringdowns that follow them, present subtleties in that they are generically, strictly speaking, type I, but in many regions approximately, in some sense, type D. To provide meaning to any claims of “approximate” Petrov class, one must define a measure of degeneracy on the space of null rays at a point. We will investigate such a measure, used recently to argue that certain binary black hole merger simulations ring down to the Kerr geometry, after hanging up for some time in Petrov type II. In particular, we argue that this hangup in Petrov type II is an artefact of the particular measure being used, and that a geometrically better-motivated measure shows a black hole merger produced by our group settling directly to Petrov type D.

Figure 1

Behavior of the degeneracy measure ${\Delta }_{ij}$ under the tetrad rotation in Eq. (23) for a particular (though essentially arbitrary) choice of roots, stated in the text. Under a rotation through 180 degrees, the root pair that originally seemed more degenerate becomes less degenerate, and the pair that originally seemed less degenerate becomes more degenerate.

Figure 2

Profiles of the behavior of the degeneracy measure ${\Delta }_{ij}$ under tetrad rotations of the form in Eqs. (23) from a quasi-Kinnersley tetrad, for various values of a fiducial time coordinate, assuming that this degeneracy measure grows exponentially in this fiducial time coordinate for the true quasi-Kinnersley tetrad ($\Phi =0$). The peak value grows exponentially in time, by construction, but values well outside the peak decay exponentially in time. The peak sharpens as it grows, so that values slightly offset from the peak grow initially, and decay later.

Figure 3

Behavior over time of the degeneracy measure ${\Delta }_{+-}^{\prime }(\Phi )≔|{\lambda }_{+}^{\prime }-{\lambda }_{-}^{\prime }|$, for roots ${\lambda }_{\pm }^{\prime }$ given by Eq. (30), for a few values of the tetrad rotation parameter $\Phi$. Each curve initially grows exponentially in the time parameter $\tau$ before eventually falling. The more offset the tetrad is from the quasi-Kinnersley tetrad ($\Phi =0$), the sooner this turnaround occurs.

Figure 4

The ringdown after the merger of two equal mass, initially nonspinning holes. The curves show the behavior of the two smallest values of each of the degeneracy measures ${\Delta }_{ij}$ and ${\Theta }_{ij}$, with respect to coordinate time, evaluated at $z=4.5$, $x=y=0$ in an asymptotically inertial coordinate system. The heavier curves show the two values of the ${\Theta }_{ij}$ measure, the lighter curves the ${\Delta }_{ij}$ measure. The solid curves show the smaller values of these measures, and the dashed curves the larger. The symmetries along this axis force the tetrad to satisfy the basic conditions of a “quasi-Kinnersley” tetrad, as described in Sec. 3, and the results of that section explain the initial exponential growth of the higher ${\Delta }_{ij}$ curve.

Figure 5

The two smallest values of the degeneracy measures ${\Delta }_{ij}$ and ${\Theta }_{ij}$ evaluated at the coordinate location $x=4.5$, $y=z=0$ in the ringdown after a merger of equal mass, initially nonspinning black holes, as in Fig. 4. The lighter curves, which are generally the highest and lowest curves, are the two values of the ${\Delta }_{ij}$ measure. The heavier curves represent the ${\Theta }_{ij}$ measure. Initial growth of the larger ${\Delta }_{ij}$ value is still present, but less striking than in Fig. 4. Nonetheless, the two ${\Delta }_{ij}$ values again differ by multiple orders of magnitude, while the two ${\Theta }_{ij}$ values generally differ by only one.

Figure 6

Magnification of the highest curve in Fig. 4. The sharp peaks in the bottom curve before time 8275 imply that principal null directions are occasionally crossing the tetrad legs. The upper curve corroborates this by explicitly showing that the absolute value of the Weyl scalar ${\Psi }_4$ approaches zero at times coincident with these sharp peaks.

Figure 7

$Ł2$ norms of the two smallest values of ${\Delta }_{ij}$ and ${\Theta }_{ij}$, integrated over a thick spherical shell extending from just within the apparent horizon, $r=2.2$ in code coordinates, to an outer boundary at $r=5$ code coordinates. Again, both values of ${\Theta }_{ij}$ fall quickly to zero at the same rate, while the larger ${\Delta }_{ij}$ value hangs up initially, and eventually falls only to a level over 2 orders of magnitude above its smaller counterpart.