Approximate Killing vectors on $S^2$

We present a new method for computing the best approximation to a Killing vector on closed 2-surfaces that are topologically $S^2$. When solutions of Killing’s equation do not exist, this method is shown to yield results superior to those produced by existing methods. In addition, this method appears to provide a new tool for studying the horizon geometry of distorted black holes.

Figure 1

The value of $⟨S_{ij}{S}^{ij}⟩$ for the new method is displayed as a solid (black) line and its value when using the KT method is shown as a dashed (red) line. The analogous quantity $⟨2{\Lambda }^2⟩$ when using conformal Killing vectors is shown as a dot-dot-dashed (black) line visible in the inset where a logarithmic scaling is used. The value of the Lagrange multiplier $\Theta$ obtained using the new method is also displayed as a dot-dashed (blue) line and its scale is shown on the right. All data is from one corotating black hole in an equal-mass binary.

Figure 2

The top portion of the figure displays the directions of the additional approximate symmetry axes with $\varphi =0$ in the positive $x$ direction. The axis direction of the Solution 1 is displayed as a solid (black) curve and the direction for Solution 2, with $\pi /2$ subtracted, is displayed as a dashed (red) line. The lines are visually coincident. The lower portion of the figure displays $\Delta \varphi \equiv {\varphi }_1-{\varphi }_2+\pi /2$ showing the degree to which the two axes are approximately orthogonal. All data is from one corotating black hole in an equal-mass binary.

Figure 3

The directions of the additional approximate symmetry axes from one nonspinning black hole in an equal-mass binary. See description for Fig. 2.