Conserved currents for a Kerr black hole and orthogonality of quasinormal modes

We introduce a bilinear form for Weyl scalar perturbations of Kerr. The form is symmetric and conserved, and we show that, when combined with a suitable renormalization prescription involving complex $r$ integration contours, quasinormal modes are orthogonal in the bilinear form for different $(l,m,n)$. These properties are apparently not evident consequences of standard properties for the radial and angular solutions to the decoupled Teukolsky relations and rely on the Petrov type D character of Kerr and its $t\text{−}\varphi$ reflection isometry. We show that quasinormal mode excitation coefficients are given precisely by the projection with respect to our bilinear form. These properties can make our bilinear form useful to set up a framework for nonlinear quasinormal mode coupling in Kerr. We also provide a general discussion on conserved local currents and their associated local symmetry operators for metric and Weyl perturbations, identifying a collection containing an increasing number of derivatives.

Figure 1

Left: Sketch of the complex $r$ contour $C_{*}$ defining the bilinear form on quasinormal modes. The contour cannot be pulled back to the real axis because the integrand crosses (an infinite number of) different sheets associated with the branch points $r_{-}$ and $r_{+}$. Right: Same contour, but in the complex $r_{*}$ plane. Note that this contour cannot be pulled back to the real axis due to the presence of Stokes lines along which the integrand of the bilinear form would diverge.

Figure 2

Numerical check of the orthogonality between two Kerr quasinormal modes with the same $l=m=2$ and different $n=0$, 1 (upper panel) and modes with the same $n=0$, $m=2$ and different $l=2$, 3 (lower panel). We show the result of the numerical evaluation of the bilinear form (55) along the most convergent contour (black points), integrating the complex sections of the contour up to a finite $u_{\mathrm{lower},\mathrm{upper}}$. Because of the difficulty in handling the branch cut in Mathematica, the lower integration limit $u_{\text{lower}}$ sets the overall accuracy of this numerical evaluation of the bilinear form. We also show an exponential fit converging to zero as $u_{\text{lower}}\to -\infty$ (red line). We use the mode solutions (normalized at the horizon) provided by the Black Hole Perturbation Toolkit [61]. In this example, we set $M=1$, $a=0.7$, $r_{*1}=-8$, and $r_{*2}=5$.