A radial scalar product for Kerr quasinormal modes

TL;DR

This work introduces a radial scalar product for Kerr quasinormal modes to deepen the spectral understanding of Teukolsky's radial equation under QNM boundary conditions. The scalar product is shown to be evaluable by direct integration or analytic continuation, with a practical emphasis on the latter through monomial moments and confluent hypergeometric representations. Applying the product to confluent Heun polynomials reveals orthogonality at fixed polynomial order and a link between eigenvalues and scalar-product structure, culminating in a proposal that Teukolsky's radial equation is, in principle, exactly tri-diagonalizable via a canonical confluent Heun polynomial basis. These results pave the way for canonical representations of radial functions, potential completeness in the radial domain, and spectral-decomposition approaches to QNM excitations with direct relevance to gravitational-wave ringdown analyses.

Abstract

A scalar product for quasinormal mode solutions to Teukolsky's homogeneous radial equation is presented. Evaluation of this scalar product can be performed either by direct integration, or by evaluation of a confluent hypergeometric functions. The related scalar product will be useful for better understanding analytic solutions to Teukolsky's radial equation, particularly the quasi-normal modes, their potential spatial completeness, and whether the quasi-normal mode overtone excitations may be estimated by spectral decomposition, rather than fitting. With that motivation, the scalar product is applied to confluent Heun polynomials where it is used to derive their peculiar orthogonality and eigenvalue properties. A potentially new relationship is derived between the confluent Heun polynomials' scalar products and eigenvalues. Using these results, it is shown for the first time that Teukolsky's radial equation (and perhaps similar confluent Heun equations) are, in principle, exactly tri-diagonalizable. To this end, "canonical" confluent Heun polynomials are conjectured.

Figures (7)

• Figure 1: Examples integration paths (white curves) against weight function contours (colored contours) for the $(s,\ell,m,n)=(-2,2,2,3)$ Quasi-Normal Mode (QNM) at three black hole (BH) spins: (left) $a/M=0.7$, $M\tilde{\omega}=0.4713- 0.5843i$, (center) $a/M=0.99$, $M\tilde{\omega}=0.8695-0.2058i$, and (right) $a/M=0.999$, $M\tilde{\omega}=0.9558-0.07299i$. Contours for the real and imaginary parts of the weight function, $\text{W}(\xi)=\mathrm{Re}(\text{W})+i\,\mathrm{Im}(\text{W})$, are shown. For $\mathrm{Re}(\text{W})$, colors between cyan and magenta represent dimensionless values between $-50$ and $50$ respectively. For $\mathrm{Im}(\text{W})$, colors between magenta and yellow represent values between $-50$ and $50$ respectively. Convergence of contours at spatial infinity ($\mathrm{Re}(\xi)=1$) corresponds to the divergence of the weight function. Horizontal contours along $\mathrm{Re}(\xi)<0$ and $\mathrm{Re}(\xi)>0$ are branch cuts. For left, center and right panels, coordinate parameters, $(K_0,K_2)$, are $(0.8530,23.4621)$, $(0.7105,1.4010\times 10^{-4})$ and $(0.1585,2.5088\times 10^{-4})$, respectively. Dotted horizontal and vertical lines mark the respective locations of the real and imaginary axes.
• Figure 2: Examples integration paths (white curves) against weight function contours (colored contours) for the $(s,\ell,m,n)=(-2,2,2,3)$ Quasi-Normal Mode (QNM) at black hole (BH) spin $a/M=0.99$. (top) A zoom-in of the central panel of Fig. \ref{['F1']} with axes centered around the black hole (BH) horizon at $\xi=0$. (bottom) Visualization of the coordinate transformed integration path and effective weight function for the same case. Contour formatting and coordinate parameters are identical to those in Fig. \ref{['F1']}.
• Figure 3: Example distributions of absolute monomial moments, $|\langle \xi^k\rangle|$ v.s. monomial order, $k$, for the $(\ell,m)=(2,2)$ Quasi-Normal Mode (QNM) at black hole (BH) dimensionless spin of $a/M=0.7$. Left, central and right panels show results for $n$ of $0$, $2$ and $6$, respectively. The corresponding Quasi-Normal Mode (QNM) frequencies are $M\tilde{\omega}=0.5326-0.0808i$, $M\tilde{\omega}=0.4999-0.4123i$, and $M\tilde{\omega}=0.4239-1.0954i$ respectively. At focus is the effect of negating the spin weight, $s$, for increasing overtone label. For all cases shown, $|\langle \xi\rangle|=1$ has been enforced. Closed circles show points for $s=+2$, and open circles show points for $s=-2$.
• Figure 4: The first four confluent Heun polynomials for $p=k$, spin weight $s=-2$, black hole dimensionless spin $a/M=0.86$, and $\tilde{\omega}$ defined by the $(\ell,m,n)=(3,3,1)$ Quasi-Normal Mode (QNM). The corresponding Quasi-Normal Mode (QNM) frequency is $M\tilde{\omega}=0.9840-0.2140i$. Polynomial orders $p\in \{0,1,2,3\}$ are shown as a function of the physically valued fractional radius $\xi$. The top left and right panels shows the real and imaginary parts of each polynomial, respectively. Bottom left and right panels show the corresponding polynomial amplitudes and phases, respectively.
• Figure 5: Distribution of roots for confluent Heun polynomials, $y_{pk}(\xi)$, for three cases, all with $s=-2$, $a/M=0.7$, and $\ell=m=2$. The three cases differ in polynomial order, $p$, and the frequency, $\tilde{\omega}$, used: (left) $y_{33}$, the $n=0$ Quasi-Normal Mode (QNM) frequency, $M\tilde{\omega}=0.5326-0.0808i$, was used, (center) $y_{77}$, the $n=0$ Quasi-Normal Mode (QNM) frequency, $M\tilde{\omega}=0.4713-0.5843i$, was used, and (right) $y_{20,20}$, the $n=12$ Quasi-Normal Mode (QNM) frequency, $M\tilde{\omega}=0.4155-2.5050i$, was used. Polynomial roots are marked by white open circles. Roots were calculated using numpy.rootsHorm:1999ma2020NumPy-Array. Values of polynomial phase, $\arg(y_{pk})$, are shaded from blue to yellow, indicating values between $-\pi$ and $\pi$, respectively. Dotted horizontal and vertical lines mark the respective locations of the real and imaginary axes.