Mixing of spherical and spheroidal modes in perturbed Kerr black holes

The angular dependence of the gravitational radiation emitted in compact binary mergers and gravitational collapse is usually separated using spin-weighted spherical harmonics ${{}_{s}Y}_{\ell m}$ of spin weight $s$, that reduce to the ordinary spherical harmonics $Y_{\ell m}$ when $s=0$. Teukolsky first showed that the perturbations of the Kerr black hole that may be produced as a result of these events are separable in terms of a different set of angular functions: the spin-weighted spheroidal harmonics ${{}_{s}S}_{\ell mn}$, where $n$ denotes the “overtone index” of the corresponding Kerr quasinormal mode frequency ${\omega }_{\ell mn}$. In this paper we compute the complex-valued scalar products of the ${{}_{s}S}_{\ell mn}$’s with the ${{}_{s}Y}_{\ell m}$’s (“spherical-spheroidal mixing coefficients”) and with themselves (“spheroidal-spheroidal mixing coefficients”) as functions of the dimensionless Kerr parameter $j$. Tables of these coefficients and analytical fits of their dependence on $j$ are available online for use in gravitational-wave source modeling and in other applications of black-hole perturbation theory.

Figure 1

Real (left panel) and imaginary part (right panel) of the spherical-spheroidal mixing coefficients ${\mu }_{m\ell {\ell }^{\prime }{n}^{\prime }}$ for $m=2$, $n^{\prime }=0$. Here we consider the dominant multipoles ($\ell =2,{\ell }^{\prime }=3$) (very thick lines in the upper half of each panel), ($\ell =3,{\ell }^{\prime }=2$) (thick lines in the lower half of each panel) and ($\ell ={\ell }^{\prime }=2$) (thin lines in each panel). Solid lines (black online) correspond to the numerical calculation presented in this paper; dash-dotted lines (red online, almost indistinguishable from the black lines) are the fitting relations of Eq. (11); dashed lines (blue online) are the leading-order Press-Teukolsky approximation [5]. In the case $\ell ={\ell }^{\prime }=2$ the mixing coefficients are close to unity, so we actually plot $1-{\mu }_{2220}$; furthermore, to leading order the Press-Teukolsky calculation predicts that they are exactly equal to unity, so their approximation is not shown in the plot.

Figure 2

Trajectories traced by the mixing coefficients ${\mu }_{m\ell {\ell }^{\prime }{n}^{\prime }}$ with $\ell ={\ell }^{\prime }=2$, $m=2$ (left panel) and $m=1$ (right panel) as the Kerr parameter increases from $j=0$ (where ${\mu }_{m22n^{\prime }}=1$) to the nearly extremal Kerr limit. Each curve can be thought of as a parametric plot, where the parameter is $j$. Filled circles denote the following discrete values of $j$: $j=0,0.1,0.2,\dots ,0.9,0.99$. To guide the eye, along each trajectory the dimensionless Kerr parameter $j=0.5$ is denoted by a hollow circle.

Figure 3

Trajectories traced by the mixing coefficients ${\mu }_{m\ell {\ell }^{\prime }{n}^{\prime }}$ in the complex plane as the Kerr parameter increases from $j=0$ (where ${\mu }_{m\ell {\ell }^{\prime }{n}^{\prime }}=0$ for $\ell \ne {\ell }^{\prime }$) to $j=1$. Panels in the top row refer to $(\ell ,{\ell }^{\prime })=(2,3)$, those in the bottom row to $(\ell ,{\ell }^{\prime })=(3,2)$; left panels are for $m=2$, right panels for $m=-2$.

Figure 4

Absolute values of the mixing coefficients $|{\mu }_{m\ell {\ell }^{\prime }{n}^{\prime }}|$ with $\ell =m=2$, $n^{\prime }=0$ and different values of ${\ell }^{\prime }=3,\dots ,7$, illustrating the roughly exponential decay of the mixing coefficients with $|{\ell }^{\prime }-\ell |$.

Figure 5

Trajectories described in the complex plane by the spheroidal-spheroidal mixing coefficients ${\alpha }_{m\ell {\ell }^{\prime }n{n}^{\prime }}(j)$ with $\ell ={\ell }^{\prime }=m=2$, $n=0$ and different values of the overtone index $n^{\prime }\ge 1$ as the dimensionless spin increases from $j=0$ to $j=1$.

Figure 6

Trajectories described in the complex plane by the spheroidal-spheroidal mixing coefficients ${\alpha }_{m\ell {\ell }^{\prime }n{n}^{\prime }}(j)$ with: (1) $\ell =2$, ${\ell }^{\prime }=3$, $n=n^{\prime }=0$ and $-\ell \le m\le \ell$ (left panel); (2) $\ell =2$, ${\ell }^{\prime }=3$, $|m|=2$, and either $(n,n^{\prime })=(0,1)$ or $(n,n^{\prime })=(1,0)$.