Spherical and Spheroidal Ringdown Modes

Waveforms are commonly extracted in spin-weighted spherical harmonics, while separated perturbations of a rotating black hole use spin-weighted spheroidal harmonics. The two bases agree only when the spheroidicity $c=a\omega$ vanishes. Consequently, one spherical waveform mode can contain several spheroidal quasinormal frequencies even in linear theory. This is angular-basis mixing, not by itself nonlinear mode coupling or additional excitation. This page derives the overlap dictionary and states how it changes when quasinormal frequencies make $c$ complex.

Required background. Spacetime Symmetries and Killing Fields supplies axial symmetry, the conserved azimuthal label, and separated angular eigenspaces.

Helpful background. Einstein Equations supplies Kerr perturbations and radiative curvature; Waves, Green Functions, and Geometric Optics supplies non-Hermitian mode expansions; and Gravitational Radiation and Waveforms supplies spin-$-2$ spherical waveform modes and the $e^{-i\omega t}$ convention.

Two angular eigenbases

The unit-normalized spherical basis is

$${}_{s}Y_{\ell m}(\theta,\phi), \qquad \int_{S^2}{}_{s}Y_{\ell m}\,{}_{s}\bar Y_{\ell' m'}d\Omega =\delta_{\ell\ell'}\delta_{mm'}.$$

It diagonalizes the spin-weighted spherical operator and is independent of frequency. For separated Kerr perturbations, the angular Teukolsky operator depends on

$$c=a\omega,$$

where $a=J/M$ has dimensions of length with $c_{\mathrm{light}}=1$, while the dimensionless spin is $a/(GM)$. The frequency $\omega$ follows $e^{-i\omega t}$. Write ${}_sS_{Lm}=S(\theta)e^{im\phi}$. Its angular equation is

$$\begin{aligned} 0={}&\frac1{\sin\theta}\frac{d}{d\theta} \left(\sin\theta\frac{dS}{d\theta}\right)\\ &+\left[c^2\cos^2\theta-2cs\cos\theta+s+A_{sLm}(c) -\frac{(m+s\cos\theta)^2}{\sin^2\theta}\right]S. \end{aligned}$$

Regularity at both poles selects the eigenvalues, with $A_{sLm}(0)=L(L+1)-s(s+1)$. The inner product uses $d\Omega=\sin\theta\,d\theta\,d\phi$. Axial symmetry preserves $m$, but the spheroidal label $L$ need not equal the spherical label $\ell$. This is the angular Teukolsky equation in Berti, Cardoso, and Starinets 2009, p. 16, eq. (25), PDF.

Choose the same $e^{im\phi}$, Condon–Shortley, dyad, and spin-$-2$ phases in both bases. For real $c$, normalize the spheroidal functions to unit Hermitian norm. Their spherical expansion is

$${}_{s}S_{L m}(c) =\sum_{\ell\ge\max(\lvert s\rvert,\lvert m\rvert)} \mu^m_{\ell L}(c)\,{}_{s}Y_{\ell m},$$

with overlap coefficients

$$\mu^m_{\ell L}(c) =\int_{S^2}{}_{s}\bar Y_{\ell m}\,{}_{s}S_{L m}(c)d\Omega.$$

The azimuthal integral enforces the exact $m$ selection rule. In the Schwarzschild or zero-frequency limit,

$$\mu^m_{\ell L}(0)=\delta_{\ell L}.$$

At small nonzero $c$, the term linear in $c\cos\theta$ first couples neighboring spherical $\ell$ values with the same $m$; higher orders spread the expansion farther.

How several frequencies enter one spherical mode

Write a linear ringdown formally as a sum of spheroidal contributions,

$$H(u,\theta,\phi) =\sum_{L,m,n} A_{Lmn}e^{-i\omega_{Lmn}(u-u_0)} {}_{-2}S_{L m}(a\omega_{Lmn};\theta,\phi).$$

Projecting onto the spherical basis gives

$$H_{\ell m}(u) =\sum_{L,n} \mu^m_{\ell L}(a\omega_{Lmn}) A_{Lmn}e^{-i\omega_{Lmn}(u-u_0)}.$$

Thus a fixed spherical $(\ell,m)$ coefficient can contain multiple spheroidal $L$ values and overtones $n$. Each term has its own frequency and therefore its own spheroidicity and mixing coefficient. The Kerr frequency structure and its dependence on $(L,m,n)$ and spin are summarized in Berti, Cardoso, and Starinets 2009, §5.3, pp. 38–41, PDF.

This equation is a basis projection. It does not predict the excitation amplitudes $A_{Lmn}$, the valid ringdown start time, or nonlinear sum and difference frequencies. Those require initial data, radial Green-function information, and a declared fitting regime.

Source-specific dynamical models and detector calibration are excluded here. The overlap matrix translates angular representations after the background spin, frequency, frame, and mode normalization have been supplied; it cannot determine any of those inputs from data.

Complex frequencies and orthogonality

Quasinormal frequencies are complex; stable retarded modes have $\operatorname{Im}\omega<0$. Hence $c=a\omega$ is generally complex and the angular operator is not self-adjoint under the ordinary conjugate inner product. Right spheroidal eigenfunctions at distinct labels or frequencies are not guaranteed to be orthogonal in that product.

The spherical overlap remains well defined because the spherical basis is complete and orthonormal. What changes is the inverse spheroidal expansion. A projection that treats complex-$c$ right eigenfunctions as an orthonormal basis can misassign amplitudes. For one fixed-$c$ angular operator, left and right eigenfunctions can provide a biorthogonal pairing when the spectral problem admits it. Quasinormal terms at different $c=a\omega$ belong to different angular operators, so their adjoint eigenfunctions are not automatically mutually dual. For that mixed-frequency family, retain the full spherical overlap matrix in the fit, or explicitly construct and check a dual for the selected family.

There is also no single spheroidal basis for a broadband ringdown: different overtones have different $a\omega_{Lmn}$. Choosing the fundamental-mode spheroidicity for every term is an approximation whose error must be tested, not a convention identity.

Distinguishing three kinds of mixing

Spherical–spheroidal mixing preserves $m$ in the aligned Kerr frame and couples $\ell$ through frequency-dependent angular overlaps. A rigid frame rotation instead preserves $\ell$ and mixes $m$ unitarily. A BMS translation, boost, or supertranslation can mix both angular and temporal structure. These patterns are diagnostically different.

Before interpreting an extra frequency in $H_{\ell m}$, check the remnant-aligned frame, calculate the relevant $\mu^m_{\ell L}$, repeat with additional spherical modes, and verify the $a\omega\to0$ limit. Only residual content beyond the linear overlap model is evidence for another physical contribution.

A quantitative truncation check compares the norm reconstructed from successively larger spherical ranges. Parseval’s identity gives

$$\sum_\ell\lvert\mu^m_{\ell L}(c)\rvert^2 =\int_{S^2}|{}_sS_{Lm}(c)|^2d\Omega,$$

for real or complex $c$. If each spheroidal function is assigned unit Hermitian norm, the sum is one in both cases. What fails at complex $c$ is mutual orthogonality between different right eigenfunctions, not the spherical norm identity for a single function. Report the chosen normalization and test both the omitted spherical norm and the reconstruction residual. Changing the sign convention for $a$, $m$, or $\omega$ also changes $c$ and must be accompanied by the corresponding spheroidal-function translation.

Exercise

Consider a two-mode toy mixing model. Let $Y_{2m}$ and $Y_{3m}$ abbreviate orthonormal spin-weighted spherical harmonics, and define two normalized angular patterns

$$S_{2m}=\sqrt{1-\varepsilon^2}\,Y_{2m}+\varepsilon Y_{3m},$$ $$S_{3m}=-\varepsilon Y_{2m}+\sqrt{1-\varepsilon^2}\,Y_{3m},$$

with $0\le\varepsilon\lt1$. These exact two-term patterns illustrate projection algebra; Kerr spheroidal harmonics generally have infinitely many spherical coefficients and a mode-dependent $c=a\omega$. If

$$H=Ae^{-i\omega_2u}S_{2m}+Be^{-i\omega_3u}S_{3m},$$

find the spherical $H_{2m}$ and $H_{3m}$.

Solution

Collecting the coefficients of each spherical harmonic gives

$$H_{2m} =\sqrt{1-\varepsilon^2}\,Ae^{-i\omega_2u} -\varepsilon Be^{-i\omega_3u},$$ $$H_{3m} =\varepsilon Ae^{-i\omega_2u} +\sqrt{1-\varepsilon^2}\,Be^{-i\omega_3u}.$$

Each spherical mode contains both frequencies when $\varepsilon\ne0$, even though the toy field contains only the two declared linear contributions. As $\varepsilon\to0$, the mixing disappears and the angular labels coincide. The exercise illustrates how basis mixing can place several frequencies in one spherical mode without introducing a new nonlinear oscillation.

References

• Berti, Emanuele, Vitor Cardoso, and Andrei O. Starinets. “Quasinormal Modes of Black Holes and Black Branes.” Classical and Quantum Gravity 26 (2009): 163001. DOI. Open PDF.