Frequency-Domain Separation and Boundary Conditions

A frequency-domain black-hole calculation becomes numerical only after its separated angular and radial equations are paired with boundary data that select a physical solution. The central task is therefore not merely to integrate an ordinary differential equation: it is to represent horizon regularity, outgoing radiation, and any outer timelike boundary without changing their analytic branches. This page develops that implementation contract for linear fields on stationary backgrounds; the perturbation equations and their physical interpretation remain the subject of Black-Hole Perturbation Theory.

Required background. Ordinary Differential Equations supplies fundamental solutions, boundary-value conditioning, and residual tests.

Helpful background. Formulations of General Relativity fixes the geometric variables, Black-Hole Perturbation Theory develops the separated equations, and Gravitational Radiation and Waveforms explains the radiative observables ultimately reconstructed from the modes.

A separated problem is a boundary-value problem

With the time convention $e^{-i\omega t}$, many perturbations reduce schematically to

$$\frac{d^2X}{dr_*^2}+\left[\omega^2-V(r;\lambda)\right]X=0,$$

where $r_*$ is a tortoise coordinate and $\lambda$ is an angular eigenvalue. This Schrödinger form applies directly to the Schwarzschild master variables below; rotating backgrounds generally have frequency-dependent coefficients and spin-dependent endpoint powers. The equation alone does not identify a retarded mode. For a suitably rescaled field at a nonextremal future horizon, the ingoing phase is $e^{-i(\omega-m\Omega_H)r_*}$; at asymptotically flat infinity the outgoing phase is $e^{+i\omega r_*}$. Additional powers depend on the radial field. The signs follow jointly from the Fourier convention and the direction of constant phase. Copying signs from a source that uses $e^{+i\omega t}$ is a common and consequential failure.

For the axial gravitational Schwarzschild example, define $m_B=GM_{\rm phys}$, $f=1-2m_B/r$, and

$$r_*=r+2m_B\log\!\left(\frac{r}{2m_B}-1\right), \qquad V_\ell=f\left[\frac{\ell(\ell+1)}{r^2}-\frac{6m_B}{r^3}\right], \qquad \ell\ge2.$$

The exterior is $r\gt2m_B$. Both terms in $V_\ell$ have dimensions of inverse length squared, matching $\omega^2$. The horizon and infinity correspond to $r_*\to-\infty$ and $+\infty$. For real positive frequency, constant phase makes $e^{-i\omega(t+r_*)}$ move inward and $e^{-i\omega(t-r_*)}$ move outward. The quasinormal boundary conditions are the analytic continuation of these branches into the lower frequency half-plane. A damped quasinormal mode grows spatially at both ends of a constant-$t$ slice; replacing the prescribed branch by a spatially decaying one would solve a different problem. The dimensionless form of this potential and its boundary factors appear in Leaver 1985, § 2, Eqs. 1–5, pp. 286–287.

The horizon, infinity, an AdS conformal boundary, and a de Sitter cosmological horizon require different domains and admissible data. A computation must therefore record the background parameters, spin weight, $(\ell,m)$ labels, radial variable, angular normalization, branch choices, and boundary condition. Berti, Cardoso, and Starinets 2009, §§3.1–3.3 show how these choices enter the standard black-hole resonance problem; they are inputs, not harmless presentation details.

Factor the known asymptotics

Integrating an exponentially growing and a decaying branch together is ill-conditioned. A more stable representation removes the prescribed endpoint behavior,

$$X(r)=B_H(r,\omega)B_\infty(r,\omega)y(x(r)),$$

where $B_H$ and $B_\infty$ contain the powers and exponentials, while $x$ maps the physical interval to a finite computational domain. For an asymptotically flat exterior, one may map $r\in[r_++\epsilon,R]$ to $x\in[-1,1]$ and supply truncated Frobenius or asymptotic series at both endpoints. The series order, offsets $\epsilon$ and $R$, and branch of every complex power belong in the input record.

Two implementations are useful. Shooting integrates two normalized homogeneous solutions and zeros their Wronskian. Collocation represents $y(x)$ globally and enforces both endpoint equations in a matrix system. Shooting exposes flux and Wronskian checks directly, whereas collocation avoids long one-sided integrations; neither is uniformly superior near extremality or for highly damped modes. Leaver’s construction demonstrates the importance of building the endpoint behavior into the expansion rather than imposing approximate values after discretization Leaver 1985, §§II–III.

Benchmark and convergence checks

A clean benchmark is the axial gravitational Regge–Wheeler problem on Schwarzschild. Let $M_{\rm geom}=GM_{\rm phys}$ be the mass length and scale radii and frequencies by $M_{\rm geom}$; a code may then set $G=c=M_{\rm geom}=1$ for the sustained calculation. For the fundamental $(\ell,n)=(2,0)$ mode, Leaver’s tabulated precision gives

$$GM_{\rm phys}\omega=M_{\rm geom}\omega \simeq0.3736715-0.0889625i.$$

This value, obtained by rescaling Leaver’s $2M_{\rm geom}=1$ entry, tests the complete convention package Leaver 1985, Table 1, p. 293. A flipped Fourier sign reverses the damping sign, and a wrong angular or mass normalization moves both components. Report the complex discrepancy, the differential-equation residual evaluated between collocation points, and a Wronskian or flux balance. Refine at least three independent controls—basis size or step tolerance, boundary offset, and boundary-series order. Agreement under only one refinement can hide a boundary truncation plateau.

The intended convergence table lists $(N,R/M_{\rm geom},\epsilon/M_{\rm geom},k_{\rm series})$, working precision, $M_{\rm geom}\omega$, residual norm, and reference error. Exponential decrease with $N$ is expected only while the factored solution is analytic in the mapped domain; finite differentiability, a nearby complex singularity, or roundoff eventually changes the slope. The result is accepted only when the frequency is stable under independent endpoint changes and the residual decreases in the same regime.

Environment and rerun record

A reproducible implementation pins the ODE or eigensolver library, compiler or interpreter, arbitrary-precision backend, hardware-relevant math settings, and code revision. A configuration file should contain all physical and numerical inputs; the executable sequence should resemble separated-mode --config schwarzschild-l2.toml --output run/, with machine-readable frequency, residual, endpoint coefficients, refinement sweep, and environment metadata. This command names an interface that an implementation should provide; it is not a claim that an executable is supplied on this page.

Independent evidence should change the method. For example, compare a compactified collocation root with a two-sided shooting Wronskian, using independently generated endpoint series. Reusing the same symbolic series and root-polishing routine in both paths does not fully test them. Preserve unrounded complex values and normalization data so another implementation can reconstruct the comparison.

Failure diagnosis

Wrong boundary branches can look numerically converged because the discretized problem is solving a different spectrum. Other warning signs are sensitivity to arbitrary normalization, loss of digits when the two homogeneous solutions become nearly parallel, endpoint residuals much larger than the interior residual, and roots that drift when precision increases. Near extremality, coalescing singular points demand adapted variables and longer series; blindly increasing resolution may only amplify cancellation.

Diagnostic. A solver returns $M_{\rm geom}\omega=0.3736715+0.0889625i$ with a residual of $10^{-12}$ and spectral convergence. Is this strong evidence for the physical fundamental mode?

Solution

No. With $e^{-i\omega t}$, a decaying mode has $\operatorname{Im}\omega\lt 0$. The small residual proves that the discretized equation and its implemented boundary conditions are mutually consistent, but the positive imaginary part indicates a Fourier-sign or outgoing/ingoing branch error. Check constant-phase directions at both endpoints, then compare the corrected solution with an independently normalized shooting Wronskian.

References

• Berti, E., Cardoso, V., and Starinets, A. O. (2009). “Quasinormal modes of black holes and black branes.” Classical and Quantum Gravity 26, 163001. arXiv.
• Leaver, E. W. (1985). “An analytic representation for the quasi-normal modes of Kerr black holes.” Proceedings of the Royal Society A 402, 285–298. DOI.