Green Functions, Branch Cuts, and Tails

The retarded black-hole response is not generally a sum of quasinormal modes. In the complex-frequency plane it also contains prompt propagation and, for long-range potentials, a branch-cut contribution that produces power-law tails. Numerically demonstrating this decomposition requires two genuinely different calculations: a contour-based frequency-domain reconstruction and a causal time evolution. This page treats their implementation and cross-checks; the analytic Green function and physical perturbation content remain with Black-Hole Perturbation Theory.

Required background. Ordinary Differential Equations supplies fundamental solutions, Wronskians, transforms, and boundary-value conditioning.

Helpful background. Formulations of General Relativity fixes the background variables, Black-Hole Perturbation Theory develops the Green function, and Gravitational Radiation and Waveforms supplies the radiation conventions.

Build the retarded Green function

For the frequency-domain operator

$$\left[\frac{d^2}{dr_*^2}+\omega^2-V_\ell(r)\right]\widetilde G_\ell (r_*,r_*';\omega)=\delta(r_*-r_*'),$$

let $X_{\rm in}$ be ingoing at the horizon and $X_{\rm up}$ outgoing at infinity. Then

$$\widetilde G_\ell(r_*,r_*';\omega)= \frac{X_{\rm in}(r_{\lt},\omega)X_{\rm up}(r_{\gt},\omega)}{W(\omega)},$$

Here $r_{\lt}=\min(r_*,r_*')$ and $r_{\gt}=\max(r_*,r_*')$, while primes differentiate $r_*$. The Wronskian $W=X_{\rm in}X_{\rm up}'-X_{\rm up}X_{\rm in}'$ is independent of matching radius for exact homogeneous solutions. Numerically, its radial drift is a valuable residual. A zero of $W$ gives a pole when it is not canceled by the numerator or a singular normalization; a branch point arises because the outgoing solution for a long-range potential is not single-valued around $\omega=0$.

Two short checks fix the normalization. The field is continuous at the source point and its derivative jump is

$$[\partial_{r_*}\widetilde G]_{r_*'} =\frac{X_{\rm in}X_{\rm up}'-X_{\rm in}'X_{\rm up}}{W}=1.$$

Rescaling either homogeneous solution multiplies numerator and denominator by the same factor, so $\widetilde G$ is unchanged. These tests detect a swapped Wronskian sign or an inconsistent normalization before contour quadrature begins.

For the time-domain operator $-\partial_t^2+\partial_{r_*}^2-V$, use the inverse transform

$$G(t)=\frac{1}{2\pi}\int_{-\infty+ic}^{\infty+ic} e^{-i\omega t}\widetilde G(\omega)\,d\omega, \qquad c\gt0,$$

with the line above all singularities. Deforming it into the lower half-plane for $t\gt0$ gives clockwise pole contours, so each simple pole contributes $-i\,\operatorname{Res}[e^{-i\omega t}\widetilde G]$. Spatial arguments are suppressed here. For initial field $\Psi_0$ and initial velocity $\Pi_0$, the unilateral transform of the homogeneous evolution satisfies

$$\widetilde\Psi(r_*;\omega)=\int\widetilde G(r_*,r_*';\omega) [i\omega\Psi_0(r_*')-\Pi_0(r_*')]\,dr_*'.$$

The source factor must be included when comparing a Green-function response with an initial-data evolution. Leaver 1986, §II.A–B and §II.B.3–4 gives the black-hole construction and the distinct pole and cut pieces. The cut orientation and the value on each bank must be recorded; changing them without changing signs everywhere else can reverse the tail amplitude.

Compute the pole and cut pieces

For the pole sum, locate zeros of $W$, compute residues with a complex derivative or contour integral, and retain enough modes for the requested time window. More overtones do not automatically improve early-time reconstruction because the series can be asymptotic and the prompt arc remains. Report the mode set, residue normalization, contour, and time interval.

For the cut, parameterize $\omega=-i\sigma$ with $\sigma\gt 0$ and evaluate the jump

$$\Delta\widetilde G(-i\sigma)= \widetilde G(-i\sigma+0)-\widetilde G(-i\sigma-0).$$

The $+0$ and $-0$ denote approach from positive and negative real frequency, respectively. With this right-bank-minus-left-bank convention, the cut contribution is

$$G_{\rm cut}(t)=-\frac{i}{2\pi}\int_0^\infty e^{-\sigma t}\Delta\widetilde G(-i\sigma)\,d\sigma.$$

Use logarithmic resolution near $\sigma=0$, where late times concentrate the weight. Independently refine the homogeneous ODE tolerance, cut sampling, upper cutoff, and quadrature. An erratum corrected signs and factors in the original presentation, so implementations must use the corrected record rather than silently copying the first printing Leaver 1988, erratum p. 725.

Before adding a potential, set $V=0$. Then $\widetilde G=e^{i\omega\lvert r_*-r_*'\rvert}/(2i\omega)$ and inversion gives $G=-\tfrac12\Theta(t-\lvert r_*-r_*'\rvert)$, where $\Theta$ is the Heaviside step. Its minus sign follows from the chosen time-domain operator. Together with the derivative jump, this free-wave result independently checks the Fourier factor and contour orientation.

Time-domain benchmark

A discriminating test evolves generic compact massless scalar data on Schwarzschild, using the reduced scalar field defined on the time-domain page. At a fixed timelike radius, the monopole has a $t^{-3}$ tail; at future null infinity it has a $u^{-2}$ tail. Special data can cancel the leading coefficient and decay faster, so record both initial field and velocity. Reconstruct the same signal from prompt, pole, and cut pieces and compare without arbitrary time or amplitude alignment. The pole contribution should reproduce the intermediate ringdown, while the cut controls the late power law. The sum, not either component alone, must agree with the causal evolution within stated error.

Use at least three time-domain grids and three cut quadratures. Track the local power index, waveform norm, Wronskian drift, flux balance, and decomposition residual

$$\epsilon(t)=\frac{\lvert\Psi_{\rm TD}(t)- [\Psi_{\rm prompt}(t)+\Psi_{\rm poles}(t)+\Psi_{\rm cut}(t)]\rvert} {\max_t\lvert\Psi_{\rm TD}(t)\rvert}.$$

Move the outer boundary and extraction location. A tail that changes at the boundary light-crossing time is contaminated. The benchmark’s validity is linear propagation on the declared stationary background; it does not show that a finite list of modes describes a nonlinear merger signal.

Environment and failure criteria

Pin the ODE library, complex branch implementation, precision backend, asymptotic-series order, contour geometry, quadrature rule, frequency and time grids, initial pulse, extraction radius, and code revision. A reproducible interface could be green-response --config scalar-l0.toml --output run/, emitting both contour integrands, pole residues, time waveforms, convergence sweeps, residuals, and environment metadata. This specifies a future rerun; it is not a claim that a program accompanies this prose.

Reject a decomposition if the Wronskian depends on matching radius, the pole sum agrees only after fitted alignment, the cut changes sign under an equivalent contour orientation, or the late exponent lacks a refinement plateau. Cancellation between large pole and cut contributions can exhaust machine precision; increase precision and evaluate the total by a second contour. A small time-domain truncation error cannot compensate for a wrongly defined analytic branch.

Archive the undeformed and deformed contour samples, not only the final waveform. Those intermediate values allow an independent implementation to locate missed arcs, residues, or orientation errors and to distinguish quadrature failure from an incorrect analytic continuation.

Diagnostic

The pole sum matches the waveform through ringdown, and the cut integral gives a $t^{-3}$ monopole tail. Their sum differs from the time evolution by five percent, but multiplying all residues by a fitted constant removes the discrepancy. Is the decomposition verified?

Solution

No. A fitted normalization hides an inconsistency in the Green-function jump, source normalization, or waveform variable. Check the derivative jump at $r_*=r_*'$, Wronskian convention, corrected cut signs, and Fourier inversion factor. Verification requires absolute agreement from declared inputs, plus convergence of both representations, without arbitrary alignment.

References

• Leaver, E. W. (1986). “Spectral decomposition of the perturbation response of the Schwarzschild geometry.” Physical Review D 34, 384–408. DOI.
• Leaver, E. W. (1988). “Erratum: Spectral decomposition of the perturbation response of the Schwarzschild geometry.” Physical Review D 38, 725. DOI.