Green Functions, Scattering, and Retarded Response

Black-hole response is a causal initial-boundary-value problem before it is a spectral decomposition. This chapter starts with the global retarded inverse, constructs its separated radial kernel from endpoint-normalized solutions, organizes scattering channels and fluxes, and only then continues frequency into the complex plane. The final inverse transform distinguishes prompt propagation, resonant ringing, and branch-cut tails without claiming that quasinormal poles form a complete basis. The recurring example is a massless scalar or a Schwarzschild master field; massive thresholds, rotating higher-spin flux factors, and timelike outer boundaries require the extensions identified below.

Helpful background. Black-Hole Geometry and Mechanics fixes horizons and asymptotic ends; PDEs and Green Functions for Gravitation supplies distributions and causal inverses; Formulations of General Relativity supplies separated operators and constraints; Events, Intervals, and Minkowski Geometry fixes wave and energy orientation.

From causal inverse to response

Let $P$ be the linear wave operator after any necessary gauge reduction; for a scalar, a typical choice is $P=\Box_g-U(x)$ with a smooth potential $U$. The central object is $G_{\mathrm{ret}}(x,x')$, the response at an observation event $x$ to a unit impulse at a source event $x'$. It is a bidistribution: it can contain delta-function wavefronts and must be integrated against smooth sources. It satisfies

$$P_xG_{\mathrm{ret}}(x,x')=\delta_g(x,x'), \qquad \operatorname{supp}G_{\mathrm{ret}}(\,\cdot\,,x')\subseteq J^+(x').$$

Here $J^+(x')$ is the causal future of $x'$, including its null cone, and $\delta_g$ is normalized with the invariant volume element $dV_g$. For a smooth compactly supported source $S$, the field with no incoming homogeneous part is

$$\Phi(x)=\int G_{\mathrm{ret}}(x,x')S(x')\,dV_g(x'), \qquad P\Phi=S.$$

Applying $P$ under the integral recovers $S$ through the delta function; the support condition prevents a response outside the source’s causal future. Nonzero initial data add a homogeneous solution, or can be included as a distributional source on the initial slice.

These statements need a domain. A smooth normally hyperbolic operator—one whose highest-derivative part is the metric wave operator—on a globally hyperbolic spacetime without boundary has unique causal Green operators. The key is uniqueness of the homogeneous Cauchy problem: the difference of two causal inverses has zero earlier data and must vanish. See Bär, Ginoux, and Pfäffle 2007, Corollary 3.4.3, pp. 88–89. Their future-supported operator is called “advanced.” Their principal wave sign is also opposite to ours (§ 1.5, pp. 33–34): apply the theorem to $-P$ and negate its inverse to obtain the retarded operator used here. The retarded Green-function page spells out both translations and the Cauchy construction.

For a stationary asymptotically flat black-hole exterior and a massless field, the separated retarded construction uses a solution ingoing at the future horizon and one outgoing at future null infinity. These conditions are selected first where the frequency transform converges and are then continued. In anti-de Sitter space or at a timelike worldtube, an additional boundary condition is part of the problem; the boundary-free theorem does not supply it.

Four distinctions recur. The local Hadamard form is valid within a normal neighborhood; the global retarded Green function propagates through caustics and curvature scattering. The real-frequency scattering problem is not yet its analytic continuation. A branch point cannot be removed by redrawing a cut, but cut placement, bank order, and the separate contour contributions depend on the representation. The discontinuity must be defined on that chosen cut; only the full reconstructed response is independent of a consistent redrawing. Finally, a zero of a regular endpoint Wronskian can produce a pole of the Green function, not a pole of the Wronskian itself. Pole data alone do not prove a complete time-domain expansion.

For the massless wave phases with time factor $e^{-i\omega t}$, at infinity $e^{-i\omega r_*}$ is incoming and $e^{+i\omega r_*}$ outgoing. At a Schwarzschild future horizon, $e^{-i\omega r_*}$ is ingoing. For rotation, replace the horizon frequency by $k_H=\omega-m\Omega_H$, where $m$ is the azimuthal integer and $\Omega_H$ the horizon angular velocity. Here $r_*$ is the tortoise coordinate, increasing from the horizon toward infinity. Field-dependent powers of $r$ or the horizon function can accompany these phases; they are not unit-flux normalizations.

Six pages in logical order

Read the chapter first in this sequence.

1. Retarded Green Functions on Black-Hole Exteriors states support, existence and uniqueness hypotheses, boundary dependence, and the limitation of the local Hadamard form.
2. Mode Green Functions and Wronskians assembles the radial kernel and verifies its unit derivative jump directly.
3. In, Up, Out, and Down Scattering Bases separates incidence from infinity and emergence from the past horizon, then relates past and future bases.
4. Reflection, Transmission, and Absorption translates amplitudes into signed flux ratios and derives the superradiant balance law.
5. Analytic Structure: Poles, Cuts, and Thresholds identifies resonances, instabilities, continuum cuts, and channel thresholds on specified sheets.
6. Inverse Transforms and Causal Time-Domain Response fixes the Bromwich contour and the assumptions behind prompt, pole, and cut contributions.

The ordering matters. Starting from a catalog of frequencies hides which operator, source normalization, and boundary problem those frequencies belong to. Starting from the causal inverse lets every later denominator, endpoint amplitude, and contour inherit a physical meaning.

Two reader routes

Route	Path	Main diagnostic
Causal propagation	Pages 1 → 2 → 5 → 6	Apply the operator to the product Green function, recover the delta jump, identify its sheet singularities, and close the inverse contour with the correct time orientation.
Scattering and flux	Pages 2 → 3 → 4 → 5	Fix endpoint amplitudes, derive the Wronskian relations, insert horizon/infinity wave numbers, and distinguish amplification from ordinary absorption.

The causal route is best for sourced evolution and late-time response. It keeps the local Hadamard singularity separate from global multiple scattering and asks when a residue-plus-cut representation is valid. The scattering route is best for reflection, transmission, and greybody factors. It emphasizes that unit amplitude is not generally unit flux and that the superradiant sign must not be removed by an absolute value.

For a visual guide to the two routes, inspect the contour diagram for the pole residues, two cut banks, and retained large arc, and the scattering-basis diagram for which past endpoint supplies a wave and which future endpoint receives it.

A useful cross-route check is normalization invariance. Rescale one homogeneous solution by a nonzero $a(\omega)$ and the other by a nonzero $b(\omega)$. Their product numerator and Wronskian both acquire $ab$, so the radial Green function is unchanged. Individual scattering amplitudes do change unless their unit-incidence condition is restored. Physical flux ratios and the causal propagator must not. When identifying poles, require these changes of normalization to be holomorphic and nonvanishing in the frequency neighborhood: zeros or poles inserted into a basis can create spurious Wronskian singularities.

A complete check with no barrier

The product kernel and its inverse transform can be tested before introducing black-hole curvature. Write $x=r_*$ and

$$P=-\partial_t^2+\partial_x^2-V(x),\qquad L_\omega=\frac{d^2}{dx^2}+\omega^2-V(x).$$

For solutions $f$ and $u$ obeying the left and right endpoint conditions, respectively, define

$$W[f,u]=fu'-uf',\qquad g(x,x';\omega)=\frac{f(x_<;\omega)u(x_>;\omega)}{W[f,u]}.$$

Primes denote $x$ derivatives, $x_<=\min(x,x')$, and $x_>=\max(x,x')$. There is no first-derivative term in $L_\omega$, so $W$ is constant in $x$. Continuity and the unit derivative jump give $L_\omega g=\delta(x-x')$. This is a one-dimensional kernel with measure $dx'$, not the unrescaled four-dimensional invariant kernel.

For $V=0$ on the whole line, choose $f=e^{-i\omega x}$ and $u=e^{i\omega x}$. Then

$$W=2i\omega,\qquad g_0(x,x';\omega)=\frac{e^{i\omega|x-x'|}}{2i\omega}.$$

Let $\tau=t-t'$ and $d=|x-x'|$. The causal inverse is

$$G_0(\tau;x,x') =\frac1{2\pi}\int_{\mathbb R+i\eta} e^{-i\omega\tau}g_0(x,x';\omega)\,d\omega =-\frac12\,\Theta(\tau-d),\qquad \eta>0.$$

The line runs left to right above the zero-frequency pole; $\Theta$ is the Heaviside step function. The exponential is $e^{-i\omega(\tau-d)}$, not just $e^{-i\omega\tau}$. For $\tau\lt d$ close upward and obtain zero. For $\tau\gt d$ close downward clockwise and obtain $-i$ times the residue, namely $-1/2$. At the wavefront the equality is distributional.

This single calculation checks the Fourier sign, factor of $2\pi$, Wronskian order, and finite propagation speed. The minus sign is required by the chosen $-\partial_t^2$ operator; it is not negative radiated energy. The interior support is that of a one-dimensional reduced wave equation, not evidence of a curvature-generated tail or failure of the four-dimensional flat-space Huygens property. Even this simple example shows why a lower contour cannot be used at every positive elapsed time: the source–observer travel distance matters.

Recurring example: one barrier, three regimes

Consider a Schwarzschild master equation

$$\left[-\partial_t^2+\partial_{r_*}^2-V_\ell(r)\right]\Psi=0$$

with a real potential vanishing at both ends. For example, the massless scalar has

$$V_\ell(r)=\left(1-\frac{2GM}{r}\right) \left[\frac{\ell(\ell+1)}{r^2}+\frac{2GM}{r^3}\right], \qquad \frac{dr_*}{dr}=\left(1-\frac{2GM}{r}\right)^{-1},$$

on $r\gt2GM$, with integer $\ell\geq0$. Its frequency-domain radial amplitude is the one for the rescaled field $\Phi=\Psi_{\ell m}Y_{\ell m}/r$. A real-frequency wave incident from infinity splits into a reflected wave and a future-horizon transmitted wave. The constant current gives $\mathcal R+\mathcal T=1$. The same endpoint solutions build the mode Green function, and a zero of their Wronskian after continuation marks a candidate resonance.

Now invert in time. At early causal arrival, large-frequency propagation and wavefront singularities matter. At an intermediate stage, residues of several lower-half-plane poles may dominate. At late time, the low-frequency branch cut can outlive every exponential. These regimes overlap and their separation depends on source, observer, and contour estimates. Leaver’s classic Schwarzschild analysis separates the three contributions in Leaver 1986, § II B, Eqs. (20)–(24) and Fig. 1, author-version pp. 7–8, PDF. The cut and its bank convention are specified in Casals and Ottewill 2013, arXiv v1, § II, p. 3, PDF.

For a massless scalar on subextremal Kerr, $a=J/M$ with $|a|\lt GM$, the tortoise coordinate and radial amplitude instead obey

$$\Delta=r^2-2GMr+a^2,\qquad \frac{dr_*}{dr}=\frac{r^2+a^2}{\Delta},\qquad X=\sqrt{r^2+a^2}\,R_{\mathrm{KG}}.$$

Here $R_{\mathrm{KG}}$ is the radial factor in

$$\Phi=e^{-i\omega t+im\phi}S(\theta)R_{\mathrm{KG}}(r).$$

For real $\omega$, this choice gives the same conserved-current prefactor at both ends, since

$$\Delta\,\operatorname{Im} \left(R_{\mathrm{KG}}^*\partial_rR_{\mathrm{KG}}\right) =\operatorname{Im}\left(X^*\partial_{r_*}X\right).$$

The corresponding balance is

$$\mathcal R=|R|^2,\qquad \mathcal T=\frac{k_H}{\omega}|T|^2,\qquad \mathcal R+\mathcal T=1.$$

Here $R$ and $T$ are reflected and transmitted amplitudes for unit incident amplitude at infinity. If $k_H\lt0$ with $\omega\gt0$, the transmitted asymptotic-time energy is negative and $\mathcal R>1$. The horizon wave remains future-ingoing. Thus causal direction, wave-number sign, and energy sign are related but distinct pieces of data. Raw electromagnetic or gravitational Teukolsky amplitudes need their spin-dependent physical flux factors, not just the scalar formula. Reflection, Transmission, and Absorption develops the current and those normalization distinctions.

Review the chapter

1. Verify the radial source. Let

$$g(x,x')=\frac{f(x_<)u(x_>)}{W[f,u]}, \qquad W[f,u]=fu'-uf',$$

where $f$ and $u$ solve $[d^2/dx^2+\omega^2-V]y=0$ and $V$ is smooth near $x'$. Verify the source normalization and then state why zeros of $W$ do not by themselves give a complete retarded response.

Solution

The kernel is continuous at $x=x'$. Its derivative jump is

$$[\partial_xg]=\frac{fu'-uf'}{W[f,u]}=1.$$

Integrating the radial operator across $x'$ therefore gives one delta function; the bounded potential term contributes nothing. If the endpoint solutions are holomorphic and nontrivial at $\omega_0$, a zero of $W$ makes them dependent. A simple zero gives a Green-operator pole with residue proportional to $f(x_<;\omega_0)u(x_>;\omega_0)/\partial_\omega W(\omega_0)$. A source overlap or an observation at a node can make that pole invisible in a particular response; that does not remove the operator pole. Singular basis normalizations need to be removed first. In addition, branch cuts supply non-pole response, and a large-frequency arc can carry prompt propagation. One must specify the sheet, justify contour deformation and arc decay, and establish convergence in the causal region. A pole list alone is not a completeness proof.

2. Turn initial data into a source. For the same $P$, let $P\psi=0$ at $t\gt0$, with $\psi(0,x)=\psi_0(x)$ and $\partial_t\psi(0,x)=\pi_0(x)$. Find $P[\Theta(t)\psi(t,x)]$. Use $G_0$ to recover the solution when $V=0$ and $\psi_0=0$.

Solution

Distributional differentiation gives

$$P[\Theta(t)\psi] =-\delta(t)\pi_0(x)-\delta'(t)\psi_0(x).$$

Only the time derivatives generate these terms; the spatial operator commutes with $\Theta(t)$. Convolution with the retarded kernel gives

$$\psi(t,x)=-\int\! \left[G(t;x,x')\pi_0(x') +\partial_tG(t;x,x')\psi_0(x')\right]dx', \qquad t\gt0.$$

Here $G$ is the causal time kernel for the chosen $V$. For the requested free case this becomes

$$\psi(t,x)=\frac12\int_{x-t}^{x+t}\pi_0(x')\,dx'.$$

Differentiating at $t=0$ recovers $\pi_0(x)$. This is d’Alembert’s initial-velocity solution: the two negative signs, one from the initial-slice source and one from $G_0$, cancel.

From causal response to resonances and tails

This chapter constructs causal Green functions and scattering data from normalized modes and relates analytic structure to time response. General Green-function theory remains in the mathematical prerequisites, while detector likelihoods and waveform inference require a model of the observation. Detailed quasinormal algorithms, tail exponents, and stability theorems belong to the following response chapters. Continue to Quasinormal Modes, Resonances, and Pseudospectra to compute poles and test their sensitivity, or to Late-Time Tails, Branch Cuts, and Spectral Representations to study the non-pole response.

References

• Bär, Christian, Nicolas Ginoux, and Frank Pfäffle. Wave Equations on Lorentzian Manifolds and Quantization. ESI Lectures in Mathematics and Physics. Zürich: European Mathematical Society, 2007. arXiv.
• Casals, Marc, and Adrian C. Ottewill. “Analytic Investigation of the Branch Cut of the Green Function in Schwarzschild Space-Time.” Physical Review D 87 (2013): 064010. DOI. Open PDF.
• Leaver, Edward W. “Spectral Decomposition of the Perturbation Response of the Schwarzschild Geometry.” Physical Review D 34 (1986): 384–408; erratum 38 (1988): 725. DOI, erratum DOI. Open PDF, author version.