Black-Hole Scattering Theory

Black-hole scattering identifies a finite-energy solution from its radiation entering through the past event horizon and past null infinity, then maps that data to radiation leaving through the future horizon and future null infinity. For the scalar wave equation on every four-dimensional subextremal Kerr exterior, this map is bounded and invertible in carefully chosen degenerate energy spaces. The word “degenerate” is essential: future redshift becomes backward blueshift, so a two-sided theory in fully nondegenerate horizon energy would be false.

Required background. Analysis for Mathematical Relativity supplies Hilbert-space estimates and traces, while Partial Differential Equations and Green Functions supplies well-posed evolution and radiation fields.

Helpful background. Einstein Equations fixes the vacuum background, Causality and Global Structure fixes the asymptotic boundaries, and Black-Hole Geometry and Mechanics together with Black-Hole Perturbation Theory supplies Kerr horizons and superradiance.

Radiation data at the asymptotic boundaries

Fix a Kerr metric with mass $M>0$, specific angular momentum $a=J/M$, and $\lvert a\rvert\lt GM$. Let $\psi$ solve $\Box_g\psi=0$. Along future null infinity, a suitably normalized limit of $r\psi$ defines the radiation field; along the future event horizon, the trace of $\psi$ in regular horizon coordinates defines the horizon field. Their derivatives with respect to the natural null generators carry energy flux.

The hypotheses include subextremality, the fixed vacuum Kerr geometry, and completion of smooth compactly supported data in the stated finite-energy norms. No axisymmetry is assumed. Regularity is measured through the radiation and Cauchy energies rather than through pointwise smoothness after completion.

For the full past-to-future scattering problem, take $\Sigma$ to be a spacelike Cauchy slice of the complete stationary exterior, running from the bifurcation end to spatial infinity. Compactly supported smooth Cauchy data define

$$\mathcal F^+: \mathcal E_\Sigma \longrightarrow \mathcal E_{\mathcal H^+}\oplus\mathcal E_{\mathscr I^+}.$$

The past map $\mathcal F^-$ is defined analogously. A scattering operator is then

$$\mathcal S=\mathcal F^+\circ(\mathcal F^-)^{-1},$$

provided the past radiation map has a bounded inverse on the completed energy spaces. A horizon-crossing slice used only for a future evolution problem instead maps to the part of $\mathcal H^+$ lying in its future, together with $\mathscr I^+$; that truncated problem does not by itself define the full past-to-future operator. Asymptotic completeness means that every solution in the physical energy class is exhausted by its horizon and null-infinity radiation: no additional finite-energy state remains trapped forever in the exterior.

The correct energy spaces

Let $T=\partial_t$ and $\Phi=\partial_\phi$. Three currents have different roles. A smooth future timelike multiplier $N$ gives an energy on a horizon-crossing spacelike slice that controls all first derivatives, including the transverse derivative. The stationary Killing field $T$ gives a conserved but indefinite energy on rotating Kerr. The scattering norm uses a third field $V$: it is timelike in the exterior, agrees with $T$ far away, and agrees near the horizon with its null generator

$$K=T+\Omega_H\Phi, \qquad \Omega_H=\frac{a}{r_+^2+a^2}, \qquad r_+=GM+\sqrt{(GM)^2-a^2}.$$

Thus $E_V=\int_\Sigma T_{\mu\nu}[\psi]n^\mu V^\nu\,d\mu_\Sigma$ is positive but loses transverse coercivity as the slice approaches the horizon. It is not the indefinite $T$-energy. With $v$ a parameter such that $K=\partial_v$ at fixed corotating angular coordinates, the corresponding future radiation norms are

$$\begin{aligned} \|\psi_{\mathcal H^+}\|_{\mathcal E^K}^2 &=\int_{\mathcal H^+}|K\psi_{\mathcal H^+}|^2\,dv\,dA,\\ \|\phi_{\mathscr I^+}\|_{\mathcal E^T}^2 &=\int_{\mathbb R\times S^2}|\partial_u\phi_{\mathscr I^+}|^2 \,du\,d\omega, \qquad \phi_{\mathscr I^+}=\lim_{r\to\infty}r\psi. \end{aligned}$$

Here $dA$ is the horizon cross-section area measure and $u$ is retarded time normalized by $T$ at infinity. The spaces are completions in these homogeneous flux norms. Intrinsic horizon data are a trace and its tangential derivatives; they do not independently prescribe a transverse derivative.

In this notation the isomorphism is $\mathcal F^+:\mathcal E^V_\Sigma\to \mathcal E^K_{\mathcal H^+}\oplus\mathcal E^T_{\mathscr I^+}$. Already on Schwarzschild, the future problem from a horizon-crossing slice has smooth finite-flux radiation data whose backward solution has infinite $N$-energy on that slice. Thus replacing the degenerate Cauchy norm by the stronger nondegenerate norm can destroy surjectivity, even at zero spin. The isomorphisms and this explicit blueshift obstruction are established in Dafermos, Rodnianski, and Shlapentokh-Rothman 2018, §§2.3.1–2.3.5, 3.4, and 8.1; Theorem 11.1 and §11.4.

Superradiance and bounded amplification

For rotating Kerr, stationary energy is not positive on every horizon mode. With $e^{-i\omega t}$ and azimuthal number $m$, the horizon flux contains $\omega(\omega-m\Omega_H)$. In the superradiant range, energy emerging at future null infinity can exceed the incoming energy from past null infinity; the difference is balanced by negative stationary flux through the horizon.

Thus the scattering operator is not unitary in the positive $K$-horizon and $T$-infinity flux norm. It nevertheless preserves the indefinite $T$-flux identity and is bounded in those positive Hilbert norms. Quantitative mode stability yields uniform reflection and transmission bounds, so amplification has a finite supremum over finite-energy incoming packets. Existence, uniqueness, asymptotic completeness, and bounded superradiant reflection are separated in Dafermos, Rodnianski, and Shlapentokh-Rothman 2018, Theorems 1–4 and §§8–10.

Proof architecture

The construction first proves forward boundedness and decay for Cauchy data, including redshift, trapping-degenerate Morawetz estimates, and far-field weighted estimates. These bounds show that radiation traces exist and that the forward map is bounded. One then solves backward characteristic problems from smooth compactly supported radiation data. Uniform estimates in the degenerate norm allow completion and prove surjectivity. Uniqueness follows from vanishing past radiation plus integrated decay; density then extends the maps to the full energy spaces.

Frequency analysis enters where superradiance obstructs a globally coercive stationary current. Quantitative bounds for the separated radial equation are combined with physical-space estimates, rather than assuming a discrete mode expansion. The decay input in the same full-subextremal range is stated in Dafermos, Rodnianski, and Shlapentokh-Rothman 2016, Main Theorem and §§5–13.

Scope and limitations

The theorem is for the linear scalar equation on a fixed Kerr exterior. It does not establish a scattering operator for the linearized Einstein equation, where gauge, constraints, and stationary Kerr modes must be quotiented. It also does not include extremal Kerr: vanishing surface gravity changes both the redshift estimate and the natural asymptotic data.

Some finite-energy future radiation data supply a counterexample to an inverse theorem in the stronger $N$-energy class, because backward evolution can produce infinite nondegenerate Cauchy energy. A further limitation is that asymptotic completeness here does not mean completeness of quasinormal modes. Prompt propagation, resonances, branch-cut tails, and radiation fields are all encoded by the evolution, but the proof does not express every solution as a convergent discrete resonance sum.

Check the norm

Why does a bound for forward evolution from nondegenerate $N$-energy Cauchy data not imply that every finite-energy pair of future radiation fields has finite $N$-energy initial data? Where does the $V$-energy lose coercivity?

Solution

A bounded forward map need not be onto. Forward redshift controls transverse derivatives on spacelike slices, but the inverse evolution reverses this effect. Finite future radiation flux can therefore correspond to infinite $N$-energy on a horizon-crossing slice. Since $V$ becomes null at the horizon, its slice energy allows weaker transverse control there and supports a bounded inverse. No extra transverse derivative is being freely supplied as part of the intrinsic horizon radiation field.

References

• Dafermos, Mihalis, Igor Rodnianski, and Yakov Shlapentokh-Rothman. “A Scattering Theory for the Wave Equation on Kerr Black Hole Exteriors.” Annales Scientifiques de l’École Normale Supérieure 51 (2018): 371–486. DOI.
• Dafermos, Mihalis, Igor Rodnianski, and Yakov Shlapentokh-Rothman. “Decay for Solutions of the Wave Equation on Kerr Exterior Spacetimes III: The Full Subextremal Case $|a|<M$.” Annals of Mathematics 183 (2016): 787–913. arXiv.