Scattering map to the Cauchy horizon for linearized gravity

Summary

Scattering map to the Cauchy horizon for linearized gravity

Why this matters

The Cauchy horizon is best understood through a precise data-to-asymptotics map.

Exact scope

Background / setting

Asymptotically flat four-dimensional general relativity unless the statement specifies otherwise. Cauchy horizon / interior regularity formulations of strong cosmic censorship are in play; distinguish $C^0$ vs higher regularity notions.

Equation type

PDE level: spectral-operator.

Linearity

Includes or emphasizes nonlinear dynamics.

Regularity

State intended regularity class for extensions across horizons explicitly (e.g. $C^0$, Lipschitz, $C^2$).

Parameter regime

Subextremal Kerr interior up to Cauchy horizons; specify SCC regularity class ($C^0$, Lipschitz, $C^k$) in the theorem.

Asymptotics

asymptotically-flat

Gauge / formulation

State gauge/fixing class compatible with cited stability or interior programs (e.g. generalized harmonic, double-null interior charts).

Status explanation

Entry imported without two independent bibliographic pointers in this repository; treat theorem-level claims as unverified here.

Problem statement

Construct a precise scattering map from exterior or horizon data to asymptotic data at the Kerr Cauchy horizon for gravitational perturbations.

What is already known

• $C^0$-extendibility and weak regularity across Cauchy horizons are understood in substantial $Lambda=0$ vacuum settings (Dafermos–Luk program); higher regularity and $Lambda>0$ charged models require separate hypotheses.
  Regime: Dynamical vacuum near Kerr, $Lambda=0$ baseline; contrast with $Lambda>0$ scalar scans.
  Sets what “partial” interior control means before claiming generic blow-up or extendibility.
• Linear scalar and Teukolsky-type decay on fixed subextremal Kerr exteriors is highly developed and feeds conditional interior instability heuristics.
  Regime: Linearized fields on exact Kerr/Kerr–Newman.
  Supplies quantitative decay exponents used in bridge hypotheses to inner horizons.
• Polyhomogeneous/null-infinity technology exists for nonlinear Minkowski and some linearized Kerr contexts; sharp nonlinear near-Kerr peeling is not packaged as one theorem.
  Regime: Null infinity / linearized models.
  Separates radiation asymptotics from interior SCC targets.

Progress summary: Partial progress exists in adjacent regimes;

What remains open

A complete answer must define the map in explicit norms and prove sharp asymptotic formulas, including generic leading coefficients.

Mathematical prerequisites

Teukolsky separation; physical-space asymptotics; scattering operators in non-globally-hyperbolic regions; gauge reconstruction.

Completion criteria

A complete answer must define the map in explicit norms and prove sharp asymptotic formulas, including generic leading coefficients.

Implications if solved

Would clarify how exterior information is reprocessed by the blue-shift region.

Formal verification suitability

FV: medium

Some subquestions may formalize before the full statement.

See Formal verification for how this database uses these labels.

References

• primary The interior of dynamical vacuum black holes I: $C^0$-stability of the Kerr Cauchy horizon — Dafermos, Luk (2017)
  Foundational interior/Cauchy-horizon stability in $Lambda=0$ vacuum; context for SCC-type questions.
• primary Stability of Minkowski space and polyhomogeneity of the metric — Hintz, Vasy (2017)
  Sharp null-infinity asymptotics in a nonlinear vacuum setting; template for peeling/polyhomogeneous questions.

Related problems

Related by shared tags

Heuristic matches on family, cluster, equation level, asymptotics, and relevance.

• K-405 — Inverse scattering for Kerr parameters
• K-503 — Algebraic uniqueness of quadratic symmetry operators commuting with $\Box_g$ on exact Kerr ($\mathcal{D}_{\le 2}$ class)

Editorial / maintainer notes

Partial: substantial adjacent results or special cases exist, but the statement as written is not fully settled. : replace with a precise description of what is proved vs. conjectured.