Derive the interior theorem directly from exterior data
Summary
Derive the interior theorem directly from exterior data
Why this matters
The cleanest SCC theorem should begin with ordinary exterior initial data, not data already prescribed inside the black hole.
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: full-einstein.
- 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
Remove interior auxiliary assumptions by proving the Kerr interior Cauchy-horizon picture directly from asymptotically flat exterior Cauchy data and exterior decay.
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 derive the needed horizon asymptotics from exterior estimates and then prove the interior extension or singularity conclusions.
Mathematical prerequisites
Matching gauges across the event horizon; transfer of Price-law decay into the interior; nonlinear continuation.
Completion criteria
A complete answer must derive the needed horizon asymptotics from exterior estimates and then prove the interior extension or singularity conclusions.
Implications if solved
Would connect the exterior and interior Kerr programs into one theorem chain.
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
Editorial / maintainer notes
Open: no complete theorem matching the statement is currently recorded on this site. : tighten if community consensus differs.