Genericity of lower bounds for linearized-gravity interior instability
Summary
Genericity of lower bounds for linearized-gravity interior instability
Why this matters
Instability theorems are strongest when their hypotheses are shown to hold for open dense or full-measure sets of data.
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
Prove that the nontrivial lower-bound assumptions appearing in linearized-gravity blue-shift instability results are generic for exterior initial data.
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 specify the data topology and prove generic nonvanishing of the instability-triggering quantity.
Mathematical prerequisites
Linearized-gravity scattering; horizon charges; genericity arguments in infinite-dimensional data spaces.
Completion criteria
A complete answer must specify the data topology and prove generic nonvanishing of the instability-triggering quantity.
Implications if solved
Would turn conditional interior blow-up statements into genuinely generic ones.
Formal verification suitability
FV: low
Global nonlinear PDE or open-ended dynamics; not a practical first formalization target without major scoping.
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.