Near-extremal interior scaling laws

Summary

Near-extremal interior scaling laws

Why this matters

Near-extremal interiors are where ordinary blue-shift theory meets extremal degeneracy.

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

Find universal scaling relations for interior blow-up, weak singularity structure, and regularity thresholds as the Kerr spin approaches extremality.

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 quantify how the relevant norms or exponents depend on $\varepsilon$ and identify any limiting profile.

Mathematical prerequisites

Matched asymptotics between throat and bulk interior; scaling analysis in $\varepsilon = 1 - |a|/M$; uniform energy estimates.

Completion criteria

A complete answer must quantify how the relevant norms or exponents depend on $\varepsilon$ and identify any limiting profile.

Implications if solved

Would connect interior SCC with extremal black-hole theory.

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-206 — Extremal interior SCC and Cauchy-horizon regularity
• K-623 — Build a rigorous scattering theory for NHEK matching to global Kerr exterior at near-extremality.

Editorial / maintainer notes

Open: no complete theorem matching the statement is currently recorded on this site. : tighten if community consensus differs.