Strong cosmic censorship in rotating $\Lambda>0$ black-hole interiors (conditional on a spectral–interior bridge)

Summary

Relate quasinormal-mode decay and interior extendibility for Kerr–de Sitter / Kerr–Newman–de Sitter models; conclusions depend on which SCC regularity formulation and which linear spectral input are assumed.

Why this matters

Positive $\Lambda$ changes exterior decay and interior amplification; SCC statements are sensitive to both the regularity notion and the spectral input used.

Exact scope

Background / setting

Kerr–de Sitter and Kerr–Newman–de Sitter–type solutions; interior up to Cauchy horizons.

Equation type

Typically begins with test fields (scalar EM) and spectral theory; full gravity interior statements are further out.

Linearity

Much of the sharpest $\Lambda>0$ literature is linearized around a fixed background; nonlinear interior claims require separate hypotheses.

Regularity

Must name the SCC variant ($C^0$ extendibility vs higher regularity blow-up, etc.); different variants are not interchangeable.

Parameter regime

Subextremal vs near-extremal rotation/charge; competition between slow QNM decay and blue-shift along inner horizons is regime-dependent.

Asymptotics

de Sitter / black-hole–de Sitter asymptotics as in the stated model.

Gauge / formulation

As in the cited spectral and interior PDE setups.

Status explanation

**Condition (explicit):** results hinge on (A) the chosen SCC formulation and (B) the assumed implication from linear QNM decay (or related spectral quantities) to nonlinear interior regularity. Different papers may use different pairs $(A,B)$; this entry does not collapse them without further editorial verification. Marked `editorial_revision_needed` until maintainers align wording with a pinned reference set.

Problem statement

Fix (i) a precise strong cosmic censorship regularity class ($C^k$ inextendibility, weaker $C^0$ formulations, etc.) and (ii) a quantitative bridge from linear spectral data (quasinormal decay on the chosen $\Lambda>0$ background) to interior Cauchy-horizon behavior. Under that pair, determine whether SCC holds or fails for rotating charged $\Lambda>0$ black holes—especially tracking how thresholds move near extremality compared with $\Lambda=0$.

What is already known

• Linear quasinormal-mode spectra and decay rates on Kerr–Newman–de Sitter backgrounds are used in recent scalar studies of strong cosmic censorship, illustrating parameter-dependent SCC outcomes.
  Regime: Test scalar fields on fixed KN-dS; linear spectral input.
  Shows SCC claims are not background-independent—must be tied to a precise model and decay assumptions.
• Dafermos–Luk establish $C^0$ Cauchy-horizon stability structure for dynamical vacuum black holes in $\Lambda=0$; this is the baseline contrast for $\Lambda>0$ interior heuristics.
  Regime: $\Lambda=0$ vacuum interior; theorem-level in their setting.
  Prevents importing $\Lambda=0$ SCC language verbatim into de Sitter asymptotics without new hypotheses.
• Hintz–Vasy develop global microlocal decay/normal-mode theory for linear waves on Kerr–de Sitter, supplying quantitative exterior decay templates feeding interior discussions.
  Regime: Linear waves on exact Kerr–de Sitter.
  Standard rigorous bridge from spectral/decay statements to PDE-controlled hypotheses—still not a full nonlinear gravity interior theorem without extra assumptions.
• Davey–Dias–Sola Gil systematically scan (linear scalar) SCC issues across Kerr–Newman–de Sitter parameter space, highlighting dependence on charge, spin, and $\Lambda$.
  Regime: Linear scalar QNM-based analysis on fixed KN-dS.
  Concrete reference point for how “conditional SCC” splits across moduli; not a unique SCC definition.

Progress summary: Recent literature studies SCC questions for Kerr–Newman–de Sitter using linear scalar QNM scans; a consolidated theorem matching one SCC definition is still conditional on the bridge hypotheses above.

What remains open

Turn the conditional template above into a single published-quality conjecture with matching hypotheses, or a theorem under explicitly listed spectral and regularity assumptions.

Mathematical prerequisites

Spectral-gap estimates; de Sitter horizon boundary conditions; Sobolev regularity at Cauchy horizons; coupled gravito-electromagnetic systems where relevant.

Scope / taxonomy note

Completion criteria

Connect spectral decay rates to extension regularity and prove either violation or restoration of SCC in a stated class matching $(A,B)$.

Implications if solved

Clarifies how SCC issues differ between $\Lambda=0$ and $\Lambda>0$ rotating models.

Formal verification suitability

FV: high

Stationary, algebraic, ODE/separable, or finite-dimensional substatements admit clearer formalization boundaries.

See Formal verification for how this database uses these labels.

References

• primary Global analysis of linear waves on Kerr–de Sitter space — Hintz, Vasy (2016)
  Rigorous linear wave decay and spectral/normal-mode framework on Kerr–de Sitter; standard input when relating QNM decay to PDE hypotheses.
• primary Strong Cosmic Censorship in Kerr–Newman–de Sitter — Davey, Dias, Sola Gil (2024)
  Recent QNM-based scan of SCC issues for Kerr–Newman–de Sitter; illustrates dependence on parameter space and linear scalar input.
• primary The interior of dynamical vacuum black holes I: The $C^0$-stability of the Kerr Cauchy horizon — Dafermos, Luk (2017)
  Foundational $\Lambda=0$ interior/Cauchy-horizon stability context; contrasts with $\Lambda>0$ spectral narratives and regularity formulations.

Related problems

Related by shared tags

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

• K-667 — Prove stability/instability of the Cauchy horizon for Kerr-de Sitter under linearized gravity with sharp norms.
• K-608 — Nonlinear stability of slowly rotating Kerr-de Sitter
• K-609 — Prove conditional nonlinear stability of Kerr-de Sitter in the full subextremal range under explicit mode-stability assumptions.
• K-665 — Prove quantitative mode stability for Kerr-de Sitter in full subextremal range and feed into nonlinear stability.
• K-666 — Prove SCC threshold for Kerr-de Sitter and Kerr-Newman-de Sitter with explicit dependence on spectral gap.
• K-003 — Nonlinear asymptotic completeness near Kerr