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

**Conditional (explicit):** any SCC conclusion stated for $\Lambda>0$ rotating charged backgrounds depends on (A) the chosen SCC regularity class and (B) a quantitative bridge from **linear** spectral/QNM data on the fixed background to the **nonlinear** interior regularity claim. Different references instantiate different pairs $(A,B)$; this entry records that split rather than asserting a single unconditional theorem.

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 — Prove unconditional 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