Cosmological Constant and Censorship

The cosmological constant changes the asymptotic boundary, the decay of radiation, and therefore the competition between exterior tails and interior blue shift. Asymptotically flat, de Sitter, and anti-de Sitter censorship problems are not three parameter values in one unchanged conjecture. They are different evolution problems: future infinity is null, spacelike, or timelike, and in the anti-de Sitter case a boundary condition is indispensable.

Evidence cutoff: 2026-08-12. The nonlinear exterior-stability theorems, linear inner-horizon calculations, and conditional 2025–2026 preprints below are classified separately. No universal censorship verdict is inferred from a test-field spectrum or one boundary condition.

Required background. Analysis for Mathematical Relativity supplies decay and spectral estimates, while the Einstein equation fixes $\Lambda$ and the matter system.

Helpful background. Causality and Global Structure supplies conformal boundaries; PDEs and Green Functions for Gravitation supplies initial-boundary evolution; and Black-Hole Geometry and Mechanics supplies the charged and rotating models.

Three asymptotic problems

With

$$G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi G T_{\mu\nu},$$

the sign of $\Lambda$ distinguishes the following asymptotic classes, provided the corresponding regular conformal completion exists. The sign alone does not prove such a completion for arbitrary Einstein–matter data.

Regime	Conformal boundary	Natural evolution data	Central censorship issue
$\Lambda=0$	Null $\mathscr I^+$ for isolated systems	Asymptotically flat Cauchy or characteristic data	Completeness/visibility at null infinity and polynomial tails feeding inner horizons
$\Lambda>0$	Spacelike future infinity, with cosmological horizons for local observers	Asymptotically de Sitter Cauchy data	Exterior exponential decay versus Cauchy-horizon blue shift; observer and component choice
$\Lambda<0$	Timelike conformal boundary	Initial data plus boundary conditions	Well-posedness, reflected or dissipated radiation, and boundary visibility

Weak censorship must replace the receiver $\mathscr I^+$ when it is absent or spacelike. Strong censorship must specify the same extension regularity in all three rows before any comparison is meaningful.

Positive Λ: exponential decay and a spectral ratio

For a subextremal de Sitter black hole, perturbations in the exterior often decay exponentially. At a Cauchy horizon, a mode $e^{-i\omega t}$ acquires a nonsmooth radial factor governed by

$$\beta=-\frac{\operatorname{Im}\omega}{\kappa_-},$$

where $\kappa_-\gt0$ is the magnitude of the inner-horizon surface gravity. In a regular transverse coordinate $V=-e^{-\kappa_-v}$, a decaying $e^{-\alpha v}$ envelope becomes $|V|^{\alpha/\kappa_-}$. For a nonzero nonsmooth scalar branch with real exponent $\beta$, the transverse derivative has size $|V|^{\beta-1}$ and

$$\int_0^\epsilon |\partial_V\phi|^2\,dV \sim\int_0^\epsilon V^{2\beta-2}\,dV.$$

This converges for $\beta\gt1/2$, diverges logarithmically at $\beta=1/2$, and diverges by a power below it. The test presumes a nonzero coefficient of the nonsmooth branch; a smooth constant zero-frequency mode is not an obstruction. A mode below the threshold can obstruct a finite-energy scalar extension when it is excited. Modes above it pass this local test, but an extension theorem for general data also requires control of the full solution and its remainder, not just sampled quasinormal frequencies.

Davey, Dias, and Sola Gil compute this criterion for a neutral massless scalar on fixed Kerr–Newman–de Sitter backgrounds and map a transition near the small, near-extremal Reissner–Nordström–de Sitter corner Davey–Dias–Sola Gil 2024, §2.3.3, pp. 13–14, and §5, pp. 34–40. This is linear spectral evidence. It does not evolve the Einstein–Maxwell metric, prove generic collapse, or include all gravito-electromagnetic modes.

By contrast, nonlinear exterior stability is rigorous in perturbative regimes. Hintz and Vasy prove that small nonsymmetric vacuum perturbations of slowly rotating Kerr–de Sitter settle exponentially to a nearby family member Hintz–Vasy 2018, Theorems 1.1–1.4 and §§1–2. Hintz proves the analogous slowly rotating Kerr–Newman–de Sitter result for the Einstein–Maxwell system Hintz 2018, Theorems 1.1–1.2 and §§1–2. These are exterior/global-domain stability theorems, not inner-horizon inextendibility theorems.

A 2025 preprint claims full-subextremal Kerr–de Sitter nonlinear stability conditional on mode stability Hintz–Petersen–Vasy 2025, Assumption 1.2 and Theorem 1.4, pp. 6–7, arXiv v1. The condition is part of the result. It cannot be cited as an unconditional full-range theorem, and even an unconditional exterior theorem would still need a separate nonlinear interior analysis to decide SCC at $C^0$, Lipschitz, or $C^2$.

Zero Λ: polynomial Price tails

For asymptotically flat black holes, exterior radiation decays polynomially. After the exponential coordinate blue shift at an inner horizon, such tails are expected to generate mass inflation. In the two-ended spherical Einstein–Maxwell–scalar model, continuous Cauchy horizons coexist with generic $C^2$ inextendibility: Luk and Oh prove the necessary exterior lower tail and the nonlinear interior blowup Luk–Oh 2019 I, Theorems 1.8–1.10 and Luk–Oh 2019 II, Theorems 1.1 and 1.5.

This cannot be transferred to $\Lambda>0$ merely by changing the metric function: exponential decay can compete much more effectively with blue shift. Nor can the spherical charged theorem settle rotating vacuum Kerr, whose radiation and gauge structure differ.

Negative Λ: boundary conditions are dynamics

For $\Lambda<0$, conformal infinity is timelike. Null rays reach it in finite conformal time, so initial data alone do not normally determine a solution. Reflecting boundary conditions return radiation to the bulk; dissipative conditions remove energy; mixed or matter-dependent conditions define still other systems. Each changes both global existence and any visibility statement.

A local Einstein–AdS theorem must specify conformal boundary data and their compatibility with spacelike initial data at the corner. After translating Friedrich’s signature and curvature conventions to the site’s $(-,+,+,+)$ convention, his theorem constructs AdS-type developments from compatible smoothly conformally compactifiable Cauchy data and smooth Lorentzian conformal-boundary data satisfying the corner conditions Friedrich 1995, Theorem 1.1, pp. 126–128. That local well-posedness does not prove global stability or weak censorship. Likewise, instability scenarios under reflecting boundary conditions do not establish the same outcome with transparent or dissipative boundaries.

An AdS censorship tuple must include:

$$(\text{bulk theory},\text{initial data}, \text{boundary condition},\text{corner class}, \text{receiver},\text{extension regularity}).$$

Omitting the boundary condition makes the conjecture underdetermined.

Formulation-dependent conclusions

The current evidence supports no simple monotone claim that positive $\Lambda$ “helps” or negative $\Lambda$ “hurts” censorship. Positive $\Lambda$ improves exterior decay but may thereby weaken the blue-shift singularity at a charged Cauchy horizon. Reflecting AdS boundaries can enhance repeated concentration but also change which data and conserved quantities are admissible. These mechanisms operate at different boundaries and regularities.

Unresolved open questions include nonlinear KNdS interior genericity across the full parameter range, a complete rotating-vacuum exterior-to-interior theorem, and boundary-condition-dependent large-data AdS evolution. A theorem covering one of these regimes, a proof or disproof of the conditional mode-stability assumption, or a corrected spectral threshold could change the corresponding conclusion.

Check the formulation

A neutral scalar calculation on fixed near-extremal Reissner–Nordström–de Sitter finds $\beta>1/2$ for every mode sampled. Does this disprove strong cosmic censorship for Einstein–Maxwell with $\Lambda>0$?

Solution

No. It indicates finite local energy for that linear test scalar under the sampling and boundary conditions, so it is evidence against a particular weak- scalar inextendibility criterion on those backgrounds. It does not include metric/electromagnetic backreaction, all perturbation sectors, nonlinear genericity, collapse data, or a metric extension construction. It also says nothing until the SCC extension class is named. A nonlinear Einstein–Maxwell theorem would require all of those additional ingredients.

References

• Davey, Alex, Óscar J. C. Dias, and David Sola Gil. “Strong Cosmic Censorship in Kerr–Newman–de Sitter.” Journal of High Energy Physics 2024, no. 7 (2024): 113. DOI.
• Friedrich, Helmut. “Einstein Equations and Conformal Structure: Existence of Anti-de Sitter-Type Space-Times.” Journal of Geometry and Physics 17 (1995): 125–184. DOI.
• Hintz, Peter, and András Vasy. “The Global Non-linear Stability of the Kerr–de Sitter Family of Black Holes.” Acta Mathematica 220 (2018): 1–206. arXiv.
• Hintz, Peter, Oliver Petersen, and András Vasy. “Conditional Non-linear Stability of Kerr–de Sitter Spacetimes in the Full Subextremal Range.” arXiv:2508.06620v1 (2025). arXiv v1.
• Hintz, Peter. “Non-linear Stability of the Kerr–Newman–de Sitter Family of Charged Black Holes.” Annals of PDE 4 (2018): 11. arXiv.
• Luk, Jonathan, and Sung-Jin Oh. “Strong Cosmic Censorship in Spherical Symmetry for Two-Ended Asymptotically Flat Initial Data I: The Interior of the Black Hole Region.” Annals of Mathematics 190 (2019): 1–111. arXiv.
• Luk, Jonathan, and Sung-Jin Oh. “Strong Cosmic Censorship in Spherical Symmetry for Two-Ended Asymptotically Flat Initial Data II: The Exterior of the Black Hole Region.” Annals of PDE 5 (2019): 6. arXiv.