Inner Horizons and Mass Inflation

Inner horizons turn small exterior tails into a competition between blueshift, nonlinear backreaction, and decay. A Cauchy horizon may survive as a continuous metric while Hawking mass, curvature, or connection coefficients diverge there. As of 2026-08-20, asymptotically flat spherical charged models and $C^0$ survival in vacuum Kerr have strong nonlinear theorems, while the generic rotating singularity structure remains conditional. The question is partially resolved and contested, with high confidence in the model-dependent split.

Helpful background. Black-Hole Interiors, Cauchy Horizons, and Mass Inflation defines the inner-horizon geometry, blueshift, Hawking mass, and weak null singularities. This record compares current endpoint and regularity evidence.

Which inner-horizon endpoint is being tested?

Fix a charged or rotating black-hole evolution and the late-time tail entering its event horizon. Ask whether a Cauchy horizon forms, whether Hawking mass, curvature, or connection norms blow up, and in which class the metric can be extended. These are distinct conclusions.

The decisive axes are spherical charged matter versus nonsymmetric vacuum Kerr; asymptotically flat versus de Sitter; real versus charged scalar; exterior complete data versus interior characteristic data; and $C^0$, Lipschitz, $C^1$, $C^2$, or connection-$L^2$ extendibility. “No mass inflation” means only that one mass quantity stays bounded; other curvature or regularity obstructions may remain.

Assessment at 2026-08-20	Meaning
Question status	Partially resolved
Evidence status	Contested
Confidence	High that mass inflation and no-mass-inflation occupy different, well-defined cells
Domain	Nonlinear charged spherical interiors and subextremal vacuum Kerr, with asymptotics, tail assumptions, and extension class explicit
What could change the conclusion	A nonlinear theorem or converged simulation resolving extendibility and singular structure

A starting project: identify the tail that drives the interior

Fit and preparation. Choose this direction if you can manipulate spherical double-null equations, define the charged Hawking mass, and distinguish a pointwise decay estimate from an integrated one. An analytic project can be completed without a numerical evolution. A computation inside the horizon should follow only with supervision in characteristic evolution and singularity diagnostics.

Reading and baseline. Start with the upper- and lower-tail assumptions in Dafermos 2005, §§1.2–1.5, then compare the assumptions used for $C^2$ inextendibility in Luk and Oh 2019, Theorems 1.4–1.5. Work through one mass-evolution identity in the former model, identifying where the lower bound enters. Keep the real neutral scalar, Maxwell charge, and asymptotically flat setting fixed throughout.

Deliverable. Write a calculation distinguishing the conclusion obtained from the upper bound alone from the conclusion requiring the lower bound. Retain the integration parameter, weights, and endpoint terms. Add a separate line for continuous metric extension, Hawking-mass divergence, and curvature regularity; prove only the implications the chosen argument supplies. This is a reproduction of one mechanism, not a generic Kerr-interior result.

Possible extension and stopping point. After reviewing the charged-scalar and de Sitter literature, a candidate project could test one alternative tail profile in the same characteristic problem, analytically or with a validated spherical solver. State whether the profile is prescribed or known to arise from exterior evolution. Stop if a numerical mass increase depends on coordinate compression or resolution, or if the proposed generic conclusion assumes an unproved exterior lower bound. Pivot to the conditional interior problem rather than labeling an arbitrarily prescribed tail generic. Rotation requires additional expertise and cannot be introduced by merely changing the background parameters.

What is established in the stated domain

Dafermos proves that an open set of spherical Einstein–Maxwell–real-scalar characteristic data develops a Cauchy horizon with a continuous extension and curvature blow-up main theorems in §1 and global structure in §§10–13. With quantified Price-law input, the Hawking mass inflates and $C^1$ extension fails main theorem and §§1.2–1.5. These results establish the coexistence of weak metric survival and stronger singularity.

Luk and Oh prove a generic $C^2$-inextendibility result for two-ended, asymptotically flat, spherically symmetric Einstein–Maxwell–real-scalar data while retaining $C^0$ extension Theorems 1.1–1.5 and §1.7. The theorem’s genericity and exterior lower bounds are model specific.

For vacuum rotation, Dafermos and Luk prove $C^0$ stability of a Kerr Cauchy-horizon segment from a broad interior data class main theorem and assumptions (4.2)–(4.3). They do not claim generic curvature blow-up. Gurriaran’s 2026 preprint supplies a conditional nonlinear curvature-blow-up and Lipschitz-inextendibility result near timelike infinity main theorem and introduction, assuming a nonlinear Price-law-type tail whose generic derivation is not part of that paper.

Results and open questions at 2026-08-20

Claim or evidence	What it supports	What it does not establish
Dafermos spherical theorems	Continuous horizon plus mass and curvature inflation under stated tails	Rotating vacuum behavior
Luk–Oh generic theorem	$C^2$ obstruction with $C^0$ survival for two-ended spherical data	One-ended collapse or Kerr
Dafermos–Luk Kerr theorem	Nonsymmetric vacuum $C^0$ horizon stability	Generic stronger singularity
Gurriaran preprint	Conditional Kerr curvature and Lipschitz obstruction near the endpoint	Generic exterior lower bound
Rossetti positive-$\Lambda$ theorem	Bounded mass and weak extension in a fast-decay charged cell	Smooth extension or asymptotically flat behavior

The evidence is contested only at the broad wording level. Within each cell, positive and adverse theorems are compatible and often sharp.

Competing endpoint pictures

The mass-inflation picture says generic Price-law tails are infinitely blueshifted, causing a weak null singularity with divergent mass or curvature. It is rigorous in important asymptotically flat spherical models and conditionally supported in Kerr. Its unresolved step in rotation is deriving the required lower tail from complete generic vacuum data and propagating it invariantly.

The regular-Cauchy-horizon picture emphasizes fast exterior decay, especially near extremality with positive cosmological constant. Rossetti proves a no-mass-inflation regime with $H^1$-type weak extension main stability and extension theorems. That result is genuine adverse evidence to universality, but it neither produces a smooth extension nor transfers to asymptotically flat Kerr.

Counterexamples, null results, and failed routes

Rossetti’s theorem shows that mass inflation is not automatic once cosmological constant, charge coupling, and decay rate change. Van de Moortel’s nonlinear charged-scalar analysis instead yields a mass-inflation/rigidity dichotomy main dichotomy and conclusion, warning against assuming the divergent branch without checking its hypothesis.

A finite metric plot across a numerically represented inner horizon would not determine differentiability. Coordinate components can remain bounded while a tetrad curvature or connection norm diverges. No unrestricted converged $3+1$ calculation resolving all relevant extension classes was verified.

What would change the assessment

The decisive analytic result would derive a generic nonlinear Price-law lower bound from complete asymptotically flat vacuum Cauchy data and feed it into an interior theorem establishing invariant blow-up at a named regularity. A theorem showing bounded curvature or connection norms on an open rotating data set would move the record oppositely. Two independent rotating simulations could constrain the endpoint if they resolve affine-parameter scaling and extension norms rather than only coordinate fields.

Research checklist

• Fix matter, cosmological constant, asymptotics, extremality range, and the initial-data surface.
• State the exterior upper and lower tail estimates and whether they are assumed or derived.
• Separate Cauchy-horizon existence, mass inflation, curvature blow-up, and each extension class.
• Use invariant or tetrad-robust diagnostics and report the tetrad normalization.
• For numerics, pin coordinates, refinement, constraints, horizon location, affine parameter, and fit range.
• Test alternative singular quantities before concluding that bounded Hawking mass means a regular endpoint.

Source and data map

Dafermos and Luk–Oh provide the controlling spherical theorems. Dafermos–Luk provides peer-reviewed Kerr $C^0$ stability, while Gurriaran v2 provides the current conditional stronger Kerr result. Rossetti and Van de Moortel supply adverse/dichotomy cells. Linear fixed-background blueshift was treated as mechanism evidence only.

No independent reproduction was performed, and no numerical result controls the assessment. No formal proof archive was found. The critical open verification is the derivation of the Price-law lower bound required by the Kerr preprint; arXiv:2603.17911v2 had no journal reference at the cutoff.

References

• Dafermos, Mihalis. “Stability and Instability of the Cauchy Horizon for the Spherically Symmetric Einstein–Maxwell-Scalar Field Equations.” Annals of Mathematics 158 (2003), 875–928. DOI.
• Dafermos, Mihalis. “The Interior of Charged Black Holes and the Problem of Uniqueness in General Relativity.” Communications on Pure and Applied Mathematics 58 (2005), 445–504. DOI.
• Dafermos, Mihalis, and Jonathan Luk. “The Interior of Dynamical Vacuum Black Holes I: The $C^0$-Stability of the Kerr Cauchy Horizon.” Annals of Mathematics 202 (2025), 309–630. DOI.
• Gurriaran, Sebastian. “Non-linear Instability of the Kerr Cauchy Horizon near $i_+$.” arXiv:2603.17911v2 (2026). 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. DOI.
• Rossetti, Flavio. “Strong Cosmic Censorship for the Spherically Symmetric Einstein–Maxwell-Charged-Klein–Gordon System with Positive $\Lambda$: Stability of the Cauchy Horizon and $H^1$ Extensions.” Annales Henri Poincaré 26 (2025), 675–753. DOI.
• Van de Moortel, Maxime. “Mass Inflation and the $C^2$-Inextendibility of Spherically Symmetric Charged Scalar Field Dynamical Black Holes.” Communications in Mathematical Physics 382 (2021), 1263–1341. DOI.