Criteria for Nonlinear Black-Hole Interior End States

There is no model-independent verdict for the generic interior of a dynamical rotating black hole. Different theorems and simulations control different matter systems, symmetry classes, data surfaces, decay rates, and extension regularities. As of the evidence cutoff 2026-08-12, the productive question is which measurements would distinguish a mass-inflation null boundary, a continuously extendible weak singularity, and a spacelike crushing endpoint in the same evolution.

Helpful background. Black-Hole Geometry and Mechanics supplies Kerr and inner horizons; Black-Hole Perturbation Theory supplies exterior decay; Formulations of General Relativity supplies characteristic evolution; Causality and Global Structure supplies extension and genericity language; Computational Relativity supplies convergence tests.

Proposed interior end states and mathematical evidence

Setting	Controlled conclusion	Missing step toward a generic endpoint
Exact subextremal Reissner–Nordström or Kerr	smooth stationary Cauchy horizon	no generic radiation or backreaction
Spherical Einstein–Maxwell–scalar, asymptotically flat, two-ended data	continuous Cauchy horizon in the subextremal class; generic $C^2$ inextendibility under precise hypotheses	no gravitational radiation or rotation
Small vacuum perturbations of rotating subextremal Kerr in the stated exterior-data class	Hintz v2 decay combined with Dafermos–Luk gives $C^0$ Cauchy-horizon stability	generic curvature lower bound and complete boundary classification
Fully nonlinear nonsymmetric Einstein–scalar calculations with interior Kerr boundary data	exponentially sharpening, possibly nonaxisymmetric outgoing shock in selected spins and data	incoming tail, asymptotically flat exterior-to-interior data, complete endpoint, and independent continuum tests
Large departures from stationary interiors	spacelike or mixed singular boundaries are plausible	theorem or independently converged classification across parameter space

The strongest spherical result fixes the model, two-ended asymptotic flatness, admissible-data topology, tail lower bound, and regularity class before concluding generic $C^2$ future inextendibility. Part I proves the interior implication conditional on an averaged event-horizon lower bound Luk and Oh 2019a, Theorem 5.7, p. 29, PDF; Part II establishes the required generic exterior estimate Luk and Oh 2019b, Theorems 4.1–4.3, pp. 17–18, PDF. The strongest directly relevant nonsymmetric vacuum result proves $C^0$ stability of a Cauchy-horizon segment for quantified interior data close to Kerr Dafermos and Luk 2025, Theorem 4.24, pp. 50–65, and §16, pp. 165–185, PDF. These conclusions are compatible, not contradictory, because their models, data, and regularity claims differ.

The 2026 exterior result now supplies a narrower rigorous bridge: Hintz reports that its decay estimates meet the Dafermos–Luk hypotheses, giving $C^0$ stability of the Cauchy horizon for the stated small vacuum perturbations of rotating subextremal Kerr Hintz 2026, v2, §1.1, consequence (1), p. 3, PDF. This current preprint does not prove generic curvature blowup, classify the complete singular boundary, or cover arbitrary collapse data.

A useful nonsymmetric computation evolves the Einstein–massless-scalar system only inside Kerr, with a localized interior perturbation, a Kerr outer boundary, no incoming exterior radiation, dimensionless spins $\chi=a/M=0.9$, $0.95$, and $0.99$ in $G=c=1$ source units, and angular spectral truncations. It finds an outgoing shock whose radial width shrinks as $e^{-\kappa_-v}$ and whose tidal components grow exponentially, including for nonaxisymmetric data Chesler, Curiel, and Narayan 2019, §§II–III, pp. 2–7, PDF. Restoring dimensions, $\chi=cJ/(GM^2)$. Because the setup deliberately omits the incoming Price-law tail needed for scalar-invariant mass inflation, the authors leave that two-stream problem to future work. It therefore supports an inner-horizon shock mechanism, not the complete generic endpoint.

Invariant quantities that separate outcomes

A credible classification should follow at least five quantities.

1. Areal or quasi-local scale. In spherical symmetry, $r=\sqrt{A/4\pi}$ distinguishes finite-area null termination from crushing $r\to0$. Without symmetry, use geometrically selected two-surfaces and their area rather than a radial coordinate.
2. Mass and curvature. Track Hawking mass where defined, curvature scalars, and parallel-propagated curvature components. Divergent mass at finite area supports mass inflation; it does not alone determine metric regularity.
3. Null geometry. Track expansions, shears, affine completeness, lapse degeneration, and integrated null flux on both characteristic directions.
4. Extension norms. Test convergence of metric components in regular charts, first derivatives in $L_{\rm loc}^2$, and curvature in the exact norm relevant to the claimed $C^k$ obstruction.
5. Causal character. Determine invariantly whether the limiting boundary is null, spacelike, or has mixed pieces.

For a numerical claim, coordinate components are insufficient. A resolution sequence should demonstrate stable convergence of surface area, curvature contractions in transported frames, constraint norms, and the causal character of level sets. Independent gauges or formulations should agree on those invariants. Excision, loss of hyperbolicity, under-resolved blueshift, or failure of a horizon finder can imitate physical termination. No new simulation was run for this page; published computations are evidence only within their documented formulations and resolution studies.

Competing conditional pictures

The weak-null picture says exterior tails are blueshifted, producing divergent connection or curvature while a nondegenerate metric extends continuously across a null boundary. It has rigorous support in spherical charged models and a $C^0$ stability framework near Kerr.

An analytic rotating model further shows how mass inflation can terminate at a null curvature singularity with finite areal scale, but it is a model construction rather than a generic vacuum theorem Ori 1992, pp. 2117–2120.

The spacelike-endpoint picture says sufficiently nonlinear focusing ultimately replaces much or all of the inner horizon with a strong spacelike singularity. Collapse models and numerical evidence make this credible in some parameter regimes, but the open question is the size and topology of the data set producing each boundary type.

The mixed-boundary picture allows an early spacelike portion and a late weak-null portion, or different angular sectors with different behavior. It is not settled whether this structure is generic for asymptotically flat vacuum collapse.

Changing asymptotics can change the competition. In de Sitter spacetimes, exterior decay can be exponential, so the ratio of decay rates to inner-horizon surface gravity matters. Changing matter can also alter energy conditions and characteristic speeds. Neither modification is a minor correction to an asymptotically flat vacuum conclusion.

What a nonlinear interior calculation must show

A complete endpoint classification requires generic exterior evolution with quantified horizon decay, transport through the interior, and invariant lower bounds determining the singular boundary and extension class. The Hintz–Dafermos–Luk combination supplies the upper-bound and continuous-extension part for its small-data rotating-vacuum class; the generic lower-bound and complete-boundary questions remain distinct. A theorem with broader hypotheses, a stable counterexample on an open data set, or independently converged nonsymmetric calculations of these quantities would materially change the assessment. Absent that link, the general endpoint remains an open question.

Formulation-choice diagnostic

A simulation reports that a coordinate component $g_{vv}$ remains bounded while the Kretschmann scalar grows with resolution near its final slice. Can it conclude that the endpoint is a $C^0$-extendible weak null singularity?

Solution

No. Boundedness of one component in one chart neither proves convergence of a nondegenerate metric nor identifies a null limiting boundary. Curvature growth can obstruct $C^2$ extension if convergence and frame control are established, but it does not prove $C^0$ extendibility or finite tidal distortion. The calculation must add regular-coordinate metric convergence, causal-character diagnostics, transported-frame curvature, constraint and resolution studies, and preferably a second formulation or gauge.

References

• Chesler, Paul M., Erik Curiel, and Ramesh Narayan, “Numerical Evolution of Shocks in the Interior of Kerr Black Holes”, Physical Review D 99, 084033, 2019. Open PDF.
• Dafermos, Mihalis, and Jonathan Luk, “The Interior of Dynamical Vacuum Black Holes I: The C⁰-Stability of the Kerr Cauchy Horizon”, Annals of Mathematics 202, 309–630, 2025. Open PDF.
• Hintz, Peter, “Nonlinear Stability of Subextremal Kerr Black Holes”, arXiv:2606.28253v2, 2026. Open PDF.
• 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, 1–111, 2019. Open PDF.
• 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, 6, 2019. Open PDF.
• Ori, Amos, “Structure of the Singularity Inside a Realistic Rotating Black Hole”, Physical Review Letters 68, 2117–2120, 1992.