Numerical Evidence and Failure Modes

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A numerical claim about cosmic censorship is unusually demanding because the event of interest often coincides with loss of regularity. Code failure, curvature growth, or failure to find an apparent horizon can each occur without a visible singularity in the continuum spacetime. Persuasive evidence must identify a convergent geometric signal, track its causal connection to the appropriate asymptotic region, and separate physical breakdown from gauge, boundary, and algorithmic breakdown.

Helpful background. Horizon finding and curvature diagnostics build on Black-Hole Geometry and Mechanics; perturbative benchmarks use Black-Hole Perturbation Theory; evolution systems come from Formulations of General Relativity; visibility belongs to Causality and Global Structure; and discretization principles are developed in Computational Relativity.

Continuum censorship claims and numerical data

Begin with a well-posed Einstein–matter initial-boundary value problem, a specified regularity class, and a definition of the censorship statement being tested. Weak-censorship tests must control the maximal development and the completeness of future null infinity or visibility from it, according to the formulation. Strong-censorship tests need an extendibility threshold at a Cauchy horizon. The numerical data should include evolved fields and constraints, curvature scalars, null expansions, horizon candidates, characteristic rays, and asymptotic charges or waveforms.

Useful geometric outputs include the areal radius, Hawking mass in spherical symmetry, eigenvalues of the Weyl tensor, scalar-field energy flux, and null expansions. Fluxes and the numerical values of $\theta_{(\ell)}$ and $\theta_{(n)}$ require a declared null normalization; expansion signs and zero sets survive positive boosts. Coordinate radius, lapse collapse, and termination time are not invariants. A worldline that appears to escape on a coordinate plot must be traced causally and, for weak censorship, connected to a controlled approximation of $\mathscr I^+$ .

Benchmarks and independent comparisons

Before exploring a candidate violation, evolve regular Minkowski, Schwarzschild, Kerr, and—where relevant—Reissner–Nordström data in the same formulation and gauge. Verify stationary charges, known horizons, and convergence over comparable durations. For critical scalar collapse, reproduce the established mass-scaling exponent and discrete self-similarity within a declared fitting interval; Choptuik gives the canonical numerical signatures Choptuik 1993, pp. 9–12 and Figures 2–4.

An independent implementation or formulation is especially valuable near a singular limit. Compare Cauchy and characteristic evolution where their domains overlap, two horizon finders with distinct initial guesses, and curvature computed both from evolved variables and reconstructed geometry. Exact spherical solutions can benchmark visibility logic even when the target calculation is nonspherical.

Convergence, errors, and causal reach

Use at least three well-separated resolutions and measure convergence of constraints and geometric observables before the smallest resolved scale reaches the grid cutoff. Adaptive refinement must demonstrate that refinement boundaries and interpolation do not seed the feature. Vary outer-boundary position and condition, extraction radius, dissipation, gauge parameters, excision placement, and angular resolution. Report a norm and fit range; a visually sharpening spike is not a convergence test.

For a proposed naked singularity, estimate the curvature scale and trace outgoing null rays from successively better-resolved neighborhoods. Show that the rays remain outside any event-horizon approximation and reach an extraction world tube whose relation to null infinity is controlled. For an interior test, evolve through the event horizon with coordinates regular there, resolve the blueshift region, and distinguish $C^0$ , curvature, and integrability diagnostics.

Failure modes and reproducibility boundary

One common failure mode is lapse collapse freezing coordinate evolution before the physical question is decided. Others include constraint-violating modes growing faster than curvature, an apparent-horizon finder missing a highly distorted surface, excision exposing an inflow boundary, reflected outer-boundary radiation, atmosphere floors changing the matter model, and adaptive mesh refinement following coordinate rather than geometric scale. Agreement of two gauge-dependent plots does not remove these risks.

This page specifies the continuum and validation protocol; it does not rerun a collapse simulation. Reproduction requires versioned initial-data generators, matter equations and parameters, evolution and gauge choices, boundary and excision algorithms, mesh histories, tolerances, horizon-search seeds, ray-tracing code, raw diagnostics, and scripts that produce every scaling fit. A calculation that omits failed resolutions or parameter searches cannot support a genericity claim.

Exercise

Across resolutions $h$ , $h/2$ , and $h/4$ , a curvature maximum grows as $h^{-2}$ , while the Hamiltonian-constraint norm also grows as $h^{-2}$ and outgoing rays change qualitatively with gauge. What conclusion is justified?

Solution

No continuum singularity or visibility conclusion is justified. The physical diagnostic is not converging independently of constraint violation, and the causal claim is gauge unstable. The next step is to restore constraint convergence, use geometric ray and horizon diagnostics, and show that the curvature scaling persists under gauge, boundary, and resolution changes.

References

Choptuik, Matthew W. “Universality and Scaling in Gravitational Collapse of a Massless Scalar Field.” Physical Review Letters 70 (1993): 9–12. doi:10.1103/PhysRevLett.70.9.