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Strong Cosmic Censorship for Spherical Einstein–Vlasov Collapse

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N=\{1,2,\ldots\}$; the standard number fields have their usual meanings. All additional conventions are stated locally.
Definitions and conventions.
The Einstein--Vlasov system couples $G_{\mu\nu}=8\pi T_{\mu\nu}[f]$ to collisionless transport of a nonnegative phase-space density $f$ along the spacetime geodesic flow on the particle mass shell. The stress-energy tensor is obtained by integrating particle momenta against $f$. Asymptotically flat initial data describe an isolated system; spherical symmetry reduces the metric and matter fields to radial variables. Weak cosmic censorship asks for complete future null infinity for generic admissible data. Strong cosmic censorship concerns inextendibility of the maximal globally hyperbolic development in a specified regularity class. In the source's advanced Eddington--Finkelstein coordinates $(v,r)$, $ds^2=-a(v,r)b(v,r)^2\,dv^2+2b(v,r)\,dv\,dr+r^2d\Omega^2$, with $b>0$. A sphere of constant $v,r$ is trapped when $a<0$. The cited class of initial data begins with $a>0$ and is arranged so that a trapped surface subsequently forms. A curvature singularity means blow-up of curvature in the sense to be specified by the collapse statement.
Source scope and qualifications.
The claim concerns the particular shell-like initial-data class in the report's cited collapse theorem, not arbitrary data merely satisfying a>0 somewhere. The report proves future completeness of the event-horizon generator but only conjectures the central singularity and strong censorship. A regularity class for inextendibility remains necessary.
Catalogue mathematical target.
Catalogue wording retained for traceability.
For spherically symmetric Einstein–Vlasov initial data in Eddington–Finkelstein coordinates that evolve from $a>0$ to form a trapped surface, must a curvature singularity arise at the center, and does strong cosmic censorship hold?
Saved literature context.
The supplied catalogue's assessment dated 2026-08-21 records: For the specified trapped-surface data, the report itself states only a conjecture that a central curvature singularity forms and strong cosmic censorship holds; no complete proof of this claim was found. The assessment is retained as dated provenance; it is not a new comprehensive literature or open-status certification.

Short English statement

For the prescribed spherical collapse data, prove formation of a central curvature singularity and strong cosmic censorship.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Phys-250-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on Phys-250

Strong Cosmic Censorship for Spherical Einstein–Vlasov Collapse

Definitions and mathematical statement

Shared notation.
$\mathbb N=\{1,2,\ldots\}$; the standard number fields have their usual meanings. All additional conventions are stated locally.
Definitions and conventions.
The Einstein--Vlasov system couples $G_{\mu\nu}=8\pi T_{\mu\nu}[f]$ to collisionless transport of a nonnegative phase-space density $f$ along the spacetime geodesic flow on the particle mass shell. The stress-energy tensor is obtained by integrating particle momenta against $f$. Asymptotically flat initial data describe an isolated system; spherical symmetry reduces the metric and matter fields to radial variables. Weak cosmic censorship asks for complete future null infinity for generic admissible data. Strong cosmic censorship concerns inextendibility of the maximal globally hyperbolic development in a specified regularity class. In the source's advanced Eddington--Finkelstein coordinates $(v,r)$, $ds^2=-a(v,r)b(v,r)^2\,dv^2+2b(v,r)\,dv\,dr+r^2d\Omega^2$, with $b>0$. A sphere of constant $v,r$ is trapped when $a<0$. The cited class of initial data begins with $a>0$ and is arranged so that a trapped surface subsequently forms. A curvature singularity means blow-up of curvature in the sense to be specified by the collapse statement.
Source scope and qualifications.
The claim concerns the particular shell-like initial-data class in the report's cited collapse theorem, not arbitrary data merely satisfying a>0 somewhere. The report proves future completeness of the event-horizon generator but only conjectures the central singularity and strong censorship. A regularity class for inextendibility remains necessary.
Catalogue mathematical target.
Catalogue wording retained for traceability.
For spherically symmetric Einstein–Vlasov initial data in Eddington–Finkelstein coordinates that evolve from $a>0$ to form a trapped surface, must a curvature singularity arise at the center, and does strong cosmic censorship hold?
Saved literature context.
The supplied catalogue's assessment dated 2026-08-21 records: For the specified trapped-surface data, the report itself states only a conjecture that a central curvature singularity forms and strong cosmic censorship holds; no complete proof of this claim was found. The assessment is retained as dated provenance; it is not a new comprehensive literature or open-status certification.

Short English statement

For the prescribed spherical collapse data, prove formation of a central curvature singularity and strong cosmic censorship.

Research attempt

Model, date and investigated scope.
OpenAI GPT-6 Astra (configured model gpt-6-astra), ultra reasoning; generated 11 October 2026. Relates trapping to spherical mass and derives a necessary cubic mass bound at a smooth regular center, showing why trapping alone does not force central blowup.
Sources and literature check.
The complete supplied definition, including source qualifications and the dated literature assessment, was reviewed. The calculation below is an elementary restricted attempt motivated by [S2]; it does not claim novelty or a fresh literature-status determination.
Context and investigated scope.
The complete supplied definition, including source qualifications and the dated literature assessment, was reviewed. The calculation below is an elementary restricted attempt motivated by [S2]; it does not claim novelty or a fresh literature-status determination.
A central regularity requirement.
For the source metric, inversion of the $(v,r)$ block gives $g^{rr}=a$. Define the spherical mass function by $m(v,r)=r(1-a(v,r))/2$. Trapping $a<0$ at a sphere is therefore exactly $2m/r>1$ there. At a smooth regular spherical center, the areal radius agrees to first order with proper radial distance and the radial metric in regular local coordinates has expansion $1+O(r^2)$. Under this regular-center hypothesis $a=1+O(r^2)$, hence \[ m(v,r)=O(r^3)\qquad(r\downarrow0). \] Thus proving failure of this bound along an approaching central region would obstruct that regular-center geometry.
Trapping at a positive radius is insufficient by itself.
Smooth nonnegative mass profiles can satisfy $m(r)=cr^3$ near zero and $2m(r_0)/r_0>1$ at a positive radius. For example $m(r)=cr^3$ up to $r_0$ has the latter property when $2cr_0^2>1$. This is only a mass-profile example, not asserted to solve the Einstein--Vlasov constraints or evolution; it shows that the algebraic trapped-sphere inequality does not imply a central failure of the cubic bound at that time.
Remaining dynamical step.
The source already supplies a shell class forming a trapped surface and future completeness of the event-horizon generator. One still needs estimates forcing concentration toward the center and curvature blowup, then an inextendibility argument in a specified regularity class. The displayed local mass identities do not prove either. In particular central loss of smoothness would not automatically establish the strongest low-regularity versions of cosmic censorship. Both requested conclusions remain unresolved for the actual source data.
Verification and adversarial audit.
Checked metric inversion, trapped-sphere equivalence, regular-center hypothesis and that the toy mass profile is not asserted to be Einstein–Vlasov data. Model self-audit only; no independent human proof checking. Prompts A and B were applied to this canonical record and its qualified source scope. Checked metric inversion, trapped-sphere equivalence, regular-center hypothesis and that the toy mass profile is not asserted to be Einstein–Vlasov data. Retained unresolved status and zero original-target progress; The Vlasov evolution has not been shown to force central concentration/curvature blowup or inextendibility in a specified class.

Coordinator checked the local spherical mass identity and regular-center cubic order, with the algebraic mass profile distinguished from an Einstein-Vlasov solution. This additional check was model-assisted, not independent human verification.

Tools, process and contributions.
Python 3 was used for file preparation and structural checks; no numerical experiment or theorem prover was used. Contribution prompt A was followed by prompt B. GPT-6 Astra with ultra reasoning drafted the research and performed its stated self-audit; any additional coordinator contribution is identified in the verification and audit. A GPT-6-based Codex coordinator assembled the manuscript and checked repository integration; its exact variant and reasoning configuration were not exposed. No human mathematical contribution, independent human proof check or formal proof certification is supplied. Complete derivations and the consulted source references are included here; private reasoning transcripts are not published.
Final status and first remaining gap.
UNRESOLVED. The Vlasov evolution has not been shown to force central concentration/curvature blowup or inextendibility in a specified class.

Completion Estimate

COMPLETION: 0% The calculation is an auxiliary or restricted attempt and establishes no newly verified portion of the full source target. The Vlasov evolution has not been shown to force central concentration/curvature blowup or inextendibility in a specified class.

Sources

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