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Phys-73 · physics · mathematics · Physics

Generic weak future inextendibility near a fixed rotating Kerr datum

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every fixed Kerr mass $M>0$ and nonzero subextremal rotation $|a|<M$, is there a sufficiently small neighborhood of its asymptotically flat vacuum initial datum, in the stated weighted smooth topology, in which future $C^0$ intersect $W^{1,2}_loc-extendible$ maximal developments form a meagre set? Equivalently ask for a residual set of weakly future-inextendible data.

Short English statement

For every fixed Kerr mass $M>0$ and nonzero subextremal rotation $|a|<M$, is there a sufficiently small neighborhood of its asymptotically flat vacuum initial datum, in the stated weighted smooth topology, in which future $C^0$ intersect $W^{1,2}_loc-extendible$ maximal developments form a meagre set? Equivalently ask for a residual set of weakly future-inextendible data.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-73-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-73

C1-extension exclusion leaves continuous W1,2 extensions with nonsmooth first derivatives possible. The target weaker extension class is not ruled out. In the same near-Kerr setting, data permitting a continuous nondegenerate future extension with locally square-integrable weak connection are claimed meagre. The question retains these same parameters and conventions. The main theorem excludes C2 extensions generically, a strictly narrower extension class than the requested C0$\cap$W1,2_loc extensions. Within these assumptions, this is a claimed affirmative answer to the stated question.

Generic weak future inextendibility near a fixed rotating Kerr datum

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: Near each fixed rotating subextremal two-ended Kerr bridge, a dense $G_\delta$ class of smooth vacuum data is claimed to have no future $C^1$ extension; In the same near-Kerr setting, data permitting a continuous nondegenerate future extension with locally square-integrable weak connection are claimed meagre; Near every fixed rotating subextremal Kerr bridge, data with future $C^2$-extendible full developments are claimed meagre, without a symmetry restriction. Together they give a claimed affirmative answer in the matching scope.

Mathematical statement of the claimed result

Result 1.
Fix a complete two-ended Kerr bridge datum $d_*=(h_*,K_*)$ on $\Sigma\cong\mathbb R\times S^2$, of mass $M>0$ and subextremal nonzero rotation $\mathfrak a$. With a smooth end-radius weight $\rho\ge1$, define \[ p_m(u,v)=\sum_{j=0}^m\left(\sup_\Sigma\rho^{1+j}|\nabla_*^ju|_*+ \sup_\Sigma\rho^{2+j}|\nabla_*^jv|_*\right). \] The data space $\mathcal D$ consists of smooth complete positive-definite $h$, with all $p_m(d-d_*)$ finite, satisfying \[ R(h)+(\operatorname{tr}_hK)^2-|K|_h^2=0,\qquad \operatorname{div}_hK-d(\operatorname{tr}_hK)=0. \] Use the relative topology of all these seminorms and $\mathcal U_\varepsilon=\{d:p_{10}(d-d_*)<\varepsilon\}$. An extension has a smooth time-oriented isometric embedding of the full maximal globally hyperbolic development with image open and strictly contained in a connected Lorentzian four-manifold, and a future timelike $C^1$ curve that exits its image at its endpoint. The ambient extension need not satisfy the vacuum equations or global hyperbolicity. For $0<\mathfrak a<M$, the theorem claims some $\varepsilon_0>0$ and a dense $G_\delta$ subset $\mathcal G\subset\mathcal U_{\varepsilon_0}$ whose data have no such future extension with nondegenerate $C^1$ metric. This is Baire genericity in the stated near-Kerr topology.
Result 2.
Fix a complete two-ended Kerr bridge datum $d_*=(h_*,K_*)$ on $\Sigma\cong\mathbb R\times S^2$, of mass $M>0$ and subextremal nonzero rotation $\mathfrak a$. With a smooth end-radius weight $\rho\ge1$, define \[ p_m(u,v)=\sum_{j=0}^m\left(\sup_\Sigma\rho^{1+j}|\nabla_*^ju|_*+ \sup_\Sigma\rho^{2+j}|\nabla_*^jv|_*\right). \] The data space $\mathcal D$ consists of smooth complete positive-definite $h$, with all $p_m(d-d_*)$ finite, satisfying \[ R(h)+(\operatorname{tr}_hK)^2-|K|_h^2=0,\qquad \operatorname{div}_hK-d(\operatorname{tr}_hK)=0. \] Use the relative topology of all these seminorms and $\mathcal U_\varepsilon=\{d:p_{10}(d-d_*)<\varepsilon\}$. An extension has a smooth time-oriented isometric embedding of the full maximal globally hyperbolic development with image open and strictly contained in a connected Lorentzian four-manifold, and a future timelike $C^1$ curve that exits its image at its endpoint. The ambient extension need not satisfy the vacuum equations or global hyperbolicity. For $0<|\mathfrak a|<M$, the source claims an $\varepsilon>0$ for which \[ \{d\in\mathcal U_\varepsilon:\text{a future extension with }\widetilde g\in C^0\cap W^{1,2}_{\mathrm{loc}}\text{ exists}\} \text{ is meagre}. \] The weak metric is continuous and nondegenerate, and its chart components have square-integrable distributional first derivatives. Its weak Christoffel coefficients are locally $L^2$.
Result 3.
Fix a complete two-ended Kerr bridge datum $d_*=(h_*,K_*)$ on $\Sigma\cong\mathbb R\times S^2$, of mass $M>0$ and subextremal nonzero rotation $\mathfrak a$. With a smooth end-radius weight $\rho\ge1$, define \[ p_m(u,v)=\sum_{j=0}^m\left(\sup_\Sigma\rho^{1+j}|\nabla_*^ju|_*+ \sup_\Sigma\rho^{2+j}|\nabla_*^jv|_*\right). \] The data space $\mathcal D$ consists of smooth complete positive-definite $h$, with all $p_m(d-d_*)$ finite, satisfying \[ R(h)+(\operatorname{tr}_hK)^2-|K|_h^2=0,\qquad \operatorname{div}_hK-d(\operatorname{tr}_hK)=0. \] Use the relative topology of all these seminorms and $\mathcal U_\varepsilon=\{d:p_{10}(d-d_*)<\varepsilon\}$. An extension has a smooth time-oriented isometric embedding of the full maximal globally hyperbolic development with image open and strictly contained in a connected Lorentzian four-manifold, and a future timelike $C^1$ curve that exits its image at its endpoint. The ambient extension need not satisfy the vacuum equations or global hyperbolicity. For $0<\mathfrak a<M$, the source asserts that some $\varepsilon>0$ satisfies \[ \{d\in\mathcal U_\varepsilon:\text{a future extension with nondegenerate }C^2\text{ metric exists}\} \text{ is meagre}. \] Equivalently, future $C^2$-inextendible data include a residual subset of this neighborhood. No symmetry condition is imposed on the perturbation.

Outline of the argument

Closed transport tests reduce arbitrary exits to selected interior branches. Exact constrained signed packets are propagated nonlinearly through a double-null region and attached to the full development; the observing geodesic records an obstruction to continuously differentiable extension.

Countably many finite-branch tests cover arbitrary weak exits. Exact-data preparation and a nonlinear double-null comparison produce curvature signals along matched observing states; averaged holonomy detects the weak-connection obstruction and yields the category conclusion.

Closed finite-observation tests are coupled to microlocally separated Kerr packets. Stationary gauges and causal modulation control propagation across compact and far regions, then the two-ended interior estimates produce curvature behavior incompatible with a twice differentiable extension.

Formal verification

Family 264 has no Lean scope page in this release. Its selected result is not presented as formalized. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

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