we-solve (beta)

Phys-82 · physics · mathematics · Physics

Original-data rigidity at equality in spacetime Penrose bounds

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For each $n\geq 3$ and the smooth DEC one-ended enclosing-area exterior class with connected nondegenerate outermost outer-minimizing horizon specified in the manuscript, does equality in the neutral spacetime Penrose bound imply a global spacelike embedding of the entire original exterior, including boundary, into the corresponding Schwarzschild–Tangherlini exterior and future horizon? Require the given metric and second fundamental form to be induced exactly, a future normal smooth up to the boundary, and the chosen end approaching spatial infinity.

Short English statement

For each $n\geq 3$ and the smooth DEC one-ended enclosing-area exterior class with connected nondegenerate outermost outer-minimizing horizon specified in the manuscript, does equality in the neutral spacetime Penrose bound imply a global spacelike embedding of the entire original exterior, including boundary, into the corresponding Schwarzschild–Tangherlini exterior and future horizon? Require the given metric and second fundamental form to be induced exactly, a future normal smooth up to the boundary, and the chosen end approaching spatial infinity.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-82-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-82

Equality in the neutral spacetime bound is claimed to reconstruct the original exterior as a Schwarzschild--Tangherlini slice under the precise horizon and decay assumptions. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

Original-data rigidity at equality in spacetime Penrose bounds

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Equality in the neutral spacetime bound is claimed to reconstruct the original exterior as a Schwarzschild--Tangherlini slice under the precise horizon and decay assumptions.

Mathematical statement of the claimed result

Result 1.
Let $\Omega^n$, $n\ge3$, be smooth, connected and orientable, with nonempty compact smooth boundary $S$, complete with that boundary included, and exactly one end, with Euclidean coordinates and compact complement. The tensors $g,K$ extend smoothly to $S$. With $\tau=\operatorname{tr}_gK$, use \[ 2\mu=R_g+\tau^2-|K|_g^2,\qquad J=\operatorname{div}_g(K-\tau g),\qquad \mu\ge|J|_g,\quad\mu,|J|_g\in L^1(\Omega,dV_g). \] The finite ADM fluxes use $\omega_{n-1}=|\mathbb S^{n-1}|$ and \[ E=\frac1{2(n-1)\omega_{n-1}}\lim_{R\to\infty}\int_{|x|=R} (\partial_jg_{ij}-\partial_ig_{jj})n_\delta^i\,dA_\delta,\quad P_i=\frac1{(n-1)\omega_{n-1}}\lim_{R\to\infty}\int_{|x|=R}(K_{ij}-\tau g_{ij})n_\delta^j\,dA_\delta. \] Orient boundary normals towards the end, with $\theta_+=H+\operatorname{tr}_SK$. Here $O_j(r^{-b})$ controls coordinate derivatives through order $j$, of respective sizes $O(r^{-b-i})$. Choose a connected smooth end-side region $D\subset\Omega$, closed in $\Omega$, containing the distant end and having its manifold interior in $\operatorname{int}\Omega$. A full cut consists of all of its intrinsic boundary $\Gamma=\partial D$; require this boundary to be compact, smoothly embedded and two-sided. All components and portions coinciding with $S=\partial\Omega$ count. Put $a_g(S)=\inf_\Gamma\operatorname{Area}_g(\Gamma)$. Suppose $g-\delta=O_6(r^{-q})$, $K=O_5(r^{-1-q})$, $(n-2)/2<q<n-2$, and $E>|P|$. Require connected $S$, $\theta_+(S)=0$, and $\operatorname{Area}_g(\Gamma)\ge A=|S|_g>0$ for every full cut. Exclude every compact smooth embedded two-sided full enclosing hypersurface wholly in $\operatorname{int}\Omega$ with $\theta_+\le0$, allowing disconnected excluded surfaces. If \[ m=\sqrt{E^2-|P|^2}=\frac12(A/\omega_{n-1})^{(n-2)/(n-1)}, \] then the entire original pair $(g,K)$, including the boundary and future normal, is induced by a smooth spacelike embedding into Schwarzschild--Tangherlini of mass $m$, using its regular extension across the future horizon. The interior lies in its domain of outer communication, the end reaches spatial infinity, and $S$ maps to a full future-horizon section or the bifurcation sphere. Conversely, admissible spacelike model slices with these same intrinsic hypotheses attain equality and have $A=\omega_{n-1}(2m)^{(n-1)/(n-2)}$. Time-symmetric and nonzero-$K$ examples exist in each $n\ge3$. In the three-dimensional complete-data equality theorem, use a smooth connected orientable boundaryless $M$ with complete metric, compact core and finitely many AF ends, each obeying $O_2/O_1$ decay with $q>1/2$, finite ADM limits, integrable densities and DEC. Take $S=\bigsqcup_{i=1}^{\ell}S_i$, $1\le\ell<\infty$, with disjoint connected compact smooth two-sided components bounding a connected closed exterior $\Omega$ with exactly the designated end. Require $\theta_+(S_i)=0$ and $\operatorname{Area}_g(\Gamma)\ge A=\sum_i|S_i|_g>0$ for every full enclosing cut. Exclude distinct enclosing cuts whose components each coincide with an entire $S_i$ or lie wholly outside, with at least one component of the latter kind, and with $\theta_+\le0$ on all components. If $E>|P|$ and $\sqrt{E^2-|P|^2}=\sqrt{A/(16\pi)}$, then $\ell=1$, $S\cong S^2$, and the entire original exterior admits a proper smooth spacelike Schwarzschild embedding of mass $\sqrt{E^2-|P|^2}$ inducing $g,K$ and the prescribed future normal. Its end reaches spatial infinity and its boundary is a full future-horizon section or bifurcation sphere, with smooth continuation through that boundary. For connected $S$, the connected-horizon corollary replaces the cut exclusion by exclusion of every wholly exterior weakly future-trapped enclosing surface, including disconnected ones. Thus $\ell\ge2$ forces strict inequality in the finite-component subclass.

Outline of the argument

Minimizing enclosures are treated through their area envelope. Saturation yields a constrained minimum in the original data, whose static rigidity equations produce the spacetime embedding and smooth horizon attachment; in three dimensions the analysis also rules out equality for disconnected horizons.

Formal verification

For family 260, the only selected family entry is the CKS end-replacement construction and Schwarzschild equality examples. A connected-boundary inequality is conditional on the flat Penrose input; the general Bondi or spacetime inequalities are not formalized there. No Comparator is assigned to the selected manuscript here; the family page selects other papers or supporting statements. 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

Go to contribution →