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Phys-42 · physics · mathematics · Physics

The numerical spacetime enclosing-area inequality in every dimension

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every $n\geq 3$, consider smooth complete one-ended asymptotically Euclidean initial-data exteriors with integrable constraint densities, DEC, finite ADM limits, weakly future-trapped compact boundary, $g-\delta =O_6(r^{-q})$, $K=O_5(r^{-1-q})$, $(n-2)/2<q<n-2$, $E>|P|$ and positive full minimum enclosing area $A_min$. Is $\operatorname{sqrt}(E^2-|P|^2)\geq (1/2)(A_min/\omega _{n-1})^((n-2)/(n-1))$?

Short English statement

For every $n\geq 3$, consider smooth complete one-ended asymptotically Euclidean initial-data exteriors with integrable constraint densities, DEC, finite ADM limits, weakly future-trapped compact boundary, $g-\delta =O_6(r^{-q})$, $K=O_5(r^{-1-q})$, $(n-2)/2<q<n-2$, $E>|P|$ and positive full minimum enclosing area $A_min$. Is $\operatorname{sqrt}(E^2-|P|^2)\geq (1/2)(A_min/\omega _{n-1})^((n-2)/(n-1))$?

Sources

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Contributions 1

Phys-42-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-42

The source gives boundary deformations/prepared metric comparisons in spatial dimensions three and four. It does not itself state the target all-dimensional invariant-mass numerical inequality. The proposed neutral spacetime Penrose inequality compares invariant ADM mass with minimum enclosing area in all dimensions $n\geq3$, under the listed energy, decay, and weak future-trapping hypotheses. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

The numerical spacetime enclosing-area inequality in every dimension

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: In dimensions three and four, trapped exteriors are claimed deformable to nonnegative-scalar-curvature metrics with protected enclosing area and arbitrarily small ADM-energy loss; The proposed neutral spacetime Penrose inequality compares invariant ADM mass with minimum enclosing area in all dimensions $n\geq3$, under the listed energy, decay, and weak future-trapping hypotheses. Together they give a claimed affirmative answer in the matching scope.

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 $a_g(S)=\inf_\Gamma\operatorname{Area}_g(\Gamma)$. In dimension three, suppose $g-\delta=O_2(r^{-q})$, $K=O_1(r^{-1-q})$ for $q>1/2$, and $\theta_+\le0$ on every boundary component. If $E>|P|$, the exterior theorem asserts \[ \sqrt{E^2-|P|^2}\ge\sqrt{a_g(S)/(16\pi)}. \] The preparatory output treats the stronger subclass with compactly supported $K$, $g-\delta=O_j(r^{-1})$ for every fixed $j$, $P=0$, $E>0$, $\theta_+<0$, $A_*=a_g(S)>0$, and, for a smooth radius extension $\varrho\ge1$, \[ R_g=O(r^{-3-\delta}),\qquad \mu-|J|_g\ge c\varrho^{-3-\delta}\quad(c>0,\ 0<\delta<1). \] For sufficiently small fixed $\epsilon>0$ and large $N$, a smooth one-ended subexterior has a complete boundary-included metric $\widehat g_{\epsilon,N}$ satisfying \[ R_{\widehat g}\ge0,\quad H_{\partial\Omega}(\widehat g)<0,\quad \widehat g\ge e^{-4\epsilon}g,\quad E_{\widehat g}\le E+\eta_{\epsilon,N},\quad |T|_{\widehat g}\ge e^{-4\epsilon}A_* \] for every full cut $T$. At fixed $\epsilon$, $0\le\eta_{\epsilon,N}\le C(1+N)^a(e^{-\epsilon N}+e^{-\epsilon N/2})\to0$. The output metric has $O_2(r^{-1})$ decay and scalar curvature $O(r^{-3-\delta'})$ for some $\delta'>0$.
Result 2.
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 $a_g(S)=\inf_\Gamma\operatorname{Area}_g(\Gamma)$. For $g-\delta=O_6(r^{-q})$, $K=O_5(r^{-1-q})$ with $(n-2)/2<q<n-2$, $\theta_+\le0$, $a_g(S)>0$ and $E>|P|$, the numerical bound is \[ \sqrt{E^2-|P|^2}\ge\frac12\left(\frac{a_g(S)}{\omega_{n-1}}\right)^{(n-2)/(n-1)}. \] Its endpoint corollary derives $E>|P|$ from the other hypotheses. In $n\in\{3,4\}$, the lower-regularity theorem only requires $g-\delta=O_2(r^{-q})$, $K=O_1(r^{-1-q})$ with $q>(n-2)/2$; it proves $a_g(S)>0$ and $E>|P|$ as well as the same inequality. The coefficient is sharp. Numerical assumptions do not require connected boundary, maximality, vacuum, outermostness, spin, or outer area minimization. There is also a complete-data formulation on a smooth connected orientable boundaryless three-manifold $M$, with complete $g$, a compact core and finitely many AF ends. Each end has $g-\delta=O_2(r^{-q})$, $K=O_1(r^{-1-q})$, $q>1/2$, and finite ADM limits; the constraint densities are globally integrable and satisfy DEC. Choose an end-side open region $D^+$, possibly disconnected or containing other ends, whose full boundary $S$ is nonempty, compact, smooth, embedded and two-sided, with $\theta_+\le0$ towards $D^+$. Define \[ A_{\min}(S)=\inf_{D'^+\subseteq D^+}\operatorname{Area}_g(\partial D'^+), \] where smooth open competitors contain the distant designated end and have compact smooth full boundary, permitting disconnected competitors, other retained ends, and coincidence with $S$. The conclusion is $E>|P|$ and $\sqrt{E^2-|P|^2}\ge\sqrt{A_{\min}(S)/(16\pi)}$.

Outline of the argument

Collars and weak boundary supports set the graph barrier. Elliptic estimates exclude contact with the floor and solve the graph-conformal deformation; its negative inner mean curvature supplies the boundary route to the neutral mass-area bound.

End preparation removes ADM momentum before a coupled graph and conformal scalar construction. Filled-equation estimates and height separation protect enclosing cuts, while the scalar identity bounds the change in mass; the Riemannian comparison then returns the inequality to the original data.

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.

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