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The Kerr–Newman area bound without electromagnetic-tail controls

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Do all smooth complete maximal axisymmetric source-free electrovacuum exteriors on ${|x|\geq 1}$, with a connected outermost future MOTS minimizing area among all enclosing cuts, $g-\delta =O_4(r^{-1})$, $K=O_3(r^{-3})$, electric and magnetic fields $O_3(r^{-2})$, integrable constraint densities and finite fluxes, zero ADM linear momentum, positive ADM energy and zero total electric and magnetic charges, obey $m^2\geq A/(16\pi )+4\pi $ $J^2/A$ on $A>8\pi |J|$, with every equality datum induced by a Kerr–Newman exterior slice? Here J is the gravitational angular-momentum flux at infinity without an electromagnetic surface term; nonzero exterior fields need not have Coulomb leading terms.

Short English statement

Do all smooth complete maximal axisymmetric source-free electrovacuum exteriors on ${|x|\geq 1}$, with a connected outermost future MOTS minimizing area among all enclosing cuts, $g-\delta =O_4(r^{-1})$, $K=O_3(r^{-3})$, electric and magnetic fields $O_3(r^{-2})$, integrable constraint densities and finite fluxes, zero ADM linear momentum, positive ADM energy and zero total electric and magnetic charges, obey $m^2\geq A/(16\pi )+4\pi $ $J^2/A$ on $A>8\pi |J|$, with every equality datum induced by a Kerr–Newman exterior slice? Here J is the gravitational angular-momentum flux at infinity without an electromagnetic surface term; nonzero exterior fields need not have Coulomb leading terms.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-65-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-65

The main theorem constructs admissible zero-total-charge exteriors with m${}^{2}$ below the requested gravitational-J expression, and equality examples not induced from Kerr–Newman. Both universal inequality and equality-rigidity assertions are refuted. Within these assumptions, this is a claimed negative answer to the stated question.

The Kerr–Newman area bound without electromagnetic-tail controls

OpenAI's claim

OpenAI presents a counterexample in the selected manuscript. With angularly varying $O(r^{-2})$ electromagnetic tails and bare gravitational angular momentum, the authors claim Penrose violations and non-Kerr--Newman equality examples.

Mathematical statement of the claimed result

Result 1.
The source constructs smooth source-free axisymmetric electrovacuum data on $\Omega=\{x\in\mathbb R^3:|x|\ge1\}$, with connected future-MOTS boundary $S$, no interior weakly future-trapped enclosing surface, and every full enclosing cut of area at least $A=|S|$. The examples are maximal, $\operatorname{tr}_gK=0$, with $g-\delta=O_4(r^{-1})$, $K=O_3(r^{-3})$, and electromagnetic fields $O_3(r^{-2})$. They obey \[ R_g-|K|^2=2(|\mathcal E|^2+|\mathcal B|^2),\quad \operatorname{div}K=2(\mathcal E\times\mathcal B)^\flat,\quad \operatorname{div}\mathcal E=\operatorname{div}\mathcal B=0. \] With $P_{\mathrm{ADM}}=0$, $m=E_{\mathrm{ADM}}>0$, gravitational ADM angular momentum $J$, and zero total electric/magnetic charge, both fields are nevertheless nonzero. One example satisfies \[ A>8\pi|J|,\qquad m^2<\frac A{16\pi}+\frac{4\pi J^2}{A}. \] Another satisfies equality in that formula but admits no smooth spacelike Kerr--Newman embedding inducing its metric, second fundamental form, and electromagnetic fields. The missing electromagnetic-tail controls are therefore material. The angular momentum convention is \[ J=\frac1{8\pi}\lim_{R\to\infty}\int_{S_R}(K-(\operatorname{tr}_gK)g)(\nu_R,\eta)\,dA_g. \] The symmetry generator $\eta$ has period $2\pi$ and agrees with standard axial rotation at infinity. This flux excludes an electromagnetic surface contribution. The counterexample does not assert failure for the corrected conserved total angular momentum.

Outline of the argument

Explicit axisymmetric electrovacuum seeds are reduced to a scalar conformal equation. Tail choices make both total charges zero while retaining nonzero fields and altering the angular-momentum balance; the conformal solution preserves the required outer horizon and realizes the inequality and rigidity failures.

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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