The Kerr–Newman extension of the angular-momentum Penrose inequality

Consider initial data satisfying the appropriate energy conditions, with horizon area $A$ , angular momentum $J$ , and electric charge $Q$ . Kerr–Newman extension conjecture. The ADM mass should satisfy

M_{\mathrm{ADM}} \geq \sqrt{\frac{A}{16\pi} + \frac{4\pi J^2}{A} + \frac{Q^2}{4}},

with equality for Kerr–Newman spacetime. This is the full charged-and-rotating extension; the source notes that the complete Kerr–Newman case remains conjectural while pure rotation and pure charge cases are treated separately.

References

Primary source

Da Xu, “The Angular Momentum Penrose Inequality”, arXiv:2512.06918 (2026).

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RemarkAI-assistedClaimed by OpenAI. Claims the sharp Kerr–Newman Penrose inequality under the stated smooth, one-ended, axisymmetric electrovacuum, topology, decay, zero ADM momentum, connected outermost outer-area-minimizing future marginally outer trapped boundary, Coulomb electromagnetic asymptotics, and physical-area hypotheses, using conserved total angular momentum including its electromagnetic contribution.See full solutionHide full solution

Claimed by OpenAI. Claims the sharp Kerr–Newman Penrose inequality under the stated smooth, one-ended, axisymmetric electrovacuum, topology, decay, zero ADM momentum, connected outermost outer-area-minimizing future marginally outer trapped boundary, Coulomb electromagnetic asymptotics, and physical-area hypotheses, using conserved total angular momentum including its electromagnetic contribution.

Scope relative to this problem: The manuscript claims m^2 >= A/(16pi) + Q^2/2 + pi(Q^4+4J^2)/A, where Q^2=Q_e^2+Q_b^2 and J includes the conserved electromagnetic contribution. Its smooth one-ended axisymmetric electrovacuum hypotheses include zero ADM momentum, a connected outermost outer-area-minimizing future MOTS, Coulomb electromagnetic tails and A >= 4pisqrt(Q^4+4J^2). Equality is classified on the strict area branch. The target displayed expression has different charge terms; this attachment does not certify that expression or its stated equality characterization.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Kerr-Newman-Penrose-Inequality-for-Axisymmetric-Electrovacuum-Exteriors-October-5-2026/paper.pdf

OpenAI-260-04-The-Kerr-Newman-Penrose-Inequality-for-Axisymmetric-Electrovacuum-Exteriors.pdf825,467 bytesOpen

RemarkAI-assistedClaimed by OpenAI. Claims to construct smooth axisymmetric electrovacuum exteriors that violate the Kerr–Newman Penrose inequality when J is the bare gravitational ADM angular momentum and the electromagnetic fields have only O(r−2 ) decay, allowing angularly varying leading tails. The examples have zero total electric and magnetic charge even though both electromagnetic fields are nonzero.See full solutionHide full solution

Claimed by OpenAI. Claims to construct smooth axisymmetric electrovacuum exteriors that violate the Kerr–Newman Penrose inequality when J is the bare gravitational ADM angular momentum and the electromagnetic fields have only O(r−2 ) decay, allowing angularly varying leading tails. The examples have zero total electric and magnetic charge even though both electromagnetic fields are nonzero.

Scope relative to this problem: The examples have zero total electric and magnetic charge but nonzero fields and angularly varying O(r^-2) tails. They violate a Kerr-Newman-form inequality when J is bare gravitational ADM angular momentum. This is a convention-sensitive counterexample, not a refutation of the source companion using conserved electromagnetic total J and Coulomb tails, nor a blanket refutation of the target without choosing its J convention.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Electromagnetic-tails-and-the-Kerr-Newman-Penrose-inequality-October-5-2026/paper.pdf

OpenAI-260-05-Electromagnetic-tails-and-the-Kerr-Newman-Penrose-inequality.pdf377,252 bytesOpen