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The spacetime Penrose inequality with charge and original-data rigidity

OpenAI · October 5, 2026

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Abstract

Under the stated energy, decay, and trapping hypotheses, we prove the sharp charged spacetime Penrose upper-area inequality for one-ended three-dimensional initial data with source-free electric and magnetic fields. The theorem allows arbitrary second fundamental form, nonzero ADM momentum, and disconnected boundary. Writing m for invariant ADM mass, Q for total charge magnitude, and rA for the minimum-enclosing-area radius, the bound is m ≥ Q and $r_A\le m+\sqrt{m^2-Q^2}$. The polynomial mass bound $m\ge(r_A+Q^2/r_A)/2$ is asserted only when $r_A\gt Q$. When m > Q, equality under the stated connected, outermost, outer-area-minimizing future-horizon hypotheses identifies the original data as a smooth spacelike slice of dyonic Reissner–Nordström, including smooth attachment at the future horizon.

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Formalization

Possibly relevant literature (found by the agent, not checked)

  • T. Dolmen, The Penrose inequality with charge for 2-convex initial data sets (arXiv:2607.29447, 31 Jul 2026). Proves a sharp charged Penrose inequality for Einstein–Maxwell data under Dong's 2-convexity condition and a connected outermost horizon with area at least 4πq², with Reissner–Nordström rigidity; this corresponds to this paper's charged upper-area bound and dyonic Reissner–Nordström rigidity under different hypotheses.
  • E. Ng, The suboptimal Penrose inequality for charged initial data sets (arXiv:2609.22418, 18 Sep 2026). Proves a charged Penrose-type bound with a non-sharp constant for three-dimensional Einstein–Maxwell data with zero magnetic field, a non-sharp counterpart of this paper's main inequality.

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Cite

@misc{OAI:The-spacetime-Penrose-inequality-with-charge-and-original-data-rigidity-October-5-2026,
  author = {{OpenAI}},
  title = {{The spacetime Penrose inequality with charge and original-data rigidity}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-spacetime-Penrose-inequality-with-charge-and-original-data-rigidity-October-5-2026/paper.pdf}{OAI:The-spacetime-Penrose-inequality-with-charge-and-original-data-rigidity-October-5-2026}},
  year = {2026}
}

Lean formalization

OpenAI provides Lean files for this family: lean/docs/260.md. The Lean statements have not been checked by people against the paper's claims.

Family 260: Spacetime Penrose inequalities: enclosing area, charge, rotation, and anti-de Sitter extensions

Proves the sharp enclosing-area spacetime Penrose inequality for smooth one-ended asymptotically flat initial data in every spatial dimension n ≥ 3, under dominant energy, weak future trapping, positive enclosing area, and the stated decay assumptions. It bounds invariant ADM mass below using minimum enclosing area, with equality rigidity under additional horizon hypotheses. Charged upper-area bounds treat dyonic three-dimensional data; the higher-dimensional purely electric extension uses the matched neutral theorem.

Other papers in this family: