Math Conjectures

General spacetime Penrose inequality

OPENLandmarkConjectureProposed 1973 · Standard version

Canonical statement

Let (M3,g,k)(M^3,g,k) be a smooth, connected, orientable, complete asymptotically flat initial data set. Define its energy and momentum densities by
16πμ=Rg+(tr⁡gk)2−∣k∣g2,8πJ=div⁡g ⁣(k−(tr⁡gk)g), 16\pi\mu=R_g+(\operatorname{tr}_gk)^2-|k|_g^2,\qquad 8\pi J=\operatorname{div}_g\!\left(k-(\operatorname{tr}_gk)g\right),
assume the dominant energy condition μ≥∣J∣g\mu\ge |J|_g, and fix an asymptotically flat end. Let Σ\Sigma be an outermost apparent horizon relative to that end, allowing a disjoint union of marginally outer trapped components H+tr⁡Σk=0H+\operatorname{tr}_{\Sigma}k=0 and marginally inner trapped components H−tr⁡Σk=0H-\operatorname{tr}_{\Sigma}k=0. Define AA to be the infimum of the total gg-areas of smooth closed surfaces enclosing Σ\Sigma relative to the chosen end. If the ADM energy-momentum of the end is (E,P)(E,P) and mADM=E2−∣P∣2m_{\mathrm{ADM}}=\sqrt{E^2-|P|^2}, then
mADM≥A16π. m_{\mathrm{ADM}}\ge\sqrt{\frac{A}{16\pi}}.
Equality should occur only when the exterior data arise from a spacelike slice of the Schwarzschild spacetime.
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Let \((M^3,g,k)\) be a smooth, connected, orientable, complete asymptotically flat initial data set. Define its energy and momentum densities by \[ 16\pi\mu=R_g+(\operatorname{tr}_gk)^2-|k|_g^2,\qquad 8\pi J=\operatorname{div}_g\!\left(k-(\operatorname{tr}_gk)g\right), \] assume the dominant energy condition \(\mu\ge |J|_g\), and fix an asymptotically flat end. Let \(\Sigma\) be an outermost apparent horizon relative to that end, allowing a disjoint union of marginally outer trapped components \(H+\operatorname{tr}_{\Sigma}k=0\) and marginally inner trapped components \(H-\operatorname{tr}_{\Sigma}k=0\). Define \(A\) to be the infimum of the total \(g\)-areas of smooth closed surfaces enclosing \(\Sigma\) relative to the chosen end. If the ADM energy-momentum of the end is \((E,P)\) and \(m_{\mathrm{ADM}}=\sqrt{E^2-|P|^2}\), then \[ m_{\mathrm{ADM}}\ge\sqrt{\frac{A}{16\pi}}. \] Equality should occur only when the exterior data arise from a spacelike slice of the Schwarzschild spacetime.
MATH-PHYS-009:[Penrose1973NakedSingularities][Mars2009PenroseStatus][AllenBrydenKazarasKhuri2025SuboptimalPenrose][Xu2025ConditionalSpacetimePenrose]
Mathematical physicsGeometry and topologyPartial differential equations

Notes

The Penrose inequality is a conjectured quantitative strengthening of the positive mass theorem for initial data containing black holes. For a complete asymptotically flat initial data set (M3,g,k)(M^3,g,k) satisfying the dominant energy condition, with outermost apparent horizon Σ\Sigma, the ADM mass should satisfy mADM≥A/(16π)m_{\mathrm{ADM}}\ge\sqrt{A/(16\pi)}, where AA is the least area of a surface enclosing Σ\Sigma, with equality only for slices of the Schwarzschild spacetime. Penrose proposed the inequality in 1973 as a consistency test for cosmic censorship: data violating it would be a candidate counterexample to the standard picture of gravitational collapse [Penrose1973NakedSingularities].

The least-enclosing-area formulation is essential: versions using the raw area of an arbitrary trapped surface, or naively of the outermost horizon itself, admit counterexamples. The time-symmetric (Riemannian) case k=0k=0 is a theorem, and sharp results hold under substantial symmetry or other extra hypotheses; see [Mars2009PenroseStatus] for a survey. For fully general data, a spacetime Penrose inequality is now known with a universal but far-from-sharp constant [AllenBrydenKazarasKhuri2025SuboptimalPenrose], while recent sharp statements remain conditional, assuming jump, area-maximizing, or quasi-final-state hypotheses [Xu2025ConditionalSpacetimePenrose].

The sharp inequality for arbitrary asymptotically flat initial data remains open.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.