Penrose inequality and positive mass theorem under averaged dominant energy conditions beyond spherical symmetry

Hafemann and Kontou (Class. Quantum Grav. 42 (2025) 195016) proved that in spherical symmetry a modified Penrose inequality — with an explicit correction term quantifying how much negative energy an integral energy bound permits — and a positive mass theorem follow from averaged (integral) dominant-energy-type conditions rather than pointwise ones, and extended the method to three special non-spherical classes of initial data (graphical, inverse-mean-curvature-flow-adapted, and SU(d+1)-invariant) using technique-dependent measures. The open problem is the removal of the symmetry restriction: prove a Penrose-type inequality and the associated positive mass theorem for general, non-spherically-symmetric asymptotically flat initial data sets under a genuinely averaged dominant-energy condition that permits pointwise violations and reduces to the spherical condition of the source work, or demonstrate that a stated class of averaged formulations cannot suffice. No progress beyond the special classes settled in the source is known in the literature, so the problem remains open.

Background

The Penrose inequality is a conjectured lower bound on the total mass of a black-hole spacetime, proposed by Penrose (1973) as a consistency test of the weak cosmic censorship conjecture — the expectation that singularities of gravitational collapse remain hidden inside event horizons. In its initial-data form it concerns a spacelike slice of spacetime, described by an initial data set $(M^d, g, K)$: a $d$-dimensional manifold with a Riemannian metric $g$ and extrinsic curvature $K$ (the second fundamental form measuring how the slice sits in spacetime), asymptotically flat, i.e. approaching Euclidean space at large distance. The ADM mass $m_{\mathrm{ADM}}$ is the total gravitational mass defined at spatial infinity of such a slice. Writing $\omega_{d-1}$ for the area of the unit $(d-1)$-sphere, the inequality asserts

$$m_{\mathrm{ADM}} \ge \frac{1}{2} \left( \frac{|\partial M|}{\omega_{d-1}} \right)^{\frac{d-2}{d-1}},$$

where $|\partial M|$ is the area of the slice's outermost black-hole boundary, taken to be an outermost marginally outer trapped surface (MOTS) — a closed surface whose outgoing null expansion vanishes, which is the quasi-local signature of an apparent horizon. In the time-symmetric case ($K = 0$, so the boundary is a minimal surface) this was proved for $d = 3$ by Huisken and Ilmanen using weak inverse mean curvature flow — an outward deformation of surfaces whose speed is the reciprocal of their mean curvature, whose Geroch monotonicity converts nonnegative scalar curvature into a mass–area bound — and by Bray using a conformal flow argument that rests on the positive mass theorem, the foundational result that nonnegative scalar curvature (more generally the dominant energy condition) forces $m_{\mathrm{ADM}} \ge 0$. The general spacetime (non-time-symmetric) version remains open; the strongest general result to date, by Allen, Bryden, Kazaras and Khuri (2025), yields it only with a suboptimal constant under the pointwise dominant energy condition. Every one of these proofs consumes a pointwise energy condition. On an initial data set, the matter content enters through the energy density $\mu$ and momentum density $J$ induced by the stress-energy tensor in the Einstein constraint equations, and the pointwise dominant energy condition (DEC) is the requirement $\mu \ge |J|_g$ everywhere. Quantum fields violate every pointwise energy condition — local renormalized energy densities can be negative — so the classical proofs do not reach semiclassical gravity. The quantum-compatible replacements are quantum energy inequalities (QEIs): integral lower bounds on the energy density averaged against test functions along null or timelike curves, such as the smeared null energy condition (SNEC) proposed by Freivogel and Krommydas. Crucially for what follows, Ford, Helfer and Roman showed that no such spatially averaged lower bound exists for generic quantum-field states in four-dimensional spacetime: averaging over a spacelike region does not tame negative quantum energy. Hafemann and Kontou (2025) initiated the study of the Penrose inequality under precisely such integral hypotheses. They define the averaged dominant energy inequality (d$\nu$-ADEI): the integral of $\mu - |J|_g$ over the initial-data manifold against a chosen measure $d\nu$ is bounded below by a constant $\kappa$ (the constant is written $K$ in the source work; it is renamed here because $K$ already denotes the extrinsic curvature). In spherical symmetry, writing the metric as $ds^2 + r(s)^2 d\Omega^2$ with areal radius $r$ (the coordinate in which a symmetry sphere has area $\omega_{d-1} r^{d-1}$), the condition becomes the spherical averaged dominant energy inequality (S-ADEI, called the S-DEC when $\kappa = 0$):

$$\int_{r_0}^{\infty} r^{d-1} \left( \mu - |J|_g \right) \, dr \ge \kappa,$$

with $r_0$ the radius of the outermost MITS (marginally inner trapped surface — a closed surface whose ingoing null expansion vanishes), or of $\partial M$ when no MITS is present, as in the source's theorem. Their main theorem (for $d < 8$) proves the modified Penrose inequality

$$m_{\mathrm{ADM}} - \frac{\kappa}{d-1} \ge \frac{1}{2} \left( \frac{A_{\mathrm{min}}(\partial M)}{\omega_{d-1}} \right)^{\frac{d-2}{d-1}},$$

where $A_{\mathrm{min}}(\partial M)$ is the infimum of the areas of surfaces enclosing $\partial M$; if no outermost MITS lies outside $\partial M$, the bound strengthens to one on the boundary area $|\partial M|$ itself. The inequality reduces to the classical Penrose inequality when $\kappa = 0$ and yields the positive mass theorem as a corollary; the correction term $\kappa/(d-1)$ quantifies exactly how much negative energy the averaged bound permits. They also prove a modified global Penrose inequality for evaporating black holes from a QEI-inspired null condition combined with the S-DEC, and they verify the S-DEC on quantum-corrected spherical black-hole models that violate the pointwise DEC. Beyond spherical symmetry they establish the d$\nu$-ADEI version only for three special classes, each with a measure adapted to the proof technique: graphical manifolds (following Lam), time-symmetric data adapted to inverse mean curvature flow (following Bray and Khuri), and SU(d+1)-invariant data (following Khuri and Kunduri). Their discussion explicitly poses the general case — and a positive mass theorem under an averaged rather than pointwise DEC — as open, noting that the cohomogeneity-one time-symmetric argument fails to generalize precisely because it relies on the positive mass theorem with pointwise nonnegative scalar curvature. The open problem is therefore: extend these integral-energy-condition results to initial data sets with no symmetry restriction whatsoever.

Problem Statement

Remove the spherical-symmetry restriction from the Penrose inequality under integral energy conditions. For general, complete, asymptotically flat initial data sets $(M^d, g, K)$, $d < 8$ (the physically central case being $d = 3$, i.e. four-dimensional spacetime), whose boundary $\partial M$ is an outermost marginally outer trapped surface, prove a Penrose-type inequality

$$m_{\mathrm{ADM}} - (\text{explicit correction term controlled by } \kappa) \ge \frac{1}{2} \left( \frac{A_{\mathrm{min}}(\partial M)}{\omega_{d-1}} \right)^{\frac{d-2}{d-1}}$$

under an averaged dominant-energy-type condition — a lower bound $\int (\mu - |J|_g) \, d\nu \ge \kappa$ on the integral of the energy density minus the norm of the momentum density over the exterior of the boundary — in place of the pointwise dominant energy condition. The correction term is part of the answer: it must be given explicitly as a function of $\kappa$ that reduces to the source's $\kappa/(d-1)$ in spherical symmetry and vanishes when $\kappa = 0$. Here, as in the source's spherical theorem, $A_{\mathrm{min}}(\partial M)$ denotes the infimum of the areas of surfaces enclosing $\partial M$; it becomes the boundary area $|\partial M|$ itself when no outermost MITS (marginally inner trapped surface) lies outside $\partial M$. In spherical symmetry the source work proves this with correction $\kappa/(d-1)$ (recovering the classical Penrose inequality $m_{\mathrm{ADM}} \ge \frac{1}{2} (|\partial M|/\omega_{d-1})^{(d-2)/(d-1)}$ when $\kappa = 0$ and no outermost MITS lies outside $\partial M$), and for three special non-spherical classes with technique-adapted measures; the required theorem has no symmetry or special-structure assumption at all. The formulation of the hypothesis is part of the problem: the condition and its measure must (i) reduce to the spherical S-ADEI/S-DEC of Hafemann and Kontou in spherical symmetry, (ii) allow pointwise violations of the dominant energy condition (otherwise it is not a genuine integral generalization), and (iii) be satisfied by nontrivial examples, such as the quantum-corrected spherical models studied in the source work. In particular, the corresponding positive mass theorem in the same generality reads: for boundary-free asymptotically flat data satisfying the averaged condition (with $\kappa = 0$, or with the corresponding correction), $m_{\mathrm{ADM}} \ge 0$; this is the ingredient the source authors explicitly identify as the obstruction blocking all current approaches, since existing positive mass theorem techniques consume pointwise scalar-curvature or energy conditions. Alternatively, a complete answer may refute the program's general (non-symmetric) claim: state a class of averaged spacelike bounds that contains the source's d$\nu$-ADEI and exhibit an explicit asymptotically flat initial data set that satisfies one non-symmetric member of that class — a member that reduces to the S-ADEI/S-DEC in spherical symmetry — and for which the corresponding corrected Penrose-type inequality fails; refuting any one such non-symmetric formulation resolves the task. (The spherically symmetric S-ADEI/S-DEC is excluded as a refutation target, since the source already proves both conclusions from it.)

References

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