The Penrose inequality for general (non-time-symmetric) asymptotically flat initial data sets

For a complete, asymptotically flat initial data set $(M^3,g,k)$ for the Einstein equations satisfying the dominant energy condition, with ADM mass $m$ and with an outermost future or past apparent horizon whose outermost minimal area enclosure has area $A$, the Penrose inequality asserts $m \ge \sqrt{A/16\pi}$, with equality only for slices of the Schwarzschild spacetime. The time-symmetric case ($k=0$) was proved by Huisken–Ilmanen and Bray, but the inequality for general extrinsic curvature $k$, together with its equality characterization, remains one of the central open problems of mathematical general relativity. The stronger Bray–Khuri 'generalized Penrose conjecture', stated for generalized trapped surfaces, was refuted by Carrasco–Mars (2010), so the standard apparent-horizon formulation is the correct open target. Known partial results cover spherical symmetry, extrinsic curvature proportional to the metric, 2-convex data, cohomogeneity-one data, and bounds with non-sharp constants.

Background

In general relativity, a spacelike hypersurface of a spacetime is described intrinsically by an initial data set $(M^3,g,k)$: a Riemannian metric $g$ (the induced metric) and a symmetric 2-tensor $k$ (the extrinsic curvature, measuring how the hypersurface bends in spacetime). The Einstein constraint equations define the energy density $\mu$ and momentum density $J$ by $16\pi\mu = R + (\operatorname{tr}_g k)^2 - |k|_2^2$ and $8\pi J = \operatorname{div}_g(k - (\operatorname{tr}_g k)g)$, where $R$ is the scalar curvature of $g$. The dominant energy condition $\mu \ge |J|_g$ expresses that matter propagates no faster than light. On an asymptotically flat end — an end where $g$ approaches the Euclidean metric and $k$ decays suitably — the ADM (Arnowitt–Deser–Misner) mass $m$ is a surface integral at infinity, built from the first-order decay of $g$, and measures the total gravitational energy.

For a closed surface $\Sigma \subset M$, let $p$ be the mean curvature of $\Sigma$ in $(M,g)$ with respect to a chosen outer normal and $q = \operatorname{tr}_\Sigma k$ the trace of $k$ along $\Sigma$. The two null expansions — the rates of change of the area of $\Sigma$ under the two future-directed null normal directions — are $p - q$ and $p + q$. A surface is future outer trapped if $p < q$, and a marginally outer trapped surface (MOTS), also called a future apparent horizon, if $p = q$; a past apparent horizon satisfies $p = -q$. Penrose (1973) argued heuristically that if cosmic censorship holds, a spacetime containing trapped surfaces settles down to a Kerr black hole, and combining Hawking's area theorem with the loss of mass to gravitational radiation predicts, on the initial slice itself,

$m \ge \sqrt{\frac{A}{16\pi}},$

where $A$ is an area read off from the horizon on the slice. Since the event horizon is defined teleologically (by escape to infinity) it cannot be located from the initial data, so the inequality must be formulated with locally defined surfaces: one takes $\Sigma$ to be an (outermost) apparent horizon, and — because the event horizon may enclose $\Sigma$ with smaller area — $A$ to be the area of the outermost minimal area enclosure of $\Sigma$, the infimum of areas of surfaces enclosing it in the chosen end. The conjectural rigidity statement is that equality forces the data outside the enclosure to be a slice of the exterior Schwarzschild spacetime. A counterexample to the inequality would signal serious trouble for the cosmic censorship picture; a proof, by the same heuristic, would be the strongest mathematical support for it. The positive mass theorem ($m \ge 0$), proved for all such data by Schoen–Yau and Witten, is the massless-horizon limiting case.

The time-symmetric case $k=0$ (the Riemannian Penrose inequality) is a theorem: Huisken–Ilmanen proved it via inverse mean curvature flow — a flow that evolves a surface outward at speed equal to the reciprocal of its mean curvature — whose associated Geroch mass (a geometric quantity interpolating between horizon area and ADM mass) is monotone when $R \ge 0$ (single connected horizon), and Bray proved it in full generality via a conformal flow of metrics. For nonzero $k$ the monotonicity breaks and the problem is open in natural generality. The most developed attack is due to Bray–Khuri: a generalized Jang equation (a deformation of the equation Schoen–Yau used for the positive mass theorem), coupled with an inverse-mean-curvature-flow level-set equation and a warping factor, would reduce the conjecture to a Riemannian-type inequality on an auxiliary Jang graph; they proved the reduction, but the required existence and regularity theory for the coupled degenerate elliptic system is unknown, and one coupled version (Jang plus a zero-divergence equation) was later shown by Jaracz to admit no smooth solutions with the desired asymptotics even in a spherically symmetric example. Bray–Khuri also proposed a stronger 'generalized Penrose conjecture', replacing apparent horizons by generalized trapped surfaces (surfaces with $p \le |q|$, a time-orientation-free condition). That stronger version is false: Carrasco–Mars constructed axially symmetric slices of the Kruskal (extended Schwarzschild) spacetime — vacuum data, hence satisfying the dominant energy condition — whose outermost generalized apparent horizon has area strictly larger than $16\pi m^2$; since Eichmair proved the outermost generalized apparent horizon is strictly outer area minimizing, the minimal-area-enclosure wording does not evade the counterexample. The standard apparent-horizon formulation above is therefore the correct surviving target. Beyond the time-symmetric case it is known only in special settings: spherical symmetry (Hayward; Iriondo–Malec–Murchadha; a Jang-equation proof by Bray–Khuri), extrinsic curvature proportional to the metric (Dong, via the $\sigma$-inverse mean curvature flow, a modification of inverse mean curvature flow whose speed is corrected by the extrinsic curvature), 2-convex data with $\mathbf{P}=(\operatorname{tr}_g k)g - k \ge 0$ (equivalently, the sum of the two smallest eigenvalues of $k$ is nonnegative; Dong, and a charged version by Dolmen), cohomogeneity-one data (Khuri–Kunduri), and, for completely general data, only versions with a non-sharp constant: Khuri's bound of the ADM energy below by a constant times $\sqrt{A}$, and the Allen–Bryden–Kazaras–Khuri bound $m \ge c\sqrt{A/16\pi}$ for a universal but suboptimal constant $c$.

Problem Statement

Let $(M^3,g,k)$ be a complete, connected initial data set for the Einstein equations which is asymptotically flat in a chosen end, with ADM mass $m$, and satisfies the dominant energy condition $\mu \ge |J|_g$ everywhere. Suppose $M$ contains a future or past apparent horizon — a closed surface $\Sigma$ with mean curvature $p_\Sigma = \operatorname{tr}_\Sigma k$ or $p_\Sigma = -\operatorname{tr}_\Sigma k$ — and let $\tilde{\Sigma}$ be the outermost minimal area enclosure of $\Sigma$ in the chosen end, with area $A$. Prove that

$m \ge \sqrt{\frac{A}{16\pi}},$

and that equality holds if and only if the region exterior to $\tilde{\Sigma}$, with its induced data, is the pullback of Cauchy data on a spacelike slice of the exterior region of a Schwarzschild spacetime, with $\tilde{\Sigma}$ mapping to an apparent horizon. The statement must be established for all such data: no symmetry assumption, no topology restriction beyond compactness of the enclosure, and no algebraic condition on $k$ (such as $k=0$, proportionality to $g$, or 2-convexity) beyond the dominant energy condition. Equivalently, and with the same content, one may take $\Sigma$ to be the outermost future (or past) apparent horizon — the boundary of the trapped region — since the minimal area enclosure of any horizon is bounded by that of the outermost one.

References

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