Foundational open problems for the Wang–Yau quasi-local mass: positivity without admissibility, solvability of the optimal embedding equation, monotonicity, and horizons

The Wang–Yau quasi-local mass (Wang and Yau, Phys. Rev. Lett. 102, 021101 (2009)) assigns to a closed spacelike 2-surface $\Sigma$ in spacetime the infimum of a Hamiltonian surface energy over isometric embeddings of $\Sigma$ into Minkowski space, taken in a canonical gauge that matches the expansions of the physical and reference surfaces. It is nonnegative under the dominant energy condition for 'admissible' reference data, vanishes on surfaces in flat spacetime, and has the correct ADM, Bondi, and small-sphere limits, making it the leading rigorous candidate for the long-sought quasi-local mass of general relativity. The originating paper explicitly left open the removal of its admissibility hypotheses from the positivity theorem (the authors expected a general result under a convexity condition involving the Hamiltonian surface-density 4-vector and the time function $\tau$), the characterization of surfaces with nonempty admissible sets, the attainment, uniqueness, and regularity of the minimizing embedding, monotonicity properties, and the fully general treatment of apparent horizons. Later work removed the Jang-solvability admissibility condition, extended positivity to strictly stable apparent horizons, solved the optimal embedding equation perturbatively (near infinity, near $\tau=0$, and near embeddable horizons), and classified near-horizon limits — but the core questions remain open. The problem is to complete the analytic foundations of this definition: positivity under minimal hypotheses, general solvability and attainment of the defining variational problem, monotonicity, and the horizon/trapped-surface cases.

Background

In general relativity the equivalence principle forbids a local energy density for the gravitational field, and total energy-momentum is defined only relative to an asymptotic regime: the ADM (Arnowitt–Deser–Misner) energy-momentum at spatial infinity and the Bondi energy-momentum at null infinity, both proven nonnegative for physically reasonable spacetimes by Schoen–Yau and Witten. A quasi-local mass instead aims to assign an energy to a finite region bounded by a closed spacelike 2-surface $\Sigma$ in spacetime, depending only on the data of $\Sigma$ (induced metric, second fundamental form, and normal-bundle connection). In the Hamilton–Jacobi approach of Brown and York one subtracts a reference configuration from the gravitational Hamiltonian; using an isometric embedding of $\Sigma$ into flat Euclidean space $\mathbb{R}^3$ as reference gives the Brown–York energy, which is gauge-dependent and not nonnegative in general. Liu and Yau defined a gauge-independent mass and proved it positive in spacetimes satisfying the dominant energy condition — the requirement that the matter energy-momentum tensor $T$ satisfy $T(e_0,e_0)\ge 0$ with $T(e_0,\cdot)$ non-spacelike for every future timelike vector $e_0$ — but Ó Murchadha, Szabados, and Tod exhibited surfaces in flat Minkowski spacetime whose Liu–Yau mass is strictly positive: momentum information is missing. Wang and Yau (2009) proposed to repair this by using general isometric embeddings of $\Sigma$ into Minkowski space $\mathbb{R}^{3,1}$ as references. Such spacelike embeddings are parametrized by their time function $\tau$ (the restriction of the Minkowski time coordinate); positivity of the Gauss curvature of the projected metric $\sigma+d\tau\otimes d\tau$ guarantees existence and uniqueness of the embedding (their Theorem 1). The quasilocal energy $E(\Sigma,\tau)$ is the difference of the physical and reference Hamiltonian surface terms evaluated in the canonical gauge — the unique future-timelike observer field $\bar u$ along $\Sigma$ whose expansion matches that of the reference surface, i.e. $\langle h,\bar u\rangle=\langle h_0,u_0\rangle$ for the mean curvature vectors $h$ of $\Sigma$ and $h_0$ of the reference image. Positivity (Theorem 2) is proved only for 'admissible pairs' $(i,t_0)$ of a reference embedding $i$ and a constant timelike observer $t_0$: (A) $i$ has convex shadow, meaning the projection of $i(\Sigma)$ onto the $\mathbb{R}^3$ orthogonal to $t_0$ is convex; (B) $\Sigma$ bounds a spacelike region $\Omega$ on which Jang's equation — the quasilinear elliptic equation introduced by Jang and used by Schoen and Yau in the positive mass theorem, whose solution's graph over $\Omega$ is the comparison surface matching the time data $\tau$ at the boundary — is solvable with Dirichlet boundary data $\tau$, with possible apparent horizons (marginally outer trapped surfaces, where one null expansion vanishes) in the interior; and (C) a boundary inequality $k'N_0-N_0^\nu v'^\mu K_{\mu\nu}>0$ holds for the gauge boosted off the Jang solution. The quasi-local mass is the infimum of $E$ over admissible pairs; it is nonnegative when the admissible set is nonempty (Theorem 3, proven when $\Sigma$ has positive Gauss curvature), vanishes on surfaces in Minkowski space with convex shadow, and its small-sphere limits recover the matter energy-momentum tensor (in vacuum, the Bel–Robinson tensor, the basic quadratic curvature functional that plays the role of gravitational energy density there) while large-sphere limits recover the ADM and Bondi masses. The Euler–Lagrange equation of the variational problem, derived in the companion paper, is the optimal embedding equation $\nabla\cdot j=0$ with current $j=\rho\nabla\tau-\alpha_{v_0}+\alpha_v$ ($\rho$ the mass surface density, $\alpha$ the normal-bundle connection forms) — a fourth-order fully nonlinear elliptic equation coupled to the isometric embedding of $\Sigma$ into $\mathbb{R}^3$. Its solvability is known only in special regimes. What remains open is exactly what the authors flagged: positivity under a minimal convexity condition instead of the admissibility hypotheses, the characterization of surfaces admitting admissible data, attainment/uniqueness/regularity of the minimizing embedding, monotonicity, and the general treatment of horizons and trapped surfaces.

Problem Statement

For a closed spacelike 2-surface $\Sigma$ with spacelike mean curvature vector in a time-orientable spacetime satisfying the dominant energy condition (this hypothesis governs items 1–3 below; item 4 additionally concerns closed spacelike 2-surfaces whose mean curvature vector is timelike or null, in a time-orientable spacetime satisfying the dominant energy condition), complete the analytic foundations of the Wang–Yau quasi-local mass, along the items explicitly left open by Wang and Yau (Phys. Rev. Lett. 102, 021101 (2009)) and still open:

1. Positivity under minimal hypotheses. Prove that the quasi-local energy $E(\Sigma,\tau)$ and the quasi-local mass are nonnegative under a minimal, natural convexity condition — the heuristic condition involving the Hamiltonian surface-density 4-vector $ku_\nu+v^\mu(K_{\mu\nu}-K\delta_{\mu\nu})$ and the time function $\tau$ anticipated by the authors (expression (2) of the letter, where $u$ and $v$ are the future-timelike and outward-spacelike unit normals of $\Sigma$, with $v$ tangent to a locally defined spacelike hypersurface $\Omega$ with $\partial\Omega=\Sigma$, $k$ is the mean curvature of $\Sigma$ in $\Omega$, and $K_{\mu\nu}$ and $K$ are the extrinsic curvature of $\Omega$ in spacetime and its trace), or another explicit hypothesis that is strictly weaker than the admissibility conditions below and under which the nonnegativity holds — without the admissibility conditions: convex shadow (equivalently, for a closed surface, positive Gauss curvature of the projected metric $\sigma+d\tau\otimes d\tau$), solvability of Jang's equation on a spacelike region bounded by $\Sigma$ with Dirichlet boundary data $\tau$, and the boundary inequality for the gauge determined by the Jang solution. In particular, settle whether the mass is nonnegative for general surfaces regardless of the sign of the intrinsic curvature of $\Sigma$, and determine geometric conditions on an arbitrary $\Sigma$ that are necessary and sufficient for the admissible set to be nonempty.

2. Well-posedness of the variational definition. Prove or disprove that the infimum defining the mass is achieved: establish existence, uniqueness, and regularity of minimizing isometric embeddings — embeddings at which the infimum is attained, as distinct from critical points of the energy — and, separately, the related question of general solvability of the optimal embedding equation $\nabla\cdot j=0$ with current $j=\rho\nabla\tau-\alpha_{v_0}+\alpha_v$ (where $\rho$ is the mass surface density of $\Sigma$ and $\alpha_v$, $\alpha_{v_0}$ are the normal-bundle connection forms of the physical and the reference surface), coupled to the isometric embedding of $\Sigma$ into $\mathbb{R}^3$ — including for surfaces near configurations that cannot be isometrically embedded into $\mathbb{R}^3$.

3. Monotonicity. Establish, or refute by explicit examples, the monotonicity of the mass under inclusion of surfaces (the mass of a surface bounding the larger region being at least that of a nested inner surface) or along geometric flows.

4. Horizons and trapped surfaces. Extend the positivity proof and the variational framework to interiors of $\Sigma$ containing marginally stable apparent horizons, and give a theorem-level extension of the mass to surfaces with timelike or null mean curvature vector, such as trapped surfaces.

Each of the four items is an independent, self-contained open problem, and the four items together constitute the full objective of establishing the Wang–Yau mass as a fully rigorous, well-posed quasi-local mass for general surfaces. Known partial resolutions — removal of the Jang-solvability condition, positivity with strictly stable horizons, perturbative solvability of the optimal embedding equation near infinity, near $\tau=0$, and near embeddable horizons, and near-horizon limit classifications — refine these questions but leave them open.

References

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