Unconditional positivity of the Bondi mass in four-dimensional Bondi-Sachs radiating spacetimes (the positive mass conjecture at null infinity)

An isolated radiating system in general relativity should not be able to lose more mass to gravitational waves than it has: the Bondi mass measured at any cut of future null infinity should be nonnegative. Schoen and Yau (Phys. Rev. Lett. 48, 369 (1982)) announced a proof by extending their Jang-equation positive mass theorem to null infinity, and contemporaries (Ludvigsen-Vickers, Israel-Nester, Horowitz-Perry, Horowitz-Tod) announced spinor-based proofs; all are outline-level or conditional. Huang, Yau and Zhang (2006) later gave the only complete proofs in Bondi's radiating vacuum metric class, and only under extra conditions: constant mass aspect at some retarded time (Jang route) or vanishing asymptotic shear (Witten route); their Proposition 4.1 shows that a Jang solution with the required null asymptotics forces the mass aspect to be constant in the angular variables, which is the precise gap the Schoen-Yau sketch left. The open problem is an unconditional proof, or an explicit counterexample, of nonnegativity of the Bondi energy-momentum in the classical four-dimensional vacuum Bondi-Sachs class with arbitrary, angularly dependent news and mass aspect and with black holes allowed. This core remains open: later literature resolved the smooth-conformal-compactification case in even dimensions (Hollands-Thorne 2014) and the initial-data (hyperboloidal) versions (Zhang, Chrusciel-Jezierski-Leski, Sakovich, Lundberg, Hirsch-Khuri-Lesourd-Zhang), but no unconditional result in the spacetime-level Bondi-Sachs class is known to date.

Background

General relativity models isolated gravitating systems by asymptotically flat spacetimes, and two complementary notions of total mass live at the two ends of the asymptotic regime. At spatial infinity, the ADM mass (the Arnowitt-Deser-Misner total energy, read off from the $1/r$ coefficient of the spatial metric on an asymptotically flat spacelike hypersurface) is governed by the positive mass theorem proved by Schoen and Yau and by Witten: for initial data satisfying the dominant energy condition (the standard physical condition that matter energy-momentum flux vectors are causal), the ADM energy-momentum is future-directed non-spacelike, and the mass is zero only for flat Minkowski data. At future null infinity - the asymptotic region approached along outgoing light rays, where an isolated system's gravitational radiation is measured - the corresponding statement is the subject of this problem.

Bondi, van der Burg and Metzner (1962) analyzed vacuum spacetimes in retarded coordinates $(u,r,\theta,\psi)$ adapted to a family of outgoing null hypersurfaces, where $u$ is the retarded time and $r$ an areal radius on each null cone. The asymptotic field equations force the metric to carry a term $-2M(u,\theta,\psi)/r$, where $M$ is the mass aspect, and the Bondi energy-momentum of the cut $u=u_0$ of null infinity is $m_\nu(u_0)=\frac{1}{4\pi}\int_{S^2}M(u_0,\theta,\psi)\,n_\nu\,dS$ with $n_0=1$ and $(n_1,n_2,n_3)$ the coordinate functions of the unit sphere; its time component $m_0(u)$ is the Bondi mass. The outgoing gravitational radiation is encoded in the news functions $c_{,0},d_{,0}$, the $u$-derivatives of the shear functions that describe the aspherical distortion of the outgoing null cones, and the Bondi mass-loss law $\frac{dm_\nu}{du}=-\frac{1}{4\pi}\int_{S^2}\big[(c_{,0})^2+(d_{,0})^2\big]\,n_\nu\,dS$ shows the mass decreases exactly by the radiated energy. The law gives only monotonicity from an initial value, so the physical content - an isolated system cannot radiate away more energy than it has, the positive mass conjecture at null infinity - is precisely the missing positivity of $m_\nu$ at every cut.

The spatial-infinity theorem does not transfer directly. Schoen and Yau's method reduces general initial data to the time-symmetric case by the Jang equation, a nonlinear elliptic equation whose solution $f$ is used as a graph function converting the data into Riemannian data to which the time-symmetric positive mass theorem applies (the general case then following by conformal deformation of scalar curvature). In the 1982 announcement, Schoen and Yau outlined the extension to null infinity: on a spacelike hypersurface asymptotic to the null cone, the asymptotic expansion of a Jang-type graph $f=\sqrt{1+r^2}+p(\theta,\psi)\ln r+q$ makes the ADM mass of the graph equal to $p$, a positive multiple of the Bondi mass coefficient, and the deformed data is run through the spatial-infinity machinery (their 1979 and 1981 Communications in Mathematical Physics papers, the latter of which also supplies the linear-momentum and angular-dependence input). The announcement, and the parallel spinor-based announcements of Ludvigsen-Vickers, Israel-Nester, Horowitz-Perry and Horowitz-Tod, all left the analytic details at null infinity incomplete: the carried-out cases assume the mass aspect constant in the angular variables, or assume an auxiliary nonsingular spacelike hypersurface reaching the given cut on which the dominant energy condition holds (the conditional form in which Szabados's Living Review records the theorem), or assume a smooth Penrose conformal compactification of null infinity (the Penrose picture in which null infinity is the boundary of a conformally rescaled smooth compactified spacetime).

The precise state of the art was fixed by Huang, Yau and Zhang (2006), who worked directly in Bondi's radiating vacuum metric class and proved complete but conditional theorems: Theorem 4.1 (the Schoen-Yau/Jang route) assumes Condition A (axis regularity of the metric functions), Condition B (a vanishing-integral condition on the news along the axis), and that the mass aspect $M(u_0,\theta,\psi)$ is constant at some retarded time $u_0$, concluding $m_0(u)\ge\sqrt{\sum_{1\le i\le3}m_i^2(u)}$ for all $u\le u_0$ with flatness rigidity; Theorem 5.1 (the Witten spinor route) assumes instead, together with Conditions A and B, that the asymptotic shear functions satisfy $c|_{u=u_0}=d|_{u=u_0}=0$; Theorem 6.1 drops only Condition B, for a modified energy functional, and still requires either a constant mass aspect or vanishing asymptotic shear at some retarded time. Their Proposition 4.1 is the central obstruction: a lengthy computation gives $J(f)\approx\frac{\ln r}{r^3}\Delta_{S^2}p+\frac{p-2M(0,\theta,\psi)}{r^3}$, so a Jang solution with the required asymptotics forces $\Delta_{S^2}p=0$ and $p=2M$, i.e. forces $p$ and the mass aspect to be constant on the sphere - identifying angular dependence of the mass aspect as the exact analytic gap behind the Schoen-Yau deferral. They also record as open whether the vacuum Einstein equations always develop logarithmic terms at null infinity (polyhomogeneity, i.e. expansions allowed logarithmic terms, versus smooth conformal compactification).

Two neighboring bodies of work must be distinguished from the source question. First, initial-data-level results: for asymptotically hyperboloidal (asymptotically null) initial data sets - spacelike Cauchy data whose asymptotic end approaches null rather than spatial infinity - the Trautman-Bondi mass (the null-infinity analogue of the ADM mass for such data) is proven nonnegative (Zhang 2004; Chrusciel-Jezierski-Leski 2004), and the Jang reduction for such data has been carried out by Sakovich (2021) in three dimensions and Lundberg (2023) in dimensions four through seven, with Hirsch-Khuri-Lesourd-Zhang (2026) proving the spacetime positive mass theorem for such data in all dimensions; these authors explicitly describe the Schoen-Yau Bondi application as a sketch and their own results as the hyperboloidal initial-data version. These are statements about Cauchy data, not about the spacetime-level Bondi-Sachs class with general news; the reduction from spacetime to data is exactly where the unresolved analytic issues live. Second, Hollands and Thorne (2014) prove Bondi-mass nonnegativity in all even spacetime dimensions $d\ge4$ for vacuum spacetimes with a smooth Penrose conformal null infinity and cross-sections admitting a real Killing spinor (including the round sphere), which resolves the smooth-conformal formulation but not the classical Bondi-Sachs class with the asymptotic behavior the vacuum equations actually force. Adjacent quasilocal routes - Penrose's 1982 observation that Bondi positivity would follow from the still-open Penrose inequality bounding total mass by horizon area, and Mondal-Yau's 2024 spinor positivity for quasilocal masses (masses assigned to closed two-surfaces, of the type introduced by Wang and Yau, whose null-infinity limits recover Bondi-type masses) - do not currently close the gap.

Problem Statement

Prove, or refute by an explicit example, the positive mass conjecture at null infinity in its classical four-dimensional form. Let $(M,g)$ be a smooth vacuum spacetime in Bondi's radiating class (the metric class of Bondi, van der Burg and Metzner, as formalized in Section 2 of Huang-Yau-Zhang) - asymptotically flat at future null infinity, written in retarded Bondi coordinates $(u,r,\theta,\psi)$ with the standard Bondi falloff of the metric coefficients (smooth in the angular variables, with at most the logarithmic terms the vacuum Einstein equations force), with arbitrary smooth news functions $c_{,0},d_{,0}$ and arbitrary angularly dependent mass aspect $M(u,\theta,\psi)$, and allowing black-hole-type singularities bounded by apparent horizons (marginally outer trapped surfaces, i.e. closed spacelike two-surfaces on which the outgoing null expansion vanishes). Prove that the Bondi energy-momentum $m_\nu(u)=\frac{1}{4\pi}\int_{S^2}M(u,\theta,\psi)\,n_\nu\,dS$ of every cut $u=\text{const}$ is future-directed non-spacelike; in particular $m_0(u)\ge0$ for all retarded times $u$, with the rigidity that $m_0(u)=0$ only when the exterior region is flat (no news and no energy flux), as in the conditional form of Szabados's review. A proof must be unconditional in this class: it may not assume the mass aspect is constant in the angular variables at any retarded time (the hypothesis of Huang-Yau-Zhang Theorem 4.1), that the asymptotic shear functions vanish at any retarded time (the hypothesis of Theorem 5.1), a smooth Penrose conformal compactification of null infinity beyond what the vacuum equations themselves enforce, or the existence of a special auxiliary spacelike hypersurface through the cut beyond what the spacetime provides. A complete treatment must discharge whichever of the recognized sub-obstructions its own route encounters; in the Schoen-Yau/Jang route these are the four the 1982 announcement left: (i) the angular dependence of the mass aspect in the null asymptotics of the Jang equation, which Huang-Yau-Zhang Proposition 4.1 shows forces constancy under the sketched construction and which therefore requires a modified construction; (ii) embeddability of the deformed (Jang-graph, conformally deformed) initial data set in the original spacetime; (iii) independence of the choice of spacelike hypersurface asymptotic to the null cone, including behavior under the supertranslation freedom of the Bondi-Metzner-Sachs group of asymptotic symmetries that moves the cuts; and (iv) regularity of the Jang-equation analysis at and across apparent horizons. A refutation must instead exhibit one explicit spacetime of this class, satisfying the vacuum equations and the Bondi falloff, whose Bondi mass $m_0$ is strictly negative at some retarded time; since the claim is universal over the class, a single such instance settles it.

References

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