The causal fate of apparent horizons: the surviving open core of the Gibbons–Hawking–Horowitz–Perry positive mass theorems for black holes

G. W. Gibbons, S. W. Hawking, G. T. Horowitz, and M. J. Perry, 'Positive mass theorems for black holes' (Communications in Mathematical Physics 88 (1983) 295–308), extended Witten's spinor proof of the positive mass theorem to asymptotically flat initial data that are regular outside an apparent horizon and left open the causal relation between the apparent horizon and the event horizon. That question, attached in the source to its time-symmetric case, is: for smooth asymptotically flat initial data satisfying the dominant energy condition on a maximal slice ( $K = \mathrm{tr}\, k = 0$ ) whose compact inner boundary $H$ is a minimal 2-surface, prove, or refute by explicit construction, that in the maximal Cauchy development $H$ lies behind the future or the past event horizon — that is, $H$ is not visible from future or past null infinity. Of this assertion the authors wrote that 'no proof of this is known to us'; it remains open.

Background

In general relativity, an isolated gravitational system is described at an instant of time by an initial data set $(M,g,k)$ : a Riemannian 3-manifold $M$ with metric $g$ (the induced spatial metric on a spacelike slice) and symmetric 2-tensor $k$ (its extrinsic curvature, describing how the slice bends in spacetime). The matter and field content is encoded in the energy density $\mu = \tfrac{1}{2}(R_g + (\mathrm{tr}\, k)^2 - |k|_g^2)$ and the momentum density $J = \mathrm{div}\, k - d(\mathrm{tr}\, k)$ , where $R_g$ is the scalar curvature. The dominant energy condition (DEC), $\mu \ge |J|_g$ , expresses that local energy-momentum is future-causal. If $M$ is asymptotically flat, meaning it agrees with Euclidean space outside a compact set up to suitable decay, each end carries the ADM energy-momentum vector $(E,P)$ (introduced by Arnowitt, Deser, and Misner), defined as limits of surface integrals at infinity. The positive mass theorem asserts $E \ge |P|$ , with equality only for flat data; it was first proved by Schoen and Yau using minimal hypersurfaces and then by Witten using spinors. In Witten's proof one assumes $M$ is spin (admits spinor fields, the square-root of vectors carrying the Dirac operator), solves the Witten equation $\gamma^{a(3)}\nabla_a \varepsilon = -\tfrac{1}{2} K \gamma^{\hat 0}\varepsilon$ for a spinor $\varepsilon$ (here $K = \mathrm{tr}\, k$ , $\gamma$ -matrices contract frame vectors, and $\hat 0$ is the timelike normal direction) with $\varepsilon$ approaching a constant spinor $\varepsilon_0$ at infinity, and integrates a Weitzenböck-type identity (an integration-by-parts formula for the Dirac operator): the interior integral has the sign of $\mu - |J|$ under the DEC, and the boundary term at infinity equals $4\pi\,\varepsilon_0^\dagger (P^{\hat 0} + P^{\hat a}\gamma^{\hat a}\gamma^{\hat 0})\varepsilon_0$ , which is nonnegative for all $\varepsilon_0$ exactly when $E \ge |P|$ .

Gibbons, Hawking, Horowitz, and Perry (1983) extended this argument to initial data on an asymptotically flat slice $\Sigma$ that is regular only outside an apparent horizon $H$ . An apparent horizon, in modern terminology a marginally outer trapped surface (MOTS), is a closed 2-surface in $M$ whose outward null expansion $\theta^+ = H_{\mathrm{mean}} + \mathrm{tr}_H k$ vanishes (with $H_{\mathrm{mean}}$ the mean curvature of the surface); $\theta^+ < 0$ means outer-trapped; a surface whose two null expansions are both negative is future-trapped, the classical signal of a strong-gravity region. Since such data say nothing about the black-hole interior, Witten's boundary term at infinity is supplemented by a boundary term over $H$ , and Gibbons–Hawking–Horowitz–Perry impose the horizon boundary condition $\gamma^{\hat 1}\gamma^{\hat 0}\varepsilon = \varepsilon$ on $H$ (with $\hat 1$ the unit normal direction of $H$ in $\Sigma$ ; in the source's tetrad convention the one-vector orthogonal to $H$ points into $\Sigma$ , i.e. away from the excised interior), under which the horizon term has a favorable sign. They thereby argued positivity of the ADM mass, the bound $M \ge |Q|$ for the total electric plus magnetic charge (using a supercovariant connection that adds the electromagnetic field strength to the spin connection), an analogous statement for the Bondi mass at null infinity (the mass defined on cuts of future null infinity, which decreases with retarded time by the Bondi mass-loss formula as radiation escapes), positivity in the time-symmetric setting of a maximal initial surface $\Sigma$ ( $K = 0$ ) regular outside a minimal 2-surface $H$ (a case first proved by Schoen and Yau), and, for a negative cosmological constant $\Lambda = -3\mu^2$ (anti-de Sitter asymptotics, where spatial infinity is asymptotically hyperbolic and the relevant spinor at infinity is asymptotically supercovariantly constant), positivity of a suitably defined mass for data regular outside an apparent horizon.

The authors were explicit that the analytic core was incomplete. Their existence argument for the spinor proceeded by conformally compactifying $\Sigma$ (rescaling the metric by a factor $\Omega^4$ with $\Omega = 1/r + O(1/r^2)$ so that infinity becomes a point), replacing the Witten operator by a compactified operator, and appealing to skew-adjointness, absence of zero modes, and a Green's function whose existence they said 'probably can be rigourously proved by the methods of Parker and Taubes' (who had made Witten's original argument rigorous) without supplying the proof; the compactification is smooth only when the ADM spatial momentum $P^a = 0$ , so the direct argument excluded boosted slices; for the anti-de Sitter case the existence of asymptotically supercovariantly constant spinor solutions with the horizon condition was asserted partly on the basis of private communications; and the Bondi-mass statement was claimed by analogy with a reference rather than written out. Most distinctively, in the time-symmetric case of a maximal slice with minimal inner boundary they noted that 'the fact that one can prove that the mass is positive in this situation regardless of what lies inside $H$ suggests that $H$ must lie behind the future or past event horizon although no proof of this is known to us.'

The causal question requires global notions. The maximal Cauchy development of initial data is the largest spacetime determined by those data through the Einstein equations. Future null infinity $\mathcal{I}^+$ is the ideal null hypersurface where outgoing light rays terminate. Writing $\mathcal{M}$ for the maximal Cauchy development, the black hole region is $B = \mathcal{M} \setminus J^-(\mathcal{I}^+)$ , the complement of the causal past $J^-(\mathcal{I}^+)$ of future null infinity, and the future event horizon is the boundary of $B$ : equivalently, it is the boundary of the set of points from which one can still escape to $\mathcal{I}^+$ in the future direction. The past event horizon and the past black hole region $\mathcal{M} \setminus J^+(\mathcal{I}^-)$ are defined analogously from past null infinity $\mathcal{I}^-$ , the ideal null hypersurface where ingoing light rays originate. Unlike an apparent horizon, which is computable from a single slice, the event horizon is teleological: it depends on the entire future (or the entire past). Penrose's weak cosmic censorship conjecture asserts that, generically, asymptotically flat initial data possess a complete future null infinity, so that singularities are hidden inside black hole regions; spacetimes with an incomplete $\mathcal{I}^+$ exhibit naked singularities. The containment statement that Gibbons–Hawking–Horowitz–Perry could not prove — that a minimal boundary of a maximal slice ends up behind the future or the past event horizon of the development — is exactly a weak-censorship-type assertion for data that already contain a marginal trapped boundary, and it is the premise that makes every positive-mass or Penrose-inequality statement for 'data regular outside an apparent horizon' a statement about black holes rather than merely about slices.

Problem Statement

The source paper bundles five self-identified gaps in Gibbons–Hawking–Horowitz–Perry's positive mass theorems for black holes. Four of them — the rigorous existence theory for the Witten spinor with the horizon boundary condition, the removal of the vanishing-ADM-momentum restriction, the self-contained existence theory for asymptotically anti-de Sitter data with an apparent horizon, and the detailed Bondi-mass adaptation — have been resolved or superseded by the later literature (see, e.g., References 5–9 and 12–19), up to adjacent residual conditions. The remaining open core, stated precisely, is the causal question, which the source attaches to its time-symmetric case (Sect. 3, p. 300):

Let $(M,g,k)$ be a smooth initial data set for the Einstein equations (vacuum, or coupled to matter satisfying the dominant energy condition) that is asymptotically flat, with compact inner boundary $H$ — the setting of an asymptotically flat slice regular outside an apparent horizon — and suppose that the slice $M$ is maximal, $K = \mathrm{tr}\, k = 0$ , and that $H$ is a minimal 2-surface in $M$ , i.e. its mean curvature $H_{\mathrm{mean}}$ in $M$ vanishes (denoted $J = 0$ in the source paper). This is the situation in which Gibbons–Hawking–Horowitz–Perry prove positivity of the ADM mass 'regardless of what lies inside $H$ '. At each point of $H$ the future- or past-directed outgoing null geodesics orthogonal to $H$ are non-diverging, but $H$ need not be strictly future or past trapped: the future-directed null geodesics may be converging over part of $H$ and the past-directed ones over the rest. Prove, or refute by an explicit construction, the following statement: in the maximal Cauchy development $\mathcal{M}$ of $(M,g,k)$ , the surface $H$ lies behind the future or the past event horizon, i.e. $H$ is not visible from future null infinity $\mathcal{I}^+$ or $H$ is not visible from past null infinity $\mathcal{I}^-$ (equivalently, $H \cap J^-(\mathcal{I}^+) = \emptyset$ or $H \cap J^+(\mathcal{I}^-) = \emptyset$ ).

The question is unconditional at the level of the initial data: its hypotheses are the data-level ones listed above (asymptotic flatness, dominant energy condition, maximality of the slice, minimality of the boundary). This is the source's own framing: in Sect. 5 the authors record that, 'with the usual weak energy condition on $T_{\mu\nu}$ , and assuming cosmic censorship, any future trapped surface must lie behind the event horizon', so what they could not prove is the containment asserted without such a hypothesis. A refutation consists of smooth asymptotically flat data in the class above whose maximal Cauchy development contains a point of $H$ visible from future null infinity and a point of $H$ visible from past null infinity.

This is a weak-censorship-type assertion for data whose slice already carries a marginal trapped boundary, and it is the premise under which every positive mass theorem 'with apparent horizon boundary' — including the source paper's own results and their modern successors — is interpreted as a statement about black holes rather than about slices. The later positive-mass theorems for boundary data with arbitrary extrinsic curvature and weakly outer trapped boundary (Lee–Lesourd–Unger and their successors) are generalizations of the mass inequality, not of this causal question, whose data class is the source's time-symmetric one.

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