The Black-Hole Region and Future Null Infinity

For a specified asymptotically flat end, a black hole is the region that cannot send any future-directed causal signal to that end’s future null infinity. This definition is coordinate independent and does not assume a stationary metric, a particular exact solution, or large local curvature. Its precision comes with a cost: the escape question refers to the complete future development and to a boundary added at infinity, so the black-hole region is a global property of the spacetime.

Helpful background. Exact Solutions of Einstein’s Equations supplies examples and extensions; Causality and Global Structure supplies causal boundaries; The Einstein Equation supplies the dynamics used in later horizon theorems; Hypersurface Normals and Tangential Projectors supplies the geometry needed when this global boundary is compared with slice-based horizons.

Escape to future null infinity

Let $(\mathcal M,g_{ab})$ be a time-oriented physical spacetime admitting an asymptotically flat conformal completion $(\widetilde{\mathcal M},\tilde g_{ab})$. Future null infinity $\mathscr I^+$ is the boundary reached by future-complete outgoing null curves in the asymptotic region. Causal pasts are taken in the completion and then restricted to the physical spacetime. For the chosen component of $\mathscr I^+$, define

$$\mathcal B =\mathcal M\setminus J^-_{\widetilde{\mathcal M}}(\mathscr I^+), \qquad \mathcal H^+ =\partial_{\mathcal M}\mathcal B =\partial_{\mathcal M}J^-_{\widetilde{\mathcal M}}(\mathscr I^+).$$

A point $p$ lies outside $\mathcal B$ precisely when at least one future-directed causal curve from $p$ reaches $\mathscr I^+$. It lies inside $\mathcal B$ when no such curve exists. The boundary $\mathcal H^+$ is the future event horizon relative to that asymptotic end. Booth develops this construction and its physical motivation in Booth 2005, §2.1, pp. 1075–1078.

The reference to a chosen component matters. A spacetime with several asymptotic ends can have a black-hole region relative to one end that differs from the region relative to another. If the intended exterior includes all acceptable ends, one instead uses the causal past of their union. The asymptotic structure is therefore part of the definition, not background decoration.

What is assumed, and what is not

Three ingredients are needed for the displayed definition to answer a physical escape question.

First, the spacetime must have a time orientation, so “future-directed” is defined. Second, it must have a sufficiently regular asymptotic completion with an identified future null infinity. Third, the conformal completion and chosen boundary component must specify what it means for a causal curve to escape. In the standard asymptotically-flat setting, outgoing radiation reaches $\mathscr I^+$. In a spacetime with a timelike conformal boundary, the geometric causal past is still fixed by the metric and the selected component, while boundary conditions separately complete the initial–boundary-value problem for fields.

Neither the Einstein equation nor an energy condition is logically required to form the set $\mathcal B$. They enter when one proves further results, such as focusing, generator regularity, or area monotonicity. Global hyperbolicity and cosmic-censorship-type predictability assumptions are likewise not part of the set-theoretic definition, although they are essential to many useful theorems about its boundary. Keeping these layers separate prevents a common error: treating every property proved for a predictable Einstein spacetime as part of the word “horizon.” The theorem-level causal development is given on Event Horizons as Global Boundaries.

Why the boundary is a horizon

The set $J^-(\mathscr I^+)$ is a past set: once a point can influence future null infinity, every point in its causal past can also do so. The boundary of a past set is achronal—no two of its points can be joined by a timelike curve lying across the boundary; the causal-boundary argument is developed on Event Horizons as Global Boundaries. Where it is smooth, this achronal boundary is a null hypersurface ruled by null geodesic generators. The smooth null-hypersurface and generator geometry is developed in Gourgoulhon and Jaramillo 2006, §§2.2–2.5, pp. 169–176.

This statement allows nonsmooth portions. In a collapse or merger, generators can join the horizon at past endpoints forming crease or caustic sets. A smooth horizon cross-section is therefore a useful local object, not a license to assume that the entire event horizon is a smooth cylinder. The event horizon also need not be a Killing horizon: nullness follows from causal boundary structure, whereas a Killing horizon requires an appropriate spacetime symmetry.

Four discriminating examples

Minkowski spacetime. Every point can send an outgoing null signal to $\mathscr I^+$, so $J^-(\mathscr I^+)=\mathcal M$ and $\mathcal B$ is empty. The presence of light cones alone does not create a black hole.

Eternal Schwarzschild. Relative to either exterior infinity, the corresponding black-hole region contains the future interior from which no causal curve reaches that infinity. The horizon is regular even though the usual static coordinates become singular there. Its location is not selected by a curvature divergence.

Gravitational collapse. The horizon can extend into a region before the collapsing matter reaches it. Its early location is fixed by whether outgoing null rays eventually escape or are later captured. This is the simplest expression of the definition’s dependence on the future.

de Sitter or anti-de Sitter asymptotics. The standard asymptotically-flat $\mathscr I^+$ is absent or has a different causal character. One must first state an observer patch, conformal boundary, or permitted boundary behavior; only then can an escape region be defined. Reusing the formula without that replacement leaves $\mathcal B$ undefined rather than generalized.

What the definition does not imply

Membership in $\mathcal B$ is not a local scalar observable. An observer crossing a sufficiently large, regular event horizon can encounter small curvature and no locally distinguished surface. Conversely, a region of large curvature need not be inside an event horizon.

The definition also does not assert that a trapped surface exists, that a spacetime is geodesically incomplete, or that curvature diverges. Those conclusions require separate hypotheses and, in the case of singularity theorems, the conclusion is generally causal geodesic incompleteness rather than a singular point. Nor does the definition produce an apparent horizon on an arbitrary slice. Global, quasi-local, and singularity statements must be related by theorems, not by vocabulary.

Check the relative asymptotic end

Suppose a maximally extended two-ended black-hole spacetime has future null infinities $\mathscr I^+_R$ and $\mathscr I^+_L$. A point can send a causal signal to $\mathscr I^+_L$ but not to $\mathscr I^+_R$. Is it in the black-hole region?

Solution

The answer depends on the declared exterior. Relative to the right end, $\mathcal B_R=\mathcal M\setminus J^-(\mathscr I^+_R)$, the point is inside the black-hole region. Relative to the left end it is outside. If escape to either end counts, the relevant set is $\mathcal M\setminus J^-(\mathscr I^+_R\cup\mathscr I^+_L)$, and the point is outside. This is why the asymptotic end or union of ends must be named.

Why Event Horizons Are Global now turns the escape definition into its principal physical consequences: dependence on the entire future, teleology, and local undetectability.

References

• Booth, Ivan. “Black-Hole Boundaries.” Canadian Journal of Physics 83 (2005): 1073–1099. DOI.
• Gourgoulhon, Éric, and José Luis Jaramillo. “A 3+1 Perspective on Null Hypersurfaces and Isolated Horizons.” Physics Reports 423 (2006): 159–294. DOI.