What Is a Black Hole?

A black hole is not identified by large curvature, darkness during a finite experiment, or the zero of a coordinate coefficient. Its standard global definition is an inaccessible region relative to a specified future receiver; Killing, marginal, isolated, and dynamical horizons answer related but different questions. This chapter builds the causal definition, explains why it depends on the full future, adapts it to non-flat asymptotics, separates it from trapped surfaces and singularities, and ends with a procedure for choosing the right boundary in analytic, mathematical, and numerical work.

Helpful background. Exact Solutions of Einstein’s Equations supplies concrete geometries; Causality and Global Structure supplies causal boundaries and completeness; The Einstein Equation connects matter to geometry; Hypersurface Normals and Tangential Projectors supplies the local geometry of candidate boundaries.

Enter this chapter

Begin by naming the question that “black hole” is meant to answer. If it is whether a causal signal ever reaches an asymptotic receiver, the defining set is global. For a suitable future receiver $R^+$,

$$\mathcal B[R^+]=M\setminus J^-(R^+), \qquad \mathcal H[R^+]=\partial_MJ^-(R^+).$$

In an asymptotically flat spacetime, $R^+=\mathscr I^+$. In de Sitter or AdS settings, the conformal boundary has a different causal character and the receiver must be stated anew. For one observer or finite screen, the same causal construction gives an observer-relative horizon or finite-time shadow, not automatically a global black-hole event horizon.

If instead the task is to find a boundary from data on one spacelike slice, calculate surface gravity in an exactly stationary geometry, or measure flux through a growing surface, other definitions are deliberately better suited. Booth’s comparison makes these distinct data requirements explicit in Booth 2005, §§2–5, pp. 1075–1095.

Global and quasi-local questions

An event horizon is invariant under a change of slicing, but locating it requires the completed future. Its teleology is therefore a final-boundary selection property, not a signal propagating from the future into the past. Two isometric local neighborhoods can have different event-horizon membership when their later global developments differ.

A trapped or marginal surface reverses the information balance. Given a specified compact spacelike two-surface $S$ and its future null normals, the signs of the two expansions can be found from nearby geometry. That makes these surfaces useful in initial data and simulations, while selecting an outermost marginal surface or apparent horizon imports a slice. An isolated horizon adds quasi-local equilibrium; a dynamical horizon adds a spacelike marginal foliation and flux. The next chapter develops that taxonomy.

The distinction is logical as well as practical. A trapped surface does not by definition equal an event horizon. With the Penrose theorem’s noncompact Cauchy surface, null convergence, global hyperbolicity, and regularity hypotheses, it implies future null geodesic incompleteness. It does not alone imply curvature blow-up or visibility behind an event horizon; the original compactness proof is on Penrose 1965, pp. 57–59.

Routes for different readers

For a causal or mathematical route, read the first four pages in order, then the theorem-boundary comparison. This route tracks the quantifier “can ever reach the receiver,” exposes dependence on conformal completion, and shows where a global conclusion needs energy and topology hypotheses.

For numerical or quasi-local work, read the teleology page first to see why one slice cannot locate an event horizon in real time. Then study the trapped-surface comparison and selection procedure before entering Horizon Taxonomy. The equilibrium and dynamical definitions later in the volume will then appear as purpose-built structures, not approximations to be ranked by a single notion of accuracy.

For stationary black-hole mechanics, read the global definition and the final selection page. Continue to the equilibrium examples and Killing horizons, where event, Killing, and marginal structures coincide under additional symmetry. Keep that coincidence as a check rather than importing it into a dynamical setting.

Contents in scientific order

1. The Black-Hole Region and Future Null Infinity states the asymptotically flat causal definition and the assumptions that make it meaningful.
2. Why Event Horizons Are Global unpacks the full-future quantifier using collapse and escape.
3. Teleology and Local Undetectability proves the operational distinction between local geometry and global membership.
4. Black Holes Beyond Asymptotic Flatness replaces future null infinity carefully in de Sitter, AdS, and finite-boundary problems.
5. Horizons, Trapped Surfaces, and Singularities states the valid theorem implication and tests the invalid converses.
6. Choosing a Black-Hole Definition selects a boundary from the target quantity, symmetry, foliation, future data, and flux.

Each page leaves detailed conformal-boundary proofs in Global Structure and exact metric construction in Exact Solutions. Here those results are used to make physically correct choices about horizons.

Three recurring checks

Collapse versus stationarity. In settled Schwarzschild geometry, the event horizon, Killing horizon, and standard marginal tube occupy $r=2GM$. In collapse they need not coincide: the event horizon can extend into a regular region before a trapped round sphere forms. Any proposed identity among definitions must survive this contrast.

Complete versus finite receivers. Failure to reach a detector before it switches off establishes only a finite-time shadow. Event-horizon language requires a future-complete receiver appropriate to the asymptotics.

Incompleteness versus curvature. Removing a regular point from Minkowski spacetime produces incomplete geodesics with zero curvature. Schwarzschild’s curvature divergence follows from its exact invariant, not from the word “incomplete.” These two cases test whether a conclusion has silently become stronger than its theorem.

Review the chapter

A numerical evolution with AdS asymptotics reports a smooth outermost surface with $\theta_{(\ell)}=0$ and $\theta_{(n)}\lt0$ on the current slice. No future data have been evolved, and no field boundary condition has been stated. Which claims can be made now, and what additional information is needed to call the surface an event horizon and to predict whether a scalar pulse returns from the boundary?

Solution

The calculation identifies a marginally outer trapped surface on the stated slice; if it is the outer boundary of the relevant trapped region, it may be called the apparent horizon for that slice. Its location can change with the foliation. To identify the event horizon one needs the full future causal development and the chosen timelike conformal-boundary component, then must construct $\partial J^-(\mathscr I)$. Predicting the scalar pulse additionally requires an admissible AdS field boundary condition. That condition completes the field evolution but does not redefine the causal cones of the fixed metric. None of these missing statements is supplied by the two expansion signs alone.

Horizon Taxonomy is the next step: it defines the global, stationary, slice-based, isolated, and dynamical boundary classes and shows exactly when they coincide.

References

• Booth, Ivan. “Black-Hole Boundaries.” Canadian Journal of Physics 83 (2005): 1073–1099. DOI.
• Penrose, Roger. “Gravitational Collapse and Space-Time Singularities.” Physical Review Letters 14 (1965): 57–59. DOI.