Choosing a Black-Hole Definition

There is no best black-hole boundary independently of the question being asked. Event horizons classify ultimate escape, Killing horizons encode exact symmetry, marginal surfaces support slice-by-slice diagnostics, isolated horizons describe quasi-local equilibrium, and dynamical horizons track quasi-local flux. A reliable choice begins with the observable or theorem one needs, then states the asymptotic receiver, future-data requirements, foliation, symmetry, and normalization that the chosen definition imports.

Helpful background. Exact Solutions of Einstein’s Equations supplies stationary and dynamical tests; Causality and Global Structure supplies causal boundaries; The Einstein Equation governs matter and geometry; Hypersurface Normals and Tangential Projectors supplies null expansions and surface geometry.

Start with the quantity to be determined

The first question is not “Where is the black hole?” but “Which conclusion must the boundary support?” Six common targets select different structures.

Target	Appropriate structure	Information required
Ultimate inability to signal a chosen asymptotic receiver	Event horizon	Complete future causal development and receiver set
Horizon tied to a stationary symmetry	Killing horizon	A specified Killing field and its normalization
Black-hole candidate on one numerical slice	Outermost marginally outer trapped surface or apparent horizon	Slice, outward direction, null expansions, and outermost rule
Equilibrium mechanics without a stationary exterior	Weakly isolated or isolated horizon	Intrinsic null-boundary conditions and equivalence class of generators
Growth and flux in a dynamical regime	Dynamical horizon or suitable trapping horizon	Foliation by marginal leaves, signature, orientation, and flux law
Accessibility to one observer or screen	Observer- or receiver-relative causal horizon	Worldline or worldtube and its future duration

The entries are not interchangeable measurements of one hidden surface. Booth compares their data requirements and principal uses in Booth 2005, §§2–5, pp. 1075–1095. Coincidence in a stationary example is a consistency check; separation during collapse is expected.

Global escape requires a receiver and the full future

If the question is whether an event can ever communicate with infinity, use

$$\mathcal H[R^+]=\partial J^-(R^+),$$

where $R^+$ is future null infinity, a chosen conformal-boundary component, or another explicitly stated complete receiver. This definition is invariant under coordinate and slicing changes, but it is teleological: locating it requires the complete future development. It is therefore appropriate for global causal theorems and retrospective analysis of a finished numerical spacetime, not for real-time horizon finding from one Cauchy slice.

If the receiver exists only for a finite interval, report an operational causal shadow rather than an event horizon. In AdS, also state the boundary conditions when making claims about field propagation. The metric causal past and the physical initial–boundary-value problem are related but distinct.

Exact symmetry selects a Killing horizon

Suppose a Killing field $\chi^a$ becomes null on a null hypersurface and is normal to it. That hypersurface is a Killing horizon. In a stationary rotating black hole one normally uses

$$\chi^a=t^a+\Omega_H\phi^a,$$

with $t^a$ normalized by the chosen asymptotic time and $\phi^a$ by closed $2\pi$ orbits. This structure is the right one for surface gravity, bifurcation surfaces, stationary charges, and equilibrium first laws.

Do not use the definition merely because a dynamical numerical horizon looks nearly steady. Exact Killing symmetry is a spacetime condition. A slowly evolving or isolated-horizon approximation may be useful, but its error and boundary conditions must then be stated instead of upgrading approximate symmetry to an exact Killing field.

Slice diagnostics use null expansions

On a spacelike slice, let $S$ be a closed two-surface with future outgoing and ingoing null normals $\ell^a$ and $n^a$. A standard black-hole candidate satisfies

$$\theta_{(\ell)}=0, \qquad \theta_{(n)}\lt0.$$

An apparent horizon is usually the outer boundary, within that slice, of an appropriate trapped region. This makes it valuable in initial-data and evolution codes: the calculation uses current-slice geometry rather than the unknown future. It also makes the answer foliation-dependent. Record the slice, the meaning of outward, whether the surface is outermost or merely one solution, and the stability criterion used to reject nearby branches.

A marginal surface is not automatically an event horizon. During spherical accretion, for example, the marginal tube can respond locally to incoming flux while the event horizon has already expanded in anticipation of that future flux. In stationary Schwarzschild, both lie at $r=2GM$; that limiting agreement validates conventions but does not prove equivalence in dynamics.

Equilibrium and flux call for quasi-local horizons

A nonexpanding horizon is a null hypersurface whose generators have vanishing expansion, supplemented by field-equation and regularity conditions. A weakly isolated horizon adds time-independence of the relevant normal-bundle connection for an equivalence class of null generators. It supports a quasi-local zeroth and first law without demanding that the entire exterior be stationary. Use it when equilibrium at the boundary, rather than a global Killing field, is the controlled statement.

A dynamical horizon is instead spacelike and foliated by marginal surfaces with the ingoing expansion negative. Its area and charges obey flux balances driven by matter and gravitational radiation. It is suited to finite growth, but the foliation, orientation, and signature assumptions matter. Ashtekar and Krishnan develop the hierarchy and corresponding balance laws in Ashtekar and Krishnan 2004, §§2–4.

Null isolated and spacelike dynamical segments may meet in a physical evolution, yet neither definition alone supplies the globally selected event horizon. Choose a quasi-local structure when its intrinsic geometry or flux is the object of interest; add the event horizon separately if ultimate escape must also be discussed.

Apply a five-step reporting rule

For any calculation or theorem, a short sequence makes the choice reproducible:

1. State the physical question: escape, symmetry, slice detection, equilibrium, flux, or observer access.
2. Name the receiver, slice, worldtube, or null boundary and its time domain.
3. State the imported structure: full future, Killing symmetry, foliation, null-normal equivalence class, energy condition, or boundary condition.
4. Give the defining equation and all orientation or normalization choices.
5. Test one coincidence limit and one separating example.

This rule also prevents overclaiming. “The code found a marginally outer trapped surface (MOTS)” is precise; “the event horizon formed at this simulation time” requires a separate global reconstruction. “The boundary is isolated” is quasi-local; “the exterior is stationary” requires symmetry beyond the boundary. “No signal reached our screen before shutdown” is finite-time evidence; it does not prove permanent inaccessibility.

Choose definitions in four scenarios

Select the primary boundary for each task: computing surface gravity in an analytic Kerr solution; detecting a newly formed common black-hole boundary during a binary simulation; measuring energy flux into a growing quasi-local boundary; and determining after a completed collapse which events can never signal future null infinity. State one limitation of each choice.

Solution

For analytic Kerr, use the Killing horizon generated by $\chi=t+\Omega_H\phi$; its surface gravity depends on the normalization of $t$. During the simulation, use an outermost stable MOTS or apparent horizon on each specified slice; its location depends on the foliation and does not give the event horizon. For local growth, use a dynamical horizon or a trapping horizon with the needed signature and orientation conditions; it gives a flux balance but not ultimate escape. After the full collapse spacetime is known, use $\partial J^-(\mathscr I^+)$; this gives the event horizon but cannot be located from a single time slice in real time. In stationary equilibrium the four may coincide, providing a useful cross-check without merging their definitions.

Horizon Taxonomy develops the individual notions and their transformation, foliation, and flux properties in detail.

References

• Ashtekar, Abhay, and Badri Krishnan. “Isolated and Dynamical Horizons and Their Applications.” Living Reviews in Relativity 7 (2004): article 10. DOI.
• Booth, Ivan. “Black-Hole Boundaries.” Canadian Journal of Physics 83 (2005): 1073–1099. DOI.