An Event Horizon Is Not Every Black-Hole Boundary

Several boundaries coincide in stationary textbook black holes, but their definitions are not interchangeable. An event horizon uses the complete causal future; a Killing horizon a symmetry; an apparent horizon a slice; and isolated or dynamical horizons conditions on extended boundaries. Ask what data define the boundary and which question it answers.

A map of the horizon notions

Let $S$ be a closed spacelike two-surface with future-directed null normals $\ell^a$ and $n^a$, normalized by $\ell\!\cdot n=-1$. If $q_{ab}$ is the induced two-metric, their expansions are

$$\theta_{(\ell)}=q^{ab}\nabla_a\ell_b, \qquad \theta_{(n)}=q^{ab}\nabla_a n_b.$$

The zero and sign of an expansion survive positive reciprocal rescalings of the null normals. They classify marginal and trapped surfaces, but they do not by themselves define every object called a horizon.

Notion	Defining object or condition	Information and dependence
Event horizon	For an asymptotic end $E$, the boundary $\partial J^-(\mathscr I_E^+)$	Global: it depends on the complete future and the chosen asymptotic end.
Killing horizon	A null hypersurface to which a Killing field is normal and on which that field is null	Requires a symmetry; constant rescaling leaves the horizon unchanged, while surface gravity depends on normalization. It need not bound a black hole.
MOTS and apparent horizon	A marginally outer trapped surface satisfies $\theta_{(\ell)}=0$; in the future marginal case used here, $\theta_{(n)}\lt0$. An apparent horizon is the outer boundary of a trapped region on a specified spacelike slice	Quasi-local on each leaf, but the outermost selection is foliation dependent and can jump.
Future outer trapping horizon	A tube foliated by marginal leaves with $\theta_{(\ell)}=0$, $\theta_{(n)}\lt0$, and $\mathcal L_n\theta_{(\ell)}\lt0$	Uses an extended tube and an outer stability sign; its signature follows from additional field and energy assumptions.
Nonexpanding, weakly isolated, or isolated horizon	Under the standard topology, regularity, field-equation, and energy hypotheses, a nonexpanding horizon is null with zero outgoing expansion, $\theta_{(\ell)}=0$. Weakly isolated and isolated horizons impose successively stronger time independence on compatible connection data	Describes equilibrium at the boundary without requiring a stationary neighborhood.
Dynamical horizon	In the original definition, a spacelike tube foliated by leaves with $\theta_{(\ell)}=0$ and $\theta_{(n)}\lt0$	Quasi-local and adapted to growth; Einstein’s equation supplies area and flux-balance laws.
Acceleration horizon	A boundary of the region accessible to a specified accelerated observer or family	Observer dependent; it can occur in flat spacetime with no black-hole region.

These distinctions, including the extra structure needed to pass from a marginal surface to a trapping, isolated, or dynamical horizon, are developed in Booth 2005, §§2.1–3.3 and Ashtekar and Krishnan 2004, §§2.1–2.2, Definitions 1–5.

The global boundary cannot be found from one slice

Assume a future conformal completion appropriate to an asymptotically flat spacetime, choose an asymptotic end $E$, and take the causal past in that completion. The black-hole region and its future event horizon are

$$\mathcal B_E=\mathcal M\setminus J^-(\mathscr I_E^+), \qquad \mathcal H_E^+=\partial\mathcal B_E =\partial J^-(\mathscr I_E^+).$$

The definition asks whether a future-directed causal signal reaches the chosen future null infinity. It cannot be decided from curvature at one event, the expansion of one two-surface, or data on one incomplete time slice. In particular, a regular neighborhood can later lie inside an event horizon after collapse. Finding that boundary in a completed numerical spacetime is usually a backward-in-time calculation.

A MOTS answers a different question: its outgoing future null congruence has zero instantaneous area expansion. Under the usual regularity assumptions, the trapped-region boundary on a slice is represented by an outermost MOTS and called the apparent horizon. Another foliation can select different leaves or none, while the completed spacetime’s event horizon is unchanged.

Why equilibrium hides the distinction

In the maximally extended Schwarzschild solution, consider the regular future surface $r=2GM$ and symmetry-compatible slices. The asymptotically normalized stationary Killing field is null there, the outgoing expansion of the standard spherical sections vanishes, and the intrinsic null geometry is time independent. With the usual outermost choice, the event, Killing, apparent, trapping, nonexpanding, and isolated horizons all occupy the same surface.

This coincidence assumes stationarity, regularity, the chosen asymptotic end, and a compatible foliation. Killing-generator normalization affects surface gravity, not the horizon’s location. “The horizon at $r=2GM$” is efficient Schwarzschild shorthand, not a definition for a dynamical spacetime.

Rindler and Vaidya separate the definitions

The first counterexample removes gravity entirely. In Minkowski spacetime, the null boundaries of a Rindler wedge are Killing horizons of the boost Killing field and acceleration horizons for uniformly accelerated observers. Yet the Riemann tensor vanishes, every event can communicate with the full Minkowski future null infinity, and the black-hole region relative to that infinity is empty. Thus a Killing or observer horizon is not automatically a black-hole event horizon.

The second counterexample keeps a black hole but removes equilibrium. For ingoing Vaidya spacetime,

$$ds^2=-\left(1-\frac{2Gm(v)}r\right)dv^2 +2\,dv\,dr+r^2d\Omega^2,$$

choose $n^a=-\partial_r$ and $\ell^a=\partial_v+\tfrac12(1-2Gm(v)/r)\partial_r$. Then $\ell\!\cdot n=-1$, and round spheres have

$$\theta_{(\ell)}= \frac1r\left(1-\frac{2Gm(v)}r\right), \qquad \theta_{(n)}=-\frac2r.$$

The marginal tube is therefore $r=2Gm(v)$. During positive-energy accretion, $dm/dv\gt0$, this tube is spacelike and growing: it is a dynamical horizon in the original sense. Its location follows from the mass function at that advanced time.

The event horizon instead follows the outgoing radial-null equation

$$\frac{dr_{\rm EH}}{dv} =\frac12\left(1-\frac{2Gm(v)}{r_{\rm EH}}\right),$$

with a future boundary condition fixed after the accretion history is known. For the standard positive-energy accretion profile that settles to a stationary black hole, it generally lies outside $r=2Gm(v)$ while the mass is increasing. The Vaidya comparison and its causal diagrams are given in Ashtekar and Krishnan 2004, §2.2.2, Figure 4 and surrounding discussion. The two surfaces coincide again after equilibrium is reached, but their different construction remains visible in the calculation.

Choose the boundary by the question

Use the name only after specifying the job:

• For escape to a chosen asymptotic region, use the event horizon and accept its global future dependence.
• For stationary mechanics and surface gravity, use a Killing horizon with a declared generator and normalization.
• For slice-by-slice numerical location or excision, report an outermost MOTS or apparent horizon and name the foliation.
• For local growth, flux, and area balance, use a future outer trapping or dynamical horizon and state the expansion, stability, signature, field, and energy conditions.
• For equilibrium on a finite null boundary, use a nonexpanding or isolated horizon and state which intrinsic data are time independent.
• For what a particular observer can receive, use that observer’s causal or acceleration horizon without relabeling it a black-hole boundary.

For the complete derivations, implication map, and numerical caveats, continue to Event, Killing, Apparent, Trapping, Isolated, and Dynamical Horizons.

What the familiar shorthand retains

“The horizon” remains useful when stated hypotheses make the notions coincide or the context names one definition. A simulation may likewise “track the horizon” after identifying an outermost MOTS. Do not infer global causal status from a local surface, equilibrium from nullness, black-hole status from a Killing horizon, or foliation independence from one slicing.

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.