Event Horizons in the Horizon Taxonomy

An event horizon answers an escape question about the completed spacetime; the other horizons in this chapter answer questions about symmetry, null expansion, or equilibrium on finite surfaces. They can coincide in a stationary black hole, but no general implication makes them interchangeable. The safest comparison therefore begins by asking what information each definition uses and only then states the hypotheses under which two resulting boundaries agree.

Helpful background. Exact Solutions of Einstein’s Equations supplies the named solutions used as checks; Causality and Global Structure supplies causal boundaries; The Einstein Equation connects focusing to matter; Timelike Congruences and Spatial Projection supplies expansion and shear; Hypersurface Normals and Tangential Projectors supplies slice geometry; and Black-Hole Geometry and Mechanics locates this comparison in the volume.

The event horizon as a causal boundary

For a specified asymptotically flat end with future null infinity $\mathscr I^+$, the black-hole region and its future event horizon are

$$\mathcal B=\mathcal M\setminus J^-(\mathscr I^+), \qquad \mathcal H^+=\partial J^-(\mathscr I^+).$$

This exact global construction is developed on Event Horizons as Global Boundaries. Here its important feature is the quantifier: a point is outside the black hole if some future-directed causal curve from it reaches the chosen $\mathscr I^+$. Determining the answer can require the entire future metric. No spacelike slice, curvature scalar, or local measurement alone supplies that information.

Under suitable regularity and predictability assumptions, a smooth portion of $\mathcal H^+$ is a null hypersurface ruled by null geodesic generators. Those properties follow from the achronal-boundary construction, not from a Killing symmetry. Further conclusions, such as nonnegative expansion and area monotonicity, additionally require a focusing condition such as the null energy condition, sufficient generator completeness, and control of future endpoints. Booth separates the definition from these extra hypotheses in Booth 2005, §2.1, pp. 1075–1078.

What the other definitions know

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

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

These scalars measure the fractional area change along the two null normal directions. They allow several finite-region definitions, each with a distinct information requirement.

Horizon notion	Defining information	Dependence that remains
Event horizon	Causal past of the chosen future boundary	Full future and asymptotic end
Killing horizon	A Killing field normal to a null hypersurface	Choice and normalization of symmetry generator
Apparent horizon	Outermost marginally outer trapped surface on one slice	Spacelike slicing and outermost convention
Future outer trapping horizon	A tube with $\theta_{(\ell)}=0$, $\theta_{(n)}\lt0$, and inward decrease of $\theta_{(\ell)}$	Choice of marginal foliation and null orientation
Nonexpanding or isolated horizon	A null boundary with zero expansion and progressively time-independent intrinsic data	Null-normal equivalence class and imposed boundary structure
Dynamical horizon	A spacelike tube foliated by surfaces with $\theta_{(\ell)}=0$ and $\theta_{(n)}\lt0$	Marginal foliation and existence of a spacelike tube

A Killing horizon need not bound a black hole: the Rindler horizon in Minkowski spacetime is a Killing horizon for accelerated observers but is not the boundary of $J^-(\mathscr I^+)$. Conversely, a collapse event horizon need not admit any Killing field. An apparent horizon can move when the spacetime is resliced, whereas the event horizon as a subset of spacetime does not. Ashtekar and Krishnan give an especially discriminating example: two null isolated boundaries can have the same quasi-local equilibrium character while only one lies on the event horizon, because a later infalling shell changes the global causal boundary Ashtekar and Krishnan 2004, §1, Figures 1–2 and surrounding discussion.

Coincidence in equilibrium

In the maximally extended Schwarzschild solution, choose a round cross-section of the future branch $\mathcal H^+$ away from the bifurcation sphere and a compatible spherically symmetric horizon-penetrating slice whose outermost MOTS is that section. With the stationary Killing field normalized to unit time translation at the chosen infinity, $r=2GM$ is simultaneously a future event horizon and a Killing horizon. Its future outgoing expansion vanishes, its future ingoing expansion is negative, and its null intrinsic geometry is time independent. On this stated section and slicing, the event, Killing, future outer trapping, nonexpanding, isolated, and apparent-horizon notions select the same familiar location.

The bifurcation sphere is an important exception to the expansion statement. On the time-reflection-symmetric Einstein–Rosen-bridge slice it is a minimal surface with $\theta_{(\ell)}=\theta_{(n)}=0$. It is still the intersection of the future and past Killing/event-horizon branches and is a MOTS, but it fails the strict future condition $\theta_{(n)}\lt0$ used for a future outer trapping horizon. Thus even Schwarzschild requires the branch, cross-section, and slicing to be specified before claiming that every horizon definition coincides.

That coincidence is an example, not a chain of definitions. It uses an asymptotically flat completion, stationarity, the correctly normalized Killing generator, regular horizon geometry, and a compatible slicing. In stationary Kerr, the relevant generator is

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

not the asymptotic time translation $t^a$ by itself. On suitable stationary slices the same coincidence holds, but the distinction between $t^a$ and $\chi^a$ is essential inside the ergoregion. In more general stationary settings, proving that an event horizon is a Killing horizon requires rigidity hypotheses; it is not a consequence of the word “stationary.”

Separation in ingoing Vaidya spacetime

The ingoing Vaidya metric provides a direct calculation. With

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

choose $\ell^a=\partial_v+(f/2)\partial_r$ and $n^a=-\partial_r$, so $\ell\!\cdot n=-1$. Round spheres have

$$\theta_{(\ell)}=\frac{f}{r}, \qquad \theta_{(n)}=-\frac{2}{r}.$$

Their marginal tube is therefore $r=2Gm(v)$. When $dm/dv\gt0$, this tube is spacelike and is a dynamical horizon in the older definition. The event horizon instead follows the outgoing-null equation

$$\frac{dr_E}{dv}=\frac12\left(1-\frac{2Gm(v)}{r_E}\right)$$

with a future boundary condition. During accretion it generally lies outside $r=2Gm(v)$ and begins responding before a locally measured flux reaches a given generator. If $m(v)$ becomes exactly constant after a finite advanced time, the marginal tube becomes null and coincides there with the equilibrium event horizon; if the mass settles only asymptotically, the two surfaces approach one another. The explicit comparison and its limitations are given in Ashtekar and Krishnan 2004, §2.2.2, especially Figures 3–4.

The calculation supplies the decisive logic. A quasi-local horizon can locate strong-field evolution from data near a slice, but it does not encode the ultimate escape of light rays. An event horizon encodes that escape exactly, but it does not by itself distinguish instantaneous equilibrium from ongoing flux.

Check your understanding

In a spherically symmetric Vaidya region with $dm/dv\gt0$, a student claims that the surface $r=2Gm(v)$ must be the event horizon because its outgoing expansion vanishes. Identify the missing step and determine the sign of $dr_E/dv$ if the event horizon satisfies $r_E\gt2Gm(v)$.

Solution

Vanishing outgoing expansion identifies a marginal surface; it does not solve the global escape problem. The event horizon must be obtained from the outgoing-null equation with a condition in the future stationary region. If $r_E\gt2Gm(v)$, then $1-2Gm(v)/r_E\gt0$, so $dr_E/dv\gt0$. The event horizon is null and expanding while the marginal tube $r=2Gm(v)$ is spacelike for positive accretion. Their later coincidence does not make their definitions identical.

Next, Killing Horizons isolates the symmetry-based notion and shows exactly what the normal Killing field contributes.

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. arXiv:gr-qc/0508107.