Event, Killing, Apparent, Trapping, Isolated, and Dynamical Horizons

The phrase “black-hole horizon” can refer to a global causal boundary, a null surface generated by a symmetry, an outermost marginal surface on one slice, or a quasi-local boundary in equilibrium or growth. These structures often coincide for an eternal stationary black hole, which is precisely why they are so easily conflated. This chapter separates their definitions, information requirements, invariances, and valid implication arrows before later chapters use them in mechanics.

Enter the horizon taxonomy

Every horizon notion answers a particular question.

• Event horizon: Can a causal signal reach the chosen future asymptotic boundary?
• Killing horizon: Does a spacetime symmetry become null and normal to a hypersurface?
• Apparent horizon: What is the outer boundary of the trapped region on this spacelike slice?
• Trapping horizon: How do marginal sections separate trapped from untrapped surfaces as they move through spacetime?
• Nonexpanding or isolated horizon: Which intrinsic data on a null boundary are time independent?
• Dynamical horizon: Which spacelike marginal tube represents local black-hole growth and supports flux balance?

No answer can be substituted for another without hypotheses. Global escape requires future causal information. Apparent horizons require a slice and an outermost choice. Killing horizons require symmetry and generator normalization. Quasi-local horizons require an extended two-surface or tube, not merely a pointwise curvature measurement. Booth compares these boundary notions and their different input data in Booth 2005, §§2–5, pp. 1075–1095.

The recurring convention is a pair of future null normals $\ell^a,n^a$ to a closed spacelike two-surface $S$, normalized by $\ell\!\cdot n=-1$. Their expansions are

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

Positive reciprocal rescalings change their magnitudes but preserve their zeros and signs. A future marginal surface has $\theta_{(\ell)}=0$ and $\theta_{(n)}\lt0$. This one equation is the common thread through apparent, trapping, isolated, and dynamical horizons, but the extra structure attached to it differs in every case; see Gourgoulhon and Jaramillo 2006, §§5.5–5.8, Open PDF.

Check your preparation

The shortest useful preparation is geometric rather than encyclopedic. You should be able to project tensors onto a two-surface, distinguish a spacetime hypersurface from its cross-sections, and interpret expansion as fractional area change. If that decomposition is unfamiliar, review Hypersurface Normals and Tangential Projectors. For the focusing argument, review Timelike Congruences and Spatial Projection and transfer the expansion–shear decomposition to null congruences.

You should also distinguish a local light-cone statement from a global causal past. Causality and Global Structure provides that repair route. Named metrics are used only as checks, so Exact Spacetimes is helpful but not needed as a metric-derivation prerequisite. The Einstein equation enters only when expansion signs are connected to matter flux and focusing.

Two routes through the chapter

For a conceptual route, begin with Event Horizons in the Horizon Taxonomy, which compares the information used by all six notions. Continue to Killing Horizons to separate symmetry from global causal status. Then read Nonexpanding and Isolated Horizons to see how equilibrium can be defined on a null boundary without imposing a stationary exterior. Finish with Dynamical Horizons and Cross-Definition Maps for the conditional implications and open mapping questions.

For a computational and numerical route, begin with Apparent Horizons and Marginally Trapped Surfaces. It derives the marginally outer trapped surface (MOTS) equation from slice data and makes slicing dependence explicit. Continue to Trapping Horizons and Marginal Tubes for outer/inner signs, causal signature, and area change. The cross-definition page then explains branch bifurcations, apparent-horizon jumps, and the boundary between established results and generic merger questions.

The later Slicing, Foliation, and Quasi-Local Ambiguity reuses the necessary theorems on a different axis: this chapter asks which horizon definition applies, while that page asks which surface, slice, or tube is held fixed in a quantitative comparison.

Both routes use Schwarzschild as the equilibrium coincidence test and ingoing Vaidya as the dynamical separation test. A reader who can explain both examples has the essential taxonomy.

Schwarzschild coincidence and Vaidya separation

On a regular round cross-section of the future Schwarzschild branch $\mathcal H^+$ away from the bifurcation sphere, the asymptotically normalized stationary Killing field is null and normal. With a compatible spherically symmetric horizon-penetrating slice whose outermost MOTS is that section, $\theta_{(\ell)}=0$ and $\theta_{(n)}\lt0$, while the null intrinsic geometry is time independent. Event, Killing, apparent, future outer trapping, nonexpanding, and isolated notions then agree at $r=2GM$.

That agreement depends on the stated branch, section, slicing, equilibrium, regularity, and normalization. At the bifurcation sphere on the time-reflection-symmetric bridge slice, both future expansions vanish. The sphere remains the intersection of the event/Killing branches and a MOTS, but it does not satisfy the strict future condition $\theta_{(n)}\lt0$.

For ingoing Vaidya,

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

round spheres have

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

The marginal tube is $r=2Gm(v)$. For $dm/dv\gt0$ it is spacelike and grows; its location is determined quasi-locally. The event horizon is instead a null surface found from outgoing rays with a future boundary condition, and it generally lies outside the marginal tube during accretion. Ashtekar and Krishnan work through this separation in Ashtekar and Krishnan 2004, §2.2.2, Figure 4 and surrounding discussion. The equilibrium limit makes the surfaces coincide again without erasing the difference between their definitions.

Distinctions to retain

Null does not mean isolated. An event horizon is null even while its area and intrinsic geometry evolve. Isolation requires vanishing expansion and time-independent connection data.

Marginal does not mean apparent. A MOTS is one two-surface with vanishing outgoing expansion. An apparent horizon additionally selects the outer boundary of a trapped region on a specified slice.

Killing does not mean black hole. Minkowski’s Rindler horizons are Killing horizons for boost symmetry but not event horizons of a black-hole region.

Quasi-local does not mean pointwise. Expansions and stability depend on an extended surface. No scalar measured at one event decides that a MOTS exists.

Area growth does not identify the boundary. A dynamical horizon can have a local positive-flux law, while an event horizon can grow before the flux arrives. The areas answer different questions.

Terminology can evolve. The original dynamical horizon is spacelike. A recent formulation uses “dynamical horizon segment” for a broader class that can also be timelike and includes an explicit nondegeneracy condition. The 2025 definition and its comparison with the older framework are given in Ashtekar and Krishnan 2025, §2.1, Definitions 1–4; a theorem must retain the definition under which it was proved.

Review the chapter

Consider an accreting, asymptotically flat black hole evolved on a chosen Cauchy slicing. A code finds an outermost MOTS on every slice, and the resulting tube is spacelike. Answer three questions: What may the code call each leaf? What extra condition makes the tube a future outer trapping horizon? Can the code conclude, before completing the evolution, that the tube is the event horizon?

Solution

Each leaf is a marginally outer trapped surface and, because it is outermost on the specified slice, is the numerical apparent horizon on that slice. The spacelike tube with $\theta_{(\ell)}=0$ and $\theta_{(n)}\lt0$ is an original dynamical horizon. To call it future outer, one must additionally verify the inward stability sign $\mathcal L_n\theta_{(\ell)}\lt0$ for the stated null orientation. The code cannot infer event-horizon identity from these facts: the event horizon is the boundary of the causal past of the chosen future infinity and requires global future information. Late-time coincidence in a stationary remnant is a check, not a proof that the two tubes agreed during accretion.

After this chapter, continue to Equilibrium Black Holes and Horizon Anatomy to apply the definitions to the principal stationary families, or proceed later to Quasi-Local and Isolated-Horizon Mechanics for charges, multipoles, and first laws on finite boundaries.

References

• Ashtekar, Abhay, and Badri Krishnan. “Isolated and Dynamical Horizons and Their Applications.” Living Reviews in Relativity 7 (2004), article 10. DOI.
• Ashtekar, Abhay, and Badri Krishnan. “Quasi-Local Black Hole Horizons: Recent Advances.” Living Reviews in Relativity 28 (2025), article 8. DOI.
• Booth, Ivan. “Black-Hole Boundaries.” Canadian Journal of Physics 83 (2005): 1073–1099. arXiv:gr-qc/0508107.
• Gourgoulhon, Éric, and José Luis Jaramillo. “A 3+1 Perspective on Null Hypersurfaces and Isolated Horizons.” Physics Reports 423 (2006): 159–294. DOI.