Event Horizons, Apparent Horizons, and Trapping Boundaries

Event, apparent, trapping, and dynamical horizons answer different questions. The event horizon identifies the global black-hole boundary after the complete future is known. The others locate marginal surfaces or tubes quasi-locally, but depend on a slicing, an “outward” direction, or additional signature and stability conditions. Agreement in a settled stationary black hole does not make the definitions interchangeable during collapse.

Required background. Hypersurface normals and tangential projectors supplies the causal and extrinsic geometry used by every definition.

Helpful background. Black-Hole Geometry and Mechanics develops causal boundaries and quasi-local mechanics; Black-Hole Perturbation Theory describes near-equilibrium deformations; Formulations of General Relativity supplies spacelike foliations; and Computational Relativity explains surface finding in evolved data.

Event, apparent, and trapping horizons

In an asymptotically flat spacetime with a suitable future null infinity $\mathscr I^+$, the future event horizon is

$$\mathcal H^+=\partial J^-(\mathscr I^+).$$

It is a null, slicing-independent causal boundary, but locating it requires knowing which events can eventually signal infinity. It may pass through a region of weak local curvature and can begin growing before infalling matter reaches it.

On a chosen spacelike slice $\Sigma_t$, define the trapped region as the union of regions bounded by outer trapped surfaces. Its boundary, when sufficiently regular, is the apparent horizon of that slice and is a marginally outer trapped surface. Numerical practice often calls the outermost MOTS “the apparent horizon,” but outermost among which surface topology and which asymptotic end must be declared.

A future outer trapping horizon is a hypersurface foliated by closed two-surfaces satisfying

$$\theta_{(\ell)}=0,\qquad \theta_{(k)}\lt 0,\qquad \delta_k\theta_{(\ell)}\lt0,$$

for declared future outward $\ell^a$ and future inward $k^a$, in a normalization for which the displayed outer condition holds. Hayward’s future/past and outer/inner classification defines the outer condition by this transverse derivative and derives its immediate trapping interpretation Hayward 1994, § III, pp. 6469–6470. A dynamical horizon is a smooth spacelike hypersurface foliated by surfaces with the first two properties; its spacelike signature is part of the definition. A marginally trapped tube imposes the expansion conditions but need not have the outer condition or a fixed signature. Ashtekar and Krishnan compare these notions in Ashtekar and Krishnan 2004, §§2.1–2.3, pp. 9–20, PDF.

The expansion signs are unchanged by angle-dependent positive boosts, but the transverse surface variation contains derivatives of the boost. Its pointwise sign in one arbitrary scaling is not itself the invariant stability test. The existence of a suitable outer deformation, or an equivalent positive-principal-eigenvalue condition, is the robust statement; see Booth and Fairhurst 2007, §III, Eqs. (3.1)–(3.7), pp. 11–12, PDF.

Comparison in a dynamical spacetime

Object	Data needed to locate it	Invariance and dependence	Characteristic use
Event horizon	Complete future development and $\mathscr I^+$	Spacetime invariant, teleological	Global black-hole region and asymptotic area theorem
Apparent horizon	One spacelike slice and its trapped region	Foliation dependent	Quasi-local boundary on initial or evolved data
Future outer trapping horizon	Foliated tube, null expansions, transverse variation	Depends on the tube foliation and null scaling; strict outer stability is geometric	Local signature and area laws under energy conditions
Dynamical horizon	Spacelike marginal tube with negative ingoing expansion	Quasi-local; distinct tubes may occur, while a fixed dynamical horizon has a preferred marginal foliation	Positive flux and area balances

In stationary Schwarzschild, the standard cross-sections at $r=2GM$ lie on the event horizon, apparent horizon, and a null future outer trapping horizon. They are the zero-flux boundary of the dynamical-horizon framework, but the null hypersurface is not itself a dynamical horizon because spacelike signature is part of that definition. During ingoing positive-energy Vaidya collapse, the event horizon generally lies outside the marginal tube and anticipates later accretion, while the marginal tube is at $r=2Gm(v)$ and is spacelike when $m'(v)\gt 0$. After flux ceases, that dynamical portion becomes null and joins the stationary horizon.

The absence of a MOTS on one slice does not prove the absence of an event horizon. Slicings can avoid trapped surfaces while entering a black-hole region, and distorted slices can present multiple MOTSs. Conversely, finding a MOTS provides strong local information but does not reconstruct the global causal boundary without assumptions on the future.

Which horizon should answer which question?

Use an event horizon to ask whether signals reach infinity, to define the global black-hole region, or to perform a retrospective balance after a complete simulation. Thornburg explains backward event-horizon tracking and its need for late-time data in Thornburg 2007, §2, pp. 11–24, PDF.

Use an apparent horizon or outermost MOTS to locate a black-hole interior on a current slice, guide excision, or provide a prompt quasi-local diagnostic. Use a smooth trapping or dynamical horizon when its leaves and signature have been verified and a local flux or area law is required. The horizon label does not excuse checking stability, branch identity, topology, and orientation.

Three comparisons are especially dangerous. First, coordinate radii of different horizons have no invariant meaning; compare areas, intrinsic multipoles, or causal relations. Second, an apparent-horizon “jump” can be a change of outermost branch rather than a discontinuous world tube. Third, a null segment of a marginal tube is not automatically an event horizon: nullness and $\theta_{(\ell)}=0$ are local conditions, whereas $\partial J^-(\mathscr I^+)$ is global.

Check your understanding

A simulation stops shortly after a MOTS forms. Which statements are justified: (a) a quasi-local black-hole boundary has been found on that slice; (b) the event horizon first formed at the same coordinate time; (c) a later complete evolution may locate the event horizon outside that MOTS?

Solution

Statement (a) is justified if the surface is verified as the relevant outermost MOTS and its orientation is declared. Statement (b) is not: event-horizon location is teleological and coordinate time is slicing dependent. Statement (c) is possible and typical in accreting collapse; retrospective null-surface tracking uses the later spacetime to locate it.

References

• Ashtekar, Abhay, and Badri Krishnan. “Isolated and Dynamical Horizons and Their Applications.” Living Reviews in Relativity 7 (2004): 10. doi:10.12942/lrr-2004-10.
• Booth, Ivan, and Stephen Fairhurst. “Isolated, Slowly Evolving, and Dynamical Trapping Horizons: Geometry and Mechanics from Surface Deformations.” Physical Review D 75 (2007): 084019. doi:10.1103/PhysRevD.75.084019.
• Hayward, Sean A. “General Laws of Black-Hole Dynamics.” Physical Review D 49 (1994): 6467–6474. doi:10.1103/PhysRevD.49.6467.
• Thornburg, Jonathan. “Event and Apparent Horizon Finders for 3+1 Numerical Relativity.” Living Reviews in Relativity 10 (2007): 3. doi:10.12942/lrr-2007-3.