Dynamical Horizons and Cross-Definition Maps

A dynamical horizon in the original classical framework is a spacelike tube foliated by future marginal surfaces. It represents local black-hole growth and supports flux balance laws, but it is neither an event horizon nor a universal replacement for one. The mapping among horizon definitions depends on stationarity, energy conditions, stability, asymptotics, and slicing; in mergers it can also depend on which connected marginal branch is followed.

Helpful background. Exact Solutions of Einstein’s Equations supplies equilibrium and collapse checks; Causality and Global Structure supplies the event-horizon definition; The Einstein Equation connects flux to geometry; Timelike Congruences and Spatial Projection supplies optical scalars; Hypersurface Normals and Tangential Projectors supplies foliation geometry; and Black-Hole Geometry and Mechanics supplies the later balance laws.

Spacelike dynamical horizons

Let $H$ be a smooth spacelike three-manifold foliated by closed spacelike two-surfaces $S_v$. With future outgoing and ingoing null normals $\ell^a$ and $n^a$ normalized by $\ell\!\cdot n=-1$, the original dynamical-horizon conditions are

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

The vector tangent to $H$ and normal to its leaves can be written $h^a=\ell^a-Cn^a$. Since

$$h^ah_a=2C,$$

spacelike signature is equivalent to $C\gt0$. Its area deformation is

$$\mathcal L_h(dA) =\left(\theta_{(\ell)}-C\theta_{(n)}\right)dA =-C\theta_{(n)}\,dA\gt0.$$

Area growth follows from the defining signs; the field equations enter when that growth is related quantitatively to matter and gravitational fluxes. The original definition, area result, and Vaidya example are given in Ashtekar and Krishnan 2004, §§2.2–3.

A dynamical horizon is quasi-local in spacetime: recognizing a leaf uses nearby geometry rather than the full future. It is not local to a single point, because expansion is a property of an extended two-surface and the world tube requires a family of such surfaces. It is also not automatically the apparent horizon. A leaf may be a marginally outer trapped surface (MOTS) without being outermost on the chosen Cauchy slice; apparent-horizon selection can jump between branches even when the union of marginal tubes remains continuous.

A conditional cross-definition map

The useful statements are implications with hypotheses, not a ladder of synonyms.

Starting structure	Additional hypotheses	Consequence	What is not implied
Stationary black-hole event horizon	Suitable rigidity assumptions, regularity, and normalized stationary symmetries	A Killing-horizon structure on the relevant component	No conclusion for generic dynamical event horizons
Regular Killing horizon	For nonexpansion, none beyond a regular symmetry-adapted section; for an isolated-horizon structure, the required topology, field/matter conditions, and compatible null-normal class	The Killing flow gives zero expansion and shear; the added boundary data can make the horizon isolated	It need not be a black-hole event horizon; Rindler is the counterexample
Isolated horizon	A separate global causal identification	It may coincide with an event-horizon portion	Isolation alone does not determine escape to infinity
Smooth closed MOTS $S$ on one leaf $\Sigma_0$ of a prescribed smooth spacelike foliation $\{\Sigma_t\}$	$S$ is strictly stably outermost, equivalently its stability operator has principal eigenvalue $\lambda_0\gt0$	A local smooth marginally outer trapped tube $H=\bigcup_t S_t$ exists with $S_t\subset\Sigma_t$, for as long as strict stability persists	No preferred tube across other spacetime foliations; no automatic dynamical-horizon or future-outer-trapping-horizon status
Future outer trapping horizon (FOTH)	Null energy condition and nondegenerate outer derivative	Spacelike or null signature, with nondecreasing area	It need not coincide with the event horizon
Original dynamical horizon	Negative inward variation of outgoing expansion	A spacelike future outer trapping horizon	It need not be outermost on every Cauchy slice

In equilibrium Schwarzschild, a round section of the future horizon away from the bifurcation sphere, together with a compatible spherically symmetric horizon-penetrating slice, can make the event, Killing, nonexpanding, isolated, future outer trapping, and apparent horizons coincide at $r=2GM$. The time-symmetric bifurcation sphere is different: $\theta_{(\ell)}=\theta_{(n)}=0$, so it fails the strict future condition even though it is a MOTS and the intersection of the Killing/event-horizon branches. In accreting Vaidya, the event horizon is null and selected by a future condition, while the spherical tube $r=2Gm(v)$ is spacelike for $dm/dv\gt0$ and is selected by $\theta_{(\ell)}=0$. The two approach each other. If accretion ends at finite advanced time, they eventually coincide; if the influx merely decays, coincidence may be only asymptotic. These examples distinguish genuine implications from slicing- and section-dependent coincidence.

What is lost under substitution

Replacing one definition with another changes the question that can be answered.

• Replacing the event horizon by an apparent or dynamical horizon loses the exact statement about causal escape to the chosen infinity.
• Replacing a dynamical horizon by the event horizon loses local flux and instantaneous growth information; the event horizon can expand in a vacuum region because of future infall.
• Replacing an isolated horizon by a Killing horizon adds an unnecessary demand for spacetime symmetry near the boundary and can exclude a black hole in equilibrium inside a dynamical exterior.
• Replacing a marginal tube by the apparent horizon discards inner branches and can turn a change of outermost component into an apparent discontinuity.
• Replacing a MOTS stability analysis by the sign $\mathcal L_n\theta_{(\ell)}$ can discard nonspherical deformation modes; the full stability operator is elliptic and its principal eigenvalue carries stronger information.

The last point matters in mergers. When a common marginal surface first appears, an outer stable and an inner unstable branch can emerge together. Tracking only the outermost surface makes the apparent horizon “jump,” while tracking all marginal surfaces reveals the branch structure. Current analytic and numerical results for such transitions are summarized in Ashtekar and Krishnan 2025, §4, especially §§4.1–4.2.

Three different existence and uniqueness questions

On one prescribed smooth foliation by spacelike Cauchy surfaces, a smooth closed MOTS with principal stability eigenvalue $\lambda_0\gt0$ continues to nearby slices while strict stability persists. This is a local continuation theorem, not a choice of tube across different spacetime slicings Andersson, Mars, and Simon 2005, Lemma 2 and Theorems 1–2, pp. 2–3, Open PDF.

A second theorem starts after a tube has been chosen: one fixed smooth spacelike dynamical horizon with smooth closed connected marginal leaves has a unique marginal foliation Ashtekar and Galloway 2005, Corollary 3.2, pp. 9–10, Open PDF. It does not require an energy condition or asymptotic flatness, and it does not say that the spacetime contains only one dynamical horizon. Thus local continuation on one ambient foliation, uniqueness of the leaves on one selected tube, and unique tube selection are three different questions. Slicing, Foliation, and Quasi-Local Ambiguity develops the stability operator, fixed-object comparison, and numerical protocol in detail.

Evolving terminology and unsettled maps

Evidence cutoff: 2026-08-31. The terminology itself is a formulation choice. The 2004 framework reserves “dynamical horizon” for a spacelike tube with $\theta_{(\ell)}=0$ and $\theta_{(n)}\lt0$. The 2025 review introduces “dynamical horizon segments” for connected, nowhere-null portions of a quasi-local horizon, allowing spacelike or timelike signature, requiring the other expansion to be nonzero, and excluding certain zero-flux degeneracies. The precise modern definitions and their relation to the older usage appear in Ashtekar and Krishnan 2025, §2.1, Definitions 1–4.

In the four-dimensional, axisymmetric balance-law formulation of Ashtekar, Paraizo, and Shu, the ambient quasi-local horizon has topology $S^2\times\mathbb R$; a dynamical-horizon segment is nowhere null, the other null expansion is nowhere zero, and a leafwise flux-genericity scalar is not identically zero on any marginal leaf. The published derivation treats spacelike segments; a timelike extension is stated with sign changes Ashtekar, Paraizo, and Shu 2026, pp. 2–4, Open PDF.

Neither language should be silently inserted into a theorem proved in the other. Timelike segments can occur for some classical matter data and are especially relevant to semiclassical evaporation models, whereas the original spacelike definition was tailored to classical black-hole growth. The matter model, energy condition, regularity, boundary condition, and data class must therefore accompany any claimed signature rule.

Several parts of the cross-definition map are not settled generically. A fixed smooth spacelike dynamical horizon has a unique foliation by smooth closed connected marginal leaves, but a spacetime can contain more than one marginal tube. A chosen Cauchy slicing may reveal different outermost MOTSs. Near binary mergers, detailed studies show intricate inner branches and continuous quasi-local structures, but their qualitative persistence for generic spinning and precessing data is an open question. The 2025 review explicitly identifies the extension beyond head-on collisions, including stability spectra, as unresolved Ashtekar and Krishnan 2025, §4.2, Open Issue 7 and surrounding discussion.

What would change this assessment is concrete: a theorem controlling existence, uniqueness, signature changes, and branch transitions for a broad regular initial-data class, or a reproducible multi-code merger comparison that separates slicing-invariant features from gauge and horizon-finder choices. Until then, no universal one-to-one map among event, apparent, trapping, and dynamical horizons is justified.

Theorems and numerical examples

The entries below answer different questions and therefore cannot be pooled into a single uniqueness claim.

Result	Evidence class and exact domain	What it establishes	Boundary of the claim
Wald–Iyer slicing construction	Exact analytic counterexample in maximally extended vacuum Schwarzschild	Smooth Cauchy slicings can penetrate the black-hole region while the covered past contains no outer trapped surface	The event horizon is unchanged; the result is not a proof that every slicing lacks marginal surfaces
Andersson–Mars–Simon continuation	Local theorem in a smooth four-dimensional spacetime with a prescribed smooth foliation by spacelike Cauchy surfaces	A smooth closed MOTS with $\lambda_0\gt0$ continues locally to MOTSs on that same foliation, while strict stability persists	It does not select a tube across other foliations or continue a branch through $\lambda_0=0$
Ashtekar–Galloway foliation uniqueness	Theorem for one fixed smooth spacelike dynamical horizon with smooth closed connected marginal leaves	The marginal foliation of that selected tube is unique	It does not prove uniqueness of the tube in the spacetime and does not transfer to a null isolated horizon
Pook-Kolb–Hennigar–Booth merger branches	Resolution-tested numerics for one head-on, nonspinning, axisymmetric vacuum Brill–Lindquist merger family	Multiple marginal branches, folds, bifurcation or annihilation connections, and stability-eigenvalue crossings occur in the computed family	Some curves terminate numerically and the proposed $S_1$–$S_1^\ast$ annihilation remains unresolved; no universality across spinning, precessing, matter-coupled, gauge-varied, or multi-code families
Generic merger map	Open problem identified in the 2025 review	No theorem currently supplies foliation-independent branch selection and transition control for a broad regular data class	Open Issue 7 explicitly includes extensions beyond head-on collisions and their stability spectra

The later Slicing, Foliation, and Quasi-Local Ambiguity page reorganizes these results by the geometric object held fixed. That is an operational comparison, whereas this matrix classifies the evidence behind the implication map.

The exact Schwarzschild construction is in Wald and Iyer 1991, pp. R3719–R3722. The merger evidence, including its unresolved endpoints, is in Pook-Kolb, Hennigar, and Booth 2021, Figs. 2–5, pp. 3–5, Open PDF.

A formulation-choice diagnostic

A simulation must report the black-hole area during the run and later decide whether any emitted signal escaped to infinity. Which horizon should be used for each task, and what metadata must accompany the area curve?

Solution

During the run, use a quasi-local marginal structure—normally the outermost MOTS on each specified Cauchy slice—and track its connected marginal tube when possible. Each reported leaf is slice-relative; calling the tube a dynamical horizon or FOTH requires the additional expansion, signature, or outer-variation conditions in the definition. This supplies an operational area curve, not an exact escape boundary. After the future development and asymptotics are controlled, reconstruct the event horizon to answer whether a signal escaped. The later slicing and ambiguity page gives the branch, stability, gauge, finder, and convergence metadata needed for a quantitative simulation comparison.

For a later operational comparison of fixed surfaces, slices, and tubes, use Slicing, Foliation, and Quasi-Local Ambiguity. Continue now to Equilibrium Black Holes and Horizon Anatomy to apply the taxonomy to Schwarzschild, Reissner–Nordström, Kerr, and Kerr–Newman.

References

• Andersson, Lars, Marc Mars, and Walter Simon. “Local Existence of Dynamical and Trapping Horizons.” Physical Review Letters 95 (2005), 111102. DOI. Open PDF.
• 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.
• Ashtekar, Abhay, and Gregory J. Galloway. “Some Uniqueness Results for Dynamical Horizons.” Advances in Theoretical and Mathematical Physics 9 (2005): 1–30. DOI. Open PDF.
• Ashtekar, Abhay, Daniel E. Paraizo, and Jonathan Shu. “Thermodynamics of Black Holes, Far from Equilibrium.” Physical Review Letters 136 (2026), 251405. DOI. Open PDF.
• Pook-Kolb, Daniel, Robie A. Hennigar, and Ivan Booth. “What Happens to Apparent Horizons in a Binary Black Hole Merger?” Physical Review Letters 127 (2021), 181101. DOI. Open PDF.
• Wald, Robert M., and Vivek Iyer. “Trapped Surfaces in the Schwarzschild Geometry and Cosmic Censorship.” Physical Review D 44 (1991): R3719–R3722. DOI.