Dynamical and Quasi-Local de Sitter Horizons

Without stationarity, an event horizon, a marginally trapped tube, an isolated horizon, and a charge at de Sitter infinity answer different questions. They can coincide for a stationary horizon, whereas boundary charges become comparable only after their asymptotic component and normalization have been matched. The horizon structures generally separate during evolution. For $\Lambda\gt0$, cosmological horizons are observer-dependent, conformal infinity is spacelike, and a slice can locate marginal surfaces without the entire future. Evidence cutoff: 2026-08-12.

Helpful background. Exact Solutions of Einstein’s Equations supplies stationary and Vaidya–de Sitter benchmarks; Causality and Global Structure supplies observer-dependent event horizons and spacelike conformal infinity; The Einstein Equation supplies positive-$\Lambda$ evolution; and Black-Hole Geometry and Mechanics supplies null expansions and horizon laws.

Four boundary structures and their data

Let $S$ be a closed spacelike two-surface with future null normals $\ell^a$ and $n^a$, normalized by $\ell\mathbin{\cdot}n=-1$. Its null expansions are

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

A future marginally outer trapped surface has $\theta_{(\ell)}=0$ and $\theta_{(n)}\lt0$. A future outer trapping horizon adds an outer condition such as $\mathcal L_n\theta_{(\ell)}\lt0$, after the directions have been declared. A marginally trapped tube is a three-surface foliated by such leaves. The outermost marginal surface on one Cauchy slice is an apparent horizon and can change when the slicing changes.

An isolated horizon is instead a null quasi-local boundary. A nonexpanding horizon has vanishing expansion and, under the field equations and suitable energy condition, vanishing shear. Weakly isolated and isolated structures add invariance conditions on the intrinsic connection, providing zeroth and first laws without requiring a Killing field throughout the exterior. A dynamical horizon is spacelike and foliated by marginal surfaces; its area change is tied to local fluxes.

An event horizon is global: it is a boundary of the causal past of a selected future endpoint or asymptotic region. In de Sitter, different observer worldlines or portions of the spacelike conformal boundary can define different cosmological event horizons. Finally, a conformal-boundary charge is not a horizon at all; it measures asymptotic data on a cut of spacelike $\mathcal I$ under additional symmetry and boundary assumptions.

Definitions and results are reviewed in Ashtekar and Krishnan 2025, §§2.1–4 and §7 and Booth 2005, §§2–5.

Comparison by formulation

A comparison must fix dimension, gravitational and matter theory, asymptotics, regularity, initial or characteristic data class, boundary conditions, observable, and whether the result is exact, perturbative, or numerical.

Structure	What fixes it	Established use	Dependence or open boundary
Observer event horizon	Entire causal future and chosen observer or asymptotic endpoint	Exact causal boundary once the completed spacetime is fixed	Teleological; changes with observer domain and future completion
Apparent or trapping horizon	Null expansions on a foliation and an outer/inner sign choice	Local existence and tracking under stability hypotheses; numerical horizon finding	Depends on slicing, leaf selection, and null orientation
Dynamical horizon	A spacelike marginally trapped tube with flux data	Exact area and energy-balance laws in its stated theory	Multiple tubes or signature changes can occur; energy normalization is extra data
Isolated horizon	Null boundary with time-independent intrinsic horizon geometry	Quasi-local equilibrium mechanics and charges	Does not select a unique exterior or guarantee global stationarity
Conformal-boundary charge	Asymptotic fields on a cut of spacelike de Sitter infinity	Charge and balance laws under specified conformal and symmetry conditions	Boundary restrictions and conformal frame affect the available symmetry and flux content

These formulations compete only when asked to represent one evolving boundary. Event horizons answer causal-exclusion questions in a completed spacetime; stable marginal tubes support local flux accounting and live numerical evolution; isolated horizons describe quasi-local equilibrium. Their different roles do not make the alternatives incorrect.

Positive $\Lambda$ does not remove the outer/inner labels: black-hole and cosmological marginal horizons can satisfy different stability signs. Reversing which side is outward changes the written classification, so every area law must state the component and null-normal convention.

Vaidya–de Sitter separates local and global horizons

Consider the ingoing spherical metric

$$ds^2=-f(v,r)\,dv^2+2\,dv\,dr+r^2d\Omega_2^2,$$ $$f(v,r)=1-\frac{2GM(v)}r-\frac{\Lambda r^2}{3}.$$

Choose future null normals

$$\ell^a\partial_a=\partial_v+\frac f2\partial_r, \qquad n^a\partial_a=-\partial_r, \qquad \ell\cdot n=-1.$$

Because a radial null normal changes the area $4\pi r^2$ at twice its fractional rate of change of $r$, their expansions are

$$\theta_{(\ell)}=\frac fr, \qquad \theta_{(n)}=-\frac2r.$$

The marginal radii are therefore the instantaneous positive roots of $f(v,r)=0$. They respond locally when $M(v)$ changes. A radial outgoing null curve instead obeys

$$\frac{dr}{dv}=\frac12 f(v,r).$$

The event-horizon solution of this equation is selected by a condition in the future. If additional influx is scheduled at a later advanced time, its location can already differ from the instantaneous marginal root before that flux arrives. This is not a numerical error: it is the event horizon’s global definition.

This exact spherical calculation does not prefer a foliation in a generic spacetime. It checks identification claims: stationarity makes the standard horizons coincide, while $dM/dv\ne0$ separates global and local constructions.

Flux laws and boundary charges

Along a suitable dynamical horizon, the change of a quasi-local energy between two leaves equals matter flux plus a nonnegative gravitational contribution built from shear and normal-connection data. With $\Lambda\gt0$, one must state whether the cosmological term is geometric or an effective stress tensor and translate the energy and volume term consistently.

Isolated-horizon mechanics takes a different limit: the horizon is null and its flux vanishes, while Hamiltonian integrability yields a first law. Neither construction needs conformal infinity. The distinctions and corresponding laws appear in Ashtekar and Krishnan 2025, §§2.3, 3.1–3.4, and 4.

At spacelike de Sitter infinity, the generic asymptotic symmetry group does not by itself provide a preferred time translation. Imposing conformal flatness of the boundary geometry reduces the symmetry but can also eliminate gravitational flux. Ashtekar, Bonga, and Kesavan 2015, §§II, IV, and V.3 derives the charge balance law and explains that tradeoff. A horizon flux and an asymptotic charge can be compared only after an evolution region, normalization, and boundary conditions connect them.

Comparing horizons in dynamical de Sitter spacetimes

At the evidence cutoff, there is no formulation-independent rule that selects one boundary for every dynamical positive-$\Lambda$ problem while preserving causal meaning, slicing independence, local flux laws, and operational computability. That is an unresolved synthesis problem, not a dispute over the validity of the individual definitions in their domains. Existing theorems usually fix a stability class, foliation, energy condition, asymptotic structure, or perturbative regime; numerical examples fix gauge, initial data, and horizon-finding choices.

A formulation-independent comparison or theorem would need a fixed asymptotic or observer structure, together with reproducible dynamical examples. A new numerical horizon track is not enough unless the slicing, convergence, marginal-surface stability, and comparison with a globally reconstructed event horizon are all documented.

Diagnose a simulation claim

A positive-$\Lambda$ simulation finds the outermost surface with $\theta_{(\ell)}=0$ on one time slice and reports that it has located the event horizon and measured the total radiated energy. What has actually been found, and what additional information is needed for each claim?

Solution

On that slice, the algorithm has found an apparent-horizon candidate: an outermost marginal surface relative to the chosen foliation and null orientation. To establish a trapping or dynamical horizon, one must follow the leaves through time, check the outer/stability condition and signature, and verify the relevant flux law. A different slicing may produce a different outermost marginal surface.

An event horizon requires the completed future and a declared observer or asymptotic endpoint. It must be reconstructed globally, for example by tracing the appropriate null boundary backward after the evolution is known. Total radiated energy additionally requires a charge and flux prescription at a specified boundary, its asymptotic conditions, conformal frame or patch generator, and normalization. One marginal surface alone supplies none of those global energy data.

Black Holes in Anti-de Sitter Spacetimes next changes the sign of $\Lambda$ and replaces spacelike de Sitter infinity by a timelike boundary with explicit boundary conditions.

References

• Ashtekar, Abhay, and Badri Krishnan. “Quasi-Local Black Hole Horizons: Recent Advances.” Living Reviews in Relativity 28 (2025): 8. DOI. Open manuscript.
• Ashtekar, Abhay, Béatrice Bonga, and Aruna Kesavan. “Asymptotics with a Positive Cosmological Constant: I. Basic Framework.” Classical and Quantum Gravity 32 (2015): 025004. DOI. Open manuscript.
• Booth, Ivan. “Black-Hole Boundaries.” Canadian Journal of Physics 83 (2005): 1073–1099. DOI. Open manuscript.