---
title: Dynamical Horizons in General Relativity
url: https://www.emergentmind.com/topics/dynamical-horizon
type: topic
---

# Dynamical Horizons in General Relativity

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A dynamical horizon is a quasi-local black-hole boundary defined, in its standard Ashtekar–Krishnan form, as a spacelike 3-surface foliated by closed spacelike 2-surfaces whose outgoing null expansion vanishes and whose ingoing null expansion is negative, $\theta_{(\ell)}=0$ and $\theta_{(n)}<0$. In dynamical space-times this notion replaces the global event horizon as the practically relevant object, because the event horizon is a teleological construction requiring knowledge of the entire future geometry, whereas dynamical horizons and related apparent or trapping horizons can be located from local or quasi-local data on a given slice [1207.3673] [1511.07775]. The literature surrounding dynamical horizons also includes the closely related notions of marginally trapped tubes, future outer trapping horizons, isolated horizons, and cosmological dynamical horizons; their precise relation depends on signature, foliation, and physical context [1204.1530] [1001.0288].

## 1. Geometric definition and neighboring concepts

The standard black-hole definition is local and foliation based. A dynamical horizon is a spacelike hypersurface foliated by marginally trapped surfaces, usually 2-spheres in four dimensions, with $\Theta_{+}=0$ or $\theta_{(\ell)}=0$ and $\Theta_{-}<0$ or $\theta_{(n)}<0$ according to notation [2312.01136] [2506.10190]. In the closely related Hayward framework, a future outer trapping horizon adds an outer condition, written for example as $\ell_-(\theta_+)<0$ or $\delta_{\beta n}\theta_{(\ell)}<0$, thereby distinguishing an outer black-hole-type boundary from other marginal tubes [1204.1530] [1207.6955].

In spherical symmetry these conditions simplify substantially. The apparent horizon is located by
\[
\nabla^c R \nabla_c R = 0,
\]
with $R$ the areal radius, and this equation becomes the operational horizon-tracking condition in many exact cosmological and collapse models [1511.07775]. This is one reason spherical symmetry is so prominent in the dynamical-horizon literature: it permits invariant localization without appeal to asymptotic structure.

The same basic idea appears in cosmology, but with a different physical interpretation. In the flat FRW setting studied in loop quantum cosmology, a cosmological dynamical horizon is a three-dimensional hypersurface foliated by spheres such that one orthogonal null congruence has vanishing expansion. For the expanding branch, the horizon radius is
\[
r_H=\frac{1}{\dot a},
\qquad
A=4\pi a^2 r_H^2=4\pi H^{-2},
\]
so the horizon is tied directly to the Hubble flow rather than to a compact black-hole region [1001.0288].

A persistent source of confusion is the relation between apparent horizons and dynamical horizons. Apparent horizons are foliation dependent and can even be timelike in some dynamical solutions, while a dynamical horizon in the Ashtekar–Krishnan sense is specifically spacelike [1401.1189]. This suggests that “dynamical horizon” is not merely a synonym for “any nonstationary horizon,” but a more restrictive geometric structure within the broader quasi-local horizon taxonomy.

## 2. Signature, causal character, and local space-time structure

The causal character of a marginally trapped tube is central. In the horizon-adapted notation of the near-horizon expansion formalism, the evolution vector tangent to the horizon and normal to the leaves is
\[
\mathcal{V}^a=\ell^a-C\,n^a,
\]
with $C>0$ corresponding to a spacelike hypersurface and $C=0$ to the null isolated-horizon limit [1207.6955]. The area law then takes the local form
\[
\mathcal{L}_{\mathcal V}\epsilon = -C\,\theta_{(n)}\epsilon \ge 0,
\]
so spacelikeness and area increase are directly linked.

A complementary causal criterion is developed for spherically symmetric marginally trapped tubes in $D=n+2$ dimensions. There the invariant quantity
\[
\beta_k=-\left[(\pounds_k\Theta_k)(\pounds_l\Theta_k)\right]_{\Theta_k=0}
\]
determines signature: $\beta_k<0$ gives a spacelike tube, $\beta_k>0$ a timelike tube, and $\beta_k=0$ a null transition [2304.05609]. In that formulation the causal nature depends only on the local invariants $R,\Lambda,\rho,p,n$, which makes causal transitions diagnosable without solving the full global problem.

This distinction matters physically. In the local Hawking-radiation treatment of a future outer trapping horizon, the Einstein-equation relation
\[
\partial_+\theta_+ = -h\,\partial_-\theta_+ = -8\pi T_{++}
\]
implies that positive ingoing energy flux makes the horizon spacelike, while the negative energy flux associated with evaporation makes it timelike [1204.1530]. Thus a black-hole boundary can change character during evolution; a spacelike dynamical horizon need not remain spacelike indefinitely.

The local geometry near such horizons can be reconstructed systematically. In ingoing Gaussian null coordinates, the near-horizon metric admits an expansion in the inward affine parameter $\rho$, and for spacelike dynamical trapping horizons the intrinsic and extrinsic geometry of the horizon is sufficient to determine the nearby space-time to second order [1207.6955]. By contrast, for a null isolated horizon extra characteristic data are required. This establishes a sharp structural difference: a spacelike dynamical horizon behaves much more like an initial-value surface than a null equilibrium horizon.

## 3. Thermodynamics, surface gravity, and semiclassical behavior

Dynamical horizons support a thermodynamic description, but not by simply transplanting stationary formulas unchanged. In spherical symmetry the Kodama vector provides the preferred time flow when no timelike Killing vector exists, and from it one can define Kodama surface gravity and Kodama temperature as the natural thermodynamic quantities for dynamical horizons [1401.1189]. A detailed comparison of surface-gravity proposals shows that the Kodama–Hayward definition is causally sensitive: under reasonable energy conditions it is positive on spacelike dynamical horizons, negative on timelike tubes, and zero at null transitions, whereas Hayward trapping gravity becomes imaginary on timelike tubes and Fodor or Booth–Fairhurst definitions do not track causal transitions in the same clean way [2304.05609].

A central semiclassical result is that Hawking radiation can be derived locally from a dynamical horizon. For a future outer trapping horizon, mode analysis with respect to the Kodama vector yields the thermal factor
\[
P(\text{emission}|\text{incident}) = e^{-2\pi\omega/k},
\]
and therefore the local Hawking temperature
\[
T_H=\frac{k}{2\pi}.
\]
The associated flux law gives
\[
F = -\frac12 \Delta r,
\]
so the horizon radius shrinks as radiation is emitted [1204.1530]. The same general theme appears in the K-essence emergent Vaidya setting, where the Hawking temperature becomes
\[
T_H=\frac{1}{8\pi k_B}\frac{(1-\phi_v^2)^2}{M},
\]
and the resulting evaporation deviates from the standard Vaidya behavior [2007.16053].

The entropy side of the subject has both local and covariant versions. In Einstein–Gauss–Bonnet theory one can assign a local entropy density and spatial entropy current to a dynamical black-hole horizon, and although both depend nontrivially on reparametrizations of the null generators, the net entropy production given by the time derivative of the density plus the divergence of the spatial current is reparametrization covariant; this was checked explicitly in the small-amplitude regime [2204.08447]. In cosmology, a different question is whether a dynamical-horizon entropy bound survives strong-curvature evolution. For a flat FRW universe with photon gas, the conjectured bound
\[
\frac{S}{A(t')-A(t)}\leq \frac{1}{4l_p^2}
\]
fails near the big bang in classical Einstein cosmology but is restored in loop quantum cosmology by the modified Friedmann equation
\[
H^2=\frac{8\pi }3 \rho \left( 1-\frac \rho {\rho _c}\right),
\]
with the quantum geometry effects rendering the relevant monotonicity condition positive throughout the evolution [1001.0288].

A further subtlety is that horizon thermodynamic quantities are not invariant under common space-time mappings in the naive dimensional sense. Under conformal and Kerr–Schild transformations, the Misner–Sharp–Hernandez mass, Kodama vector, Kodama surface gravity, and apparent-horizon location transform nontrivially, so one must compute them explicitly rather than infer simple rescalings [1401.1189].

## 4. Evolution patterns, multiplicity, and numerical realizations

One of the most developed parts of the subject concerns how dynamical or apparent horizons evolve in exact solutions and simulations. For cosmological black holes that are not asymptotically flat, two generic apparent-horizon behaviors have been identified. In the fold-type or “C-curve” behavior, a black-hole horizon and a cosmological horizon appear or disappear in a pair. In the cusp-type or “S-curve” behavior, a single horizon is present initially, then a pair appears, one branch grows while another shrinks, and two later annihilate. A key result is that the C-curve is the $\omega_0\to+\infty$ General Relativity limit of the S-curve in the Clifton–Mota–Barrow Brans–Dicke family, with the lower bend of the S-curve pushed to $t_2\to+\infty$ and replaced by a finite-radius singularity [1511.07775].

Strong-field numerical relativity reveals even richer local structure. In simulations of a Kerr black hole perturbed by a pulse of ingoing gravitational radiation, up to five concentric marginally outer trapped surfaces can exist simultaneously. These surfaces appear and disappear in pairs, so the total number of marginally outer trapped surfaces at any given time is odd, and the corresponding marginally trapped tubes are spacelike during the highly dynamical regime while approaching a null hypersurface at early and late times [1011.2601]. This is a concrete demonstration that a “horizon” in dynamical general relativity may consist of several intertwined quasi-local branches rather than a single smooth world tube.

Binary black-hole merger simulations supply a different manifestation. After merger a common dynamical horizon forms, is initially highly distorted, and then settles toward a Kerr isolated horizon as the shear decays [2312.01136]. The shear of this common horizon is strongly correlated with the outgoing gravitational-wave news extracted far away, and after time-and-phase alignment the magnitudes satisfy the phenomenological fit
\[
\left|\frac{\sigma}{\mathcal{N}}\right| = 24.81 q^2 - 30.64 q + 10.62 .
\]
The shear is also well fit by fundamental quasi-normal modes and carries information about the remnant mass, remnant spin, and even the progenitor binary parameters [2312.01136]. This suggests a direct strong-field/weak-field linkage between horizon dynamics and radiation at infinity.

Rigorous mathematical results on collapse reinforce the same picture. In $3+1$ Einstein vacuum gravity with general anisotropic initial data, a smooth and spacelike apparent horizon emerges as a family of marginally outer trapped surfaces, thereby producing a dynamical horizon in the sense of Ashtekar–Krishnan/Ashtekar–Galloway [2010.12524]. The areas obey
\[
\lim_{u\to 0}\operatorname{Area}(M_u)=0, \qquad \operatorname{Area}(M_{u'})>\operatorname{Area}(M_u)\quad \text{for }u'>u,
\]
and the construction yields explicit finger-type single- and multi-valley anisotropic horizons [2010.12524]. This suggests that highly nonspherical dynamical horizons are not numerical artifacts but mathematically controlled outcomes of collapse.

## 5. Topology and generalized horizon frameworks

Although much of the literature is spherical, the concept is not restricted to spherical topology. An explicit anti-de Sitter Vaidya-type solution exhibits a spacelike dynamical horizon with toroidal cross-sections $T^2$, located by
\[
f(v,r)=0,
\qquad
r_{\mathcal H}(v)=\left(\frac{4m(v)}{\alpha^{3}}\right)^{1/3},
\]
with $\alpha^2=-\Lambda/3>0$ [1207.3673]. In that model the ordinary matter flux through the horizon is nonzero, the gravitational flux vanishes, and the total flux including the cosmological-constant sector vanishes. The example demonstrates that dynamical-horizon area balance does not require spherical cross-sections and can accommodate negative cosmological constant.

The subject also has analogues in Lorentz-violating theories. There the relevant causal boundary is the universal horizon rather than the null horizon of general relativity. In spherical symmetry a dynamical universal horizon is defined by
\[
u_\lambda \zeta^\lambda=0,
\]
with $\zeta^\mu$ the geometrically defined vector orthogonal to constant-areal-radius hypersurfaces, and this horizon lies inside the apparent horizon [1501.04134]. In collapsing-shell and finite-thickness-star models, universal horizons form dynamically and can block even infinitely fast preferred-frame signals, with the characteristic radius $r=3M/2$ in the thin-shell cuscuton analysis and $R_{UH}^{\text{Sch.}}=3M/2$ in the Einstein-aether collapse model [1310.4143] [1501.04134]. These constructions are not dynamical horizons in the Ashtekar–Krishnan sense, but they show how the quasi-local-horizon program generalizes once the causal structure of the theory itself changes.

The K-essence emergent Vaidya analysis provides yet another generalization. There the paper adopts Sawayama’s modified notion in which the horizon hypersurface may be spacelike or timelike while retaining the marginally trapped conditions $\Theta_{(l)}=0$ and $\Theta_{(n)}<0$, and the resulting horizon radius is modified by the scalar kinetic term and can be written using the Wright Omega function [2007.16053]. A plausible implication is that dynamical-horizon techniques are robust enough to survive substantial modifications of the effective geometry, although the precise thermodynamic interpretation becomes model dependent.

## 6. Extremality, symmetry constraints, and horizon transitions

Dynamical horizons are generically sub-extremal. A local extremality parameter
\[
e := \frac{1}{4\pi}\int_{S_v} d^2x\,\sqrt{\tilde q}\left(8\pi G\,T_{ab}\ell^a n^b + \|\tilde\omega\|^2\right)
\]
was introduced to generalize the Kerr notion of extremality to isolated and dynamical horizons, with $e<1$ sub-extremal, $e=1$ extremal, and $e>1$ super-extremal [0708.2209]. For generic dynamical horizons, the trapped-surface characterization implies
\[
e<1.
\]
This makes extremality a quasi-local geometric condition rather than merely a statement about global mass and angular momentum.

Space-time symmetries near a dynamical horizon are sharply restricted. Recent work on dynamical horizon segments shows that no timelike or causal Killing field can exist in a neighborhood containing a complete marginally trapped leaf, and that generically any surviving Killing field must be spacelike and rotational, leaving each marginally trapped 2-sphere invariant [2506.10190]. If such a rotational Killing field is hypersurface orthogonal, then all spin multipoles vanish on every leaf, so the entire spin structure is indistinguishable from that of a spherically symmetric dynamical horizon [2506.10190]. This suggests that strong dynamical area evolution is fundamentally incompatible with stationary symmetry.

Transitions between isolated-horizon phases can also carry horizon memory. A preferred slicing of an isolated horizon is selected by
\[
\mathcal D^A \omega_A = 0,
\]
and evolution through a dynamical horizon induces a supertranslation precisely when this condition is not preserved [2009.09270]. In the spherically symmetric examples studied there is no induced supertranslation for truly spherically symmetric collapse, while angular structure in the flux can source a nontrivial transition [2009.09270]. This places dynamical horizons in the broader context of horizon symmetry, soft hair, and memory-type effects.

Taken together, these results portray the dynamical horizon as a local geometric structure that unifies several themes: quasi-local black-hole boundaries, causal transitions of marginally trapped tubes, non-equilibrium black-hole thermodynamics, strong-field numerical relativity, and mathematically rigorous collapse theory. The concept is narrow enough to admit precise geometric theorems, yet flexible enough to interface with AdS topology, higher-derivative entropy currents, cosmological entropy bounds, and preferred-foliation analogues.

Source: https://www.emergentmind.com/topics/dynamical-horizon