Quasi-local Horizons in Black Hole Theory

• Quasi-local horizons are black-hole boundary concepts defined solely by local spacetime geometry, avoiding the need for teleological event horizons.
• They are identified using closed spacelike 2-surfaces and null expansion conditions, making them ideal for equilibrium and dynamical analyses in numerical relativity.
• The framework extends to alternative gravity theories through generalized charges, balance laws, and entropy measures, enhancing black-hole mechanics.

Quasi-local horizons are black-hole boundary concepts defined from local or quasi-local geometric data rather than from the global causal structure required for event horizons. In this framework, the basic objects are closed spacelike 2-surfaces, typically of spherical topology, equipped with future-directed null normals whose expansions determine whether the surface is trapped, marginally trapped, or untrapped. A central motivation is that event horizons are both global and teleological: they are defined only in spacetimes admitting a suitable future conformal boundary and their location depends on the entire future development of the spacetime, whereas quasi-local horizons can be identified from geometry in a finite spacetime region and are therefore central in numerical relativity, black-hole mechanics, and dynamical strong-field problems (Engle et al., 2011, Krishnan, 2013, Ashtekar, 2023).

1. Foundational geometry and motivation

The quasi-local program begins with a smooth closed spacelike 2-surface $S$ in spacetime, with induced metric $q_{ab}$, area form $\epsilon$, and two future-directed null normals, usually denoted $\ell^a$ and $n^a$, normalized by

$\ell \cdot n = -1.$

The null expansions are

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

A future trapped surface satisfies

$\Theta_{(\ell)}<0,\qquad \Theta_{(n)}<0,$

while a marginally future trapped surface satisfies

$\Theta_{(\ell)}=0,\qquad \Theta_{(n)}<0.$

A marginally outer trapped surface (MOTS) is defined by

$\Theta_{(\ell)}=0,$

without any further condition on $q_{ab}$0 (Krishnan, 2013).

The underlying critique of event horizons is twofold. First, an event horizon is defined through the black-hole region

$q_{ab}$1

so one must know the full future of spacetime to determine it. Second, the event horizon can respond to future infall before any local signal is present; the Vaidya spacetime provides the standard example in which the event horizon extends into a flat region prior to collapse (Engle et al., 2011, Krishnan, 2013, Ashtekar, 2023). This is why quasi-local horizons are used instead in fully nonlinear and numerically evolved spacetimes.

The quasi-local viewpoint also distinguishes between slice-dependent and spacetime notions. On a spatial hypersurface $q_{ab}$2, the outermost connected component of the boundary of trapped surfaces is an apparent horizon. By contrast, a marginally trapped tube (MTT) is a spacetime 3-surface foliated by MOTSs. This distinction is essential in dynamical settings because the trapped surfaces found on a given slice depend on the slicing, whereas a horizon world tube carries additional geometric structure (Krishnan, 2013, Ashtekar, 2023).

2. Equilibrium and non-equilibrium horizon frameworks

The equilibrium sector is described by isolated horizons. A non-expanding horizon (NEH) is a null hypersurface $q_{ab}$3 such that $q_{ab}$4 is topologically $q_{ab}$5, the expansion of any null normal vanishes,

$q_{ab}$6

the field equations hold at $q_{ab}$7, and the stress-energy tensor satisfies the causal energy condition that $q_{ab}$8 is future-directed and causal for any future-directed null normal $q_{ab}$9 (Engle et al., 2011). On a NEH, vanishing expansion together with the energy condition and the Raychaudhuri equation imply vanishing shear, so the intrinsic metric is time independent along the generators: $\epsilon$0

A weakly isolated horizon (WIH) is a NEH together with an equivalence class $\epsilon$1 of null normals, related by constant positive rescalings, such that

$\epsilon$2

where $\epsilon$3 is the induced normal connection defined by

$\epsilon$4

The associated surface gravity

$\epsilon$5

then satisfies

$\epsilon$6

which is the isolated-horizon version of the zeroth law (Engle et al., 2011). A strongly isolated horizon (SIH) adds

$\epsilon$7

for every vector field tangent to $\epsilon$8, freezing the entire intrinsic connection. The hierarchy is

$\epsilon$9

The non-equilibrium sector is described by dynamical horizons and related trapping-horizon notions. A dynamical horizon (DH) is a smooth spacelike 3-manifold $\ell^a$0 foliated by compact 2-surfaces $\ell^a$1 satisfying

$\ell^a$2

A trapping horizon is a 3-surface foliated by such marginal surfaces together with an outer condition, usually written as

$\ell^a$3

in the Vaidya analysis (Nielsen et al., 2010). Hayward’s terminology is also used in inhomogeneous cosmological models, where a trapping horizon is the closure of a hypersurface foliated by marginally trapped surfaces satisfying

$\ell^a$4

or the corresponding version with $\ell^a$5, depending on the future/past convention (Polášková et al., 2018).

The causal character distinguishes equilibrium from growth. Null quasi-local horizons correspond to equilibrium. Spacelike quasi-local horizons correspond to growth under infalling matter or gravitational radiation. Timelike marginally trapped tubes can occur, but they are not standard dynamical horizons in the Ashtekar–Krishnan sense (Nielsen et al., 2010, Krishnan, 2013, Ashtekar, 2023).

3. Mechanics: area, angular momentum, mass, and multipoles

A principal achievement of the quasi-local horizon program is that black-hole mechanics can be formulated without assuming global stationarity. For isolated horizons, the basic charges are the area

$\ell^a$6

the electric charge

$\ell^a$7

and the angular momentum. In the axisymmetric case, the gravitational-plus-electromagnetic horizon angular momentum is

$\ell^a$8

while the purely gravitational piece can also be written as

$\ell^a$9

A horizon energy $n^a$0 exists precisely when the Hamiltonian variation satisfies

$n^a$1

or, in thermodynamic notation,

$n^a$2

with

$n^a$3

In Einstein–Maxwell theory, the canonical horizon mass is fixed by the Kerr–Newman family and equals

$n^a$4

This provides a quasi-local generalization of the Kerr–Newman mass formula (Engle et al., 2011).

For dynamical horizons, the area increase law is local and flux-balanced. If $n^a$5 is the portion of a dynamical horizon between cuts $n^a$6 and $n^a$7, then

$n^a$8

where the two terms on the right are interpreted as matter flux and gravitational flux (Krishnan, 2013). In axisymmetry, the angular momentum of a cut $n^a$9 is

$\ell \cdot n = -1.$0

and finite-transition first-law-type balance relations can be written using the Kerr expressions $\ell \cdot n = -1.$1, $\ell \cdot n = -1.$2, and $\ell \cdot n = -1.$3 evaluated on each cut (Ashtekar, 2023).

The quasi-local framework also supports horizon multipoles. On axisymmetric isolated horizons, one defines

$\ell \cdot n = -1.$4

with $\ell \cdot n = -1.$5 encoding shape information and $\ell \cdot n = -1.$6 encoding spin information. On dynamical horizons, these become time-dependent multipoles on the leaves $\ell \cdot n = -1.$7, defined through a complex seed field built from the scalar curvature of the cut and the curl of the pulled-back rotation 1-form (Ashtekar, 2023). This suggests a hierarchy of horizon-balance laws far richer than the familiar zeroth, first, and second laws.

A significant refinement concerns non-axisymmetric horizons. For a generic MOTS $\ell \cdot n = -1.$8, the standard angular-momentum functional

$\ell \cdot n = -1.$9

still applies, but the challenge is to select the appropriate axial field $$\Theta_{(\ell)} = q^{ab}\nabla_a \ell_b,\qquad \Theta_{(n)} = q^{ab}\nabla_a n_b.$$0. A geometrically preferred choice can be extracted from the conformal decomposition of the intrinsic 2-metric and the Möbius group of the conformal sphere. In conformally spherical coordinates, the six conformal generators split into three rotations $$\Theta_{(\ell)} = q^{ab}\nabla_a \ell_b,\qquad \Theta_{(n)} = q^{ab}\nabla_a n_b.$$1 and three proper conformal generators $$\Theta_{(\ell)} = q^{ab}\nabla_a \ell_b,\qquad \Theta_{(n)} = q^{ab}\nabla_a n_b.$$2, and one defines charge triples

$$\Theta_{(\ell)} = q^{ab}\nabla_a \ell_b,\qquad \Theta_{(n)} = q^{ab}\nabla_a n_b.$$3

The Lorentzian mixing of $$\Theta_{(\ell)} = q^{ab}\nabla_a \ell_b,\qquad \Theta_{(n)} = q^{ab}\nabla_a n_b.$$4 and $$\Theta_{(\ell)} = q^{ab}\nabla_a \ell_b,\qquad \Theta_{(n)} = q^{ab}\nabla_a n_b.$$5 under proper conformal transformations leads to the invariants

$$\Theta_{(\ell)} = q^{ab}\nabla_a \ell_b,\qquad \Theta_{(n)} = q^{ab}\nabla_a n_b.$$6

and hence to an invariant quasi-local angular momentum

$$\Theta_{(\ell)} = q^{ab}\nabla_a \ell_b,\qquad \Theta_{(n)} = q^{ab}\nabla_a n_b.$$7

for all nondegenerate cases $$\Theta_{(\ell)} = q^{ab}\nabla_a \ell_b,\qquad \Theta_{(n)} = q^{ab}\nabla_a n_b.$$8. This agrees with the standard isolated-horizon or dynamical-horizon expression in axisymmetry and supplies a canonical spin for generic non-axisymmetric MOTSs (0707.2824).

4. Foliation dependence, numerical relativity, and quasi-local detection

A persistent issue is foliation dependence. Apparent horizons and more general MOTSs depend on the chosen spacelike slices, and therefore quasi-local quantities assigned to them can vary with the slicing. This is analyzed explicitly in Vaidya spacetime with metric

$$\Theta_{(\ell)} = q^{ab}\nabla_a \ell_b,\qquad \Theta_{(n)} = q^{ab}\nabla_a n_b.$$9

for which the radial null expansions are

$\Theta_{(\ell)}<0,\qquad \Theta_{(n)}<0,$0

The spherically symmetric MOTS lies at

$\Theta_{(\ell)}<0,\qquad \Theta_{(n)}<0,$1

but non-spherical slicings

$\Theta_{(\ell)}<0,\qquad \Theta_{(n)}<0,$2

produce distorted axisymmetric MOTSs whose location and area differ from the spherical one (Nielsen et al., 2010).

The dependence is real but, in the slowly evolving regime studied, small. For the linear mass function with $\Theta_{(\ell)}<0,\qquad \Theta_{(n)}<0,$3, the area variation across the family of slicings examined is about

$\Theta_{(\ell)}<0,\qquad \Theta_{(n)}<0,$4

This suggests that, while the ambiguity is conceptually fundamental, the induced change in area can be modest in near-equilibrium situations (Nielsen et al., 2010). The same paper emphasizes that even event-horizon cross-sectional areas vary with slicing, so foliation dependence is not unique to quasi-local horizons.

In practice, quasi-local horizons are indispensable in numerical relativity because they can be located slice by slice. In the Vaidya study, the numerical procedure is explicitly: rewrite the metric in $\Theta_{(\ell)}<0,\qquad \Theta_{(n)}<0,$5 form on a Cartesian grid, place each $\Theta_{(\ell)}<0,\qquad \Theta_{(n)}<0,$6 const slice on the grid, and use AHFinderDirect in the Cactus framework to solve $\Theta_{(\ell)}<0,\qquad \Theta_{(n)}<0,$7 for the axisymmetric MOTS (Nielsen et al., 2010). This operational role underlies the broader emphasis in black-hole merger studies: quasi-local horizons, not event horizons, are the objects that can be tracked during the evolution (Ashtekar, 2023).

The detection problem has also motivated invariant alternatives. One proposal defines geometric horizons as hypersurfaces on which the curvature tensor or its derivatives become more algebraically special than in the surrounding spacetime. In four dimensions, necessary type II/D conditions can be written as discriminant constraints on scalar polynomial curvature invariants, such as

$\Theta_{(\ell)}<0,\qquad \Theta_{(n)}<0,$8

$\Theta_{(\ell)}<0,\qquad \Theta_{(n)}<0,$9

for the Weyl tensor. This is intended as a foliation-independent quasi-local characterization, especially relevant for numerical relativity, though the fully dynamical program remains conjectural (Coley et al., 2017).

A different line of work uses boundary quasi-local mass to infer the presence of a horizon inside a compact domain. For an admissible initial data set $\Theta_{(\ell)}=0,\qquad \Theta_{(n)}<0.$0, with outer boundary $\Theta_{(\ell)}=0,\qquad \Theta_{(n)}<0.$1, a MOTS is defined by

$\Theta_{(\ell)}=0,\qquad \Theta_{(n)}<0.$2

The Liu–Yau mass

$\Theta_{(\ell)}=0,\qquad \Theta_{(n)}<0.$3

and Wang–Yau mass

$\Theta_{(\ell)}=0,\qquad \Theta_{(n)}<0.$4

then obey comparison theorems against Hawking masses of strictly minimizing hulls in Jang graphs. The resulting localized Penrose inequalities imply, for suitable interior surfaces $\Theta_{(\ell)}=0,\qquad \Theta_{(n)}<0.$5,

$\Theta_{(\ell)}=0,\qquad \Theta_{(n)}<0.$6

and provide sufficient conditions for the existence or nonexistence of a MOTS inside $\Theta_{(\ell)}=0,\qquad \Theta_{(n)}<0.$7 (Alaee et al., 2019).

5. Extensions beyond general relativity and alternative quasi-local horizon notions

The quasi-local horizon program extends beyond Einstein gravity in several distinct directions. In Einstein–Gauss–Bonnet gravity with an $\Theta_{(\ell)}=0,\qquad \Theta_{(n)}<0.$8-dimensional Einstein horizon space satisfying the Dotti–Gleiser Weyl condition

$\Theta_{(\ell)}=0,\qquad \Theta_{(n)}<0.$9

one can define a generalized Misner–Sharp mass

$\Theta_{(\ell)}=0,$0

which satisfies a unified first law and supports quasi-local trapping-horizon mechanics, including monotonicity, a signature law, an area law in the GR branch, and a dynamical entropy law (Maeda, 2010). This suggests that much of the trapping-horizon framework survives higher-curvature corrections, though branch structure and non-GR behavior introduce major caveats.

In scalar-tensor and $\Theta_{(\ell)}=0,$1 gravity, the relevant horizon quantity is not area but generalized entropy. For entropy 2-form

$\Theta_{(\ell)}=0,$2

with $\Theta_{(\ell)}=0,$3 in scalar-tensor theory and $\Theta_{(\ell)}=0,$4 in $\Theta_{(\ell)}=0,$5 gravity, the standard future outer trapping-horizon conditions

$\Theta_{(\ell)}=0,$6

are replaced by entropy-based conditions

$\Theta_{(\ell)}=0,$7

This modification is motivated by the fact that ordinary future outer trapping horizons are not conformally invariant, whereas the generalized entropy is the conformally meaningful quantity. The resulting quasi-local horizons obey an entropy increase law under the appropriate positivity conditions (Faraoni et al., 2011).

There are also quasi-local horizon notions adapted to theories with a preferred foliation. In such settings, the standard null event horizon is not the relevant causal barrier because arbitrarily fast signals may exist. The proposed quasilocal universal horizon is defined by the vanishing of the optical scalar

$\Theta_{(\ell)}=0,$8

where $\Theta_{(\ell)}=0,$9 is the preferred flow, $q_{ab}$00 is the preferred spatial unit normal orthogonal to the codimension-two foliation, and

$q_{ab}$01

In spherical symmetry this reproduces the usual universal-horizon condition, shows that such horizons can occur only in trapped or antitrapped regions, and implies that there are no universal analogues of cosmological horizons in FLRW models for any scale factor (Maciel, 2015).

A further extension is the quasi-local conformal Killing horizon. In its non-rotating form, a null inner boundary $q_{ab}$02 is required to have vanishing shear, nonzero expansion, a scalar field obeying

$q_{ab}$03

and a Lie-dragged conformal analogue of surface gravity,

$q_{ab}$04

This allows a covariant phase-space construction and a first law even though the horizon area changes. The rotating extension adds an axial conformal Killing vector $q_{ab}$05, distorted $q_{ab}$06 cross-sections, and a Hamiltonian angular momentum

$q_{ab}$07

leading to a differential first law

$q_{ab}$08

for this restricted class of growing null horizons (Chatterjee et al., 2014, Chatterjee et al., 2015).

These generalizations indicate that “quasi-local horizon” is not a single definition but a family of structures adapted to different theories and physical questions. This suggests that the common core lies in the use of finite-region geometry—null expansions, induced connections, entropy densities, preferred flows, or curvature discriminants—rather than in one universal kinematical criterion.

6. Applications, inner horizons, and unresolved structure

Quasi-local horizons are increasingly used to analyze transient and fully dynamical black-hole phenomena. In gauge-invariant reduced-phase-space perturbation theory with backreaction, the apparent horizon on a preferred Gullstrand–Painlevé foliation is defined by

$q_{ab}$09

with the canonical form

$q_{ab}$10

For a deformed horizon $q_{ab}$11, the second-order perturbative solution yields a remarkably simple area law: $q_{ab}$12 and hence a quasi-local mass

$q_{ab}$13

Classically,

$q_{ab}$14

while the authors argue that the quantum theory may permit $q_{ab}$15 to decrease, as expected in Hawking evaporation (Neuser et al., 13 May 2026).

Inner horizons provide another arena where the quasi-local perspective alters the standard interpretation. In spherical symmetry, with metric

$q_{ab}$16

and two quasi-local horizons given by

$q_{ab}$17

a slowly-evolving non-extremal inner trapping horizon with

$q_{ab}$18

drives an exponential approach of outgoing null rays to the inner horizon,

$q_{ab}$19

This produces finite but potentially large mass inflation even without a Cauchy horizon. The stationary divergence is recovered only when the drift of the inner trapping horizon vanishes. This reinterprets mass inflation as a quasi-local instability of inner trapping horizons rather than a phenomenon intrinsically tied to global Cauchy horizons (Carballo-Rubio et al., 2024).

In inhomogeneous cosmology, Hayward trapping horizons can also be analyzed explicitly. In Lemaître spacetime, future and past horizons are determined by

$q_{ab}$20

and both are null exactly when the Misner–Sharp mass is constant along them, which is equivalent on the horizon to

$q_{ab}$21

In the non-symmetric Szekeres–Szafron spacetime, horizon existence depends not only on collapse or expansion but also on the sign of

$q_{ab}$22

showing directly how inhomogeneous shell motion enters the trapping-horizon conditions (Polášková et al., 2018).

Across these applications, several structural issues remain unresolved. Quasi-local horizons in dynamical spacetimes are not unique; different slicings can generate different MOTSs and different trapping or dynamical horizons (Nielsen et al., 2010, Krishnan, 2013). Apparent horizons remain slicing-dependent, while more invariant alternatives such as geometric horizons are still partly conjectural (Coley et al., 2017). The equilibrium sector, by contrast, is substantially cleaner: isolated horizons support a consistent action principle, covariant phase space, horizon charges, and both classical and quantum treatments, including the loop-quantum-gravity description in which the horizon boundary term becomes that of an $q_{ab}$23 Chern–Simons theory (Engle et al., 2011).

A plausible implication is that quasi-local horizon theory is best viewed as a layered framework. Isolated horizons provide the equilibrium limit; dynamical and trapping horizons encode local growth and flux; more specialized constructions adapt the notion to conformal frames, preferred foliations, or non-Einstein dynamics; and invariant detection schemes seek to reduce foliation dependence without returning to teleological definitions. Within that layered picture, quasi-local horizons have become the primary language for black-hole mechanics in the fully nonlinear regime and for any setting in which the global event horizon is either inaccessible or conceptually inadequate (Ashtekar, 2023).