Nonexpanding and Isolated Horizons

An isolated horizon describes a black hole in quasi-local equilibrium even when radiation or matter remains dynamical elsewhere in the spacetime. The construction begins with a null boundary whose cross-sectional area does not change, then adds successively stronger time-independence conditions on its intrinsic connection. It does not assume a Killing field in a neighborhood and does not, by itself, identify the global event horizon.

Helpful background. Exact Solutions of Einstein’s Equations supplies stationary examples; Causality and Global Structure separates null boundaries from event horizons; The Einstein Equation connects null focusing to stress–energy; Timelike Congruences and Spatial Projection supplies expansion and shear; Hypersurface Normals and Tangential Projectors supplies induced geometry; and Black-Hole Geometry and Mechanics supplies the later mechanics route.

Three levels of boundary equilibrium

In four-dimensional Einstein gravity, a nonexpanding horizon (NEH) $\Delta$ is a smooth null three-manifold, conventionally with topology $S^2\times\mathbb R$, whose future null normal satisfies

$$\theta_{(\ell)}=0$$

everywhere on $\Delta$. The field equations hold at the boundary, and $-T^a{}_b\ell^b$ is required to be future causal there. Compact sections and regularity complete the finite-area black-hole interpretation. This is a condition on an entire null hypersurface, not on one marginal two-surface Ashtekar and Krishnan 2004, §2.1.1, Definition 1.

The null hypersurface inherits an intrinsic derivative operator $D_a$. Its action on a null generator defines the rotation one-form $\omega_a$ by

$$D_a\ell^b=\omega_a\ell^b.$$

The next two levels add conditions on these connection data:

Structure	Additional data and condition	What becomes time independent
NEH $\Delta$	$\theta_{(\ell)}=0$ plus the field and flux conditions	Cross-sectional metric and area
WIH $(\Delta,[\ell])$	A class under $\ell\mapsto c\ell$, $c\gt0$ constant, with $[\mathcal L_\ell,D_a]\ell^b=0$	The rotation one-form for the chosen class; surface gravity is constant
IH $(\Delta,[\ell])$	$[\mathcal L_\ell,D_a]V^b=0$ for every tangent $V^b$	The full intrinsic derivative operator

Thus, as specified structures,

$$\mathrm{IH}\Longrightarrow\mathrm{WIH} \Longrightarrow\mathrm{NEH}.$$

The reverse wording needs care. Every NEH admits suitable WIH structures, but generally more than one. The bare NEH definition does not specify which WIH class $[\ell]$ is included as part of the horizon structure. A specified WIH need not admit the stronger IH structure. Ashtekar and Krishnan give the hierarchy and the zeroth-law implication in Ashtekar and Krishnan 2004, §§2.1.2–2.1.4.

The compact focusing check

For a possibly nonaffine hypersurface-orthogonal generator,

$$\mathcal L_\ell\theta_{(\ell)} =\kappa_{(\ell)}\theta_{(\ell)} -\frac12\theta_{(\ell)}^2 -\sigma^{(\ell)}_{ab}\sigma_{(\ell)}^{ab} -8\pi G T_{ab}\ell^a\ell^b.$$

On an NEH, both $\theta_{(\ell)}$ and its derivative along $\Delta$ vanish. The remaining two terms are nonpositive under the energy hypothesis, so each must vanish:

$$\sigma^{(\ell)}_{ab}=0, \qquad T_{ab}\ell^a\ell^b=0.$$

The deformation equation then gives $\mathcal L_\ell q_{ab}=0$. This one argument explains why NEH equilibrium means fixed intrinsic area geometry and no shear or null matter flux through the boundary. It would fail if expansion vanished only on one leaf or if negative null energy could balance shear. The complete derivation, including rescaling and induced-connection details, is on Nonexpanding Horizons; see also Gourgoulhon and Jaramillo 2006, §6.2 and §7.1.3, pp. 210–211 and 219.

Why weak isolation needs a normal class

The rotation one-form is normalization dependent, and its contraction with the generator gives the associated surface gravity, $\kappa_{(\ell)}=\omega_a\ell^a$.

Under a positive rescaling $\ell'^a=f\ell^a$,

$$\omega'_a=\omega_a+D_a\ln f, \qquad \kappa_{(\ell')}=f\kappa_{(\ell)}+\mathcal L_\ell f.$$

Zero expansion survives every such rescaling, but weak isolation retains only the constant-rescaling class. Requiring $\mathcal L_\ell\omega_a=0$ then makes $\kappa_{(\ell)}$ constant on $\Delta$ without fixing its numerical normalization. The full proof and the distinction between WIH and IH are developed on Weakly Isolated and Isolated Horizons.

What the hierarchy does not identify

A regular stationary black-hole Killing horizon supplies an isolated-horizon example after a compatible generator class is chosen, but the converse is false. Isolated-horizon conditions constrain fields at $\Delta$; they do not produce a Killing field in a surrounding neighborhood. Radiation may remain dynamical outside without crossing the boundary.

Isolation also does not determine global causal status. An event horizon is selected by the entire future and a chosen asymptotic receiver, whereas an isolated horizon is recognized from finite-boundary data. A quiet horizon segment before later infall can therefore satisfy isolated-boundary conditions without those conditions locating the event horizon. A dynamical horizon differs again: it is spacelike, carries flux, and has increasing area.

This division is the practical point of the taxonomy. Use intrinsic null-boundary data to ask whether the boundary is in equilibrium; use a Killing field to ask whether it is generated by a spacetime symmetry; use the event-horizon definition to ask whether signals escape. The later quasi-local chapter constructs the charges and first laws that require the extra WIH data.

Check your understanding

Suppose a null boundary satisfies the NEH conditions and admits a class $[\ell]$ with $\mathcal L_\ell\omega_a=0$. Gravitational radiation is present nearby but does not cross the boundary. Which horizon labels follow, and which do not?

Solution

The boundary is an NEH, and the specified class makes it a WIH. Full isolation does not follow without showing that $[\mathcal L_\ell,D_a]V^b=0$ for every tangent vector field. Nearby radiation does not contradict weak isolation when it does not cross $\Delta$, but it prevents any inference of a stationary exterior. No spacetime Killing field has been supplied, so Killing-horizon status does not follow; no global escape calculation has been made, so event-horizon status does not follow either.

Next, Dynamical Horizons and Cross-Definition Maps compares this null equilibrium hierarchy with spacelike and timelike evolving horizon segments.

References

• Ashtekar, Abhay, and Badri Krishnan. “Isolated and Dynamical Horizons and Their Applications.” Living Reviews in Relativity 7 (2004), article 10. DOI.
• Gourgoulhon, Éric, and José Luis Jaramillo. “A 3+1 Perspective on Null Hypersurfaces and Isolated Horizons.” Physics Reports 423 (2006): 159–294. DOI.