Weakly Isolated and Isolated Horizons

A nonexpanding horizon (NEH) freezes its intrinsic two-metric but leaves part of its connection free to evolve and leaves its null normal arbitrarily rescalable. A weakly isolated horizon (WIH) selects a constant-rescaling class of generators and requires the associated normal-bundle connection to be time independent; this is the minimum structure that supports constant surface gravity and a horizon first law. A fully isolated horizon (IH) freezes the entire intrinsic connection. Neither upgrade demands a Killing field in the surrounding spacetime.

Required background. Hypersurface Normals and Tangential Projectors supplies null-boundary geometry. Helpful background. Exact Spacetimes supplies Killing-horizon examples; Causality and Global Structure separates intrinsic and global notions; Einstein Equations controls horizon curvature; Black-Hole Geometry and Mechanics supplies the hierarchy; Variational and Symplectic Geometry prepares the horizon Hamiltonian.

Connection data left by nonexpansion

Let $\Delta$ be a nonexpanding horizon with induced degenerate metric $q_{ab}$ and derivative operator $D_a$. For any future null normal $\ell^a$,

$$D_a\ell^b=\omega_a\ell^b, \qquad \kappa_{(\ell)}=\ell^a\omega_a.$$

The rotation one-form $\omega_a$ records the connection in the null normal bundle. Under an arbitrary positive rescaling $\ell'=f\ell$,

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

Nonexpansion gives $\mathcal L_\ell q_{ab}=0$ but places no general condition on $\mathcal L_\ell\omega_a$. Consequently it does not yet supply a preferred normalization class or constant surface gravity. Gourgoulhon and Jaramillo derive the induced connection in Gourgoulhon and Jaramillo 2006, §§7.2 and 7.5, pp. 219–223 and give the rescaling law in Gourgoulhon and Jaramillo 2006, §5.8, Table 1, p. 209.

Weak isolation and the zeroth law

A weakly isolated horizon is a pair $(\Delta,[\ell])$. Here $\Delta$ is an NEH and

$$\ell'^a\sim\ell^a \quad\Longleftrightarrow\quad \ell'^a=c\ell^a, \qquad c\gt0\ \text{constant}.$$

This constant-rescaling equivalence class and the additional weak-isolation condition are formulated in Gourgoulhon and Jaramillo 2006, §8.2, pp. 231–233.

For every representative, require

$$[\mathcal L_\ell,D_a]\ell^b=0,$$

equivalently

$$\mathcal L_\ell\omega_a=0.$$

The restriction to constant rescalings is essential. It preserves the condition and sends $\kappa_{(\ell)}\mapsto c\kappa_{(\ell)}$. An arbitrary function $f$ adds $D_a\ln f$ to $\omega_a$ and can destroy its time independence.

On an NEH, the curvature of the normal bundle is horizontal: its contraction with $\ell^a$ vanishes under the field and energy conditions. Cartan’s identity then gives

$$\mathcal L_\ell\omega =\ell\mathbin{\cdot}d\omega+d(\ell\mathbin{\cdot}\omega) =d\kappa_{(\ell)}.$$

Weak isolation therefore implies

$$d\kappa_{(\ell)}=0.$$

Surface gravity is constant over the entire horizon, although its numerical value still changes by the allowed constant rescaling. This is the quasi-local zeroth law. Ashtekar and Krishnan state the WIH conditions and proof in Ashtekar and Krishnan 2004, §§2.1.2–2.1.3.

The proof does not normalize $\ell^a$. In an asymptotically flat stationary spacetime, matching to unit time translation at infinity does so. For a purely quasi-local boundary, a Hamiltonian prescription or comparison family selects a normalization appropriate to the mechanics question.

Full isolation freezes the intrinsic connection

An isolated horizon strengthens the commutator condition to

$$[\mathcal L_\ell,D_a]V^b=0$$

for every vector field $V^b$ tangent to $\Delta$. Thus the complete intrinsic connection—not only its action on the null generator—is invariant along $\ell^a$. The logical hierarchy is

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

This is a hierarchy of structures, not a claim that an NEH can never support weak isolation. In fact every NEH admits suitable WIH structures, generally more than one; the bare NEH definition does not specify which constant-rescaling class $[\ell]$ is included as part of the horizon structure. By contrast, a specified WIH can retain time dependence in transverse components of $D_a$ acting on other tangent fields and need not admit the stronger IH structure. The IH condition rules out that remaining intrinsic evolution.

Every nondegenerate stationary black-hole Killing horizon with regular compact sections that also satisfies the NEH topology, field-equation, and causal energy-flux hypotheses can furnish an isolated-horizon structure after its generator class is fixed. The converse is not a theorem: intrinsic connection symmetry on one null boundary does not extend automatically to a spacetime Killing field in an open neighborhood.

Equilibrium boundary, dynamical exterior

The boundary conditions prohibit flux across $\Delta$, not dynamics everywhere outside it. Initial data can contain gravitational radiation in the exterior whose support does not intersect the horizon during the interval under study. The horizon can be weakly isolated while no Killing vector exists throughout that exterior region.

This provides the relevant counterexample to a common overstatement. A WIH is not a stationary spacetime truncated at a null surface. It is a finite equilibrium boundary embedded in a possibly dynamical solution. It need not coincide with the event horizon either, because event-horizon location depends on the full future and chosen infinity.

The distinction also separates WIH from a dynamical horizon. A dynamical horizon is spacelike, has growing area, and carries matter or gravitational flux. When flux ceases it may join a null isolated segment, but the change of signature and boundary conditions must be analyzed; one does not relabel the same flux-carrying segment “isolated.”

Data fixed at each level

Structure	Invariant data along generators	Additional freedom retained
NEH	Cross-sectional metric and area	Normal class and connection may evolve
WIH	Metric plus rotation one-form for $[\ell]$	Other intrinsic connection components may evolve
IH	Full intrinsic metric and derivative operator	Exterior fields may still be dynamical

The table describes boundary data, not increasing claims about the exterior. The isolated-horizon first law needs WIH-level mechanics data and a chosen evolution field; horizon multipoles can require additional symmetry on the cross-sections.

Test the rescaling class

Let $(\Delta,[\ell])$ be weakly isolated with constant $\kappa_{(\ell)}\ne0$. Compare $\ell'=c\ell$, with constant $c\gt0$, to $\ell'=f(v)\ell$, where $\mathcal L_\ell v=1$ and $f$ is nonconstant. Which transformation stays in $[\ell]$, and why can the second spoil the zeroth law?

Solution

The constant rescaling stays in the equivalence class. It sends $\omega_a$ to itself and $\kappa$ to $c\kappa$, which remains constant on $\Delta$. For nonconstant $f(v)$,

$$\kappa_{(f\ell)}=f\kappa_{(\ell)}+\frac{df}{dv}.$$

This expression generally depends on $v$, while $\omega'_a=\omega_a+D_a\ln f$ generally fails $\mathcal L_{f\ell}\omega'_a=0$. The null direction and zero expansion are unchanged, so the rescaled field still describes the same NEH direction, but it is not generally a representative of the specified weakly isolated structure.

Quasi-Local Horizon Angular Momentum and Charge uses the rotation one-form and gauge flux to construct finite-boundary charges.

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.