Isolated horizon

An advantage of isolated horizons over event horizons is that while one needs the entire spacetime history to locate an event horizon, isolated horizons are defined using local spacetime structures only. The laws of black hole mechanics, initially proved for event horizons, are generalized to isolated horizons.

An isolated horizon ${\displaystyle (\mathrm{\Delta },[\ell ])}$ refers to the quasilocal definition of a black hole which is in equilibrium with its exterior, and both the intrinsic and extrinsic structures of an isolated horizon (IH) are preserved by the null equivalence class ${\displaystyle [\ell ]}$. The concept of IHs is developed based on the ideas of non-expanding horizons (NEHs) and weakly isolated horizons (WIHs): A NEH is a null surface whose intrinsic structure is preserved and constitutes the geometric prototype of WIHs and IHs, while a WIH is a NEH with a well-defined surface gravity and based on which the black-hole mechanics can be quasilocally generalized.

Definition of IHs

A three-dimensional submanifold ${\displaystyle \mathrm{\Delta }}$ equipped with an equivalence class ${\displaystyle [\ell ]}$ is defined as an IH if it respects the following conditions:

(i) ${\displaystyle \mathrm{\Delta }}$ is null and topologically ${\displaystyle S^2\times \mathbb{R}}$;
(ii) Along any null normal field ${\displaystyle l}$ tangent to ${\displaystyle \mathrm{\Delta }}$, the outgoing expansion rate ${\displaystyle {\displaystyle {\theta }_{(l)}:={\hat{h}}^{ab}{\hat{\mathrm{\nabla }}}_a{l}_b}}$ vanishes;
(iii) All field equations hold on ${\displaystyle \mathrm{\Delta }}$, and the stress–energy tensor ${\displaystyle T_{ab}}$ on ${\displaystyle \mathrm{\Delta }}$ is such that ${\displaystyle V^a:=-T_b^a{l}^b}$ is a future-directed causal vector (${\displaystyle V^a{V}_a\le 0}$) for any future-directed null normal ${\displaystyle l^a}$.
(iv) The commutator ${\displaystyle [{\mathcal{L}}_{\ell },{\mathcal{D}}_a]=0}$, where ${\displaystyle {\mathcal{D}}_a}$ denotes the induced connection on the horizon.

Note: Following the convention set up in refs., "hat" over the equality symbol ${\displaystyle \widehat{=}}$ means equality on the black-hole horizons (NEHs), and "hat" over quantities and operators (${\displaystyle {\hat{h}}^{ab}}$, ${\displaystyle \hat{\mathrm{\nabla }}}$, etc.) denotes those on the horizon or on a foliation leaf of the horizon (this makes no difference for IHs).