Isolated horizon

A three-dimensional submanifold ${\displaystyle \Delta }$ equipped with an equivalence class ${\displaystyle [\ell ]}$ is defined as an IH if it respects the following conditions:[2][3][4]

(i) ${\displaystyle \Delta }$ is null and topologically ${\displaystyle S^{2}\times \mathbb {R} }$;
(ii) Along any null normal field ${\displaystyle l}$ tangent to ${\displaystyle \Delta }$, the outgoing expansion rate ${\displaystyle \displaystyle \theta _{(l)}:={\hat {h}}^{ab}{\hat {\nabla }}_{a}l_{b}}$ vanishes;
(iii) All field equations hold on ${\displaystyle \Delta }$, and the stress–energy tensor ${\displaystyle T_{ab}}$ on ${\displaystyle \Delta }$ is such that ${\displaystyle V^{a}:=-T_{b}^{a}l^{b}}$ is a future-directed causal vector (${\displaystyle V^{a}V_{a}\leq 0}$) 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.,[2][3][4] "hat" over the equality symbol ${\displaystyle {\hat {=}}}$ means equality on the black-hole horizons (NEHs), and "hat" over quantities and operators (${\displaystyle {\hat {h}}^{ab}}$, ${\displaystyle {\hat {\nabla }}}$, etc.) denotes those on the horizon or on a foliation leaf of the horizon (this makes no difference for IHs).

The properties of a generic IH manifest themselves as a set of boundary conditions expressed in the language of Newman–Penrose formalism,

${\displaystyle \kappa \,{\hat {=}}\,0}$ (geodesic), ${\displaystyle {\text{Im}}(\rho )\,{\hat {=}}\,0}$ (twist-free, hypersurface orthogonal), ${\displaystyle {\text{Re}}(\rho )\,{\hat {=}}\,0}$ (expansion-free), ${\displaystyle \sigma \,{\hat {=}}\,0}$ (shear-free),

${\displaystyle \Phi _{00}\,{\hat {=}}\,0\,,\quad \Phi _{10}={\overline {\Phi _{01}}}\,{\hat {=}}\,0}$ (no flux of any kinds of matter charges across the horizon),

${\displaystyle \Psi _{0}\,{\hat {=}}\,0\,,\quad \Psi _{1}\,{\hat {=}}\,0}$ (no gravitational waves across the horizon).

In addition, for an electromagnetic IH,

${\displaystyle \phi _{0}\,{\hat {=}}\,0\,,\quad \Phi _{02}={\overline {\Phi _{20}}}=\,2\,\phi _{0}\,{\overline {\phi _{2}}}\,{\hat {=}}\,0\,.}$

Moreover, in a tetrad adapted to the IH structure,[3][4] we have

${\displaystyle \pi \,{\hat {=}}\,\alpha +{\bar {\beta }}\,,\quad \varepsilon \,{\hat {=}}\,{\bar {\varepsilon }}\,,\quad {\bar {\mu }}\,{\hat {=}}\,\mu \,.}$

Remark: In fact, these boundary conditions of IHs just inherit those of NEHs.

Full analysis of the geometry and mechanics of an IH relies on the on-horizon adapted tetrad.[3][4] However, a more comprehensive view of IHs often requires investigation of the near-horizon vicinity and off-horizon exterior.[5][6][7][8][9][10] The adapted tetrad on an IH can be smoothly extended to the following form which cover both the horizon and off-horizon regions,

${\displaystyle l^{a}\partial _{a}=\partial _{v}+U\partial _{r}+X^{3}\partial _{y}+X^{4}\partial _{z}\,,}$
${\displaystyle n^{a}\partial _{a}=-\partial _{r}\,,}$
${\displaystyle m^{a}\partial _{a}=\Omega \partial _{r}+\xi ^{3}\partial _{y}+\xi ^{4}\partial _{z}\,,}$
${\displaystyle {\bar {m}}^{a}\partial _{a}={\bar {\Omega }}\partial _{r}+{\bar {\xi }}^{3}\partial _{y}+{\bar {\xi }}^{4}\partial _{z}\,.}$

where ${\displaystyle \{y,z\}}$ are either real isothermal coordinates or complex stereographic coordinates labeling the cross-sections of { v=constant, r=constant}, and the gauge conditions in this tetrad are
${\displaystyle \nu =\tau =\gamma =0\,,\quad \mu ={\bar {\mu }}\,,\quad \pi =\alpha +{\bar {\beta }}\,.}$

The local nature of the definition of an isolated horizon makes it more convenient for numerical studies.

The local nature makes the Hamiltonian description viable. This framework offers a natural point of departure for non-perturbative quantization and derivation of black hole entropy from microscopic degrees of freedom.[11]

• Non-expanding horizon
• Newman–Penrose formalism