Non-expanding horizon

A non-expanding horizon (NEH) is an enclosed null surface whose intrinsic structure is preserved. An NEH is the geometric prototype of an isolated horizon which describes a black hole in equilibrium with its exterior from the quasilocal perspective. It is based on the concept and geometry of NEHs that the two quasilocal definitions of black holes, weakly isolated horizons and isolated horizons, are developed.

A three-dimensional submanifold ∆ is defined as a generic (rotating and distorted) NEH if it respects the following conditions:

(i) ∆ is null and topologically ${\displaystyle S^{2}\times \mathbb {R} }$;
(ii) Along any null normal field ${\displaystyle l}$ tangent to ∆, the outgoing expansion rate ${\displaystyle \displaystyle \theta _{(l)}:={\hat {h}}^{ab}{\hat {\nabla }}_{a}l_{b}}$ vanishes;
(iii) All field equations hold on ∆, and the stress–energy tensor ${\displaystyle T_{ab}}$ on ∆ 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 ${\displaystyle l^{a}}$.

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