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:[1][2][3]
vanishes; (iii) All field equations hold on ∆, and the stress–energy tensor
Condition (i) is fairly trivial and just states the general fact that from a 3+1 perspective[4] an NEH ∆ is foliated by spacelike 2-spheres ∆'=S2, where S2 emphasizes that ∆' is topologically compact with genus zero (
The signature of ∆ is (0,+,+) with a degenerate temporal coordinate, and the intrinsic geometry of a foliation leaf ∆'=S2 is nonevolutional.
in condition (ii) plays a pivotal role in defining NEHs and the rich implications encoded therein will be extensively discussed below.
Note: In this article, following the convention set up in refs.,[1][2][3] "hat" over the equality symbol
means equality on the black-hole horizons (NEHs), and "hat" over quantities and operators (
Now let's work out the implications of the definition of NEHs, and these results will be expressed in the language of NP formalism with the convention[5][6]
, this is the usual one employed in studying trapped null surfaces and quasilocal definitions of black holes[10]).
As a summary, Thus, the isolated horizon ∆ is nonevolutional and all foliation leaves ∆'=S2 look identical with one another.
Also, there should be no gravitational waves crossing the horizon; however, gravitational waves are propagation of perturbations of the spacetime continuum rather than flows of charges, and therefore depicted by four Weyl-NP scalars
To sum up, we have which means that,[5] geometrically, a principal null direction of Weyl's tensor is repeated twice and
is aligned with the principal direction; physically, no gravitational waves (transverse component
This result is consistent with the physical scenario defining NEHs.
For a better understanding of the previous section, we will briefly review the meanings of relevant NP spin coefficients in depicting null congruences.
[7] The tensor form of Raychaudhuri's equation[12] governing null flows reads where
The quantities in Raychaudhuri's equation are related with the spin coefficients via[5][13][14] Moreover, a null congruence is hypersurface orthogonal if
This is the simplest extension of vacuum NEHs, and the nonvanishing energy-stress tensor for electromagnetic fields reads
The boundary conditions derived in the previous section are applicable to generic NEHs.
[15] Usually, null tetrads adapted to spacetime properties are employed to achieve the most succinct NP descriptions.
For example, a null tetrad can be adapted to principal null directions once the Petrov type is known; also, at some typical boundary regions such as null infinity, timelike infinity, spacelike infinity, black hole horizons and cosmological horizons, tetrads can be adapted to boundary structures.
Similarly, a preferred tetrad[1][2][3] adapted to on-horizon geometric behaviors is employed in the literature to further investigate NEHs.
As indicated from the 3+1 perspective from condition (i) in the definition, an NEH ∆ is foliated by spacelike hypersurfaces ∆'=S2 transverse to its null normal along an ingoing null coordinate
, where we follow the standard notation of ingoing Eddington–Finkelstein null coordinates and use
should be purely intrinsic; thus in the commutator the coefficients for the directional derivatives
Based on NEHs, WIHs which have valid surface gravity can be defined to generalize the black hole mechanics.
WIHs are sufficient in studying the physics on the horizon, but for geometric purposes,[2] stronger restrictions can be imposed to WIHs so as to introduce IHs, where the equivalence class of null normals