Non-expanding horizon


A non-expanding horizon 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.

Definition of NEHs

A three-dimensional submanifold ∆ is defined as a generic NEH if it respects the following conditions:


∆ is null and topologically ;

Along any null normal field tangent to ∆, the outgoing expansion rate vanishes;

All field equations hold on ∆, and the stress–energy tensor on ∆ is such that is a future-directed causal vector for any future-directed null normal.


Condition is fairly trivial and just states the general fact that from a 3+1 perspective an NEH ∆ is foliated by spacelike 2-spheres ∆'=S2, where S2 emphasizes that ∆' is topologically compact with genus zero. The signature of ∆ is with a degenerate temporal coordinate, and the intrinsic geometry of a foliation leaf ∆'=S2 is nonevolutional. The property in condition plays a pivotal role in defining NEHs and the rich implications encoded therein will be extensively discussed below. Condition makes one feel free to apply the Newman–Penrose formalism of Einstein-Maxwell field equations to the horizon and its near-horizon vicinity; furthermore, the very energy inequality is motivated from the dominant energy condition and is a sufficient condition for deriving many boundary conditions of NEHs.


Note: In this article, following the convention set up in refs., "hat" over the equality symbol means equality on the black-hole horizons, and "hat" over quantities and operators denotes those on a foliation leaf of the horizon. Also, ∆ is the standard symbol for both an NEH and the directional derivative ∆ in NP formalism, and we believe this won't cause an ambiguity.

Boundary conditions implied by the definition

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 . Being a null normal to ∆, is automatically geodesic,, and twist free,. For an NEH, the outgoing expansion rate along is vanishing,, and consequently. Moreover, according to the Raychaudhuri-NP expansion-twist equation,
it follows that on ∆
where is the NP-shear coefficient. Due to the assumed energy condition, we have , and therefore is nonnegative on ∆. The product is of course nonnegative, too. Consequently, and must be simultaneously zero on ∆, i.e. and. As a summary,
Thus, the isolated horizon ∆ is nonevolutional and all foliation leaves ∆'=S2 look identical with one another. The relation implies that the causal vector in condition is proportional to and is proportional to on the horizon ∆; that is, and,. Applying this result to the related Ricci-NP scalars, we get, and, thus
The vanishing of Ricci-NP scalars signifies that, there is no energy–momentum flux of any kind of charge across the horizon, such as electromagnetic waves, Yang–Mills flux or dilaton flux. 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 rather than Ricci-NP quantities. According to the Raychaudhuri-NP shear equation
or the NP field equation on the horizon
it follows that. Moreover, the NP equation
implies that. To sum up, we have
which means that, geometrically, a principal null direction of Weyl's tensor is repeated twice and is aligned with the principal direction; physically, no gravitational waves enter the black hole. This result is consistent with the physical scenario defining NEHs.

Remarks: Spin coefficients related to Raychaudhuri's equation

For a better understanding of the previous section, we will briefly review the meanings of relevant NP spin coefficients in depicting null congruences. The tensor form of Raychaudhuri's equation governing null flows reads
where is defined such that. The quantities in Raychaudhuri's equation are related with the spin coefficients via
Moreover, a null congruence is hypersurface orthogonal if.

Constraints from electromagnetic fields

NEHs on which are the simplest types of NEHs, but in general there can be various physically meaningful fields surrounding an NEH, among which we are mostly interested in electrovacuum fields with. This is the simplest extension of vacuum NEHs, and the nonvanishing energy-stress tensor for electromagnetic fields reads


where refers to the antisymmetric electromagnetic field strength, and is trace-free by definition and respects the dominant energy condition..
The boundary conditions derived in the previous section are applicable to generic NEHs. In the electromagnetic case, can be specified in a more particular way. By the NP formalism of Einstein-Maxwell equations, one has
where denote the three Maxwell-NP scalars. As an alternative to Eq, we can see that the condition also results from the NP equation
It follows straightforwardly that


These results demonstrate that, there are no electromagnetic waves across or along the NEH except the null geodesics generating the horizon. It is also worthwhile to point out that, the supplementary equation in Eq is only valid for electromagnetic fields; for example, in the case of Yang–Mills fields there will be where are Yang–Mills-NP scalars.

Adapted tetrad on NEHs and further properties

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 adapted to on-horizon geometric behaviors is employed in the literature to further investigate NEHs.
As indicated from the 3+1 perspective from condition 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 to label the 2-dimensional leaves at ; that is, ∆=∆'×=S2×. is set to be future-directed and choose the first tetrad covector as, and then there will be a unique vector field as null normals to satisfying the cross-normalization and affine parametrization ; such choice of would actually yields a preferred foliation of ∆. While are related to the extrinsic properties and null generators, the remaining two complex null vectors are to span the intrinsic geometry of a foliation leaf, tangent to ∆ and transverse to ; that is,.
Now let's check the consequences of this kind of adapted tetrad. Since
with, we have
Also, in such an adapted frame, the derivative on ∆'×=S2× should be purely intrinsic; thus in the commutator
the coefficients for the directional derivatives and ∆ must be zero, that is
so the ingoing null normal field is twist-free by, and equals the ingoing expansion rate.

Discussion

So far, the definition and boundary conditions of NEHs have been introduced. The boundary conditions include those for an arbitrary NEH, specific characteristics for Einstein-Maxwell NEHs, as well as further properties in an adapted tetrad. 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, stronger restrictions can be imposed to WIHs so as to introduce IHs, where the equivalence class of null normals fully preserves the induced connection on the horizon.