Evolving Null Horizons Near an Isolated Black Hole

Totally geodesic null hypersurfaces have been widely used as models of time-independent event and isolated black hole horizons. However, in reality black hole being surrounded by a local mass distribution there is significant difference in the structure of the surrounding region of isolated black holes. In this paper, we use metric conformal symmetry which provides a class of a family of totally umbilical null hypersurfaces (Theorem 4), supported by a physical model and an example of time-dependent evolving null horizons (see Definition 6) conformally related to an isolated black hole. We establish an interrelation between the spacelike dynamical horizons (see Definition 7), isolated and evolving null horizons. Finally we propose further study on null geometry and physics of the surface closer to an isolated horizon.


Introduction
Considerable work has been done on time-independent isolated black hole physics of asymptotically flat spacetimes which is still an active area of research.Such isolated black holes deal with the concept of an event and isolated horizons briefly explained as follows: Event horizons.A boundary of a spacetime is called an event horizon beyond which events cannot affect the observer.An event horizon is intrinsically a global concept since its definition requires the knowledge of the whole spacetime to determine whether null geodesics can reach null infinity.For basic information on event horizons we refer (Hawking, 1972) and three papers of (Há jičeke, 1973-74).However, in practice an event horizon is generally not very useful since to actually locate a black hole one needs to know the full spacetime metric up to the infinite future.Moreover, even if one locates the event horizon, using it to calculate the physical parameters is extremely difficult.Therefore, attempts have been made to find a quasilocal concept of a horizon which requires only minimum number of conditions to detect a black hole and study its properties.For this purpose, (Asktekar et al., 1999) introduced following three notions of "Isolated Horizons".Let (H, q) be a null hypersurface of a 4-dimensional spacetime (M, g) where q is the degenerate metric induced by the metric g of M. We assume that the null normal,say ℓ, is null geodesic future directed and is defined in some subset of M around H. This will permit to well-define the spacetime covarient derivative ∇ℓ where ∇ denotes the Levi-Civita connection on M. The expansion θ (ℓ) is defined by θ (ℓ) = q ab ∇ a ℓ b and the vorticity-free Raychaudhuri equation is where σ ab , s and R ab are shear tensor, a pseudo-arc parameter and Ricci tensor, respectively.
(3) All equations of motion hold at H and the stress energy tensor T ab is such that −T a b ℓ b is future-causal for any future directed null normal ℓ a , The condition (1) implies that marginally trapped surfaces (Hayward, 1994) are related to a black hole spacetime.The conditions (2) and (3) imply from the Raychaudhuri equation that £ ℓ q = 0 on H, which further implies that the metric q is time-independent and H is totally geodesic in M. In general, there does not exist a unique induced connection on H due to degenerate q.However, on an NEH, the property £ ℓ q = 0 implies that the spacetime connection ∇ induces a unique (torsion-free) connection, say D, on H which is compatible with q.We say that two null normals ℓ and l belong to the same equivalence class [ℓ] if ℓ = c l for some positive constant c.Definition 2. The pair (H, [ℓ]) is called a weakly isolated horizon (WIH) if it is a NEH and each normal ℓ ∈ [ℓ] satisfies (£ ℓ D a − D a £ ℓ )ℓ a = 0, i. e., D a ℓ b is also time-independent.Definition 3. A WIH (H, q, [ℓ]) is called an Isolated Horizon (IH) if the full connection D is time-independent, that is, if (£ ℓ D a − D a £ ℓ )V = 0 for arbitrary vector fields V a tangent to H.
For information and examples on isolated horizons we refer (Lewandowski, 2000), (Asktekar-Krishnan, 2002-3) , (Gourgoulhon-Jaramillo, 2006) and several others listed in these papers.However, in reality the present day research indicates that black hole has a cosmological background or it is surrounded by a local mass distribution.Therefore, there is significant difference in the structure and properties of the surrounding dynamical region of isolated black holes.The purpose of this paper is to use a conformal symmetry on an isolated black hole spacetime (M, g) which brings in a family of totally umbilical null hypersurfaces representing time-dependent null horizons near an isolated black hole.We also explain how this family of null horizons may evolve (for some cases) into a black hole isolated horizon.

Method
Let (M, g) be a spacetime with a conformal symmetry defined by a map ϕ : M → M such that the where ϕ ⋆ is the differential (tangent) map of ϕ and Ω s is a scalar function on M for some parameter value of s.
The set of all conformal maps, satisfying (1), form a group of conformal motions under composition of mappings.Let each V s be a smooth vector field on M and U denote a neighborhood of each p ∈ M with local coordinate system.Let the integral curve of each V s , through any point p in U, be defined on an open interval (−ϵ, ϵ) for ϵ > 0. For each t in this interval we define a map ϕ t on U such that for p in U, ϕ t (p) is that point with parameter value t on the integral curve of V s through p.Then, each V s generates a local 1-parameter group of transformations ϕ t : x a → x a + tV a s .If ϕ t satisfies the conformal symmetry equation ( 1), then, we say that V s is a conformal vector field, briefly denoted by CKV.In local coordinates, V s conformal implies that Expanding G s ab (x + tV s ) up to first order in t, and then using the Lie derivative operator £ V s , we get As t is small, so is Ω s .Setting Ω s = t σ s and expanding e tσ s up to first order in t, we get Above equations are well-known as conformal Killing equations.In particular, V s is homothetic or a Killing vector field according as σ s is a no-zero constant or zero.Let M be the space of all smooth Lorentzian metrics on M. Consider a family C = (G s ) ⊂ M whose each member is conformally related to the metric g of (M, g) with conformal symmetry defined by (1).Denote by a family of spacetimes conformally related to (M, g) and G s ∈ (G s ) for some parameter value of s.

Results
Let S = ((H u ), (h u ), (ℓ u )) be a family of null hypersurfaces of (M, (G s )) where u is corresponding parameter induced by the parameter s.For simplicity, we consider (H, h, ℓ) a member of the family S for some parameter value of u.
The "bending" of H in M is described by the Weingarten map: W ℓ associates each X of H the variation of ℓ along X, with respect to the spacetime connection ∇.The second fundamental form, say B, of H is the symmetric bilinear form and is related to the W ℓ by In particular, H or a portion of H is called totally geodesic if and only if B vanishes, i.e., if and only if f vanishes on H or some portion of H.
Theorem 4. Let S = ((H u ), (h u ), (ℓ u )) be a family of null hypersurfaces of a family F = (M, (G s )) of spacetimes defined by (3) whose each memeber is conformally related to an isolated black hole spacetime (M, g), with an isolated horizon (H, q) as defined by ( 1), which lies to the future of each (H u , h u ).Then, (a) there exists a family of maps ϕ u : ((H u ), (h u ), (ℓ u )) → (H, q) such that each metric h u transforms to h u = e Ω u q, where Ω u is the induced conformal factor of Ω s .
(b) Each (H u , h u ) is totally umbilical in (M, (G s )).
(c) (H u , h u ) may coincide with (H, q) on a portion of H u only if Ω u vanishes on that portion and, then, this common null hypersurface (H, q) has null mean curvature θ = 0.
Proof.Part (a) of the theorem follows easily since conformal transformations preserve causal structure and, therefore, each member of S is conformal to (H, q), with the induced conformal factor Ω u for some value of u.To prove (b) we observe that, as explained in Section 2, the conformal structure in (a) will induce a family of null conformal Killing vector (CKV) fields, say (ℓ u ) of the family (H u ) such that for each value of u Express the left side of above equation in the form £ ℓ u h u (X, Y) = h u (∇ X ℓ u , Y) + h u (∇ Y ℓ u , X).Then, using (4) and (5) with B(X, Y) symmetric it follows that which is well defined up to conformal rescaling (related to the choice of ℓ u ).Thus, each (H u , h u ) is totally umbilical , which proves (b).For the case (c) observe that (H u , h u ) approaches (H, q) for some value of u only if h u = q for that value of u, which further means that only if Ω u vanishes for that value of u.This proves the first part of (c).Now, as per definition of isolated horizons, (H u , h u ) = (H, q) is totally geodesic.Moreover, It is well-known that any null hypersurface of a semi-Riemannian manifold has zero mean curvature if and only if it is totally geodesic, which proves (c).
Now we address the question of how the Theorem 4 can be used to show the existence of a family of time-dependent null horizons near an isolated black hole.For this purpose we recall that (Perlick, 2005) proved following general result for a totally umbilical submanifold (also holds for totally geodesic case) H of a semi-Riemannian manifold M. "A null geodesic vector field of M that starts tangential to H remains within H for some parameter interval around the starting point".
Above result satisfies a requirement for the existence of a null horizon in relativity.Since we assume that each null normal ℓ u of the family of hypersurfaces S is null geodesic, using above result of (Perlick, 2005) we state the following corollary as a physical consequence of the Theorem 4 (proof is easy).
Corollary 5. Let (M, g) be a null geodesically complete spacetime obeying the null energy condition Ric(X, X) ≥ 0 for all null vectors X and the hypothesis of Theorem 4 holds.Then, each null geodesic vector ℓ u of S is contained in its respective smooth totally umbilical null hypersurface (H u , ℓ u ) of (M, G s ) In particular, this property will also hold for the totally geodesic hypersurface (H, q) of (M, g) .
In support of Theorem 4, we now present following physical model and an example.
Physical Model.In two recent papers of (Duggal, 2012(Duggal, , 2014) ) a new class of null hypersurfaces of a spacetime was studied using the following definition: Definition 6.A null hypersurface (H, h, ℓ) of a spacetime (M, g) is called an Evolving Null Horizon, briefly denoted by (ENH), if (i) H is totally umbilical in (M, g) and may include a totally geodesic portion.
(ii) All equations of motion hold at H and energy tensor T i j is such that T a b ℓ b is future-causal for any future directed null normal ℓ.
Comparing the two conditions of above definition with Theorem 4, we notice that the first part of condition (i) is same as the conclusion (2) of Theoren 4 and for the second part we first observe that the energy condition of (ii) requires R ab ℓ a ℓ b non-negative for any ℓ, which implies from page 95 of (Hawking & Ellis, 1973) that θ (ℓ) monotonically decreases in time along ℓ, that is, M obeys the null convergence condition, which further means that the null hypersurface (H, h) is time-dependent in the region where θ (ℓ) is non-zero and may evolve into a time-independent totally geodesic hypersurface as a model of an isolated horizon.Thus, above two conditions of the Definition 6 clearly show that there exists a Physical Model of a class S = ((H u ), (h u ), (ℓ u )) of a family of totally umbilical null hypersurfaces of a family F = (M, (G s )), satisfying the hypothesis and three conclusions of Theorem 4, such that its each member is an evolving null horizon(ENH) which may evolve into an isolated horizon.Simple example is a family of null cones non of which evolves into an isolated horizon.We refer (Duggal, 2012(Duggal, , 2014(Duggal, , 2015) ) for this and some more examples with details on the geometry and physics of ENHs.
Physical example.To construct an example we first recall that (Asktekar-Krishnan, 2003) studied the following quasi-local concept of dynamical horizons (briefly denoted by DH) which model the present day evolving black holes and their asymptotic states are isolated horizons.Definition 7. A smooth, 3-dimensional spacelike submanifold (possibly with boundary) Σ of a spacetime is said to be a dynamical horizon (DH) if it can be foliated by a family of closed 2-manifolds such that 1. on each leaf L its future directed null normal ℓ has zero expansion, θ (ℓ) = 0, 2. and the other null normal, k, has negative expansion θ (k) < 0.
Above definition requires that Σ be spacelike except for a special case in which portions of marginally trapped surfaces lie on a spacelike horizon and the remainder on a null horizon.Recall that the concept of marginally trapped surfaces was first introduced by (Hayward, 1994) as an attempt to describe the surface of an evolving black hole.In the null case, Σ reaches equilibrium for which the shear and the matter flux vanish and this portion is represented by a weakly isolated horizon.Since in this paper we only focus on null horizons, we refer (Asktekar-Krishnan, 2003) for details on DHs and their properties.Here, in order to construct a new physical example of an ENH satisfying the three conclusions of Theorem 4, we use the following Vaidya metric of a spacetime (M, g) which is an explicit example of dynamical horizons with their equilibrium states-the weakly isolated horizons (WIS).Let (v, r, θ, ϕ) be the Eddington-Frinkelstein coordinates (Hawking-Ellis, 1973) of the metric g given by Based on above relation we propose further work on null version of spacelike results proved using DHs.