A Report on Null Horizons in Relativity

any provided the and Abstract This is a review paper on isolated, distorted and time-dependent null horizons by providing up-date information on their role in black hole physics. Geometry of totally umbilical null hyper surfaces has been used to establish an interrelation between these three types of horizons in a unified manner. Distorted horizons describe the near isolated black holes which are dis- torted by the presence of faraway matter. On the other hand, time-dependent null horizons are modeled by a family of totally umbilical null hyper surfaces. A sketch of the proofs of the most important results is presented together with sufficient related references.


Introduction
It is well-known that null hyper surfaces play an important role in the study of a variety of black hole horizons. Shortly after Einstein's first version of the theory of gravitation was published, in 1916 Karl Schwarzschild computed the gravitational fields of stars using Einstein's field equations. He assumed that the star is spherical, gravitationally collapsed and non-rotating. His solution is called a Schwarzschild solution which is an exact solution of static vacuum fields of the pointmass. Since then, considerable work has been done on black hole physics of time-independent and time-dependent space times. The purpose of this paper is to report up-to-date information on this most active area of black hole physics. To include a large number of results and at the same time not to repeat the known material, we quote all the main results with either sketch or reference for their proof. For easily readable to a large audience, technical details are minimized. Hopefully, this paper will serve as a reference for those working on some aspects of geometry and physics of null horizons and also stimulate further research in black hole physics.

Event Horizons
A boundary of a space time is called an event horizon (briefly denoted by EH) beyond which events cannot affect the observer. An EH is intrinsically a global concept since its definition requires the knowledge of the whole space time to determine whether null geodesics can reach null infinity. EHs have played a key role and this includes Hawking's area increase theorem, black hole thermodynamics, black hole perturbation theory and the topological censorship results. Most important family is the Kerr-Newman black holes. Moreover, an EH always exists in black hole asymptotically flat space time under a weak cosmic censorship condition. We refer Hawking's paper on "Event Horizon" [1], three papers of Hajiceke's work [2][3][4] on "Perfect Horizons" (later called "non-expanding horizons" by Ashtekar et al. [5]. Galloway [6] has shown that the null hyper surfaces which arise most naturally in space time geometry and general relativity, such as black hole EHs, are in general C 0 but not C 1 . His approach has its roots in the well-known geometric maximum principle of E. Hopf, a powerful analytic tool which is often used in the theory of minimal or constant mean curvature hyper surfaces. This principle implies that two different minimal hyper surfaces in a Riemannian manifold cannot touch each other from one side. A published proof of this fact is not available, however, for a special case of Euclidean spaces [7]. To understand Galloway's work, we first recall some features of the intrinsic geometry of a 3-dimensional null hyper surface, say Σ, of a space time (M, g) where the metric g ij has signature (−, +, +, +). Denote by ← = ij i j q g the intrinsic degenerate induced metric on Σ which is the pull back of g ij , where an under arrow denotes the pullback to Σ. Degenerate q ij has signature (0, +, +) and does not have an inverse in the standard sense, but, in the weaker sense it admits an inverse q ij if it satisfies q ik q jm q km = q ij . Using this, the expansion ( )     (a) µ ≤ ν on Ω and µ(x 0 ) for some x 0 ∈ Ω, and (b) Ω(ν) ≤ Ω(µ) on Ω.
In year 2000, Galloway [6] proved the following result for smooth null hyper surfaces restricted to the zero mean curvature case and suitable to asymptotically flat space times.
Then Σ 1 and Σ 2 coincide near p and this common null hyper surface has mean curvature θ = 0.
Sketch of the proof: Let Σ 1 and Σ 2 have a common null direction at p and let P be a time like hyper surface in M passing through p and transverse to this direction. Take P so small such that the intersections H 1 = Σ 1 ∩ P and H 2 = Σ 2 ∩ P are smooth space like hyper surfaces in P, with H 2 to the future side of H 1 near p. These two hyper surfaces may be expressed as graphs over a fixed space like hyper surface H in P, with respect to normal coordinates around H. Precisely, let 1 where Q is the mean curvature operator on space like graph over H . Thus θ is a second order quasi-linear elliptic operator. In this case we have: (1) u 1 ≤ u 2 , and u 1 (p) = u 2 (p).
Then above quoted Alexandrov's Theorem 1 implies that u 1 =u 2 . Thus H 1 and H 2 agree near p. The null normal geodesics to Σ 1 and Σ 2 in M will then also agree. Consequently, Σ 1 and Σ 2 agree near p, which completes the proof.

Remark:
Although above maximum principle theorem is for smooth null hyper surfaces, in reality null hyper surfaces occurring in relativity are the null portions of achronal boundaries as the sets ( ), ± = ∂ ⊂ I A A M which are always C 0 hyper surfaces and contain non-differentiable points. For example [6], consider one such set ( ) − Σ = ∂I A where A consists of two disjoint closed disks in the t=0 slice of a Minkowski 3-space. This set can be represented as a merging surface of two truncated cones having a curve of non-differentiable points corresponding to the intersection of the two cones, but, otherwise it is a smooth null hyper surface. The most important feature of these C 0 null hyper surfaces is that they are ruled by null geodesics which are either past or future in extendible and contained in the hyper surface.
Precisely, a C 0 future null hyper surface is a locally achronal topological hyper surface Σ of M which is ruled by in extendible null geodesics. These null geodesics (entirely contained in Σ) are the null geodesic generators of Σ. The important use of Galloway's Theorem 2 is that it extends to an appropriate manner [6] to C 0 null hyper surfaces. Based on this we quote the following physically meaningful Null Splitting Theorem for null hyper surfaces arising in space time geometry: Theorem 3: [6] Let (M, g) be a null geodesically complete space time which obeys the null energy condition, ( , ) 0 ≥ Ric X X for all null vectors X. If M admits a null line η, then, η is contained in a smooth closed achronal totally geodesic null hyper surface Σ.
Physical model: Consider a 4-dimensional stationary space time (M, g) which is chronological, that is, M admits no closed time like curves. It is known [9] that a stationary M admits a smooth 1-parameter group, say G, of isometries whose orbits are time like curves in M. A static space time is stationary with the condition that its time like Killing vector field, say T , is hyper surface orthogonal, that is, there exists a space like hyper surface orthogonal to T. Our model will be applicable to these both types. Denote by ′ M the Hausdorff and para compact 3-dimensional Riemannian orbit space of the action G. The projection : M is a principal R-bundle, with the time like fiber G. Let T = ∂t be the non -vanishing time like Killing vector field, where t is a global time coordinate on M ′ . Then, g induces a Riemannian where η is a connection 1 -form for the R-bundle π. It is known that a stationary space-time (M, g) uniquely determines the orbit data Thus, Anderson's above result implies that only a non-flat M will have a non-empty metric boundary of its orbit space. It turns out that asymptotically flat space times are best physical systems for the non-flat stationary space times, many of them do have Killing horizons.

Distorted horizons in vacuum
Black holes which retain the time-independent character but are non-isolated due to the presence of some external distribution of matter in the neighborhood of a static or stationary black hole are called "Distorted Horizons" which have played an important role in problems involving black holes immersed in external fields or surrounded by matter rings or black hole collisions. The presence of external matter allows the event horizon to be distorted. In 1982, Geroch-Hartle [11] obtained all exact solutions of Einstein's equations that represent static, axisymmetric distorted horizons using a spacetime (M, g) with Weyl metric (see equation (1)). Topologically, the black holes of these distorted horizons are either S 2 × R for the spherical case or S 1 × R × R for the toroidal case. Geroch-Hartle discussed as follows: R g be a space time with Weyl metric g [12] given by where ψ and γ are functions of ρ and z only and Σ is a connected orientable 3-dimensional Riemannian manifold. It is known that all static axisymmetric solutions to Einstein's equations can be expressed by the above form where Σ is orthogonal to a static Killing field. Geroch-Hartle worked on distorted black holes in vacuum, static axisymmetric space times for which the Einstein's equation for ψ is the Laplace equation They observed that spherical and toroidal are the only two possibilities for the topology of horizon cross section C, that is, a φ either has a zero-point in a sphere and Σ must be topologically 2 × R S or it does not vanish on C (resulting in a torus) and Σ must be topologically 1 1 × × R S S . In the spherical case, when ρ2 → 0 on the horizon one can show that Σ = Σ − H , where H is a segment of the axis in Σ with axial Killing field and Σ is an open neighborhood of this H.
The function ψ may be singular in Σ at H. The distorted black hole in the neighborhood of the spherical horizon has the same features as the Schwarzschild holes. On the other hand the geometry of distorted black holes of the toroidal horizon is that of a twisted torus with Killing field φ a and in Σ we have flat tori of constant ρ, each with Killing field of period 2π such that these tori converge when ρ → 0 on the horizon. Again we have Σ = Σ − H where ψ in general singular in Σ at H. However, it is easy to see from the Weyl   . Then, with respect to the system (t, r, θ, φ) above metric transforms into representing the distorted horizon H at r = 2m. In [11] the authors explained how the above axisymmetric metric would evolve into the metric of a 2-sphere through a sequence of equilibrium states, examined the local and global structures of distorted and undistorted holes.
Open problem: "Geroch-Hartle [11] proposed following problem (still remains unsolved): Is there any analogous result for more complicated topology? Suppose the matter were slowly moved from the vicinity of the hole to distant region. How would the hole, which could not, presumably, permit this to happen while retaining its horizon-topology, react? Similarly, what would happen if a spinning gyroscope were dropped into a hole with other than spherical or toroidal topology? The hole could not become rotating consistent with its horizon-topology. To what equilibrium state would it finally settle, and how would it radiate to achieve this state?"

Distorted charged black holes
In 2001, Fairhurst-Krishnan [13] extended the vacuum case in [11] to the solutions of Einstein-Maxwell equations for distorted charged black holes using the Weyl metric (1). We review this case as follows: Let the electromagnetic potential be in the form where Φ and β are the electromagnetic and magnetic potentials, respectively. They assumed β=0 (results also hold for 0 β ≠ ). The field equations are Above field equations being non-linear in ψ, this prevents from "distorting" the known black hole solutions as in the vacuum case of linear Laplace equation. Therefore, they used a mapping [12] which transforms a distorted vacuum solution of a Schwarzschild family to the Reissner-Nordstrom family. Let (ψ, γ, b, c)   ) ( ) Then it follows from the first two equations in (3) that Using (4) and (5), it is easy to verify that ( , , ) ψ γ Φ is a solution to the Einstein-Maxwell equations (3) where B and η are function of ρ and z given by where we choose b = m/e and c a free parameter. It follows from (6)  The horizon is a line segment H on the z-axis with |z| ≤ A. Although the Weyl coordinates cover only the region outside the horizon at r = m + A, the spacetime can be extended through this horizon by transforming (7) into Eddington-Finkelstein coordinates for which the metric is regular at ∆(r = m + A) = 0. To do this transformation we set r . This is the metric regular at the horizon ∆(r) = 0 whose horizon geometry is given by the metric These solutions represent a charged black hole distorted by external matter. Also, these solutions (although regular at the horizon) are not asymptotically flat and so the notion of infinity and an event horizon is not applicable for this case. However, these solutions do admit locally defined isolated horizons [14] (see details on isolated horizons in next section). Nevertheless, there is a way [13] to show that (under some reasonable conditions) this solution can be extended to be asymptotically flat, in which case the horizon will be the event horizon.
They discussed the zeroth and first law for these black holes and proved the first law in two different forms, one using the isolated horizon framework and the other using normalization at infinity. They also suggested that the isolated horizon framework provides a clearer interpretation of the first law for these black holes.

Isolated Horizons
Since to actually locate a black hole one needs to know the full space time metric up to the infinite future and even if one locates the event horizon (EH), using it to calculate the physical parameters is extremely difficult. Also EH is too global to be useful in a number of physical situations ranging from quantum gravity to numerical relativity and to astrophysics. In particular, since it refers to infinity, it cannot be used in especially compact space time. Therefore, attempts were made to find a quasi-local concept of a horizon which requires only minimum number of conditions to detect a black hole and study its properties. To achieve this objective, in a 1999 paper [5] Asktekar et al. introduced the following three notions of isolated horizons, namely, non-expanding, weakly and stronger isolated horizons, respectively: on H does not necessarily imply that l is a Killing vector of the full metric g ij . In general, there does not exist a unique induced connection on H due to degenerate q ij . However, on an NEH, the property £ 0 = l ij q implies that the space time connection∇induces a unique (torsion-free) connection say D, on H which is compatible with q ij .

Definition 7: A WIH ( ,[ ])
l H is called an isolated horizon (IH) if the full connection D is time-independent, that is, if for arbitrary vector fields V tangent to H.
An IH is stronger notion of isolation as its above condition cannot always be satisfied by a choice of null normal. IHs are quasi-local and do not require the knowledge of the whole space time. Any Killing horizon which is topologically 2 × R S is a trivial example of an IH [15][16][17] for examples and their physical use.
On the other hand, we know that the isolated horizons model specifically quasi-local equilibrium regimes of black hole space times. However, in nature, black holes are rarely in equilibrium. This led to research on a quasi-local frame work to describe the geometry of the surface of the black hole, not just at its equilibrium state. First attempt in this direction was made by Hayward [18], in 1994, using the frame work of (2 + 2) -formalism, based on the notion of trapped surfaces. He proposed the following notion of future, outer, trapping horizons (FOTH). . After this, Asktekar-Krishnan [19] observed that in dynamical situations Hayward's condition (iii) is not required for most of the key physical results. For this reason, they studied the following quasilocal concept of dynamical horizons (briefly denoted by DH) which model the evolving black holes and their asymptotic states are isolated horizons. They first required that H be space like everywhere and then studied the case in which portions of marginally trapped surfaces lie on a space like horizon and the remainder on a null horizon. In the null case, H reaches equilibrium for which the shear and the matter flux vanish and this portion is represented by a weakly isolated horizon. The Vaidya metrics are explicit examples of dynamical horizons with their equilibrium states-the isolated horizons. The horizon geometry of DHs is time-dependent. Compared to Hayward's (2 + 2)-formalism, the DH frame work is based on the standard (1 + 3)-formulism and has the advantage that it only refers to the intrinsic structure of H, without any conditions on the evolution of fields in transverse directions to H. DH has provided a new perspective covering all areas of black holes, that is, quantum gravity, mathematical physics, numerical relativity and gravitational wave phenomenology, leading to the underlying unity of the subject. Since in this paper we only focus on null horizons, we refer [19] for a review on dynamical horizons.

Time-Dependent Null Horizons
We know from discussion so far that the event and isolated horizons have a common condition that their future null normal has vanishing expansion, that is, their underlying null hyper surface is totally geodesic in the corresponding space time. Moreover, these horizons are timeindependent. However, in reality the black hole has a cosmological background or it is surrounded by a local mass distribution. Also, black hole grows by swallowing star and galactic debris and electromagnetic as well as gravitational radiation. Therefore, it may cease to be timeindependent. Thus, there is significant difference in the structure and properties of such fully dynamical black holes from the well-known event and isolated black holes. For this reason we now review some works done on finding realistic models of non-isolated and timedependent black hole space times. Among other approaches to get such a model, this raises the possibility of using a metric symmetry from Killing to conformal Killing symmetry. Recall a space time (M, g) admits a conformal Killing vector field (briefly denoted by CKV) l if £ 2 φ = l g g, where φ is a function on M. Using this metric symmetry, we first review two papers of Sultana and Dyer [20,21] as follows:

Conformal killing horizons
In their first paper [20], Sultana and Dyer have studied this problem for those space times which admit a CKV field. They considered a conformal transformation, 2 = Ω G g, to stationary asymptotically flat black hole space times which admits a Killing horizon, say 0 Σ , generated by the Killing vector field l , where Ω is a conformal function. Under such a conformal transformation l is mapped to a CKV. Although space times are asymptotically conformally flat, nevertheless, there can be non-asymptotically flat space times. In this particular paper [20] Sultana-Dyer considered space times admitting a time like CKV which becomes null on a boundary called the conformal stationary limit hyper surface and locally described the time-dependent event horizon by using this boundary, provided that it is a null geodesic hyper surface. Such a horizon is called a conformal Killing horizon, briefly denoted by CKH). They have shown that such a hyper surface of M is null geodesic if and only if the twist of the conformal Killing trajectories on M vanishes. Following is their main result on the extension of the Hawking's strong rigidity theorem [9]. Theorem 10: [20] Let (M, G) be a space time which is conformally related to an analytic black hole space time (M, g), with a Killing horizon Σ 0 , such that the conformal factor in 2 = Ω G ggoes to a constant at null infinity. Then the CKH in (M, G) is globally equivalent to the event horizon, provided that the stress energy tensor satisfies the weak energy condition.

Sketch of proof:
The global definition of the event horizon and the properties of conformal transformations imply that the global definition of an event is conformally invariant, provided the conformal factor tends to a constant at null infinity. This means that, at the null infinity state, the CKV field reduces to the homothetic vector (HV) field. Since, as opposed to the proper conformal symmetry, the Einstein equations are invariant with respect to homothetic symmetry, the structure of homothetic infinity in (M, G) is preserved. This means that in the manifold (M, g), the event horizon is a Killing horizon; while in the conformal manifold (M, G) it is a CKH which reduces to an event horizon at null infinity that completes the proof. This paper also contains the case as to what happens when the conformal stationary limit hyper surface does not coincide with the event horizon at infinity. For this case, they have proved a generalized weak In another paper [21], they gave an example of a dynamical cosmological black hole, which is a space time that describes an expanding black hole in the asymptotic background of the Einstein-de Sitter universe. For this case, the black hole is primordial in the sense that it forms Ab Initio with the big bang singularity and its expanding event horizon is represented by a CKH whose conformal factor goes to a constant at null infinity. The metric representing the black hole space time is obtained by applying a time-dependent conformal transformation on the Schwarzschild metric, such that the result is an exact solution with the matter content described by a perfect fluid and the other a null fluid. They have also studied properties of several physical quantities related to black holes.

Maximum principle for time-dependent null horizons
Observe that Galloway's maximum principle (see Theorem2) was designed for totally geodesic null hyper surfaces suitable only for asymptotically flat space times. Recently, the present author has modified maximum principle suitable for a family of time-dependent totally umbilical null horizons of a large variety of space times not necessarily restricted to asymptotically flat space times, which we present in this section. Recall that a hyper surface (Σ, h) of a space time manifold (M, g) is totally umbilical if and only if there is a smooth function ρ on Σ such that B = ρh, where B denotes the second fundamental form of Σ and h is its induced metric. Σ is proper totally umbilical if and only if ρ is non-zero on entire Σ. In particular, Σ is totally geodesic if and only if B vanishes, i.e., if and only if ρ vanishes on Σ. Let (M, g) be an n-dimensional space time manifold evolved from a space like hyper surface H t at a coordinate time t to another space like hyper surface H t+dt at coordinate time t + dt whose metric g is given by where 0 = x t , and ( 1, , 1) = −  a x a n are spatial coordinates of the hyper surface H t with γ ab its (n − 1)-metric induced from g, 1 1 ( , , , ) is the lapse function and U is a space like shift vector.
The coordinate time vector ∂ = ∂t t is such that ( , ) 1 = g dt t . We write where n is the future time like unit vector field. In general, each space like hyper surface H t intersects a null hyper surface say Σ on some (n − 2)-dimensional sub manifold t S , that is, . Consider a family ( ) , is an element of the family S a foliation of ( ) Σ u , the coordinate t can be used as a nonaffine parameter along each null geodesic generating each Σ u . We normalize each member l of F such that it is tangent vector associated with this parameterization of the null generators, i.e., . This means that l is a vector field "dual" to the 1-form dt. Equivalently, the function t can be regarded as a coordinate compatible with l, i.e.,  S . Then, we define a transversal vector field k u of p T M not belonging to F expressed as another suitable linear combination of n u and s u such that it represents the light rays emitted in the opposite direction, called the ingoing direction, satisfying: We now state and reproduce complete proof of the following modified maximum principle. h of totally umbilical smooth null hyper surfaces of a space time (M, g) whose metric g is given by (8). Suppose x are its coordinates centered on p. Express t S as a graph over t V , that is, S t = graph µ = {(µ(x), x) ∈ Σ ∩ H t : x ∈ V t }, where µ ∈ C ∞ (V t ). Let Q(µ) be the mean curvature of S t = graph µ and G be the Riemannian metric on t V whose components are given by where µ is a 2 C U -admissible function, αβ a , The operator Q will also satisfy as elliptic operator and, therefore, ( ) µ = Q Q is a second order quasi-linear elliptic operator. Take ( ) = + l n s a future null normal vector field on Σ. Denote by B H and B S the second fundamental forms of H t and S t , respectively. Then, space times and in some cases their relation with the event and isolated horizons. Also examining the similarity and difference between the Theorem 12 with the two papers of (Sultana and Dyer) (see Subsection 4.1) related to common issue of time-dependent null horizons, it is clear that although their result on time-dependent null conformal Killing horizons is similar with the two conclusions of Theorem 12, but, it is only limited to null hyper surfaces of asymptotically flat space times whereas Theorem 12 is applicable to a variety of space times admitting a time like conformal Killing vector field.

Some Related Results on Black Hole Physics
In view of a very large number of excellent papers appearing in this field we present here a brief on some selected papers closely related to the material presented so far.
(1) On the global structure of solutions (primarily related to event horizons) we refer a review article by Chrusciel [25]. His work included quasi-local mass, strong cosmic censorship, non-linear stability, new construction of solutions of the constraint equations, improved under-standing of asymptotic properties of the solutions, existence of solutions with low regularity, and construction of initial data with trapped surfaces or apparent horizons.
(2) Attempts have been made to extend the black hole mechanics by replacing the use of event horizons in stationary space times with isolated horizons in some dynamical space times and in some cases those space times which admit radiation close to black holes. In a paper by Ashtekar-Beetle-Lewandowski [26] they stated that so far such an extension is restricted to non-rotating black holes. In their paper they have filled this important gap by extending the first law to the rotating case.
(3) It is well-known that a maximally rotating Kerr black hole is said to be external. On this issue we refer a paper (with some related references cited therein) of Booth and Fairhurst [27]. These authors have studied three characterizations of extremality. They presented a way how the standard notions for Kerr black hole do not require the horizon to be either stationary or rotationally symmetric. They studied physical implications and applications of these results. In particular, they have examined how close a horizon is to extremality and should be calculable in numerical simulations.
(4) Kunduri and Lucietti [28] have recently studied a new infinite class of near-horizon geometries of null horizons which satisfy Einstein's equations for all odd dimensions greater than five. The symmetry and topology of these solutions is compatible with those of black holes. They have studied those horizon cross manifolds which all possess Sasakian horizon metrics.