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2.1 Isolated horizons

In this part, we provide the basic definitions and discuss geometrical properties of non-expanding, weakly isolated, and isolated horizons which describe black holes which are in equilibrium in an increasingly stronger sense.

These horizons model black holes which are themselves in equilibrium, but in possibly dynamical space-times [1213Jump To The Next Citation Point25Jump To The Next Citation Point15Jump To The Next Citation Point]. For early references with similar ideas, see [156106]. A useful example is provided by the late stage of a gravitational collapse shown in Figure 1View Image. In such physical situations, one expects the back-scattered radiation falling into the black hole to become negligible at late times so that the ‘end portion’ of the event horizon (labelled by Δ in the figure) can be regarded as isolated to an excellent approximation. This expectation is borne out in numerical simulations where the backscattering effects typically become smaller than the numerical errors rather quickly after the formation of the black hole (see, e.g., [33Jump To The Next Citation Point46Jump To The Next Citation Point]).

2.1.1 Definitions

The key idea is to extract from the notion of a Killing horizon the minimal conditions which are necessary to define physical quantities such as the mass and angular momentum of the black hole and to establish the zeroth and the first laws of black hole mechanics. Like Killing horizons, isolated horizons are null, 3-dimensional sub-manifolds of space-time. Let us therefore begin by recalling some essential features of such sub-manifolds, which we will denote by Δ. The intrinsic metric qab on Δ has signature (0,+,+), and is simply the pull-back of the space-time metric to Δ, qab = g←a−b, where an underarrow indicates the pullback to Δ. Since qab is degenerate, it does not have an inverse in the standard sense. However, it does admit an inverse in a weaker sense: qab will be said to be an inverse of qab if it satisfies qamqbnqmn = qab. As one would expect, the inverse is not unique: We can always add to ab q a term of the type (a b) ℓ V, where a ℓ is a null normal to Δ and b V any vector field tangential to Δ. All our constructions will be insensitive to this ambiguity. Given a null normal ℓa to Δ, the expansion Θ (ℓ) is defined as

ab Θ (ℓ) := q ∇a ℓb. (3 )
(Throughout this review, we will assume that ℓa is future directed.) We can now state the first definition:

Definition 1: A sub-manifold Δ of a space-time (ℳ, gab) is said to be a non-expanding horizon (NEH) if

  1. Δ is topologically 2 S × ℝ and null;
  2. any null normal ℓa of Δ has vanishing expansion, Θ (ℓ) = 0; and
  3. all equations of motion hold at Δ and the stress energy tensor Tab is such that − T aℓb b is future-causal for any future directed null normal a ℓ.

The motivation behind this definition can be summarized as follows. Condition 1 is imposed for definiteness; while most geometric results are insensitive to topology, the S2 × ℝ case is physically the most relevant one. Condition 3 is satisfied by all classical matter fields of direct physical interest. The key condition in the above definition is Condition 2 which is equivalent to requiring that every cross-section of Δ be marginally trapped. (Note incidentally that if Θ(ℓ) vanishes for one null normal ℓa to Δ, it vanishes for all.) Condition 2 is equivalent to requiring that the infinitesimal area element is Lie dragged by the null normal a ℓ. In particular, then, Condition 2 implies that the horizon area is ‘constant in time’. We will denote the area of any cross section of Δ by aΔ and define the horizon radius as ∘ ------- R Δ := aΔ ∕4π.

Because of the Raychaudhuri equation, Condition 2 also implies

a b ab Rabℓ ℓ + σabσ = 0, (4 )
where σab is the shear of ℓa, defined by σab := ∇ (aℓb) − 1Θ (ℓ)qab ← 2, where the underarrow denotes ‘pull-back to Δ’. Now the energy condition 3 implies that R ℓaℓb ab is non-negative, whence we conclude that each of the two terms in the last equation vanishes. This in turn implies that ←Ta−−bℓ−b−−= 0 and ∇←−(−a−−ℓ−b) = 0 on Δ. The first of these equations constrains the matter fields on Δ in an interesting way, while the second is equivalent to ℒ ℓqab = 0 on Δ. Thus, the intrinsic metric on an NEH is ‘time-independent’; this is the sense in which an NEH is in equilibrium.

The zeroth and first laws of black hole mechanics require an additional structure, which is provided by the concept of a weakly isolated horizon. To arrive at this concept, let us first introduce a derivative operator 𝒟 on Δ. Because q ab is degenerate, there is an infinite number of (torsion-free) derivative operators which are compatible with it. However, on an NEH, the property ←∇ −(a−−ℓ−b−) = 0 implies that the space-time connection ∇ induces a unique (torsion-free) derivative operator 𝒟 on Δ which is compatible with qab [25Jump To The Next Citation Point137Jump To The Next Citation Point]. Weakly isolated horizons are characterized by the property that, in addition to the metric q ab, the connection component 𝒟 ℓb a is also ‘time independent’.

Two null normals ℓa and &tidle;ℓa to an NEH Δ are said to belong to the same equivalence class [ℓ] if &tidle;ℓa = cℓa for some positive constant c. Then, weakly isolated horizons are defined as follows:

Definition 2: The pair (Δ, [ℓ]) is said to constitute a weakly isolated horizon (WIH) provided Δ is an NEH and each null normal ℓa in [ℓ] satisfies

(ℒ ℓ𝒟a − 𝒟a ℒ ℓ)ℓb = 0. (5 )

It is easy to verify that every NEH admits null normals satisfying Equation (5View Equation), i.e., can be made a WIH with a suitable choice of [ℓ]. However the required equivalence class is not unique, whence an NEH admits distinct WIH structures [15Jump To The Next Citation Point].

Compared to conditions required of a Killing horizon, conditions in this definition are very weak. Nonetheless, it turns out that they are strong enough to capture the notion of a black hole in equilibrium in applications ranging from black hole mechanics to numerical relativity. (In fact, many of the basic notions such as the mass and angular momentum are well-defined already on NEHs although intermediate steps in derivations use a WIH structure.) This is quite surprising at first because the laws of black hole mechanics were traditionally proved for globally stationary black holes [184], and the definitions of mass and angular momentum of a black hole first used in numerical relativity implicitly assumed that the near horizon geometry is isometric to Kerr [5].

Although the notion of a WIH is sufficient for most applications, from a geometric viewpoint, a stronger notion of isolation is more natural: The full connection 𝒟 should be time-independent. This leads to the notion of an isolated horizon.

Definition 3: A WIH (Δ, [ℓ]) is said to constitute an isolated horizon (IH) if

(ℒ 𝒟 − 𝒟 ℒ )V b = 0 (6 ) ℓ a a ℓ
for arbitrary vector fields V a tangential to Δ.

While an NEH can always be given a WIH structure simply by making a suitable choice of the null normal, not every WIH admits an IH structure. Thus, the passage from a WIH to an IH is a genuine restriction [15Jump To The Next Citation Point]. However, even for this stronger notion of isolation, conditions in the definition are local to Δ. Furthermore, the definition only uses quantities intrinsic to Δ; there are no restrictions on components of any fields transverse to Δ. (Even the full 4-metric gab need not be time independent on the horizon.) Robinson–Trautman solutions provide explicit examples of isolated horizons which do not admit a stationary Killing field even in an arbitrarily small neighborhood of the horizon [66Jump To The Next Citation Point]. In this sense, the conditions in this definition are also rather weak. One expects them to be met to an excellent degree of approximation in a wide variety of situations representing late stages of gravitational collapse and black hole mergers2.

2.1.2 Examples

The class of space-times admitting NEHs, WIHs, and IHs is quite rich. First, it is trivial to verify that any Killing horizon which is topologically S2 × ℝ is also an isolated horizon. This in particular implies that the event horizons of all globally stationary black holes, such as the Kerr–Newman solutions (including a possible cosmological constant), are isolated horizons. (For more exotic examples, see [155].)

View Image

Figure 3: Set-up of the general characteristic initial value formulation. The Weyl tensor component Ψ0 on the null surface Δ is part of the free data which vanishes if Δ is an IH.

But there exist other non-trivial examples as well. These arise because the notion is quasi-local, referring only to fields defined intrinsically on the horizon. First, let us consider the sub-family of Kastor–Traschen solutions [126152] which are asymptotically de Sitter and admit event horizons. They are interpreted as containing multiple charged, dynamical black holes in presence of a positive cosmological constant. Since these solutions do not appear to admit any stationary Killing fields, no Killing horizons are known to exist. Nonetheless, the event horizons of individual black holes are WIHs. However, to our knowledge, no one has checked if they are IHs. A more striking example is provided by a sub-family of Robinson–Trautman solutions, analyzed by Chrusciel [66]. These space-times admit IHs whose intrinsic geometry is isomorphic to that of the Schwarzschild isolated horizons but in which there is radiation arbitrarily close to Δ.

More generally, using the characteristic initial value formulation [91161], Lewandowski [141] has constructed an infinite dimensional set of local examples. Here, one considers two null surfaces Δ and 𝒩 intersecting in a 2-sphere S (see Figure 3View Image). One can freely specify certain data on these two surfaces which then determines a solution to the vacuum Einstein equations in a neighborhood of S bounded by Δ and 𝒩, in which Δ is an isolated horizon.

2.1.3 Geometrical properties

Rescaling freedom in ℓa
 
As we remarked in Section 2.1.1, there is a functional rescaling freedom in the choice of a null normal on an NEH and, while the choice of null normals is restricted by the weakly isolated horizon condition (5View Equation), considerable freedom still remains. That is, a given NEH Δ admits an infinite number of WIH structures (Δ, [ℓ]) [15Jump To The Next Citation Point].

On IHs, by contrast, the situation is dramatically different. Given an IH (Δ, [ℓ]), generically the Condition (6View Equation) in Definition 3 can not be satisfied by a distinct equivalence class of null normals [ℓ′]. Thus on a generic IH, the only freedom in the choice of the null normal is that of a rescaling by a positive constant [15Jump To The Next Citation Point]. This freedom mimics the properties of a Killing horizon since one can also rescale the Killing vector by an arbitrary constant. The triplet a (qab,𝒟a,[ℓ ]) is said to constitute the geometry of the isolated horizon.

Surface gravity
 
Let us begin by defining a 1-form ωa which will be used repeatedly. First note that, by Definition 1, a ℓ is expansion free and shear free. It is automatically twist free since it is a normal to a smooth hypersurface. This means that the contraction of ∇a ℓb with any two vectors tangent to Δ is identically zero, whence there must exist a 1-form ω a on Δ such that for any V a tangent to Δ,
V a∇ ℓb = V aω ℓb. (7) a a
Note that the WIH condition (5View Equation) requires simply that ωa be time independent, ℒℓωa = 0. Given ωa, the surface gravity κ(ℓ) associated with a null normal ℓa is defined as
κ(ℓ) := ℓaωa. (8)
Thus, κ(ℓ) is simply the acceleration of a ℓ. Note that the surface gravity is not an intrinsic property of a WIH (Δ, [ℓ]). Rather, it is a property of a null normal to Δ: κ (cℓ) = cκ (ℓ). An isolated horizon with κ(ℓ) = 0 is said to be an extremal isolated horizon. Note that while the value of surface gravity refers to a specific null normal, whether a given WIH is extremal or not is insensitive to the permissible rescaling of the normal.

Curvature tensors on Δ
 
Consider any (space-time) null tetrad (ℓa,na,ma, m¯a ) on Δ such that ℓa is a null normal to Δ. Then, it follows from Definition 1 that two of the Newman–Penrose Weyl components vanish on Δ: a b c d Ψ0 := Cabcdℓ m ℓ m = 0 and a b c d Ψ1 := Cabcdℓ m ℓ n = 0. This in turn implies that Ψ2 := Cabcdℓamb ¯mcnd is gauge invariant (i.e., does not depend on the specific choice of the null tetrad satisfying the condition stated above.) The imaginary part of Ψ 2 is related to the curl of ωa,
d ω = 2 (Im Ψ2 )ε, (9)
where εab is the natural area 2-form on Δ. Horizons on which Im Ψ2 vanishes are said to be non-rotating: On these horizons all angular momentum multipoles vanish [23Jump To The Next Citation Point]. Therefore, Im Ψ2 is sometimes referred to as the rotational scalar and ωa as the rotation 1-form of the horizon.

Next, let us consider the Ricci-tensor components. On any NEH Δ we have: Φ00 := 12Rabℓaℓb := 0, Φ01 := 12Rabℓamb = 0. In the Einstein–Maxwell theory, one further has: On Δ, Φ := 1R mamb = 0 02 2 ab and Φ := 1R m¯a ¯mb = 0 20 2 ab.

Free data on an isolated horizon
 
Given the geometry (qab,𝒟, [ℓ]) of an IH, it is natural to ask for the minimum amount of information, i.e., the free data, required to construct it. This question has been answered in detail (also for WIHs) [15Jump To The Next Citation Point]. For simplicity, here we will summarize the results only for the non-extremal case. (For the extremal case, see [15Jump To The Next Citation Point143Jump To The Next Citation Point].)

Let S be a spherical cross section of Δ. The degenerate metric qab naturally projects to a Riemannian metric &tidle;qab on S, and similarly the 1-form ωa of Equation (7View Equation) projects to a 1-form &tidle;ω a on S. If the vacuum equations hold on Δ, then given (&tidle;q , &tidle;ω ) ab a on S, there is, up to diffeomorphisms, a unique non-extremal isolated horizon geometry (qab,𝒟, [ℓ]) such that &tidle;qab is the projection of qab, &tidle;ωa is the projection of the ωa constructed from 𝒟, and κ (ℓ) = ωaℓa ⁄= 0. (If the vacuum equations do not hold, the additional data required is the projection on S of the space-time Ricci tensor.)

The underlying reason behind this result can be sketched as follows. First, since qab is degenerate along ℓa, its non-trivial part is just its projection &tidle;qab. Second, &tidle;qab fixes the connection on S; it is only the quantity Sab := 𝒟anb that is not constrained by &tidle;qab, where na is a 1-form on Δ orthogonal to S, normalized so that a ℓ na = − 1. It is easy to show that Sab is symmetric and the contraction of one of its indices with ℓa gives ω a: ℓaS = ω ab a. Furthermore, it turns out that if ω ℓa ⁄= 0 a, the field equations completely determine the angular part of Sab in terms of &tidle;ωa and &tidle;qab. Finally, recall that the surface gravity is not fixed on Δ because of the rescaling freedom in ℓa; thus the ℓ-component of ωa is not part of the free data. Putting all these facts together, we see that the pair (&tidle;qab,ω&tidle;a ) enables us to reconstruct the isolated horizon geometry uniquely up to diffeomorphisms.

Rest frame of a non-expanding horizon
 
As at null infinity, a preferred foliation of Δ can be thought of as providing a ‘rest frame’ for an isolated horizon. On the Schwarzschild horizon, for example, the 2-spheres on which the Eddington–Finkelstein advanced time coordinate is constant – which are also integral manifolds of the rotational Killing fields – provide such a rest frame. For the Kerr metric, this foliation generalizes naturally. The question is whether a general prescription exists to select such a preferred foliation.

On any non-extremal NEH, the 1-form ωa can be used to construct preferred foliations of Δ. Let us first examine the simpler, non-rotating case in which Im Ψ2 = 0. Then Equation (9View Equation) implies that ω a is curl-free and therefore hypersurface orthogonal. The 2-surfaces orthogonal to ωa must be topologically 2 S because, on any non-extremal horizon, a ℓ ωa ⁄= 0. Thus, in the non-rotating case, every isolated horizon comes equipped with a preferred family of cross-sections which defines the rest frame [25Jump To The Next Citation Point]. Note that the projection &tidle;ωa of ωa on any leaf of this foliation vanishes identically.

The rotating case is a little more complicated since ωa is then no longer curl-free. Now the idea is to exploit the fact that the divergence of the projection &tidle;ωa of ωa on a cross-section is sensitive to the choice of the cross-section, and to select a preferred family of cross-sections by imposing a suitable condition on this divergence [15Jump To The Next Citation Point]. A mathematically natural choice is to ask that this divergence vanish. However, (in the case when the angular momentum is non-zero) this condition does not pick out the v = const. cuts of the Kerr horizon where v is the (Carter generalization of the) Eddington–Finkelstein coordinate. Pawlowski has provided another condition that also selects a preferred foliation and reduces to the v = const. cuts of the Kerr horizon:

div &tidle;ω = − &tidle;Δ ln|Ψ |1,∕3 (10) 2
where Δ&tidle; is the Laplacian of &tidle;qab. On isolated horizons on which |Ψ2 | is nowhere zero – a condition satisfied if the horizon geometry is ‘near’ that of the Kerr isolated horizon – this selects a preferred foliation and hence a rest frame. This construction is potentially useful to numerical relativity.

Symmetries of an isolated horizon
 
By definition, a symmetry of an IH (Δ, [ℓ]) is a diffeomorphism of Δ which preserves the geometry (q ,𝒟, [ℓ]) ab. (On a WIH, the symmetry has to preserve (q ,ω ,[ℓ]) ab a. There are again three universality classes of symmetry groups as on an IH.) Let us denote the symmetry group by G Δ. First note that diffeomorphisms generated by the null normals in [ℓa] are symmetries; this is already built into the very definition of an isolated horizon. The other possible symmetries are related to the cross-sections of Δ. Since we have assumed the cross-sections to be topologically spherical and since a metric on a sphere can have either exactly three, one or zero Killing vectors, it follows that G Δ can be of only three types [14Jump To The Next Citation Point]:

In the asymptotically flat context, boundary conditions select a universal symmetry group at spatial infinity, e.g., the Poincaré group, because the space-time metric approaches a fixed Minkowskian one. The situation is completely different in the strong field region near a black hole. Because the geometry at the horizon can vary from one space-time to another, the symmetry group is not universal. However, the above result shows that the symmetry group can be one of only three universality classes.


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