These horizons model black holes which are themselves in equilibrium, but in possibly dynamical space-times [12, 13, 25, 15]. For early references with similar ideas, see [156, 106]. A useful example is provided by the late stage of a gravitational collapse shown in Figure 1. 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 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., [33, 46]).
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 . The intrinsic metric on has signature (0,+,+), and is simply the pull-back of the space-time metric to , , where an underarrow indicates the pullback to . Since is degenerate, it does not have an inverse in the standard sense. However, it does admit an inverse in a weaker sense: will be said to be an inverse of if it satisfies . As one would expect, the inverse is not unique: We can always add to a term of the type , where is a null normal to and any vector field tangential to . All our constructions will be insensitive to this ambiguity. Given a null normal to , the expansion is defined as
(Throughout this review, we will assume that is future directed.) We can now state the first definition:Definition 1: A sub-manifold of a space-time is said to be a non-expanding horizon (NEH) if
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 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 be marginally trapped. (Note incidentally that if vanishes for one null normal to , it vanishes for all.) Condition 2 is equivalent to requiring that the infinitesimal area element is Lie dragged by the null normal . In particular, then, Condition 2 implies that the horizon area is ‘constant in time’. We will denote the area of any cross section of by and define the horizon radius as .
Because of the Raychaudhuri equation, Condition 2 also implies
where is the shear of , defined by , where the underarrow denotes ‘pull-back to ’. Now the energy condition 3 implies that is non-negative, whence we conclude that each of the two terms in the last equation vanishes. This in turn implies that and on . The first of these equations constrains the matter fields on in an interesting way, while the second is equivalent to on . 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 on . Because is degenerate, there is an infinite number of (torsion-free) derivative operators which are compatible with it. However, on an NEH, the property implies that the space-time connection induces a unique (torsion-free) derivative operator on which is compatible with [25, 137]. Weakly isolated horizons are characterized by the property that, in addition to the metric , the connection component is also ‘time independent’.
Two null normals and to an NEH are said to belong to the same equivalence class if for some positive constant . Then, weakly isolated horizons are defined as follows:
Definition 2: The pair is said to constitute a weakly isolated horizon (WIH) provided is an NEH and each null normal in satisfies
It is easy to verify that every NEH admits null normals satisfying Equation (5), i.e., can be made a WIH with a suitable choice of . However the required equivalence class is not unique, whence an NEH admits distinct WIH structures [15].
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 should be time-independent. This leads to the notion of an isolated horizon.
Definition 3: A WIH is said to constitute an isolated horizon (IH) if
for arbitrary vector fields tangential to .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 [15]. However, even for this stronger notion of isolation, conditions in the definition are local to . Furthermore, the definition only uses quantities intrinsic to ; there are no restrictions on components of any fields transverse to . (Even the full 4-metric 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 [66]. 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 mergers^{2}.
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 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].)
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 [126, 152] 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 .
More generally, using the characteristic initial value formulation [91, 161], Lewandowski [141] has constructed an infinite dimensional set of local examples. Here, one considers two null surfaces and intersecting in a 2-sphere (see Figure 3). One can freely specify certain data on these two surfaces which then determines a solution to the vacuum Einstein equations in a neighborhood of bounded by and , in which is an isolated horizon.
On IHs, by contrast, the situation is dramatically different. Given an IH , generically the Condition (6) in Definition 3 can not be satisfied by a distinct equivalence class of null normals . Thus on a generic IH, the only freedom in the choice of the null normal is that of a rescaling by a positive constant [15]. This freedom mimics the properties of a Killing horizon since one can also rescale the Killing vector by an arbitrary constant. The triplet is said to constitute the geometry of the isolated horizon.
Next, let us consider the Ricci-tensor components. On any NEH we have: , . In the Einstein–Maxwell theory, one further has: On , and .
Let be a spherical cross section of . The degenerate metric naturally projects to a Riemannian metric on , and similarly the 1-form of Equation (7) projects to a 1-form on . If the vacuum equations hold on , then given on , there is, up to diffeomorphisms, a unique non-extremal isolated horizon geometry such that is the projection of , is the projection of the constructed from , and . (If the vacuum equations do not hold, the additional data required is the projection on of the space-time Ricci tensor.)
The underlying reason behind this result can be sketched as follows. First, since is degenerate along , its non-trivial part is just its projection . Second, fixes the connection on ; it is only the quantity that is not constrained by , where is a 1-form on orthogonal to , normalized so that . It is easy to show that is symmetric and the contraction of one of its indices with gives : . Furthermore, it turns out that if , the field equations completely determine the angular part of in terms of and . Finally, recall that the surface gravity is not fixed on because of the rescaling freedom in ; thus the -component of is not part of the free data. Putting all these facts together, we see that the pair enables us to reconstruct the isolated horizon geometry uniquely up to diffeomorphisms.
On any non-extremal NEH, the 1-form can be used to construct preferred foliations of . Let us first examine the simpler, non-rotating case in which . Then Equation (9) implies that is curl-free and therefore hypersurface orthogonal. The 2-surfaces orthogonal to must be topologically because, on any non-extremal horizon, . Thus, in the non-rotating case, every isolated horizon comes equipped with a preferred family of cross-sections which defines the rest frame [25]. Note that the projection of on any leaf of this foliation vanishes identically.
The rotating case is a little more complicated since is then no longer curl-free. Now the idea is to exploit the fact that the divergence of the projection of 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 [15]. 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 cuts of the Kerr horizon where is the (Carter generalization of the) Eddington–Finkelstein coordinate. Pawlowski has provided another condition that also selects a preferred foliation and reduces to the cuts of the Kerr horizon:
where is the Laplacian of . On isolated horizons on which 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.
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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