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1 Introduction

Research inspired by black holes has dominated several areas of gravitational physics since the early seventies. The mathematical theory turned out to be extraordinarily rich and full of surprises. Laws of black hole mechanics brought out deep, unsuspected connections between classical general relativity, quantum physics, and statistical mechanics [34Jump To The Next Citation Point42Jump To The Next Citation Point43Jump To The Next Citation Point44]. In particular, they provided a concrete challenge to quantum gravity which became a driving force for progress in that area. On the classical front, black hole uniqueness theorems [67119] took the community by surprise. The subsequent analysis of the detailed properties of Kerr–Newman solutions [63] and perturbations thereof [64Jump To The Next Citation Point] constituted a large fraction of research in mathematical general relativity in the seventies and eighties. Just as the community had come to terms with the uniqueness results, it was surprised yet again by the discovery of hairy black holes [37Jump To The Next Citation Point47]. Research in this area continues to be an active branch of mathematical physics [182Jump To The Next Citation Point]. The situation has been similar in numerical relativity. Since its inception, much of the research in this area has been driven by problems related to black holes, particularly their formation through gravitational collapse [153Jump To The Next Citation Point], the associated critical phenomenon [65105], and the dynamics leading to their coalescence (see, e.g., [1127160562139Jump To The Next Citation Point]). Finally, black holes now play a major role in relativistic astrophysics, providing mechanisms to fuel the most powerful engines in the cosmos. They are also among the most promising sources of gravitational waves, leading to new avenues to confront theory with experiments [77Jump To The Next Citation Point].

Thus there has been truly remarkable progress on many different fronts. Yet, as the subject matured, it became apparent that the basic theoretical framework has certain undesirable features from both conceptual and practical viewpoints. Nagging questions have persisted, suggesting the need of a new paradigm.

Dynamical situations
 
For fully dynamical black holes, apart from the ‘topological censorship’ results which restrict the horizon topology [11089], there has essentially been only one major result in exact general relativity. This is the celebrated area theorem proved by Hawking in the early seventies [111Jump To The Next Citation Point113Jump To The Next Citation Point]: If matter satisfies the null energy condition, the area of the black hole event horizon can never decrease. This theorem has been extremely influential because of its similarity with the second law of thermodynamics. However, it is a qualitative result; it does not provide an explicit formula for the amount by which the area increases in physical processes. Now, for a black hole of mass M, angular momentum J, area a, surface gravity κ, and angular velocity Ω, the first law of black hole mechanics,
κ δM = ----δa + Ω δJ, (1) 8πG
does relate the change in the horizon area to that in the energy and angular momentum, as the black hole makes a transition from one equilibrium state to a nearby one [34184Jump To The Next Citation Point]. This suggests that there may well be a fully dynamical version of Equation (1View Equation) which relates the change in the black hole area to the energy and angular momentum it absorbs in fully dynamical processes in which the black hole makes a transition from a given state to one which is far removed. Indeed, without such a formula, one would have no quantitative control on how black holes grow in exact general relativity. Note however that the event horizons can form and grow even in a flat region of space-time (see Figure 4View Image and Section 2.2.2 for illustrations). During this phase, the area grows in spite of the fact that there is no flux of energy or angular momentum across the event horizon. Hence, in the standard framework where the surface of the black hole is represented by an event horizon, it is impossible to obtain the desired formula. Is there then a more appropriate notion that can replace event horizons?

Equilibrium situations
 
The zeroth and first laws of black hole mechanics refer to equilibrium situations and small departures therefrom. Therefore, in this context, it is natural to focus on isolated black holes. It was customary to represent them by stationary solutions of field equations, i.e, solutions which admit a time-translational Killing vector field everywhere, not just in a small neighborhood of the black hole. While this simple idealization was natural as a starting point, it is overly restrictive. Physically, it should be sufficient to impose boundary conditions at the horizon which ensure only that the black hole itself is isolated. That is, it should suffice to demand only that the intrinsic geometry of the horizon be time independent, whereas the geometry outside may be dynamical and admit gravitational and other radiation. Indeed, we adopt a similar viewpoint in ordinary thermodynamics; while studying systems such as a classical gas in a box, one usually assumes that only the system under consideration is in equilibrium, not the whole world. In realistic situations, one is typically interested in the final stages of collapse where the black hole has formed and ‘settled down’ or in situations in which an already formed black hole is isolated for the duration of the experiment (see Figure 1View Image). In such cases, there is likely to be gravitational radiation and non-stationary matter far away from the black hole. Thus, from a physical perspective, a framework which demands global stationarity is too restrictive.
View Image

Figure 1: Left panel: A typical gravitational collapse. The portion Δ of the event horizon at late times is isolated. Physically, one would expect the first law to apply to Δ even though the entire space-time is not stationary because of the presence of gravitational radiation in the exterior region. Right panel: Space-time diagram of a black hole which is initially in equilibrium, absorbs a finite amount of radiation, and again settles down to equilibrium. Portions Δ1 and Δ2 of the horizon are isolated. One would expect the first law to hold on both portions although the space-time is not stationary.

Even if one were to ignore these conceptual considerations and focus just on results, the framework has certain unsatisfactory features. Consider the central result, the first law of Equation (1View Equation). Here, the angular momentum J and the mass M are defined at infinity while the angular velocity Ω and surface gravity κ are defined at the horizon. Because one has to go back and forth between the horizon and infinity, the physical meaning of the first law is not transparent1. For instance, there may be matter rings around the black hole which contribute to the angular momentum and mass at infinity. Why is this contribution relevant to the first law of black hole mechanics? Shouldn’t only the angular momentum and mass of the black hole feature in the first law? Thus, one is led to ask: Is there a more suitable paradigm which can replace frameworks based on event horizons in stationary space-times?

Entropy calculations
 
The first and the second laws suggest that one should assign to a black hole an entropy which is proportional to its area. This poses a concrete challenge to candidate theories of quantum gravity: Account for this entropy from fundamental, statistical mechanical considerations. String theory has had a remarkable success in meeting this challenge in detail for a subclass of extremal, stationary black holes whose charge equals mass (the so-called BPS states) [120]. However, for realistic black holes the charge to mass ratio is less than 10 −18. It has not been possible to extend the detailed calculation to realistic cases where charge is negligible and matter rings may distort the black hole horizon. From a mathematical physics perspective, the entropy calculation should also encompass hairy black holes whose equilibrium states cannot be characterized just by specifying the mass, angular momentum and charges at infinity, as well as non-minimal gravitational couplings, in presence of which the entropy is no longer a function just of the horizon area. One may therefore ask if other avenues are available. A natural strategy is to consider the sector of general relativity containing an isolated black hole and carry out its quantization systematically. A pre-requisite for such a program is the availability of a manageable action principle and/or Hamiltonian framework. Unfortunately, however, if one attempts to construct these within the classical frameworks traditionally used to describe black holes, one runs into two difficulties. First, because the event horizon is such a global notion, no action principle is known for the sector of general relativity containing geometries which admit an event horizon as an internal boundary. Second, if one restricts oneself to globally stationary solutions, the phase space has only a finite number of true degrees of freedom and is thus ‘too small’ to adequately incorporate all quantum fluctuations. Thus, again, we are led to ask: Is there a more satisfactory framework which can serve as the point of departure for a non-perturbative quantization to address this problem?

Global nature of event horizons
 
The future event horizon is defined as the future boundary of the causal past of future null infinity. While this definition neatly encodes the idea that an outside observer can not ‘look into’ a black hole, it is too global for many applications. First, since it refers to null infinity, it can not be used in spatially compact space-times. Surely, one should be able to analyze black hole dynamics also in these space-times. More importantly, the notion is teleological; it lets us speak of a black hole only after we have constructed the entire space-time. Thus, for example, an event horizon may well be developing in the room you are now sitting in anticipation of a gravitational collapse that may occur in this region of our galaxy a million years from now. When astrophysicists say that they have discovered a black hole in the center of our galaxy, they are referring to something much more concrete and quasi-local than an event horizon. Is there a satisfactory notion that captures what they are referring to?
View Image

Figure 2: A spherical star of mass M undergoes collapse. Much later, a spherical shell of mass δM falls into the resulting black hole. While Δ1 and Δ2 are both isolated horizons, only Δ2 is part of the event horizon.

The teleological nature of event horizons is also an obstruction to extending black hole mechanics in certain physical situations. Consider for example, Figure 2View Image in which a spherical star of mass M undergoes a gravitational collapse. The singularity is hidden inside the null surface Δ1 at r = 2M which is foliated by a family of marginally trapped surfaces and would be a part of the event horizon if nothing further happens. Suppose instead, after a million years, a thin spherical shell of mass δM collapses. Then Δ1 would not be a part of the event horizon which would actually lie slightly outside Δ1 and coincide with the surface r = 2(M + δM ) in the distant future. On physical grounds, it seems unreasonable to exclude Δ1 a priori from thermodynamical considerations. Surely one should be able to establish the standard laws of mechanics not only for the equilibrium portion of the event horizon but also for Δ1.

Next, let us consider numerical simulations of binary black holes. Here the main task is to construct the space-time containing evolving black holes. Thus, one needs to identify initial data containing black holes without the knowledge of the entire space-time and evolve them step by step. The notion of a event horizon is clearly inadequate for this. One uses instead the notion of apparent horizons (see Section 2.2). One may then ask: Can we use apparent horizons instead of event horizons in other contexts as well? Unfortunately, it has not been possible to derive the laws of black hole mechanics using apparent horizons. Furthermore, as discussed in section 2, while apparent horizons are ‘local in time’ they are still global notions, tied too rigidly to the choice of a space-like 3-surface to be directly useful in all contexts. Is there a truly quasi-local notion which can be useful in all these contexts?

Disparate paradigms
 
In different communities within gravitational physics, the intended meaning of the term ‘black hole’ varies quite considerably. Thus, in a string theory seminar, the term ‘fundamental black holes’ without further qualification generally refers to the BPS states referred to above – a sub-class of stationary, extremal black holes. In a mathematical physics talk on black holes, the fundamental objects of interest are stationary solutions to, say, the Einstein–Higgs–Yang–Mills equations for which the uniqueness theorem fails. The focus is on the ramifications of ‘hair’, which are completely ignored in string theory. In a numerical relativity lecture, both these classes of objects are considered to be so exotic that they are excluded from discussion without comment. The focus is primarily on the dynamics of apparent horizons in general relativity. In astrophysically interesting situations, the distortion of black holes by external matter rings, magnetic fields and other black holes is often non-negligible [8698Jump To The Next Citation Point87Jump To The Next Citation Point]. While these illustrative notions seem so different, clearly there is a common conceptual core. Laws of black hole mechanics and the statistical mechanical derivation of entropy should go through for all black holes in equilibrium. Laws dictating the dynamics of apparent horizons should predict that the final equilibrium states are those represented by the stable stationary solutions of the theory. Is there a paradigm that can serve as an unified framework to establish such results in all these disparate situations?

These considerations led to the development of a new, quasi-local paradigm to describe black holes. This framework was inspired by certain seminal ideas introduced by Hayward [116Jump To The Next Citation Point117115118] in the mid-nineties and has been systematically developed over the past five years or so. Evolving black holes are modelled by dynamical horizons while those in equilibrium are modelled by isolated horizons. Both notions are quasi-local. In contrast to event horizons, neither notion requires the knowledge of space-time as a whole or refers to asymptotic flatness. Furthermore, they are space-time notions. Therefore, in contrast to apparent horizons, they are not tied to the choice of a partial Cauchy slice. This framework provides a new perspective encompassing all areas in which black holes feature: quantum gravity, mathematical physics, numerical relativity, and gravitational wave phenomenology. Thus, it brings out the underlying unity of the subject. More importantly, it has overcome some of the limitations of the older frameworks and also led to new results of direct physical interest.

The purpose of this article is to review these developments. The subject is still evolving. Many of the key issues are still open and new results are likely to emerge in the coming years. Nonetheless, as the Editors pointed out, there is now a core of results of general interest and, thanks to the innovative style of Living Reviews, we will be able to incorporate new results through periodic updates.

Applications of the quasi-local framework can be summarized as follows:

Black hole mechanics
 
Isolated horizons extract from the notion of Killing horizons, just those conditions which ensure that the horizon geometry is time independent; there may be matter and radiation even nearby [66Jump To The Next Citation Point]. Yet, it has been possible to extend the zeroth and first laws of black hole mechanics to isolated horizons [25Jump To The Next Citation Point74Jump To The Next Citation Point14Jump To The Next Citation Point]. Furthermore, this derivation brings out a conceptually important fact about the first law. Recall that, in presence of internal boundaries, time evolution need not be Hamiltonian (i.e., need not preserve the symplectic structure). If the inner boundary is an isolated horizon, a necessary and sufficient condition for evolution to be Hamiltonian turns out to be precisely the first law! Finally, while the first law has the same form as before (Equation (1View Equation)), all quantities which enter the statement of the law now refer to the horizon itself. This is the case even when non-Abelian gauge fields are included.

Dynamical horizons allow for the horizon geometry to be time dependent. This framework has led to a quantitative relation between the growth of the horizon area and the flux of energy and angular momentum across it [29Jump To The Next Citation Point30Jump To The Next Citation Point]. The processes can be in the non-linear regime of exact general relativity, without any approximations. Thus, the second law is generalized and the generalization also represents an integral version of the first law (1View Equation), applicable also when the black hole makes a transition from one state to another, which may be far removed.

Quantum gravity
 
The entropy problem refers to equilibrium situations. The isolated horizon framework provides an action principle and a Hamiltonian theory which serves as a stepping stone to non-perturbative quantization. Using the quantum geometry framework, a detailed theory of the quantum horizon geometry has been developed. The horizon states are then counted to show that the statistical mechanical black hole entropy is indeed proportional to the area [9Jump To The Next Citation Point10Jump To The Next Citation Point83Jump To The Next Citation Point149Jump To The Next Citation Point24Jump To The Next Citation Point]. This derivation is applicable to ordinary, astrophysical black holes which may be distorted and far from extremality. It also encompasses cosmological horizons to which thermodynamical considerations are known to apply [99Jump To The Next Citation Point]. Finally, the arena for this derivation is the curved black hole geometry, rather than a system in flat space-time which has the same number of states as the black hole [175146]. Therefore, this approach has a greater potential for analyzing physical processes associated with the black hole.

The dynamical horizon framework has raised some intriguing questions about the relation between black hole mechanics and thermodynamics in fully dynamical situations [53Jump To The Next Citation Point]. In particular, they provide seeds for further investigations of the notion of entropy in non-equilibrium situations.

Mathematical physics
 
The isolated horizon framework has led to a phenomenological model to understand properties of hairy black holes [20Jump To The Next Citation Point19Jump To The Next Citation Point]. In this model, the hairy black hole can be regarded as a bound state of an ordinary black hole and a soliton. A large number of facts about hairy black holes had accumulated through semi-analytical and numerical studies. Their qualitative features are explained by the model.

The dynamical horizon framework also provides the groundwork for a new approach to Penrose inequalities which relate the area of cross-sections of the event horizon Ae on a Cauchy surface with the ADM mass M ADM at infinity [157Jump To The Next Citation Point]: ∘A---∕16π-≤ M e ADM. Relatively recently, the conjecture has been proved in time symmetric situations. The basic monotonicity formula of the dynamical horizon framework could provide a new avenue to extend the current proofs to non-time-symmetric situations. It may also lead to a stronger version of the conjecture where the ADM mass is replaced by the Bondi mass [30Jump To The Next Citation Point].

Numerical relativity
 
The framework has provided a number of tools to extract physics from numerical simulations in the near-horizon, strong field regime. First, there exist expressions for mass and angular momentum of dynamical and isolated horizons which enable one to monitor dynamical processes occurring in the simulations [30Jump To The Next Citation Point] and extract properties of the final equilibrium state [14Jump To The Next Citation Point84Jump To The Next Citation Point]. These quantities can be calculated knowing only the horizon geometry and do not pre-suppose that the equilibrium state is a Kerr horizon. The computational resources required in these calculations are comparable to those employed by simulations using cruder techniques, but the results are now invariant and interpretation is free from ambiguities. Recent work [33Jump To The Next Citation Point] has shown that these methods are also numerically more accurate and robust than older ones.

Surprisingly, there are simple local criteria to decide whether the geometry of an isolated horizon is that of the Kerr horizon [142]. These criteria have already been implemented in numerical simulations. The isolated horizon framework also provides invariant, practical criteria to compare near-horizon geometries of different simulations [11Jump To The Next Citation Point] and leads to a new approach to the problem of extracting wave-forms in a gauge invariant fashion. Finally, the framework provides natural boundary conditions for the initial value problem for black holes in quasi-equilibrium [70Jump To The Next Citation Point125Jump To The Next Citation Point80Jump To The Next Citation Point], and to interpret certain initial data sets [137Jump To The Next Citation Point]. Many of these ideas have already been implemented in some binary black hole codes [84Jump To The Next Citation Point33Jump To The Next Citation Point46Jump To The Next Citation Point] and the process is continuing.

Gravitational wave phenomenology
 
The isolated horizon framework has led to a notion of horizon multipole moments [23Jump To The Next Citation Point]. They provide a diffeomorphism invariant characterization of the isolated horizon geometry. They are distinct from the Hansen multipoles in stationary space-times [107Jump To The Next Citation Point] normally used in the analysis of equations of motion because they depend only on the isolated horizon geometry and do not require global stationarity. They represent source multipoles rather than Hansen’s field multipoles. In Kerr space-time, while the mass and angular momenta agree in the two regimes, quadrupole moments do not; the difference becomes significant when a ∼ M, i.e., in the fully relativistic regime. In much of the literature on equations of motion of black holes, the distinction is glossed over largely because only field multipoles have been available in the literature. However, in applications to equations of motion, it is the source multipoles that are more relevant, whence the isolated horizon multipoles are likely to play a significant role.

The dynamical horizon framework enables one to calculate mass and angular momentum of the black hole as it evolves. In particular, one can now ask if the black hole can be first formed violating the Kerr bound a ≤ M but then eventually settle down in the Kerr regime. Preliminary considerations fail to rule out this possibility, although the issue is still open [30Jump To The Next Citation Point]. The issue can be explored both numerically and analytically. The possibility that the bound can indeed be violated initially has interesting astrophysical implications [88Jump To The Next Citation Point].

In this review, we will outline the basic ideas underlying dynamical and isolated horizon frameworks and summarize their applications listed above. The material is organized as follows. In Section 2 we recall the basic definitions, motivate the assumptions and summarize their implications. In Section 3 we discuss the area increase theorem for dynamical horizons and show how it naturally leads to an expression for the flux of gravitational energy crossing dynamical horizons. Section 4 is devoted to the laws of black hole mechanics. We outline the main ideas using both isolated and dynamical horizons. In the next three sections we review applications. Section 5 summarizes applications to numerical relativity, Section 6 to black holes with hair, and Section 7 to the quantum entropy calculation. Section 8 discusses open issues and directions for future work. Having read Section 2, Sections 3, 4, 5, 6, and 7 are fairly self contained and the three applications can be read independently of each other.

All manifolds will be assumed to be k+1 C (with k ≥ 3) and orientable, the space-time metric will be Ck, and matter fields Ck −2. For simplicity we will restrict ourselves to 4-dimensional space-time manifolds ℳ (although most of the classical results on isolated horizons have been extended to 3-dimensions space-times [22Jump To The Next Citation Point], as well as higher dimensional ones [144]). The space-time metric gab has signature (− ,+, +, +) and its derivative operator will be denoted by ∇. The Riemann tensor is defined by d Rabc Wd := 2∇ [a∇b ]Wc, the Ricci tensor by Rab := Racbc, and the scalar curvature by R := gabRab. We will assume the field equations

1 Rab − --R gab + Λgab = 8 πGTab. (2 ) 2
(With these conventions, de Sitter space-time has positive cosmological constant Λ.) We assume that Tab satisfies the dominant energy condition (although, as the reader can easily tell, several of the results will hold under weaker restrictions.) Cauchy (and partial Cauchy) surfaces will be denoted by M, isolated horizons by Δ, and dynamical horizons by H.


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