Isolated and Dynamical Horizons and Their Applications
Over the past three decades, black holes have played an important role in quantum gravity, mathematical physics, numerical relativity and gravitational wave phenomenology. However, conceptual settings and mathematical models used to discuss them have varied considerably from one area to another. Over the last five years a new, quasi-local framework was introduced to analyze diverse facets of black holes in a unified manner. In this framework, evolving black holes are modelled by dynamical horizons and black holes in equilibrium by isolated horizons. We review basic properties of these horizons and summarize applications to mathematical physics, numerical relativity, and quantum gravity. This paradigm has led to significant generalizations of several results in black hole physics. Specifically, it has introduced a more physical setting for black hole thermodynamics and for black hole entropy calculations in quantum gravity, suggested a phenomenological model for hairy black holes, provided novel techniques to extract physics from numerical simulations, and led to new laws governing the dynamics of black holes in exact general relativity.
black holes, generalized thermodynamics, numerical relativity, quantum gravity
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 [35, 44, 45, 46]. 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 [69, 119] took the community by surprise. The subsequent analysis of the detailed properties of Kerr–Newman solutions  and perturbations thereof  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 [38, 49]. Research in this area continues to be an active branch of mathematical physics . 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 , the associated critical phenomenon [67, 105], and the dynamics leading to their coalescence (see, e.g., [1, 126, 160, 58, 2, 138]). 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 .
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 [110, 90], 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 [111, 113]: 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 , angular momentum , area , surface gravity , and angular velocity , the first law of black hole mechanics,
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 [35, 182]. This suggests that there may well be a fully dynamical version of Equation (1) 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 4 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 1). 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.
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 (1). Here, the angular momentum and the mass are defined at infinity while the angular velocity and surface gravity 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 transparent\epubtkFootnoteThe situation is even more puzzling in the Einstein–Yang–Mills theory where the right side of Equation (1) acquires an additional term, . In treatments based on stationary space-times, not only the Yang–Mills charge , but also the potential (the analog of and , is evaluated at infinity .. 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) . However, for realistic black holes the charge to mass ratio is less than . 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?
The teleological nature of event horizons is also an obstruction to extending black hole mechanics in certain physical situations. Consider for example, Figure 2 in which a spherical star of mass undergoes a gravitational collapse. The singularity is hidden inside the null surface at 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 collapses. Then would not be a part of the event horizon which would actually lie slightly outside and coincide with the surface in the distant future. On physical grounds, it seems unreasonable to exclude 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 .
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 [87, 98, 88]. 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 [117, 118, 115, 116] 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 . Yet, it has been possible to extend the zeroth and first laws of black hole mechanics to isolated horizons [26, 75, 15]. 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 (1)), 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 [30, 31]. 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 (1), 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 [10, 11, 84, 149, 25]. 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 . 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 [174, 145]. 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 . 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 [21, 20]. 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 on a Cauchy surface with the ADM mass at infinity : . 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 .
- 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  and extract properties of the final equilibrium state [15, 85]. 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  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 . 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  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 [72, 124, 81], and to interpret certain initial data sets . Many of these ideas have already been implemented in some binary black hole codes [85, 34, 48] and the process is continuing.
- Gravitational wave phenomenology
The isolated horizon framework has led to a notion of horizon multipole moments . They provide a diffeomorphism invariant characterization of the isolated horizon geometry. They are distinct from the Hansen multipoles in stationary space-times  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 , 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 but then eventually settle down in the Kerr regime. Preliminary considerations fail to rule out this possibility, although the issue is still open . The issue can be explored both numerically and analytically. The possibility that the bound can indeed be violated initially has interesting astrophysical implications .
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 (with ) and orientable, the space-time metric will be , and matter fields . For simplicity we will restrict ourselves to 4-dimensional space-time manifolds (although most of the classical results on isolated horizons have been extended to 3-dimensions space-times , as well as higher dimensional ones ). The space-time metric has signature and its derivative operator will be denoted by . The Riemann tensor is defined by , the Ricci tensor by , and the scalar curvature by . We will assume the field equations
(With these conventions, de Sitter space-time has positive cosmological constant .) We assume that 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 , isolated horizons by , and dynamical horizons by .
2 Basic Notions
This section is divided into two parts. The first introduces isolated horizons, and the second dynamical horizons.
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 [13, 14, 26, 16]. 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., [34, 48]).
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
is topologically and null;
any null normal of has vanishing expansion, ; and
all equations of motion hold at and the stress energy tensor is such that is future-causal for any future directed null normal .
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 [26, 136]. 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 .
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 , 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 .
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 . 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 . 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\epubtkFootnoteHowever, Condition (6) may be too strong in some problems, e.g., in the construction of quasi-equilibrium initial data sets, where the notion of WIH is more useful  (see Section 5.2)..
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 .)
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 [125, 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 . 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 [92, 161], Lewandowski  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.
2.1.3 Geometrical properties
- Rescaling freedom in
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 (5), considerable freedom still remains. That is, a given NEH admits an infinite number of WIH structures .
On IHs, by contrast, the situation is dramatically different. Given an IH , generically the Condition (6) in Definition 2.1.1 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 . 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.
- Surface gravity
Let us begin by defining a 1-form which will be used repeatedly. First note that, by Definition 2.1.1, 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 with any two vectors tangent to is identically zero, whence there must exist a 1-form on such that for any tangent to ,
Note that the WIH condition (5) requires simply that be time independent, . Given , the surface gravity associated with a null normal is defined as
Thus, is simply the acceleration of . Note that the surface gravity is not an intrinsic property of a WIH . Rather, it is a property of a null normal to : . An isolated horizon with 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
Consider any (space-time) null tetrad on such that is a null normal to . Then, it follows from Definition 2.1.1 that two of the Newman–Penrose Weyl components vanish on : and . This in turn implies that 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 is related to the curl of ,
where is the natural area 2-form on . Horizons on which vanishes are said to be non-rotating: On these horizons all angular momentum multipoles vanish . Therefore, is sometimes referred to as the rotational scalar and as the rotation 1-form of the horizon.
Next, let us consider the Ricci-tensor components. On any NEH we have: , . In the Einstein–Maxwell theory, one further has: On , and .
- Free data on an isolated horizon
Given the geometry 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) . For simplicity, here we will summarize the results only for the non-extremal case. (For the extremal case, see [16, 143].)
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.
- Rest frame of a non-expanding horizon
As at null infinity, a preferred foliation of 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 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 . 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 . 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.
- Symmetries of an isolated horizon
By definition, a symmetry of an IH is a diffeomorphism of which preserves the geometry . (On a WIH, the symmetry has to preserve . There are again three universality classes of symmetry groups as on an IH.) Let us denote the symmetry group by . First note that diffeomorphisms generated by the null normals in 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 . 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 can be of only three types :
Type I: The pair is spherically symmetric; is four dimensional.
Type II: The pair is axisymmetric; is two dimensional.
Type III: Diffeomorphisms generated by are the only symmetries of the pair ; is one dimensional.
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.
2.2 Dynamical horizons
This section is divided into three parts. In the first, we discuss basic definitions, in the second we introduce an explicit example, and in the third we analyze the issue of uniqueness of dynamical horizons and their role in numerical relativity.
To explain the evolution of ideas and provide points of comparison, we will introduce the notion of dynamical horizons following a chronological order. Readers who are not familiar with causal structures can go directly to Definition 2.2.1 of dynamical horizons (for which a more direct motivation can be found in ).
As discussed in Section 1, while the notion of an event horizon has proved to be very convenient in mathematical relativity, it is too global and teleological to be directly useful in a number of physical contexts ranging from quantum gravity to numerical relativity to astrophysics. This limitation was recognized early on (see, e.g., , page 319) and alternate notions were introduced to capture the intuitive idea of a black hole in a quasi-local manner. In particular, to make the concept ‘local in time’, Hawking [111, 113] introduced the notions of a trapped region and an apparent horizon, both of which are associated to a space-like 3-surface representing ‘an instant of time’. Let us begin by recalling these ideas.
Hawking’s outer trapped surface is a compact, space-like 2-dimensional sub-manifold in such that the expansion of the outgoing null normal to is non-positive. Hawking then defined the trapped region in a surface as the set of all points in through which there passes an outer-trapped surface, lying entirely in . Finally, Hawking’s apparent horizon is the boundary of a connected component of . The idea then was to regard each apparent horizon as the instantaneous surface of a black hole. One can calculate the expansion of knowing only the intrinsic 3-metric and the extrinsic curvature of . Hence, to find outer trapped surfaces and apparent horizons on , one does not need to evolve away from even locally. In this sense the notion is local to . However, this locality is achieved at the price of restricting to lie in . If we wiggle even slightly, new outer trapped surfaces can appear and older ones may disappear. In this sense, the notion is still very global. Initially, it was hoped that the laws of black hole mechanics can be extended to these apparent horizons. However, this has not been possible because the notion is so sensitive to the choice of .
To improve on this situation, in the early nineties Hayward proposed a novel modification of this framework . The main idea is to free these notions from the complicated dependence on . He began with Penrose’s notion of a trapped surface. A trapped surface a la Penrose is a compact, space-like 2-dimensional sub-manifold of space-time on which , where and are the two null normals to . We will focus on future trapped surfaces on which both expansions are negative. Hayward then defined a space-time trapped region. A trapped region a la Hayward is a subset of space-time through each point of which there passes a trapped surface. Finally, Hayward’s trapping boundary is a connected component of the boundary of an inextendible trapped region. Under certain assumptions (which appear to be natural intuitively but technically are quite strong), he was able to show that the trapping boundary is foliated by marginally trapped surfaces (MTSs), i.e., compact, space-like 2-dimensional sub-manifolds on which the expansion of one of the null normals, say , vanishes and that of the other, say , is everywhere non-positive. Furthermore, is also everywhere of one sign. These general considerations led him to define a quasi-local analog of future event horizons as follows:
Definition 4: A future, outer, trapping horizon (FOTH) is a smooth 3-dimensional sub-manifold of space-time, foliated by closed 2-manifolds , such that
the expansion of one future directed null normal to the foliation, say , vanishes, ;
the expansion of the other future directed null normal is negative, ; and
the directional derivative of along is negative, .
In this definition, Condition 2 captures the idea that is a future horizon (i.e., of black hole rather than white hole type), and Condition 3 encodes the idea that it is ‘outer’ since infinitesimal motions along the ‘inward’ normal makes the 2-surface trapped. (Condition 3 also serves to distinguish black hole type horizons from certain cosmological ones  which are not ruled out by Condition 2). Using the Raychaudhuri equation, it is easy to show that is either space-like or null, being null if and only if the shear of as well as the matter flux across vanishes. Thus, when is null, it is a non-expanding horizon introduced in Section 2.1. Intuitively, is space-like in the dynamical region where gravitational radiation and matter fields are pouring into it and is null when it has reached equilibrium.
In truly dynamical situations, then, is expected to be space-like. Furthermore, it turns out that most of the key results of physical interest [30, 31], such as the area increase law and generalization of black hole mechanics, do not require the condition on the sign of . It is therefore convenient to introduce a simpler and at the same time ‘tighter’ notion, that of a dynamical horizon, which is better suited to analyze how black holes grow in exact general relativity [30, 31]:
Definition 5: A smooth, three-dimensional, space-like sub-manifold (possibly with boundary) of space-time is said to be a dynamical horizon (DH) if it can be foliated by a family of closed 2-manifolds such that
on each leaf the expansion of one null normal vanishes; and
the expansion of the other null normal is negative.
Note first that, like FOTHs, dynamical horizons are ‘space-time notions’, defined quasi-locally. They are not defined relative to a space-like surface as was the case with Hawking’s apparent horizons nor do they make any reference to infinity as is the case with event horizons. In particular, they are well-defined also in the spatially compact context. Being quasi-local, they are not teleological. Next, let us spell out the relation between FOTHs and DHs. A space-like FOTH is a DH on which the additional condition holds. Similarly, a DH satisfying is a space-like FOTH. Thus, while neither definition implies the other, the two are closely related. The advantage of Definition 2.2.1 is that it refers only to the intrinsic structure of , without any conditions on the evolution of fields in directions transverse to . Therefore, it is easier to verify in numerical simulations. More importantly, as we will see, this feature makes it natural to analyze the structure of using only the constraint (or initial value) equations on it. This analysis will lead to a wealth of information on black hole dynamics. Reciprocally, Definition 2.2.1 has the advantage that, since it permits to be space-like or null, it is better suited to analyze the transition to equilibrium .
A DH which is also a FOTH will be referred to as a space-like future outer horizon (SFOTH). To fully capture the physical notion of a dynamical black hole, one should require both sets of conditions, i.e., restrict oneself to SFOTHs. For, stationary black holes admit FOTHS and there exist space-times  which admit dynamical horizons but no trapped surfaces; neither can be regarded as containing a dynamical black hole. However, it is important to keep track of precisely which assumptions are needed to establish specific results. Most of the results reported in this review require only those conditions which are satisfied on DHs. This fact may well play a role in conceptual issues that arise while generalizing black hole thermodynamics to non-equilibrium situations\epubtkFootnoteIndeed, the situation is similar for black holes in equilibrium. While it is physically reasonable to restrict oneself to IHs, most results require only the WIH boundary conditions. The distinction can be important in certain applications, e.g., in finding boundary conditions on the quasi-equilibrium initial data at inner horizons..
Let us begin with the simplest examples of space-times admitting DHs (and SFOTHs). These are provided by the spherically symmetric solution to Einstein’s equations with a null fluid as source, the Vaidya metric [179, 137, 186]. (Further details and the inclusion of a cosmological constant are discussed in .) Just as the Schwarzschild–Kruskal solution provides a great deal of intuition for general static black holes, the Vaidya metric furnishes some of the much needed intuition in the dynamical regime by bringing out the key differences between the static and dynamical situations. However, one should bear in mind that both Schwarzschild and Vaidya black holes are the simplest examples and certain aspects of geometry can be much more complicated in more general situations. The 4-metric of the Vaidya space-time is given by
where is any smooth, non-decreasing function of . Thus, are the ingoing Eddington–Finkelstein coordinates. This is a solution of Einstein’s equations, the stress-energy tensor being given by
where . Clearly, satisfies the dominant energy condition if , and vanishes if and only if . Of special interest to us are the cases illustrated in Figure 4: is non-zero until a certain finite retarded time, say , and then grows monotonically, either reaching an asymptotic value as tends to infinity (panel a), or, reaching this value at a finite retarded time, say , and then remaining constant (panel b). In either case, the space-time region is flat.
Let us focus our attention on the metric 2-spheres, which are all given by and . It is easy to verify that the expansion of the outgoing null normal vanishes if and only if ( and) . Thus, these are the only spherically symmetric marginally trapped surfaces MTSs. On each of them, the expansion of the ingoing normal is negative. By inspection, the 3-metric on the world tube of these MTSs has signature when is non-zero and if is zero. Hence, in the left panel of Figure 4 the surface is the DH . In the right panel of Figure 4 the portion of this surface is the DH , while the portion is a non-expanding horizon. (The general issue of transition of a DH to equilibrium is briefly discussed in Section 5.) Finally, note that at these MTSs, . Hence in both cases, the DH is an SFOTH. Furthermore, in the case depicted in the right panel of Figure 4 the entire surface is a FOTH, part of which is dynamical and part null.
This simple example also illustrates some interesting features which are absent in the stationary situations. First, by making explicit choices of , one can plot the event horizon using, say, Mathematica  and show that they originate in the flat space-time region , in anticipation of the null fluid that is going to fall in after . The dynamical horizon, on the other hand, originates in the curved region of space-time, where the metric is time-dependent, and steadily expands until it reaches equilibrium. Finally, as Figures 4 illustrate, the dynamical and event horizons can be well separated. Recall that in the equilibrium situation depicted by the Schwarzschild space-time, a spherically symmetric trapped surface passes through every point in the interior of the event horizon. In the dynamical situation depicted by the Vaidya space-time, they all lie in the interior of the DH. However, in both cases, the event horizon is the boundary of . Thus, the numerous roles played by the event horizon in equilibrium situations get split in dynamical contexts, some taken up by the DH.
What is the situation in a more general gravitational collapse? As indicated in the beginning of this section, the geometric structure can be much more subtle. Consider 3-manifolds which are foliated by marginally trapped compact 2-surfaces . We denote by the normal whose expansion vanishes. If the expansion of the other null normal is negative, will be called a marginally trapped tube (MTT). If the tube is space-like, it is a dynamical horizon. If it is time-like, it will be called time-like membrane. Since future directed causal curves can traverse time-like membranes in either direction, they are not good candidates to represent surfaces of black holes; therefore they are not referred to as horizons.
In Vaidya metrics, there is precisely one MTT to which all three rotational Killing fields are tangential and this is the DH . In the Oppenheimer–Volkoff dust collapse, however, the situation is just the opposite; the unique MTT on which each MTS is spherical is time-like [180, 47]. Thus we have a time-like membrane rather than a dynamical horizon. However, in this case the metric does not satisfy the smoothness conditions spelled out at the end of Section 1 and the global time-like character of is an artifact of the lack of this smoothness. In the general perfect fluid spherical collapse, if the solution is smooth, one can show analytically that the spherical MTT is space-like at sufficiently late times, i.e., in a neighborhood of its intersection with the event horizon . For the spherical scalar field collapse, numerical simulations show that, as in the Vaidya solutions, the spherical MTT is space-like everywhere . Finally, the geometry of the numerically evolved MTTs has been examined in two types of non-spherical situations: the axi-symmetric collapse of a neutron star to a Kerr black hole and in the head-on collision of two non-rotating black holes . In both cases, in the initial phase the MTT is neither space-like nor time-like all the way around its cross-sections . However, it quickly becomes space-like and has a long space-like portion which approaches the event horizon. This portion is then a dynamical horizon. There are no hard results on what would happen in general, physically interesting situations. The current expectation is that the MTT of a numerically evolved black hole space-time which asymptotically approaches the event horizon will become space-like rather soon after its formation. Therefore most of the ongoing detailed work focuses on this portion, although basic analytical results are available also on how the time-like membranes evolve (see Appendix A of ).
Even in the simplest, Vaidya example discussed above, our explicit calculations were restricted to spherically symmetric marginally trapped surfaces. Indeed, already in the case of the Schwarzschild space-time, very little is known analytically about non-spherically symmetric marginally trapped surfaces. It is then natural to ask if the Vaidya metric admits other, non-spherical dynamical horizons which also asymptote to the non-expanding one. Indeed, even if we restrict ourselves to the 3-manifold , can we find another foliation by non-spherical, marginally trapped surfaces which endows it with another dynamical horizon structure? These considerations illustrate that in general there are two uniqueness issues that must be addressed.
First, in a general space-time , can a space-like 3-manifold be foliated by two distinct families of marginally trapped surfaces, each endowing it with the structure of a dynamical horizon? Using the maximum principle, one can show that this is not possible . Thus, if admits a dynamical horizon structure, it is unique.
Second, we can ask the following question: How many DHs can a space-time admit? Since a space-time may contain several distinct black holes, there may well be several distinct DHs. The relevant question is if distinct DHs can exist within each connected component of the (space-time) trapped region. On this issue there are several technically different uniqueness results . It is simplest to summarize them in terms of SFOTHs. First, if two non-intersecting SFOTHs and become tangential to the same non-expanding horizon at a finite time (see the right panel in Figure 4), then they coincide (or one is contained in the other). Physically, a more interesting possibility, associated with the late stages of collapse or mergers, is that and become asymptotic to the event horizon. Again, they must coincide in this case. At present, one can not rule out the existence of more than one SFOTHs which asymptote to the event horizon if they intersect each other repeatedly. However, even if this were to occur, the two horizon geometries would be non-trivially constrained. In particular, none of the marginally trapped surfaces on can lie entirely to the past of .
A better control on uniqueness is perhaps the most important open issue in the basic framework for dynamical horizons and there is ongoing work to improve the existing results. Note however that all results of Sections 3 and 5, including the area increase law and the generalization of black hole mechanics, apply to all DHs (including the ‘transient ones’ which may not asymptote to the event horizon). This makes the framework much more useful in practice.
The existing results also provide some new insights for numerical relativity . First, suppose that a MTT is generated by a foliation of a region of space-time by partial Cauchy surfaces such that each MTS is the outermost MTS in . Then can not be a time-like membrane. Note however that this does not imply that is necessarily a dynamical horizon because may be partially time-like and partially space-like on each of its marginally trapped surfaces . The requirement that be space-like – i.e., be a dynamical horizon – would restrict the choice of the foliation of space-time and reduce the unruly freedom in the choice of gauge conditions that numerical simulations currently face. A second result of interest to numerical relativity is the following. Let a space-time admit a DH which asymptotes to the event horizon. Let be any partial Cauchy surface in which intersects in one of the marginally trapped surfaces, say . Then, is the outermost marginally trapped surface – i.e., apparent horizon in the numerical relativity terminology – on .
3 Area Increase Law
As mentioned in the introduction, the dynamical horizon framework has led to a monotonicity formula governing the growth of black holes. In this section, we summarize this result. Our discussion is divided into three parts. The first spells out the strategy, the second presents a brief derivation of the basic formula, and the third is devoted to interpretational issues.
The first law of black hole mechanics (1) tells us how the area of the black hole increases when it makes a transition from an initial equilibrium state to a nearby equilibrium state. The question we want to address is: Can one obtain an integral generalization to incorporate fully dynamical situations? Attractive as this possibility seems, one immediately encounters a serious conceptual and technical problem. For, the generalization requires, in particular, a precise notion of the flux of gravitational energy across the horizon. Already at null infinity, the expression of the gravitational energy flux is subtle: One needs the framework developed by Bondi, Sachs, Newman, Penrose, and others to introduce a viable, gauge invariant expression of this flux [54, 33, 185]. In the strong field regime, there is no satisfactory generalization of this framework and, beyond perturbation theory, no viable, gauge invariant notion of the flux of gravitational energy across a general surface.
Yet, there are at least two general considerations that suggest that something special may happen on DHs. Consider a stellar collapse leading to the formation of a black hole. At the end of the process, one has a black hole and, from general physical considerations, one expects that the energy in the final black hole should equal the total matter plus gravitational energy that fell across the horizon. Thus, at least the total integrated flux across the horizon should be well defined. Indeed, it should equal the depletion of the energy in the asymptotic region, i.e., the difference between the ADM energy and the energy radiated across future null infinity. The second consideration involves the Penrose inequality  introduced in Section 1. Heuristically, the inequality leads us to think of the radius of a marginally trapped surface as a measure of the mass in its interior, whence one is led to conclude that the change in the area is due to influx of energy. Since a DH is foliated by marginally trapped surfaces, it is tempting to hope that something special may happen, enabling one to define the flux of energy and angular momentum across it. This hope is borne out.
In the discussion of DHs (Sections 3 and 4.2) we will use the following conventions (see Figure 5). The DH is denoted by and marginally trapped surfaces that foliate it are referred to as cross-sections. The unit, time-like normal to is denoted by with . The intrinsic metric and the extrinsic curvature of are denoted by and , respectively. is the derivative operator on compatible with , its Ricci tensor, and its scalar curvature. The unit space-like vector orthogonal to and tangent to is denoted by . Quantities intrinsic to are generally written with a tilde. Thus, the two-metric on is and the extrinsic curvature of is ; the derivative operator on is and its Ricci tensor is . Finally, we fix the rescaling freedom in the choice of null normals to cross-sections via and (so that ). To keep the discussion reasonably focused, we will not consider gauge fields with non-zero charges on the horizon. Inclusion of these fields is not difficult but introduces a number of subtleties and complications which are irrelevant for numerical relativity and astrophysics.
3.2 Area increase law
The qualitative result that the area of cross-sections increases monotonically on follows immediately from the definition,
since and . Hence increases monotonically in the direction of . The non-trivial task is to obtain a quantitative formula for the amount of area increase.
To obtain this formula, one simply uses the scalar and vector constraints satisfied by the Cauchy data on :
and is the matter stress-energy tensor. The strategy is entirely straightforward: One fixes two cross-sections and of , multiplies and with appropriate lapse and shift fields and integrates the result on a portion which is bounded by and . Somewhat surprisingly, if the cosmological constant is non-negative, the resulting area balance law also provides strong constraints on the topology of cross sections .
Specification of lapse and shift is equivalent to the specification of a vector field with respect to which energy-flux across is defined. The definition of a DH provides a preferred direction field, that along . Hence it is natural set . We will begin with this choice and defer the possibility of choosing more general vector fields until Section 4.2.
The object of interest now is the flux of energy associated with across . We denote the flux of matter energy across by :
By taking the appropriate combination of Equations (14) and (15) we obtain
Since is foliated by compact 2-manifolds , one can perform a 2 + 1 decomposition of various quantities on . In particular, one first uses the Gauss–Codazzi equation to express in terms of , , and a total divergence. Then, one uses the identity
to simplify the expression. Finally one sets
(Note that is just the shear tensor since the expansion of vanishes.) Then, Equation (18) reduces to
To simplify this expression further, we now make a specific choice of the lapse . We denote by the area-radius function; thus is constant on each and satisfies . Since we already know that area increases monotonically, is a good coordinate on , and using it the 3-volume on can be decomposed as , where denotes the gradient on . Therefore calculations simplify if we choose
We will set . Then, the integral on the left side of Equation (21) becomes
where and are the (geometrical) radii of and , and is the Gauss–Bonnet topological invariant of the cross-sections . Substituting back in Equation (21) one obtains
This is the general expression relating the change in area to fluxes across . Let us consider its ramifications in the three cases, being positive, zero, or negative:
If , the right side is positive definite whence the Gauss–Bonnet invariant is positive definite, and the topology of the cross-sections of the DH is necessarily that of .
If , then is either spherical or toroidal. The toroidal case is exceptional: If it occurs, the matter and the gravitational energy flux across vanishes (see Section 3.3), the metric is flat, (so can not be a FOTH), and . In view of these highly restrictive conditions, toroidal DHs appear to be unrelated to the toroidal topology of cross-sections of the event horizon discussed by Shapiro, Teukolsky, Winicour, and others [121, 167, 139]. In the generic spherical case, the area balance law (24) becomes
If , there is no control on the sign of the right hand side of Equation (24). Hence, a priori any topology is permissible. Stationary solutions with quite general topologies are known for black holes which are asymptotically locally anti-de Sitter. Event horizons of these solutions are the potential asymptotic states of these DHs in the distant future.
For simplicity, the remainder of our discussion of DHs will be focused on the zero cosmological constant case with 2-sphere topology.
3.3 Energy flux due to gravitational waves
Let us interpret the various terms appearing in the area balance law (25).
The left side of this equation provides us with the change in the horizon radius caused by the dynamical process under consideration. Since the expansion vanishes, this is also the change in the Hawking mass as one moves from the cross section to . The first integral on the right side of this equation is the flux of matter energy associated with the vector field . The second term is purely geometrical and accompanies the term representing the matter energy flux. Hence it is interpreted as the flux of -energy carried by the gravitational radiation:
A priori, it is surprising that there should exist a meaningful expression for the gravitational energy flux in the strong field regime where gravitational waves can no longer be envisaged as ripples on a flat space-time. Therefore, it is important to subject this interpretation to viability criteria analogous to the ‘standard’ tests one uses to demonstrate the viability of the Bondi flux formula at null infinity. It is known that it passes most of these tests. However, to our knowledge, the status is still partially open on one of these criteria. The situation can be summarized as follows:
- Gauge invariance
Since one did not have to introduce any structure, such as coordinates or tetrads, which is auxiliary to the problem, the expression is obviously gauge invariant. This is to be contrasted with definitions involving pseudo-tensors or background fields.
The energy flux (26) is manifestly non-negative. In the case of the Bondi flux, positivity played a key role in the early development of the gravitational radiation theory. It was perhaps the most convincing evidence that gravitational waves are not coordinate artifacts but carry physical energy. It is quite surprising that a simple, manifestly non-negative expression can exist in the strong field regime of DHs. One can of course apply our general strategy to any space-like 3-surface , foliated by 2-spheres. However, if is not a DH, the sign of the geometric terms in the integral over can not be controlled, not even when lies in the black hole region and is foliated by trapped (rather than marginally trapped) surfaces . Thus, the positivity of is a rather subtle property, not shared by 3-surfaces which are foliated by non-trapped surfaces, nor those which are foliated by trapped surfaces; one needs a foliation precisely by marginally trapped surfaces. The property is delicately matched to the definition of DHs .
All fields used in Equation (26) are defined by the local geometrical structures on cross-sections of . This is a non-trivial property, shared also by the Bondi-flux formula. However, it is not shared in other contexts. For example, the proof of the positive energy theorem by Witten  provides a positive definite energy density on Cauchy surfaces. But since it is obtained by solving an elliptic equation with appropriate boundary conditions at infinity, this energy density is a highly non-local function of geometry. Locality of enables one to associate it with the energy of gravitational waves instantaneously falling across any cross section .
- Vanishing in spherical symmetry
The fourth criterion is that the flux should vanish in presence of spherical symmetry. Suppose is spherically symmetric. Then one can show that each cross-section of must be spherically symmetric. Now, since the only spherically symmetric vector field and trace-free, second rank tensor field on a 2-sphere are the zero fields, and .
- Balance law
The Bondi–Sachs energy flux also has the important property that there is a locally defined notion of the Bondi energy associated with any 2-sphere cross-section of future null infinity, and the difference equals the Bondi–Sachs flux through the portion of null infinity bounded by and . Does the expression (26) share this property? The answer is in the affirmative: As noted in the beginning of this section, the integrated flux is precisely the difference between the locally defined Hawking mass associated with the cross-section. In Section 5 we will extend these considerations to include angular momentum.
Taken together, the properties discussed above provide a strong support in favor of the interpretation of Equation (26) as the -energy flux carried by gravitational waves into the portion of the DH. Nonetheless, it is important to continue to think of new criteria and make sure that Equation (26) passes these tests. For instance, in physically reasonable, stationary, vacuum solutions to Einstein’s equations, one would expect that the flux should vanish. However, on DHs the area must increase. Thus, one is led to conjecture that these space-times do not admit DHs. While special cases of this conjecture have been proved, a general proof is still lacking. Situation is similar for non-spherical DHs in spherically symmetric space-times.
We will conclude this section with two remarks:
The presence of the shear term in the integrand of the flux formula (26) seems natural from one’s expectations based on perturbation theory at the event horizon of the Kerr family [108, 66]. But the term is new and can arise only because is space-like rather than null: On a null surface, the analogous term vanishes identically. To bring out this point, one can consider a more general case and allow the cross-sections to lie on a horizon which is partially null and partially space-like. Then, using a 2 + 2 formulation  one can show that flux on the null portion is given entirely by the term . However, on the space-like portion, the term does not vanish in general. Indeed, on a DH, it cannot vanish in presence of rotation: The angular momentum is given by the integral of , where is the rotational symmetry.
The flux refers to a specific vector field and measures the change in the Hawking mass associated with the cross-sections. However, this is not a good measure of the mass in presence of angular momentum (see, e.g.,  for numerical simulations). Generalization of the balance law to include angular momentum is discussed in Section 4.2.
4 Black Hole Mechanics
As mentioned in the introduction, the discovery of the laws of black hole mechanics has led to fundamental insights in both classical and quantum gravity. In this section we discuss how the standard framework tied to stationary space-times can be extended using WIHs and DHs.
4.1 Mechanics of weakly isolated horizons
The isolated horizon framework has not only extended black hole mechanics, but it has also led to a deeper insight into the ‘origin’ of the laws of black hole mechanics. In this section we will summarize these developments using WIHs. Along the way we shall also obtain formulas for the mass and angular momentum of a WIH. For simplicity, in the main part of the discussion, we will restrict ourselves to type II (i.e., axi-symmetric) WIHs on which all matter fields vanish. Generalizations including various types of matter field can be found in [19, 26, 15, 75, 76, 77].
4.1.1 The zeroth law
The zeroth law of thermodynamics says that the temperature of a system in thermodynamic equilibrium is constant. Its counterpart for black hole mechanics says that surface gravity of a weakly isolated horizon is constant. This result is non-trivial because the horizon geometry is not assumed to be spherically symmetric; the result holds even when the horizon itself is highly distorted so long as it is in equilibrium. It is established as follows.
Recall from Section 2.1.3 that the notion of surface gravity is tied to the choice of a null normal of the isolated horizon: . Now, using Equation (5) in Definition 2.1.1 (of WIHs), we obtain:
Next, recall from Equation (9) that the curl of is related to the imaginary part of :
where is the natural area 2-form on satisfying and . Hence we conclude which in turn implies that is constant on the horizon:
This completes the proof of the zeroth law. As the argument shows, given an NEH, the main condition (5) in the definition of a WIH is equivalent to constancy of surface gravity. Note that no restriction has been imposed on which determines the mass and angular momentum multipoles : as emphasized above, the zeroth law holds even if the WIH is highly distorted and rapidly rotating.
If electromagnetic fields are included, one can also show that the electric potential is constant on the horizon . Finally, there is an interesting interplay between the zeroth law the action principle. Let us restrict ourselves to space-times which admit a non expanding horizon as inner boundary. Then the standard Palatini action principle is not well defined because the variation produces a non-vanishing surface term at the horizon. The necessary and sufficient condition for this surface term to vanish is precisely that the gravitational (and the electromagnetic) zeroth laws hold . Consequently, the standard action principle is well-defined if inner boundaries are WIHs.
4.1.2 Phase space, symplectic structure, and angular momentum
In field theories, conserved quantities such as energy and angular momentum can be universally defined via a Hamiltonian framework: they are the numerical values of Hamiltonians generating canonical transformations corresponding to time translation and rotation symmetries. In absence of inner boundaries, it is this procedure that first led to the notion of the ADM energy and angular momentum at spatial infinity . At null infinity, it can also be used to define fluxes of Bondi energy and angular momentum across regions of , and values of these quantities associated with any cross-section of [18, 185].
This procedure can be extended to allow inner boundaries which are WIHs. The first ingredient required for a Hamiltonian framework is, of course, a phase space. The appropriate phase space now consists of fields living in a region of space-time outside the black hole, satisfying suitable boundary conditions at infinity and horizon. Let be the region of space-time that we are interested in. The boundary of consists of four components: the time-like cylinder at spatial infinity, two space-like surfaces and which are the future and past boundaries of , and an inner boundary which is to be the WIH (see Figure 6). At infinity, all fields are assumed to satisfy the fall-off conditions needed to ensure asymptotic flatness. To ensure that is a type II horizon, one fixes a rotational vector field on and requires that physical fields on are such that the induced geometry on is that of a type II horizon with as the rotational symmetry.
Two Hamiltonian frameworks are available. The first uses a covariant phase space which consists of the solutions to field equations which satisfy the required boundary conditions [26, 15]. Here the calculations are simplest if one uses a first order formalism for gravity, so that the basic gravitational variables are orthonormal tetrads and Lorentz connections. The second uses a canonical phase space consisting of initial data on a Cauchy slice of . In the gravitational sector, this description is based on the standard ADM variables. Since the conceptual structure underlying the main calculation and the final results are the same, the details of the formalism are not important. For definiteness, in the main discussion, we will use the covariant phase space and indicate the technical modifications needed in the canonical picture at the end.
The phase space is naturally endowed with a (pre-)symplectic structure – a closed 2-form (whose degenerate directions correspond to infinitesimal gauge motions). Given any two vector fields (i.e., infinitesimal variations) and on , the action of the symplectic 2-form on them provides a function on . A vector field on is said to be a Hamiltonian vector field (i.e., to generate an infinitesimal canonical transformation) if and only if . Since the phase space is topologically trivial, it follows that this condition holds if and only if there is a function on such that for all vector fields . The function is called a Hamiltonian and its Hamiltonian vector field; alternatively, is said to generate the infinitesimal canonical transformation .
Since we are interested in energy and angular momentum, the infinitesimal canonical transformations will correspond to time translations and rotations. As in any generally covariant theory, when the constraints are satisfied, values of Hamiltonians generating such diffeomorphisms can be expressed purely as surface terms. In the present case, the relevant surfaces are the sphere at infinity and the spherical section of the horizon. Thus the numerical values of Hamiltonians now consist of two terms: a term at infinity and a term at the horizon. The terms at infinity reproduce the ADM formulas for energy and angular momentum. The terms at the horizon define the energy and angular momentum of the WIH.
Let us begin with angular momentum (see  for details). Consider a vector field on which satisfies the following boundary conditions: (i) At infinity, coincides with a fixed rotational symmetry of the fiducial flat metric; and, (ii) on , it coincides with the vector field . Lie derivatives of physical fields along define a vector field on . The question is whether this is an infinitesimal canonical transformation, i.e., a generator of the phase space symmetry. As indicated above, this is the case if and only if there exists a phase space function satisfying:
for all variations . If such a phase space function exists, it can be interpreted as the Hamiltonian generating rotations.
Now, a direct calculation  shows that, in absence of gauge fields on , one has:
As expected, the expression for consists of two terms: a term at the horizon and a term at infinity. The term at infinity is the variation of the familiar ADM angular momentum associated with . The surface integral at the horizon is interpreted as the variation of the horizon angular momentum . Since variations are arbitrary, one can recover, up to additive constants, and from their variations, and these constants can be eliminated by requiring that both of these angular momenta should vanish in static axi-symmetric space-times. One then obtains:
where the function on is related to by . In the last step we have used Equation (28) and performed an integration by parts. Equation (32) is the expression of the horizon angular momentum. Note that all fields that enter this expression are local to the horizon and is not required to be a Killing field of the space-time metric even in a neighborhood of the horizon. Therefore, can be calculated knowing only the horizon geometry of a type II horizon.
We conclude our discussion of angular momentum with some comments:
- The Hamiltonian
It follows from Equation (31) and (32) that the total Hamiltonian generating the rotation along is the difference between the ADM and the horizon angular momenta (apart from a sign which is an artifact of conventions). Thus, it can be interpreted as the angular momentum of physical fields in the space-time region outside the black hole.
- Relation to the Komar integral
If happens to be a space-time Killing field in a neighborhood of , then agrees with the Komar integral of . If is a global, space-time Killing field, then both as well as agree with the Komar integral, whence the total Hamiltonian vanishes identically. Since the fields in the space-time region are all axi-symmetric in this case, this is just what one would expect from the definition of .
- for general axial fields
If the vector field is tangential to cross-sections of , continues to the generator of the canonical transformation corresponding to rotations along , even if its restriction to does not agree with the axial symmetry of horizon geometries of our phase space fields. However, there is an infinity of such vector fields and there is no physical reason to identify the surface term arising from any one of them with the horizon angular momentum.
- Inclusion of gauge fields
If non-trivial gauge fields are present at the horizon, Equation (32) is incomplete. The horizon angular momentum is still an integral over ; however it now contains an additional term involving the Maxwell field. Thus contains not only the ‘bare’ angular momentum but also a contribution from its electromagnetic hair (see  for details).
- Canonical phase space
The conceptual part of the above discussion does not change if one uses the canonical phase space  in place of the covariant. However, now the generator of the canonical transformation corresponding to rotations has a volume term in addition to the two surface terms discussed above. However, on the constraint surface the volume term vanishes and the numerical value of the Hamiltonian reduces to the two surface terms discussed above.
4.1.3 Energy, mass, and the first law
To obtain an expression of the horizon energy, one has to find the Hamiltonian on generating diffeomorphisms along a time translation symmetry on . To qualify as a symmetry, at infinity must approach a fixed time translation of the fiducial flat metric. At the horizon, must be an infinitesimal symmetry of the type II horizon geometry. Thus, the restriction of to should be a linear combination of a null normal and the axial symmetry vector ,
where and the angular velocity are constants on .
However there is subtlety: Unlike in the angular momentum calculation where is required to approach a fixed rotational vector on , the restriction of to can not be a fixed vector field. For physical reasons, the constants and should be allowed to vary from one space-time to another; they are to be functions on phase space. For instance, physically one expects to vanish on the Schwarzschild horizon but not on a generic Kerr horizon. In the terminology of numerical relativity, unlike , the time translation must be a live vector field. As we shall see shortly, this generality is essential also for mathematical reasons: without it, evolution along will not be Hamiltonian!
At first sight, it may seem surprising that there exist choices of evolution vector fields for which no Hamiltonian exists. But in fact this phenomenon can also happen in the derivations of the ADM energy for asymptotically flat space-times in the absence of any black holes. Standard treatments usually consider only those that asymptote to the same unit time translation at infinity for all space-times included in the phase space. However, if we drop this requirement and choose a live which approaches different asymptotic time-translations for different space-times, then in general there exists no Hamiltonian which generates diffeomorphisms along such a . Thus, the requirement that the evolution be Hamiltonian restricts permissible