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 [34, 42, 43, 44]. 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 [67, 119] took the community by surprise. The subsequent
analysis of the detailed properties of Kerr–Newman solutions [63] and perturbations thereof [64]
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 [37, 47]. Research in this area continues to be
an active branch of mathematical physics [182]. 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 [153], the
associated critical phenomenon [65, 105], and the dynamics leading to their coalescence (see, e.g.,
[1, 127, 160, 56, 2, 139]). 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 [77].
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, 89], 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 [34, 184]. 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 spacetime (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 timetranslational 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 nonstationary 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.
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 spacetimes?

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 socalled BPS states) [120].
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 nonminimal 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 prerequisite 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
nonperturbative 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 spacetimes. Surely, one should be able to analyze black
hole dynamics also in these spacetimes. More importantly, the notion is teleological; it lets us
speak of a black hole only after we have constructed the entire spacetime. 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 quasilocal 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 spacetime containing evolving black holes. Thus, one needs to identify initial
data containing black holes without the knowledge of the entire spacetime 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 spacelike 3surface to be directly
useful in all contexts. Is there a truly quasilocal 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
subclass 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 nonnegligible [86, 98, 87]. 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, quasilocal paradigm to describe black holes. This
framework was inspired by certain seminal ideas introduced by Hayward [116, 117, 115, 118] in the
midnineties 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 quasilocal. In contrast to event horizons, neither notion requires the knowledge of spacetime as a
whole or refers to asymptotic flatness. Furthermore, they are spacetime 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 quasilocal 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 [66]. Yet, it has been possible to extend the zeroth and first laws of black
hole mechanics to isolated horizons [25, 74, 14]. 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 nonAbelian 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 [29, 30]. The processes can be in the nonlinear 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
nonperturbative 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 [9, 10, 83, 149, 24]. 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 [99]. Finally, the arena for this
derivation is the curved black hole geometry, rather than a system in flat spacetime which has
the same number of states as the black hole [175, 146]. 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 [53].
In particular, they provide seeds for further investigations of the notion of entropy in
nonequilibrium situations.

Mathematical physics

The isolated horizon framework has led to a phenomenological model to understand properties
of hairy black holes [20, 19]. 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 semianalytical 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 crosssections of the event horizon on a Cauchy surface
with the ADM mass at infinity [157]: . 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
nontimesymmetric situations. It may also lead to a stronger version of the conjecture where
the ADM mass is replaced by the Bondi mass [30].

Numerical relativity

The framework has provided a number of tools to extract physics from numerical simulations
in the nearhorizon, 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 [30] and extract properties of the final equilibrium state [14, 84].
These quantities can be calculated knowing only the horizon geometry and do not presuppose
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 [33] 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 nearhorizon geometries of different simulations [11] and leads to a new
approach to the problem of extracting waveforms in a gauge invariant fashion. Finally, the
framework provides natural boundary conditions for the initial value problem for black holes in
quasiequilibrium [70, 125, 80], and to interpret certain initial data sets [137]. Many of these
ideas have already been implemented in some binary black hole codes [84, 33, 46] and the
process is continuing.

Gravitational wave phenomenology

The isolated horizon framework has led to a notion of horizon multipole moments [23]. They
provide a diffeomorphism invariant characterization of the isolated horizon geometry. They
are distinct from the Hansen multipoles in stationary spacetimes [107] 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 spacetime, 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 [30].
The issue can be explored both numerically and analytically. The possibility that the bound
can indeed be violated initially has interesting astrophysical implications [88].
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 spacetime metric will
be , and matter fields . For simplicity we will restrict ourselves to 4dimensional
spacetime manifolds (although most of the classical results on isolated horizons have
been extended to 3dimensions spacetimes [22], as well as higher dimensional ones [144]).
The spacetime 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 spacetime 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 .