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 [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 [65] and perturbations thereof [66
] 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 [181
]. 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 [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 [78
].
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. For instance, there may be
matter rings around the black hole which contribute to the angular
momentum and mass at infinity. Why is this contribution relevant to
the first law of black hole mechanics? Shouldn’t only the angular
momentum and mass of the black hole feature in the first law? Thus,
one is led to ask: Is there a more suitable paradigm which can
replace frameworks based on event horizons in stationary
space-times?
- Entropy
calculations
-
The first and the second laws suggest that one should assign to a
black hole an entropy which is proportional to its area. This poses
a concrete challenge to candidate theories of quantum gravity:
Account for this entropy from fundamental, statistical mechanical
considerations. String theory has had a remarkable success in
meeting this challenge in detail for a subclass of extremal,
stationary black holes whose charge equals mass (the so-called BPS
states) [120].
However, for realistic black holes the charge to mass ratio is less
than
. 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 [68
]. 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 [99
]. 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 [56
]. 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 [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
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 [31
].
- 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 [31
] 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 [34
] has shown that
these methods are also numerically more accurate and robust than
older ones.
Surprisingly, there are simple local criteria to decide whether the
geometry of an isolated horizon is that of the Kerr
horizon [142]. These criteria have already been
implemented in numerical simulations. The isolated horizon
framework also provides invariant, practical criteria to compare
near-horizon geometries of different
simulations [12
] 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 [136
]. 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 [24
]. They provide a
diffeomorphism invariant characterization of the isolated horizon
geometry. They are distinct from the Hansen multipoles in
stationary space-times [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 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 [31
]. The issue can be
explored both numerically and analytically. The possibility that
the bound can indeed be violated initially has interesting
astrophysical implications [89
].
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 [23
], as well as higher
dimensional ones [141]). 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
.