8 Outlook
In the last six sections, we summarized the isolated and dynamical horizon frameworks and their
applications. These provide a quasilocal and more physical paradigm for describing black holes both in the
equilibrium and dynamical regimes. One of the most pleasing aspects of the paradigm is that it provides a
unified approach to a variety of problems involving black holes, ranging from entropy calculations in
quantum gravity, to analytical issues related to numerical simulations, to properties of hairy black holes, to
laws of black hole mechanics. More importantly, as summarized in Section 1, these frameworks enable one
to significantly extend the known results based on Killing and event horizons, and provide brand new laws
in the dynamical regime.
In this section, we will discuss some of the open issues whose resolution would lead to significant
progress in various areas.

Isolated horizons

This is the best understood part of the new paradigm. Nonetheless, several important issues still
remain. We will illustrate these with a few examples:

Black hole mechanics

Throughout, we assumed that the spacetime metric is (with ) and the
topology of is . These assumptions rule out, by fiat, the presence of a NUT
charge. To incorporate a nonzero NUT charge in black hole mechanics, one must either
allow to be topologically , or allow for ‘wire singularities’ in the rotation 1form
on . The zeroth law goes through in these more general situations. What about
the first law? Arguments based on Euclidean methods [114, 147, 96] show that entropy is
no longer given by the horizon area but there is also contribution due to ‘Misner strings’.
However, to our knowledge, a systematic derivation in the Lorentzian regime is not yet
available. Such a derivation would provide a better understanding of the physical origin of
the extra terms. The covariant phase space methods used in the isolated horizon framework
should be applicable in this case.

Application to numerical relativity

We saw in Section 5.4 that, in the IH framework, one can introduce an approximate
analog of future null infinity and invariant coordinate systems and tetrads in its
neighborhood. With this structure, it is feasible to extract waveforms and energy fluxes
in a reliable manner within the standard 3 + 1 Cauchy evolution of numerical relativity,
without having to do a Cauchy characteristic matching or use conformal field equations.
The challenge here is to develop, on the approximate , the analog of the basics of the
Bondi framework [52, 32, 185] at null infinity.

Colored black holes

As discussed in Section 6, black hole uniqueness theorems of the Einstein–Maxwell theory
fail once nonAbelian fields are included. For example, there are black hole solutions to
the Einstein–Yang–Mills equations with nontrivial Yang–Mills fields whose only nonzero
charge at infinity is the ADM mass. Thus, from the perspective of infinity, they are
indistinguishable from the Schwarzschild solution. However, their horizon properties are
quite different from those of the Schwarzschild horizon. This example suggests that perhaps
the uniqueness theorems fail because of the insistence on evaluating charges at infinity.
Corichi, Nucamendi, and Sudarsky [74] have conjectured that the uniqueness theorems
could be restored if they are formulated in terms of all the relevant horizon charges.
This is a fascinating idea and it has been pursued numerically. However, care is needed
because the list of all relevant charges may not be obvious a priori. For example, in
the case of static but not necessarily spherical Yang–Mills black holes, the conjecture
seemed to fail [133] until one realized that, in addition to the standard Yang–Mills charges
at the horizon, one must also include a topological, ‘winding charge’ in the list [73].
Once this charge is included, uniqueness is restored not only in the static sector of
the Einstein–Yang–Mills theory, but also when Higgs fields [109] and dilatons [132] are
included. The existence of these seminumerical proofs suggests that it should be possible
to establish uniqueness completely analytically. A second set of problems involves the
surprising relations, e.g., between the soliton masses and horizon properties of colored
black holes, obtained using isolated horizons. Extensions of these results to situations with
nonzero angular momentum should be possible and may well provide yet new insights.

Quantum black holes

In the approach based on isolated horizons, the microscopic degrees of freedom responsible
for the statistical mechanical entropy of black holes are directly related to the quantum
geometry of horizons. Therefore, their relation to the curved spacetime geometry is clearer
than in, say, the string theory calculations based on Dbranes. Therefore, one can now
attempt to calculate the Hawking radiation directly in the physical spacetime as a process
in which quanta of area are converted to quanta of matter. However, such a calculation is
yet to be undertaken. A direct approach requires quantum field theory (of matter fields) on
a quantum geometry state which is approximated by the classical black hole spacetime.
Elements of this theory are now in place. The concrete open problem is the calculation of
the absorption crosssection for quantum matter fields propagating on this ‘background
state’. If this can be shown to equal the classical absorption crosssection to the leading
order, it would follow [28] that the spectrum of the outgoing particles would be thermal
at the Hawking temperature. Another, perhaps more fruitful, avenue is to introduce an
effective model whose Hamiltonian captures the process by which quanta of horizon area
are converted to quanta of matter. Both these approaches are geared to large black holes
which can be regarded as being in equilibrium for the process under consideration, i.e.,
when . However, this approximation would fail in the Planck regime whence the
approaches can not address issues related to ‘information loss’. Using ideas first developed
in the context of quantum cosmology, effects of the quantum nature of geometry on the
black hole singularity have recently been analyzed [16]. As in the earlier analysis of the
bigbang singularity, it is found that the black hole singularity is resolved, but the classical
spacetime dissolves in the Planck regime. Therefore, the familiar Penrose diagrams are no
longer faithful representations of the physical situation. Suppose that, evolving through
what was singularity in the classical theory, the quantum state becomes semiclassical
again on the ‘other side’. Then, the indications are that information would not be lost:
It would be recovered on full , although observers restricted to lie in the part of
spacetime which is completely determined by the data on would see an approximate
Hawking radiation. If on the other hand the evolved state on the ‘other side’ never becomes
semiclassical, information would not be recovered on the available . An outstanding
open issue is which of this possibility is actually realized.

Dynamical horizons

The DH framework is less developed and the number of open issues is correspondingly
higher. At least for the classical applications of the framework, these problems are more
important.

Free data and multipoles

Since is spacelike, to find the fields which constitute the DH geometry,
one has to solve just the initial value equations, subject to the condition that admits a
foliation by marginally trapped surfaces. A general solution of this problem would provide
the ‘free data’ on DHs. In the case when the marginally trapped surfaces are round spheres,
this problem has been analyzed by Bartnik and Isenberg [36]. As noted in Section 2.2,
in this case there are no DHs in absence of matter sources. In presence of matter, one
can freely specify the trace of the extrinsic curvature and the (radial component of
the) momentum density , and determine the geometry by solving a nonlinear
ordinary differential equation whose solutions always exist locally. It would be interesting
to analyze the necessary and sufficient conditions on the free data which guarantee that
global solutions exist and the DH approaches the Schwarzschild horizon asymptotically.
From the point of view of numerical relativity, a more pressing challenge is to solve
the constraint equations in the vacuum case, assuming only that the marginally trapped
surfaces are axisymmetric. Using the free data, as in the case of isolated horizons [23],
one could define multipoles. Since is again defined unambiguously, a natural starting
point is to use it as the key geometrical object as in the IH case. However, just as the
Bondi mass aspect acquires shear terms in presence of gravitational radiation [52], it is
likely that, in the transition from isolated to dynamical horizons, would have to be
supplemented with terms involving, e.g., (and perhaps also ). For instance, by
adding a suitable combination of such terms, one may be able to relate the rate of change
of the mass quadrupole moment with the flux of energy across .

Geometric analysis

The dynamical horizon framework provides new inputs for the proof of Penrose inequalities
which, when applied to time symmetric data (i.e., when the extrinsic curvature vanishes),
say that the total (ADM) mass of spacetime must be greater than half the radius
of the apparent horizon on any Cauchy slice. This conjecture was recently proved by
Bray [57], and by Huisken and Ilmamen [122]. Recently, for the nontime symmetric case,
Ben–Dov has constructed an example where the apparent horizon does not satisfy this
inequality [45]. This is not a contradiction with the original Penrose inequality which
referred to the area of crosssections of the event horizon, however it does show that
extending the results beyond timesymmetry would be quite nontrivial. The ‘flows’ which
led to the area law in Section 3 and balance equations in Section 4.2.2 may be potentially
useful for this purpose. This approach could lead to an inequality relating the area of
certain marginally trapped surfaces (the ones connected to future time like infinity via a
dynamical horizon) to the future limit of the Bondi mass. The framework also suggests
a program which could shed much light on what John Wheeler called ‘the issue of the
final state’: what are the final equilibrium states of a dynamical black hole and how,
in detail, is this equilibrium reached? From a mathematical perspective an important
step in addressing this issue is to analyze the nonlinear stability of Kerr black holes.
Consider, then, a neighborhood of the initial data of the Kerr solution in an appropriate
Sobolev space. One would expect the spacetime resulting from evolution of this data to
admit a dynamical horizon which, in the distant future, tends to an isolated horizon with
geometry that is isomorphic to that of a Kerr horizon. Can one establish that this is what
happens? Can one estimate the ‘rate’ with which the Kerr geometry is approached in the
asymptotic future? One avenue is to first establish that the solution admits a Kerr–Schild
type foliation, each leaf of which admits an apparent horizon, then show that the world
tube of these apparent horizons is a DH, and finally study the decay rates of fields along
this DH.

Angular momentum

As we saw in Section 5, most of the current work on IHs and DHs assumes the presence
of an axial symmetry on the horizon. A natural question arises: Can one weaken
this requirement to incorporate situations in which there is only an approximate – rather
than an exact – symmetry of the horizon geometry? The answer is in the affirmative
in the following sense. Recall first that the Newman–Penrose component is gauge
invariant on IHs and DHs. Let . (While any geometric field could be used here,
is the most natural candidate because, for IHs, encodes the horizon geometry.)
If the horizon geometry admits a symmetry, the orbits of the symmetry field are
the level surfaces of . More generally, let us suppose that the level surfaces of
provide a foliation of each good cut (minus two points) of the horizon. Then, using the
procedure of section 2.1 of [23], one can introduce on a vector field , tangential
to the foliation, which has the property that it agrees with the symmetry vector field
whenever the horizon geometry admits a symmetry. The procedure fails if the metric on
is at of a round sphere but should work generically. One can then use to define
angular momentum. On IHs this angular momentum is conserved; on DHs it satisfies
the balance law of Section 4.2.2; and in terms of the initial data, angular momentum
has the familiar form (59). IS this proposal viable for numerical simulations? In
cases ready analyzed, it would be interesting to construct d compare it with the
symmetry vector field obtained via Killing transport. A better test would be provided by
nonaxisymmetric Brill waves. A second issue associated with angular momentum is ether
the Kerr inequality can be violated in the early stages of black hole formation
or merger, particularly in a nonaxisymmetric context. Equations that must hold on DHs
provide no obvious obstruction [30]. Note that such an occurrence is not incompatible
with the DH finally settling down to a Kerr horizon. For, there is likely to be radiation
trapped between the DH and the ‘peak’ of the effective gravitational potential that could
fall into the DH as time elapses, reducing its angular momentum and increasing its mass.
The issue of whether a black hole can violate the Kerr inequality when it is first formed is
of considerable interest to astrophysics [88]. Again, numerical simulations involving, say,
Brill waves would shed considerable light on this possibility.

Black hole thermodynamics

The fact that an integral version of the first law is valid even for nonequilibrium
processes, during which the horizon makes a transition from a given state to one
which is far removed, has interesting thermodynamic ramifications. In nonequilibrium
thermodynamical processes, in general the system does not have time to come to
equilibrium, whence there is no canonical notion of its temperature. Therefore, while
one can still interpret the difference as the heat absorbed by
the system, in general there is no longer a clean split of this term into a
temperature part and a change in entropy part. If the process is such that the system
remains close to equilibrium throughout the process, i.e., can be thought of as making
continuous transitions between a series of equilibrium states, then the difference can be
expressed as , where the temperature varies slowly during the transition.
The situation on dynamical horizons is analogous. It is only when the horizon geometry
is changing slowly that the effective surface gravity of Section 4.2.2 would be a good
measure of temperature, and the horizon area a good measure of entropy (see Section 5.3
of [30]). These restricted situations are nonetheless very interesting. Can the black hole
entropy derivations based on counting of microstates, such as those of [9], be extended
to such DHs? In the case of event horizons one would not expect such a procedure to be
meaningful because, as we saw in Equation 2.2.2, an event horizon can be formed and
grow in a flat space region in anticipation of a future gravitational collapse. It is difficult
to imagine how a quasilocal counting of microstates can account for this phenomenon.
Perhaps the most surprising aspect of the current status of the theory of black holes is that so little is
known about their properties in the fully dynamical and nonlinear regime of general relativity. Indeed, we
do not even have a fully satisfactory definition of a dynamical black hole. Traditionally, one uses
event horizons. But as we discussed in detail, they have several undesirable features. First, they
are defined only in spacetimes where one can unambiguously identify infinity. Even in these
restricted contexts, event horizons are teleological, can form in a flat region of spacetime and
grow even though there is no flux of energy of any kind across them. When astronomers tell us
that there is a black hole in the center of Milky Way, they are certainly not referring to event
horizons.
Numerical simulations [46, 102] suggest that the outermost marginally trapped worldtubes become
dynamical horizons soon after they are formed. As we saw, dynamical horizons have a number of attractive
properties that overcome the limitations of event horizons: they are defined quasilocally, can not be formed
in flat spacetime, and their growth is dictated by balance laws with direct physical interpretation.
Physically, then, dynamical horizons satisfying the additional physical condition (i.e.,
SFOTHs) appear to be good candidates to represent the surface of a black hole. But so far, our
understanding of their uniqueness is rather limited. If a canonical dynamical horizon could be singled out by
imposing physically reasonable conditions, one could use it as the physical representation of an evolving
black hole.
A plausibility argument for the existence of a canonical dynamical horizon was given by Hayward. Note
first that on physical grounds it seems natural to associate a black hole with a connected, trapped region
in spacetime (see Section 2.2.1). Hayward [116] sketched a proof that, under seemingly natural but
technically strong conditions, the dynamical portion of its boundary, , would be a dynamical horizon
. This could serve as the canonical representation of the surface of an evolving black hole.
However, it is not clear whether Hayward’s assumptions are not too strong. To illustrate the concern, let
us consider a single black hole. Then, Hayward’s argument implies that there are no trapped
surfaces outside . On the other hand there has been a general expectation in the community
that, given any point in the interior of the event horizon, there passes a (marginally) trapped
surface through it (see, e.g., [85]). This would imply that the boundary of the trapped region is
the event horizon which, being null, can not qualify as a dynamical horizon. However, to our
knowledge, this result has not been firmly established. But it is clear that this expectation
contradicts the conclusion based on Hayward’s arguments. Which of these two expectations is
correct? It is surprising that such a basic issue is still unresolved. The primary reason is that
very little is known about trapped and marginally trapped surfaces which fail to be spherical
symmetric. Because of this, we do not know the boundary of the trapped region even in the Vaidya
solution.
If it should turn out that the second expectation is correct, one would conclude that Hayward’s
assumptions on the properties of the boundary of the trapped region are not met in physically interesting
situations. However, this would not rule out the possibility of singling out a canonical dynamical horizon
through some other conditions (as, e.g., in the Vaidya solution). But since this dynamical horizon would not
be the boundary of the trapped region, one would be led to conclude that in the dynamical and fully
nonlinear regime, one has to give up the idea that there is a single 3manifold that can be interpreted as
the black hole surface without further qualifications. For certain questions and in certain situations, the
dynamical horizon may be the appropriate concept, while for other questions and in different
situations, the boundary of the trapped region (which may be the event horizon) may be more
appropriate.