8 Outlook
In the last six sections, we summarized the isolated and dynamical
horizon frameworks and their applications. These provide a
quasi-local 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 space-time metric is
(with
) and the topology of
is
. These assumptions rule out,
by fiat, the presence of a NUT charge. To incorporate a non-zero
NUT charge in black hole mechanics, one must either allow
to be topologically
, or allow for
‘wire singularities’ in the rotation 1-form
on
. The zeroth law goes through in these
more general situations. What about the first law? Arguments based
on Euclidean methods [114, 146, 147] 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 [54
, 33, 185] at null infinity.
-
Colored black holes
-
As discussed in Section 6, black hole uniqueness theorems of the
Einstein-Maxwell theory fail once non-Abelian fields are included.
For example, there are black hole solutions to the
Einstein-Yang-Mills equations with non-trivial Yang-Mills fields
whose only non-zero 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 [75] 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 [132] 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 [9]. 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 [131]
are included. The existence of these semi-numerical 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 non-zero 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 space-time
geometry is clearer than in, say, the string theory calculations
based on D-branes. Therefore, one can now attempt to calculate the
Hawking radiation directly in the physical space-time 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
space-time. Elements of this theory are now in place. The concrete
open problem is the calculation of the absorption cross-section for
quantum matter fields propagating on this ‘background state’. If
this can be shown to equal the classical absorption cross-section
to the leading order, it would follow [29] 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 [17]. As in the earlier analysis of the
big-bang singularity, it is found that the black hole singularity
is resolved, but the classical space-time 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 semi-classical 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
space-time 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
semi-classical, 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 space-like, 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 [37]. 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 non-linear 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
axi-symmetric. Using the free data, as in the case of isolated
horizons [24
], 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 [54], 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 space-time must be greater than half the radius of
the apparent horizon on any Cauchy slice. This conjecture was
recently proved by Bray [59], and by Huisken and Ilmamen [122]. Recently, for the non-time symmetric
case, Ben-Dov has constructed an example where the apparent horizon
does not satisfy this inequality [47]. This is not a contradiction with the
original Penrose inequality which referred to the area of
cross-sections of the event horizon, however it does show that
extending the results beyond time-symmetry would be quite
non-trivial. 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 non-linear 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 space-time 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 [24], 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 non-axisymmetric 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 non-axisymmetric context.
Equations that must hold on DHs provide no obvious
obstruction [31
]. 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 [89]. 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 non-equilibrium processes, during which the horizon makes a
transition from a given state to one which is far removed, has
interesting thermodynamic ramifications. In non-equilibrium 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 [31]). These restricted
situations are nonetheless very interesting. Can the black hole
entropy derivations based on counting of micro-states, such as
those of [10], 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
quasi-local counting of micro-states 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 non-linear 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 space-times where one can unambiguously
identify infinity. Even in these restricted contexts, event
horizons are teleological, can form in a flat region of space-time
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 [48, 102] suggest that the outermost
marginally trapped world-tubes 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 quasi-locally, can not be formed in flat
space-time, 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 space-time (see
Section 2.2.1). Hayward [117] 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., [86]). 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 non-linear regime, one has to give up the idea that there
is a single 3-manifold 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.