6.2 Structure of colored,
static black holes
We will briefly summarize research in three areas in which the
isolated horizon framework has been used to illuminate the
structure of static, colored black holes and associated
solitons.
6.2.1
Horizon mass
Let us begin with Einstein-Yang-Mills theory
considered in the last Section 6.1. As we saw, the ADM mass fails to be a
good measure of the horizon mass for colored black holes. The
failure of black hole uniqueness theorems also prevents the
isolated horizon framework from providing a canonical notion of
horizon mass on the full phase space. However, one can repeat the
strategy used for dilatonic black holes to define horizon mass
unambiguously for the static solutions [77
, 26
].
Consider a connected component of the known
static solutions, labelled by
. Using
for
the surface gravity of the properly normalized
static Killing vector, in this sector one can construct a live
vector field
and obtain a first law. The energy
is well-defined on the full phase space and can be
naturally interpreted as the horizon mass
for colored black holes with ‘quantum number’
. The explicit expression is given by
where, following a convention in the literature on hairy black
holes [77
], we have used
rather than the surface gravity
of static
solutions. Now, as in the Einstein-Maxwell case, the Hamiltonian
generating evolution along
contains only surface terms:
and is constant on each
connected, static sector if
coincides with the static
Killing field on that sector. By construction, our
has this property for the
-sector under consideration. Now, in the
Einstein-Maxwell case, since there is no constant with the
dimension of energy, it follows that the restriction of
to the static sector must vanish. The situation is
quite different in Einstein-Yang-Mills theory where the Yang-Mills
coupling constant
provides a scale. In
units,
. Therefore, we can only conclude that
for some
-dependent constant
. As the horizon radius shrinks to zero, the static
solution [127, 181
] under consideration
tends to the solitonic solution with the same ‘quantum numbers’
. Hence, by taking this limit, we conclude
. Therefore, on any
-static sector, we have the following interesting
relation between the black hole and solitonic solutions:
Thus, although the main motivation behind the isolated horizon
framework was to go beyond globally time-independent situations, it
has led to an interesting new relation between the ADM masses of
black holes and their solitonic analogs already in the static
sector.
The relation (80) was first proposed
for spherical horizons in [77], verified in [75
], and extended to
distorted horizons in [26]. It provided impetus for
new work by mathematical physicists working on colored black holes.
The relation has been confirmed in three more general and
non-trivial cases:
- Non-spherical, non-rotating black holes
parameterized by two quantum numbers [132
].
- Non-spherical solutions to the more general
Einstein-Yang-Mills-Higgs theory [109
], where distortions
are caused by ‘magnetic dipole hair’ [130].
- Static solutions in the Born-Infeld
theory [62].
6.2.2
Phenomenological
model of colored black holes
Isolated horizon considerations suggested the
following simple heuristic model of colored black holes [21
]: A colored black hole
with quantum numbers
should
be thought of as ‘bound states’ of a ordinary (colorless)
black hole and a soliton with color quantum
numbers
, where
can be more general
than considered so far. Thus the idea is that an uncolored black
hole is ‘bare’ and becomes ‘colored’ when ‘dressed’ by the
soliton.
The mass formula (80) now suggests that the
total ADM mass
has three components: the
mass
of the bare horizon, the mass
of the colored soliton, and a binding energy given
by
If this picture is correct, being gravitational binding energy,
would have to be negative. This expectation is borne
out in explicit examples. The model has several predictions on the
qualitative behavior of the horizon mass (77), the surface gravity,
and the relation between properties of black holes and solitons. We
will illustrate these with just three examples (for a complete list
with technical caveats, see [21
]):
- For any fixed value of
and of all quantum numbers except
, the horizon mass and surface gravity decrease
monotonically with
.
- For fixed values of all quantum numbers
, the horizon mass
is
non-negative, vanishing if and only if
vanishes, and
increases monotonically with
.
is positive and
bounded above by
.
- For fixed
, the binding energy
decreases as the horizon area increases.
The predictions for fixed
have recently been verified beyond spherical
symmetry: for the distorted, axially symmetric Einstein-Yang-Mills
solutions in [132
] and for the
distorted ‘dipole’ solutions in Einstein-Yang-Mills-Higgs solutions
in [109
]. Taken together,
the predictions of this model can account for all the qualitative
features of the plots of the horizon mass and surface gravity as
functions of the horizon radius and quantum numbers. More
importantly, they have interesting implications on the stability
properties of colored black holes.
One begins with an observation about solitons and deduces
properties of black holes. Einstein-Yang-Mills solitons are known
to be unstable [173]; under small
perturbations, the energy stored in the ‘bound state’ represented
by the soliton is radiated away to future null infinity
. The phenomenological model suggests that colored
black holes should also be unstable and they should decay into
ordinary black holes, the excess energy being radiated away to
infinity. In general, however, even if one component of a bound
system is unstable, the total system may still be stable if the
binding energy is sufficiently large. An example is provided by the
deuteron. However, an examination of energetics reveals that this
is not the case for colored black holes, so instability should
prevail. Furthermore, one can make a few predictions about the
nature of instability. We summarize these for the simplest case of
spherically symmetric, static black holes for which there is a
single quantum number
:
- All the colored black holes on the
th branch have the same number (namely,
) of unstable modes as the
th soliton. (The detailed features of these unstable
modes can differ especially because they are subject to different
boundary conditions in the two cases.)
- For a given
, colored black holes
with larger horizon area are less unstable. For a given horizon
area, colored black holes with higher value of
are more unstable.
- The ‘available energy’ for the process is
given by
Part of it is absorbed by the black hole so that its horizon area
increases and the rest is radiated away to infinity. Note that
can be computed knowing just the initial configuration.
- In the process the horizon area necessarily
increases. Therefore, the energy radiated to infinity is strictly
less than
.
Expectation 1 of the model is known to be
correct [172].
Prediction 2 has been shown to be correct in the
, colored black holes in the sense that the frequency
of all unstable modes is a decreasing function of the area, whence
the characteristic decay time grows with area [181
, 51]. To our
knowledge a detailed analysis of instability, needed to test
Predictions 3 and 4 are yet to be made.
Finally, the notion of horizon mass and the
associated stability analysis has also provided an ‘explanation’ of
the following fact which, at first sight, seems puzzling. Consider
the ‘embedded Abelian black holes’ which are solutions to
Einstein-Yang-Mills equations with a specific magnetic charge
. They are isometric to a family of magnetically
charged Reissner-Nordström solutions and the isometry maps the
Maxwell field strength to the Yang-Mills field strength. The only
difference is in the form of the connection; while the Yang-Mills
potential is supported on a trivial
bundle, the
Maxwell potential requires a non-trivial
bundle. Therefore, it comes as an initial surprise
that the solution is stable in the Einstein-Maxwell theory but
unstable in the Einstein-Yang-Mills theory [52, 61]. It turns out that this difference is
naturally explained by the WIH framework. Since the solutions are
isometric, their ADM mass is the same. However, since the horizon
mass arises from Hamiltonian considerations, it is theory
dependent: It is lower in the
Einstein-Yang-Mills theory than in the Einstein-Maxwell theory!
Thus, from the Einstein-Yang-Mills perspective, part of the ADM
mass is carried by the soliton and there is positive
which can be radiated away to infinity. In the
Einstein-Maxwell theory,
is zero.
The stability analysis sketched above therefore implies that the
solution should be unstable in the Einstein-Yang-Mills theory but
stable in the Einstein-Maxwell theory. This is another striking
example of the usefulness of the notion of the horizon mass.
6.2.3
More general
theories
We will now briefly summarize the most
interesting result obtained from this framework in more general
theories. When one allows Higgs or Proca fields in addition to
Yang-Mills, or considers Einstein-Skyrme theories, one acquires
additional dimensionful constants which trigger new
phenomena [50, 178, 181]. One of the most
interesting is the ‘crossing phenomena’ of Figure 11 where curves in the
‘phase diagram’ (i.e., a plot of the ADM mass versus horizon
radius) corresponding to the two distinct static families cross.
This typically occurs in theories in which there is a length scale
even in absence of gravity, i.e., even when Newton’s constant is
set equal to zero [154, 21
].
In this case, the notion of the horizon mass acquires further
subtleties. If, as in the Einstein-Yang-Mills theory considered
earlier, families of static solutions carrying distinct quantum
numbers do not cross, there is a well-defined notion of horizon
mass for each static solution, although, as the example of
‘embedded Abelian solutions’ shows, in general its value is theory
dependent. When families cross, one can repeat the previous
strategy and use Equation (77) to define a mass
along each branch. However, at the intersection
point
of the
th and
th branches, the mass is discontinuous. This
discontinuity has an interesting implication. Consider the closed
curve
in the phase diagram, starting at the intersection
point and moving along the
th branch in
the direction of decreasing area until the area becomes zero, then
moving along
to the
th branch and moving
up to the intersection point along the
th branch (see
Figure 11). Discontinuity in
the horizon mass implies that the integral of
along this closed curve is non-zero. Furthermore,
the relation between the horizon and the soliton mass along each
branch implies that the value of this integral has a direct
physical interpretation:
This is a striking prediction because it relates differences in
masses of solitons to the knowledge of
horizon properties of the corresponding black
holes! Because of certain continuity properties which hold
as one approaches the static Einstein-Yang-Mills sector in the
space to static solutions to Einstein-Yang-Mills-Higgs
equations [76
], one can also
obtain a new formula for the ADM masses of Einstein-Yang-Mills
solitons. If
for the black hole solutions
of this theory is integrable over the entire positive half line,
one has [21
]:
Both these predictions of the phenomenological model [21] have been verified numerically in the
spherically symmetric case [76], but the axi-symmetric
case is still open.