Consider the problem of constructing initial
data on
, representing a binary black
hole system. This problem has two distinct aspects. The first has
to do with the inner boundaries. A natural avenue to handle the
singularities is to ‘excise’ the region contained in each black
hole (see, e.g., [3, 73]). However,
this procedure creates inner boundaries on
and one must specify suitable boundary conditions
there. The boundary conditions should be appropriate for the
elliptic system of constraint equations, and they should also
capture the idea that the excised region represents black holes
which have certain physical parameters and which are in quasi-equilibrium at the
‘instant’ represented by
. The second aspect of the
problem is the choice of the free data in the bulk. To be of
physical interest, not only must the free data satisfy the
appropriate boundary conditions, but the values in the bulk must
also have certain properties. For example, we might want the black
holes to move in approximately circular orbits around each other,
and require that there be only a minimal amount of spurious
gravitational radiation in the initial data. The black holes would
then remain approximately isolated for a sufficiently long time,
and orbit around each other before finally coalescing.
While a fully satisfactory method of prescribing
such initial data is still lacking, there has been significant
progress in recent years. When the black holes are far apart and
moving on approximately circular orbits, one might expect the
trajectory of the black holes to lie along an approximate helical
Killing vector [53, 91, 103, 104, 4]. Using concepts
from the helical Killing vector approximation and working in the
conformal thin-sandwich decomposition of the initial
data [192], Cook has
introduced the ‘quasi-equilibrium’ boundary conditions which
require that each of the black holes be in instantaneous
equilibrium [72
]; see
also [190
, 74] for a similar approach. The relation
between these quasi-equilibrium boundary conditions and the
isolated horizon formalism has also been recently
studied [124
].
In this section, we consider only the first
aspect mentioned above, namely the inner boundaries. Physically,
the quasi-equilibrium approximation ought to be valid for time
intervals much smaller than other dynamical time scales in the
problem, and the framework assumes only that the approximation
holds infinitesimally ‘off ’. So, in this section, the
type II NEH
will be an infinitesimal world tube of apparent
horizons. We assume that there is an axial symmetry vector
on the horizon, although, as discussed in
Section 8, this assumption can be weakened.
Depending on the degree of isolation one wants to
impose on the individual black holes, the inner boundary may be
taken to be the cross-section of either an NEH, WIH, or an IH. The
strategy is to first start with an NEH and impose successively
stronger conditions if necessary. Using the local geometry of
intersections of
with
, one can easily calculate the area radius
, the angular momentum
given by
Equation (59
), the canonical
surface gravity
given by Equation (38
), and angular velocity
given by Equation (39
). (Note that while a
WIH structure is used to arrive at the expressions for
,
, and
, the expressions
themselves are unambiguously defined also on a NEH.) These
considerations translate directly into restrictions on the shift
vector
at the horizon. If, as in Section 4.1.3, one requires that the restriction
of the evolution vector field
to
be of the form
Next, one imposes the condition that the infinitesimal world-tube of apparent horizons is an ‘instantaneous’ non-expanding horizon. This requirement is equivalent to
onUp to this point, the considerations are general
in the sense that they are not tied to a particular method of
solving the initial value problem. However, for the
quasi-equilibrium problem, it is the conformal thin-sandwich method [192, 70] that appears to
be best suited. This approach is based on the conformal
method [144, 191] where we write the
3-metric as . The free data consists of
the conformal 3-metric
, its time derivative
, the lapse
, and the trace of the
extrinsic curvature
. Given this free data, the remaining
quantities, namely the conformal factor and the shift, are
determined by elliptic equations provided appropriate boundary
conditions are specified for them on the horizon6. It turns out [124] that the horizon
conditions (63
) are well-tailored for
this purpose. While the issue of existence and uniqueness of
solutions using these boundary conditions has not been proven, it
is often the case that numerical calculations are convergent and
the resulting solutions are well behaved. Thus, these conditions
might therefore be sufficient from a practical point of view. In
the above discussion, the free data consisted of
and one solved elliptic equations for
. However, it is common to consider an enlarged
initial value problem by taking
as part of the
free data (usually set to zero) and solving an elliptic equation
for
. We now need to prescribe an additional boundary
condition for
. It turns out that this can be done by
using WIHs, i.e., by bringing in surface gravity, which did not
play any role so far. From the definition of surface gravity in
Equations (7
, 8
), it is clear that the
expression for
will involve a time derivative; in
particular, it turns out to involve the time derivative of
. It can be shown that by choosing
on
(e.g., by taking
) and requiring surface gravity to be constant on
and equal to
, one obtains a suitable
boundary condition for
. (The freedom to choose
freely the function
mirrors the fact that fixing
surface gravity does not uniquely fix the rescaling freedom of the
null normal.) Note that
is required to be constant
only on
, not on
. To ask it to be constant on
would require
, which in turn
would restrict the second time
derivative; this necessarily involves the evolution equations, and
they are not part of the initial data scheme.
One may imagine using the yet stronger notion of an IH, to completely fix the value of the lapse at the horizon. But this requires solving an elliptic equation on the horizon and the relevant elliptic operator has a large kernel [16, 72]. Nonetheless, the class of initial data on which its inverse exists is infinite dimensional so that the method may be useful in practice. However, this condition would genuinely restrict the permissible initial data sets. In this sense, while the degree of isolation implied by the IH boundary condition is likely to be met in the asymptotic future, for quasi-equilibrium initial data it is too strong in general. It is the WIH boundary conditions that appear to be well-tailored for this application.
Finally, using methods introduced by
Dain [80], a variation of the above procedure was
recently introduced to establish the existence and uniqueness of
solutions and to ensure that the conformal factor is everywhere positive [81]. One again imposes Equation (63
). However, in place of
Dirichlet boundary conditions (62
) on the shift, one now
imposes Neumann-type conditions on certain components of
. This method is expected to be applicable all
initial data constructions relying on the conformal method.
Furthermore, the result might also be of practical use in numerical
constructions to ensure that the codes converge to a well behaved
solution.
For initial data representing a binary black
hole system, the quantity
is called the effective binding
energy, where
is the ADM mass, and
are the individual masses of the two black holes.
Heuristically, even in vacuum general relativity, one would expect
to have several components. First there is the
analog of the Newtonian potential energy and the spin-spin
interaction, both of which may be interpreted as contributing to
the binding energy. But
also contains contributions
from kinetic energy due to momentum and orbital angular momentum of
black holes, and energy in the gravitational radiation in the
initial data. It is only when these are negligible that
is a good measure of the physical binding
energy.
The first calculation of binding energy was made
by Brill and Lindquist in such a context. They considered two
non-spinning black holes initially at rest [63]. For large
separations, they found that, in a certain mathematical sense, the
leading contribution to binding energy comes just from the usual
Newtonian gravitational potential. More recently, Dain [79
] has extended this
calculation to the case of black holes with spin and has shown that
the spin-spin interaction energy is correctly incorporated in the
same sense.
In numerical relativity, the notion of binding
energy has been used to locate sequences of quasi-circular orbits.
The underlying heuristic idea is to minimize with respect to the proper separation between the
holes, keeping the physical parameters of the black holes fixed.
The value of the separation which minimizes
provides an estimate of sizes of stable ‘circular’
orbits [71, 39, 159]. One finds that these
orbits do not exist if the orbital angular momentum is smaller than
a critical value (which depends on other parameters) and uses this
fact to approximately locate the ‘inner-most stable circular orbit’
(ISCO). In another application, the binding energy has been used to
compare different initial data sets which are meant to describe the
same physical system. If the initial data sets have the same values
of the black hole masses, angular momenta, linear momenta, orbital
angular momenta, and relative separation, then any differences in
should be due only to the different radiation
content. Therefore, minimization of
corresponds to
minimization of the amount of radiation in the initial
data [158].
In all these applications, it is important that
the physical parameters of the black holes are calculated
accurately. To illustrate the potential problems, let us return to
the original Brill-Lindquist calculation [63]. The topology of
the spatial slice is
with two points (called
‘punctures’) removed. These punctures do not represent curvature
singularities. Rather, each of them represents a spatial infinity
of an asymptotically flat region which is hidden behind an apparent
horizon. This is a generalization of the familiar Einstein-Rosen
bridge in the maximally extended Schwarzschild solution. The black
hole masses
and
are taken to be the
ADM masses of the corresponding hidden asymptotic regions.
(Similarly, in [79
], the angular
momentum of each hole is defined to be the ADM angular momentum at
the corresponding puncture.) Comparison between
and the Newtonian binding energy requires us to
define the distance between the holes. This is taken to be the
distance between the punctures in a fiducial
flat background metric; the physical distance between the
two punctures is infinite since they represent asymptotic ends of
the spatial 3-manifold. Therefore, the sense in which one recovers
the Newtonian binding energy as the leading term is physically
rather obscure.
Let us re-examine the procedure with an eye to extending it to a more general context. Let us begin with the definition of masses of individual holes which are taken to be the ADM masses in the respective asymptotic regions. How do we know that these are the physically appropriate quantities for calculating the potential energy? Furthermore, there exist initial data sets (e.g., Misner data [150, 151]) in which each black hole does not have separate asymptotic regions; there are only two common asymptotic regions connected by two wormholes. For these cases, the use of ADM quantities is clearly inadequate. The same limitations apply to the assignment of angular momentum.
A natural way to resolve these conceptual issues
is to let the horizons, rather than the punctures, represent black
holes. Thus, in the spirit of the IH and DH frameworks, it is more
appropriate to calculate the mass and angular momentum using
expressions (60, 59
) which involve the
geometry of the two apparent horizons. (This requires the apparent
horizons to be axi-symmetric, but this limitation could be overcome
following the procedure suggested in Section 8.)
Similarly, the physical distance between the black holes should be
the smallest proper distance between the two apparent horizons. To
test the viability of this approach, one can repeat the original
Brill-Lindquist calculation in the limit when the black holes are
far apart [136
]. One first
approximately locates the apparent horizon, finds the proper
distance
between them, and then calculates the horizon masses
(and thereby
) as a power series in
. The leading-order term does turn out to be the
usual Newtonian gravitational potential energy but the higher order
terms are now different from [63]. Similarly, it would be interesting to
repeat this for the case of spinning black holes and recover the
leading order term of [79] within this more physical paradigm
using, say, the Bowen-York initial data. This result would
re-enforce the physical ideas and the approach can then be used as
a well defined method for calculating binding energy in more
general situations.