The first approach to be developed generalized the Misner
approach [78]. It was attractive because an isometry condition relating the
two asymptotically flat universes provides two useful things.
First, because the two universes are identical, finding a
solution in one universe means that you have the full solution.
Second, because the throats are fixed-point sets of the isometry,
we can construct boundary conditions on any quantity there. This
allows us to excise the region interior to the spherical throats
from one of the Euclidean background spaces and solve for the
initial data in the remaining volume.
The generalization of the Misner approach seemed preferable to
trying to solve the constraints on
N
+1 Euclidean manifolds stitched together smoothly at the throats
of
N
black holes. However, Brandt and Brügmann [27] realized it was possible to factor out analytically the
behavior of the singular points in the Euclidean manifold of the
N
+1 sheeted approach. Referred to as the ``puncture'' method, this
approach allows us to rewrite the constraint equations for
functions on an
N
+1 sheeted manifold as constraint equations for new functions on
a simple Euclidean manifold.
Another approach tried, which we will not discuss in detail, avoided the issue of the topology of the initial-data slice entirely. Developed by Thornburg [100], this approach was based on the idea that only the domain exterior to the apparent horizon of a black hole is relevant. The equation describing the location of an apparent horizon can be rewritten in a form that can be used as a boundary condition for the conformal factor in the Hamiltonian constraint equation. Thus, given a compatible solution to the momentum constraints, this boundary condition can be used to construct a solution of the Hamiltonian constraint in the domain exterior to the apparent horizons of any black holes, with no reference at all to the topology of the full manifold.
Here,
represents a flat metric in any suitable coordinate system. The
assumption of conformal flatness means that the differential
operators in the constraints are the familiar flat-space
operators. More importantly, if we use the conformal
transverse-traceless decomposition (32
), we find that in vacuum the momentum constraints completely
decouple from the Hamiltonian constraint.
The importance of this last property stems from the fact that
York and Bowen were able to find analytic solutions of this
version of the momentum constraints, solutions that represent a
black hole with both linear momentum and spin [26,
23,
24]. If we choose
, then the momentum constraints (31
) become
A solution of this equation is
Here,
and
are vector parameters,
r
is a coordinate radius, and
is the outward-pointing unit normal of a sphere in the flat
conformal space (
).
is the 3-dimensional Levi-Civita tensor.
This solution of the momentum constraints yields the tracefree part of the extrinsic curvature,
Remarkably, using this solution (70) and the assumptions in (67
), we can determine the physical values for the linear and
angular momentum of any initial data we can construct. The
momentum contained in an asymptotically flat initial-data
hypersurface can be calculated from the integral [112
]
where
is a Killing vector of the 3-metric
. Since we are not likely to have true Killing vectors, we make
use of the asymptotic translational and rotational Killing
vectors of the flat conformal space. We find from (71
), (70
), and (67
) that the
physical
linear momentum of the initial-data hypersurface is
and the
physical
angular momentum of the slice is
. Furthermore, because the momentum constraints are linear, we
can add any number of solutions of the form of Eq. (70
) to represent a collection of linear and angular momentum
sources. The total physical linear momentum of the initial-data
slice will simply be the vector sum of the individual linear
momenta. The total physical angular momentum cannot be obtained
by simply summing the individual spins because this neglects the
orbital angular momentum of the various sources. However, the
total angular momentum can still be computed without having to
solve the Hamiltonian constraint [114].
The Bowen-York solution for the extrinsic curvature is the
starting point for all the general multi-hole initial-data sets
we have discussed in Section
3.2
. However, this solution is not
inversion symmetric
. That is, it does not satisfy the isometry condition that any
field must satisfy to exist on a two-sheeted manifold like that
of Misner's solution. Fortunately, there is a method of images,
similar to that used in electrostatics but applicable to tensors,
that can be used to make any tensor inversion symmetric [78,
26,
66,
65,
113].
For the conformal extrinsic curvature of a single black hole,
there are two inversion-symmetric solutions [26],
Here, a is the radius of the coordinate 2-sphere that is the throat of the black hole. Of course, this coordinate 2-sphere is the fixed-point set of the isometry and is the surface on which we can impose boundary conditions. Notice that this radius enters the solutions only when we make it inversion symmetric.
When the extrinsic curvature represents more than one black hole, the process for making the solution inversion symmetric is rather complex and results in an infinite-series solution. However, in most cases of interest, the solution converges rapidly and it is straightforward to evaluate the solution numerically [38].
Given an inversion-symmetric conformal extrinsic curvature, it
is possible to find an inversion-symmetric solution of the
Hamiltonian constraint [26]. Given our assumptions (67
), the Hamiltonian constraint becomes
The isometry condition imposes a condition on the conformal
factor at the throat of each hole. This condition takes the
form [26]
where
is the outward-pointing unit-normal vector to the
throat and
is the coordinate radius of that throat. This condition can be
used as a boundary condition when solving (73
) in the region exterior to the throats.
In addition to boundary conditions on the throats, a boundary
condition on the outer boundary of the domain is needed before
the quasilinear elliptic equation in (73) can be solved as a well-posed boundary-value problem. This
final boundary condition comes from the fact that we want an
asymptotically flat solution. This implies that the solution
behaves as
where
E
is the total ADM energy content of the initial-data
hypersurface. Equation (75) can be used to construct appropriate boundary conditions either
at infinity or at a large, but finite, distance from the black
holes [116].
Based on the time-symmetric solution, it is reasonable to assume that the conformal factor will take the form
If
u
is sufficiently smooth, (76) implies that the manifold will have the topology of
N
+1 asymptotically flat regions just as in the Brill-Lindquist
solution. In this case, asymptotic flatness requires that
.
Substituting (76) into the Hamiltonian constraint (73
) yields
where
Near each singular point, or ``puncture'', we find that
. From (70
), we see that
behaves no worse than
, so
vanishes at the punctures at least as fast as
.
With this behavior, Brandt and Brügmann [27] have shown the existence and uniqueness of
solutions of the modified Hamiltonian constraint (77
). The resulting scheme for constructing multiple black hole
initial data is very simple. The mass and position of each black
hole are parameterized by
and
, respectively. Their linear momenta and spin are parameterized
by
and
in the conformal extrinsic curvature (70
) used for each hole. Finally, the solution for
u
is found on a simple Euclidean manifold, with no need for any
inner boundaries to avoid singularities. This is a great
simplification over the conformal-imaging approach, where proper
handling of the inner boundary is the most difficult aspect of
solving the Hamiltonian constraint numerically [41].
These choices for the freely specifiable data are not always commensurate with the desired physical solution. For example, if we choose to use either method to construct a single spinning black hole, we will not obtain the Kerr solution. The Kerr-Newman solution can be written in terms of a quasi-isotropic radial coordinate on a K =0 time slice [28]. Let r denote the usual Boyer-Lindquist radial coordinate and make the standard definitions
A quasi-isotropic radial coordinate
can be defined via
The interval then becomes
with
We see immediately that the 3-geometry associated with a
hypersurface of (81
) is not conformally flat. In fact, Garat and Price [54] have shown that in general there is no spatial slicing of the
Kerr spacetime that is axisymmetric, conformally flat, and
smoothly goes to the Schwarzschild solution as the spin parameter
.
Since the Kerr solution is stationary, the inescapable conclusion is that conformally flat initial data for a single rotating black hole must also contain some nonvanishing dynamical component. When we evolve such data, the system will emit gravitational radiation and eventually settle down to the Kerr geometry [26, 29]. But, it cannot be the Kerr geometry initially, and it is unlikely that the spurious gravitational radiation content of the initial data has any desirable physical properties. Conformally flat initial data for spinning holes contain some amount of unphysical ``junk'' radiation. A similar conclusion is reached for conformally flat data for a single black hole with linear momentum [112].
The choice of a conformally flat 3-geometry was originally
made for convenience. Combined with the choice of maximal
slicing, these simplifying assumptions allowed for an analytic
solution of the momentum constraints which vastly simplified the
process of constructing black-hole initial data. Yet there has
been much concern about the possible adverse physical effects
that these choices (especially the choice of conformal flatness)
will have in trying to study black-hole spacetimes [40,
70,
87,
60,
86
]. While these conformally flat data sets may still be useful for
tests of black-hole evolution codes, it is becoming widely
accepted that the unphysical initial radiation will significantly
contaminate any gravitational waveforms extracted from evolutions
of these data. In short, these data are not astrophysically
realistic.
The various initial-data decompositions outlined in Section 2.2 and Section 2.3 are all capable of producing completely general black-hole initial data sets. The only limitation of these schemes is our understanding of what choices to make for the freely specifiable data and the boundary conditions to apply when solving the sets of elliptic equations. All these choices will have a critical impact on the astrophysical significance of the data produced. It is also important to remember that similar choices for the freely specifiable data will result in physically different solutions when applied to the different schemes.
The first studies of black-hole initial data that are not
conformally flat were carried out by Abrahams
et al.
[1]. They looked at the superposition of a gravitational wave and a
black hole. Using a form of the conformal metric that allows for
so-called Brill waves, they constructed time-symmetric initial
data that were not conformally flat and yet satisfied the
isometry condition (65) used in the conformal-imaging method of Section
3.2.1
. These data were further generalized by Brandt and Seidel [30] to include rotating black holes with a superimposed
gravitational wave. In this case, the data are no longer
time-symmetric yet they satisfy a generalized form of the
isometry condition so that the solution is still represented on
two isometric, asymptotically flat hypersurfaces.
Matzner
et al.
[77] have begun to move beyond conformally flat initial data for
binary black holes Their proposal is to use boosted versions of
the Kerr metric written in the Kerr-Schild form to represent each
black hole. Thus, an isolated black hole will have no spurious
radiation content in the initial data. To construct solutions
with multiple black holes, they propose, essentially, to use a
linear combination of the single-hole solutions. The resulting
metric can be used as the
conformal
3-metric, the trace of the resulting extrinsic curvature can be
used for
K, and the tracefree part of the resulting extrinsic curvature can
be used for
. Their scheme uses York's conformal transverse-traceless
decomposition outlined in Section
2.2.1, with the boundary conditions of
and
on the horizons of the black holes and conditions appropriate
for asymptotic flatness at large distances from the holes. A
related method has been proposed by Bishop
et al.
[15], but their approach is much different and outside the current
scope of this review.
The approach outlined by Matzner should certainly yield
``cleaner'' data than the conformally flat data currently
available. For the task of specifying data for astrophysical
black-hole binaries in nearly circular orbits, it is still true
that these new data will not contain the correct initial
gravitational wave content. Because the black holes are in orbit,
they must be producing a continuous wave-train of gravitational
radiation. This radiation will not be included in the method
proposed by Matzner
et al
. Also, it is clear that the boundary conditions being used do
not correctly account for the tidal distortion of each black hole
by its companion. When the black holes are sufficiently far
apart, the radiation from the orbital motion can be computed
using post-Newtonian techniques. One possibility for producing
astrophysically realistic, binary black-hole initial data is to
use information from these post-Newtonian calculations to obtain
better guesses for
,
, and
K
.
![]() |
Initial Data for Numerical Relativity
Gregory B. Cook http://www.livingreviews.org/lrr-2000-5 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |