If we choose an isotropic radial coordinate, then the interval is written as
In both (59) and (60
),
M
represents the mass of the black hole as measured at spacelike
infinity. Both of these solutions exist on the same foliation of
slices. But, notice that the 3-geometry of the slice associated
with (60
) is conformally flat, while the 3-geometry associated with (59
) is not.
The solution given in (60) is easily generated by any of the methods in Section
2.2
or Section
2.3
. By choosing a time-symmetric initial-data hypersurface, we
immediately get
, which eliminates the need to solve the momentum constraints.
If we choose the conformal 3-geometry to be given by a flat
metric (in spherical coordinates in this case), then the vacuum
Hamiltonian constraint (18
) becomes
where
is the flat-space Laplace operator. For the solution
to yield an asymptotically flat physical 3-metric, we have the
boundary condition that
. The simplest solution of this equation is
where we have chosen the remaining integration constant to give a mass at infinity of M .
We now have full Cauchy initial data representing a single
black hole. If we want to generate a full solution of Einstein's
equations, we must choose a lapse and a shift vector and
integrate the evolution equations (12) and (13
). In this case, a reasonable approach for specifying the lapse
is to demand that the time derivative of
K
vanish. For the case of
K
=0, this yields the so-called maximal slicing equation which, for
the current situation, takes the form
If we choose boundary conditions so that the lapse is frozen
on the event horizon () and goes to one at infinity, we find that the solution is
If we now choose
, we find that the left-hand sides of the evolution
equations (12
) and (13
) vanish identically, and we have found the static solution of
Einstein's equations given in (60
). We can, of course, recover the usual Schwarzschild coordinate
solution (59
) by using the purely spatial coordinate transformation
.
It is interesting to examine the differences in these two
representations of the Schwarzschild solution. The isotropic
radial coordinate representation is well behaved everywhere
except, it seems, at
. However, even here, the solution is well behaved. The
3-geometry is invariant under the coordinate transformation
The event horizon at
is a fixed-point set of the isometry condition (65
) which identifies points in two causally disconnected,
asymptotically flat universes. We see that
is simply an image of infinity in the other universe [31
].
Given our choice for the lapse (64), which is frozen on the event horizon, we find that the
solution can cover only the exterior of the black hole. To cover
any of the interior with the lapse pinned to zero at the horizon
would require we use a slice that is not spacelike everywhere.
This is exactly what happens when the usual Schwarzschild
areal-radial coordinate is used. At the event horizon,
r
=2
M, there is a coordinate singularity, and inside this radius the
hypersurface is no longer spacelike. It is impossible to perform
a Cauchy evolution interior to the event horizon using the
areal-radial coordinate and the given time slicing.
We find that a Cauchy evolution, using the usual Schwarzschild time slicing that is frozen at the horizon, is capable of evolving only the region exterior to the black hole's event horizon. Portions of the interior of the black hole can be covered by an evolution that begins with data on a standard Schwarzschild time slice, but the result is not a time-independent solution. As we will see later, there are other slicings of the Schwarzschild spacetime that cover the interior of the black hole and yield time-independent solutions.
One approach for generating a time-symmetric multi-hole
solution is straightforward. Brill-Lindquist initial data [31,
69] again assume a flat conformal 3-geometry, and the only
non-trivial constraint equation is the Hamiltonian constraint,
which again takes the form given in (61). But this time, we use the linearity of the Hamiltonian
constraint and choose the solution to be a superposition of
solutions with the form of (62
). More precisely, we choose the solution
Here,
is a coordinate distance from the point
in the Euclidean conformal space, and the
are constants related to the masses of the black holes. Assuming
the points
are sufficiently far apart, this solution of the initial-value
equations represents
N
black holes momentarily at rest in ``our'' asymptotically flat
universe. As was the case for a single black hole, each singular
point in the solution,
, represents infinity in a different, causally disconnected
universe. In fact, each black hole connects ``our'' universe to a
different universe, so that there are
N
+1 asymptotically flat hypersurfaces connected together at the
throats of
N
black holes. While we started with a 3-dimensional Euclidean
manifold, the requirement that we delete the singular points,
, results in a manifold that is not simply connected. This
solution is often referred to as having a topology with
N
+1 ``sheets''.
Brill-Lindquist initial data are very similar to the
Schwarzschild initial data in isotropic coordinates except for
one major difference: The solution does not represent two
identical universes that have been joined together. The
coordinate transformation (65) can still be used to show that each pole in the solution
corresponds to infinity on an asymptotically flat hypersurface.
But, the solution has
N
+1 different asymptotically flat universes connected together,
not two, and ``our'' universe containing
N
black holes cannot be isometric to any of the other universes
which contain just one. Interestingly, Misner [78
] found that it is possible to construct a solution of the
vacuum, time-symmetric Hamiltonian constraint (61
) that has two isometric asymptotically flat hypersurfaces
connected by
N
black holes. The case of two black holes that satisfies this
isometry condition (often called inversion symmetry) is usually
referred to as ``Misner initial data''. It has an analytic
representation in terms of an infinite series expansion. The
construction is tedious, and there are several representations of
the solution [93,
38
]
. Like the Brill-Lindquist data, the non-simply connected
topology of the full manifold is represented on a Euclidean
3-manifold by the presence of singular points that must be
removed. Brill-Lindquist data have
N
singular points, each representing an
image
of infinity as seen through the throat of the black hole
connecting our universe to that hole's other universe. Misner's
data contain an infinite number of singular points for each black
hole, each representing an image of one of two asymptotic
infinities. The result is seen to represent two identical
asymptotically flat universes joined by
N
black holes. The two universes join together at the throats of
the
N
black holes, with each throat being a coordinate 2-sphere in the
conformal space. Data at any point in one universe are related to
data at the corresponding point in the alternate universe via the
same isometry condition (65
) found in the Schwarzschild case. And, as in the Schwarzschild
case, the 2-sphere throat of each black hole forms a fixed-point
set of the isometry condition.
Both the Brill-Lindquist data and Misner's data represent N black holes at a moment of time-symmetry (i. e., all the black holes are momentarily at rest). Both are conformally flat and the difference in the topology of the two solutions is hidden from an observer outside the black holes. Yet solutions where the holes are chosen to have the same size and separation yield similar, but physically distinct, solutions [36, 3].
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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 |