It is possible to cover all, or part, of the interior of a single black hole with a time-independent slicing. However, doing so seems to require that we give up the maximal-slicing condition. To cover the interior of the black hole, we need a slicing that passes smoothly through the event horizon. A convenient way to generate such solutions is to begin with the metric in standard ingoing-null coordinates. If we want to consider a rotating and charged black hole, then we use the Kerr-Newman geometry in Kerr coordinates:
where
and
are defined by Eq. (79
), and
is the ingoing-null coordinate. This metric is regular at
, where
and
are the locations of the event horizon and the Cauchy horizon,
respectively.
This metric can be put into a form suitable for producing time-independent Cauchy initial data by making coordinate transformations of the general form
where
f
and
g
are suitably chosen functions of the radial coordinate
r
. There are a few particularly significant solutions for the
general Kerr-Newman geometry, and I will outline these below,
listing the nonzero components of the metric in the standard
format.
The nonzero components of the lapse, shift, and 3-metric are then given by:
Cartesian coordinate components can be obtained from these via the standard Kerr-Schild coordinate transformations [79]
This yields the implicit definition of r from
with
r
> 0 and
r
=0 on the disk described by
z
=0 and
.
This equation is satisfied by using the coordinate choice
The nonzero components of the lapse, shift, and 3-metric are then given by:
Cartesian coordinate components can be obtained from these via
the standard Kerr-Schild coordinate transformations (92) and (93
). However, for the harmonic slicing, the
hypersurface is spacelike only outside the Cauchy horizon at
.
Fully harmonic coordinates () can be defined when Cartesian spatial coordinates are used by
employing a variation of the standard Kerr-Schild coordinate
transformations [42]
This yields the implicit definition of r from
Fully harmonic coordinates are useful because applying a boost
to a harmonically sliced black hole yields a solution that
satisfies (94) only if the black hole is written in fully harmonic
coordinates. In this case, the boosted solution also satisfies
the fully harmonic coordinate conditions.
A generalization of the Painlevé-Gullstrand gauge derived by Doran [48] includes the Kerr spacetime, and the extension of this solution to the full Kerr-Newman spacetime is trivial. In the limit that a and Q vanish, this solution reduces to the Painlevé-Gullstrand gauge. The coordinate transformation is written most easily as
The nonzero components of the lapse, shift, and 3-metric are then given by:
Notice that the lapse remains one, but the 3-geometry is no
longer flat when the black hole is spinning. Cartesian coordinate
components can be obtained from these via the standard
Kerr-Schild coordinate transformations (92) and (93
). Like the Kerr-Schild time slicing, a
slice of the generalized Painlevé-Gullstrand gauge remains
spacelike for all
.
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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 |