3.4 Quasicircular Binary Data3 Black Hole Initial Data3.2 General Multi-Hole Solutions

3.3 Horizon-Penetrating Solutions 

We noted in Section  3.1.1 that the time-independent maximal slicing of Schwarzschild with isotropic coordinates covers only the exterior of the black hole. This is because the time independence of this gauge requires that the lapse vanish on the horizon. It is possible to evolve into the black hole's interior when starting from initial data constructed in this gauge, but it requires a choice for the lapse that yields a time-dependent solution [94]. The time dependence of such a solution is purely gauge, of course, since the spacetime is static.

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:

  equation1202

where tex2html_wrap_inline3067 and tex2html_wrap_inline3523 are defined by Eq. (79Popup Equation), and tex2html_wrap_inline3525 is the ingoing-null coordinate. This metric is regular at tex2html_wrap_inline3527, where tex2html_wrap_inline3529 and tex2html_wrap_inline3531 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

  equation1229

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 tex2html_wrap_inline2971 format.

3.3.1 Kerr-Schild Coordinates 

A spherical coordinate version of the standard Kerr-Schild coordinate system is obtained from (83Popup Equation) by using the coordinate choice (cf. Refs. [73, 42Jump To The Next Citation Point In The Article])

  equation1243

The nonzero components of the lapse, shift, and 3-metric are then given by:

  equation1248

  equation1254

  equation1259

  equation1265

  equation1271

  equation1275

Cartesian coordinate components can be obtained from these via the standard Kerr-Schild coordinate transformations [79]

  equation1282

This yields the implicit definition of r from

  equation1287

with r > 0 and r =0 on the disk described by z =0 and tex2html_wrap_inline3549 .

3.3.2 Harmonic Coordinates 

Harmonic time slicing is integral to some hyperbolic formulations of general relativity, and a time-independent harmonic slicing of the Kerr-Newman geometry does exist [17, 42Jump To The Next Citation Point In The Article]. The harmonic time slicing condition is tex2html_wrap_inline3551, which can be written

  equation1294

This equation is satisfied by using the coordinate choice

  equation1300

The nonzero components of the lapse, shift, and 3-metric are then given by:

  equation1306

  equation1318

  equation1323

  equation1330

  equation1340

  equation1348

  equation1352

Cartesian coordinate components can be obtained from these via the standard Kerr-Schild coordinate transformations (92Popup Equation) and (93Popup Equation). However, for the harmonic slicing, the tex2html_wrap_inline3103 hypersurface is spacelike only outside the Cauchy horizon at tex2html_wrap_inline3555 .

Fully harmonic coordinates (tex2html_wrap_inline3557) can be defined when Cartesian spatial coordinates are used by employing a variation of the standard Kerr-Schild coordinate transformations [42]

  equation1363

This yields the implicit definition of r from

  equation1368

Fully harmonic coordinates are useful because applying a boost to a harmonically sliced black hole yields a solution that satisfies (94Popup Equation) only if the black hole is written in fully harmonic coordinates. In this case, the boosted solution also satisfies the fully harmonic coordinate conditions.

3.3.3 Generalized Painlevé-Gullstrand Coordinates 

The Painlevé-Gullstrand gauge choice for the Schwarzschild geometry has been rediscovered many times because of its simple form (cf. Refs. [85, 59, 88, 64, 67]). It is another time-independent solution, but the 3-geometry is completely flat (not simply conformally flat). The lapse is one in this gauge, and all of the information regarding the curvature of spacetime is contained in the shift. The Painlevé-Gullstrand gauge also has an intuitive physical interpretation [74]. An observer starting at rest at infinity and freely falling will trace out a world line that is everywhere orthogonal to the tex2html_wrap_inline3103 hypersurfaces in Painlevé-Gullstrand coordinates.

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

  equation1383

The nonzero components of the lapse, shift, and 3-metric are then given by:

  equation1399

  equation1403

  equation1410

  equation1416

  equation1424

  equation1428

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 (92Popup Equation) and (93Popup Equation). Like the Kerr-Schild time slicing, a tex2html_wrap_inline3103 slice of the generalized Painlevé-Gullstrand gauge remains spacelike for all tex2html_wrap_inline3569 .



3.4 Quasicircular Binary Data3 Black Hole Initial Data3.2 General Multi-Hole Solutions

image Initial Data for Numerical Relativity
Gregory B. Cook
http://www.livingreviews.org/lrr-2000-5
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