3.2 General Multi-Hole Solutions3 Black Hole Initial Data3 Black Hole Initial Data

3.1 Classic Solutions 

3.1.1 Schwarzschild 

Of course, the simplest black-hole solution is the Schwarzschild solution. It represents a static spacetime containing a single black hole that connects two causally disconnected, asymptotically flat universes. There are actually many different coordinate representations of the Schwarzschild solutions. The simplest representations are time-symmetric (tex2html_wrap_inline3329), and so exist on a ``maximally embedded'' spacelike hypersurface (K =0). These choices fix the foliation tex2html_wrap_inline3001 . Spherical symmetry fixes two of the three spatial gauge choices. If we choose an ``areal-radial coordinate'' Popup Footnote, then the interval is written as

  equation826

If we choose an isotropic radial coordinate, then the interval is written as

  equation837

In both (59Popup Equation) and (60Popup Equation), M represents the mass of the black hole as measured at spacelike infinity. Both of these solutions exist on the same foliation of tex2html_wrap_inline3103 slices. But, notice that the 3-geometry of the slice associated with (60Popup Equation) is conformally flat, while the 3-geometry associated with (59Popup Equation) is not.

The solution given in (60Popup Equation) 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 tex2html_wrap_inline3329, 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 (18Popup Equation) becomes

  equation864

where tex2html_wrap_inline3345 is the flat-space Laplace operator. For the solution tex2html_wrap_inline3107 to yield an asymptotically flat physical 3-metric, we have the boundary condition that tex2html_wrap_inline3349 . The simplest solution of this equation is

  equation868

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 (12Popup Equation) and (13Popup Equation). 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

  equation875

If we choose boundary conditions so that the lapse is frozen on the event horizon (tex2html_wrap_inline3357) and goes to one at infinity, we find that the solution is

  equation879

If we now choose tex2html_wrap_inline3359, we find that the left-hand sides of the evolution equations (12Popup Equation) and (13Popup Equation) vanish identically, and we have found the static solution of Einstein's equations given in (60Popup Equation). We can, of course, recover the usual Schwarzschild coordinate solution (59Popup Equation) by using the purely spatial coordinate transformation tex2html_wrap_inline3361 .

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 tex2html_wrap_inline3363 . However, even here, the solution is well behaved. The 3-geometry is invariant under the coordinate transformation

  equation894

The event horizon at tex2html_wrap_inline3365 is a fixed-point set of the isometry condition (65Popup Equation) which identifies points in two causally disconnected, asymptotically flat universes. We see that tex2html_wrap_inline3363 is simply an image of infinity in the other universe [31Jump To The Next Citation Point In The Article].

Given our choice for the lapse (64Popup Equation), 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 tex2html_wrap_inline3103 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.

3.1.2 Time-Symmetric Multi-Hole Solutions 

As we saw in Section  3.1.1, the simplest approach for generating initial data is to assume time symmetry and let the conformal 3-geometry be flat. We could have generated the exterior solution for Schwarzschild with the areal-radial coordinate by a clever choice for the conformal 3-geometry. The Hamiltonian constraint is still linear, even if tex2html_wrap_inline2989 is not flat. But there is no obvious motivation for the correct tex2html_wrap_inline2989 that would yield the desired solution.

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 (61Popup Equation). 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 (62Popup Equation). More precisely, we choose the solution

  equation915

Here, tex2html_wrap_inline3377 is a coordinate distance from the point tex2html_wrap_inline3379 in the Euclidean conformal space, and the tex2html_wrap_inline3381 are constants related to the masses of the black holes. Assuming the points tex2html_wrap_inline3379 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, tex2html_wrap_inline3387, 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, tex2html_wrap_inline3379, 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 (65Popup Equation) 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 [78Jump To The Next Citation Point In The Article] found that it is possible to construct a solution of the vacuum, time-symmetric Hamiltonian constraint (61Popup Equation) 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, 38Jump To The Next Citation Point In The Article] Popup Footnote . 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 (65Popup Equation) 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].



3.2 General Multi-Hole Solutions3 Black Hole Initial Data3 Black Hole Initial Data

image 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