Classic solutions

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 ( $Kij = 0$ ), and so exist on a “maximally embedded” spacelike hypersurface ( $K = 0$ ). These choices fix the foliation ${Σ }$ . Spherical symmetry fixes two of the three spatial gauge choices. If we choose an “areal-radial coordinate”⁷ , then the interval is written as

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 $t = const.$ 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 $Kij = 0$ , 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 $&tidle; 2 ∇$ is the flat-space Laplace operator. For the solution $ψ$ to yield an asymptotically flat physical 3-metric, we have the boundary condition that $ψ(&tidle;r → ∞ ) = 1$ . 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 ( $α(&tidle;r = M ∕2) = 0$ ) and goes to one at infinity, we find that the solution is

If we now choose $βi = 0$ , 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 $M r = &tidle;r(1 + 2r&tidle;)2$ .

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 $r&tidle;= 0$ . However, even here, the solution is well behaved. The 3-geometry is invariant under the coordinate transformation

The event horizon at $&tidle;r = M- 2$ is a fixed-point set of the isometry condition (65

) which identifies points in two causally disconnected, asymptotically flat universes. We see that $&tidle;r = 0$ 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, $r = 2M$ , there is a coordinate singularity, and inside this radius the $t = const.$ 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 $&tidle;γij$ is not flat. But there is no obvious motivation for the correct $&tidle;γij$ that would yield the desired solution.

One approach for generating a time-symmetric multi-hole solution is straightforward. Brill–Lindquist initial data [31 , 70 ] 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, $|x − Cσ |$ is a coordinate distance from the point $Cσ$ in the Euclidean conformal space, and the $μ σ$ are constants related to the masses of the black holes. Assuming the points $Cσ$ 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, $x = Cσ$ , 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, $C σ$ , 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 [79 ] 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 ]⁸ . 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 ].