3.2 General multi-hole solutions

Time-symmetric black-hole solutions of the constraint equations such as those described in Section 3.1 are useful as test cases because they have analytic representations. However, they have very little physical relevance. General time-asymmetric solutions are needed to represent black holes that are moving and spinning9. A few approaches for generating general multi-hole solutions have been explored and, below, we will look at the two approaches which are direct generalizations of the Misner and Brill–Lindquist data. Generalizing these two approaches was the most natural first step toward constructing general multi-hole solutions.

The first approach to be developed generalized the Misner approach [79Jump To The Next Citation Point]. It was attractive because an isometry condition relating the two asymptotically flat universes provides two useful things. First, because the two universes are identical, finding a solution in one universe means that you have the full solution. Second, because the throats are fixed-point sets of the isometry, we can construct boundary conditions on any quantity there. This allows us to excise the region interior to the spherical throats from one of the Euclidean background spaces and solve for the initial data in the remaining volume.

The generalization of the Misner approach seemed preferable to trying to solve the constraints on N + 1 Euclidean manifolds stitched together smoothly at the throats of N black holes. However, Brandt and Brügmann [27Jump To The Next Citation Point] realized it was possible to factor out analytically the behavior of the singular points in the Euclidean manifold of the N + 1 sheeted approach. Referred to as the “puncture” method, this approach allows us to rewrite the constraint equations for functions on an N + 1 sheeted manifold as constraint equations for new functions on a simple Euclidean manifold.

Another approach tried, which we will not discuss in detail, avoided the issue of the topology of the initial-data slice entirely. Developed by Thornburg [100], this approach was based on the idea that only the domain exterior to the apparent horizon of a black hole is relevant. The equation describing the location of an apparent horizon can be rewritten in a form that can be used as a boundary condition for the conformal factor in the Hamiltonian constraint equation. Thus, given a compatible solution to the momentum constraints, this boundary condition can be used to construct a solution of the Hamiltonian constraint in the domain exterior to the apparent horizons of any black holes, with no reference at all to the topology of the full manifold.

3.2.1 Bowen–York data

The generalization of the Misner approach was developed by York and his collaborators [25Jump To The Next Citation Point23Jump To The Next Citation Point24Jump To The Next Citation Point67Jump To The Next Citation Point113Jump To The Next Citation Point66Jump To The Next Citation Point2638Jump To The Next Citation Point]. The approach is often called the “conformal-imaging” method, and the data are usually referred to as “Bowen–York” data. This approach begins with a set of simplifying assumptions that is common to all three of the approaches described above. These assumptions are that

K = 0, maximal slicing, γ&tidle;ij = fij, conformal flatness, (67 ) ψ | = 1, asymptotic flatness. ∞
Here, fij represents a flat metric in any suitable coordinate system. The assumption of conformal flatness means that the differential operators in the constraints are the familiar flat-space operators. More importantly, if we use the conformal transverse-traceless decomposition (32View Equation), we find that in vacuum the momentum constraints completely decouple from the Hamiltonian constraint.

The importance of this last property stems from the fact that York and Bowen were able to find analytic solutions of this version of the momentum constraints, solutions that represent a black hole with both linear momentum and spin [25Jump To The Next Citation Point2324]. If we choose M&tidle;ij = 0, then the momentum constraints (31View Equation) become

( ) &tidle;2 i 1&tidle; i &tidle; j ∇ V + 3∇ ∇jV = 0. (68 )
A solution of this equation is
[ ] Vi = − -1- 7P i + ninjP j + 1-𝜖ijknjSk. (69 ) 4r r2
Here, P i and Si are vector parameters, r is a coordinate radius, and ni is the outward-pointing unit normal of a sphere in the flat conformal space (ni ≡ xi∕r). 𝜖ijk is the 3-dimensional Levi-Civita tensor.

This solution of the momentum constraints yields the tracefree part of the extrinsic curvature,

[ ] A&tidle;ij = -3-- Pinj + Pjni − (fij − ninj)P knk 2r2 (70 ) + 3- [𝜖 Sℓnkn + 𝜖 S ℓnkn ]. r3 kiℓ j kjℓ i
Remarkably, using this solution (70View Equation) and the assumptions in (67View Equation), we can determine the physical values for the linear and angular momentum of any initial data we can construct. The momentum contained in an asymptotically flat initial-data hypersurface can be calculated from the integral [112Jump To The Next Citation Point]
∮ ii 1-- ( j j ) i 2 Π ξ(k) = 8π K i − δiK ξ(k)d Sj, (71 ) ∞
where ξi(k) is a Killing vector of the 3-metric γij10. Since we are not likely to have true Killing vectors, we make use of the asymptotic translational and rotational Killing vectors of the flat conformal space. We find from (71View Equation), (70View Equation), and (67View Equation) that the physical linear momentum of the initial-data hypersurface is P i and the physical angular momentum of the slice is Si. Furthermore, because the momentum constraints are linear, we can add any number of solutions of the form of Eq. (70View Equation) to represent a collection of linear and angular momentum sources. The total physical linear momentum of the initial-data slice will simply be the vector sum of the individual linear momenta. The total physical angular momentum cannot be obtained by simply summing the individual spins because this neglects the orbital angular momentum of the various sources. However, the total angular momentum can still be computed without having to solve the Hamiltonian constraint [114].

The Bowen-York solution for the extrinsic curvature is the starting point for all the general multi-hole initial-data sets we have discussed in Section 3.2. However, this solution is not inversion symmetric. That is, it does not satisfy the isometry condition that any field must satisfy to exist on a two-sheeted manifold like that of Misner’s solution. Fortunately, there is a method of images, similar to that used in electrostatics but applicable to tensors, that can be used to make any tensor inversion symmetric [7925Jump To The Next Citation Point6766113].

For the conformal extrinsic curvature of a single black hole, there are two inversion-symmetric solutions [25Jump To The Next Citation Point],

3 [ ] A&tidle;±ij = --2-Pinj + Pjni − (fij − ninj)Pknk 2r 2 ∓ 3a--[P n + P n + (f − 5n n )Pkn ] (72 ) 2r4 i j j i ij i j k 3 [ ℓ k ℓ k ] + r3 𝜖kiℓS n nj + 𝜖kjℓS n ni .
Here, a is the radius of the coordinate 2-sphere that is the throat of the black hole. Of course, this coordinate 2-sphere is the fixed-point set of the isometry and is the surface on which we can impose boundary conditions. Notice that this radius enters the solutions only when we make it inversion symmetric.

When the extrinsic curvature represents more than one black hole, the process for making the solution inversion symmetric is rather complex and results in an infinite-series solution. However, in most cases of interest, the solution converges rapidly and it is straightforward to evaluate the solution numerically [38].

Given an inversion-symmetric conformal extrinsic curvature, it is possible to find an inversion-symmetric solution of the Hamiltonian constraint [25Jump To The Next Citation Point]. Given our assumptions (67View Equation), the Hamiltonian constraint becomes

2 1 − 7 ij &tidle;∇ ψ + 8ψ &tidle;Aij &tidle;A = 0. (73 )
The isometry condition imposes a condition on the conformal factor at the throat of each hole. This condition takes the form [25Jump To The Next Citation Point]
| || ni &tidle;∇iψ || = − -ψ-| , (74 ) σ aσ 2rσ|aσ
where i nσ is the outward-pointing unit-normal vector to the th σ throat and aσ is the coordinate radius of that throat. This condition can be used as a boundary condition when solving (73View Equation) in the region exterior to the throats.

In addition to boundary conditions on the throats, a boundary condition on the outer boundary of the domain is needed before the quasilinear elliptic equation in (73View Equation) can be solved as a well-posed boundary-value problem. This final boundary condition comes from the fact that we want an asymptotically flat solution. This implies that the solution behaves as

E ψ = 1 + ---+ 𝒪 (r−2), (75 ) 2r
where E is the total ADM energy content of the initial-data hypersurface. Equation (75View Equation) can be used to construct appropriate boundary conditions either at infinity or at a large, but finite, distance from the black holes [116].

3.2.2 Puncture data

The generalization of the Brill–Lindquist data developed by Brandt and Brügmann [27Jump To The Next Citation Point] begins with the same set of assumptions (67View Equation) as the conformal-imaging approach outlined in Section 3.2.1. We immediately have Eq. (70View Equation) from the solution of the momentum constraints, and we must solve the Hamiltonian constraint, which again takes the form of Eq. (73View Equation). At this point, however, the method of solution differs from the conformal-imaging approach.

Based on the time-symmetric solution, it is reasonable to assume that the conformal factor will take the form

∑N ψ = -1 + u, 1-≡ ----μσ----. (76 ) χ χ σ=12 |x − C σ|
If u is sufficiently smooth, (76View Equation) implies that the manifold will have the topology of N + 1 asymptotically flat regions just as in the Brill–Lindquist solution. In this case, asymptotic flatness requires that −1 u = 1 + 𝒪 (r ).

Substituting (76View Equation) into the Hamiltonian constraint (73View Equation) yields

2 −7 ∇&tidle; u + η(1 + χu ) = 0, (77 )
where
η = 1χ7 &tidle;Aij &tidle;Aij. (78 ) 8
Near each singular point, or “puncture”, we find that χ ≈ |x − Cσ |. From (70View Equation), we see that A&tidle;ijA&tidle;ij behaves no worse than |x − C σ|− 6, so η vanishes at the punctures at least as fast as |x − C | σ.

With this behavior, Brandt and Brügmann [27] have shown the existence and uniqueness of 2 C solutions of the modified Hamiltonian constraint (77View Equation). The resulting scheme for constructing multiple black hole initial data is very simple. The mass and position of each black hole are parameterized by μ σ and C σ, respectively. Their linear momenta and spin are parameterized by P σ and S σ in the conformal extrinsic curvature (70View Equation) used for each hole. Finally, the solution for u is found on a simple Euclidean manifold, with no need for any inner boundaries to avoid singularities. This is a great simplification over the conformal-imaging approach, where proper handling of the inner boundary is the most difficult aspect of solving the Hamiltonian constraint numerically [41].

3.2.3 Problems with these data

Both the conformal-imaging and puncture methods for generating multiple black hole initial data allow for completely general configurations of the relative sizes of the black holes, as well as their linear and angular momenta. This does not mean that these schemes allow for the generation of all desired black-hole initial data. The two schemes rely on specific assumptions about the freely specifiable gravitational data. In particular, they assume K = 0, ij M&tidle; = 0, and, most importantly, that the 3-geometry is conformally flat.

These choices for the freely specifiable data are not always commensurate with the desired physical solution. For example, if we choose to use either method to construct a single spinning black hole, we will not obtain the Kerr solution. The Kerr–Newman solution can be written in terms of a quasi-isotropic radial coordinate on a K = 0 time slice [28]. Let r denote the usual Boyer–Lindquist radial coordinate and make the standard definitions

2 2 2 2 2 2 2 ρ ≡ r + a cos 𝜃 and Δ ≡ r − 2M r + a + Q . (79 )
A quasi-isotropic radial coordinate &tidle;r can be defined via
( ∘ -------) ( ∘ --------) M + a2 + Q2 M − a2 + Q2 r = &tidle;r 1 + --------------- 1 + --------------- . (80 ) 2 &tidle;r 2&tidle;r
The interval then becomes
[ ] ds2 = − α2dt2 + ψ4 e2μ∕3(dr&tidle;2 + &tidle;r2d𝜃2) + &tidle;r2sin2𝜃e −4μ∕3(dϕ + β ϕdt)2 , (81 )
with
2 ---------ρ2Δ---------- α = (r2 + a2 )2 − Δa2 sin2𝜃 , 2 2 β ϕ = − a-----r-+--a-−--Δ------, (r2 + a2)2 − Δa2 sin2𝜃 (82 ) ρ2∕3((r2 + a2)2 − Δa2 sin2 𝜃)1∕3 ψ4 = ---------------2--------------, &tidle;r 4 e2μ = ----------ρ-----------. (r2 + a2 )2 − Δa2 sin2𝜃

We see immediately that the 3-geometry associated with a t = const. hypersurface of (81View Equation) is not conformally flat. In fact, Garat and Price [54] have shown that in general there is no spatial slicing of the Kerr spacetime that is axisymmetric, conformally flat, and smoothly goes to the Schwarzschild solution as the spin parameter a → 0.

Since the Kerr solution is stationary, the inescapable conclusion is that conformally flat initial data for a single rotating black hole must also contain some nonvanishing dynamical component. When we evolve such data, the system will emit gravitational radiation and eventually settle down to the Kerr geometry [2529]. But, it cannot be the Kerr geometry initially, and it is unlikely that the spurious gravitational radiation content of the initial data has any desirable physical properties. Conformally flat initial data for spinning holes contain some amount of unphysical “junk” radiation. A similar conclusion is reached for conformally flat data for a single black hole with linear momentum [112].

The choice of a conformally flat 3-geometry was originally made for convenience. Combined with the choice of maximal slicing, these simplifying assumptions allowed for an analytic solution of the momentum constraints which vastly simplified the process of constructing black-hole initial data. Yet there has been much concern about the possible adverse physical effects that these choices (especially the choice of conformal flatness) will have in trying to study black-hole spacetimes [407188Jump To The Next Citation Point6187Jump To The Next Citation Point]. While these conformally flat data sets may still be useful for tests of black-hole evolution codes, it is becoming widely accepted that the unphysical initial radiation will significantly contaminate any gravitational waveforms extracted from evolutions of these data. In short, these data are not astrophysically realistic.

The various initial-data decompositions outlined in Section 2.2 and Section 2.3 are all capable of producing completely general black-hole initial data sets. The only limitation of these schemes is our understanding of what choices to make for the freely specifiable data and the boundary conditions to apply when solving the sets of elliptic equations. All these choices will have a critical impact on the astrophysical significance of the data produced. It is also important to remember that similar choices for the freely specifiable data will result in physically different solutions when applied to the different schemes.

The first studies of black-hole initial data that are not conformally flat were carried out by Abrahams et al. [1]. They looked at the superposition of a gravitational wave and a black hole. Using a form of the conformal metric that allows for so-called Brill waves, they constructed time-symmetric initial data that were not conformally flat and yet satisfied the isometry condition (65View Equation) used in the conformal-imaging method of Section 3.2.1. These data were further generalized by Brandt and Seidel [30] to include rotating black holes with a superimposed gravitational wave. In this case, the data are no longer time-symmetric yet they satisfy a generalized form of the isometry condition so that the solution is still represented on two isometric, asymptotically flat hypersurfaces.

Matzner et al. [78] have begun to move beyond conformally flat initial data for binary black holes Their proposal is to use boosted versions of the Kerr metric written in the Kerr–Schild form to represent each black hole. Thus, an isolated black hole will have no spurious radiation content in the initial data. To construct solutions with multiple black holes, they propose, essentially, to use a linear combination of the single-hole solutions. The resulting metric can be used as the conformal 3-metric, the trace of the resulting extrinsic curvature can be used for K, and the tracefree part of the resulting extrinsic curvature can be used for M&tidle; ij. Their scheme uses York’s conformal transverse-traceless decomposition outlined in Section 2.2.1, with the boundary conditions of ψ = 1 and i V = 0 on the horizons of the black holes and conditions appropriate for asymptotic flatness at large distances from the holes. A related method has been proposed by Bishop et al. [15], but their approach is much different and outside the current scope of this review.

The approach outlined by Matzner should certainly yield “cleaner” data than the conformally flat data currently available. For the task of specifying data for astrophysical black-hole binaries in nearly circular orbits, it is still true that these new data will not contain the correct initial gravitational wave content. Because the black holes are in orbit, they must be producing a continuous wave-train of gravitational radiation. This radiation will not be included in the method proposed by Matzner et al. Also, it is clear that the boundary conditions being used do not correctly account for the tidal distortion of each black hole by its companion. When the black holes are sufficiently far apart, the radiation from the orbital motion can be computed using post-Newtonian techniques. One possibility for producing astrophysically realistic, binary black-hole initial data is to use information from these post-Newtonian calculations to obtain better guesses for &tidle;γij, &tidle;M ij, and K.


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