Initial data

5.2 Initial data

In order to accurately calculate the gravitational waveforms for, say, the coalescence of binary black holes, one must begin with astrophysically relevant initial data. While there has been some progress on the construction of such data, the general problem is yet to be solved (see, e.g., [69

] for a recent review). However, there has been notable progress in the case of black holes which are in quasi-equilibrium. Physically, this situation arises when the black holes are sufficiently far apart for their orbits to be quasi-periodic. In his section, we will first summarize this development [70

, 125

] and then review an approach to the calculation of the binding energy in the initial data. The discussion is based on the IH framework.

5.2.1 Boundary conditions at the inner boundary

Consider the problem of constructing initial data $(¯qab,K¯ab )$ on $M$ , representing a binary black hole system. This problem has two distinct aspects. The first has to do with the inner boundaries. A natural avenue to handle the singularities is to ‘excise’ the region contained in each black hole (see, e.g., [3 , 71 ]). However, this procedure creates inner boundaries on $M$ and one must specify suitable boundary conditions there. The boundary conditions should be appropriate for the elliptic system of constraint equations, and they should also capture the idea that the excised region represents black holes which have certain physical parameters and which are in quasi-equilibrium at the ‘instant’ represented by $M$ . The second aspect of the problem is the choice of the free data in the bulk. To be of physical interest, not only must the free data satisfy the appropriate boundary conditions, but the values in the bulk must also have certain properties. For example, we might want the black holes to move in approximately circular orbits around each other, and require that there be only a minimal amount of spurious gravitational radiation in the initial data. The black holes would then remain approximately isolated for a sufficiently long time, and orbit around each other before finally coalescing.

While a fully satisfactory method of prescribing such initial data is still lacking, there has been significant progress in recent years. When the black holes are far apart and moving on approximately circular orbits, one might expect the trajectory of the black holes to lie along an approximate helical Killing vector [51 , 90 , 103 , 104 , 4 ]. Using concepts from the helical Killing vector approximation and working in the conformal thin-sandwich decomposition of the initial data [192 ], Cook has introduced the ‘quasi-equilibrium’ boundary conditions which require that each of the black holes be in instantaneous equilibrium [70 ]; see also [190 , 72 ] for a similar approach. The relation between these quasi-equilibrium boundary conditions and the isolated horizon formalism has also been recently studied [125 ].

In this section, we consider only the first aspect mentioned above, namely the inner boundaries. Physically, the quasi-equilibrium approximation ought to be valid for time intervals much smaller than other dynamical time scales in the problem, and the framework assumes only that the approximation holds infinitesimally ‘off $M$ ’. So, in this section, the type II NEH $Δ$ will be an infinitesimal world tube of apparent horizons. We assume that there is an axial symmetry vector $ϕa$ on the horizon, although, as discussed in Section 8, this assumption can be weakened.

Depending on the degree of isolation one wants to impose on the individual black holes, the inner boundary may be taken to be the cross-section of either an NEH, WIH, or an IH. The strategy is to first start with an NEH and impose successively stronger conditions if necessary. Using the local geometry of intersections $S$ of $M$ with $Δ$ , one can easily calculate the area radius $R Δ$ , the angular momentum $J Δ$ given by Equation (59 ), the canonical surface gravity $κ = κKerr(R Δ,JΔ )$ given by Equation (38 ), and angular velocity $Ω = ΩKerr(R Δ, JΔ)$ given by Equation (39 ). (Note that while a WIH structure is used to arrive at the expressions for $JΔ$ , $κ$ , and $Ω$ , the expressions themselves are unambiguously defined also on a NEH.) These considerations translate directly into restrictions on the shift vector $βa$ at the horizon. If, as in Section 4.1.3, one requires that the restriction of the evolution vector field $ta$ to $Δ$ be of the form

where $a a a ℓ0 = T + R$ , then the values of the lapse and shift fields at the horizon are given by

We are free to choose the lapse $α$ arbitrarily on $S$ , but this fixes the shift $βa$ . The tangential part of $βa$ can, if desired, be chosen differently depending on the asymptotic value of $βa$ at infinity which determines the angular velocity of inertial observers at infinity.

Next, one imposes the condition that the infinitesimal world-tube of apparent horizons is an ‘instantaneous’ non-expanding horizon. This requirement is equivalent to

on $Δ$ , which imply that the shear and expansion of $ℓa$ vanish identically. These can be easily translated to conditions that the Cauchy data $(¯q ,K¯ ) ab ab$ must satisfy at the horizon [70

, 125

]. These boundary conditions are equivalent to the quasi-equilibrium boundary conditions developed by Cook [70

]. However, the isolated horizon formalism allows greater control on the physical parameters of the black hole. In particular, we are not restricted just to the co-rotational or irrotational cases. Furthermore, the use of IHs streamlines the procedure: Rather than adding conditions one by one as needed, one begins with a physically well-motivated notion of horizons in equilibrium and systematically derives its consequences.

Up to this point, the considerations are general in the sense that they are not tied to a particular method of solving the initial value problem. However, for the quasi-equilibrium problem, it is the conformal thin-sandwich method [192 , 69 ] that appears to be best suited. This approach is based on the conformal method [145 , 191 ] where we write the 3-metric as $4 ¯qab = ψ ˆqab$ . The free data consists of the conformal 3-metric $ˆqab$ , its time derivative $ℒtˆqab$ , the lapse $α$ , and the trace of the extrinsic curvature $¯ K$ . Given this free data, the remaining quantities, namely the conformal factor and the shift, are determined by elliptic equations provided appropriate boundary conditions are specified for them on the horizon⁶ . It turns out [125 ] that the horizon conditions (63 ) are well-tailored for this purpose. While the issue of existence and uniqueness of solutions using these boundary conditions has not been proven, it is often the case that numerical calculations are convergent and the resulting solutions are well behaved. Thus, these conditions might therefore be sufficient from a practical point of view.

In the above discussion, the free data consisted of $(ˆqab,ℒt ˆqab,α, ¯K )$ and one solved elliptic equations for $a (ψ,β )$ . However, it is common to consider an enlarged initial value problem by taking $¯ ℒtK$ as part of the free data (usually set to zero) and solving an elliptic equation for $α$ . We now need to prescribe an additional boundary condition for $α$ . It turns out that this can be done by using WIHs, i.e., by bringing in surface gravity, which did not play any role so far. From the definition of surface gravity in Equations (7 , 8 ), it is clear that the expression for $κ$ will involve a time derivative; in particular, it turns out to involve the time derivative of $α$ . It can be shown that by choosing $ℒtα$ on $S$ (e.g., by taking $ℒtα = 0$ ) and requiring surface gravity to be constant on $S$ and equal to $κKerr$ , one obtains a suitable boundary condition for $α$ . (The freedom to choose freely the function $ℒt α$ mirrors the fact that fixing surface gravity does not uniquely fix the rescaling freedom of the null normal.) Note that $κ$ is required to be constant only on $S$ , not on $Δ$ . To ask it to be constant on $Δ$ would require $ℒtκ = 0$ , which in turn would restrict the second time derivative; this necessarily involves the evolution equations, and they are not part of the initial data scheme.

One may imagine using the yet stronger notion of an IH, to completely fix the value of the lapse at the horizon. But this requires solving an elliptic equation on the horizon and the relevant elliptic operator has a large kernel [15 , 70 ]. Nonetheless, the class of initial data on which its inverse exists is infinite dimensional so that the method may be useful in practice. However, this condition would genuinely restrict the permissible initial data sets. In this sense, while the degree of isolation implied by the IH boundary condition is likely to be met in the asymptotic future, for quasi-equilibrium initial data it is too strong in general. It is the WIH boundary conditions that appear to be well-tailored for this application.

Finally, using methods introduced by Dain [79 ], a variation of the above procedure was recently introduced to establish the existence and uniqueness of solutions and to ensure that the conformal factor $ψ$ is everywhere positive [80 ]. One again imposes Equation (63 ). However, in place of Dirichlet boundary conditions (62 ) on the shift, one now imposes Neumann-type conditions on certain components of $βa$ . This method is expected to be applicable all initial data constructions relying on the conformal method. Furthermore, the result might also be of practical use in numerical constructions to ensure that the codes converge to a well behaved solution.

5.2.2 Binding energy

For initial data representing a binary black hole system, the quantity $Eb = MADM − M1 − M2$ is called the effective binding energy, where $MADM$ is the ADM mass, and $M1,2$ are the individual masses of the two black holes. Heuristically, even in vacuum general relativity, one would expect $Eb$ to have several components. First there is the analog of the Newtonian potential energy and the spin-spin interaction, both of which may be interpreted as contributing to the binding energy. But $Eb$ also contains contributions from kinetic energy due to momentum and orbital angular momentum of black holes, and energy in the gravitational radiation in the initial data. It is only when these are negligible that $Eb$ is a good measure of the physical binding energy.

The first calculation of binding energy was made by Brill and Lindquist in such a context. They considered two non-spinning black holes initially at rest [61 ]. For large separations, they found that, in a certain mathematical sense, the leading contribution to binding energy comes just from the usual Newtonian gravitational potential. More recently, Dain [78 ] has extended this calculation to the case of black holes with spin and has shown that the spin-spin interaction energy is correctly incorporated in the same sense.

In numerical relativity, the notion of binding energy has been used to locate sequences of quasi-circular orbits. The underlying heuristic idea is to minimize $Eb$ with respect to the proper separation between the holes, keeping the physical parameters of the black holes fixed. The value of the separation which minimizes $Eb$ provides an estimate of sizes of stable ‘circular’ orbits [68 , 38 , 159 ]. One finds that these orbits do not exist if the orbital angular momentum is smaller than a critical value (which depends on other parameters) and uses this fact to approximately locate the ‘inner-most stable circular orbit’ (ISCO). In another application, the binding energy has been used to compare different initial data sets which are meant to describe the same physical system. If the initial data sets have the same values of the black hole masses, angular momenta, linear momenta, orbital angular momenta, and relative separation, then any differences in $Eb$ should be due only to the different radiation content. Therefore, minimization of $Eb$ corresponds to minimization of the amount of radiation in the initial data [158 ].

In all these applications, it is important that the physical parameters of the black holes are calculated accurately. To illustrate the potential problems, let us return to the original Brill–Lindquist calculation [61 ]. The topology of the spatial slice is $ℝ3$ with two points (called ‘punctures’) removed. These punctures do not represent curvature singularities. Rather, each of them represents a spatial infinity of an asymptotically flat region which is hidden behind an apparent horizon. This is a generalization of the familiar Einstein–Rosen bridge in the maximally extended Schwarzschild solution. The black hole masses $M1$ and $M2$ are taken to be the ADM masses of the corresponding hidden asymptotic regions. (Similarly, in [78 ], the angular momentum of each hole is defined to be the ADM angular momentum at the corresponding puncture.) Comparison between $Eb$ and the Newtonian binding energy requires us to define the distance between the holes. This is taken to be the distance between the punctures in a fiducial flat background metric; the physical distance between the two punctures is infinite since they represent asymptotic ends of the spatial 3-manifold. Therefore, the sense in which one recovers the Newtonian binding energy as the leading term is physically rather obscure.

Let us re-examine the procedure with an eye to extending it to a more general context. Let us begin with the definition of masses of individual holes which are taken to be the ADM masses in the respective asymptotic regions. How do we know that these are the physically appropriate quantities for calculating the potential energy? Furthermore, there exist initial data sets (e.g., Misner data [150 , 151 ]) in which each black hole does not have separate asymptotic regions; there are only two common asymptotic regions connected by two wormholes. For these cases, the use of ADM quantities is clearly inadequate. The same limitations apply to the assignment of angular momentum.

A natural way to resolve these conceptual issues is to let the horizons, rather than the punctures, represent black holes. Thus, in the spirit of the IH and DH frameworks, it is more appropriate to calculate the mass and angular momentum using expressions (60 , 59 ) which involve the geometry of the two apparent horizons. (This requires the apparent horizons to be axi-symmetric, but this limitation could be overcome following the procedure suggested in Section 8.) Similarly, the physical distance between the black holes should be the smallest proper distance between the two apparent horizons. To test the viability of this approach, one can repeat the original Brill–Lindquist calculation in the limit when the black holes are far apart [137 ]. One first approximately locates the apparent horizon, finds the proper distance $d$ between them, and then calculates the horizon masses (and thereby $Eb$ ) as a power series in $1∕d$ . The leading-order term does turn out to be the usual Newtonian gravitational potential energy but the higher order terms are now different from [61 ]. Similarly, it would be interesting to repeat this for the case of spinning black holes and recover the leading order term of [78 ] within this more physical paradigm using, say, the Bowen–York initial data. This result would re-enforce the physical ideas and the approach can then be used as a well defined method for calculating binding energy in more general situations.

Remark: In the examples mentioned above, we focussed on mass and angular momentum; linear momentum was not considered. A meaningful definition for the linear momentum of a black hole would be useful for several problems: It would enable one to define the orbital angular momentum, locate quasi-circular orbits and enable a comparison between different initial data sets. However, just as angular momentum is associated with rotational symmetry, linear momentum ought to be associated with a space translational symmetry. In a general curved space-time, one does not have even an approximate notion of translations except at infinity. Thus, whenever one speaks of linear momentum of initial data (e.g., in the Bowen–York construction [55 ]) the quantity is taken to be the appropriate component of the ADM momentum at infinity. For reasons discussed above, it is more appropriate to associate black hole parameters with the respective horizons. So, a question naturally arises: Can one define linear momentum as an integral over the apparent horizon? If $a z$ is a vector field on $M$ which, for some reason, can be regarded as an approximate translational symmetry, and $S$ the apparent horizon, then one might naively define the linear momentum associated with $za$ as

Unfortunately, this definition is not satisfactory. For example, in the boosted Kerr–Schild metric, if one lets $za$ to be the translational vector field of the background Minkowski space, Equation (64 ) does not yield the physically expected linear momentum of the boosted black hole. See [137 ] for further discussion and for other possible ways to define linear momentum. Further work is needed to address this question satisfactorily.