5.9 The binary black-hole inner boundary

CCM also offers a new approach to singularity excision in the binary black-hole problem in the manner described in Section 5.5.3 for a single spherically-symmetric black hole. In a binary system, there are computational advantages in posing the Cauchy evolution in a frame, which is co-rotating with the orbiting black holes. In this co-orbiting description, the Cauchy evolution requires an inner boundary condition inside the black holes and also an outer boundary condition on a worldtube outside of which the grid rotation is likely to be superluminal. An outgoing characteristic code can routinely handle such superluminal gauge flows in the exterior [53Jump To The Next Citation Point]. Thus, successful implementation of CCM could solve the exterior boundary problem for this co-orbiting description.

CCM also has the potential to handle the two black holes inside the Cauchy region. As described earlier with respect to Figure 6View Image, an ingoing characteristic code can evolve a moving black hole with long term stability [141, 139]. This means that CCM might also be able to provide the inner boundary condition for Cauchy evolution once stable matching has been accomplished. In this approach, the interior boundary of the Cauchy evolution is located outside the apparent horizon and matched to a characteristic evolution based upon ingoing null cones. The inner boundary for the characteristic evolution is a trapped or MTS, whose interior is excised from the evolution.

In addition to restricting the Cauchy evolution to the region outside the black holes, this strategy offers several other advantages. Although finding an MTS on the ingoing null hypersurfaces remains an elliptic problem, there is a natural radial coordinate system (r,𝜃,ϕ ) to facilitate its solution. Motion of the black hole through the grid reduces to a one-dimensional radial problem, leaving the angular grid intact and thus reducing the computational complexity of excising the inner singular region. (The angular coordinates can even rotate relative to the Cauchy coordinates in order to accommodate spinning black holes.) The chief danger in this approach is that a caustic might be encountered on the ingoing null hypersurface before entering the trapped region. This is a gauge problem whose solution lies in choosing the right location and geometry of the surface across which the Cauchy and characteristic evolutions are matched. There is a great deal of flexibility here because the characteristic initial data can be posed without constraints. This global strategy is tailor-made to treat two black holes in the co-orbiting gauge, as illustrated in Figure 8View Image. Two disjoint characteristic evolutions based upon ingoing null cones are matched across worldtubes to a central Cauchy region. The interior boundaries of each of these interior characteristic regions border a trapped surface. At the outer boundary of the Cauchy region, a matched characteristic evolution based upon outgoing null hypersurfaces propagates the radiation to infinity.

View Image

Figure 8: CCM for binary black holes, portrayed in a co-rotating frame. The Cauchy evolution is matched across two inner worldtubes Γ 1 and Γ 2 to two ingoing null evolutions whose inner boundaries excise the individual black holes. The outer Cauchy boundary is matched across the worldtube Γ to an outgoing null evolution extending to + ℐ.

Present characteristic and Cauchy codes can handle the individual pieces of this problem. Their unification offers a new approach to simulating the inspiral and merger of two black holes. The individual pieces of the fully nonlinear CCM module, as outlined in Section 5.2, have been implemented and tested for accuracy. The missing ingredient is long term stability in the nonlinear gravitational case, which would open the way to future applications.

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