The conformal metric of is provided by the conformal horizon model for a binary black-hole horizon [197, 173], which treats the horizon in stand-alone fashion as a three-dimensional manifold endowed with a degenerate metric and affine parameter along its null rays. The metric is obtained from the conformal mapping of the intrinsic metric of a flat space null hypersurface emanating from a convex surface embedded at constant time in Minkowski space. The horizon is identified with the null hypersurface formed by the inner branch of the boundary of the past of , and its extension into the future. The flat space null hypersurface expands forever as its affine parameter (Minkowski time) increases, but the conformal factor is chosen to stop the expansion so that the cross-sectional area of the black hole approaches a finite limit in the future. At the same time, the Raychaudhuri equation (which governs the growth of surface area) forces a nonlinear relation between the affine parameters and . This is what produces the nontrivial topology of the affine -slices of the black-hole horizon. The relative distortion between the affine parameters and , brought about by curved space focusing, gives rise to the trousers shape of a binary black-hole horizon.
An embedding diagram of the horizon for an axisymmetric head-on collision, obtained by choosing to be a prolate spheroid, is shown in Figure 3 . The black-hole event horizon associated with a triaxial ellipsoid reveals new features not seen in the degenerate case of the head-on collision , as depicted in Figure 4. If the degeneracy is slightly broken, the individual black holes form with spherical topology but as they approach, an effective tidal distortion produces two sharp pincers on each black hole just prior to merger. At merger, the two pincers join to form a single temporarily toroidal black hole. The inner hole of the torus subsequently closes up to produce first a peanut shaped black hole and finally a spherical black hole. No violation of topological censorship  occurs because the hole in the torus closes up superluminally. Consequently, a causal curve passing through the torus at a given time can be slipped below the bottom of a trouser leg to yield a causal curve lying entirely outside the hole . In the degenerate axisymmetric limit, the pincers reduce to a point so that the individual holes have teardrop shape and they merge without a toroidal phase. Animations of this merger can be viewed at .
The conformal horizon model determines the data on and . The remaining data necessary to evolve the exterior spacetime are given by the conformal geometry of , which constitutes the outgoing radiation waveform. The determination of the merger-ringdown waveform proceeds in two stages. In the first stage, this outgoing waveform is set to zero and the spacetime is evolved backward in time to calculate the incoming radiation entering from . (This incoming radiation is eventually absorbed by the black hole.) From a time-reversed point of view, this evolution describes the outgoing waveform emitted in the fission of a white hole, with the physically-correct initial condition of no ingoing radiation. Preliminary calculations show that at late times the waveform is entirely quadrupolar () but that a strong octopole mode () exists just before fission. In the second stage of the calculation, this waveform could be used to generate the physically-correct outgoing waveform for a black-hole merger. The passage from the first stage to the second is the nonlinear equivalent of first determining an inhomogeneous solution to a linear problem and then adding the appropriate homogeneous solution to satisfy the boundary conditions. In this context, the first stage supplies an advanced solution and the second stage the homogeneous retarded minus advanced solution. When the evolution is carried out in the perturbative regime of a Kerr or Schwarzschild background, as in the close approximation , this superposition of solutions is simplified by the time reflection symmetry . The second stage has been carried out in the perturbative regime of the close approximation using a characteristic code, which solves the Teukolsky equation, as described in Section 4.4. More generally, beyond the perturbative regime, the merger-ringdown waveform must be obtained by a more complicated inverse scattering procedure, which has not yet been attempted.
There is a complication in applying the PITT code to this double null evolution because a dynamic horizon does not lie precisely on -grid points. As a result, the -derivative of the null data, i.e., the ingoing shear of , must also be provided in order to initiate the radial hypersurface integrations. The ingoing shear is part of the free data specified at . Its value on can be determined by integrating (backward in time) a sequence of propagation equations involving the horizon’s twist and ingoing divergence. A horizon code that carries out these integrations has been tested to give accurate data even beyond the merger .
The code has revealed new global properties of the head-on collision by studying a sequence of data for a family of colliding black holes that approaches a single Schwarzschild black hole. The resulting perturbed Schwarzschild horizon provides global insight into the close limit , in which the individual black holes have joined in the infinite past. A marginally anti-trapped surface divides the horizon into interior and exterior regions, analogous to the division of the Schwarzschild horizon by the bifurcation sphere. In passing from the perturbative to the strongly nonlinear regime there is a rapid transition in which the individual black holes move into the exterior portion of the horizon. The data pave the way for the PITT code to calculate whether this dramatic time dependence of the horizon produces an equally dramatic waveform. See Section 4.4.2 for first stage results.
Living Rev. Relativity 15, (2012), 2
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