4.3 Characteristic treatment of binary black holes

An important application of characteristic evolution is the calculation of the waveform emitted by binary black holes, which is possible during the very interesting nonlinear domain from merger to ringdown [197Jump To The Next Citation Point, 305Jump To The Next Citation Point]. The evolution is carried out along a family of ingoing null hypersurfaces, which intersect the horizon in topological spheres. It is restricted to the period following the merger, as otherwise the ingoing null hypersurfaces would intersect the horizon in disjoint pieces corresponding to the individual black holes. The evolution proceeds backward in time on an ingoing null foliation to determine the exterior spacetime in the post-merger era. It is an example of the characteristic initial value problem posed on an intersecting pair of null hypersurfaces [259, 159], for which existence theorems apply in some neighborhood of the initial null hypersurfaces [213, 115, 114]. Here one of the null hypersurfaces is the event horizon ℋ+ of the binary black holes. The other is an ingoing null hypersurface J+, which intersects ℋ+ in a topologically spherical surface 𝒮+, approximating the equilibrium of the final Kerr black hole, so that J + approximates future null infinity + ℐ. The required data for the analytic problem consists of the degenerate conformal null metrics of + ℋ and + J and the metric and extrinsic curvature of their intersection 𝒮+.

The conformal metric of ℋ+ is provided by the conformal horizon model for a binary black-hole horizon [197Jump To The Next Citation Point, 173Jump To The Next Citation Point], which treats the horizon in stand-alone fashion as a three-dimensional manifold endowed with a degenerate metric γab and affine parameter t along its null rays. The metric is obtained from the conformal mapping 2 γab = Ω ˆγab of the intrinsic metric ˆγab of a flat space null hypersurface emanating from a convex surface 𝒮0 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 𝒮 0, and its extension into the future. The flat space null hypersurface expands forever as its affine parameter ˆ t (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 t and ˆt. This is what produces the nontrivial topology of the affine t-slices of the black-hole horizon. The relative distortion between the affine parameters t and ˆ t, brought about by curved space focusing, gives rise to the trousers shape of a binary black-hole horizon.

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Figure 3: Trousers shaped event horizon obtained by the conformal model.

An embedding diagram of the horizon for an axisymmetric head-on collision, obtained by choosing 𝒮0 to be a prolate spheroid, is shown in Figure 3View Image [197]. The black-hole event horizon associated with a triaxial ellipsoid reveals new features not seen in the degenerate case of the head-on collision [173], as depicted in Figure 4View Image. 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 [113] 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 [267]. 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 [198].

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Figure 4: Upper left: Tidal distortion of approaching black holes Upper right: Formation of sharp pincers just prior to merger. Middle left: Temporarily toroidal stage just after merger. Middle right: Peanut shaped black hole after the hole in the torus closes. Lower: Approach to final equilibrium.

The conformal horizon model determines the data on ℋ+ and 𝒮+. The remaining data necessary to evolve the exterior spacetime are given by the conformal geometry of J+, 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 ℐ−. (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 (ℓ = 2) but that a strong octopole mode (ℓ = 4) 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 [239Jump To The Next Citation Point], this superposition of solutions is simplified by the time reflection symmetry [305]. 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 r-grid points. As a result, the r-derivative of the null data, i.e., the ingoing shear of ℋ, must also be provided in order to initiate the radial hypersurface integrations. The ingoing shear is part of the free data specified at + 𝒮. Its value on ℋ 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 [137].

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 [239], 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 r = 2 M 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.


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