4 Cauchy-Characteristic Matching3 Characteristic Evolution Codes3.5 3D Characteristic Evolution

3.6 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 [100Jump To The Next Citation Point In The Article, 154Jump To The Next Citation Point In The Article]. 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, for 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 [123, 85], for which existence theorems apply in some neighborhood of the initial null hypersurfaces [157, 60, 59]. Here one of the null hypersurfaces is the event horizon tex2html_wrap_inline2155 of the binary black holes. The other is an ingoing null hypersurface tex2html_wrap_inline2157 which intersects tex2html_wrap_inline2155 in a topologically spherical surface tex2html_wrap_inline2161 approximating the equilibrium of the final Kerr black hole, so that tex2html_wrap_inline2157 approximates future null infinity tex2html_wrap_inline1655 . The required data for the analytic problem consists of the degenerate conformal null metrics of tex2html_wrap_inline2155 and tex2html_wrap_inline2157 and the metric and extrinsic curvature of tex2html_wrap_inline2161 .

The conformal metric of tex2html_wrap_inline2155 is provided by the conformal horizon model for a binary black hole horizon [100Jump To The Next Citation Point In The Article, 89Jump To The Next Citation Point In The Article], which treats the horizon in stand-alone fashion as a three-dimensional manifold endowed with a degenerate metric tex2html_wrap_inline2175 and affine parameter t along its null rays. The metric is obtained from the conformal mapping tex2html_wrap_inline2179 of the intrinsic metric tex2html_wrap_inline2181 of a flat space null hypersurface emanating from a convex surface tex2html_wrap_inline2183 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 tex2html_wrap_inline2183, and its extension into the future. The flat space null hypersurface expands forever as its affine parameter tex2html_wrap_inline2187 (given by 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 tex2html_wrap_inline2187 which produces the nontrivial topology of the affine slices of the black hole horizon. The relative distortion between the affine parameters t and tex2html_wrap_inline2187, 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 tex2html_wrap_inline2183 to be a prolate spheroid, is shown in Fig.  3  [100]. The black hole event horizon associated with a triaxial ellipsoid reveals new features not seen in the degenerate case of the head-on collision [89], as depicted in Fig.  4 . If the degeneracy is slightly broken, the individual black holes form with spherical topology but as they approach, 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 (superluminally) to produce first a peanut shaped black hole and finally a spherical black 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. No violation of the topological censorship [58] 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 [126]. Details of this merger can be viewed at [150].

  

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

The conformal horizon model determines the data on tex2html_wrap_inline2155 and tex2html_wrap_inline2161 . The remaining data necessary to evolve the exterior spacetime is the conformal geometry of tex2html_wrap_inline2157, 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 tex2html_wrap_inline2205 . (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 quadrupole (tex2html_wrap_inline2207) but that a strong tex2html_wrap_inline2145 mode exists just before fission. In the second stage of the calculation, which has not yet been carried out, this waveform is 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 [117Jump To The Next Citation Point In The Article], this superposition of solutions is simplified by the time reflection symmetry [154]. More generally, beyond the perturbative regime, the merger-ringdown waveform must be obtained by a more complicated inverse scattering procedure.

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 tex2html_wrap_inline2155, must also be provided in order to initiate the radial hypersurface integrations. The ingoing shear is part of the free data specified at tex2html_wrap_inline2161 . Its value on tex2html_wrap_inline2155 can be determined by integrating (backward in time) a sequence of propagation equations involving the horizon's twist and ingoing divergence. A horizon code which carries out these integrations has been tested to give accurate data even beyond the merger [68].

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 which approaches a single Schwarzschild black hole. The resulting perturbed Schwarzschild horizon provides global insight into the close limit [117], 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 paves the way for the PITT code to calculate whether this dramatic time dependence of the horizon produces an equally dramatic waveform.



4 Cauchy-Characteristic Matching3 Characteristic Evolution Codes3.5 3D Characteristic Evolution

image Characteristic Evolution and Matching
Jeffrey Winicour
http://www.livingreviews.org/lrr-2001-3
© Max-Planck-Gesellschaft. ISSN 1433-8351
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