The conformal metric of
is provided by the conformal horizon model for a binary black
hole horizon [100,
89], which treats the horizon in stand-alone fashion as a
three-dimensional manifold endowed with a degenerate metric
and affine parameter
*t*
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
(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
which produces the nontrivial topology of the affine slices of
the black hole horizon. The relative distortion between the
affine parameters
*t*
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 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].

The conformal horizon model determines the data on and . The remaining data necessary to evolve the exterior spacetime is 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 quadrupole () but that a strong 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 [117], 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
, 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 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.

Characteristic Evolution and Matching
Jeffrey Winicour
http://www.livingreviews.org/lrr-2001-3
© Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |