The Binary Black Hole Grand Challenge has fostered striking
progress in developing a 3D characteristic code. At the outset of
the Grand Challenge, the Pittsburgh group had just completed
calibration of their axisymmetric characteristic code. Now, this
has not only been extended to a full 3D code which calculates
waveforms at infinity [59,
60], it has also been supplied with a horizon finder to
successfully move distorted black holes [61,
62] on a computational grid. This has been accomplished with
unlimited long term stability and demonstrated second order
accuracy, in the harshest nonlinear physical regimes
corresponding to radiation powers of a galactic rest mass per
second, and with the harshest gauge conditions, corresponding to
superluminal coordinate rotation.
The waveforms are initially calculated in arbitrary
coordinates determined by the ``3+1'' gauge conditions on an
inner worldtube. An important feature for the binary black hole
problem is that these coordinates can be rigidly rotating, so
that the evolution near infinity is highly superluminal, without
affecting code performance. The waveforms are converted to the
standard ``plus and cross'' inertial polarization modes by
numerically carrying out the transformation to an inertial frame
at infinity.
Spherical coordinates and spherical harmonics are standard
analytic tools in the description of radiation, but, in
computational work, spherical coordinates have mainly been used
in axisymmetric systems, where polar singularities may be
regularized by standard tricks. In the absence of symmetry, these
techniques do not generalize and would be especially prohibitive
to develop for tensor fields. A crucial ingredient of the 3D
characteristic code is a module which allows use of spherical
coordinates by implementing a computational version of the
NewmanPenrose eth formalism [64]. The eth module covers the sphere with two overlapping
stereographic coordinate grids (North and South). It provides
everywhere regular, second order accurate, finite difference
expressions for tensor fields on the sphere and their covariant
derivatives [65].
Although the eth calculus both simplifies the equations and
avoids spurious coordinate singularities, there is a large
proliferation of angular derivatives of vector and tensor fields
in reexpressing Einstein's equations in eth form. MAPLE scripts
have been developed which greatly facilitate reliable coding of
the curvature and other tensors entering the problem. The
translation of a formula from tensor to eth formalism is ideal
for computer algebra: It is straightforward and algorithmic  but
lengthy.
The code ran stably in all regimes of radiating, single black
hole spacetimes, including extremely nonlinear systems with the
Bondi news as large as
N
=400 (in dimensionless geometric units). This means that the code
can cope with an enormous power output
in conventional units. This exceeds the power that would be
produced if, in 1 second, the whole Galaxy were converted to
gravitational radiation.
Code tests verified second order accuracy of the 3D code in an
extensive number of testbeds:
 Linearized waves on a Minkowski background in null cone
coordinates
 Boost and rotation symmetric solutions
 Schwarzschild in rotating coordinates
 Polarization symmetry of nonlinear twistfree axisymmetric
waveforms
 RobinsonTrautman waveforms from perturbed Schwarzschild
black holes.
The simulations of nonlinear RobinsonTrautman spacetimes
showed gross qualitative differences with perturbative waveforms
once radiative mass losses rose above 3% of the initial
energy.
The chief physical application of the code has been to the
nonlinear version of the classic problem of scattering off a
Schwarzschild black hole, first solved perturbatively by Price [35]. Here the inner worldtube for the initial value problem
consists of the ingoing
r
=2
m
surface (the past horizon), where Schwarzschild data is
prescribed. The nonlinear problem of a gravitational wave
scattering off a Schwarzschild black hole is then posed in terms
of data on an outgoing null cone consisting of an incoming pulse
with compact support.
The news function for this problem was studied as a function
of incoming pulse amplitude. Here the computational eth formalism
smoothly handles the complicated time dependent transformation
between the noninertial computational frame at
and the inertial (Bondi) frame necessary to obtain the standard
``plus'' and ``cross'' polarization modes. In the perturbative
regime, the news corresponds to the backscattering of the
incoming pulse off the effective Schwarzschild potential.
However, for higher amplitudes the waveform behaves quite
differently. Not only is its amplitude greater, but it also
reveals the presence of extra oscillations. In the very high
amplitude case, the mass of the system is dominated by the
incoming pulse, which essentially backscatters off itself in a
nonlinear way.
The 3D characteristic code was extended to handle evolution
based upon a foliation by ingoing null hypersurfaces [62]. This code incorporates a null hypersurface version of an
apparent horizon finder, which is used to excise black hole
interiors from the computation. The code accurately evolves and
tracks moving, distorted, radiating black holes. Test cases
include moving a boosted Schwarzschild black hole across a 3D
grid. A black hole wobbling relative to an orbiting
characteristic grid has been evolved and tracked for over 10,000
M, corresponding to about 200 orbits, with absolutely no sign of
instability. These results can be viewed online. [63]. The surface area of distorted black holes is calculated and
shown to approach the equilibrium value of the final
Schwarzschild black hole which is built into the boundary
conditions.
The code excises the singular region and evolves black holes
forever with second order accuracy. It has attained
the Holy Grail of numerical relativity
as originally specified by Teukolsky and Shapiro. [13]
This exceptional performance opens a promising new approach to
handle the inner boundary condition for Cauchy evolution of black
holes by the matching methods reviewed below.

Characteristic Evolution and Matching
Jeffrey Winicour
http://www.livingreviews.org/lrr19985
© MaxPlanckGesellschaft. ISSN 14338351
Problems/Comments to
livrev@aeipotsdam.mpg.de
