3.5 Three-D Characteristic Evolution

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.

3.5.1 The eth Module

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 Newman-Penrose 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].

3.5.2 Computer Algebra Scripts

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.

3.5.3 Code Tests

The code ran stably in all regimes of radiating, single black hole space-times, 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 twist-free axisymmetric waveforms
• Robinson-Trautman waveforms from perturbed Schwarzschild black holes.

The simulations of nonlinear Robinson-Trautman space-times showed gross qualitative differences with perturbative waveforms once radiative mass losses rose above 3% of the initial energy.

3.5.4 Nonlinear Scattering Off a Schwarzschild Black Hole

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 non-inertial 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.

3.5.5 Moving Black Holes

The 3-D 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/lrr-1998-5 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de