Richard Matzner, University of Texas at Austin

richard@ricci.ph.utexas.edu

The Binary Black Hole Grand Challenge has completed more than four years of existence. A large fraction of that time has been devoted to developing a coherent infrastructure for assault on the two-black-hole problem. The Alliance approach involves a central Cauchy strong field region, a boundary (matching) module, and an outer module (perturbative, or strong-field characteristic) which carries the radiation to infinity. The interior module is an ADM (Arnowitt-Deser-Misner [1] ) ``'' code, with fundamental variables the 3-metric and extrinsic curvature ; the extrinsic curvature is the momentum of the 3-metric. An extensive investigation has been made of the Choquet-Bruhat/York [2] version of a ``hyperbolic'' Cauchy formulation, but the more traditional ADM form has the advantage of more mature development. Hence in 1996 the Alliance focused on the ADM version.

Important infrastructure features include DAGH, which allows a single-processor unigrid code to be distributed on a parallel machine, and supports adaptive mesh refinement. This system is now in use for the very largest of our black-hole runs. Another very important tool is RNPL, which takes a high-level description of the physics and the differencing scheme and generates C or FORTRAN code. Another tool, on the collaborative level, is SCIVIZ, which allows researchers to collaborate to manipulate and visualize computational results. A new file format, SDF, has been developed, which overcomes efficiency and size limitations of some other formats, for large parallel applications.

All of the Alliance codes are, and continue to be, demonstrated second order convergent. In all of our models, black holes are handled by excising the domain inside the apparent horizon. We have not yet begun (we are about to begin) carrying out multiple black hole evolutions. For single black holes (Schwarzschild or Kerr, and their strongly perturbed forms), where we expect a stationary final state, we use the analytic solution as an outer boundary condition.

* Black Holes*
The characteristic module evolves the strong-field Einstein equations
in a characteristic formation which has a very rigid coordinate gauge,
and which therefore has a simpler equation set. Unfortunately it
cannot be used alone for the binary black hole problem, since
gravitational focusing causes caustics in the rays generating the null
surfaces of the coordinatization. Thus the basic approach of the
Alliance code is centered on a Cauchy strong field module. However,
the characteristic code can handle * single* black hole
spacetimes.

The characteristic module can exist in two forms based on either
ingoing or outgoing characteristic surfaces. In its form based on
ingoing null hypersurfaces it shows unlimited long-term stability for
evolving single, isolated (distorted) black holes. In these cases data
are set on an ingoing initial null hypersurface. The inner edge of the
domain is set at a marginally trapped surface; no boundary condition is
needed there since it is inside the horizon. The outer boundary is set
analytically to the black hole solution. These problems evolve to the
stationary black hole form. They have been evolved to times of **60,000
M**, at which time differences are on order of machine precision, the
operational definition of running forever. In some cases the
coordinates are deliberately ``wobbled'' producing a time dependence in
the description at late times, but producing a stationary geometry
nonetheless [3]. In the 1970s and 1980s, the difficulty of
stably simulating even a single black hole in strictly spherical
symmetry (one spatial dimension) led to the formulation of ``the Holy
Grail of numerical relativity" - requirements for a hypothetical
``code that simultaneously:

Avoids singularities

Handles black holes

Maintains high accuracy

Runs forever.'' [4]

It is clear that the characteristic code has achieved the grail in the 3-dimensional single-black-hole case, a dramatic improvement over the state of the art only a few years ago. However, goals recede, and from the viewpoint of the Binary Black Hole Alliance, this is a step along the way, important because it validates the stability and accuracy of the characteristic code.

The interior code, the Cauchy code, has not yet shown the very
long-term stability of the characteristic code. With fixed Dirichlet
boundaries, the code runs for a maximum [5] of **100M** for
isolated Schwarzschild data written in Kerr-Schild [6] form. With
blended outer boundary conditions, the code has been evolved beyond
**500M**. (The blended outer conditions are applied gradually by mixing
the computed results with the analytic ones over a shell of a few
computational zones' thickness; see also the discussion of this
technique for perturbative matching below.) In this case there is still
some influence from the outer boundary and there are additional modes
(small oscillations in the supposedly static solution) which are not
fully understood. What is apparent is that inaccurate outer boundary
setting disturbs the code substantially (which is why the matching
algorithm is so important), but the inner edge of the domain, handled
with causal differencing (hence ``no boundary condition'') is well
behaved, and this free evolution shows (at worst) controllable
constraint drifting.

The Kerr-Schild data are represented by two fields on a background flat
space: a scalar function ( for Schwarzschild), and a null vector
(ingoing, unit for Schwarzschild). Because of this very simple
structure, * boosting* these data is trivial, and we have used
such boosted initial data to start evolutions of black holes moving
across the computational domain. So far as we know, only the Alliance
has achieved this. The characteristic code has demonstrated a linearly
moving black hole [7]. However (because of the caustic
problem), the characteristic module cannot evolve a black hole moving
farther than one diameter. The Cauchy module can do so, and has been
demonstrated to do so for **60M** in time at , hence a translation
through **6M** in distance [5]. The boundary conditions for
this moving case are analytical Dirichlet with no blending. (Since we
know the analytic form for the boosted black hole as a function of
time, we compute new outer boundaries as a function of time for the
evolution.)

The black hole interior is excised in all our evolutions. At the
resolutions we use (typically 60 to 100 grid zones in each direction),
there is room for only a few () points interior to the
black hole. We find the best behavior when the hole is excised with a
buffer zone zones wide for both the moving and the stationary
evolutions. Thus the excision of the interior occurs zones
inside the apparent horizon location. This is probably relevant to the
fact that we do not lock the horizon coordinate location. Rather, the
excision is based on the analytically expected coordinate location of
the horizon; and all our code crashes seem to be related to the
excised, un-evolved, region eventually extending beyond the horizon.
(This can happen because coordinate drift, which we do not attempt to
control, changes the coordinate location of the horizon, while our
excision domain has a * fixed* coordinate location.) To our
knowledge only Daues [8] has demonstrated active horizon locking
in 3-dimensional black holes. Daues achieved non-moving
Schwarzschild black hole evolutions. Implementing this tracking in the
Alliance code is a high priority and holds out the hope of even longer
evolutions.

** Exterior Modules and Matching **

The * perturbative* exterior module is written in explicitly
Cauchy form. The terms neglected in this perturbative module are
wave-wave interactions, while the background is explicitly modeled
(Kerr or Schwarzschild). The matching to the Cauchy interior
* works* in this case; this matching has been demonstrated for
linear waves with very long evolution [9]; some more recent
results are at the Alliance web site (see below). The matching is
accomplished in a way that correctly treats the outgoing nature of the
solution; in fact, the Sommerfeld condition is modified on its
right-hand side from **0**, to a contribution arising from the
perturbative outer evolution, so there is a strong similarity between
the perturbative and the characteristic boundary application.

In practice, the perturbative outer boundary match is handled in a
``thick'' shell. At some radius , the inner solution is
sampled. These data are used for a perturbative evolution to a very
large radius . At a finite radius begins the boundary region . The computed
inner solution is merged in this region with the value determined from
the exterior module. This provides a merged boundary condition on the
interior solution: that the Sommerfeld condition properly reflect the
terms describing backscatter, derived from the perturbative evolution.
* For the weak wave case this is a successful complete
expression of the inner-module/boundary/outer-module paradigm of the
Alliance philosophy.*

To match the * characteristic* module to the Cauchy inner module,
the outgoing characteristic form must be used. (This * match*
has not yet been achieved.) For outgoing radiation near the coordinate
outgoing null surfaces, the wave variables have slow variation, and the
system can be compactified so that infinity is a finite distance away
while still maintaining finite derivatives. Hence, a characteristic
code can compute the whole exterior spacetime in a finite domain. For
nonlinear scalar radiation [10], for spherical general
relativity [11], for cylindrically symmetric
relativity [12], and as we saw, for the weak field problem in full
3-d general relativity, the match has been carried out. But so far a
stable match between the full 3-d strong-field Cauchy and
characteristic modules has not been achieved. We are now attempting
such a match through blending, as in the successful perturbative case,
and there is hope that such an approach will work to match the Cauchy
and the characteristic codes.

Immediate future work involves setting data and beginning 2-hole evolutions. Because of the apparently better behavior of Kerr-Schild formulated single holes, the initial data is being recomputed for this case. (These slices differ macroscopically from the ``standard'' conformally flat data that were solved completely prior to the beginning of the Alliance [13].) This work will proceed while further runs for single holes continue. The Cauchy module requires standardization, validation against known behavior of distorted black holes, and an explicit demonstration of its ability to evolve rotating (Kerr) black holes.

Recent developments, including the points discussed here, are frequently posted to the Los Alamos preprint archive, and can also be found at the Alliance Web page:

http://www.npac.syr.edu/projects/bh/

Select ``New developments.''

Richard Matzner is the Lead PI of the Binary Back Hole Grand Challenge Alliance, NSF ASC/PHY 9318152 (arpa supplemented), which supported this work.

** References:**

[1] R. Arnowitt, S. Deser, C. W. Misner, in
* Gravitation, an Introduction to Current Research*,
L. Witten, ed. (Wiley, New York, 1962).

[2] Y. Choquet-Bruhat and J.W. York, ``Geometrical Well Posed
Systems for the Einstein Equations,'' C.R. Acad. Sci. Paris,
** 321** 1089 (1995).

[3] The Binary Black Hole Grand Challenge Alliance,
``Stable characteristic evolution of generic

3-dimensional single-black-hole spacetimes", submitted to * Physical
Review Letters* (1998). gr-qc/9801069

[4] S.L. Shapiro, S.A. Teukolsky, in * Dynamical Spacetimes and
Numerical Relativity*, ed. J. Centrella (Cambridge UP, Cambridge, 1986)
p. 74.

[5] The Binary Black Hole Grand Challenge Alliance
``Boosted three-dimensional black-hole evolutions with
singularity excision", * Physical Review Letters* (in press, 1998).
gr-qc/9711078

[6] R.P. Kerr and A. Schild, ``Some Algebraically Degenerate
Solutions of Einstein's Gravitational Field Equations,''
* Applications of Nonlinear Partial Differential Equations in
Mathematical Physics*, Proc. of Symposia B Applied Math., Vol. XV11
(1965).
R.P. Kerr and A. Schild,``A New Class of Vacuum Solutions of the Einstein
Field Equations,''
* Atti del Convegno Sulla Relativita Generale: Problemi
Dell'Energia E Onde Gravitazionale*, G. Barbera, ed. (1965).

[7] R. Gomez, L. Lehner, R.L. Marsa, J. Winicour,
``Moving Black Holes in 3D", * The Physical Review* ** D56**, 6310 (1997).
gr-qc/9710138

[8] G. Daues, Ph.D. dissertation, Washington University,
Saint Louis (1996).

[9] The Binary Black Hole Grand Challenge Alliance
``Gravitational wave extraction and outer boundary conditions by
perturbative matching", * Physical Review Letters* (in press, 1998).
gr-qc/9709082

[10] Nigel T. Bishop, Roberto Gomez, Paulo R.
Holvorcem, Richard A. Matzner, Philippos Papadopoulos, and Jeffrey
Winicour, ``Cauchy-characteristic matching: A new approach to
radiation boundary conditions,'' * Physical Review Letters*
** 76** 4303 (1996).

[11] R. Gomez, R. Marsa and J. Winicour, ``Black hole
excision with matching,'' * Physical Review D* ** 56**,
(November 1997), gr-qc/9708002.

[12] C. Clarke, R. d'Inverno, and J. Vickers, * Physical
Review D* ** 52**, 6863 (1995).

M. Dubal, R. d'Inverno, and C. Clarke,
* Physical Review D* ** 52**, 6868 (1995).

[13] G.B. Cook, M. W. Choptuik, M. R. Dubal, S. Klasky,
Richard A. Matzner and S.R. Oliveira, ``Three-Dimensional Initial
Data for the Collision of Two Black Holes,'' * Physical Review
D* ** 47 ** 1471-1490 (February 1993).

Sun Feb 8 20:46:09 EST 1998