These instabilities came as a rude shock and led to a retreat to the simpler problem of axisymmetric scalar waves propagating in Minkowski space, with the metric
in outgoing null cone coordinates. A null cone code for this problem was constructed using an algorithm based upon Equation (8 Aware now of the subtleties of the CFL condition near the vertices, the Pittsburgh group returned to
the Bondi problem, i.e. to evolve the Bondi metric [54]
In the computational implementation of this system by the Pittsburgh group [120], the null
hypersurfaces were chosen to be complete null cones with nonsingular vertices, which (for simplicity) trace
out a geodesic worldline
. The smoothness conditions at the vertices were formulated in local
Minkowski coordinates.
The vertices of the cones were not the chief source of difficulty. A null parallelogram marching algorithm,
similar to that used in the scalar case, gave rise to another instability that sprang up throughout the grid.
In order to reveal the source of this instability, physical considerations suggested looking at the linearized
version of the Bondi equations, where they can be related to the wave equation. If this relationship were
sufficiently simple, then the scalar wave algorithm could be used as a guide in stabilizing the evolution of
. A scheme for relating
to solutions
of the wave equation had been formulated in the original
paper by Bondi, Metzner, and van der Burgh [54]. However, in that scheme, the relationship of the
scalar wave to
was nonlocal in the angular directions and was not useful for the stability
analysis.
A local relationship between and solutions of the wave equation was found [120
]. This
provided a test bed for the null evolution algorithm similar to the Cauchy test bed provided
by Teukolsky waves [239]. More critically, it allowed a simple von Neumann linear stability
analysis of the finite difference equations, which revealed that the evolution would be unstable if
the metric quantity
was evaluated on the grid. For a stable algorithm, the grid points
for
must be staggered between the grid points for
,
, and
. This unexpected
feature emphasizes the value of linear stability analysis in formulating stable finite difference
approximations.
It led to an axisymmetric code [186, 120
] for the global Bondi problem which ran stably,
subject to a CFL condition, throughout the regime in which caustics and horizons did not form.
Stability in this regime was verified experimentally by running arbitrary initial data until it
radiated away to
. Also, new exact solutions as well as the linearized null solutions were
used to perform extensive convergence tests that established second order accuracy. The code
generated a large complement of highly accurate numerical solutions for the class of asymptotically
flat, axisymmetric vacuum spacetimes, a class for which no analytic solutions are known. All
results of numerical evolutions in this regime were consistent with the theorem of Christodoulou
and Klainerman [75] that weak initial data evolve asymptotically to Minkowski space at late
time.
An additional global check on accuracy was performed using Bondi’s formula relating mass loss to the time integral of the square of the news function. The Bondi mass loss formula is not one of the equations used in the evolution algorithm but follows from those equations as a consequence of a global integration of the Bianchi identities. Thus it not only furnishes a valuable tool for physical interpretation but it also provides a very important calibration of numerical accuracy and consistency.
An interesting feature of the evolution arises in regard to compactification. By construction, the
-direction is timelike at the origin where it coincides with the worldline traced out by the vertices of the
outgoing null cones. But even for weak fields, the
-direction generically becomes spacelike at large
distances along an outgoing ray. Geometrically, this reflects the property that
is itself a null
hypersurface so that all internal directions are spacelike, except for the null generator. For a flat space time,
the
-direction picked out at the origin leads to a null evolution direction at
, but this direction
becomes spacelike under a slight deviation from spherical symmetry. Thus the evolution generically
becomes “superluminal” near
. Remarkably, this leads to no adverse numerical effects. This
remarkable property apparently arises from the natural way that causality is built into the marching
algorithm so that no additional resort to numerical techniques, such as “causal differencing” [78], is
necessary.
Stewart has implemented a characteristic evolution code which handles the Bondi problem by
a null tetrad, as opposed to metric, formalism [231]. The geometrical algorithm underlying
the evolution scheme, as outlined in [233, 99], is Friedrich’s [94
] conformal-null description
of a compactified spacetime in terms of a first order system of partial differential equations.
The variables include the metric, the connection, and the curvature, as in a Newman–Penrose
formalism, but in addition the conformal factor (necessary for compactification of
) and its
gradient. Without assuming any symmetry, there are more than 7 times as many variables as in a
metric based null scheme, and the corresponding equations do not decompose into as clean a
hierarchy. This disadvantage, compared to the metric approach, is balanced by several advantages:
The code was intended to study gravitational waves from an axisymmetric star. Since only the vacuum
equations are evolved, the outgoing radiation from the star is represented by data ( in
Newman–Penrose notation) on an ingoing null cone forming the inner boundary of the evolved domain. This
inner boundary data is supplemented by Schwarzschild data on the initial outgoing null cone, which models
an initially quiescent state of the star. This provides the necessary data for a double-null initial value
problem. The evolution would normally break down where the ingoing null hypersurface develops
caustics. But by choosing a scenario in which a black hole is formed, it is possible to evolve the
entire region exterior to the horizon. An obvious test bed is the Schwarzschild spacetime for
which a numerically satisfactory evolution was achieved (although convergence tests were not
reported).
Physically interesting results were obtained by choosing data corresponding to an outgoing quadrupole
pulse of radiation. By increasing the initial amplitude of the data , it was possible to evolve into a
regime where the energy loss due to radiation was large enough to drive the total Bondi mass
negative. Although such data is too grossly exaggerated to be consistent with an astrophysically
realistic source, the formation of a negative mass was an impressive test of the robustness of the
code.
Papadopoulos [183] has carried out an illuminating study of mode mixing by computing the evolution
of a pulse emanating outward from an initially Schwarzschild white hole of mass
. The
evolution proceeds along a family of ingoing null hypersurfaces with outer boundary at
.
The evolution is stopped before the pulse hits the outer boundary in order to avoid spurious
effects from reflection and the radiation is inferred from data at
. Although gauge
ambiguities arise in reading off the waveform at a finite radius, the work reveals interesting
nonlinear effects: (i) modification of the light cone structure governing the principal part of the
equations and hence the propagation of signals; (ii) modulation of the Schwarzschild potential by
the introduction of an angular dependent “mass aspect”; and (iii) quadratic and higher order
terms in the evolution equations which couple the spherical harmonic modes. A compactified
version of this study [256
] was later carried out with the 3D PITT code, which confirms these
effects as well as new effects which are not present in the axisymmetric case (see Section 4.5 for
details).
Oliveira and Rodrigues [81] have presented the first code based upon the Galerkin spectral method to
evolve the axisymmetic Bondi problem. The strength of spectral methods is to provide high accuracy
relative to computational effort. The spectral decomposition reduces the partial differential evolution system
to a system of ordinary differential equations for the spectral coefficients. Several numerical tests were
performed to verify stability and convergence, including linearized gravitational waves and the global energy
momentum conservation law relating the Bondi mass to the radiated energy flux. The main feature of the
Galerkin method is that each basis function is chosen to automatically satisfy the boundary conditions, in
this case the regularity conditions on the Bondi variables on the axes of symmetry and at the
vertices of the outgoing null cones and the asymptotic flatness condition at infinity. Although
is not explicitly compactified, the choice of radial basis functions allows verification of the
asymptotic relations governing the coefficients of the leading gauge dependent terms of the
metric quantities. The results promise a new approach to study spacetimes with gravitational
waves. It will be interesting to see whether the approach can be generalized to the full 3D
case.
The Southampton group, as part of its goal of combining Cauchy and characteristic evolution,
has developed a code [85, 86
, 194
] which extends the Bondi problem to full axisymmetry, as
described by the general characteristic formalism of Sachs [211]. By dropping the requirement
that the rotational Killing vector be twist-free, they were able to include rotational effects,
including radiation in the “cross” polarization mode (only the “plus” mode is allowed by twist-free
axisymmetry). The null equations and variables were recast into a suitably regularized form to allow
compactification of null infinity. Regularization at the vertices or caustics of the null hypersurfaces was
not necessary, since they anticipated matching to an interior Cauchy evolution across a finite
worldtube.
The code was designed to insure standard Bondi coordinate conditions at infinity, so that the metric has
the asymptotically Minkowskian form corresponding to null-spherical coordinates. In order to
achieve this, the hypersurface equation for the Bondi metric variable must be integrated
radially inward from infinity, where the integration constant is specified. The evolution of the
dynamical variables proceeds radially outward as dictated by causality [194
]. This differs from
the Pittsburgh code in which all the equations are integrated radially outward, so that the
coordinate conditions are determined at the inner boundary and the metric is asymptotically flat
but not asymptotically Minkowskian. The Southampton scheme simplifies the formulae for
the Bondi news function and mass in terms of the metric. It is anticipated that the inward
integration of
causes no numerical problems because this is a gauge choice which does not
propagate physical information. However, the code has not yet been subject to convergence
and long term stability tests so that these issues cannot be properly assessed at the present
time.
The matching of the Southampton axisymmetric code to a Cauchy interior is discussed in Section 5.6.
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