It is my pleasure to review progress in numerical relativity based upon characteristic evolution. In the spirit of Living Reviews in Relativity, I invite my colleagues to continue to send me contributions and comments at winicour@pitt.edu.
We are now in an era in which Einstein’s equations can effectively be considered solved at the
local level. Several groups, as reported here and in other Living Reviews, have developed 3D
Cauchy evolution codes, which are stable and accurate in some sufficiently-bounded domain. The
pioneering works [235] (based upon a harmonic formulation) and [78
, 21] (based upon BSSN
formulations [268
, 38
]) have initiated dramatic progress in the ability of these codes to simulate the inspiral
and merger of binary black holes, the premier problem in classical relativity. Global solutions of
binary black holes are another matter. Characteristic evolution codes have been successful in
treating the exterior region of asymptotically-flat spacetimes extending to future null infinity. Just
as several coordinate patches are necessary to describe a spacetime with nontrivial topology,
the most effective attack on the binary black-hole waveform might involve a global solution
patched together from pieces of spacetime handled by a combination of different codes and
techniques.
Most of the effort in numerical relativity has centered about Cauchy codes based upon the {3 + 1}
formalism [308], which evolve the spacetime inside an artificially-constructed outer boundary. It has been
common practice in Cauchy simulations of binary black holes to compute the waveform from data on a
finite extraction worldtube inside the outer boundary, using perturbative methods based upon introducing a
Schwarzschild background in the exterior region [1, 3
, 2
, 4
, 255
, 251
, 214
]. In order to properly
approximate the waveform at null infinity the extraction worldtube must be sufficiently large but at the
same time causally and numerically isolated from errors propagating in from the outer boundary.
Considerable improvement in this approach has resulted from efficient methods for dealing with a very large
outer boundary and from techniques to extrapolate the extracted waveform to infinity. However,
this is not an ideally efficient approach and is especially impractical to apply to simulations of
stellar collapse. A different approach, which is specifically tailored to study radiation at null
infinity, can be based upon the characteristic initial-value problem. This eliminates error due to
asymptotic approximations and the gauge effects introduced by the choice of a finite extraction
worldtube.
In the 1960s, Bondi [62, 63
] and Penrose [227
] pioneered the use of null hypersurfaces to describe
gravitational waves. The characteristic initial-value problem did not receive much attention before its
importance in general relativity was recognized. Historically, the development of computational physics has
focused on hydrodynamics, where the characteristics typically do not define useful coordinate surfaces and
there is no generic outer boundary behavior comparable to null infinity. But this new approach has
flourished in general relativity. It has led to the first unambiguous description of gravitational radiation in a
fully nonlinear context. By formulating asymptotic flatness in terms of characteristic hypersurfaces
extending to infinity, it was possible to reconstruct, in a nonlinear geometric setting, the basic properties of
gravitational waves, which had been developed in linearized theory on a Minkowski background. The major
new nonlinear features were the Bondi mass and news function, and the mass loss formula
relating them. The Bondi news function is an invariantly-defined complex radiation amplitude
, whose real and imaginary parts correspond to the time derivatives
and
of
the “plus” and “cross” polarization modes of the strain
incident on a gravitational wave
antenna. The corresponding waveforms are important both for the design of detection templates
for a binary black-hole inspiral and merger and for the determination of the resulting recoil
velocity.
The recent success of Cauchy evolutions in simulating binary black holes emphasizes the need to apply
global techniques to accurate waveform extraction. This has stimulated several attempts to increase the
accuracy of characteristic evolution. The Cauchy simulations have incorporated increasingly sophisticated
numerical techniques, such as mesh refinement, multi-domain decomposition, pseudo-spectral collocation
and high-order (in some cases eighth-order) finite difference approximations. The initial characteristic codes
were developed with unigrid second-order accuracy. One of the prime factors affecting the accuracy of any
characteristic code is the introduction of a smooth coordinate system covering the sphere, which
labels the null directions on the outgoing light cones. This is also an underlying problem in
meteorology and oceanography. In a pioneering paper on large-scale numerical weather prediction,
Phillips [229] put forward a list of desirable features for a mapping of the sphere to be useful for
global forecasting. The first requirement was the freedom from singularities. This led to two
distinct choices, which had been developed earlier in purely geometrical studies: stereographic
coordinates (two coordinate patches) and cubed-sphere coordinates (six patches). Both coordinate
systems have been rediscovered in the context of numerical relativity (see Section 4.1). The
cubed-sphere method has stimulated two new attempts at improved codes for characteristic
evolution (see Section 4.2.4). An ingenious third treatment, based upon a toroidal map of the
sphere, was devised in developing a characteristic code for Einstein equations [36] (see Section
4.1.3).
Another issue affecting code accuracy is the choice between a second or first differential order reduction
of the evolution system. Historically, the predominant importance of computational fluid dynamics has
favored first-order systems, in particular the reduction to symmetric hyperbolic form. However, in acoustics
and elasticity theory, where the natural treatment is in terms of second-order wave equations, an
effective argument for the second-order form has been made [187, 188]. In general relativity, the
question of whether first or second-order formulations are more natural depends on how Einstein’s
equations are reduced to a hyperbolic system by some choice of coordinates and variables. The
second-order form is more natural in the harmonic formulation, where the Einstein equations
reduce to quasilinear wave equations. The first-order form is more natural in the Friedrich–Nagy
formulation [118
], which includes the Weyl tensor among the evolution variables, and was used in the
first demonstration of a well-posed initial-boundary value problem for Einstein’s equations.
Investigations of first-order formulations of the characteristic initial-value problem are discussed in
Section 4.2.3.
The major drawback of a stand-alone characteristic approach arises from the formation
of caustics in the light rays generating the null hypersurfaces. In the most ambitious scheme
proposed at the theoretical level such caustics would be treated “head-on” as part of the evolution
problem [283]. This is a profoundly attractive idea. Only a few structural stable caustics can arise in
numerical evolution, and their geometrical properties are well-enough understood to model their
singular behavior numerically [120
], although a computational implementation has not yet been
attempted.
In the typical setting for the characteristic initial-value problem, the domain of dependence of a single
smooth null hypersurface is empty. In order to obtain a nontrivial evolution problem, the null hypersurface
must either be completed to a caustic-crossover region where it pinches off, or an additional inner boundary
must be introduced. So far, the only caustics that have been successfully evolved numerically in general
relativity are pure point caustics (the complete null cone problem). When spherical symmetry is not
present, the stability conditions near the vertex of a light cone place a strong restriction on the
allowed time step [146]. Nevertheless, point caustics in general relativity have been successfully
handled for axisymmetric vacuum spacetimes [142]. Progress toward extending these results
to realistic astrophysical sources has been made by coupling an axisymmetric characteristic
gravitational-hydro code with a high-resolution shock-capturing code for the relativistic hydrodynamics, as
initiated in the thesis of Siebel [269
]. This has enabled the global characteristic simulation of the
oscillation and collapse of a relativistic star in which the emitted gravitational waves are computed
at null infinity (see Sections 7.1 and 7.2). Nevertheless, computational demands to extend
these results to 3D evolution would be prohibitive using current generation supercomputers,
due to the small timestep required at the vertex of the null cone (see Section 3.3). This is an
unfortunate feature of present-day finite-difference codes, which might be eliminated by the
use, say, of a spectral approach. Away from the caustics, characteristic evolution offers myriad
computational and geometrical advantages. Vacuum simulations of black-hole spacetimes, where the
inner boundary can be taken to be the white-hole horizon, offer a scenario where both the
timestep and caustic problems can be avoided and three-dimensional simulations are practical (as
discussed in Section 4.5). An early example was the study of gravitational radiation from the
post-merger phase of a binary black hole using a fully-nonlinear three-dimensional characteristic
code [311
, 312
].
At least in the near future, fully three-dimensional computational applications of characteristic evolution
are likely to be restricted to some mixed form, in which data is prescribed on a non-singular but incomplete
initial null hypersurface and on a second inner boundary
, which together with the initial null
hypersurface determines a nontrivial domain of dependence. The hypersurface
may be either (i) null,
(ii) timelike or (iii) spacelike, as schematically depicted in Figure 1
. The first two possibilities give rise to
(i) the double null problem and (ii) the nullcone-worldtube problem. Possibility (iii) has more than one
interpretation. It may be regarded as a Cauchy initial-boundary value problem where the outer boundary is
null. An alternative interpretation is the Cauchy-characteristic matching (CCM) problem, in which the
Cauchy and characteristic evolutions are matched transparently across a worldtube W, as indicated in
Figure 1
.
In CCM, it is possible to choose the matching interface between the Cauchy and characteristic regions to be a null hypersurface, but it is more practical to match across a timelike worldtube. CCM combines the advantages of characteristic evolution in treating the outer radiation zone in spherical coordinates, which are naturally adapted to the topology of the worldtube with the advantages of Cauchy evolution in treating the inner region in Cartesian coordinates, where spherical coordinates would break down.
In this review, we trace the development of characteristic algorithms from model 1D problems to a 2D axisymmetric code, which computes the gravitational radiation from the oscillation and gravitational collapse of a relativistic star, to a 3D code designed to calculate the waveform emitted in the merger to ringdown phase of a binary black hole. And we trace the development of CCM from early feasibility studies to successful implementation in the linear regime and through current attempts to treat the binary black-hole problem.
CCM eliminates the need of outer boundary data for the Cauchy evolution and supplies the waveform at
null infinity via a characteristic evolution. At present, the only successful 3D application of CCM in
general relativity has been to the linearized matching problem between a 3D characteristic code
and a 3D Cauchy code based upon harmonic coordinates [287] (see Section 5.8). Here the
linearized Cauchy code satisfies a well-posed initial-boundary value problem, which seems to be
a critical missing ingredient in previous attempts at CCM in general relativity. Recently, a
well-posed initial-boundary value problem has been established for fully nonlinear harmonic
evolution [192
] (see Section 5.3), which should facilitate the extension of CCM to the nonlinear
case.
Cauchy-characteristic extraction (CCE), which is one of the pieces of the CCM strategy, also supplies
the waveform at null infinity by means of a characteristic evolution. However, in this case the artificial outer
Cauchy boundary is left unchanged and the data for the characteristic evolution is extracted from
Cauchy data on an interior worldtube. Since my last review, the most important development
has been the application of CCE to the binary black-hole problem. Beginning with the work
in [243], CCE has become an important tool for gravitational-wave data analysis (see Section 6.2).
The application of CCE to this problem was developed as a major part of the PhD thesis of
Reisswig [241].
In previous reviews, I tried to include material on the treatment of boundaries in the computational
mathematics and fluid dynamics literature because of its relevance to the CCM problem. The
fertile growth of this subject has warranted a separate Living Review on boundary conditions,
which is presently under construction and will appear soon [261]. In anticipation of this, I will
not attempt to keep this subject up to date except for material of direct relevance to CCM.
See [260
, 250
] for independent reviews of boundary conditions that have been used in numerical
relativity.
The well-posedness of the associated initial-boundary value problem, i.e., that there exists a unique
solution, which depends continuously on the data, is a necessary condition for a successful numerical
treatment. In addition to the forthcoming Living Review [261], this subject is covered in the review [119]
and the book [185].
If well-posedness can be established using energy estimates obtained by integration by parts with respect
to the coordinates defining the numerical grid, then the analogous finite-difference estimates obtained by
summation by parts [191] provide guidance for a stable finite-difference evolution algorithm. See the
forthcoming Living Review [261
] for a discussion of the application of summation by parts to numerical
relativity.
The problem of computing the evolution of a neutron star in close orbit about a black hole is of clear
importance to the new gravitational wave detectors. The interaction with the black hole could be strong
enough to produce a drastic change in the emitted waves, say by tidally disrupting the star, so that a
perturbative calculation would be inadequate. The understanding of such nonlinear phenomena requires
well-behaved numerical simulations of hydrodynamic systems satisfying Einstein’s equations. Several
numerical relativity codes for treating the problem of a neutron star near a black hole have
been developed, as described in the Living Review on “Numerical Hydrodynamics in General
Relativity” by Font [109]. Although most of these efforts concentrate on Cauchy evolution, the
characteristic approach has shown remarkable robustness in dealing with a single black hole or
relativistic star. In this vein, axisymmetric studies of the oscillation and gravitational collapse
of relativistic stars have been achieved (see Section 7.2) and progress has been made in the
3D simulation of a body in close orbit about a Schwarzschild black hole (see Sections 4.6 and
7.3).
http://www.livingreviews.org/lrr-2012-2 |
Living Rev. Relativity 15, (2012), 2
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