One of the earliest characteristic evolution codes, constructed by Corkill and Stewart [79, 229], treated
spacetimes with two Killing vectors using a grid based upon double null coordinates, with the null
hypersurfaces intersecting in the surfaces spanned by the Killing vectors. They simulated colliding plane
waves and evolved the Khan–Penrose [157] collision of impulsive (-function curvature) plane waves to
within a few numerical zones from the final singularity, with extremely close agreement with the analytic
results. Their simulations of collisions with more general waveforms, for which exact solutions are not
known, provided input to the understanding of singularity formation which was unforeseen in the analytic
treatments of this problem.
Many {1 + 1}-dimensional characteristic codes have been developed for spherically symmetric systems. Here matter must be included in order to make the system non-Schwarzschild. Initially the characteristic evolution of matter was restricted to simple cases, such as massless Klein–Gordon fields, which allowed simulation of gravitational collapse and radiation effects in the simple context of spherical symmetry. Now, characteristic evolution of matter is progressing to more complicated systems. Its application to hydrodynamics has made significant contributions to general relativistic astrophysics, as reviewed in Section 7.
The synergy between analytic and computational approaches has led to dramatic results in the massless
Klein–Gordon case. On the analytic side, working in a characteristic initial value formulation based upon
outgoing null cones, Christodoulou made a penetrating study of the spherically symmetric
problem [69, 70, 71, 72, 73, 74]. In a suitable function space, he showed the existence of an open ball
about Minkowski space data whose evolution is a complete regular spacetime; he showed that an evolution
with a nonzero final Bondi mass forms a black hole; he proved a version of cosmic censorship for generic
data; and he established the existence of naked singularities for non-generic data. What this analytic
tour-de-force did not reveal was the remarkable critical behavior in the transition to the black hole
regime, which was discovered by Choptuik [67, 68] in simulations using Cauchy evolution. This
phenomenon has now been understood in terms of the methods of renormalization group theory
and intermediate asymptotics, and has spawned a new subfield in general relativity, which is
covered in the Living Review in Relativity on “Critical Phenomena in Gravitational Collapse” by
Gundlach [127].
The characteristic evolution algorithm for the spherically symmetric Einstein–Klein–Gordon problem provides a simple illustration of the techniques used in the general case. It centers about the evolution scheme for the scalar field, which constitutes the only dynamical field. Given the scalar field, all gravitational quantities can be determined by integration along the characteristics of the null foliation. This is a coupled problem, since the scalar wave equation involves the curved space metric. It illustrates how null algorithms lead to a hierarchy of equations which can be integrated along the characteristics to effectively decouple the hypersurface and dynamical variables.
In a Bondi coordinate system based upon outgoing null hypersurfaces and a surface area
coordinate
, the metric is
Because any two-dimensional geometry is conformally flat, the surface integral of over a null
parallelogram
gives exactly the same result as in a flat 2-space, and leads to an integral identity upon
which a simple evolution algorithm can be based [122
]. Let the vertices of the null parallelogram be labeled
by
,
,
, and
corresponding, respectively, to their relative locations (North, East, South, and
West) in the 2-space, as shown in Figure 2
. Upon integration of Equation (7
), curvature introduces an
integral correction to the flat space null parallelogram relation between the values of
at the vertices:
This identity, in one form or another, lies behind all of the null evolution algorithms that have been
applied to this system. The prime distinction between the different algorithms is whether they are based
upon double null coordinates, or upon Bondi coordinates as in Equation (3). When a double null
coordinate system is adopted, the points
,
,
, and
can be located in each computational
cell at grid points, so that evaluation of the left hand side of Equation (8
) requires no interpolation. As
a result, in flat space, where the right hand side of Equation (8
) vanishes, it is possible to
formulate an exact evolution algorithm. In curved space, of course, there is a truncation error
arising from the approximation of the integral, e.g., by evaluating the integrand at the center of
.
The identity (8) gives rise to the following explicit marching algorithm, indicated in Figure 2
. Let the
null parallelogram lie at some fixed
and
and span adjacent retarded time levels
and
. Imagine for now that the points
,
,
, and
lie on the spatial
grid, with
. If
has been determined on the entire initial cone
, which contains the points
and
, and
has been determined radially outward
from the origin to the point
on the next cone
, then Equation (8
) determines
at the next radial grid point
in terms of an integral over
. The integrand can be
approximated to second order, i.e. to
, by evaluating it at the center of
. To
this same accuracy, the value of
at the center equals its average between the points
and
, at which
has already been determined. Similarly, the value of
at the
center of
can be approximated to second order in terms of values of
at points where
it can be determined by integrating the hypersurface equations (5
, 6
) radially outward from
.
After carrying out this procedure to evaluate at the point
, the procedure can then be iterated
to determine
at the next radially outward grid point on the
level, i.e. point
in Figure 2
. Upon completing this radial march to null infinity, in terms of a compactified
radial coordinate such as
, the field
is then evaluated on the next null cone
at
, beginning at the vertex where smoothness gives the startup condition that
.
In the compactified Bondi formalism, the vertices ,
,
, and
of the null parallelogram
cannot be chosen to lie exactly on the grid because, even in Minkowski space, the velocity of light in
terms of a compactified radial coordinate
is not constant. As a consequence, the fields
,
,
and
at the vertices of
are approximated to second order accuracy by interpolating
between grid points. However, cancellations arise between these four interpolations so that
Equation (8
) is satisfied to fourth order accuracy. The net result is that the finite difference
version of Equation (8
) steps
radially outward one zone with an error of fourth order in
grid size,
. In addition, the smoothness conditions (4
) can be incorporated
into the startup for the numerical integrations for
and
to insure no loss of accuracy
in starting up the march at
. The resulting global error in
, after evolving a finite
retarded time, is then
, after compounding errors from
number of
zones.
When implemented on a grid based upon the coordinates, the stability of this algorithm is
subject to a Courant–Friedrichs–Lewy (CFL) condition requiring that the physical domain of dependence
be contained in the numerical domain of dependence. In the spherically symmetric case, this condition
requires that the ratio of the time step to radial step be limited by
, where
. This condition can be built into the code using the value
,
corresponding to the maximum of
at
. The strongest restriction on the time step
then arises just before the formation of a horizon, where
at
. This infinite
redshift provides a mechanism for locating the true event horizon “on the fly” and restricting
the evolution to the exterior spacetime. Points near
must be dropped in order to evolve
across the horizon due to the lack of a nonsingular compactified version of future time infinity
.
The situation is quite different in a double null coordinate system, in which the vertices of the null
parallelogram can be placed exactly on grid points so that the CFL condition is automatically satisfied. A
characteristic code based upon double null coordinates was developed by Goldwirth and Piran in a study
of cosmic censorship [109] based upon the spherically symmetric gravitational collapse of a
massless scalar field. Their early study lacked the sensitivity of adaptive mesh refinement (AMR)
which later enabled Choptuik to discover the critical phenomena appearing in this problem.
Subsequent work by Marsa and Choptuik [171
] combined the use of the null related ingoing
Eddington–Finklestein coordinates with Unruh’s strategy of singularity excision to construct a
1D code that “runs forever”. Later, Garfinkle [104
] constructed an improved version of the
Goldwirth–Piran double null code which was able to simulate critical phenomena without using adaptive
mesh refinement. In this treatment, as the evolution proceeds on one outgoing null cone to
the next, the grid points follow the ingoing null cones and must be dropped as they cross the
origin at
. However, after half the grid points are lost they are then “recycled” at new
positions midway between the remaining grid points. This technique is crucial for resolving the
critical phenomena associated with an
size horizon. An extension of the code [105]
was later used to verify that scalar field collapse in six dimensions continues to display critical
phenomena.
Hamadé and Stewart [135] also applied a double null code to study critical phenomena. In order to
obtain the accuracy necessary to confirm Choptuik’s results they developed the first example of a
characteristic grid with AMR. They did this with both the standard Berger and Oliger algorithm and their
own simplified version, with both versions giving indistinguishable results. Their simulations of critical
collapse of a massless scalar field agreed with Choptuik’s values for the universal parameters
governing mass scaling and displayed the echoing associated with discrete self-similarity. Hamadé,
Horne, and Stewart [134] extended this study to the spherical collapse of an axion/dilaton
system and found in this case that self-similarity was a continuous symmetry of the critical
solution.
Brady, Chambers, and Gonçalves [55] used Garfinkle’s [104] double null algorithm to investigate the
effect of a massive scalar field on critical phenomena. The introduction of a mass term in the scalar wave
equation introduces a scale to the problem, which suggests that the critical point behavior might differ from
the massless case. They found that there are two different regimes depending on the ratio of the Compton
wavelength
of the scalar mass to the radial size
of the scalar pulse used to induce collapse. When
, the critical solution is the one found by Choptuik in the
case, corresponding to a
type II phase transition. However, when
, the critical solution is an unstable soliton star
(see [216
]), corresponding to a type I phase transition where black hole formation turns on at a finite
mass.
A code based upon Bondi coordinates, developed by Husa and his collaborators [149], has
been successfully applied to spherically symmetric critical collapse of a nonlinear -model
coupled to gravity. Critical phenomena cannot be resolved on a static grid based upon the Bondi
-coordinate. Instead, the numerical techniques of Garfinkle were adopted by using a dynamic grid
following the ingoing null rays and by recycling radial grid points. They studied how coupling
to gravity affects the critical behavior previously observed by Bizoń [51] and others in the
Minkowski space version of the model. For a wide range of the coupling constant, they observe
discrete self-similarity and typical mass scaling near the critical solution. The code is shown to
be second order accurate and to give second order convergence for the value of the critical
parameter.
The first characteristic code in Bondi coordinates for the self-gravitating scalar wave problem was
constructed by Gómez and Winicour [122]. They introduced a numerical compactification of
for the
purpose of studying effects of self-gravity on the scalar radiation, particularly in the high amplitude limit of
the rescaling
. As
, the red shift creates an effective boundary layer at
which causes the Bondi mass
and the scalar field monopole moment
to be related
by
, rather than the quadratic relation of the weak field limit [122]. This
could also be established analytically so that the high amplitude limit provided a check on
the code’s ability to handle strongly nonlinear fields. In the small amplitude case, this work
incorrectly reported that the radiation tails from black hole formation had an exponential decay
characteristic of quasinormal modes rather than the polynomial
or
falloff expected
from Price’s [198
] work on perturbations of Schwarzschild black holes. In hindsight, the error
here was not having confidence to run the code sufficiently long to see the proper late time
behavior.
Gundlach, Price, and Pullin [129, 130] subsequently reexamined the issue of power law tails using a
double null code similar to that developed by Goldwirth and Piran. Their numerical simulations verified the
existence of power law tails in the full nonlinear case, thus establishing consistency with analytic
perturbative theory. They also found normal mode ringing at intermediate time, which provided reassuring
consistency with perturbation theory and showed that there is a region of spacetime where the results of
linearized theory are remarkably reliable even though highly nonlinear behavior is taking place
elsewhere. These results have led to a methodology that has application beyond the confines of
spherically symmetric problems, most notably in the “close approximation” for the binary black hole
problem [199]. Power law tails and quasinormal ringing have also been confirmed using Cauchy
evolution [171
].
The study of the radiation tail decay of a scalar field was subsequently extended by Gómez, Schmidt,
and Winicour [125] using a characteristic code. They showed that the Newman–Penrose constant [181] for
the scalar field determines the exponent of the power law (and not the static monopole moment as often
stated). When this constant is non-zero, the tail decays as
on
, as opposed to the
decay
for the vanishing case. (They also found
corrections, in addition to the exponentially decaying
contributions of the quasinormal modes.) This code was also used to study the instability of a topological
kink in the configuration of the scalar field [23]. The kink instability provides the simplest
example of the turning point instability [152, 226] which underlies gravitational collapse of static
equilibria.
Brady and Smith [57] have demonstrated that characteristic evolution is especially well adapted to
explore properties of Cauchy horizons. They examined the stability of the Reissner–Nordström Cauchy
horizon using an Einstein–Klein–Gordon code based upon advanced Bondi coordinates (where the
hypersurfaces
are ingoing null hypersurfaces). They studied the effect of a spherically
symmetric scalar pulse on the spacetime structure as it propagates across the event horizon. Their
numerical methods were patterned after the work of Goldwirth and Piran [109
], with modifications of the
radial grid structure that allow deep penetration inside the black hole. In accord with expectations from
analytic studies, they found that the pulse first induces a weak null singularity on the Cauchy horizon,
which then leads to a crushing spacelike singularity as
. The null singularity is weak in the sense
that an infalling observer experiences a finite tidal force, although the Newman–Penrose Weyl component
diverges, a phenomenon known as mass inflation [193]. These results confirm the earlier
result of Gnedin and Gnedin [108] that a central spacelike singularity would be created by the
interaction of a charged black hole with a scalar field, in accord with a physical argument by
Penrose [188] that a small perturbation undergoes an infinite redshift as it approaches the Cauchy
horizon.
Burko [60] has confirmed and extended these results, using a code based upon double null coordinates
which was developed with Ori [61] in a study of tail decay. He found that in the early stages the
perturbation of the Cauchy horizon is weak and in agreement with the behavior calculated by perturbation
theory.
Brady, Chambers, Krivan, and Laguna [56] have found interesting effects of a non-zero cosmological
constant on tail decay by using a characteristic Einstein–Klein–Gordon code to study the effect of a
massless scalar pulse on Schwarzschild–de Sitter and Reissner–Nordström–de Sitter spacetimes. First, by
constructing a linearized scalar evolution code, they show that scalar test fields with
have
exponentially decaying tails, in contrast to the standard power law tails for asymptotically flat
spacetimes. Rather than decaying, the monopole mode asymptotes at late time to a constant, which
scales linearly with
, in contrast to the standard no-hair result. This unusual behavior for
the
case was then independently confirmed with a nonlinear spherical characteristic
code.
Using a combination of numerical and analytic techniques based upon null coordinates,
Hod and Piran have made an extensive series of investigations of the spherically symmetric
charged Einstein–Klein–Gordon system dealing with the effect of charge on critical gravitational
collapse [142] and the late time tail decay of a charged scalar field on a Reissner–Nordström black
hole [143, 146, 144, 145]. These studies culminated in a full nonlinear investigation of horizon formation
by the collapse of a charged massless scalar pulse [147]. They track the formation of an apparent horizon
which is followed by a weakly singular Cauchy horizon which then develops a strong spacelike
singularity at
. This is in complete accord with prior perturbative results and nonlinear
simulations involving a pre-existing black hole. Oren and Piran [182] increased the late time accuracy
of this study by incorporating an adaptive grid for the retarded time coordinate
, with a
refinement criterion to maintain
. The accuracy of this scheme is confirmed
through convergence tests as well as charge and constraint conservation. They were able to
observe the physical mechanism which prohibits black hole formation with charge to mass ration
. Electrostatic repulsion of the outer parts of the scalar pulse increases relative to the
gravitational attraction and causes the outer portion of the charge to disperse to larger radii
before the black hole is formed. Inside the black hole, they confirm the formation of a weakly
singular Cauchy horizon which turns into a strong spacelike singularity, in accord with other
studies.
Hod extended this combined numerical-analytical double null approach to investigate higher order
corrections to the dominant power law tail [140], as well as corrections due to a general spherically
symmetric scattering potential [139] and due to a time dependent potential [141]. He found
modifications to the leading order tail behavior for a Schwarzschild black hole, in accord with earlier
results of Gómez et al. [125]. These modifications fall off at a slow rate so that a very long
numerical evolution (
)is necessary to cleanly identify the leading order power law
decay.
The foregoing numerical-analytical work based upon characteristic evolution has contributed to a very
comprehensive classical treatment of spherically symmetric gravitational collapse. Sorkin and Piran [225]
have investigated the question of quantum corrections due to pair creation on the gravitational collapse of a
charged scalar field. For observers outside the black hole, several analytic studies have indicated that such
pair-production can rapidly diminish the charge of the black hole. Sorkin and Piran apply the same
double-null characteristic code used in studying the classical problem [147] to evolve across the event
horizon and observe the quantum effects on the Cauchy horizon. The quantum electrodynamic
effects are modeled in a rudimentary way by a nonlinear dielectric constant that limits the
electric field to the critical value necessary for pair creation. The back-reaction of the pairs
on the stress-energy and the electric current are ignored. They found that quantum effects
leave the classical picture of the Cauchy horizon qualitatively intact but that they shorten
its “lifetime” by hastening the conversion of the weak null singularity into a strong spacelike
singularity.
The Southampton group has constructed a {1 + 1}-dimensional characteristic code for spacetimes with
cylindrical symmetry [77, 87
]. The original motivation was to use it as the exterior characteristic code in a
test case of CCM (see Section 5.5.1 for the application to matching). Subsequently, Sperhake, Sjödin, and
Vickers [223, 227] modified the code into a global characteristic version for the purpose of studying
cosmic strings, represented by massive scalar and vector fields coupled to gravity. Using a Geroch
decomposition [106] with respect to the translational Killing vector, they reduced the global problem to a
{2 + 1}-dimensional asymptotically flat spacetime, so that
could be compactified and
included in the numerical grid. Rather than the explicit scheme used in CCM, the new version
employs an implicit, second order in space and time, Crank–Nicholson evolution scheme. The code
showed long term stability and second order convergence in vacuum tests based upon exact
Weber–Wheeler waves [247
] and Xanthopoulos’ rotating solution [252], and in tests of wave scattering
by a string. The results show damped ringing of the string after an incoming Weber–Wheeler
pulse has excited it and then scattered to
. The ringing frequencies are independent of
the details of the pulse but are inversely proportional to the masses of the scalar and vector
fields.
Frittelli and Gómez [101] have cast the spherically symmetric Einstein-Klein-Gordon problem in
symmetric hyperbolic form, where in a Bondi-Sachs gauge the fundamental variables are the scalar field,
lapse and shift. The Bondi-Sachs gauge conditions relate the usual ADM variables (the 3-metric
and extrinsic curvature) to the lapse and shift, which obey simpler evolution equations. The
resulting Cauchy problem is well-posed and the outer boundary condition is constraint preserving
(although whether the resulting IBVP is well-posed is not addressed, i.e. whether the boundary
condition is dissipative). A numerical evolution algorithm based upon the system produces a
stable simulation of a scalar pulse scattering off a black hole. The initial data for the pulse
satisfies
so, as expected, it contains an ingoing part, which crosses the horizon, and
an outgoing part, which leaves the grid at the outer boundary with a small amount of back
reflection.
The goal of computing waveforms from relativistic binaries, such as a neutron star or stellar mass back hole spiraling into a supermassive black hole, requires more than a stable convergent code. It is a delicate task to extract a waveform in a spacetime in which there are multiple length scales: the size of the supermassive black hole, the size of the star, the wavelength of the radiation. It is commonly agreed that some form of mesh refinement is essential to attack this problem. Mesh refinement was first applied in characteristic evolution to solve specific spherically symmetric problems regarding critical phenomena and singularity structure [104, 135, 60].
Pretorius and Lehner [197] have presented a general approach for AMR to a generic characteristic code.
Although the method is designed to treat 3D simulations, the implementation has so far been restricted to
the Einstein–Klein–Gordon system in spherical symmetry. The 3D approach is modeled after the Berger
and Oliger AMR algorithm for hyperbolic Cauchy problems, which is reformulated in terms of null
coordinates. The resulting characteristic AMR algorithm can be applied to any unigrid characteristic code
and is amenable to parallelization. They applied it to the problem of a spherically symmetric massive
Klein–Gordon field propagating outward from a black hole. The non-zero rest mass restricts the
Klein–Gordon field from propagating to infinity. Instead it diffuses into higher frequency components which
Pretorius and Lehner show can be resolved using AMR but not with a comparison unigrid
code.
http://www.livingreviews.org/lrr-2009-3 | ![]() This work is licensed under a Creative Commons License. Problems/comments to |