They then tackled the gravitational problem. In their first paper in a series, they set up the machinery necessary for investigating cylindrically symmetric vacuum space-times . Although the problem involves only one spatial dimension, there are two independent modes of polarization. The Cauchy metric was treated in the Jordan-Ehlers-Kompaneets canonical form, using coordinates adapted to the cylindrical symmetry. The advantage here is that u = t - r is then a null coordinate which can be used for the characteristic evolution. They successfully recast the equations in a suitably regularized form for the compactification of in terms of the coordinate . The simple analytic relationship between Cauchy coordinates (t, r) and characteristic coordinates (u, y) facilitates the translation between Cauchy and characteristic variables on the matching worldtube, given by r = const .
The next paper in this series implements the scheme as a numerical code . The interior Cauchy evolution is carried out using an unconstrained leap frog scheme. It is notable that they report no problems with instability, which have arisen in other attempts at unconstrained leapfrog evolution in general relativity. The characteristic evolution also uses a leap frog scheme for the evolution between retarded time levels u, while numerically integrating the hypersurface equations outward along the characteristics.
The matching interface is located at points common to both the Cauchy and characteristic grids. In order to update these points by Cauchy evolution, it is necessary to obtain field values at the Cauchy ``guard'' points which lie outside the worldtube in the characteristic region. These values are obtained by interpolation from characteristic grid points (lying on three levels of null hypersurfaces in order to ensure second order accuracy). Similarly, the boundary data for starting up the characteristic integration is obtained by interpolation from Cauchy grid values inside the worldtube.
The code was tested using exact solutions for cylindrical waves, which come in from , pass through the symmetry axis and expand out to . The numerical errors are oscillatory with low growth rate, and second order convergence was confirmed. Of special importance, little numerical noise was introduced by the interface.
Comparisons of CCM were made with Cauchy evolutions using a standard outgoing radiation boundary condition . At high amplitudes the standard condition developed a large error very quickly and was competitive only for weak waves with a large outer boundary. In contrast, the matching code performed well even with a small matching radius.
Some interesting simulations were presented in which an outgoing wave in one polarization mode collides with an incoming wave in the other mode, a problem studied earlier by pure Cauchy evolution . The behavior of the collision was qualitatively similar in these two studies.
The geometrical setup is analogous to the cylindrically symmetric problem. Initial data are specified on the union of a spacelike hypersurface and a null hypersurface. The evolution used a 3-level Cauchy scheme in the interior and a 2-level characteristic evolution in the compactified exterior. A fully constrained Cauchy evolution was adopted because of its earlier success in providing accurate calculations of scalar wave collapse . Characteristic evolution was based upon the null parallelogram marching algorithm Eq. (8). The matching between the Cauchy and characteristic foliations was achieved by imposing continuity conditions on the metric, extrinsic curvature and scalar field variables, ensuring smoothness of fields and their derivatives across the matching interface. The extensive analytical and numerical studies of this system in recent years aided the development of CCM in a non-trivial geometrical setting without exact solutions by providing basic knowledge of the expected physical and geometrical behavior.
The CCM code exhibited accurate simulation of wave propagation and black hole formation for all values of M / R at the matching radius, with no symptoms of instability or back reflection. Second order accuracy was established by checking energy conservation.
In the CCM strategy, the interior black hole region is evolved using an ingoing null algorithm whose inner boundary is a marginally trapped surface and whose outer boundary lies outside the black hole and forms the inner boundary of a region evolved by the Cauchy algorithm. In turn, the outer boundary of the Cauchy region is handled by matching to an outgoing null evolution extending to . Data is passed between the inner characteristic and central Cauchy regions using a CCM procedure similar to that already described for an outer Cauchy boundary. The main difference is that, whereas the outer Cauchy boundary is matched to an outgoing null hypersurface, the inner Cauchy boundary is matched to an ingoing null hypersurface which enters the event horizon and terminates at a marginally trapped surface.
The translation from an outgoing to an incoming null evolution algorithm can be easily carried out. The substitution in the 3D version of the Bondi metric Eq. (3) provides a simple formal recipe for switching from an outgoing to an ingoing null formalism.
In order to ensure that trapped surfaces exist on the ingoing null hypersurfaces, initial data is chosen which guarantees black hole formation. Such data can be obtained from initial Cauchy data for a black hole. However, rather than extending the Cauchy hypersurface inward to an apparent horizon, it is truncated sufficiently far outside the apparent horizon to avoid computational problems with the Cauchy evolution. The initial Cauchy data is then extended into the black hole interior as initial null data until a marginally trapped surface is reached. Two ingredients are essential in order to arrange this. First, the inner matching surface must be chosen to be convex, in the sense that its outward null normals uniformly diverge and its inner null normals uniformly converge. (This is trivial to satisfy in the spherically symmetric case). Given any physically reasonable matter source, the focusing theorem then guarantees that the null rays emanating inward from the matching sphere continue to converge until reaching a caustic. Second, the initial null data must lead to a trapped surface before such a caustic is encountered. This is a relatively easy requirement to satisfy because the initial null data can be posed freely, without any elliptic or algebraic constraints other than continuity with the Cauchy data.
A code was developed which implemented CCM at both the inner and outer boundaries . Its performance showed that CCM provided as good a solution to the black hole excision problem in spherical symmetry as any previous treatment [107, 108, 33, 109]. CCM was computationally more efficient than these pure Cauchy approaches (fewer variables) and much easier to implement. Achieving stability with a purely Cauchy scheme in the region of an apparent horizon is trickier, involving much trial and error in choosing difference schemes. There were no complications with stability of the null evolution at the marginally trapped surface.
The Cauchy evolution was carried out in ingoing Eddington-Finklestein (IEF) coordinates. The initial Cauchy data consisted of a Schwarzschild black hole with an ingoing Gaussian pulse of scalar radiation. Since IEF coordinates are based on ingoing null cones, it is possible to construct a simple transformation between the IEF Cauchy metric and the ingoing null metric. Initially there was no scalar field present on either the ingoing or outgoing null patches. The initial values for the Bondi variables and V were determined by matching to the Cauchy data at the matching surfaces and integrating the hypersurface equations (5) and (6).
As the evolution proceeds, the scalar field passes into the black hole, and the marginally trapped surface (MTS) grows outward. The MTS could be easily located in the spherically symmetric case by an algebraic equation. In order to excise the singular region, the grid points inside the marginally trapped surface were identified and masked out of the evolution. The backscattered radiation propagated cleanly across the outer matching surface to . The strategy worked smoothly, and second order accuracy of the approach was established by comparison with an independent numerical solution obtained using a second order accurate, purely Cauchy code. As discussed below, this inside-outside application of CCM has potential application to the binary black hole problem.
|Characteristic Evolution and Matching
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