Application of multigrid to general relativity

Numerical solutions CNakamura, 1981 and reference therein, Piran and Stark, 1984) of Einsteinrs equations following the collapse of axially symmetric rotating bodies show, for some value of the initial ratio a/m, the formation of structure with a toroidal shape around centrally condensed core. If by some mechanisms, the aim ratio of the core is reduced Csee Miller and De Felice, 1985 and De Felice et al., 1985 for a discussion of such mechanisms) to a value less than unity then a rotating black hole may be formed, which will be surrounded by a massive toroidal structure. Other situations in which toroidal structures may be relevant are in the modelling of quasars, active galactic nuclei and other similar objects which most probably contain black holes and thick accretion disks. So far the general relativistic description of these situations had been restricted to the case in which the self-gravity of the disk was negligible CAbramowicz et al., 1978 and Kozlowski et al. 1978) or to the case when both the self-gravity of the disk and the rotation of the black hole could be considered as perturbations CUill, 1974, 19751 to the Schwarzschild black hole. However there may be situations in which the mass of the disk or tori is comparable with that of a rotating hole CUiita, 1985). In this case the full Einstein equations should be solved for the perturbations induced by the self-gravity of the matter to the Kerr black hole. Moreover it is not clear yet whether the self gravity of the disk induces runaway instability. Abramowicz et al. C1980), using a very simple model of the black hole accretion disk system suggested that this kind of instability could act in a few dynamical time scales so that the disk itself could eventually be eaten by the black hole. This instability occurs because the growing black hole changes its gravitational field and therefore the location of the cusp through which matter is accreted changes. On the other hand, Uilson C1984), using models of non-self-gravitating disks in the Kerr metric, concluded that there is no such kind of instability. A final answer to this problem can be given only after sequences of equilibrium configurations of self-gravitating disks or tori around black holes for different masses ratio have been constructed. This means solving numerically Einstein equations consistently with the given distribution of matter. Since the problem is quite complicated in structure and a standard numerical method will not easily cope with it, we have decided to use the Multigrid method CBrandt, 1977) which although is complicated to program will deal naturally with the difficulties of the model. As one of the first applications of the method in general relativity Csee Choptuik and Unruh, 1986, for a different one), we decided to solve few representative test problems before solving the entire one. The plan of the thesis is as follows. In Chapter I we review the general theory of figure of equilibrium in Newtonian and relativistic theories. Chapter II contains a discussion of stationary and axisymmetric space-times and a derivation of Einstein 1 s equations with the relative boundary conditions. Also, included is a discussion and deriv~tion of the equations governing the fluid configuration. Chapter III reviews some numerical techniques used to solve Einstein's equations for stationary and axisymmetric configurations, with particular emphasis to the Multigrid which is the method applied by us. Chapter IV contains an application of Hultigrid in general relativity in the case of vacuum stationary and axisymmetric space-times. In Chapter V we write down the equations for an infinitesimally thin disk around a black hole in Newtonian and relativistic theories. Also, an outline on how to apply the Multigrid in this case, is given. Finally, Chapter VI contains the outline of the application to the case of a self-gravitating toroidal structure around a rapidly rotating black hole.