Anfangswerte für einen späteren Schwarzes-Loch-Kollaps von kugelsymmetrischen relativistischen Flüssigkeiten - Existenzsätze und Numerik

Abstract

Initial boundary value problems for quasilinear, partial differential equations of first order $\partial_t u+{\bf A}(u)\cdot \partial_x u={\bf b}(u)$ in two unknowns of hyperbolic type are considered. An astrophysically interesting and challenging (1) example hereof is a spherically symmetric perfect fluid space time, whose later collapse can be achieved by suitable choice of inital values. By that a timely global statement 'occurence of an event horizon from innocuous initial data' is won from a timely local existence theorem. The initial data must hold physical conditions which prevent a smooth solution in the star boundary (2). The proof stays clearly arranged since the system is brought into diagonal form. The diagonalisation method is used on the one hand, on the above mentioned Einstein equations and on the other hand, on the equations of one-dimensional gas dynamics (written in comoving and after transformation also in Eulerian coordinates) A new hybride algorithm is numerically testet for a single equation with creasing inital data. footnotes (1) Apart from the non-linearity of the equations system, which is not given in divergent form, the coordinates are concatenated with the underlying geometry, where the components of the metric are unknowns of the partial differential system. (2) where a vacuum space time is connected and the mass energy is positive (surface of a fluid) (3) new in that method is that it works also if the matrix ${\bf A$ is not invertible, because some eigenvalues disappear identically, provided that the system is written in such a way that one found so called constraints for all trivially propagated unknowns