Analytische Methoden bei Gravitationslinsenphänomenen: Schatten Schwarzer Löcher

This doctoral thesis is about an analytic way to describe the shadow of black holes. As introduction, a survey of the attempts to observe the shadow of the black holes in our Galaxy near Sgr A* and in the neighbouring galaxy M87 is presented. Black holes are described by metrics of the general Plebanski-Demianski class of space-times. All these metrics are axially symmetric and stationary type D solutions to the Einstein-Maxwell equations with a cosmological constant. The space-times are characterized by seven parameters: mass, spin, electric and magnetic charge, gravitomagnetic NUT charge, a so-called acceleration parameter and the cosmological constant. Based on a detailed discussion of the metrics, analytical formulas are derived for the photon regions (regions that contain spherical lightlike geodesics) and for the boundary curve of the shadow as it is seen by an observer at given Boyer-Lindquist coordinates in the domain of outer communication. They make it possible to analyze the dependency of the shadow of a Kerr black hole on the motion of the observer. For all cases, the photon regions and shadows are visualized for various values of the parameters. The analytical formulas are used to find explicit expressions for the horizontal and vertical angular diameters of the shadow. Finally, these values are estimated for the black holes at the centers of our Galaxy and of M87.