Construction of vacuum radiating space-times

Finding solutions to Einstein`s (vacuum) field equations with physically relevant properties is one of the most important tasks to improve our understanding of the implications of the general theory of relativity. A systematic method to do that is to provide suitable data on certain initial hypersurfaces from which the existence of such a solution can be inferred. In view of the current efforts to detect gravitational waves, it is of key importance to understand space-times describing isolated gravitational systems. The current understanding of such models is based on the hypothesis of a specific fall-off behavior of the gravitational field far away from sources. In spite of decades of efforts by many researchers, the verification of this hypothesis for physically relevant initial data is still lacking. One of the approaches to the issues at hand is provided by the so-called asymptotic initial value problems. Here the space-time is rescaled in such a way that infinity is represented by a set of finite points, which form hypersurfaces. Prescribing radiation data on such a hypersurface, which correspond to gravitational waves coming in from infinity, one is led to vacuum solutions of Einstein`s field equations where the gravitational field shows such a fall-off behavior, at least into the past. The main object of this project is to analyze under which conditions the emerging space-time exhibits the radiative behavior into the future as well. This would provide a powerful tool to construct physically relevant global solutions of the Einstein equations, and it would give new insights to what extent the required fall-off behavior of the gravitational field is compatible with Einstein`s field equations.

Einsteins (vacuum) field equations provide the heart of general relativity. Finding solutions to these equations with physically relevant properties is one of the most important aspects of research. With regard to the detection of gravitational waves, it is of key importance to understand asymptotically flat space-times which describe isolated gravitational systems. The notion of asymptotic flatness is based on the hypothesis of a specific fall-off behavior of the gravitational field far away from sources, although a verification of this hypothesis for physically relevant initial data is still lacking. An elegant approach to this issue at hand is to require that the spacetime can be rescaled in such a way that infinity is represented by a set of finite points, which form regular surfaces J - and J + , where gravitational waves enter and leave the spacetime, respectively. A powerful method to generate solutions to Einsteins field equations is in terms of initial value problems where suitable data are provided on a certain initial surface from which the existence of a solution can be inferred. Prescribing radiation data on J - , which correspond to gravitational waves coming in from infinity, one is led to vacuum solutions of Einsteins field equations where the gravitational field shows an asymptotically flat radiative fall-off behavior into the past. An important question is under which conditions the emerging space-time exhibits such a radiative behavior into the future, as well. The main focus of this project was to analyze an intermediate step towards this goal, which is to analyze the intermediate region spatial infinity I , which connects J - and J + , and where one wants the gravitational field to show an asymptotically flat behavior, as well. The main result of this project is that this behavior is deeply intertwined with the behavior of the radiation field when approaching I.