Eindeutige Fortsetzung durch kompakte Hyperflächen
Zusammenfassung der Projektergebnisse
My research during the DFG financed Postdoc at Stanford University concerns the theory of black holes in Einstein’s theory of General Relativity. General Relativity actually has a completely satisfactory mathematical formulation in the language of Differential Geometry, which is one of the main research fields in mathematics. Many parts of Differential Geometry have in fact been influenced by Einstein’s work. One talks about “Einstein spaces” even in geometry which is far away from General Relativity. Black holes are arguably (perhaps together with the Big Bang) the most intriguing predictions of General Relativity. In recent years, black holes have been one of the main focuses of modern physics, with emphasis on the observation of gravitational waves emanated from spiralling black hole mergers and the recently taken pictures of a black hole. My research at Stanford University was about fundamental theoretical questions about black holes. It is widely believed that black holes are unique in the sense that there are no other reasonable models than the famous Kerr and Kerr-de Sitter models which can describe black holes in a non-expanding and expanding universe, respectively. Together with Klaus Kr¨ncke o at the University of Hamburg, we proved that one can compute the full asymptotic expansion of the gravitational field at the event horizon of a black hole in terms of simple geometric data at the horizon. This contributes to our understanding of how a black hole must look near an event horizon and is therefore relevant to the uniqueness question for black holes. It is also widely believed that black holes are dynamically stable, i.e. that small perturbations of the gravitational field radiate either into the black hole or off to infinity and that the black hole therefore settles down to a stationary black hole after a long time. My joint work with a Andr´s Vasy at Stanford University is a contribution to this question in the a case of expanding universes. In fact, Peter Hintz and Andr´s Vasy have already proven that slowly rotating black holes in expanding spacetimes are stable. We have now taken the first main step towards generalizing their results to fast rotating black holes. The research findings at Stanford University, joint with my collaborators, are based on the following three novel surprises: • There is in fact a choice in how to think about the time-invariance in a Kerr-de Sitter spacetimes. • The second main idea are certain new identities, which surprisingly well describe the trapped photons in subextremal Kerr-de Sitter spacetimes. This is crucial for all analysis of such black holes (previous results required restrictions on the angular momentum or on the expansion of the black hole). • One can compute the full asymptotic expansion of the gravitational field at the event horizon of a black hole in terms of simple geometric data at the horizon. I believe that these surprising observations are central for the theory of black holes and will be a central ingredient in forthcoming research.
