Stationary Black Holes and Penrose Inequalities

The "singularity theorems" in General Relativity, proven by S. Hawking and R. Penrose at around 1970, read essentially as follows: A gravitational field considered at a certain time, which contains a "trapped surface", will necessarily develop a "singularity" within a finite time. This is a state in which the theory does not allow further predictions about the future for principal reasons. This "breakdown of physics" should be avoided by the "cosmic censor" - a hypothesis which says that the singularities themselves, as well as their entire domain of influence will remain within a certain domain of spacetime ("black holes"). For a suitable model of the universe (in particular for an "asymptotically flat" one, which approaches flat space at large distances from all trapped surfaces) physics can then continue to pay its usual role exterior to the black holes. These facts entail the following questions: How do trapped surfaces form ? Do they appear at a given time only inside a restricted spatial domain (bounded by "marginally trapped surfaces") ? If yes, what is the evolution in time of these surfaces (called the "apparent horizons") ? How are these apparent horizon located with respect to the boundaries of the black holes (the "event horizons") ? These problems have been investigated in particular under the simplifying assumption that the universe which contains marginally trapped surfaces is stationary (i.e. all fields are time-independent). In absence of matter one believes that the general stationary gravitational field is represented by the model of an axially symmetric, rotating black hole found by R. Kerr in 1963 which could, however, so far only be proven under further simplifying assumptions. A much more realistic requirement is that the time evolution of the vacuum universe with marginally trapped surfaces only approaches a stationary state. In this case one conjectures, in particular, an evolution towards the Kerr spacetime, and would like to describe in particular the location of the aforementioned horizons. Both issues (uniqueness of stationary black holes and approach to the stationary state) are dealt with in this project. Another problem in connection with cosmic censorship is the "Penrose inequality", a hypothesis formulated by R. Penrose in 1973. It reads that the square of the mass of any marginally trapped surface (respectively, black hole) must be greater or equal to its area divided by 16 pi. Under more restrictive requirements, in particular with general marginally trapped surfaces replaced by minimal ones, the inequality was proven by H. Bray and by G. Huisken and T. Ilmanen in 2001. In its general version the Penrose inequality is at present under investigation by several groups and also subject of this project. Here we will primarily focus on an idea developed in 2006 by H. Bray, S. Hayward, M. Mars and the present applicant. One also conjectures a more general version of the Penrose inequality, which should apply to rotating minimal surfaces or rotating black holes and involve their angular momenta explicitly as well, with equality holding only for the Kerr spacetime. We will try to prove this version as well in the present project. Since mass, area and angular momentum are the most important (and at least theoretically measurable) parameters of marginally trapped sufarces and black holes, such simple relations between them are of direct physical interest. For notions like "the mass of a marginally trapped surface" there is no unique definition, however - we plan here to apply the concepts developed by R. Bartnik in 1989.