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.

This is a project on Mathematical Relativity meaning that we are dealing with mathematical results which are applied to Einsteins field equations, or with mathematical methods which are specially developed to solve these equations. The results of the project consist in essence of the following two parts which will be explained below:1.) Construction of a class of initial data for rotating cosmological models.2.) Proofs of inequalities between the area of a black hole and other characteristic parameters, in particular its angular momentum and its electric and magnetic charges.1.) The present picture of the cosmic evolution is based on a concentrated initial state (big bang) which has led to the present state in about 14 billion years. The most probable further evolution is a gradually accelerating, eternal expansion. The simplest model which explains most observations, in particular this accelerated expansion, are homogeneous solutions of Einsteins equations with a so-called cosmological constant. Since little is known about the big bang, it is simpler to start evolution - in the model - from later initial data, in particular from the present time. This part of the project is concerned with the construction of a class of initial data in the framework of the so-called conformal method, whose foundations are due to Lichnerowicz (1944). We focus on rotating cosmologies. In the simplified model of so-called maximal data the problem reduces to solving a single, semilinear elliptic equation. However, the latter is so complicated that a satisfactory proof of existence of its solutions was only achieved in 2014 by Premoselli.2.) The objects which we call here black holes are actually so-called stable marginally trapped surfaces. Still simplified, these are boundaries of domains in which the gravitational field, at a certain moment of time, is so strong that neither matter nor light can escape. The most important parameter of such an object is its surface area. Moreover, for axially symmetric, rotating black holes one can define an angular momentum, and if an electromagnetic field is present, the hole can carry electric and magnetic charges. These parameters have to satisfy inequalities. Intuitively, one can imagine this by noting that the presence of angualar mometum and charges require a corresponding amount of energy, which in turn entails a certain minimal size, in particular a minimal surface area. The contributions of the present project now focus on the influence of the cosmological constant (as explained in point 1.) on black holes in general, and on such inequalities in particular.