Marginally outer trapped tubes

Einsteins Theory of General Relativity (GR) is the best physical theory of space, time, matter and energy up to date. It explains gravity via the curvature of spacetime, and it is formulated mathematically via Einsteins equations a system of partial differential equations. Among other effects, GR predicts the existence of black holes regions of space where gravity is so strong that neither material bodies nor light can escape. The boundaries of such regions are called horizons, but in the sequel we use the precise mathematical term marginally outer trapped surfaces (MOTS). In its time-evolution a MOTS forms a marginally outer trapped tube (MOTT). Upon approach, two black holes merge, which was registered for the first time in 2015 via gravitational waves which are emitted in this process. Recording this on earth very weak signal was only possible because one knew what to look for via model calculations (which make use of MOTS). These calculations show in particular the following: When two MOTS approach each other, a new enclosing MOTS forms even before the original ones touch. This new MOTS splits instantaneously into two concentric MOTS, both enclosing the two original MOTS. The the sequel there arises a complicated plaint of MOTS (including ones with self-intersections) which (probably) all merge in the long run. The main aim of this project is to improve the understanding of the time-evolution of MOTS. A MOTS is described mathematically via a quasilinear elliptic differential equation. To determine the signal of gravitational waves which is emitted upon merger it suffices to solve this equation numerically - roughly speaking, this is intelligent guesswork via a computer, which can only lead to approximate statements. In contrast, this project aims at proving rigorous, generally valid statements which should substantiate the numerical simulations. This requires, in particular, the application of bifurcation theory, which describes qualitative changes of the behaviour of solutions of differential equations depending on parameters, when the latter reach a treshold a suitable such parameter in the merger case is the distance between the two MOTS. A further part of the project concerns the mass of MOTS. In General Relativity there is no unique, universally applicable definition of the mass of an extended domain in space (which can contain matter and/or black holes). An interesting definition of mass, which can also be applied to black holes, was suggested by the mathematician Robert Bartnik. We plan to investigate this definition in particular for axially symmetric MOTS, applying mainly methods of functional analysis.