Critical Phenomena and Singularity Formation in Gravity

Consider the collapse of spherical shell of matter under its own weight. The dynamics of this process, modeled by Einstein`s equations, can be understood intuitively in terms of the competition between gravitational attraction and repulsive internal forces (due, for instance, to kinetic energy of matter or pressure). If the initial configuration is dilute, then the repulsive forces "win" and the collapsing matter will rebound or implode through the center, and eventually will disperse. On the other hand, if the density of matter is sufficiently large, some fraction of the initial mass will form a black hole. Critical gravitational collapse occurs when the attracting and repulsive forces governing the dynamics of this process are almost in balance, or in other words, the initialconfiguration is near the threshold for black hole formation. The systematic studies of critical gravitational collapse were launched in the early nineties by the seminal paper by Matt Choptuik and since then have been one of the most active areas of research in classical general relativity. Although more than one hundred papers have been devoted to this topic, the mathematical understanding of the observed numerical phenomenology remains a great challenge. In broad terms the purpose of this project is toget a deeper insight into some key aspects of critical gravitational collapse. Technically speaking, the concrete problems we intend to study lie at the interface of classical general relativity, geometric PDE theory, dynamical systems, and bifurcation theory.

Stars, like our sun, have a finite lifetime. After nuclear reactions have come to an end in the interior, the further evolution is complicated, but the final stage is gravitational collapse. i.e. the contraction of matter under its own gravitational force. There are essentially three end states from this evolution: white dwarfs, neutron stars or black holes. The research project studied with the help of simple mathematical matter models collapse to black holes. An important phenomena is called critical collapse, which occurs when the gravitational attracting and the repulsive forces of matter almost balance. In these cases the evolution approaches the singularity in a universal way, independent of the special form of initial data. This universality is often characterized by self-similarity, where all dynamical quantities e.g. the matter density, retain their shape in time except for an overall scale. There are two forms of self-similarity, continuous and discrete, where for the latter the shape returns after a fixed time interval. For the matter model studied (the non-linear su(2) sigma model), we have shown that both type of self-similarity are observed, depending on the strength of the coupling to gravity (coupling constant). One of the main results is that in the transition region, from continuous to discrete self-similarity we found new fine structure. By this it was possible clarify how the discrete solution splits off (bifurcates) from the continues self-similar solution. This finding gave way to an understanding in terms of a bifurcation scenario in dynamical system theory. Another main aspect of this project was to study global aspects of critical collapse. Since collapse process is concentrated in a small region of space, it is important to find out how it would look for "fare away" observers? It was shown numerically the self-similar behavior manifest itself in the outgoing radiation. Moreover, the typical spectrum, i.e. the quasi-normal modes of the object, could be identified in this highly dynamical process. Finally, we were able to find the characteristic late stage behavior for dynamical system i.e. "tails" in the outgoing radiation. Further, analytical and numerical studies focused on the existence and on properties of self-similar solutions in various spacetime dimensions.