Dynamics Of Nonlinear Matter Fields in General Relativity

The aim of the proposed project is to study the dynamics of nonlinear matter fields coupled to gravity via a combination of analytical and numerical techniques, focusing on the existence of solitons and the formation of black holes. This includes a comparison of the qualitatively different forms of dynamical behavior, dependent on whether the cosmological constant is positive or zero. In the last years, following numerical work of M. Choptuik, a burst of activity has concentrated on critical phenomena at the threshold of black hole formation. Two different types of critical behavior have been discovered so far. Formally these correspond to type I and type II phase transitions of statistical physics. In this project, we plan to mainly concentrate on nonlinear sigma-models coupled to gravity. These models are interesting due to the presence of a dimensionless parameter, which enters into the theory nontrivially. Open questions, that we want to deal with in the example of our models are the genericity of the phenomena found so far, in particular regarding the coexistence of type I and II critical behavior, the appearance of discrete versus continuous self-similarity in type I, and the possible discovery of further types of critical solutions. Technically, this requires the numerical solution of nonlinear partial differential equations. The plan of this project is as follows: 1. In the first step we plan to extend our previous work on solitons of nonlinear sigma-model on the de Sitter background to a detailed analysis of static solutions of the coupled "Einstein-sigma-model" system with a positive cosmological constant. 2. In the next step we plan to turn to our main issue, namely the full time evolution problem in spherical symmetry. Our previous results suggest that both type I and type II critical behavior on the coupling constant. The appearance of bifurcation structures is likely, possibly leading to the discovery of new types of critical behavior. 3. Partly as a further step, partly as work done in parallel with the precious steps, we would like to extend our investigations to compare with different matter modes, e.g. sigma-models with a potential term or perfect fluids. What would happen for fluids with equations of state which do admit stable static regular solutions, like the ideal Fermi gas? It would be interesting to extend the work done on the formation of black holes to this case. One interesting question is whether black holes can form below the Chandrasekhar limit. The results are expected to be relevant to a wide range of central questions in general relativity, such as the existence of soliton solutions, critical phenomena at the threshold of black hole formation, the interior structure of black holes, the cosmic censorship conjecture and the cosmic no hair conjecture.

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 a phenomena called critical collapse, where one analyses the boundary at which matter undergoes collapse to a black hole and matter configurations which do not collapse. Since the work of M. Choptuik it is known, that at the threshold of black hole formation the system shows similarities to phenomena observed in phase transitions: universality, scaling and self-similarity. Of special interest is the so called 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). In the transition region, from continuous to discrete self-similarity we found a new phenomena which we call "episodic self-similarity": continuous self-similar phases are followed by discrete ones for several stages in the evolution. We were able to identify this phenomenon as a global bifurcation, where the discrete self-similar solution splits (bifurcates) from the continuous solution as a function of the coupling. We expect that this outcome, which was obtained by analytic methods as well as numerical simulations, will lead to a better understanding of critical phenomena for systems undergoing gravitational collapse in the universe.