r - qc / 0 50 70 09 v 1 2 J ul 2 00 5 Staticity , Self - Similarity and Critical Phenomena in a Self

The aim of this work is to study certain aspects of the dynamics of a class of self-gravitating non-linear matter fields. In particular we concentrate on soliton solutions (in the presence of a positive cosmological constant), self-similar solutions and the formation of black holes. The dynamics of initial data at the threshold of black hole formation are characterized by phenomena (so-called critical phenomena) including scaling, selfsimilarity (resp. staticity) and universality. In this work we concentrate on SU(2) σ models coupled to gravity in spherical symmetry. These models are interesting due to a dimensionless parameter – the coupling – which enters the theory non-trivially. The aim is to investigate how critical phenomena – in particular the critical solution and the scaling exponent – depend on the coupling. We use two essentially different methods to study the threshold behavior: Making use of the symmetry both discrete (DSS) and continuous (CSS) self-similar solutions are constructed (numerically) by solving boundary value problems (Chapter 4). The stability of these solutions is studied. For small couplings, reproducing results of Bizon et al., we find a discrete one-parameter family of CSS solutions. Of particular interest is the first CSS excitation. For large couplings we construct a DSS solution. Both solutions have one unstable mode. We conjecture, that the DSS solution bifurcates from the CSS solution in a homoclinic loop bifurcation at some value of the coupling constant. The second method consists of evolving one parameter families of initial data numerically (Chapter 5). By a bisection search the initial data are fine-tuned such that they are close to the threshold. The critical solution then is determined by the intermediate asymptotics of near-critical data. The scaling exponent is determined from the black hole mass as a function of the parameter in the initial data. Our results are in very good agreement with the results on the self-similar solutions we obtained above. For small couplings the critical solution is CSS, for large couplings it is DSS and for intermediate couplings we find a new transition from CSS to DSS in the critical solution, which shows “episodic CSS”. This transition is consistent with the hypothesis of the homoclinic loop bifurcation. In addition this work also contains results on static solutions of the model in the presence of a positive cosmological constant Λ (Chapter 3).