Abstract:
We investigate spherically symmetric continuously self-similar (CSS) solutions in the SU(2) sigma model coupled to gravity. Using mixed numerical and analytical methods, we provide evidence for the existence (for small coupling) of a countable family of regular CSS solutions. This fact is argued to have important implications for the ongoing studies of black hole formation in the model.

Abstract:
We present a detailed analytical study of spherically symmetric self-similar solutions in the SU(2) sigma model coupled to gravity. Using a shooting argument we prove that there is a countable family of solutions which are analytic inside the past self-similarity horizon. In addition, we show that for sufficiently small values of the coupling constant these solutions possess a regular future self-similarity horizon and thus are examples of naked singularities. One of the solutions constructed here has been recently found as the critical solution at the threshold of black hole formation.

Abstract:
We prove existence of a countable family of spherically symmetric self-similar wave maps from 3+1 Minkowski spacetime into the 3-sphere. These maps can be viewed as excitations of the ground state wave map found previously by Shatah. The first excitation is particularly interesting in the context of the Cauchy problem since it plays the role of a critical solution sitting at the threshold of singularity formation. We analyze the linear stability of our wave maps and show that the number of unstable modes about a given map is equal to its excitation index. Finally, we formulate a condition under which these results can be generalized to higher dimensions.

Abstract:
The stability of transparent spherically symmetric thin shells (and wormholes) to linearized spherically symmetric perturbations about static equilibrium is examined. This work generalizes and systematizes previous studies and explores the consequences of including the cosmological constant. The approach shows how the existence (or not) of a domain wall dominates the landscape of possible equilibrium configurations.

Abstract:
In recent years, a number of alternative theories of gravity have been proposed as possible resolutions of certain cosmological problems or as toy models for possible but heretofore unobserved effects. However, the implications of such theories for the stability of structures such as stars have not been fully investigated. We use our "generalized variational principle", described in a previous work, to analyze the stability of static spherically symmetric solutions to spherically symmetric perturbations in three such alternative theories: Carroll et al.'s f(R) gravity, Jacobson & Mattingly's "Einstein-aether theory", and Bekenstein's TeVeS. We find that in the presence of matter, f(R) gravity is highly unstable; that the stability conditions for spherically symmetric curved vacuum Einstein-aether backgrounds are the same as those for linearized stability about flat spacetime, with one exceptional case; and that the "kinetic terms" of vacuum TeVeS are indefinite in a curved background, leading to an instability.

Abstract:
This dissertation deals with singularity formation in spherically symmetric solutions of the hyperbolic Yang Mills equations in (4+1) dimensions and in spherically symmetric solutions of C P^1 wave maps in (2+1) dimensions. These equations have known moduli spaces of time-independent (static) solutions. Evolution occurs close to the moduli space of static solutions. The evolution is modeled numerically using an iterative finite differencing scheme, and modeling is done close to the adiabatic limit, i.e., with small velocities. The stability of the numerical scheme is analyzed and growth is shown to be bounded, yielding a convergence estimate for the numerical scheme. The trajectory of the approach is characterized, as well as the shape of the profile at any given time during the evolution.

Abstract:
A stability analysis of a spherically symmetric star in scalar-tensor theories of gravity is given in terms of the frequencies of quasi-normal modes. The scalar-tensor theories have a scalar field which is related to gravitation. There is an arbitrary function, the so-called coupling function, which determines the strength of the coupling between the gravitational scalar field and matter. Instability is induced by the scalar field for some ranges of the value of the first derivative of the coupling function. This instability leads to significant discrepancies with the results of binary-pulsar-timing experiments and hence, by the stability analysis, we can exclude the ranges of the first derivative of the coupling function in which the instability sets in. In this article, the constraint on the first derivative of the coupling function from the stability of relativistic stars is found. Analysis in terms of the quasi-normal mode frequencies accounts for the parameter dependence of the wave form of the scalar gravitational waves emitted from the Oppenheimer-Snyder collapse. The spontaneous scalarization is also discussed.

Abstract:
In this paper, we present two observations about static spherically symmetric solutions of the Einstein-Klein-Gordon equations. The first is a comment extending the well-known result of the existence of static states (i.e. standing wave solutions) of the Einstein-Klein-Gordon equations. The second more important observation shows that, in the low field limit, the mass profiles of these static states lie along hyperbolas of constant $\Upsilon$, the fundamental constant of the Einstein-Klein-Gordon equations.

Abstract:
Certain steady states of the Vlasov-Poisson system can be characterized as minimizers of an energy-Casimir functional, and this fact implies a nonlinear stability property of such steady states. In previous investigations by Y. Guo and the author stability was obtained only with respect to spherically symmetric perturbations. In the present investigation we show how to remove this unphysical restriction.

Abstract:
We discuss spherically symmetric solutions for point-like sources in Lorentz-breaking massive gravity theories. This analysis is valid for St\"uckelberg's effective field theory formulation, for Lorentz Breaking Massive Bigravity and general extensions of gravity leading to an extra term $-Sr^{\gamma}$ added to the Newtonian potential. The approach consists in analyzing the stability of the geodesic equations, at the first order (deviation equation). The main result is a strong constrain in the space of parameters of the theories. This motivates higher order analysis of geodesic perturbations in order to understand if a class of spherically symmetric Lorentz-breaking massive gravity solutions, for self-gravitating systems, exists. Stable and phenomenologically acceptable solutions are discussed in the no-trivial case $S\neq 0$.