Cordas cosmicas e vortices em relatividade geral : pertubações radiais de soluções estaticas e estacionarias




A linearized theory of the dynamics of small perturbations of certain cylin­drically symmetric systems in general relativity is presented. Firstly, we deal with straight, static cosmic strings of finite thickness in the U(l) gauge theory coupled to the Einstein equations. Taking into ac­count the asymptotic behavior of the solution and certain integral identities, we develop specific numerical methods to solve the system of ordinary dif­ferential equations which describes this type of string. A method based on quintic spline collocation turns out to be very effective, allowing us to refine the published values for the string angular deficit and linear energy density. Assuming that the perturbations have cylindrical symmetry, the field equa­tions are then linearized about the static string solution. Using the method of normal modes, the problem is reduced to a system of ordinary differential equations for the spatial structure of the normal modes. Algorithms for the numerical solution of generalized value problems provide evidence that the string is stable against radial perturbations. We show that the string has a continuous spectrum of neutral modes, in which the perturbations of the metric and the fields which make up the string behave as stationary waves. The spectrum may be divided in regions of high, intermediate and low frequencies the number of independent normal modes per frequency is different in each region. Employing the static solutions previously obtained in terms of splines for the computation of the coefficients in the perturbation equati­ons, we obtain numerical solutions for the neutral modes by the Runge-Kutta method. It is argued that the concepts of quasi-normal modes and complex outgoing modes, commonly used in the literature on stellar models, are not particularly useful in the description of the perturbations of U (1) strings. In the second part of this work, we introduce a new class of stationary solutions of the Einstein equations, which represent self-gravitating vortices in an ideal relativistic gas with an uniform temperature distribution and an arbitrary distribution of angular velocities about the axis of symmetry. Examples of such solutions are determined numerically by a shooting method. With the assumption of uniform temperature, it is possible to derive an inte­gral expression for the linear density of the vortex, wherein the rotation contributions appear explicitly. Furthermore, we show that the non-linear boundary-value problem which defines the vortex is invariant under a certain scaling of the coordinates and fields. In the study of radial perturbations of these systems, we restrict our attention to the simplest case, namely non­rotating polytrope. Performing an analysis which is analogous to the one for U (1) strings, we find numerical evidence that the static polytrope is sta­ble with respect to radial perturbations. The structure of the spectrum of neutral modes is much simpler than that of the U(l) string, consisting of a single mode per frequency; each mode exhibits coupled stationary gravitati­onal and acoustic waves. The curves of asymptotic amplitude and phase of the gravitational component of the normalized normal mode as a function of frequency display characteristics which are normally associated with the existence of a discrete spectrum of complex outgoing modes; however, it is argued that there are no modes in which both the acoustic and the gravitati­onal perturbations are outgoing. A limitation of the present linearized theory is its inability to describe the final evolution of a perturbation which propa­gates outwards; this limitation is explained by the formation of shock waves (a non-linear phenomenon) in the outer, rarefied layers of the poly trope


einstein relatividade geral (fisica) equções de equções diferenciais - soluções numericas perturbação (matematica)

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