This PhD thesis is devoted to the study of the non-radial oscillations of slowly rotating neutron stars, in the framework of General Relativity. We studied these oscillations using the linear perturbation theory. At first we constructed the background stellar model. Then we introduced small perturbations, linearized the Einstein equations and studied the response of the perturbed system by solving the equations describing it, as an initial value problem or as it is usually said by many authors, in the time domain. The basic assumptions that we made in this study are the following, ² The star is a perfect fluid and has zero temperature, ² The star is rotating uniformly with angular velocity and this rotation is a considered as perturbation of the non-rotating stellar model, ² The magnetic field of the star is negligible. All the above assumptions follow from observational facts. Subsequently we introduced small perturbations on this stellar model both on the fluid of the star and the spacetime around it. The perturbation functions are assumed to be of small order ± and in general are function of all the variables of the problem, i.e. t, r, µ, A. We then splitted the perturbation functions into radial and angular parts using spherical harmonics. This is allowed because of the spherical symmetry of the background. We also introduced a dimensionless parameter " = =K, that is the ratio of the angular velocity of the star, over the angular velocity at the mass shedding limit. By assuming that this parameter is small " << 1, i.e. the star is rotating slowly, we calculated the Einstein equations (3.33) and the equations of motion of the fluid (3.38) and linearized them to both parameters " and ±. By integrating the above equations over solid angles we eliminated the angular dependence of the perturbation functions. This way we arrived to a system of partial differential equations (PDEs) of time t and space r that describes the small non-radial perturbations of a slowly rotating neutron star. Having the above system of equations in hand, we tried to solve it numerically and calculate the eigenfrequencies of the system. In order to understand them better we have splitted them into two basic parts, as is common in the bibliography. The part that describes the fluid perturbations and the part that describes the spacetime perturbations. In the literature is common to use the term “Cowling Approximation” when the spacetime perturbations are neglected, and the term “Inverse Cowling Approximation (ICA)” when the perturbations of the stellar fluid are neglected. The first step was to study the part that describes the fluid perturbations, and extract the stellar oscillation modes. By studying this part of the problem we have gained useful information about f, p and r modes of slowly rotating neutron stars. We also studied a interesting phenomenon that appears in this level of approximation, i.e. existence of a continuous spectrum. The continuous spectrum has significant influence on the appearance and the life of the normal modes of the star, for different spherical harmonic indices l. As a second step we have re-written the equations that describe the perturbations of a slowly rotating neutron star, in a new gauge, that has been used up to now only for non-rotating stars. The motivation was that the already existing equations were not very well posed for numerical evolution, due to the existence of mixed second order spatial and temporal derivatives of the perturbation functions. Indeed the equations that we produced in the new gauge seemed more appropriate for numerical evolutions, and they could be rather easily transformed into a first order system. Subsequently we have turned to the old system of equations in the widely used Regge-Wheeler gauge. By redefinition of new variables and lengthy calculations we have managed to re-write it in first order form of evolution equations. As a test for the numerical stability of this system we evolved the part that describes the spacetime perturbations and showed that is numerically stable. For the first time we also calculated frequencies of w-modes for both polytropic and uniform density equations of state. Finally, in order to check the limits of our linear slow rotation approximation for the fluid modes, we added to the perturbed equations of motion of the fluid (3.38) the second order terms in rotation O("2). We then studied the improvement of the accuracy in the calculation of the background model i.e. the mass and the radius. We concluded our study by examining the way the eigenfrequencies of the various oscillation modes is influenced by the inclusion of the second order terms.
