On the quasinormal modes in generalized Nariai Spacetimes

Abstract

Quasinormal modes are eigenmodes of dissipative systems. For instance, if a spacetime with an event or cosmological horizon is perturbed from its equilibrium state, quasinormal modes arise as damped oscillations with a spectrum of complex frequencies, called quasinormal frequencies, that does not depend on the details of the excitation. In fact, these frequencies depend just on the charges which define the geometry of the spacetime in which the perturbation is propagating, such as the mass, electric charge, and angular momentum. Quasinormal modes have been studied for a long time and the interest in this topic has been renewed by the recent detection of gravitational waves, inasmuch as these are the configurations that are generally measured by experiments. Mathematically, this discrete spectrum of quasinormal modes stems from the fact that certain boundary conditions must be imposed to the physical fields propagating in such a spacetime. In this thesis, we shall consider a higher-dimensional generalization of the charged Nariai spacetime that is comprised of the direct product of the two-dimensional de Sitter space, dS2, with an arbitrary number of two-spheres, S2, and investigate the dynamics of spin-s field perturbations for s = 0, 1/2, 1 and 2. As a first step, we shall attain the separability of the equations of motion for each perturbation type in such a geometry and its reduction into a Schrödinger-like differential equation whose potential is contained in the Rosen-Morse class of integrable potentials, which has the so-called Pöschl-Teller potential as a particular case. A key step in order to attain this separability is to use a suitable basis for the angular functions depending on the rank of the tensorial degree of freedom that one needs to describe. Here we define such a basis, which is a generalization of the tensor spherical harmonics that is suited for spaces that are the product of several spaces of constant curvature. Finally, with the integration of the Schrödinger-like differential equation at hand, the boundary conditions leading to quasinormal modes are analyzed and the quasinormal frequencies are analytically obtained.