Propagation of Axial and Polar Gravitational Waves in Kantowski-Sachs Universe

Abstract

In this paper, we apply the Regge-Wheeler formalism in our study of axial and polar gravitational waves in Kantowski-Sachs universe. The background field equations and the linearised perturbation equations for axial and polar modes are derived in presence of matter. To find the analytical solutions, we analyse the propagation of waves in vacuum spacetime. The background field equations in absence of matter are first solved by assuming that the expansion scalar Θ to be proportional to the shear scalar σ (so that the metric coefficients are given by the relation a = b n , where n is an arbitrary constant). Using the method of separation of variables, the axial perturbation parameter h 0 (t, r) is obtained from its wave equation. The other perturbation h 1 (t, r) is then determined from h 0 (t, r). We observe that the anisotropy of the background spacetime is responsible for the damping of the axial waves propagating through it. On the other hand, the polar perturbation equations are much more involved compared to their FLRW counterparts, as well as to the axial perturbations in Kantowski-Sachs background. The polar equations contain complicated couplings among the perturbation variables. In both the axial and polar cases, the radial and temporal solutions for the perturbations separate out as product. The perturbation equations in presence of matter show that the axial waves can cause perturbations only in the azimuthal velocity of the fluid without deforming the matter field. But the polar waves must perturb the energy density, the pressure and also the non-azimuthal components of the fluid velocity. The propagation of axial and polar gravitational waves in Kantowski-Sachs and Bianchi I spacetimes is found to be more or less similar in nature.