We employ the minimal geometric deformation approach to gravitational decoupling (MGDdecoupling) in order to generate an exact anisotropic and non-uniform version of the ultracompact Schwarzschild star, or gravastar', proposed by Mazur and Mottola. This new system represents an ultracompact configuration of radius RS = 2 M whose interior metric can be matched smoothly to a conformally deformed Schwarzschild exterior. Remarkably, the model satisfies some of the basic requirements to describe a stable stellar model, such as a positive density everywhere and decreasing monotonously from the centre, as well as a non-uniform and monotonic pressure.

AcknowledgmentsI am grateful to the universe, for its harmony. The project is the study of integral and surface properties of slowly rotating homogeneous masses in the gravastar limit R → R s , where R s is the Schwarzschild radius. For this purpose we followed the perturbative method proposed by Hartle in 1967. In this model, the relativistic equations of structure for a slowly rotating star were derived at second order in the angular velocity Ω. An interesting, and educational, application of this model was investigated by Chandrasekhar and Miller. I am indebted to the Department of Physics andIn their approach, they solved numerically the structure equations of a homogeneous star (constant energy density) up to the Buchdahl bound (9/8)R s . Based on this work, our objective was to investigate the interesting region below the Buchdahl bound R s < R < (9/8)R s , which has not been studied previously in the literature.Our results were astonishing. We found that the surface properties and quadrupole mass moment approach the values corresponding to those of the Kerr metric when expanded at second order in angular momentum. This remarkable result provides a long sought solution to the problem of the source of rotation in the Kerr spacetime.iv

The Schwarzschild star is an ultracompact object beyond the Buchdahl limit, which has Schwarzschild geometry outside its surface and positive pressure in the external layer which vanishes at the surface. Recently it has been shown that the Schwarzschild star is stable against spherically symmetric perturbations. Here we study arbitrary axial nonspherical perturbations, and show that the observable quasinormal modes can be as close to the Schwarzschild limit as one wishes, what makes the Schwarzschild star a very good mimicker of a black hole. The decaying time-domain profiles prove that the Schwarzschild star is stable against nonspherical perturbations as well. Another peculiar feature is the absence of echoes at the end of the ringdown. Instead we observe a nonoscillating mode which might belong to the class of algebraically special modes. At asymptotically late times, Schwarzschildian power-law tails dominate in the signal.