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.

In this paper we used the theory of adiabatic radial oscillations developed by Chandrasekhar to study the conditions for dynamical stability of constant energydensity stars, or Schwarzschild stars, in the unstudied ultra compact regime beyond the Buchdahl limit, that is, for configurations with radius R in the range R S < R < (9/8)R S , where R S is the Schwarzschild radius of the star. These recently found analytical solutions exhibit a negative pressure region in their centre and, in the limit when R → R S , the full interior region of the star becomes filled with negative pressure. Here we present a systematic analysis of the stability of these configurations against radial perturbations. We found that, contrary to the usual expectation found in many classical works, the ultra compact Schwarzschild star is stable against radial oscillations. We computed values of the critical adiabatic index γ c for several stellar models with varying radius R/R S and found that it also approaches a finite value as R/R S → 1.

One of the macroscopically measurable effects of gravity is the tidal deformability of astrophysical objects, which can be quantified by their tidal Love numbers. For planets and stars, these numbers measure the resistance of their material against the tidal forces, and the resulting contribution to their gravitational multipole moments. According to general relativity, nonrotating deformed black holes, instead, show no addition to their gravitational multipole moments, and all of their Love numbers are zero. In this paper we explore different configurations of nonrotating compact and ultracompact stars to bridge the compactness gap between black holes and neutron stars and calculate their Love number k 2. We calculate k 2 for the first time for uniform density ultracompact stars with mass M and radius R beyond the Buchdahl limit (compactness M/R > 4/9), and we find that k 2 → 0+ as M/R → 1/2, i.e., the Schwarzschild black hole limit. Our results provide insight on the zero tidal deformability limit and we use current constraints on the binary tidal deformability Λ ̃ from GW170817 (and future upper limits from binary black hole mergers) to propose tests of alternative models.

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We complete the stability study of general relativistic spherically symmetric polytropic perfect fluid spheres, concentrating attention to the newly discovered polytropes containing region of trapped null geodesics. We compare the methods of treating the dynamical stability based on the equation governing infinitesimal radial pulsations of the polytropes and the related Sturm-Liouville eigenvalue equation for the eigenmodes governing the pulsations, to the methods of stability analysis based on the energetic considerations. Both methods are applied to determine the stability of the polytropes governed by the polytropic index n in the whole range 0 < n < 5, and the relativistic parameter σ given by the ratio of the central pressure and energy density, restricted by the causality limit. The critical values of the adiabatic index for stability are determined, together with the critical values of the relativistic parameter σ. For the dynamical approach we implemented a numerical method which is independent on the choice of the trial function, and compare its results with the standard trial function approach. We found that the energetic and dynamic method give nearly the same critical values of σ. We found that all the configurations having trapped null geodesics are unstable according to both methods.