Slowly rotating super-compact Schwarzschild stars

Abstract

The Schwarzschild interior solution, or `Schwarzschild star', which describes a spherically symmetric homogeneous mass with constant energy density, shows a divergence in pressure when the radius of the star reaches the Schwarzschild-Buchdahl bound. Recently Mazur and Mottola showed that this divergence is integrable through the Komar formula, inducing non-isotropic transverse stresses on a surface of some radius $R_{0}$. When this radius approaches the Schwarzschild radius $R_{s}=2M$, the interior solution becomes one of negative pressure evoking a de Sitter spacetime. This gravitational condensate star, or gravastar, is an alternative solution to the idea of a black hole as the ultimate state of gravitational collapse. Using Hartle's model to calculate equilibrium configurations of slowly rotating masses, we report results of surface and integral properties for a Schwarzschild star in the very little studied region $R_{s}<R<(9/8)R_{s}$. We found that in the gravastar limit, the angular velocity of the fluid relative to the local inertial frame tends to zero, indicating rigid rotation. Remarkably, the normalized moment of inertia $I/MR^2$ and the mass quadrupole moment $Q$ approach to the corresponding values for the Kerr metric to second order in $\Omega$. These results provide a solution to the problem of the source of a slowly rotating Kerr black hole.