Defrosting frozen stars: spectrum of non-radial oscillations

The frozen star model describes a type of black hole mimicker; that is, a regular, horizonless, ultracompact object that behaves just like a Schwarzschild black hole from an external-observer's perspective. In particular, the frozen star is bald, meaning that it cannot be excited. To mimic the possible excitations of the frozen star, it needs to be "defrosted" by allowing deviations from the maximally negative radial pressure and vanishing tangential pressure of the fluid sourcing the star. Here, we extend a previous study on non-radial oscillations of the defrosted star by considering, in addition to the fluid modes, the even-parity metric perturbations and their coupling to the fluid modes. At first, general equations are obtained for the perturbations of the energy density and pressure along with the even-parity perturbations of the metric for a static, spherically symmetric but otherwise generic background with an anisotropic fluid. This formal framework is then applied to the case of a defrosted star. The spectrum of non-radial oscillations is obtained to leading order in an expansion in terms of $\gamma$, which is the small relative deviation away from maximally negative radial pressure. We find that the sound velocity of the modes is non-relativistic, and proportional to $\gamma$, while their lifetime is parametrically long, proportional to $1/\gamma^2$. This result was anticipated by previous discussions on the collapsed polymer model, whose strongly non-classical interior is argued to provide a microscopic description of the frozen and defrosted star geometries. Our results will serve as a starting point for calculating the spectrum of emitted gravitational waves from an excited frozen star.