Defrosting frozen stars: Spectrum of nonradial 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 nonradial 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 nonradial oscillations is obtained to leading order in an expansion in terms of $\ensuremath{\gamma}$, which is the small relative deviation away from maximally negative radial pressure. We find that the sound velocity of the modes is nonrelativistic, and proportional to $\ensuremath{\gamma}$, while their lifetime is parametrically long, proportional to $1/{\ensuremath{\gamma}}^{2}$. This result was anticipated by previous discussions on the collapsed polymer model, whose strongly nonclassical 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.