Gauge-invariant and coordinate-independent perturbations of stellar collapse: The interior

Small nonspherical perturbations of a spherically symmetric but time-dependent background spacetime can be used to model situations of astrophysical interest, for example the production of gravitational waves in a supernova explosion. We allow for perfect fluid matter with an arbitrary equation of state p=p(ρ,s), coupled to general relativity. Applying a general framework proposed by Gerlach and Sengupta, we obtain covariant field equations, in a 2+2 reduction of the spacetime, for the background and a complete set of gauge-invariant perturbations, and then scalarize them using the natural frame provided by the fluid. Building on previous work by Seidel, we identify a set of true perturbation degrees of freedom admitting free initial data for the axial and for the $l>~2$ polar perturbations. The true degrees of freedom are evolved among themselves by a set of coupled wave and transport equations, while the remaining degrees of freedom can be obtained by quadratures. The polar $l=0,1$ perturbations are discussed in the same framework. They require gauge fixing and do not admit an unconstrained evolution scheme.