Spherically symmetric models with pressure: separating expansion from contraction and generalizing TOV condition

We investigate spherically symmetric perfect fluid spacetimes and discuss the existence and stability of a dividing shell separating expanding and collapsing regions. We perform a 3+1 splitting and obtain gauge invariant conditions relating the intrinsic spatial curvature of the shells to the ADM mass and to a function of the pressure which we introduce and that generalises the Tolman-Oppenheimer-Volkoff equilibrium condition. We analyse the particular cases of the Lema\^itre-Tolman-Bondi dust models with a cosmological constant as an example of a $\Lambda$-CDM model and its generalization to contain a central perfect fluid core. These models provide simple, but physically interesting illustrations of our results.