Separating expansion from contraction and generalizing TOV condition in spherically symmetric models with pressure

We investigate spherically symmetric solutions with pressure and discuss the existence of a dividing shell separating expanding and collapsing regions. We perform a 3+1 splitting and obtain gauge invariant conditions relating not only the intrinsic spatial curvature of the shells to the ADM mass, but also a function of the pressure which we introduce that generalises the Tolman-Oppenheimer-Volkoff equilibrium condition. We consider the particular case of a Lemaître-Tolman dust models with a cosmological constant (a $Λ$-CDM model) as an example of our results.

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