Local conditions separating expansion from collapse in spherically
  symmetric models with anisotropic pressures
  We investigate spherically symmetric spacetimes with an anisotropic fluid and
discuss the existence and stability of a dividing shell separating expanding
and collapsing regions. We resort to a 3+1 splitting and obtain gauge invariant
conditions relating intrinsic spacetimes quantities to properties of the matter
source. We find that the dividing shell is defined by a generalization of the
Tolman-Oppenheimer-Volkoff equilibrium condition. The latter establishes a
balance between the pressure gradients, both isotropic and anisotropic, and the
strength of the fields induced by the Misner-Sharp mass inside the separating
shell and by the pressure fluxes. This defines a local equilibrium condition,
but conveys also a non-local character given the definition of the Misner-Sharp
mass. By the same token, it is also a generalized thermodynamical equation of
state as usually interpreted for the perfect fluid case, which now has the
novel feature of involving both the isotropic and the anisotropic stress. We
have cast the governing equations in terms of local, gauge invariant quantities
which are revealing of the role played by the anisotropic pressures and
inhomogeneous electric part of the Weyl tensor. We analyse a particular
solution with dust and radiation that provides an illustration of our
conditions. In addition, our gauge invariant formalism not only encompasses the
cracking process from Herrera and coworkers but also reveals transparently the
interplay and importance of the shear and of the anisotropic stresses.
