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 separating shell dividing expanding and collapsing regions. We resort to a $3+1$ splitting and obtain gauge invariant conditions relating intrinsic spacetime quantities to properties of the matter source. We find that the separating 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 nonlocal 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 stresses. We have cast the governing equations in terms of local, gauge invariant quantities that are revealing of the role played by the anisotropic pressures and inhomogeneous electric part of the Weyl tensor. We analyze 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 co-workers but also reveals transparently the interplay and importance of the shear and of the anisotropic stresses.

Figure 1

Dynamical analysis of a local $W<0$ shell. The dynamic for a given shell (fixed $M$, $W$ and $E$ without shell crossing) obeys Eq. (4.10). It then behaves as a one dimensional particle in an effective potential, following 14. We draw a qualitative energy diagram to illustrate the definition of the critical curvature/energy $E_{\lim }$, when it lies in a region of $W<0$. The various cases of $E_{>}>E_{\lim }$, $E_{<}<E_{\lim }$ and $E=E_{\lim }$ yield unbound, bound and marginally bound behaviors.

Figure 2

Global, qualitative, analysis yielding the separating shell at the intersection of $E$ with $E_{\lim }$. Following 14, we construct initial cosmological conditions with a $W<0$ region such that Eqs. (4.10, 4.12) are 0 simultaneously, that is $E=E_{\lim }$ and $W=-M$, so the intersection of the curves gives a global dynamical separation.