José P. Mimoso, Morgan Le Delliou, Filipe C. Mena
We investigate spherically symmetric spacetimes with an anisotropic ?uid and discuss the exis- tence 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 characteristics of the matter source. We ?nd that the dividing shell is de?ned by a generalization of the Tolman-Oppenheimer-Volko? equilibrium condition. The latter establishes a balance between the pressure gradients, both isotropic and anisotropic, and the strength of the ?elds induced by the Misner-Sharp mass inside the separating shell and by the pressure ?uxes. This de?nes a local equi- librium condition, but conveys also a non-local character given the de?nition of the Misner-Sharp mass. By the same token it is also a generalized thermodynamical equation of state as usually interpreted for the perfect ?uid case, which now has the novel feature of involving both the isotropic pressure P 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 et al. [18] but also reveals transparently the interplayand importance of the shear and of the anisotropic stresses.
View original:
http://arxiv.org/abs/1302.6186
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