We study the role played by local anisotropy and radiation (in the free-streaming regime) in the onset of dynamical instability. In the case of anisotropy we find that, for Newtonian and post-Newtonian approximations, the relevant quantity is the anisotropy of the unperturbed fluid, and that small anisotropies may, in principle, drastically change the stability of the system. The influence of the free-streaming radiation upon stability appears to be qualitatively of the same type as that of radiative diffusion. Some speculations and prospective applications to astrophysical scenarios are presented.

An algorithm recently presented by Lake to obtain all static spherically symmetric perfect fluid solutions is extended to the case of locally anisotropic fluids (principal stresses unequal). As expected, the new formalism requires the knowledge of two functions (instead of one) to generate all possible solutions. To illustrate the method some known cases are recovered.

We put forward a new definition of complexity, for static and spherically symmetric self-gravitating systems, based on a quantity, hereafter referred to as complexity factor, that appears in the orthogonal splitting of the Riemann tensor, in the context of general relativity. We start by assuming that the homogeneous (in the energy density) fluid, with isotropic pressure is endowed with minimal complexity. For this kind of fluid distribution, the value of complexity factor is zero. So, the rationale behind our proposal for the definition of complexity factor stems from the fact that it measures the departure, in the value of the active gravitational mass (Tolman mass), with respect to its value for a zero complexity system. Such departure is produced by a specific combination of energy density inhomogeneity and pressure anisotropy. Thus, zero complexity factor may also be found in self-gravitating systems with inhomogeneous energy density and anisotropic pressure, provided the effects of these two factors, on the complexity factor, cancel each other. Some exact interior solutions to the Einstein equations satisfying the zero complexity criterium are found, and prospective applications of this newly defined concept, to the study of the structure and evolution of compact objects, are discussed.

The full set of equations governing the evolution of self-gravitating spherically symmetric dissipative fluids with anisotropic stresses is deployed and used to carry out a general study on the behavior of such systems, in the context of general relativity. Emphasis is given to the link between the Weyl tensor, the shear tensor, the anisotropy of the pressure, and the density inhomogeneity. In particular we provide the general, necessary, and sufficient condition for the vanishing of the spatial gradients of energy density, which in turn suggests a possible definition of a gravitational arrow of time. Some solutions are also exhibited to illustrate the discussion.