A unified approach to regular interiors of black holes with smooth matter distributions in the core region is given. The approach is based on a class of Kerr-Schild metrics representing minimal deformations of the Kerr-Newman solution, and allows us to give a common treatment for (charged and uncharged) rotating and nonrotating black holes. It is shown that the requirement of smoothness of the source constraints the structure of the core region in many respects: in particular, for Schwarzschild holes a de Sitter core can be selected, which is surrounded by a smooth shell giving a leading contribution to the total mass of the source. In the rotating, noncharged case the source has a similar structure, taking the form of a (anisotropic and rotating) de Sitter-like core surrounded by a rotating elliptic shell. The Kerr singular ring is regularized by anisotropic matter rotating in the equatorial plane, so that the negative sheet of the Kerr geometry is absent. In the charged case the sources take the form of "bags", which can have de Sitter or anti de Sitter interiors and a smooth domain wall boundary, with a tangential stress providing charge confinement. The ADM and Tolman relations are used to calculate the total mass of the sources.

The gravitational dynamics of anisotropic elastic spheres supported only by tangential stresses and satisfying an equation of state is analysed, and a fairly large class of nonstatic, spherically symmetric solutions of the Einstein field equations is found by quadratures. The solutions contain three arbitrary functions. Two such functions are immediately recognized as the initial distributions of mass and energy, familiar from the Tolman-Bondi (dust) models, while the third is the elastic internal energy per unit volume. If this function is a constant, the energy density becomes proportional to the matter density and therefore the metric reduces to the Tolman-Bondi one. In the general case, however, the solutions contain oscillating models as well as finite-bouncing models.

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The general exact solution describing the dynamics of anisotropic elastic spheres supported only by tangential stresses is reduced to a quadrature using Ori's mass-area coordinates. This leads to the explicit construction of the root equation governing the nature of the central singularity. Using this equation, we formulate and motivate on physical grounds a conjecture on the nature of this singularity. The conjecture covers a large sector of the space of initial data; roughly speaking, it asserts that addition of a tangential stress cannot undress a covered dust singularity. The root equation also allows us to analyze the case of self-similar spacetimes and to get some insight on the role of stresses in deciding the nature of the singularities in this case.Class. Quantum Gravity 15, (1998), p. 3215-3228The present status of research in black hole formation and cosmic censorship is quite intriguing. Indeed, both analytical results in dust collapse (see e.g. Jhingan & Joshi 1997 and references therein) and numerical results in scalar field collapse (see e.g. Gundlach 1997 and references therein) indicate the existence of a critical behaviour governing the formation of black holes or naked singularities. In principle, this behaviour should be the "remnant" of some hypothesis of a -still unknown -cosmic censorship theorem and, as such, should be related to the properties of the collapsing matter like fulfilment of energy conditions and local stability. If we want to understand the physics of such phenomena in the case of "ordinary" matter (i.e. not boson stars), we are enforced to approach analytically the gravitational collapse of non pressureless matter, since the dust equation of state is "trivial" from the viewpoint of matter properties: all energy conditions, causality conditions and stability conditions reduce to positivity of energy. However, this problem is extremely difficult if approached in full generality (see e.g. Joshi 1996 and references therein).Recently (Magli 1997, to be referred towards as [I]), we discussed a class of solutions of the Einstein field equations describing spherically symmetric, non-static elastic spheres supported only by tangential stresses (elastic matter is of interest in strongly collapsed situations; for instance neutron stars typically have solid regions -see e.g. Haensel 1995). Previous investigations on systems having vanishing radial stresses trace back to Einstein (1939) and Florides (1974) in the static case, while non-static models have been considered by Datta (1970), Bondi (1971), and Herrera & Santos (1995) (for a general review on anisotropic systems in General Relativity see .From the viewpoint of cosmic censorship, understanding the nature of singularities for such solutions can be considered as a first step toward a full understanding of spherical gravitational collapse with general stresses. Indeed the investigation of some particular models carried out recently by Singh and Witten (1997) already shows behaviours which can be drastically different from the dust ...

A new class of solutions of the Einstein field equations in spherical symmetry is found. The new solutions are mathematically described as the metrics admitting separation of variables in area-radius coordinates. Physically, they describe the gravitational collapse of a class of anisotropic elastic materials. Standard requirements of physical acceptability are satisfied, in particular, existence of an equation of state in closed form, weak energy condition, and existence of a regular Cauchy surface at which the collapse begins. The matter properties are generic in the sense that both the radial and the tangential stresses are non vanishing, and the kinematical properties are generic as well, since shear, expansion, and acceleration are also non-vanishing. As a test-bed for cosmic censorship, the nature of the future singularity forming at the center is analyzed as an existence problem for o.d.e. at a singular point using techniques based on comparison theorems, and the spectrum of endstates -blackholes or naked singularities -is found in full generality. Consequences of these results on the Cosmic Censorship conjecture are discussed.