Gravitational collapse with non-vanishing tangential stresses: II. A laboratory for cosmic censorship experiments

Abstract

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 ...

Spherical universes with anisotropic pressure

“…They cannot be solved explicitly in the coordinates t and R. Magli [10] found a solution by using mass-area coordinates, that is r and R.…”

“…He discussed a general class of spherically symmetric solutions to Einstein's equations with vanishing radial pressure. He was able to solve for the metric in terms of mass-area coordinates [10], based on the method used for charged dust collapse by Ori [11]. Using his solution, Harada et al [12] investigated naked singularity formation in the model, and gave a particular case in which the metric could be written in terms of elementary functions.…”

Nakedness and curvature strength of a shell-focusing singularity in spherically symmetric spacetime with vanishing radial pressure

“…We have introduced a function h = h(r, R) ≥ 0 as 8) where the comma denotes the partial derivative. We should note that the definition of h is slightly different from Magli (1997Magli ( , 1998)'s notation. The dust limit is given by h = h(r).…”

“…Using these expressions, we finally obtain the radial null geodesic equation for m → 0 in the explicit form 17) where the ordinary derivative is taken along R = 2y 0 m β . In evaluating the right hand side of Eq.…”

Erratum to: General spherically symmetric elastic stars in relativity

“…At this point, it is worth mentioning that general (i.e., not restricted by the shearfree condition) non-static solutions with vanishing radial pressure have been studied in detail by Magli [18] using Ori's massfbarea coordinates, although in a rather different context (the cosmic censorship conjecture), solving the problem up to a quadrature.…”