In 1969 R. Penrose formulated what is considered to be the most important open problem in general relativity, The cosmic censorship conjecture, i.e. the idea that the formation of event horizons predicted in some models of gravitational collapse is consequence of a fundamental principle of nature that acts to hide the classically predicted singularity derived from such collapse. This conjecture has not just remained open, but its own formulation is still controversial. This is a very puzzling aspect linked to the limitations of the classical formalism.

We have studied the main component of the classical dominant versions of the conjecture, i.e. the global hyperbolic property. The cosmic censorship conjecture has been normally formulated in terms of the validity of this property, the reason for this choice is the relation between global hyperbolicity and classical determinism, this relation has led to formulate the conjecture hypothesizing that the principle acting in gravitational collapse is acting to conserve classical determinism to some degree.

This choice is highly biased by the ideas of Penrose and others about determinism in quantum mechanics, Penrose advocates the idea that gravitation is related with a “dynamical collapse” formulation of quantum mechanics that in some sense will transform quantum theory in a classical deterministic theory ; in this context the con- servation of classical determinism via global hyperbolicity can be considered as fundamental even in the context of a more general quantum gravity theory.

The stability of Kerr black holes provides4 some strong support in favour of the conjecture, the fact that this fundamental principle of nature has to be formulated in terms of the validity of the global hyperbolic property is less convincing.  We have study the stability of the global hyperbolic property in the interval topology of metrics. We have shown that the global hyperbolic property is stable in this topology, a result studied for the first time by R. Geroch ; although the proof he provided was not entirely correct without some non trivial amendments. We have given an alternative more simple proof and the corrected version of Geroch’s proof. From this work we saw that even if the global hyperbolic property results stable in this topology, the proof of this result shows that this property is far from having the character of a generic funda- mental property of spacetimes. This was our first motivation to think that the breakdown of classicality linked to the formation of event horizons will probably have to be understood from a wider perspective than that related to the breakdown of the global hyperbolic property. With this idea in mind we tried to understand if causality violations, which destroy the global hyperbolic property and are predicted in important gravitational collapse models as the Kerr’s solution, could be related to this breakdown of classicality in a fundamental way.

Documents:

Global hyperbolicity is stable in the interval topology