  We analyze the stability of the Cauchy horizon associated with a globally
naked, shell-focussing singularity arising from the complete gravitational
collapse of a spherical dust cloud. In previous work, we have studied the
dynamics of spherical test scalar fields on such a background. In particular,
we proved that such fields cannot develop any divergences which propagate along
the Cauchy horizon. In the present work, we extend our analysis to the more
general case of test fields without symmetries and to linearized gravitational
perturbations with odd parity. To this purpose, we first consider test fields
possessing a divergence-free stress-energy tensor satisfying the dominant
energy condition, and we prove that a suitable energy norm is uniformly bounded
in the domain of dependence of the initial slice. In particular, this result
implies that free-falling observers co-moving with the dust particles measure a
finite energy of the field, even as they cross the Cauchy horizon at points
lying arbitrarily close to the central singularity. Next, for the case of
Klein-Gordon fields, we derive point-wise bounds from our energy estimates
which imply that the scalar field cannot diverge at the Cauchy horizon, except
possibly at the central singular point. Finally, we analyze the behaviour of
odd-parity, linear gravitational and dust perturbations of the collapsing
spacetime. Similarly to the scalar field case, we prove that the relevant
gauge-invariant combinations of the metric perturbations stay bounded away from
the central singularity, implying that no divergences can propagate in the
vacuum region. Our results are in accordance with previous numerical studies
and analytic work in the self-similar case.
