Cauchy horizon stability in a collapsing spherical dust cloud: I. Geometric optics approximation and spherically symmetric test fields

A spherical dust cloud which is initially at rest and which has a monotonously decaying density profile collapses and forms a shell-focusing singularity. Provided the density profile is not too flat, meaning that its second radial derivative is negative at the centre, this singularity is visible to local, and sometimes even to global observers. According to the strong cosmic censorship conjecture, such naked singularities should be unstable under generic, non-spherical perturbations of the initial data or when more realistic matter models are considered. In an attempt to gain further insight into this stability issue, in this work we initiate the analysis of a simpler but related problem. We discuss the stability of test fields propagating in the vicinity of the Cauchy horizon associated to the naked central singularity. We first study the high-frequency limit and show that the fields undergo a blueshift as they approach the Cauchy horizon. However, in contrast to what occurs at inner horizons of black holes, we show that the blueshift is uniformly bounded along incoming and outgoing null rays. Motivated by this boundedness result, we take a step beyond the geometric optics approximation and consider the Cauchy evolution of spherically symmetric test scalar fields. We prove that under reasonable conditions on the initial data a suitable rescaled field can be continuously extended to the Cauchy horizon. In particular, this result implies that the physical field is everywhere finite on the Cauchy horizon away from the central singularity.