Instability of spherical naked singularities of a scalar field under isotropic gravitational perturbations

Published in 1999, Christodoulou proved both the weak and strong cosmic censorship conjectures in the context of spherically symmetric solutions of the Einstein equations coupled with a massless scalar field, by proving certain instability properties of the naked singularities. In order to gain insights to attack the problem beyond spherical symmetry, we study in this paper a characteristic initial value problem of the Einstein equations coupled with a massless scalar field with initial data given on two intersection null cones, the incoming one of which is assumed to be spherically symmetric and singular at its vertex, and the outgoing one of which has no symmetries. We are able to find a subset of the space of the initial data sets, containing rough initial data, such that, in a certain sense, the maximal future development of the initial data in this subset has a sequence of closed trapped surfaces approaching the singularity. Moreover, fixed arbitrary initial lapse and scalar field function on the outgoing null cone, the complement of this subset is of codimension at least $1$ in the space of the initial conformal metrics on the outgoing null cone satisfying a certain isotropic condition. We may therefore say that, the spherical naked singularities of a self-gravitating scalar field are not stable under isotropic gravitational perturbations.