Discretely self-similar exterior-naked singularities for the Einstein-scalar field system

The problem of constructing naked singularities in general relativity can be naturally divided into two parts: (i) the construction of the region exterior to the past light cone of the singularity, extending all the way to (an incomplete) future null infinity and yielding the nakedness property (what we will call exterior-naked singularity regions); (ii) attaching an interior fill-in that ensures that the singularity arises from regular initial data. This problem has been resolved for the spherically symmetric Einstein-scalar field system by Christodoulou, but his construction, based on a continuously self-similar ansatz, requires that both the exterior and the interior regions are mildly irregular on the past cone of the singularity. On the other hand, numerical works suggest that there exist naked singularity spacetimes with discrete self-similarity arising from smooth initial data. In this paper, we revisit part (i) of the problem and we construct exterior-naked singularity regions with discretely self-similar profiles which are smooth on the past cone of the singularity. We show that the scalar field remains uniformly bounded, but the singularity is characterized by the infinite oscillations of the scalar field and the mass aspect ratio. (Our examples require however that the mass aspect ratio is uniformly small, and thus the solutions are distinct from the exterior regions of the numerical examples.) It remains an open problem to smoothly attach interior fill-ins as in (ii) to our solutions, which would yield a new construction of naked singularity spacetimes, now arising from smooth initial data.