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In this paper, we initiate the study of the instability of nakedsingularities without symmetries. In a series of papers, Christodoulou provedthat naked singularities are not stable in the context of the sphericallysymmetric Einstein equations coupled with a massless scalar field. We study inthis paper the next simplest case: a characteristic initial value problem ofthis coupled system with the initial data given on two intersecting null cones,the incoming one of which is assumed to be spherically symmetric and singularat its vertex, and the outgoing one of which has no symmetries. It is shownthat, arbitrarily fixing the initial scalar field, the set of the initialconformal metrics on the outgoing null cone such that the maximal futuredevelopment does not have any sequences of closed trapped surfaces approachingthe singularity, is of first category in the whole space in which the sheartensors are continuous. Such a set can then be viewed as exceptional, althoughthe exceptionality is weaker than the at least $1$ co-dimensionality inspherical symmetry. Almost equivalently, it is also proved that, arbitrarilyfixing an incoming null cone $\underline{C}_\varepsilon$ to the future of theinitial incoming null cone, the set of the initial conformal metrics such thatthe maximal future development has at least one closed trapped surface before$\underline{C}_\varepsilon$, contains an open and dense subset of the wholespace. Since the initial scalar field can be chosen such that the singularityis naked if the initial shear is set to be zero, we may say that the sphericalnaked singularities of a self-gravitating scalar field are not stable undergravitational perturbations. This in particular gives new families ofnon-spherically symmetric gravitational perturbations different from theoriginal spherically symmetric scalar perturbations given by Christodoulou.

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