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Motivated by cosmic censorship in general relativity and string theory, weextend Christodoulou's celebrated examples of naked singularity formation inthe Einstein-massless scalar field system to include a positive or negativescalar potential of exponential forms, i.e., $V(\phi)=\pm\exp(2\phi/\kappa)$with a parameter $\kappa$. Under spherical symmetry and a self-similar ansatzdepending on $\kappa$, we derive a 3-dimensional autonomous system offirst-order ordinary differential equations, which incorporates the equationsfor massless scalar fields as a special case. Local behavior of the phase spaceis studied analytically with global solutions constructed numerically. Withinthe 3-dimensional solution manifold, we observe, for the negative potentials,naked singularity formation from nonsingular initial data for $\kappa^2<1$.Meanwhile, transitions between solutions containing naked singularities andblack holes are also identified. However, when the potential is taken positive,numerical evolutions result in formation of black holes, but not nakedsingularities.

Naked singularity and black hole formation in self-similar Einstein-scalar fields with exponential potentials