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A static spherical distribution of incoherent matter which is a source of the Klein-Gordon field is considered in equilibrium under its gravitational attraction and short-range repulsion. Numerical solutions of the full Einstein-scalar equations are obtained. The solutions are regular everywhere, and have simple physical interpretation. The stability of the system under various degrees of concentration is discussed. The impossibility of a bounded, static configuration of massless Klein-Gordon charges under its self-gravitation is deduced; furthermore, it is shown that unbounded configurations are exact solutions of the vacuum equations with cosmological constant.
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Chung et al. (1977) studied this question.
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