Static spherically-symmetric scalar-field theory in general relativity

A spherically-symmetric static scalar field in general relativity is considered. The field equations are defined by $$begin{gathered} R_{ik} = - mu varphi _i varphi _k ,varphi _i = frac{{partial varphi }}{{partial x^i }}, varphi ^i = g^{ik} varphi _k , hfill hfill end{gathered} $$ where ?=?(r,t) is a scalar field. In the past, the same problem was considered by Bergmann and Leipnik (1957) and Buchdahl (1959) with the assumption that ?=?(r) be independent oft and recently by Wyman (1981) with the assumption ?=?(r, t). The object of this paper is to give explicit results with a different approach and under a more general condition $$phi _{;i}^i = ( - g)^{ - 1/2} frac{partial }{{partial x^i }}left[ {( - g)^{1/2} g^{ik} frac{partial }{{partial x^k }}} right] = - 4pi ( -g )^{ - 1/2} rho $$ where ?=?(r, t) is the mass or the charge density of the sources of the field.