Self-gravitating scalar field in spherically symmetric comoving system: Solutions depending on one variable

Abstract

The scalar and Einstein field equations are considered in the general symmetric comoving system. The self-gravitational interaction of the scalar field is expressed by identifying the energy momentum tensor with the one canonically associated to the scalar field. It is shown that the scalar field can depend only on the radial and time variables and that the energy momentum tensor is implicitly conserved. Two classes of solutions are then studied. A first class, that contains the static case previously considered and the complete time-dependent case, is determined by a priori assuming the unknown functions to depend on one only variable. This class is meaningful essentially for massless field. A second class of solutions refers to the a priori assumption that the scalar field alone depends on r or on t. In both cases some comments are developed for the massive scalar field and for a large class of massless fields we are able to determine the solutions.