General spherically symmetric gravity interacting with massive complex scalar field

Interaction of gravity with complex massive scalar field is considered by coupling the scalar field equation to the Einstein field equation whose source term is given by the Energy Momentum tensor of the scalar field. Assuming general spherically symmetric space time, the scalar field results to depend only on time and radial coordinate. In turn this implies consistency of the scheme that is, both the vanishing of the divergence of the Energy Momentum tensor of the scalar field and the independence of its trace from the angular variables. The explicit scheme of coupled partial differential equations is partially integrated twice on general ground. This leads to a partial differential equation that can be interpreted as a generalized Kepler like equation whose solution is difficult. To have some information about explicit solutions, a method of variable separation integration is employed by considering suitable special assumptions. The cases considered are essentially those in which the scalar field is assumed to depend on one only variable. There result situations in which, among other, a homogenization effect is recovered. Also a cosmological interpretation of the results could be possible on condition of solving systems of coupled non-linear separated differential time equations