Scalar field interpretation of the radius in spherically symmetric comoving systems

The physical radius appearing in spherically symmetry co-moving metric is interpreted in terms of a complex scalar field. The corresponding scalar field equation is formulated in the same space-time defined by the physical radius. There follows a highly non-linear scalar field equation that can be reduced to separated radial and time equations. The general integral of the separated radial equation is determined in case of real field. The same for the separated real massless time equation. Particular complex solutions are also given. It is shown that the real solutions of the scheme are not solutions of the Einstein and scalar field equations coupled in the conventional way. They can be interpreted as generating possible cosmologies with self-settling gravitational field. As to the complex solutions, they satisfy an "orthonormality" relation, similar to the one that defines the normal modes of field equation in curved space-time. This suggest the idea of possible quantization of the physical radius field. Owing to non-linearity, the procedure seems difficult to perform and to be set up completely.