On the action of the complete Brans-Dicke theory

Abstract: Recently the most general completion of Brans-Dicke theory was appeared with energy exchanged between the scalar field and ordinary matter, given that the equation of motion for the scalar field keeps the simple wave form of Brans-Dicke. This class of theories contain undetermined functions, but there exist only three theories which are unambiguously determined from consistency. Here, for the first such theory, it is found the action of the vacuum theory which arises as the limit of the full matter theory. A symmetry transformation of this vacuum action in the Jordan frame is found which consists of a conformal transformation of the metric together with a redefinition of the scalar field. Since the general family of vacuum theories is parametrized by an arbitrary function of the scalar field, the action of this family is also found. As for the full matter theory it is only found the action of the system when the matter Lagrangian vanishes on-shell, as for example for pressureless dust. Due to the interaction, this matter Lagrangian is non-minimally coupled either in the Jordan or the Einstein frame.