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.
Highlights
Scalar–tensor gravitational theories are studied extensively as an alternative to General Relativity
Relieving the standard exact conservation of matter, but still preserving the simple wave equation of motion for the scalar field sourced by the trace of the matter energy-momentum tensor, it was one recently found the most general completion of Brans–Dicke theory
Keeping a single interaction term each time to express the energy exchange between the scalar field and ordinary matter, three uniquely defined theories arise from consistency, which form the prominent and natural complete Brans–Dicke theories
Summary
Scalar–tensor gravitational theories are studied extensively as an alternative to General Relativity. We should note that if the interaction model is to be worked out at the level of an action, there are various interactions of the matter Lagrangian with the scalar field, all having as limit the Brans–Dicke action in the absence of interactions The number of such actions can increase in the presence of Newton’s constant GN or a new mass scale. C (2016) 76:686 matter stress tensor was relaxed, while preserving the equation φ = 4π λT This class of theories is parametrized by one or two free functions of the scalar field, but it was found that there are only three theories, each with a specific interaction term, which are unambiguously determined by consistency. Where Lm(gκλ, ) is the matter Lagrangian depending on some extra fields
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