The complete Brans-Dicke theories

TL;DR

This paper explores the complete set of consistent Brans-Dicke theories, identifying three unique variants with specific matter interactions, extending the classical theory by including new parameters and energy exchange mechanisms.

Contribution

It systematically classifies all consistent Brans-Dicke theories that violate energy conservation, revealing three unique theories with specific interaction terms and new parameters.

Findings

Identified three unique consistent theories with energy exchange terms.

Established the connection to classical Brans-Dicke theory when parameters are zero.

Extended the theory to include new parameters and matter interactions.

Abstract

Given that the simple wave equation of Brans-Dicke theory for the scalar field is preserved, we have investigated, through exhaustively analyzing the Bianchi identities, the consistent theories which violate the exact energy conservation equation. It is found that only three theories exist which are unambiguously determined from consistency, without imposing arbitrary functions by hand. Each of these theories possesses a specific interaction term which controls the energy exchange between the scalar field and ordinary matter. The theories contain new parameters (integration constants from the integration procedure) and when these are switched-off, Brans-Dicke theory emerges. As usually, the vacuum theories can be defined from the complete Brans-Dicke theories when the matter energy-momentum tensor vanishes.