Considering weak-field limit of power-class (R1−ε) gravity in terms of its equivalent STG-(Brans-Dicke type) theory in the vacuum-Jordan-conformal frame

Abstract

The weak-field limit of power-class type theory of gravity in terms of its equivalent STG-Brans-Dicke type theory with a vanishing Brans-Dicke parameter is considered in this paper. Through a known procedure of Legendre transformation, the cosmological function of the equivalent STG theory is defined from the lagrangian of power-class theory in the absence of matter terms, and obtain dynamical equations for the metric and Brans-Dicke scalar field from the equivalent STG lagrangian. As then naturally expected, in the weak-field limit we find that the resulting Poisson equation of vacuum STG theory is no longer the ordinary form of Poisson equation of vacuum Newtonian gravity theory. Here, we adopt Einstein's gauge to reduce the Ricci tensor, though it imposes vanishing of time derivatives of the fields, the fields would not be completely time-independent because of the presence of extra term in the resulting Poisson equation.