Conformal classes of Brans-Dicke gravity

TL;DR

This paper classifies Brans-Dicke gravity theories based on the behavior of the gravitational coupling, discusses the significance of conformal frames, and highlights the scale-invariance properties in different matter coupling scenarios.

Contribution

It provides a classification of Brans-Dicke theories by conformal class and clarifies the physical relevance of conformal frames in scalar-tensor gravity.

Findings

Both Jordan and Einstein frames can belong to the same conformal class.

Jordan frame with nonminimally coupled matter is scale-invariant.

Original Brans-Dicke theory with minimally coupled matter breaks scale-invariance.

Abstract

A classification of Brans-Dicke theories of gravitation, based on the behaviour of the dimensionless gravitational coupling constant, is given. It is noted that the discussion takes place in the current literature, about which of the two distinguished conformal frames in which scalar-tensor theories of gravity can be formulated: the Jordan frame and the Einstein frame, is the physical one, may, in most cases, be meaningless for both frames may belong to the same conformal class. It is also noted that the Jordan frame formulation of Brans-Dicke gravity with ordinary matter nonminimally coupled, that is shown to be just the Jordan frame formulation of general relativity, is scale-invariant, unlike the situation with the Jordan frame representation of Brans-Dicke gravity with matter minimally coupled (the original formulation of Brans-Dicke theory), where the presence of nonzero mass ordinary matter breaks the scale-invariance of the theory.