Hamiltonian Approach to Conformal Coupling Scalar Field in the General Relativity

TL;DR

This paper investigates the effects of conformal coupling of scalar fields in General Relativity using a Hamiltonian approach, revealing violations of standard geometrical structures and proposing a transformation to restore them, with implications for cosmology.

Contribution

It introduces a Bekenstein-type transformation to convert conformally coupled scalar fields into minimally coupled ones, restoring standard GR structure and exploring cosmological impacts.

Findings

Conformal coupling violates Einstein equations' geometry.

Scalar-scale mixing restores standard GR structure.

Early universe dynamics show a dominant rigid state contribution.

Abstract

The dynamic status of scalar fields is studied in the Hamiltonian approach to the General Relativity. We show that the conformal coupling of the scalar field violates the standard geometrical structure of the Einstein equations in GR and their solutions including the Schwarzschild one and the Newton static interaction. In order to restore the standard geometrical structure of GR, the scalar field is mixed with the scale metric component by the Bekenstein type transformation. This "scalar-scale" mixing converts the conformal coupling scalar field with conformal weight (n= -1) into the minimal coupling scalar field with zero conformal weight (n=0) called a "scalar graviton". Cosmological consequences of the "scalar-scale" mixing are considered in the finite space-time by extraction of the zero mode (homogeneous) harmonics of a "scalar graviton". The classical dynamics of "scalar graviton" testifies about a tremendous contribution of evolution at the beginning in the form of the rigid state.