Obtaining a scalar fifth force for matter via a conformal coupling to the scalar field

Abstract

In the framework of special relativity (SR), I propose that matter conformally couples to a scalar field through the Lagrangian density of matter, whether matter is characterized by classical or by quantum (statistical) mechanics. The largest interaction strength of the scalar-mediated force can achieve the order of $1/\Lambda_E^2$ with the cosmological constant $\Lambda_E\approx 2.4\,\rm{meV}$. This is about 60 orders of magnitude larger than Newtonian gravity, i.e., ${(M_{\mathrm{Pl}}/\Lambda_E)}^2\sim 10^{60}$, where $M_{\mathrm{Pl}}\approx2.4\times 10^{18}\, \rm{GeV} $ is the reduced Planck energy. However, the discrete $\mathbb{Z}_2$ symmetry of the conformal coupling does not allow a massless scalar field, satisfying the constraint that no evidence of the long range force is observed. The interaction range of the force is inversely proportional to the square root of the ambient density. Therefore, in the general situation in which the ambient density is larger than the current cosmic density, the interaction range is always smaller than $1/\Lambda_E \approx 80\, \mu\rm{m}$. The form of general relativity (GR) still holds even when the energy-momentum tensors of the quintessence and the conformal interaction are included. The main purpose here is to overcome the puzzle in \cite{hcz2}, i.e., which frame is physical.