Torsion, Scalar Field and f(\mathcal{R}) Gravity

TL;DR

This paper explores how torsion and a scalar field in Riemann-Cartan geometry relate to f(R) gravity, showing the scalar field's connection to Ricci scalar and Newton's constant, thus linking torsion, scalar fields, and modified gravity.

Contribution

It demonstrates that a specific Lagrangian with torsion and scalar field reduces to an f(R) gravity theory, revealing new insights into their interrelation.

Findings

Divergence of axial torsion yields Newton's constant.

Scalar field becomes a function of Ricci scalar R.

Lagrangian reduces to f(R) gravity form.

Abstract

The role of torsion and a scalar field $\phi$ in gravitation in the background of a particular class of the Riemann-Cartan geometry is considered here. Some times ago, a Lagrangian density with Lagrange multipliers has been proposed by the author which has been obtained by picking some particular terms from the SO(4,1) Pontryagin density, where the scalar field $\phi$ causes the de Sitter connection to have the proper dimension of a gauge field. Here it has been shown that the divergence of the axial torsion gives the Newton's constant and the scalar field becomes a function of the Ricci scalar $\mathcal{R}$. The starting Lagrangian then reduces to a Lagrangian representing the metric $f(\mathcal{R})$ gravity theory.