SO(4; 2) and derivatively coupled dRGT massive gravity

TL;DR

This paper explores the geometric structures of Lie groups like the conformal group and develops a gravity theory that combines features of quintessence and massive gravity through a geometric approach.

Contribution

It introduces a geometric framework for the conformal group leading to a novel gravity theory with derivative couplings to massive gravity components.

Findings

A geometric structure for the conformal group is derived from Maurer-Cartan equations.

A gravity theory with a scalar field derivatively coupled to massive gravity is formulated.

The resulting theory resembles quintessence models with geometric origins.

Abstract

In this paper we study the possibility of assigning a geometric structure to the Lie groups. It is shown the Poincar\'{e} and Weyl groups have geometrical structure of the Riemann-Cartan and Weyl space-time respectively. The geometric approach to these groups can be carried out by considering the most general (non)metricity conditions, or equivalently, tetrad postulates which we show that can be written in terms of the group's gauge fields. By focusing on the conformal group we apply this procedure to show that a nontrivial 3-metrics geometry may be extracted from the group's Maurer-Cartan structure equations. We systematically obtain the general characteristics of this geometry, i.e. its most general nonmetricity conditions, tetrad postulates and its connections. We then deal with the gravitational theory associated to the conformal group's geometry. By proposing an Einstein-Hilbert type action, we conclude that the resulting gravity theory has the form of quintessence where the scalar field derivatively coupled to massive gravity building blocks.