Note on the matter coupling in Exotic General Massive 3D Gravity

TL;DR

This paper analyzes matter coupling in Exotic General Massive 3D Gravity using exterior algebra, deriving source forms and consistency conditions, and enabling exploration of non-minimal couplings and higher-order extensions.

Contribution

It explicitly constructs matter source forms in EGMG, derives consistency relations for matter coupling, and provides a formalism for non-minimal matter interactions and higher-order curvature extensions.

Findings

Derived explicit source 2-forms involving quadratic and derivative terms.

Established a consistency relation for matter-coupled EGMG field equations.

Provided a formalism for non-minimal matter couplings and higher-order extensions.

Abstract

In this work, within exterior algebra formalism, we revisit and study the matter coupling in Exotic General Massive 3D Gravity (EGMG), previously introduced in \cite{ozkan_1}. Considering that the matter Lagrangian is both metric co-frame and connection dependent in general, we explicitly construct associated source 2-form by algebraically solving auxiliary 1-form fields in terms of gravitational field variables, energy-momentum and hyper-momentum 2-forms. It is seen that that the source 2-form involves terms that are quadratic in energy-momentum and hyper-momentum 2-forms as well as terms that depend on covariant derivatives of them. In addition, we derive the consistency relation in order that matter-coupled EGMG field equation be third-way consistent. In general, the consistency requires a particular relation that should be satisfied by energy-momentum and hyper-momentum 2-forms. We note that our construction and formalism allows to investigate non-minimal matter couplings with gravitational fields. It also enables to construct the higher order curvature extensions of EGMG.