# Metric formulation of the simple theory of 3d massive gravity

## TL;DR

This paper introduces a new simple first order action for 3D massive gravity that breaks Lorentz symmetry, leading to modified Einstein equations with higher derivatives and providing a novel example of third way consistent field equations.

## Contribution

It presents a new first order formulation of 3D massive gravity that disentangles metric and non-metric degrees of freedom, resulting in novel, consistent metric field equations.

## Key findings

- Reproduces minimal massive gravity as a special case
- Derives new metric equations with higher derivatives
- Provides a new example of third way consistent theories

## Abstract

We have recently introduced a new and very simple action for three-dimensional massive gravity. This action is written in a first order formulation where the triad and the connection play a manifestly symmetric role, but where internal Lorentz gauge symmetry is broken. The absence of Lorentz invariance, which in this model is the mechanism underlying the propagation of a massive graviton, does however prevent from writing a purely metric non-linear action for the theory. Nonetheless, in this letter, we explain how to disentangle, at the non-linear level, the metric and non-metric degrees of freedom in the equations of motion. Focusing on the metric part, we show that it satisfies modified Einstein equations with higher derivative terms. As a particular case, these equations reproduce a well-studied model known as minimal massive gravity. In the general case, we obtain new metric field equations for massive gravity in three dimensions starting from the simple first order action. These field equations are consistent through a mechanism known as "third way consistency", which our theory therefore provides a new example of.