Solutions of massive gravity theories in constant scalar invariant geometries

TL;DR

This paper finds solutions to massive gravity equations within specific Lorentzian geometries, identifying new Petrov type II and III CSI spacetimes and analyzing their stability against tachyonic modes.

Contribution

It provides a comprehensive classification of homogeneous and VSI Lorentzian geometries as solutions to massive gravity theories, including new explicit Petrov type II and III solutions.

Findings

Identified all homogeneous Lorentzian 3D geometries solving massive gravity equations.

Found new explicit Petrov type II and III CSI spacetime solutions.

Analyzed stability of anti-de Sitter solutions against tachyonic modes.

Abstract

We solve massive gravity field equations in the framework of locally homogenous and vanishing scalar invariant (VSI) Lorentzian spacetimes, which in three dimensions are the building blocks of constant scalar invariant (CSI) spacetimes. At first, we provide an exhaustive list of all Lorentzian three-dimensional homogeneous spaces and then we determine the Petrov type of the relevant curvature tensors. Among these geometries we determine for which values of their structure constants they are solutions of the field equations of massive gravity theories with cosmological constant. The homogeneous solutions founded are of all various Petrov types : I_C, I_R, II, III, D_t, D_s, N, O; the VSI geometries which we found are of Petrov type III. The Petrov types II and III are new explicit CSI spacetimes solutions of these types. We also examine the conditions under which the obtained anti-de Sitter solutions are free of tachyonic massive graviton modes.