On Exact Solutions and the Consistency of 3D Minimal Massive Gravity

TL;DR

This paper investigates exact solutions and the consistency of field equations in 3D Minimal Massive Gravity, demonstrating inheritance of solutions from Topologically Massive Gravity and analyzing the conditions for equation consistency with and without matter.

Contribution

It establishes the inheritance of algebraic solutions from TMG to MMG and proves the classical consistency of MMG's field equations for both source-free and matter-coupled cases.

Findings

Inheritance of algebraic solutions from TMG to MMG

Double-divergence of field equations vanishes for solutions

Consistency of MMG field equations confirmed with matter coupling

Abstract

We show that all algebraic Type-O, Type-N and Type-D and some Kundt-Type solutions of Topologically Massive Gravity are inherited by its holographically well-defined deformation, that is the recently found Minimal Massive Gravity. This construction provides a large class of constant scalar curvature solutions to the theory. We also study the consistency of the field equations both in the source-free and matter-coupled cases. Since the field equations of MMG do not come from a Lagrangian that depends on the metric and its derivatives only, it lacks the Bianchi identity valid for all non-singular metrics. But it turns out that for the solutions of the equations, Bianchi identity is satisfied. This is a necessary condition for the consistency of the classical field equations but not a sufficient one, since the the rank-two tensor equations are susceptible to double-divergence. We show that for the source-free case the double-divergence of the field equations vanish for the solutions. In the matter-coupled case, we show that the double-divergence of the left-hand side and the right-hand side are equal to each other for the solutions of the theory. This construction completes the proof of the the consistency of the field equations.