Non linear massive gravity as a gravitational $\sigma$-model

TL;DR

This paper draws an analogy between ghost-free non-linear massive gravity and $\sigma$-models, explaining the existence of different vacuum solutions based on parameter values, and highlighting vacuum degeneracy in certain cases.

Contribution

It establishes a novel analogy between non-linear massive gravity and $\sigma$-models, clarifying the structure of vacuum solutions and degeneracies in the theory.

Findings

Two families of solutions for spherically symmetric case identified.

Vacuum degeneracy occurs when parameters satisfy $eta= ext{alpha}^2$.

Single vacuum state exists for $eta< ext{alpha}^2$.

Abstract

We show the direct analogy between the ghost-free non-linear formulation of massive gravity and the standard $\sigma$-models well understood in the literature. This issue explains why there are two non-trivial family of solutions for the spherically symmetric case inside the non-linear massive gravity formulations with two free-parameters $\alpha$ and $\beta$. In general, the case $\beta<\alpha^2$ has a single physical vacuum state. On the other hand, the case $\beta=\alpha^2$ contains a natural vacuum degeneracy. This is in perfect analogy with the $\sigma$-model for scalar fields where depending on the values taken by the parameters of the theory the vacuum can be single or degenerate.