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

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.