  The basic building block for Lorentz invariant and ghost free massive gravity
is the square root of the combination $g^{-1}\eta\,$, where $g^{-1}$ is the
inverse of the physical metric and $\eta$ is a reference metric. Since the
square root of a matrix is not uniquely defined, it is possible to have
physically inequivalent potentials corresponding to different branches. We show
that around Minkowski background the only perturbatively well defined branch is
the potential proposed by de Rham, Gabadadze and Tolley. On the other hand, if
Lorentz symmetry is broken spontaneously, other potentials exist with a
standard perturbative expansion. We show this explicitly building new Lorentz
invariant, ghost-free massive gravity potentials for theories that in the
background preserve rotational invariance, but break Lorentz boosts.
