  We study the stability of fracton gravity, a variant of linearized gravity
where the gauge symmetry is restricted to longitudinal diffeomorphisms. These
transformations can be connected to a spacetime generalization of dipole
symmetry, hence the tag fracton. We find that fracton gravity features an
instability in the spin-1 sector corresponding to solutions with growing
amplitude. This dynamical instability can be removed by tuning the couplings of
the theory. Nonetheless, the Hamiltonian for the spin-1 modes remains always
unbounded from below when evaluated on the classical solutions. We find no
other sources of instability in the spin-2 or spin-0 sectors. We analyze in
detail the canonical formulation and the constraints arguing that neither
auxiliary fields nor gauge-fixing conditions can be employed to remove the
problematic vector modes or stabilize them.
