  We consider the theory of a symmetric tensor field in 4D, invariant under a
subclass of infinitesimal diffeomorphism transformations, where the vector diff
parameter is the 4-divergence of a scalar parameter. The resulting gauge
symmetry characterizes the "fracton" quasiparticles and identifies a theory
which depends on a dimensionless parameter, which cannot be reabsorbed by a
redefinition of the tensor field, despite the fact that the theory is free of
interactions. This kind of "electromagnetic gauge symmetry" is weaker that the
original diffeomorphism invariance, in the sense that the most general action
contains, but is not limited to, linearized gravity, and we show how it is
possible to switch continuously from linearized gravity to a mixed phase where
both gravitons and fractons are present, without changing the degrees of
freedom of the theory. The gauge fixing procedure is particularly rich and
rather peculiar, and leads to the computation of propagators which in the
massive case we ask to be tachyonic-free, thus constraining the domain of the
parameter of the theory. Finally, a closer contact to fractons is made by the
introduction of a parameter related to the "rate of propagation". For a
particular value of this parameter the theory does not propagate at all, and we
guess that, for this reason, the resulting theory should be tightly related to
the fracton excitations.
