  Due to their driven-dissipative nature, photonic quantum fluids present new
challenges in understanding superfluidity. Some associated effects have been
observed, and notably the report of nearly dissipationless flow for coherently
driven microcavity-polaritons was taken as a smoking gun for superflow. Here we
show that the superfluid response --- the difference between responses to
longitudinal and transverse forces --- is zero for coherently driven
polaritons. This is a consequence of the gapped excitation spectrum caused by
external phase locking. Furthermore, while a normal component exists at finite
pump momentum, the remainder forms a rigid state that is unresponsive to either
longitudinal or transverse perturbations. Interestingly, the total response
almost vanishes when the real part of the excitation spectrum has a linear
dispersion, which was the regime investigated experimentally. This suggests
that the observed suppression of scattering should be interpreted as a sign of
this new rigid state and not a superfluid.
