  In this work we consider the superfluid stiffness of a generically
non-Galilean invariant interacting system and investigate under what conditions
the stiffness may nonetheless approach the Galilean-invariant value $n/m$.
Within Eliashberg theory we find that the renormalized stiffness is
approximately given by $n/m$ in the case when the $l=0$ and $l=1$ components of
the effective Fermi-surface projected interaction are approximately equal over
a range of frequencies. This holds, in particular, when the interaction is
peaked at zero momentum transfer. We examine this result through three
complementary lenses: the $\delta (\omega)$ term in the conductivity, the phase
dependence of the Luttinger-Ward free energy, and the coupling of the amplitude
and phase sectors in the Hubbard-Stratonovich collective mode action. From
these considerations we show that the value of the stiffness is determined by
the strength of renormalization of the current vertex and that the latter can
be interpreted as the shift of the self-consistent solution due to flow of the
condensate, or alternatively as coupling of the phase mode to $l=1$
fluctuations of the order parameter. We highlight that even though the
superfluid stiffness in some non-Galilean systems approaches the Galilean
value, this is not enforced by symmetry, and in general the stiffness may be
strongly suppressed from its BCS value. As a corollary we obtain the generic
form of the phase action within Eliashberg theory and charge and spin Ward
identities for a superconductor with frequency dependent gap function.
