  Phase fluctuations of a d-wave superconducting order parameter are
theoretically studied in the context of high-T$_c$ cuprates. We consider the
$t-J$ model describing layered compounds, where the Heisenberg interaction is
decoupled by a d-wave order parameter in the particle-particle channel.
Assuming first that the equilibirum state has long-range phase order, the
effective action $\mathcal{S}_{eff}$ is derived perturbatively for small
fluctuations within a path integral formalism, in the presence of the Coulomb
and Hubbard interaction terms. In a second step, a more general derivation of
$\mathcal{S}_{eff}$ is performed in terms of a gradient expansion which only
assumes that the gradients of the order parameter are small whereas the value
of the phase may be large. We show that in the phase-only approximation the
resulting $\mathcal{S}_{eff}$ reduces in leading order in the field gradients
to the perturbative one which thus allows to treat also the case without
long-range phase order or vortices. Our result generalizes previous expressions
for $\mathcal{S}_{eff}$ to the case of interacting electrons, is explicitly
gauge invariant, and avoids problematic singular gauge transformations.
