  We develop a gauge-invariant linear response theory for relativistic
Bardeen-Cooper-Schrieffer (BCS) superfluids based on a
consistent-fluctuation-of-the order-parameter (CFOP) approach. The response
functions from the CFOP approach satisfy important generalized Ward identities.
The gauge invariance of the CFOP theory is a consequence of treating the gauge
transformation and the fluctuations of the order parameter on equal footing so
collective-mode effects are properly included. We demonstrate that the pole of
the response functions is associated with the massless Goldstone boson.
Important physical quantities such as the compressibility and superfluid
density of relativistic BCS superfluids can also be inferred from our approach.
We argue that the contribution from the massless Goldstone boson is crucial in
obtaining a consistent expression for the compressibility.
