  Applying the Helmholtz Decomposition theorem to linearized General Relativity
leads to a gauge-invariant formulation where the transverse-traceless part of
the metric perturbation describes gravitational waves in matter. Gravitational
waves incident on a superconductor can be described by a linear London-like
constituent equation characterized by a "gravitational shear modulus" and a
corresponding plasma frequency and penetration depth. Electric-like and
magnetic-like gravitational tensor fields are defined in terms of the strain
field of a gravitational wave. It is shown that in the DC limit, the
magnetic-like tensor field is expelled from the superconductor in a
gravitational Meissner-like effect. The Cooper pair density is described by the
GinzburgLandau free energy density embedded in curved spacetime. The ionic
lattice is modeled by quantum harmonic oscillators coupled to gravitational
waves and characterized by quasi-energy eigenvalues for the phonon modes. The
formulation predicts the possibility of a dynamical Casimir effect since the
zero-point energy of the ionic lattice phonons is found to be modulated by the
gravitational wave, in a quantum analog of a "Weber-bar effect." Applying
periodic thermodynamics and the Debye model in the low-temperature limit leads
to a free energy density for the ionic lattice. Lastly, we relate the
gravitational strain of space to the strain of matter to show that the response
to a gravitational wave is far less for the Cooper pair density than for the
ionic lattice. This predicts a charge separation effect in the superconductor
as a result of the gravitational wave.
