A relativistic version of the Lawrence-Doniach model is formulated to break the local U(1) gauge symmetry in analogy to the Higgs mechanism. Thereby the global U(1) invariance is spontaneously broken via the superconducting condensate. The resulting differential-difference equations for the order parameter, the in-plane and interplane components of the vector potential are of the Klein-Gordon, Proca and sine-Gordon type, respectively. A comparison with the standard sine-Gordon equation for the superconducting phase difference is given in the London limit. The present dynamical scheme is applicable to high-Tc cuprates with one layer per unit cell and weak interlayer Josephson tunneling.