  We formulate the gauge invariant Lorentz covariant Ginzburg-Landau theory
which describes nonstationary regimes: relaxation of a superconducting system
accompanied by eigen oscillations of internal degrees of freedom (Higgs mode
and Goldstone mode), and also forced oscillations under the action of an
external gauge field. The theory describes Lorentz covariant electrodynamics of
superconductors where Anderson-Higgs mechanism occurs, at the same time the
dynamics of conduction electrons remains non-relativistic. It is demonstrated
that Goldstone oscillations cannot be accompanied by oscillations of charge
density and they generate the transverse field only. In addition, we consider
Goldstone modes and features of Anderson-Higgs mechanism in two-band
superconductors. We study dissipative processes, which are caused by movement
of the normal component of electron liquid and violate the Lorentz covariance,
on the examples of the damped oscillations of the order parameter and the
skin-effect for electromagnetic waves. An experimental consequence of the
extended time-dependent Ginzburg-Landau theory regarding the penetration of the
electromagnetic field into a superconductor is proposed.
