  In this paper, we give the covariant formulation of second gradient
electrodynamics, which is a generalized electrodynamics of second order
including derivatives of higher order. The relativistic form of the field
equations, the energy-momentum tensor and the Lorentz force density are
presented. For an electric point charge, the generalized Lienard-Wiechert
potentials and the corresponding electromagnetic field strength tensor are
given as retarded integral expressions. Explicit formulas for the
electromagnetic potential vector and electromagnetic field strength tensor of a
uniformly moving point charge are found without any singularity and
discontinuity. In addition, a world-line integral expression for the self-force
of a charged point particle is given. The relativistic equation of motion of a
charged particle coupled with electromagnetic fields in second gradient
electrodynamics is derived, which is an integro-differential equation with
nonlocality in time. For a uniformly accelerated charge, explicit formulas of
the self-force and the electromagnetic mass, being non-singular, are given.
Moreover, the wave propagation and the dispersion relations in the vacuum of
second gradient electrodynamics are analyzed. Three modes of waves were found:
one non-dispersive wave as in Maxwell electrodynamics, and two dispersive waves
similar to the wave propagation in a collisionless plasma.
