  A uniform in space, oscillatory in time plasma equilibrium sustained by a
time-dependent current density is analytically and numerically studied
resorting to particle-in-cell simulations. The dispersion relation is derived
from the Vlasov equation for oscillating equilibrium distribution functions,
and used to demonstrate that the plasma has an infinite number of unstable
kinetic modes. This instability represents a new kinetic mechanism for the
decay of the initial mode of infinite wavelength (or equivalently null
wavenumber), for which no classical wave breaking or Landau damping exists. The
relativistic generalization of the instability is discussed. In this regime,
the growth rate of the fastest growing unstable modes scales with
$\gamma_T^{-1/2}$, where $\gamma_T$ is the largest Lorentz factor of the plasma
distribution. This result hints that this instability is not as severely
suppressed for large Lorentz factor flows as purely streaming instabilities.
The relevance of this instability in inductive electric field oscillations
driven in pulsar magnetospheres is discussed.
