  In the standard cosmological model, magnetic fields and vorticity are
generated during the radiation era via second-order density perturbations. In
order to clarify the complicated physics of this second-order magnetogenesis,
we use a covariant approach and present the electromagneto-dynamical equations
in the nonlinear regime. We use the tight-coupling approximation to analyze
Thomson and Coulomb scattering. At the zero-order limit of exact
tight-coupling, we show that the vorticity is zero and no magnetogenesis takes
place at any nonlinear order. We show that magnetogenesis also fails at all
orders if either protons or electrons have the same velocity as the radiation,
and momentum transfer is neglected. Then we prove a key no-go result: at
first-order in the tight-coupling approximation, magnetic fields and vorticity
still cannot be generated even via nonlinear effects. The tight-coupling
approximation must be broken at first order, for the generation of vorticity
and magnetic fields, and we derive a closed set of nonlinear evolution
equations that governs this generation. We estimate that the amplitude of the
magnetic field at recombination on the horizon scale is $\sim 10^{-27} $G.
