  We study the peculiar motion of non-relativistic matter in a fully covariant
way. The exact nonlinear equations are derived and then applied to the case of
pressure-free matter, moving relatively to a quasi-Newtonian Eulerian frame.
Our two-frame formalism facilitates the study of the nonlinear kinematics of
the matter, as the latter decouples from the background expansion and starts to
``turn around'' and collapse. Applied to second perturbative order, our
equations provide a fully covariant formulation of the Zeldovich approximation,
which by construction addresses the mildly nonlinear regime of structure
formation. Employing a dynamical system approach, we show that, just like in
the Newtonian case, the relativistic treatment also predicts that pancakes are
the natural end-structures for any generic overdensity.
