A relativistic approach to nonlinear peculiar velocities and the Zeldovich approximation

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.