Relativistic approach to nonlinear peculiar velocities and the Zeldovich approximation

We study the peculiar motion of nonrelativistic 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.