  A Lagrangian relativistic approach to the non--linear dynamics of
cosmological perturbations of an irrotational collisionless fluid is
considered. Solutions are given at second order in perturbation theory for the
relevant fluid and geometric quantities and compared with the corresponding
ones in the Newtonian approximation. Specifically, we compute the density, the
volume expansion scalar, the shear, the ``electric" part, or tide, and the
``magnetic" part of the Weyl tensor. The evolution of the shear and the tide
beyond the linear regime strongly depends on the ratio of the characteristic
size of the perturbation to the cosmological horizon distance. For
perturbations on sub--horizon scales the usual Newtonian approximation applies,
at least at the considered perturbative order; on super--horizon scales,
instead, a new picture emerges, which we call ``silent universe", as each fluid
element evolves independently of the environment, being unable to exchange
signals with the surrounding matter through either sound waves or gravitational
radiation. For perturbations inside the Hubble radius particular attention is
paid in singling out non--local effects during the non--linear evolution of
fluid elements. These non--local effects are shown to be carried by a traceless
and divergenceless tensor, contained in the magnetic part of the Weyl tensor,
which is dynamically generated as soon as the system evolves away from the
linear regime.
