S. Matarrese et al., A RELATIVISTIC APPROACH TO GRAVITATIONAL-INSTABILITY IN THE EXPANDINGUNIVERSE - 2ND-ORDER LAGRANGIAN SOLUTIONS, Monthly Notices of the Royal Astronomical Society, 271(3), 1994, pp. 513-522
A Lagrangian relativistic approach to the non-linear dynamics of cosmo
logical perturbations of an irrotational collisionless fluid is consid
ered. Solutions are given at second order in perturbation theory for t
he relevant fluid and geometric quantities and compared with the corre
sponding ones in the Newtonian approximation. Specifically, we compute
the density, the volume expansion scalar, the shear, the 'electric' p
art, or tide, and the 'magnetic' part of the Weyl tenser. The evolutio
n of the shear and the tide beyond the linear regime strongly depends
on the ratio of the characteristic size of the perturbation to the cos
mological horizon distance. For perturbations on sub-horizon scales th
e usual Newtonian approximation applies, at least at the considered pe
rturbative order; on super-horizon scales, instead, a new picture emer
ges, which we call the 'silent universe', as each fluid element evolve
s independently of the environment, being unable to exchange signals w
ith the surrounding matter through either sound waves or gravitational
radiation. For perturbations inside the Hubble radius, particular att
ention is paid to 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 divergence-free tenser, contained in the ma
gnetic part of the Weyl tenser, which is dynamically generated as soon
as the system evolves away from the linear regime.