Gravitational Lagrangian dynamics of cold matter using the deformation tensor

In this paper we present a new local Lagrangian approximation to the gravitational dynamics of cold matter. We describe the dynamics of a Lagrangian fluid element through only one quantity, the deformation tenser. We show that this tensor is clearly suited to the study of gravitational dynamics and, moreover, that knowing its evolution is enough to completely describe a fluid element. In order to determine this evolution, we make some physical approximations on the exact dynamical system and we thus obtain a closed local system of differential equations governing the evolution of the deformation tenser. Our approximate dynamics treat exactly the conservation of mass, of the velocity divergence and of the shear and is exact in the case of planar, cylindrical and spherical collapses. It also reproduces very accurately the evolution of all dynamical quantities for a very wide class of initial conditions as illustrated by a detailed comparison with the homogeneous ellipsoid model. Beside, we highlight, for the first time, the important dynamical role played by the Newtonian counterpart of the magnetic part of the Weyl tensor.