Relativistic Lagrangian displacement field and tensor perturbations
  We investigate the purely spatial Lagrangian coordinate transformation from
the Lagrangian to the basic Eulerian frame. We demonstrate three techniques for
extracting the relativistic displacement field from a given solution in the
Lagrangian frame. These techniques are (a) from defining a local set of
Eulerian coordinates embedded into the Lagrangian frame; (b) from performing a
specific gauge transformation; and (c) from a fully non-perturbative approach
based on the ADM split. The latter approach shows that this decomposition is
not tied to a specific perturbative formulation for the solution of the
Einstein equations. Rather, it can be defined at the level of the
non-perturbative coordinate change from the Lagrangian to the Eulerian
description. Studying such different techniques is useful because it allows us
to compare and develop further the various approximation techniques available
in the Lagrangian formulation. We find that one has to solve the gravitational
wave equation in the relativistic analysis, otherwise the corresponding
Newtonian limit will necessarily contain spurious non-propagating tensor
artefacts at second order in the Eulerian frame. We also derive the magnetic
part of the Weyl tensor in the Lagrangian frame, and find that it is not only
excited by gravitational waves but also by tensor perturbations which are
induced through the non-linear frame-dragging. We apply our findings to
calculate for the first time the relativistic displacement field, up to second
order, for a $\Lambda$CDM Universe in the presence of a local primordial
non-Gaussian component. Finally, we also comment on recent claims about whether
mass conservation in the Lagrangian frame is violated.
