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General Relativity and Quantum Cosmology, gr-qc,Astrophysics, Cosmology and Extragalactic Astrophysics, astro-ph.CO
Abstract:
We investigate the purely spatial Lagrangian coordinate transformation from
the Lagrangian to the fundamental 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 for gravitational
waves in the relativistic analysis, otherwise the corresponding Newtonian limit
will necessarily contain spurious tensor artefacts at second order, in both the
Lagrangian and 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 attendance of a local primordial non-Gaussian
component. Finally, we also comment on recent claims about whether mass
conservation in the Lagrangian frame is violated.
