Relativistic Lagrangian displacement field and tensor perturbations

Abstract

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.

arXiv comment19 pages, two figures, improved discussion, matches published version

References in corpus (28)

• The Dynamics of General Relativity
• Large-Scale Structure of the Universe and Cosmological Perturbation Theory
• The cosmological gravitational wave background from primordial density perturbations
• Cosmological perturbations
• Resumming Cosmological Perturbations via the Lagrangian Picture: One-loop Results in Real Space and in Redshift Space
• Relativistic second-order perturbations of the Einstein-de Sitter Universe
• General Relativistic Dynamics of Irrotational Dust: Cosmological Implications
• Newtonian Cosmology in Lagrangian Formulation: Foundations and Perturbation Theory
• General Relativistic N-body simulations in the weak field limit
• Post-Newtonian Cosmological Dynamics in Lagrangian coordinates
• Lagrangian perturbations and the matter bispectrum I: fourth-order model for non-linear clustering
• Lagrangian Evolution of the Weyl Tensor
• Time-analyticity of Lagrangian particle trajectories in ideal fluid flow
• Computing General Relativistic effects from Newtonian N-body simulations: Frame dragging in the post-Friedmann approach
• Lagrangian theory of structure formation in relativistic cosmology II: average properties of a generic evolution model
• The recursion relation in Lagrangian perturbation theory
• Lagrangian theory of structure formation in relativistic cosmology I: Lagrangian framework and definition of a nonperturbative approximation
• A Relativistic Approch to Gravitational Instability in the Expanding Universe: Second Order Lagrangian Solutions
• Non-Gaussian initial conditions in $\Lambda$CDM: Newtonian, relativistic and primordial contributions
• The Zel'dovich-type approximation for an inhomogeneous universe in general relativity: second-order solutions
• Second-order matter perturbations in a LambdaCDM cosmology and non-Gaussianity
• Lagrangian perturbations and the matter bispectrum II: the resummed one-loop correction to the matter bispectrum
• Einstein's signature in cosmological large-scale structure
• Newtonian versus relativistic cosmology
• Zel'dovich approximation and General Relativity
• Initial conditions for cold dark matter particles and General Relativity
• Frame dragging and Eulerian frames in General Relativity
• Cosmological dynamics: from the Eulerian to the Lagrangian frame - I. Newtonian approximation

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• Lagrangian theory of structure formation in relativistic cosmology. V. Irrotational fluids
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