Lagrangian theory of structure formation in relativistic cosmology. IV. Lagrangian approach to gravitational waves

Abstract

The relativistic generalization of the Newtonian Lagrangian perturbation theory is investigated. In previous works, the perturbation and solution schemes that are generated by the spatially projected gravitoelectric part of the Weyl tensor were given to any order of the perturbations, together with extensions and applications for accessing the nonperturbative regime. We here discuss more in detail the general first-order scheme within the Cartan formalism including and concentrating on the gravitational wave propagation in matter. We provide master equations for all parts of Lagrangian-linearized perturbations propagating in the perturbed spacetime, and we outline the solution procedure that allows one to find general solutions. Particular emphasis is given to global properties of the Lagrangian perturbation fields by employing results of Hodge-de Rham theory. We here discuss how the Hodge decomposition relates to the standard scalar-vector-tensor decomposition. Finally, we demonstrate that we obtain the known linear perturbation solutions of the standard relativistic perturbation scheme by performing two steps: first, by restricting our solutions to perturbations that propagate on a flat unperturbed background spacetime and, second, by transforming to Eulerian background coordinates with truncation of nonlinear terms.

15 pages, matches published version in PRD

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• On general-relativistic Lagrangian perturbation theory and its non-perturbative generalization
• Beyond relativistic Lagrangian perturbation theory. I. An exact-solution controlled model for structure formation
• On restrictions of current warp drive spacetimes and immediate possibilities of improvement
• On the maximum volume of collapsing structures
• Nonperturbative collapse models for collisionless self-gravitating flows
• Direct correspondence between Newtonian gravitation and general relativity
• Backreaction and the Role of Spatial Curvature in the Cosmic Neighborhood
• Cosmological perturbations on an averaged background