Lagrangian theory of structure formation in relativistic cosmology: Lagrangian framework and definition of a nonperturbative approximation

In this paper, we present a Lagrangian framework for the description of structure formation in general relativity, restricting attention to irrotational dust matter. As an application we present a self-contained derivation of a general-relativistic analogue of Zel'dovich's approximation for the description of structure formation in cosmology and compare it with previous suggestions in the literature. This approximation is then investigated: paraphrasing the derivation in the Newtonian framework we provide general-relativistic analogues of the basic system of equations for a single dynamical field variable and recall the first-order perturbation solution of these equations. We then define a general-relativistic analogue of Zel'dovich's approximation and investigate its implications by functionally evaluating relevant variables, and we address the singularity problem. We so obtain a possibly powerful model that, although constructed through extrapolation of a perturbative solution, can be used to put into practice nonperturbatively, e.g., problems of structure formation, backreaction problems, nonlinear properties of gravitational radiation, and light propagation in realistic inhomogeneous universe models. With this model we also provide the key building blocks for initializing a fully relativistic numerical simulation.