  In this first 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.
