Lagrangian theory of structure formation in relativistic cosmology. VI.
  Comparison with Szekeres exact solutions
  We examine the relation between the Szekeres models and relativistic
Lagrangian perturbation schemes, in particular the Relativistic Zel'dovich
Approximation (RZA). We show that the second class of the Szekeres solutions is
exactly contained within the RZA when the latter is restricted to an
irrotational dust source with a flow-orthogonal foliation of spacetime. In such
a case, the solution is governed by the first principal scalar invariant of the
deformation field, proving a direct connection with a class of Newtonian
three-dimensional solutions without symmetry. For the second class, a necessary
and sufficient condition for the vanishing of cosmological backreaction on a
scale of homogeneity is expressed through integral constraints. Domains with no
backreaction can be smoothly matched, forming a lattice model, where exact
deviations average out at a given scale of homogeneity, and the homogeneous and
isotropic background is recovered as an average property of the model. Although
the connection with the first class of Szekeres solutions is not
straightforward, this class allows for the interpretation in terms of a spatial
superposition of nonintersecting fluid lines, where each world line evolves
independently and under the RZA model equations, but with different associated
`local backgrounds'. This points to the possibility of generalizing the
Lagrangian perturbation schemes to structure formation models on evolving
backgrounds, including global cosmological backreaction.
