Approximation Scheme for Constructing a Clumpy Universe in General Relativity

We shall develop an approximation scheme to construct the metric representing a realistic clumpy universe in general relativity. Spatial averaging is used to express the fact that the expansion of the universe is generated collectively by the clumps of matter. The dynamics of the clumps is treated by the post-Newtonian-type approximation. The scheme allows one to calculate the back reaction due to the growth of clumps on the expansion and vice versa. An expression for the deviation from homogeneous and isotropic expansion is derived in terms of inhomogeneities of the gravitational field generated by clumps. It should also be used as the correct interpretation of the observation of gravitational lenses.