  The averaging problem in general relativity concerns the difficulty of
defining meaningful averages of tensor quantities and we consider various
aspects of the problem. We first address cosmological backreaction which arises
because the averaged Einstein tensor is not the same as the Einstein tensor of
the averaged metric. It has been suggested that backreaction might account for
the dark energy. We show numerically in the Buchert formalism that the
corrections from (quasi)linear perturbations are only of the order of 10^-5 and
act as a dark matter. We then focus on constructing averaged metrics and
present a generally covariant averaging process which decomposes the metric
into Vielbeins and parallel transports them with a relativistic Wegner-Wilson
operator to a single point where they can then be averaged. The Vielbeins are
chosen in a Lagrangian formalism. The functionality of the process is
demonstrated in specific examples in two and three space dimensions. This
involves the numerical solution of partial differential equations by the aid of
the simulation toolkit Gascoigne.
