  We develop a new, mathematically precise framework for treating the effects
of nonlinear phenomena occurring on small scales in general relativity. Our
approach is an adaptation of Burnett's formulation of the "shortwave
approximation", which we generalize to analyze the effects of matter
inhomogeneities as well as gravitational radiation. Our framework requires the
metric to be close to a "background metric", but allows arbitrarily large
stress-energy fluctuations on small scales. We prove that, within our
framework, if the matter stress-energy tensor satisfies the weak energy
condition (i.e., positivity of energy density in all frames), then the only
effect that small scale inhomogeneities can have on the dynamics of the
background metric is to provide an "effective stress-energy tensor" that is
traceless and has positive energy density---corresponding to the presence of
gravitational radiation. In particular, nonlinear effects produced by small
scale inhomogeneities cannot mimic the effects of dark energy. We also develop
"perturbation theory" off of the background metric. We derive an equation for
the "long-wavelength part" of the leading order deviation of the metric from
the background metric, which contains the usual terms occurring in linearized
perturbation theory plus additional contributions from the small-scale
inhomogeneities. Under various assumptions concerning the absence of
gravitational radiation and the non-relativistic behavior of the matter, we
argue that the "short wavelength" deviations of the metric from the background
metric near a point $x$ should be accurately described by Newtonian gravity,
taking into account only the matter lying within a "homogeneity lengthscale" of
$x$. Finally, we argue that our framework should provide an accurate
description of the actual universe.
