What is a mean gravitational field?

The equations of General Relativity are non-linear. This makes their averaging non-trivial. The notion of mean gravitational field is defined and it is proven that this field obeys the equations of General Relativity if the unaveraged field does. The workings of the averaging procedure on Maxwell's field and on perfect fluids in curved space-times are also discussed. It is found that Maxwell's equations are still verified by the averaged quantities but that the equation of state for other kinds of matter generally changes upon average. In particular, it is proven that the separation between matter and gravitational field is not scale-independent. The same result can be interpreted by introducing a stress-energy tensor for a mean-vacuum. Possible applications to cosmology are discussed. Finally, the work presented in this article also suggests that the signature of the metric might be scale-dependent too.