Nonlinear effects of general relativity from multiscale structure
  When do the nonlinear effects of general relativity matter in astrophysical
situations? They are obviously relevant for very compact sources of the
gravitational field, such as neutron stars or black holes. In this paper I
discuss another, less obvious situation, in which large relativistic effects
may arise due to a complicated, multiscale structure of the matter
distribution. I present an exact solution with an inhomogeneous energy density
distribution in the form of a hierarchy of nested voids and overdensities of
various sizes, extending from the homogeneity scale down to arbitrary small
scales. I show that although each of the voids and overdensities seems to be
very weakly relativistic, and thus easy to describe using the linearized
general relativity, the solution taken as a whole lies in fact in the nonlinear
regime. Its nonlinear properties are most easily seen when we compare the ADM
mass of the solution and the integral of the local mass density: the difference
between them, i.e. the relativistic mass deficit, can be significant provided
that the inhomogeneities extend to sufficiently small scales. The
non-additivity of masses implies a large backreaction effect, i.e. significant
discrepancy between the averaged, large-scale effective stress-energy tensor
and the naive average of the local energy density. I show that this is a
general relativistic effect arising due to the inhomogeneous, multiscale
structure. I also discuss the relevance of the results in cosmology and
relativistic astrophysics.
