Nonlinear effects of general relativity from multiscale structure

TL;DR

This paper demonstrates that complex multiscale inhomogeneous structures can induce significant nonlinear relativistic effects, affecting mass and energy estimates in astrophysical and cosmological contexts, even when individual components appear weakly relativistic.

Contribution

It provides an exact solution illustrating how multiscale inhomogeneities lead to nonlinear relativistic effects and backreaction, highlighting their importance beyond compact astrophysical objects.

Findings

Significant relativistic mass deficit due to small-scale inhomogeneities.

Large backreaction effects alter effective stress-energy tensors.

Nonlinear effects arise from multiscale structure, not just compact sources.

Abstract

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.