Abstract
We study the effect of small-scale inhomogeneities for Einstein clusters. We construct a spherically symmetric static spacetime with small-scale radial inhomogeneities and propose the Gedankenexperiment. An hypothetical observer at the center constructs, using limited observational knowledge, a simplified homogeneous model of the configuration. An idealization introduces side effects. The inhomogeneous spacetime and the effective homogeneous spacetime are given by simple solutions to Einstein equations. They provide a basic toy-model for studies of the effect of small-scale inhomogeneities in general relativity. We show that within our highly inhomogeneous model the effect of small-scale inhomogeneities remains small for a central observer. The homogeneous model fits very well to all hypothetical observations as long as their precision is not high enough to reveal a tension.
Highlights
96% of its energy content has not been previously known and is seen only via gravitational interactions. This apparent contradiction motivated broad studies of validity of a basic assumption of the model – exact spatial homogeneity. We take on this topic and study an effect of small-scale inhomogeneities which is especially interesting in the light of the recent presumable tension between ‘local’ and ‘early’ universe measurements of the Hubble constant [1,2,3,4]
If one assumes that Einstein equations hold, additional or missing terms are incorrectly interpreted as a contribution to the energy– momentum tensor
Since the C D M energy–momentum tensor is dominated by dark matter and dark energy – the forms of energy and matter detected so far only through their gravitational interactions, the backreaction effect has a potential to clarify our understanding of the Universe
Summary
The problem of small-scale inhomogeneities may be split into two topics: the effect of inhomogeneities on geodesics (light, gravitational waves, test bodies) which alters interpretation of our observations and the so-called backreaction effect which alters the structure of spacetime in a sense which will be explained below. Our approach is based on a class of solutions to Einstein equations called an Einstein cluster This type of solutions have interesting properties that allow for novel studies of the effect of small-scale inhomogeneities in an alternative setting. The vanishing of radial pressure allows to construct static spacetimes with small-scale inhomogeneities without introducing unphysical equation of state. We present a heuristic analysis which implies that the backreaction vanishes in all possible models constructed within an Einstein cluster class This motivate us to study the effect of inhomogeneities beyond the Green-Wald framework. Our analysis is restricted to the particular class of solutions to Einstein equations, but it illustrates what the effect of small-scale inhomogeneities could be in principle
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