The Similarity Hypothesis in General Relativity

TL;DR

This paper examines the 'similarity hypothesis' in general relativity, exploring how various cosmological and spherically symmetric models tend to evolve towards self-similar solutions under different conditions.

Contribution

It provides evidence supporting the hypothesis that solutions in general relativity often evolve towards self-similar forms, clarifying when and why this occurs across different models.

Findings

Self-similar solutions act as attractors in many cosmological models.

The hypothesis holds in spatially homogeneous and some inhomogeneous cases.

Self-similarity often appears as an intermediate attractor in dynamical evolution.

Abstract

Self-similar models are important in general relativity and other fundamental theories. In this paper we shall discuss the ``similarity hypothesis'', which asserts that under a variety of physical circumstances solutions of these theories will naturally evolve to a self-similar form. We will find there is good evidence for this in the context of both spatially homogenous and inhomogeneous cosmological models, although in some cases the self-similar model is only an intermediate attractor. There are also a wide variety of situations, including critical pheneomena, in which spherically symmetric models tend towards self-similarity. However, this does not happen in all cases and it is it is important to understand the prerequisites for the conjecture.