Choptuik scaling and the scale invariance of Einstein's equation

TL;DR

This paper investigates how Choptuik scaling relates to the scale invariance of Einstein's equations, showing how to decompose the equations into scale and scale-invariant parts, and analyzing a toy model to illustrate these concepts.

Contribution

It introduces a method to separate Einstein's equations into scale and scale-invariant components, linking Choptuik scaling to periodic self-similarity in spacetime.

Findings

Periodic self-similarity implies scale invariance in spacetime

A toy model reproduces key features of Choptuik scaling

Decomposition clarifies the role of scale in Einstein's equations

Abstract

The relationship of Choptuik scaling to the scale invariance of Einstein's equation is explored. Ordinary dynamical systems often have limit cycles: periodic orbits that are the asymptotic limit of generic solutions. We show how to separate Einstein's equation into the dynamics of the overall scale and the dynamics of the ``scale invariant'' part of the metric. Periodicity of the scale invariant part implies periodic self-similarity of the spacetime. We also analyze a toy model that exhibits many of the features of Choptuik scaling.