A Definition of Background Independence

TL;DR

This paper introduces a new definition of background independence in dynamical theories, applies it to models including gravity, and explores its implications for gauge symmetry and observable quantities.

Contribution

It provides a novel framework for defining background independence based on best-matching and applies it to derive insights into general relativity and unimodular gravity.

Findings

General relativity can be derived within this framework.

The actions of best matching theories must have local gauge symmetry.

A procedure for incorporating background time leads to unimodular gravity.

Abstract

We propose a definition for background (in)/dependence in dynamical theories of the evolution of configurations that have a continuous symmetry and test this definition on particle models and on gravity. Our definition draws from Barbour's best-matching framework developed for the purpose of implementing spatial and temporal relationalism. Among other interesting theories, general relativity can be derived within this framework in novel ways. We study the detailed canonical structure of a wide range of best matching theories and show that their actions must have a local gauge symmetry. When gauge theory is derived in this way, we obtain at the same time a conceptual framework for distinguishing between background dependent and independent theories. Gauge invariant observables satisfying Kuchar's criterion are identified and, in simple cases, explicitly computed. We propose a procedure for inserting a global background time into temporally relational theories. Interestingly, using this procedure in general relativity leads to unimodular gravity.