Absolute versus Relational Debate: a Modern Global Version

TL;DR

This paper compares absolute and relational approaches in geometry and physics, proposing a new perspective that avoids stratification issues and discusses the implications for shape theory and general relativity.

Contribution

It introduces a novel approach with a generic absolute space that eliminates stratification, and extends the discussion to Kendall-type Shape Theory, which is a new contribution.

Findings

Stratification arises in quotienting absolute space, complicating relational theories.

A generic absolute space with no Killing vectors simplifies shape and scale theories.

Relational and absolute strategies have distinct technical advantages and limitations.

Abstract

Suppose one seeks to free oneself from a symmetric absolute space by quotienting out its symmetry group. This in general however fails to erase all memory of this absolute space's symmetry properties. Stratification is one major reason for this, which is present in both a) Kendall-type Shape Theory and subsequent Relational Mechanics, and b) General Relativity configuration spaces. We consider the alternative starting point with a generic absolute space, meaning with no nontrivial generalized Killing vectors whatsoever. In this approach, generically Shape-and-Scale Theory is but trivially realized, there is no separate Shape Theory and indeed no stratification. While the GR configuration space version of these considerations was already expounded in 1996 by Fischer and Moncrief, the Kendall-type shape theory version is new to the current article. In each case, this amounts to admitting some small deformation by which symmetry's hard consequences at the level of reduced configuration spaces are warded off.We end by discussing the senses in which each of the above two strategies retain absolutist features, each's main known technical advantages and disadvantages, and the desirability of replacing Kendall-type Shape Theory with a Local-and-Approximate Shape Theory. This article is in honour of Prof. Niall \'{o} Murchadha, on the occasion of his Festschrift.