The Dynamics of Shape

TL;DR

This thesis introduces shape dynamics, a theory of gravity emphasizing the shape of space with local conformal invariance, offering a new perspective on the fundamental structure of spacetime and its gauge-theoretic formulation.

Contribution

It develops a gravity theory equivalent to general relativity but with local conformal invariance, and formulates a gauge theory for the shape of space, advancing understanding of spacetime's relational structure.

Findings

Shape dynamics provides an alternative to ADM formulation of gravity.

Gauge theory framework for shape of space elucidates relational properties.

Preliminary results indicate promising directions for quantum gravity.

Abstract

This thesis consists of two parts, connected by one central theme: the dynamics of the "shape of space". The first part of the thesis concerns the construction of a theory of gravity dynamically equivalent to general relativity (GR) in 3+1 form (ADM). What is special about this theory is that it does not possess foliation invariance, as does ADM. It replaces that "symmetry" by another: local conformal invariance. In so doing it more accurately reflects a theory of the "shape of space", giving us reason to call it \emph{shape dynamics} (SD). In the first part we will try to present some of the highlights of results so far, and indicate what we can and cannot do with shape dynamics. Because this is a young, rapidly moving field, we have necessarily left out some interesting new results which are not yet in print and were developed alongside the writing of the thesis. The second part of the thesis will develop a gauge theory for "shape of space"--theories. To be more precise, if one admits that the physically relevant observables are given by shape, our descriptions of Nature carry a lot of redundancy, namely absolute local size and absolute spatial position. This redundancy is related to the action of the infinite-dimensional conformal and diffeomorphism groups on the geometry of space. We will show that the action of these groups can be put into a language of infinite-dimensional gauge theory, taking place in the configuration space of 3+1 gravity. In this context gauge connections acquire new and interesting meanings, and can be used as "relational tools".