'Shape Dynamics': Foundations Reassessed

TL;DR

This paper reassesses the mathematical foundations of shape dynamics, focusing on volume-preserving conformal transformations and their group-theoretic modeling, proposing a shift from finite to infinitesimal implementations.

Contribution

It critically examines the modeling of VPCTs in shape dynamics, suggesting a transition from finite integral to differential infinitesimal approaches based on Lie group theory.

Findings

Analysis of group-theoretic properties of VPCTs

Identification of issues with current finite integral implementations

Proposals for differential infinitesimal modeling of VPCTs

Abstract

`Shape dynamics' is meant here in the sense of a type of conformogeometrical reformulation of GR, some of which have of late been considered as generalizations of or alternatives to GR. This note concerns in particular cases based on the notion of volume-preserving conformal transformations (VPCTs), in the sense of preserving a solitary global volume of the universe degree of freedom. The extent to which various ways of modelling VPCTs make use of group theory at all, in a congruous manner, and with minimal departure from standard Lie group theory, is considered. This points to changing conception of VPCTs from the current finite integral implementation to an infinitesimal differential implementation (or to avoiding using them at all). Some useful observations from flat-space conformal groups (well-known from CFT) concerning the existence or otherwise of VPCT groups are also provided.