Variations on the Seventh Route to Relativity

TL;DR

This paper extends Barbour, Foster, and O Murchadha's 3-space formulation of general relativity to include all known fundamental matter fields, using lapse-uneliminated actions and hypersurface frameworks, thus broadening the theory's applicability.

Contribution

It introduces a generalized 3-space approach that incorporates all fundamental matter fields, including fermions, within a covariant hypersurface framework, and addresses practical issues with lapse-eliminated actions.

Findings

All fundamental matter fields can be accommodated in the 3-space formulation.

Lapse-uneliminated actions simplify the inclusion of matter fields.

The generalized formulation supports Wheeler's thin sandwich conjecture for these theories.

Abstract

As motivated in the full abstract, this paper further investigates Barbour, Foster and O Murchadha (BFO)'s 3-space formulation of GR. This is based on best-matched lapse-eliminated actions and gives rise to several theories including GR and a conformal gravity theory. We study the simplicity postulates assumed in BFO's work and how to weaken them, so as to permit the inclusion of the full set of matter fields known to occur in nature. We study the configuration spaces of gravity-matter systems upon which BFO's formulation leans. In further developments the lapse-eliminated actions used by BFO become impractical and require generalization. We circumvent many of these problems by the equivalent use of lapse-uneliminated actions, which furthermore permit us to interpret BFO's formulation within Kuchar's generally covariant hypersurface framework. This viewpoint provides alternative reasons to BFO's as to why the inclusion of bosonic fields in the 3-space approach gives rise to minimally-coupled scalar fields, electromagnetism and Yang--Mills theory. This viewpoint also permits us to quickly exhibit further GR-matter theories admitted by the 3-space formulation. In particular, we show that the spin-1/2 fermions of the theories of Dirac, Maxwell--Dirac and Yang--Mills--Dirac, all coupled to GR, are admitted by the generalized 3-space formulation we present. Thus all the known fundamental matter fields can be accommodated. This corresponds to being able to pick actions for all these theories which have less kinematics than suggested by the generally covariant hypersurface framework. For all these theories, Wheeler's thin sandwich conjecture may be posed, rendering them timeless in Barbour's sense.