Field Theories found Geometrically from Embeddings in Flat Frame Bundles

TL;DR

This paper introduces new geometric field theories derived from embeddings of frame bundles into flat spaces, including Einstein vacuum relativity, with well-posed differential systems and explicit variational principles.

Contribution

It presents two families of exterior differential systems for embedding frame bundles, including a new formulation of Einstein vacuum relativity and integrable nonlinear PDE systems.

Findings

EDS satisfy Cartan's test and are well-posed

Explicit variational principles are derived for these theories

Includes a new geometric embedding of Einstein vacuum relativity

Abstract

We present two families of exterior differential systems (EDS) for non-isometric embeddings of orthonormal frame bundles over Riemannian spaces of dimension q = 2, 3, 4, 5.... into orthonormal frame bundles over flat spaces of sufficiently higher dimension. We have calculated Cartan characters showing that these EDS satisfy Cartan's test and are well-posed dynamical field theories. The first family includes a constant-coefficient (cc) EDS for classical Einstein vacuum relativity (q = 4). The second family is generated only by cc 2-forms, so these are integrable (but nonlinear) systems of partial differential equations. These calibrated field theories apparently are new, although the simplest case q = 2 turns out to embed a ruled surface of signature (1,1) in flat space of signature (2,1). Cartan forms are found to give explicit variational principles for all these dynamical theories.