Classic field theories of gravitation embedded in ten dimensions

TL;DR

This paper presents two classic metric gravity theories embedded in ten dimensions using Exterior Differential Systems, revealing their well-posedness and solution structures, with potential links to quantum field theoretic cosmological models.

Contribution

It introduces two constant-coefficient EDS formulations of 10D gravity theories, one related to vacuum tetrad gravity and the other to a Yang-Mills type model, with detailed analysis of their solution spaces.

Findings

Both theories are well-posed with calculable Cartan characteristic integers.

Solutions embed 4-dimensional Riemannian spaces within 10D frameworks.

The second theory's solutions are torsion-free and fibered over 3-spaces.

Abstract

Two classic field theories of metric gravitation are given as constant-coefficient Exterior Differential Systems (EDS) on the flat orthonormal frame bundle over ten dimensional space. They are derivable by variation of Cartan 4-forms, and shown to be well-posed by calculation of their Cartan characteristic integers. Their solutions are embedded Riemannian 4-spaces. The first theory is generated by torsion 2-forms and Ricci-flat 3-forms and is a constant-coefficient EDS for vacuum tetrad gravity; its Cartan character table is the same as found for an EDS recently given in terms of tetrad frame and connection variables [1] [2]. The second constant-coefficient EDS is generated solely by 2-forms, and has a Cartan form of quadratic Yang-Mills type. Its solutions lie in torsion free 6-spaces and are fibered over 3-spaces. We conjecture that these solutions may be classically related to 10-dimensional quantum field theoretic constructions of cosmological vacua [3].