Exterior Differential Systems for Einstein Vacuum and Kaluza Gravity Theories

TL;DR

This paper develops exterior differential systems for embedding Riemannian spaces into higher-dimensional flat spaces, including a new Einstein vacuum embedding and a Kaluza-type integrable system, advancing geometric formulations of gravity.

Contribution

It introduces two novel families of EDS for gravitational theories, including a new non-isometric Einstein vacuum embedding and a stringy Kaluza-type system, with explicit Cartan forms.

Findings

EDS are shown to be dynamical field theories via Cartan characters.

A new non-isometric Einstein vacuum embedding EDS is constructed.

A family of integrable Kaluza-type systems generated by 2-forms is developed.

Abstract

We present two families of exterior differential systems (EDS) for causal embeddings of orthonormal frame bundles over Riemannian spaces of dimension q = 2,3,4,5.. into orthonormal frame bundles over flat spaces of higher dimension. We calculate Cartan characters showing that these EDS are dynamical field theories. The first family includes a new non-isometric embedding EDS for classical Einstein vacuum relativity (q = 4). The second, generated only by 2-forms, is a family of classical "stringy" or Kaluza-type (q = 5) integrable systems. Cartan forms are found for all these dynamical systems.