Canonical general relativity: Matter fields in a general linear frame

TL;DR

This paper extends the general linear frame approach to general relativity by incorporating matter fields like Maxwell and Dirac fermions, revealing how these fields influence the canonical formulation and the role of frames.

Contribution

It demonstrates the inclusion of matter fields into the linear frame formalism of general relativity, analyzing their effects on the canonical structure and curvature representations.

Findings

Maxwell fields lead to extrinsic curvature involving matter and gravitational degrees of freedom.

Metric compatibility remains unaffected despite derivative couplings.

Dirac fermions introduce milder derivative couplings in the formalism.

Abstract

Building on the results of previous work, we demonstrate how matter fields are incorporated into the general linear frame approach to general relativity. When considering the Maxwell one-form field, we find that the system that leads naturally to canonical vierbein general relativity has the extrinsic curvature of the Cauchy surface represented by gravitational as well as non-gravitational degrees of freedom. Nevertheless the metric compatibility conditions are undisturbed, and this apparent derivative-coupling is seen to be an effect of working with (possibly orthonormal) linear frames. The formalism is adapted to consider a Dirac Fermion, where we find that a milder form of this apparent derivative-coupling appears.