Triad formulations of Canonical Gravity without a fixed reference frame

TL;DR

This paper proposes a simplified triad formulation of canonical gravity that avoids fixed reference frames, leading to new insights into the structure and constraints of the theory, including a straightforward approach to reality conditions.

Contribution

It introduces a novel triad formulation of canonical gravity that eliminates the need for a fixed coordinate system, simplifying the analysis of constraints and reality conditions.

Findings

Momentum naturally expressed as a 2-form in the new formulation

Elimination of second class constraints in the Palatini formalism

Equivalent to two canonical transformations in the Ashtekar theory

Abstract

One can simplify the triad formulations of canonical gravity by abandoning any relation to a fixed coordinate system. That means in case of the \ADM formalism that one can determine the momentum by direct derivation of the Lagrange-3-form w.r.t the time-derivative of the triad-1-forms, thus the momentum is most naturally a 2-form. We apply this concept to the Palatini formulation where we can closely follow Dirac's concept to find and eliminate the second class constraints. Following the same way for the Ashtekar theory it will turn out to be equivalent to two successive canonical transformations where the first makes explicit use of the spatial dimension being 3 and the second is usually hidden in the use of densities. At the end we can give a simple version of the reality constraints.