Triad Formulations of Canonical Gravity without a fixed reference frame

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. 1 Technical Preliminaries Avoiding any coordinate system will eliminate all determinants from the theory, but the associated problems will be contained in the frequently used Hodge operator. Yet the algebra of this operator is simple because our triads are normalized. To have an effective way of handling this operator we first introduce some notations and formulas where we mainly follow the treatment given in [1]. Let (M, g) be a m-dimensional pseudo-Riemannian manifold. We first define the interior multiplication i of two forms of different degree. For q ≤ p let the bilinear mapping i : Ω(M) × Ω(M) −→ Ω(M); (μ, ν) 7−→ iμν have the following properties: iμν = g (μ, ν) = μaνbg ab fr p = q = 1 (1.1) iμ(ν1 ∧ ν2) = iμν1 ∧ ν2 + (−1) ν1 ∧ iμν2 fr νi ∈ Ω i(M), μ ∈ Ω(M) (1.2) i(μ1∧μ2) = iμ2 ◦ iμ1 (1.3)