Constraints of the teleparallel equivalent of general relativity in a gauge

In Ref. 12 we assumed that $M=R×Σ$ and that the orientation of Σ is induced by the orientation of $M$ in the following way: suppose (xⁱ) is a (local) coordinate system on Σ and $t∈R$⁠. Then (t, xⁱ) is a coordinate system on $M$⁠. If (t, xⁱ) is compatible with the orientation of $M$⁠, then (xⁱ) is compatible with the orientation of Σ.

In Refs. 14 and 15 we introduced in fact a family of new variables labeled by functions defined on the set of all coframes (θ^I) and valued in the set {−1, 1}. Here we use the variables given by the function (θ^I) ↦  sgn(θ^I) defined by (2.3).

The curve (3.1) is smooth in the sense that for every point y ∈ Σ the corresponding curve $]λ1,λ2[∋λ↦a(λ)y∈Ty*Σ$⁠, where a(λ)_y is the value of a(λ) at y, is smooth.

The initial condition makes sense if λ₁ < 0 < λ₂ in (3.1), which can be assumed without loss of generality.

The original lemma in Ref. 23 is formulated in terms of a general Banach space rather than a finite dimensional vector space.

Note that our convention is to use (δ^IJ) to raise lower indices I, J, K, …. But if ξ^I = 0, then (δ^IJ) are the components of the metric inverse (dual) to q in the coframe (θ^I). Therefore the formulas (6.1) holds true.