Cartan Formalism (physics) - The Basic Ingredients

The Basic Ingredients

Suppose we are working on a differential manifold M of dimension n, and have fixed natural numbers p and q with

p + q = n.

Furthermore, we assume that we are given a SO(p, q) principal bundle B over M and a SO(p, q)-vector bundle V associated to B by means of the natural n-dimensional representation of SO(p, q). Equivalently, V is a rank n real vector bundle over M, equipped with a metric η with signature (p, q) (aka non degenerate quadratic form).

The basic ingredient of the Cartan formalism is an invertible linear map, between vector bundles over M where TM is the tangent bundle of M. The invertibility condition on e is sometimes dropped. In particular if B is the trivial bundle, as we can always assume locally, V has a basis of orthogonal sections . With respect to this basis is a constant matrix. For a choice of local coordinates on M (the negative indices are only to distinguish them from the indices labeling the ) and a corresponding local frame of the tangent bundle, the map e is determined by the images of the basis sections

They determine a (non coordinate) basis of the tangent bundle (provided e is invertible and only locally if B is only locally trivialised). The matrix is called the tetrad, vierbein, vielbein etc.. Its interpretation as a local frame crucially depends on the implicit choice of local bases.

Note that an isomorphism gives a reduction of the frame bundle, the principal bundle of the tangent bundle. In general, such a reduction is impossible for topological reasons. Thus, in general for continuous maps e, one cannot avoid that e becomes degenerate at some points of M.